The Neumann compendium

V " H

They are the measures for the squares of the differences of Q1 and Q, P» and P respectively, i.e., they must be approximately e and r\2 respectively. We therefore require

l^*n - V n " ^ 6 ' Hp*n " V J ' 1 " ' Instead of speaking of Q«, p« , it is then more appro­ priate only to seek a complete orthonormal set <J> , 4> , . . .

Quantum Mechanics

ho6

63

v.- GENERAL CONSIDERATIONS

for which, for suitable choice of a ^ a , ... and b-,b 2 ,... , the above estimates hold. Individual (with I I ♦ I J = i ) , for which (for suitable a, b ) |I |Q * | | |Q- -aao|

, - t » 0bo| 1 | = I== €€ , , | |i P diPd, | T) = i]

are known from III.k.: 1

> == o ♦p,«,r V*,r

T - i21(q - a)2 + ^ £ iq ( <(n) l ) = ((12.' e

= lf>

Hence, because of er\ = ^ € =

~

we have again

<M > i =

\}hny

(i.e., 7 = C/T) ), and we choose a = a, b = p . We now must construct a complete orthonormal set with the help of these . Since o is the Q- and p the P-expectation value, it is plausible that p, a should each run through a set of numbers independently of each other, and in fact, in such a way that the p-set has approxi­ mately the density € and the cr-set approximately the density TJ . It proves practical to choose the units &/*". e

= «/h7 -Thy

and

2N/~S7. 2N/~*. TJ =

/—

,

i.e., p = sThy [ i , o = p = sThy [ i , o = (\i,

v = 0, + 1, + 2 , . . . ) .

Jy Jy

v v

The

= 0

<$>

>7

(n, v = o, + 1, +2,...) ought then to correspond to the

The Neumann

64

4.

Compendium

TI THE MACROSCOPIC MEASUREMENT

k07

♦ (n = 1,2,...) • It Is obviously irrelevant that we have two indices \x, v in place of the one n . However, these y are not yet orthogonal. (They are normalized, however, and they satisfy ||QVv

- ^ n

V v

| |

= «, | | P

V v

- ^ v

V v

M

= i.)

If we now orthogonalize them by the E. Schmidt process (In order, cf. II.2., proof of THEOREM 8. ), then we can prove the completeness of the resulting normalized orthogonal set f' without any particular difficulties, and can also establish the estimates | |Qf

- ^/h7 n * '

I I < Ce,

| |Pi|r'

-

/ ^ v>|ff

| | < CTI

with certain fixed C . A value C -^6o has been obtained in this way, and it could probably be reduced. The proof of this fact leads to rather tedious calculations, which require no new concepts, and we shall omit them. The factors C — 6o are not important, since ei\ = h/4* measured in macroscopic (CGS) units is exceedingly small (C 10" 28 ) .

Summing up, we can then say that it is justified to assume the commutativity of all macroscopic operators, and in particular the commutativity of the macroscopic projections E introduced above. The E correspond to all macroscopically answer­ able questions @ , i.e., to all discriminations of alter­ natives in the system investigated, that can be carried out macroscopically. They are all commutative. We can conclude from I L 5 - , that 1 - E belongs to them along with E , and that EP, E + F - EF, E - EP belong along with E, F . It is reasonable to assume that there are only a finite number of them: E-,...,E . We introduce the notation E v ; = E, EK J = 1 - E and consider all vs

n

2

products

E1

l;

V b

. . . Kn

n '

(Sl,...,sn = + ) .

Any

Quantum Mechanics

hoQ

65 V. GENERAL CONSIDERATIONS

two different ones among these have the product zero: For (Sl) (s n ) (t,) (t n ) and E are two if E 1 ' * • •E n 1 * • ' ^n such, and s / t , then there appear in their product the (s ) (t ) + KV anH V , E -n, factors E i.e., E v( ') == ET? and E"R ' ~ ' = V

V

V

V

1 - E , whose product is zero. Each E several such products: Indeed,

E;a■,...E(3^')•s(<)a
7"

Ev = 1

7

/ . v-1

is the sum of

1 v+1

v-1

v

v+1

ii

n -

Among these products consider the ones which are different from zero. Call them Ej, . • . ,E^ . (Evidently m
[1

'

V

(From the latter it also follows that n = 2 .) It should be noted that E + E = E ! can never occur, unless |i V p EH = 0, E p = E 1 or Eu = pE 1 , vE = 0 . Otherwise, E\i , Ev V ' 1 1 would be sums of several E , and therefore E the sum *

P

of >_ 2 terms E^ (possibly with repetitions). By II.h., THEOREMS 15, 16., these would all differ from one another, since their number is :> 2 and all are 4 o , they also differ from E 1 — therefore their product with E 1 would be zero. Hence the product of their sum with E» would also be zero, but this contradicts the assertion that the sum is = E 1 . P The properties ^j'---*®^ corresponding to the Ej,...,EL are then macroscopic properties of the following type: None is absurd. Every two are mutually exclusive. Each macroscopic property obtains by disjunction of several of them. None of them can be resolved by disjunction into two sharper macroscopic properties, ®j>'->®™ therefore represent the furthest that we can go in macroscopic dis­ crimination, for they are macroscopically indecomposable.

66

The Neumann

if. THE MACROSCOPIC MEASUREMENT

Compendium

409

In the following, we shall not require that their number be finite, but only that there exist macroscopically indecomposable properties @j,@ 2 ,... • Let their projec­ tions be Ej,E2,... , all again different from zero, mutually orthogonal, and each macroscopic E the sum of several of them. Therefore 1 is also a sum of several of them. If an E r did not occur in this sum, it would be orthog­ onal to each term and hence to the sum, that is to 1 : E' = Ef.i = 0 , which is impossible. Therefore Ej + E£ + . • • = 1 . We drop the prime notation: (S-j,^,... and E.jEp,... • The closed linear manifolds belonging to these will be called SJ^,^, ... , and their dimension numbers s.,s2,... . If all the s = 1, i.e., all 9K one dimension­ al, then 2&n = f*n]> ^n = P r ] a n d because E 1 + E 2 + .. . = i , the 1, 4>2, .. . would form a complete orthonormal set. This would mean that macroscopic measurements would themselves make a complete determination of the state of the observed system possible. Since this is ordinarily not the case, we have in general s n > 1 , and in fact, sn » i • In addition, It should be observed that the ft , which are the elementary building blocks of the macroscopic description of the world, correspond in a certain sense to the ordinary cell division of phase space in the classical theory. We have already seen that they can reproduce the behavior of non-commutative operators in an approximate fashion, in particular, that of Q, P , which are so important for phase space. Now, what entropy does the mixture U have for a macroscopic observer whose indecomposable projections are EjjEg,... ? Or, more precisely, how much entropy can such an observer maximally obtain by transforming U into V — i.e., what entropy decrease (under suitable conditions, naturally this decrease may be -^ o) can he produce, under

Quantum

410

Mechanics

67 V.

GENERAL CONSIDERATIONS

the most favorable circumstances, in external objects as compensation for the transition U — ► V ? First, it must be emphasized that he cannot distinguish between each two ensembles U, U' , if both give the same expectation value to EL for each n = 1,2,... , that is, if Tr (UER) = Tr (U'E^ (n = 1,2,... ). After some time, of course, the discrimination may become possible, since U, U1 change according to 2., and Tr (AUA ^ J

= Tr (AU'A"*1^), A = e

2£l tH

20^

must no longer hold. But we considered only measure­ ments which are carried out immediately. Under the above conditions we may therefore regard U, U* as indistin­ guishable. Furthermore, the observer can also use only such semi-permeable walls which transmit the of some E and reflect the remainder unchanged. This possibility suffices, as can be seen without difficulty. By means of the method of V-2-, to transform a

U

= 12 V^n n=i

204T

If E commutes with H , and therefore with equality still holds because

A , the

Tr (A.UA" 1 ^) = Tr ( U A ' ^ . A ) = Tr (UA"1AEn) = Tr (UE^) • But all E , i.e., all macroscopically observable quan­ tities, are in no way-all commutative with H . Indeed, many such quantities, for example, the center of gravity of a gas in diffusion, change appreciably with t , i.e., Tr (tflS) is not constant. Since all macroscopic quantities do commute, H is never a macroscopic quantity, i.e., the

The Neumann

68 h. into

Compendium

THE MACROSCOPIC MEASUKEJVIENT

^11

a 00

V

1

n=i n=i reversibly, so that the entropy difference is still K Tr (IP In U' ) - K Tr (V l n V ) , i.e., the entropy of U» equals - K Tr (Uf In U') . To be sure, in order that such Uf with Tr U' = 1 exist in general, the Tr E Q , i.e., the numbers s , must be finite. We therefore assume that all s n are finite. U ! has the s^fold eigenvalue x 1 , the s -fold eigenvalue x2,... . There­ fore - U1 In Uf has the s^fold eigenvalue - x 1 In x 1 , the s2-fold eigenvalue - x 2 In x 2 , ... . Consequently Tr U' = 1 implies

/

i

n

n

n=i and the entropy is equal to

K

Z ] Snxn l n *n '

n=l Because of U,E

ta = £ ]

x

n E A = *mV

Tr


Tr ^

= s ^

,

n=!

Tr(U'EL) ^m

=

'

—s

therefore the

m

entropy is equal to

00

-A -

K

h n=l

Tr ( U

For arbitrary

'V

ln

Tr (U'E.)

3—— •

U (Tr U = 1 ) , the entropy must

energy is not measured macroscopically with complete preci­ sion. This is plausible without additional comment.

Quantum Mechanics

h\2

69

V. GENERAL CONSIDERATIONS

also be equal to

-

Y^i

K l^

Tr (U^) in

Tr (UEL)

g-iL. n

n=i because, if we set Tr (OT^) 11

S

r, n

'u' = ZI X A

n=i then Tr (UE^) = Tr (U'E^) , and since U, U 1 are indis­ tinguishable, they have the same entropy. We must also mention the fact that this entropy always exceeds the customary entropy: ^ -- K ^^

Tr Tr (UE^) (UE^) In In

Tr Tr (( U E ) ) --——— Tr (U (U In In U) U) — >> -- K* Tr nn

n=l n=i

and that the equality holds only for 00

■z: n=i

u =

x

n\

'

n=i

By the results of V. 3 •, this is certainly the case if A A U' U' = >

Tr Tr (UEL) (UE) EL EL qa ^

can be obtained from U by several (not necessarily macroscopic) applications of the process 1. — because on the left we have - K Tr (U1 In U f ) , and u

= ZI "IAI

n=i means the same as U = U1 . We take an orthonormal set *! >•••>♦« which spans the closed linear manifold n I belonging to E^ . Because of

The Neumann

70 k.

Compendium

k!3

THE MACROSCOPIC MEASUREMENT

E *n-1 • ~.->«..,......::.,.....o * - . « * * . orthonormal set. Let R be an operator belonging to these eigenfunctions (with only distinct eigenvalues) and 9* its physical quantity. In the measurement of SR , we get from U (by 1.)

V°=tL«\in)>

\(n>>-V (n) l

n=l V=1

v

J

Then, if we set n * (n) _ '

-g-

n

MV

Z> =1

(u = i , .• •. , s n ) ,

n V=1

n the 11n , . . .s ,i|r(n) a ^ ' form an orthonormal set which spans the same closed linear manifold as as the the ♦ ,••-,♦_ :

'' s„ ,n* "n aj^ . Therefore the t ' (n = 1,2,..-; v = l,2,...,s ) also form a complete orthonormal set, and we form an operator S with these eigenfunctions, and the corre­ sponding physical quantity @- We must note the validity of the following formulas: v

f =o

for m 4 n

/ (m) . (m)\ /pp /( nnx) #*V(m), ♦ ( m ) M ( f %( n ) ]M 'M " /

V t*v J

/ I

s sn

n TL 2 ] P [L * (n)j= H v=1 v=1

I J

= ■— for m = n , s L n | n

L

v v

In the measurement of

P P

v v= =1 1

(n)]="^^ [l + U¥(n)] l ¥

v v ''

© , therefore,

••

U" becomes (by 1. )

Quantum

Mechanics

71

41 4

oo oo

V.

GENERAL CONSIDERATIONS

m

I m, Em=i E " •*i • ♦?"»,,(.>, n=i n LV

LV

m=i n=i

S

00 00

l—

J

n

J

—i

S

m II °° °°

nn

n, l c,,,, ■i:EEE"'. = EEZW n,b,-i.*?v> '^. w1'J ;i-,,". -i- ik'" [ ,(.,I 1 m=i

=1 MM=1

" J ? r 3 ? (U* =

LL

L n = i v=i v=i Ln=i

(m)

*

(m)

vv ' '

JJ

)l

°° m

L, ZJ m=i n = l L vjL =i

-m

J

tv

J [*< n >] J

m=l n = 1 L v = l

m

J

*\x

=

l

Sm

L, L* m=l v=i m=1 V = 1

v vn

n

Tr (TIE ) ^m

P

¥ [lJm) n ] L *n J

A Tr (ra^) (ray m=i

Consequently, two processes 1. suffice to transform U into U1 — and this is all we needed for the proof. This entropy for states (U = Pr^p Tr (UE^)

(En*, ♦) = ||EnHI2 ) ,

-

K

A > iivil

MVii2 ln —h—

2

is no longer subject to the inconveniences of the "macro­ scopic" entropy: In general, it is not constant in time (i.e., in process 2.)> and not = 0 for all states U = Pr^j . In fact: that the Tr (lE^) , from which our entropy is formed, are not time constant in general, was discussed in Note 20^. It is easy to determine when the state U = Pr^-j has the entropy 0 : Since

IIV"2

a l l summands

IIE*!' ^3 3 n

> °' <

— h s— > °> < n

""

2

I'V"

1

1

~

II V I I ln

sn

The Neumann

72 4.

THE MACROSCOPIC MEASUREMENT

in the entropy expression are fore be = 0 . That is,

IIV"2

Compendium

^15

£ 0 . All these must there­

= o, 11

•

The former means that En = 0 , the latter that I lE^I | = \/"s^ , but since I l E ^ M < 1, s n > 1 this implies s n = 1, | lE^II = I U N ; i.e., E ^ = * ; or: s = 1 , * in W . The latter can certainly not hold for two different n , but also, it cannot hold at all because then E = 0 would always be true, and therefore $ = o since 00

L

%=

1

•

n=i n=i

Hence, for exactly one n , is in 2R , and then 3 = 1 . Since we determined that in general all s » 1 , this is impossible. That is, our entropy is always > 0 . Since the macroscopic entropy is time variable, the next question to be answered is this: does it behave like the entropy of the phenomenological thermodynamics in the real world, i.e., does it increase predominantly? This question is answered affirmatively in classical mechanical theory by the so-called Boltzmann H-theorem. In that, however, certain statistical assumptions, the so-called "disorder assumptions" must be made.2 ^ In quantum

20*5

^For the classical H-theorem, see Boltzmann, Vorlesungen uber Gastheorie, Leipzig, 1896, as well as the extremely instructive discussion by P. and T. Ehrenfest in the article cited in Note 185. The "disorder assumptions" which can take the place (in quantum mechanics) of those of Boltzmann

Quantum

k\6

Mechanics

73 V. GENERAL CONSIDERATIONS

mechanics, it was possible for the author to prove the corresponding theorem without such assumptions.2 Since the detailed discussion of this subject, as well as of the ergodic theorem closely connected with it (cf. the refer­ ence in Note 206, where this theorem is also proved) would go beyond the scope of this volume, we cannot report on these investigations. The reader who is interested in this problem can refer to the treatments in the references.

have been formulated by W- Pauli (Sommerfeld-Festschrift, 1928), and the H-theorem is proved there with their help. More recently, the author also succeeded in proving the classical-mechanical ergodic theorem, cf. Proc. Nat. A c , Jan. and March, 1932, as well as the improved treatment of G. D. Birkhoff, Proc Nat. A c , D e c 1931 and March, 1932. 206r Z. Physik, 57 (1929)-

The Neumann

74

C H A P T E R

Compendium

VI

THE MEASURING PROCESS 1 . FORMULATION OP THE PROELM In the discussions so far, we have treated the relation of quantum mechanics to the various causal and statistical methods of describing nature. In the course of this we found a peculiar dual nature of the quantum mechanical procedure which could not he satisfactorily ex­ plained. Namely, we found that on the one hand, a state is transformed into the state ♦' under the action of an energy operator H in the time interval 0 < T < t :

IT EH TT

^ V 3f so if we write

♦

(0 <

\

T

< t) ,

= 0, ♦+. = ♦' > then

♦' = e

*n * i tt H -- ^£± H *

which is purely causal. A mixture transformed into

U' = e U' e

- ^nn I t H

U

Is correspondingly

2^tH

Ue Ue nn

.

Therefore, as a consequence of the causal change of

Quantum Mechanics

*H8

75

VI. THE MEASURING PROCESS

into ' , the states U = P r -, go over into the states IP = P [ — which may measure a quantity with a pure dis­ crete spectrum, distinct eigenvalues and eigenfunctions < > t 1, 4>2, • • • — undergoes in a measurement a non-causal change in which each of the states 4^, 2, .. . can result, and in fact does result with the respective probabilities !(♦> ^ )\2> !(♦, ♦ 2 ) | 2 , - - - - That is, the mixture p°

*' - £ l < * , ^n)l 2p [,n ] n=1

obtains.

More generally, the mixture

U

goes over into

00

U- - Y. n=i

(U

V

*n>PU ] n

(process 1. in V.1.). Since the states go over into mix­ tures, the process is not causal. The difference between these two processes U ► U* is a very fundamental one: aside from the different behaviors in regard to the principle of causal­ ity, they are also different in that the former is (thermodynamically) reversible, while the latter is not (cf. V-3-)Let us now compare these circumstances with those which actually exist in nature or in its observation. First, it is inherently entirely correct that the measure­ ment or the related process of the subjective perception is a new entity relative to the physical environment and is not reducible to the latter. Indeed, subjective per­ ception leads us into the intellectual inner life of the individual, which is extra-observational by its very nature (since it must be taken for granted by any conceivable observation or experiment). (Cf. the discussion above.) Nevertheless, it is a fundamental requirement of the scientific viewpoint — the so-called principle of the

76

The Neumann

1 . FORMULATION OF THE PROBLEM

Compendium

4l 9

psycho-physical parallelism — that it must be possible so to describe the extra-physical process of the subjective perception as if it were in reality in the physical world — i.e., to assign to its parts equivalent physical processes in the objective environment, in ordinary space. (Of course, in this correlating procedure there arises the frequent necessity of localizing some of these processes at points which lie within the portion of space occupied by our own bodies. But this does not alter the fact of their belonging to the "world about us," the objective environment referred to above.) In a simple example, these concepts might be applied about as follows: We wish to measure a temperature. If we want, we can pursue this process numerically until we have the temperature of the environment of the mercury container of the thermometer, and then say: this temperature is measured by the thermometer. But we can carry the calculation further, and from the properties of the mercury, which can be ex­ plained in kinetic and molecular terms, we can calculate Its heating, expansion, and the resultant length of the mercury column, and then say: this length is seen by the observer. Going still further, and taking the light source into consideration, we could find out the reflection of the light quanta on the opaque mercury column, and the path of the remaining light quanta into the eye of the observer, their refraction in the eye lens, and the formation of an image on the retina, and then we would say: this image is registered by the retina of the observer. And were our physiological knowledge more precise than it is today, we could go still further, tracing the chemical reactions which produce the impression of this image on the retina, in the optic nerve tract and in the brain, and then in the end say: these chemical changes of his brain cells are perceived by the observer. But in any case, no matter how far we calculate — to the mercury vessel, to the scale of the thermometer, to the retina, or into the brain, at some

Quantum

Mechanics

420

77 VI. THE MEASURING PROCESS

time we must say: and this Is perceived by the observer. That is, we must always divide the world into two parts, the one being the observed system, the other the observer. In the former, we can follow up all physical processes (in principle at least) arbitrarily precisely. In the latter, this is meaningless. The boundary between the two is arbitrary to a very large extent. In particular we saw in the four different possibilities in the example above, that the observer in this sense needs not to become identified with the body of the actual observer: In one instance in the above example, we included even the ther­ mometer in it, while in another instance, even the eyes and optic nerve tract were not included. That this boundary can be pushed arbitrarily deeply into the interior of the body of the actual observer is the content of the principle of the psycho-physical parallelism — but this does not change the fact that in each method of descrip­ tion the boundary must be put somewhere, if the method Is not to proceed vacuously, i.e., if a comparison with ex­ periment is to be possible. Indeed experience only makes statements of this type: an observer has made a certain (subjective) observation; and never any like this: a physical quantity has a certain valueNow quantum mechanics describes the events which occur in the observed portions of the world, so long as they do not interact with the observing portion, with the aid of the process 2. (V.1.), but as soon as such an inter­ action occurs, i.e., a measurement, it requires the application of process 1. The dual form is therefore justified.207 However, the danger lies in the fact that

2 0 7'N. w

Bohr, Naturwiss. J_7 (1929), was the first to point out that the dual description which is necessitated by the formalism of the quantum mechanical description of nature is fully justified by the physical nature of things that it may be connected with the principle of the psycho-physical parallelism.

The Neumann

78

1 . FORMULATION OP THE PROBLM

Compendium

421

the principle of the psycho-physical parallelism is vio­ lated, so long as it is not shown that the boundary between the observed system and the observer can be displaced arbitrarily in the sense given above. In order to discuss this, let us divide the world into three parts: I, II, III- Let I be the system actually observed, II the measuring instrument, and III the actual observer.2 It is to be shown that the bound­ ary can just as well be drawn between I and II + III as between I + II and III . (In our example above, in the comparison of the first and second cases, I was the system to be observed, II the thermometer, and III the ligjit plus the observer; in the comparison of the second and third cases, I was the system to be observed plus the thermometer, II the light plus the eye of the observer, III the observer, from the retina on; in the comparison of the third and fourth cases, I was everything up to the retina of the observer, II his retina, nerve tracts and brain, III his abstract "ego.") That is, In one case 2. Is to be applied to I , and 1. to the interaction between I and II + III ; and in the other case, 2. to I + II , and 1. to the interaction between I + II and III . (In each case, III itself remains outside of the calculation. ) The proof of this assertion, that both procedures give the same results regarding I (this and only this belongs to the observed part of the world in both cases), is then our problem. But in order to be able to accomplish this successfully, we must first investigate more closely the process of forming the union of two physical. systems (which leads from I and II to I + II) . 208™

The discussion which is carried out in the following, as well as that in VI.3-, contains essential elements which the author owes to conversations with L. Szilard. Cf. also the similar considerations of Heisenberg, in the reference cited in Note 181.

Quantum

Mechanics

79

**22

VI- THE MEASURING PROCESS 2.

COMPOSITE SYSTEMS

As was stated at the end of the preceding sec­ tion, we consider two physical systems I, II (which do not necessarily have the meaning of the I, II above), and their combination I + II . In the classical mechan­ ical method of description, I would have k degrees of freedom, and therefore the coordinates q.1,-..,qk , in place of which we shall use the one symbol q ; corre­ spondingly, let II have 1 degrees of freedom, and the coordinates r^, .. . ,r, which shall be denoted by r . " Therefore, I + II has k + 1 degrees of freedom and the coordinates q., - . . ,q, , r 1 , .. -, r-, , or, more briefly, q, r . In quantum mechanics then, the wave functions of I have the form (q) > those of II the form $(r) and those of I + II the form $(q, r ) . In the corresponding Hilbert spaces ft1, ffiIx, $ I + I 1 , the inner product is defined by / (q)*(q; I t(r)r\{r) dr and // <&(q, r)Y(q, r)dq dr respectively. The physical quan­ tities of I, II, I + II are correspondingly the (hypermaximal ) Hermitian operators A , A, and A in m , m , 1 11 and ft -L" " respectively. Each physical quantity in I is naturally also one in I + II , and in fact its A is to be obtained from its A in this way: to obtain A <&(q, r ) consider r as a constant and apply A to the q function $(q, r ) . This rule of transformation is correct in any case for the coordinate and momentum operators Q ^ - ' - ^ Q ^ and P

1 ' •"•' p k ' i ' e '' q

„

h

d

h

s

T '••' q k'27T "3q^~ ' " ''"251 "3q^

(cf. 1-2. ), and it conforms with the principles I., II . in

209

I t can easily be shown that if A hypermaximal, A is also.

is Hermit ian or

The Neumann

80 2.

Compendium

COMPOSITE SYSTHVIS

423

IV.2. 2 1 0 We therefore postulate them generally. (This is the customary procedure in quantum mechanics.) In the same way, each physical quantity in II Is also one in I + II , and its A gives its A by the same rule: A $(q, r) equals A$(q, r) if in the latter expression, q is taken as constant, and <&(q, r) is considered as a function of r . If ^(q.) (m = 1*2,... ) is a complete orthonormal set in SR and £ n ( r ) (n = 1 > 2 >--- ) one in ® Z1 , then * m | n (q, .r) = <»>ni(qL)ln(r) (m'n = 1 ' 2 '"* > i s clearly one in 91 . The operators A , A, A can there­ fore be represented by matrices t a m i m i ^ ^ a nln f ^ ' a n d to Wi|m«n ,} respectively (m, n', n, n» = 1,2,... ) . 2 1 1 We shall make frequent use of this. The matrix representa­ tion means that 00

00

A

a A* (q) X "mlm.'V-^^ ' m l m . V ^ ' AA*| n(( r )) == ZS a n|n'*n' Vm(( 1 ) = fZ n|n'*n'( (rr)) m' m =i =l nn»=l '=1

and 00

A

V((1'

r )

= Z

°Wi|m'n'Vn'((l'

r )

>

i.e., 21 0

For I . this is clear, and for II . also, so long as only polynomials are concerned. For general functions, It can be inferred from the fact that the correspondence of a resolution of the Identity and a Hermitian operator is not disturbed in our transition A — ► A . 21 1

Because of the large number and variety of indices, we use this method of denoting the matrices, which differs somewhat from the notation used thus far.

Quantum

Mechanics

h2k

81

V I . THE MEASURING PROCESS 00

A* m (q)t n (r)=

Z amnim.n.V^in'*10 * m'n^i

In particular the correspondence A — ►

A means that

00

A* m (Q)e n (r) = < A * m ( q ) ) { n ( r ) =

£

° m | m . V (q)S n
I.e.,

r= 1 , for ^mnlm'n1

'Vim' n|nl

n = nf I \

I 5n|n» -

|^= o , for

n/n

1

D!/

In an analogous fashion, the correspondence A — ► A implies that ofmn|m „. .„. l , = a. . f,8^. . . •^ n n|n mlm* A statistical ensemble in I + II is character­ ized by its statistical operator U or by its matrix

[-o

i_tv,.} - This also determines the statistical properr r mn|m , n , ties of all quantities in I + II , and therefore the properties of the quantities in I also. Consequently there also corresponds to it a statistical ensemble in I alone. In fact, an observer who could perceive only I , and not II , would view the ensemble of systems I + II as one such of systems I . What Is now the statistical operator u or Its matrix t u m i m t) > vhich belongs to this I ensemble? We determine it as follows: The I quantity with the matrix ( a m | m t) h a s t h e matrix ( a m | m » 5 n i n t} a s an I + II quantity, and therefore, by reason of a calcula­ tion in I , It has the00expectation value 00

z /

m|m , m» m' |m |m *-> mlm* m,m T = l

;;

while the calculation in I + II gives

The Neumann

82

2.

Compendium

COMI COMPOSITE SYSTMS

oo

425

00

^ ^mnlm'n^m 1 |m n f |n f m,n,m , n ' = i

^ ^mnlm'n1 m» |m m,mSn=i 00

00

^mnlm'n 1 a m , m m,m = 1 \ n=l / f

In order that both expressions be equal, we must have u

m|m«

=

5>i mnlm'n n=i

In the same way, our I + II ensemble, If only II is considered and I is Ignored, determines a II ensemble, with a statistical operator U and matrix U n | n ,} • 3y analogy, we obtain

un|n» " X»n L umn|mn» • m=i We have thus established the rules of corre­ spondence for the statistical operators of I, II, I + II , i.e., U , U, 17- They proved to be essentially different from those which control the correspondence between the operators A , A, A of physical quantities. It should be mentioned that our u , U, u corre­ spondence depends only apparently on the choice of the complete orthonormal sets %(
Quantum

Mechanics

h2S

83 VI. THE MEASURING PROCESS

answer because all "probability dependencies" which may exist between the two systems disappear as the information is reduced to the sole knowledge of U and U , i.e., of the separated systems I and II . But if one knows the state of I precisely, as also that of II , "probability questions" do not arise, and then I + II , too, Is pre­ cisely known. An exact mathematical discussion is, how­ ever, preferable to these qualitative considerations, and we shall proceed to this. The problem is, then: For two given definite matrices tumimiJ 3n(^L ^nln 1 ^ ' find a third definite matrix {v>ran, m'n* . ,) , such that 00

00

^ ^mnlm'n " u m|m f ' 2-, n=1 m=1

v>'mnlmn* , =

mn|mn

u un|n* t

n|n

(From 00 00

00 00 DO

u = 11 u Z UUm|m m|m " n|n n | n *=1 1>> X >> ZZZ un|n n=l m= 1 m=i n=i

it then follows directly that 00

A ^mn|mn " m,n=i

'

i.e., the correct normalization is obtained.) This prob­ lem is always solvable, for example, ^ ^ ^ = u m | m iU m | n t is always a solution (it can easily be seen that this matrix is definite), but the question arises as to whether this is the only solution. We shall show that this is the case if and only if at least one of the two matrices ^ u m i m t ^ ^ u nln^ i s a state. First we prove the necessity of this condition, i.e., the existence of several solutions if both matrices correspond to mixtures. In such a case (cf. IV.2. )

The Neumann

84

2. uu

ra|m- = aa m m || m m --

Compendium

COMPOSITE SYSTEMS

1*2 7

pw ++ pw

m|mm|m' '' uun|nn | n ' == 7rvVn|nn|n'

+ + 8 w5 w

n|nn|n- **

( v m|m" w m|m- definite and v n | n , , w n | n , also, differing by more than a constant factor, OO 00

00

A vvm|m m|m A m=l a,

00

00

w

^^ Wwmm|m | m ~ £■* n| n £■*vvnn|| n ~~ £-, £-, 2 w nw|nn|n m=i m=i n=l n=l n=i n=i

= 1

p, 7, § > 0, a + p = 1, 7 + 5 = 1 ) . We easily verify that each

m!n mnlm'n

mlm 1 njn 1

mlmf1 n|n f m|m

m|m T n|n n|nff

h TW

mlm m|m'1 w nln n|n' 1

with J« C

+ + cr cr == Of, Of, pp ++ TT == PP, , it it ++ pp == 77, , acr + + xx == 66

it, , a, a, *, pp,

> 0 0 T >

,,

,,

is a solution. Then it, p, a, T can be chosen in an in­ finite number of ways: Because of a + p = 7 + 6 only three of the four equations are independent; therefore, p = 7 - j t , a = of - it, T = ( 5 - o f ) + jt, and in order that all be > 0 , we must require o r - 5 = 7 - p < j t < a , 7 , which is the case for infinitely many it . Now different *, p, a, T lead to different o . . , , because the ran|m,n, v m | m - - v n | n " ••-' w m|m'- w n|n> a r e l l n e a r l y independent, since the vmlm , 1..wm|m , . are such, as well as the v . _.. T nln* w n|n- • Next we prove the sufficiency, and here we may assume that u m | m t corresponds to a state (the other case is disposed of in the same w a y ) . Then U = P r , and since the complete orthonormal set ,2,-«- was arbitrary, we can assume = 0 . u = Pf/K . has the matrix f = 1 , for m = m' = 11 I I u

m|m'

{

1 1 = 0 ,

otherwise

Quantum

Mechanics

85

^28

VI. THE MEASURING PROCESS m = m1 = 1I

= 1 , for

Therefore I

00

I

y » i . »n i n=i

\I

-•

tv. = 0 , otherwise

In particular, for

■

m 4 1, 00

}

2-, ^mnlmn ~ n=i

but since all t>mn|mn , ^ >— o because of the definiteness of ^^i^t^t $™)1 > therefore in this case mnlm'n' C^^irv,^ mn|mn = ( U $„„> mn mn ^m^i*™ mn|mn = ° • That is, ( u *„,„, mn $„„) mn = 0 , and hence, because of the definiteness of U , ( U * , * m , n , ) also = 0 (cf. II.5-, THEOREM 19-), where m», n» are arbitrary. That is, it follows from m 4 1 that v> ._._, = o , and 7 l , ' mn|m n because of the Hennitian nature, this also follows from because of the Hermitian nature, this also follows from m1 / 1 . For m = m1 = 1 however, this gives 00

0

in|lnf

=

Zr ^mnlmn'

=

^In'

•

m=l m=l

Consequently, as was asserted, the solution ^rrmimin i s determined uniquely. We can thus summarize our result as follows: A statistical ensemble in I + II with the operator V = {^rnnimtnt} i s determined uniquely by the statistical ensembles determined by it in I and II individually, with the respective operators U = (umimi} a n d U = ( u , .) , if and only if the following two conditions are satisfied: v 1- umn|m , ,, = w i ,v .„. M ,w. n m|m n|nf . (Prom 00 00

Tr U

=

OO

00 00

X Vi|mn == XX vvm|m m|m XX vvn|n n|n == 11 >> m,n=i m=l n=i

The Neumann

86 COMPOSITE COMPO!

2.

Compendium

SYSTEMS

^2 9

it follows that, by multiplication of v m j m l and v n | n , with two reciprocal constant factors, we can obtain 00

00

X vm|m m= 1

=

1

> X vn|n n=1

But t h e n we s e e t h a t u m | m t = v ^ , , 2. E i t h e r y m | m f = x m x i n l ( I n d e e d U = P r ^ i means t h a t

=

1

u ^ , = v n | n l .) or v n | n l = xnxn .

00

* - X Via > m=1

8Lns3i and therefore u . f = J^mt correspondingly for v I , ; by analogy the same is true with U = Pr.] •) We shall call U and U the projections of u 212 in I and II respectively. We now apply ourselves to the states of I + II , C7= Pr^l . The corresponding wave functions $(q, r ) can be expanded according to the complete orthonormal set

Wq'

r)

= VlHn ( r » : *(q, r *(q, r )) =

00 00

Y, m,n=l m,n=l

ff

n n VV<
We can therefore replace them by the coefficients f (m, n = 1,2,.-.) which are subject only to the condition that 00

f 22 2 IZ llfmnl mnl == HI'*" *ll2 m,n=l m,n=l

be finite. 212,

The projections of a state of I + II are in general mixtures in I or II ; cf. above. This circumstance was discovered by Landau, Z- Physik k-5 (1927).

Quantum

Mechanics

J+30

87 V I . THE MEASURING PROCESS

We can define two operators

F, F

"by

F*(q) = J $(q, r)*(q)dq (F-) F*£(r) = J

o(q, r)|(r)dr •

These are linear, but have the peculiarity of being de­ fined in 91 and 9* respectively, and of taking on values from $ and SR respectively. Their relation is that of adjoints, since obviously (F, |) = (, F £) (the inner product on the left is to be formed in $ and that on the right is to be formed in 91 ). Since the difference of M and W is mathematically unimportant, we can apply the results of 11.11: then, since we are dealing with integral operators, z(F) and z(F ) are equal to

\ \

I*(a. v)\2dadv

=ll*ll2 = ini*ll

in

5RI+I1! ) ,

and are therefore finite. Consequently F, F* are con­ tinuous, in fact are completely continuous operators, and F F as well as FF are definite operators, Tr (F F ) = z(F) = 1, Tr (FF*) = Z(F*) = 1 . If we again consider the difference between 9* IT * and 91 then we see that F F is defined and assumes X

#

XX

values in 91 , and FF similarly in 91 . Since F (q) 00 comes out equal to 00

£

r

mn*n '

n=1 n=1 F has the matrix f ^ p ) ["by use of the complete orthonormal sets % ( C L ) an( i "^n^r^ respectively — note that the latter is a complete orthonormal set along with £ (r) ], likewise F has the matrix f ^ ™ ) (with the same complete orthonormal systems). Therefore F F, FF

The Neumann

88 2.

Compendium

COMPOSITE SYSTEMS

1+31

have the matrices

f

A

mn m'n .n=l (using the complete orthonormal set ^(0.) in 91 ) and mv r . | 1 y Zi Y mnf mn,1 f Z-i mn mn 1 n=1 n=1 (using the complete orthonormal set i n Cr) in fH ). On the other hand, u = Pr^j has the matrix tr f^.^,} (using the complete orthonormal set mn m f n f ^ ^ T+TT (qL r ) = < t > ( q ) l ( r ) *mn ' m n ^ * >' s o t h a t i t s P r o J e c tions in I and II , U and U have the matrices

i Yr f 1 I Lr mn m f n f

ln=l

J

and

I Yr f 1 1 ^» mn mn1

lm=l

J

respectively (with the complete orthonormal sets given above).2 3 Consequently ( U. )

F*F,U U= =FF* FF*• • U u= =F*F,

Note that the definitions (F.) and the equations (U.) make no use of the ♦ . | — hence they are valid independently of these. The operators U , U are completely continuous, and by 11.11. and IV.3-, they can be written in the form

J 213m The mathematical discussion is based on a paper by

E. Schmidt, Math. Ann. 83 (1907).

Quantum

Mechanics

^32

89

VI.

THE MEASURING PROCESS

00

u

- I

00

k=1 k=i

u

" k XK r

= X -kptTlk] i k=1 k=i

K

K*

in which the t k form a complete orthonormal set in 91 * , the T]k one in 91 11 and all w£, w£ > o . We now neglect the terms in each of the two formulas with w£ = 0 or w£ = 0 respectively, and number the remaining terms with k * 1,2,-.. . Then the t k and ijk again form orthonormal, but not necessarily complete sets; the sums M'

M"

1 . 1k=l

k=l k=l

k=l

appear in place of the two

I k=l where M % M" can be equal to «> or finite. Also, all wVL, w£ are now > 0 . Let us now consider a y^. u * k = ^ t ^ and therefore F*F*k = w£yk, FF*F*k = v£Ftk* UFt k = v£Fyk . Furthermore

(P+k, Ptx) = (F F*k, tx) = (u V

= v£(tk,

tl)

r

t

l)

r= w£ , for

ll k = ll

I= o

k 4 l| 1

<

2

, for

1

The | |Ftkl I = W V T—F*k IT then form an orthonormal set in R and they are eigenfunctions of U , with the same eigenvalues as the yk for u (i-e., w£ ). That is, each eigenvalue of u is

therefore, in particular,

The Neumann

90 2.

COMPOSITE SYSTEMS

Compendium

1^33

also one of U with at least the same multiplicity. Interchanging u > U shows that they have the same eigen­ values with the same multiplicities. The w£ and w£ therefore coincide except for their order. Hence M 1 = M" = M , and by re-enumeration of the w £ w e can obtain w k = w k = w k * ^n^i i f t h i s o c c u r s > then we can clearly choose

,k in general.

■ vkP,k

Then

"^=

p#t

»k

=

W~ F * F *k

WV^

k

=

w~

u

=

*k

Wk

K

+k

K

-

K

Therefore

(v.)

,k , k

• v\ FV "•' v^F%k • - ^

F V

k

*

- " ^

p % k

'

2

'

2

Let us now extend the orthonormal set ♦1^y2*«-to a complete ^ > ^2> ' ' '' *i' *?' ' * * anci l i k e w i s e Ti^Tjg, ... to Ti^Hg, ..., T)J,TJ£,... (each of the two sets *]>*£,••• and T)|,TI^,... can be empty, finite or in­ finite, and in addition each set independently of the other set). We have observed before, that (F.), (U.) make no reference to the ♦ , tn . We may therefore use (V. ), as well as the above construction, and let them determine the choice of the complete orthonormal sets . , , . . . and t^9t2'"m • Specifically we let these coincide with the ♦ ^tg*---* tj>t£>--- and T J ^ T ^ , • •., ^ ^ ^ • • • respec­ tively. Now let y v correspond to , TJV to | K K v ^k k (k = 1, . .. ,M) (^1 ,|i2, . . . different from one another, v 1 ,v 2 ,... likewise). Then F Fo *„,n k = =

T/WT 5 VSr i«v

F*m

0

=

V k

for

, vk > m ^ n^ji

.. .

Quantum

Mechanics

91

h3^

VI. THE MEASURING PROCESS

Therefore

r.

C = >/wk , for m = n k , n = v k , k = 1,2, . . . I

f

ran

| L=

>

0 , otherwise

|

or equivalently

M

• (<1, *) = X ^w k ♦ (q)s (r) . k

k=1

k

By suitable choice of the complete orthonorraal sets ^mCo.) an(^- £n^ r ^ w e ^ a v e t n u s established that each column of the matrix ( f ^ ) contains at most one element 4 o (that this is real and > 0 , namely %fwk , is un­ important for what follows). What is the physical meaning of this mathematical statement? Let A be an operator with the eigenfunctions ♦ .,, 4>2> '-' &n& with only distinct eigenvalues, say a 1 ,a 2 ,... ; likewise B with i^lg*--- aj[^L bj,b2,... . A corresponds to a physical quantity in I , B to one in II . They are therefore simultaneously measurable. It is easily seen that the statement "A has the value a. and B has the value b " determines the state <& (q, r) = n mn ^' ♦ p (q)e (r) , and that this2 has the probability ( r$mn ]•* •) = K*> % ) l = l ^ l 2 ^ the state * < c l ' r ) Consequently, our statement means that A, B are simul­ taneously measurable, and that if one of them was measured in $ , then the value of the other is determined by it uniquely. (An a. with all f = 0 cannot result, be­ cause Its total probability -mn' n=1 cannot be

0 , If

a

is ever observed —

therefore for

The Neumann

92

2.

Compendium

COMPOSITE SYSTEMS

*05

e x a c t l y one n , f 4 ° \ l i k e w i s e f o r b . ) That i s , there a r e several possible A values i n t h e s t a t e $ (namely, t h o s e a^ f o r which 00

5>mn"2> ° '

n=l

i.e., for which there exists an n with f „ 4 o — 7 mn ' of possible usually all a are such), and an equal number usually all a are such), and an equal number of possible B values (those b for which 00 00

m n ll 22>> ° Z ll ffm ° '' n=l i.e., for which there exists an m with f / 0 ), but $ establishes a one-to-one correspondence between the possible A values and the possible B values. If we call the possible m values n-,n ,... and the corresponding possible n values v-,v ", •*. , then

t

r'= = c kk 4 0 , for m = >i jikk, n = vkk, k = 1,2,.. 1,2,> •... • • II

mn

j L= 0 >

therefore

'9 otherwise

|

(M finite or » ) M

*(q, r) = Y. c kV ( c ^v v ( r ) y k=1

*

k

hence = |c v | 2 , for m = m1 = n v , k = 1,2, \y ' •

00

^^nm1

{'

Aranm 1 ]! n=i

o

, otherwise

•|

Quantum

Mechanics

93

*06

VI. THE MEASURING PROCESS Ic k | 2 , for n = n' = v k , k = 1,2, ....

r= u

nnf

=

^*mnfmnf j m=l I =

o

, otherwise

I

and therefore M U

M 2p

- I>kl [*M ] ' = Z"ck"2p[|v ] • k=i

U

k

k=i

k

Hence, when $ is projected in I or II , it in general becomes a mixture, while it Is a state in I + II only. Indeed, it involves certain information regarding I + II which cannot be made use of in I alone or in II alone, namely the one-to-one correspondence of the A and B values with each other. For each o we can therefore so choose A, B , i.e., the ♦ and the * , that our condition is satis7 m *n ' fied; for arbitrary A, B , it may of course be violated. Each state o then establishes a particular relation between I and II , while the related quantities A, B depend on $ . How far $ determines them, i.e., the 4>m and the | , is not difficult to answer. If all |ckl are different and 4 o , then U , U (which are determined by o ) determine the respective m, £ n uniquely (cf. IV.3.). The general discussion is left to the reader. Finally, let us mention the fact that for M 4 1 neither U nor U is a state (because all |ckl > 0 ). For M = 1 they both are: u = P r ) ) ) i , U = Pr j . Then $(q> r ) = c.,* ( q ) | (r) . We can absorb cin 4>n (q) . 1 \i v 1 }

T

1

Therefore U , U are states if and only if *(q, r ) has the form ■'♦(q)l(r) , and in that case they are equal to P £ 4 j and P j j respectively. On the basis of the above results, we note: If

The Neumann

94 3-

DISCUSSION OF THE MEASURING PROCESS

Compendium

^37

I is in the state <»(q) and II in the state |(r) , then I + II is in the state *(q, r) = *(q)|(r) • If on the other hand I + II is in a state <&(q, r) which is not a product *(q)i(r) , then I and II are mixtures and not states, but $ establishes a one-to-one correspond­ ence between the possible values of certain quantities in I and in II . 3 • DISCUSSION OP THE MEASURING PROCESS Before we complete the discussion of the measur­ ing process in the sense of the ideas developed in VI.i. (with the aid of the formal tools developed in VI.2.), we shall make use of the results of VI.2. to exclude a possi­ ble explanation often proposed for the statistical charac­ ter of the process 1. (V.i.)- This rests on the following idea: Let I be the observed system, II the observer. If I is in a state u = Pr.i before the measurement, while II on the other hand is in a mixture

' • i V,g , then I + II is a uniquely determined mixture U , and in fact, as we can easily calculate from VI.2., 00

U =m U

r) = # (c1 SX wwnnP P[* [* ]' ] ' *n n (<1'' r ) = **^U ( < l ^nn(r) (r) nn

n=i n=i

If now a measurement of a quantity A takes place in I , then this is to be regarded as an interaction of I and II . This is a process 2. (V.i.), with an energy operator H . If it has the time duration t , then we obtain _ 2*1

t/' = e

^>

t H

2«1

u e

T

t

Quantum

Mechanics

1*38 from

95 V I . THE MEASURING PROCESS

U , and in fact, 00

u' If now each

■-H-*"j - 3«1

tH

e

*n(q, r)

were of the form *n(q)T] (r) , where the y R are the eigenfunctions of A , and the i\ any fixed complete orthonormal set, then this intervention would have the character of a measurement. For it transforms each state ♦ of I into a mixture of the eigenfunctions y of A . The statistical character therefore arises in this way: Before the measurement I was in a (unique) state, but II was a mixture — and the mixture character of II has, in the course of the interaction, associated itself with I + II , and in particular, it has made a mixture of the projection in I . That is, the result of the measurement is indeterminate, because the state of the observer before the measurement is not known exactly. It is conceivable that such a mechanism might function, because the state of information of the observer regarding his own state could have absolute limitations, by the laws of nature. These limitations would be expressed in the values of the w , which are characteristic of the observer alone (and there­ fore independent of * ) • At this point, the attempted explanation breaks down. For quantum mechanics requires that w = (Pj *> ♦) = l(o* * n )l 2 > i-e** w n d e P e n d e n t v n There might exist another decomposition 00

n=i

on

♦l

The Neumann

96

3.

DISCUSSION OF THE MEASURING PROCESS

Compendium

^39

(the <&f(q, r) = t (q)T)n(r) are orthonormal) but this is of no use either; because the w^ are (except for order) determined uniquely by U • (IV. 3-), and are therefore 21 k equal to the w n . Therefore, the non-causal nature of the process 1. is not produced by any incomplete knowledge of the state of the observer, and we shall therefore assume in all that follows that this state is completely known. Let us now apply ourselves again to the problem formulated at the end of VI.1. I, II, III shall have the meanings given there, and, for the quantum mechanical in­ vestigation of I, II , we shall use the notation of VI.2., while III remains outside of the calculations (cf. the discussion of this in VI. 1. ). Let A be the quantity (in I) actually to be measured, 2(q), • • • its eigenfunctions. Let I be in the state 4>(q) . If I is the observed system, II + III the observer, then we must apply the process 1., and we find that the measurement transforms I from the state n (n = 1,2,...) , the probabilities for which are respectively |(«>, 4>n) | 2 (n = 1,2,...) . Now, what is the method of description if I + II is the ob­ served system, and only III the observer? In this case we must say that II is a measuring instrument which shows on a scale the value of A (in I) : the position of the pointer on this scale is a physical quantity B (in II) which is actually observed by III (if II is already within the body of the observer, we have the corresponding physiological concepts in place of the scale and pointer, e.g., retina and image on the retina, etc.) Let A have the values a-,a ,... , B the values b^bp,... , and let the numbering be such that a is associated with b . 2lK

This approach is capable of still more variants, which must be rejected for similar reasons.

Quantum Mechanics

^0

97

vi. THE MEASURING PROCESS

Initially, I is in the (unknown) state (q) and II in the (known) state |(r) , therefore I + II is in the state *(q, r) = (q)|(r) . The measurement (so far as it is performed by II on I) is, as in the earlier example, carried out by an energy operator H (in I + II) in the time t : This is the process 2., which transforms the * into

- 2gl

t|I

*' = e

* -

Viewed by the observer III , one has a measurement only if the following is the case: If III were to measure (by process 1.) the simultaneously measurable quantities A, B (in I or II respectively, or both in I + II) , then the pair of values a , b would "have the probability 0 for m 4 n , and the probability w for m = n . That is, it suffices "to look at" II , and A Is measured In 1 . Quantum mechanics then requires in addition w

*' * n ) | 2 " If this is established, then the measuring process so far as it occurs in II , is "explained" theo­ retically, i.e., the division of I | II + III discussed in VI.1. is shifted to I + II | III The mathematical problem is then the following. A complete orthonormal set ♦1,*2>--- is given in I . Such a set !,,£,>> ••• in JR as well as a state £ in T also an (energy) operator H i n * I+II, and a t , are R , also an (energy) operator H i n * , and to be found so that the following holds- If ♦ is an arbitrary state in R and n

=

l(

*(q, r) = o(q)|(r), *'(q, r) = e> then

*f(q, r) must have the form 00 00

I

n=i n=i

c n ♦n V( QH /) 5In v ( r ')

h

*(q, r) r) , *(q,

The Neumann

98

OF THE MEASURING PROCESS 3. DISCUSSION m (the

c

Compendium

^1

are naturally dependent on ). Therefore

2

|c | ='!(♦, ^n^' 2 * (T*13^ tlie latter is equivalent to the physical requirement formulated above was discussed in VI.2.) In the following we shall use a fixed set I., t , .. . and a fixed £ along with the fixed 4^ , <»2, . . . , and shall investigate the unitary operator - 2«i

A =e

tH

^

instead of H . The mathematical problem leads us back to the problem solved in VI • 2.: there the quantity corresponding to our present $ was given, and we showed the existence of c . , * . Now , e are fixed and $, c are n n n n' s n ' n given dependent on , and it remains so to determine a fixed A that for *» = A* these c . , t result. 7 7 b n n n Is indeed possible. In this case only the principle is of We shall show that such a determination of A is indeed possible. In this case only the principle is of importance to us, i.e., the existence of any such A. The further question, whether the A =e corresponding to simple and plausible measuring arrange­ ments also have this property, shall not concern us. In­ deed, we saw that our requirements coincide with a plausible intuitive criterion of the measurement character in an intervention. Furthermore the arrangements in question are to possess the characteristics of the measurement. Hence quantum mechanics, as applied to observation would be in blatant contradiction with experience, if these A did not satisfy the requirements in question (at least approx­ imately). 2 5 Therefore, in the following, only an abstract 215

Th( The corresponding calculation for the case of the posi-

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99 VI. THE MEASURING PROCESS

A which satisfies our conditions exactly, shall be given. Therefore, let the $ (m = 0, + 1, + 2, .. . ) and the 1 (n = 0, + 1, + 2,... ) respectively be two given ~ I II complete orthonormal sets in 9t and 91 respectively. (We do not let m, n run over 1,2, ... , but over 0, + 1, + 2, . .. • This is purely for technical conven­ ience, and is in principle equivalent to the former). Let the state | be, for simplicity, | . We define the operator A by 00

A

00

£ V ^ ^ n ' ^ = 2 x mn*m (< l ) 6 m+ n (l>) ' m,n=-» m,n=-»

since the * m (q)£ n ( p ) a s v e H a s the V ^ ^ m + n ^ f o r m a complete orthonormal set in 9* , this A is unitary. Now 00

♦ (q) = Y, (*' V ^ V ^ ' 6(r) = l o ( r ) ' m=-a> therefore *(q, r) = ♦(q)t(r) = £ (♦, ♦m>-*m(q)S0(r> > m=-» 00

*'(q, r) = A*(q, r) = ]T (♦, ♦m)-*m(q>*m(r) • Hence our purpose is accomplished. We have in addition c

n - (*> *n> * A better overall view of the mechanism of this process can be obtained if we exemplify it by concrete Schrodinger wave functions, and give H in place of A . The observed object, as well as the observer tion measurement discussed in III-h. is contained in a paper by Weizsacker, Z. Physik 7£ 0931 )•

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Compendium

kk3

(i.e., I and II respectively) may be characterized "by a single variable q and r respectively, running con­ tinuously from - * to + 00 . That is, let both be thougjht of as points which can move along a line. Their ■wave functions then have always the form t(q) and T](r) respectively. We assume that their masses m1 and m are so large that the kinetic energy portion of the energy operator (i.e., ^ - ( ^ J L ) 2 + - I - ( ^ ^ ) 2 ) can be neglected. Then there remains of H only the interaction energy part which is decisive for the measurement. For this we choose the particular form -^r q -^ • The Schrodinger time dependent differential equation then is (for the I + II wave functions ■ +t (q.. r)

):

^r - - ^ r * ^ * t ^

r)

>

(^ + 1 ^)t t (q, r) = 0 , i.e., tt(q, r) = f(q, r - tq) . If, for t = 0 , *0(q, r) = $(q, r) , then we have f(q, r) = $(q, r) , and therefore Vt( r) = *(q, r - tq) In particular, if the initial states of I, II are repre­ sented by (q) and |(r) respectively, then, in the sense of our calculation scheme (if the time t appearing therein is chosen to be 1 ) *(q, r) = o(qh(r) , *'(q, r) = ^ ( q , r) = 4>(q)l(r - q) .

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We now wish to show that this can be used by II for a position measurement of I , I.e., that the coordinates are tied to each other. (Since q, r have continuous spectra, they are therefore measurable with only arbitrary precision, but not with absolute precision. Hence this can be accomplished only approximately. ) For this purpose, we wish to assume that |(r) is different from o only in a very small interval -e
J ||(r)|2dr = i -

The probability therefore that q lies in the interval q 0 - & < q < q 0 + & , and r in the interval ro - 8 ! < r < ro + &« is q +5 r +& 1 Jq o -& r Q -8

q + & " r +& 2

l*'(q, r ) | d q d r =

I

\ - *

If a , r Q are to differ this is 0 , i.e., q, r other that the difference 5 + 6f + € . And for r

V5

| * ( q ) | 2 | | ( r - q)|2dqdr .

r Q -6

by more than 5 + &' + € , then are so very closely tied to each can never be greater than = c^ this is, equal to


If we choose 5» > 5 + e , because of the assumptions on £ . But since we can choose 5, 6», e arbitrarily small (they must be different from zero, however), this means that q, r are tied to each other with arbitrary close­ ness, and the probability density has the value furnished

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by quantum mechanics,

Compendium

U5

|4>(CL)|

That is, the relations of the measurement, as we had discussed them in IV. 1., and in this section, are realized. The discussion of more complicated examples, say of an analog to our four-term example of IV. 1., or the control determination of the validity of a measurement which II carried out on I , effected by a second ob­ server III , can also be carried out in this fashion. It is left to the reader.

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THE LOGIC OF QUANTUM MECHANICS BY GARRBTT BIRKHOFF AND JOHN VON NEUMANN

1. Introduction. One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it pre­ supposes. I t asserts that even a complete mathematical description of a physi­ cal system © does not in general enable one to predict with certainty the result of an experiment on ©, and that in particular one can never predict with cer­ tainty both the position and the momentum of © (Heisenberg's Uncertainty Principle). I t further asserts that most pairs of observations are incompatible, and cannot be made on @ simultaneously (Principle of Non-commutativity of Observations). The object of the present paper is to discover what logical structure one may hope to find in physical theories which, like quantum mechanics, do not con­ form to classical logic. Our main conclusion, based on admittedly heuristic arguments, is that one can reasonably expect to find a calculus of propositions which is formally indistinguishable from the calculus of linear subspaces with respect to set products, linear sums, and orthogonal complements—and resembles the usual calculus of propositions with respect to and, or, and not. In order to avoid being committed to quantum theory in its present form, we have first (in §§2-6) stated the heuristic arguments which suggest that such a calculus is the proper one in quantum mechanics, and then (in §§7-14) recon­ structed this calculus from the axiomatic standpoint. In both parts an attempt has been made to clarify the discussion by continual comparison with classical mechanics and its propositional calculi. The paper ends with a few tentative conclusions which may be drawn from the material just summarized. I. PHYSICAL BACKGROUND

2. Observations on physical systems. The concept of a physically observ­ able "physical system is present in all branches of physics, and we shall assume it. It is clear that an -"observation" of a physical system © can be described generally as a writing down of the readings from various 1 compatible measure­ ments. Thus if the measurements are denoted by the symbols MI > • • • , Mn, -then 1

If one prefers, one may regard a set of compatible measurements as a single composite "measurement"—and also admit non-numerical readings—without interfering with subse­ quent arguments. Among conspicuous observables in quantum theory are position, momentum, energy, and (non-numerical) symmetry.

105 Published in Ann. Math., Vol. 37, pp. 823-843, © 1936 Princeton University Press. Reprinted from John von Neumann Collected Works, ed. A. Taub, Vol. IV, pp. 105-125.

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an observation of © amounts to specifying numbers xif • • ,x„ corresponding to the different MA • It follows that the most general form of a prediction concerning © is that the point (a-i, • • ■ , xH) determined by actually measuring m, • • • , /x„, will lie in a subset S of (xi, • • • , x„)-spaee. Hence if we call the (xl9 • • • , z n )-spaces asso­ ciated with ©, its "observation-spaces," we may call the subsets of the observa­ tion-spaces associated with any physical system ©, the "experimental propo­ sitions" concerning ©. 3. Phase-spaces. There is one concept which quantum theory shares alike with classical mechanics and classical electrodynamics. This is the concept of a mathematical "phase-space." According to this concept, any physical system @ is at each instantly hypothetically associated with a "point" p in a fixed phase-space 2 ; this point is supposed to represent mathematically the "state" of ©, and'the "state" of © is supposed to be ascertainable by "maximal" 2 observations. Furthermore, the point po associated with © at a time Jo, together with a pre­ scribed mathematical "law of propagation," fix the point pt associated with © at any later time t; this assumption evidently embodies the principle of mathe­ matical causation} Thus in classical mechanics, each point of 2 corresponds to a choice of n position and n conjugate momentum coordinates—and the law of propagation may be Newton's inverse-square law of attraction. Hence in this case 2 is a region of ordinary 2n-dimensional space. In electrodynamics, the points of 2 can only be specified after certain functions—such as the electromagnetic and electrostatic potential—are known; hence 2 is a function-space of infinitely many dimensions. Similarly, in quantum theory the points of 2 correspond to so-called "wave-functions," and hence 2 is again a function-space—usually4 assumed to be Hilbert space. In electrodynamics, the law of propagation is contained in Maxwell's equa­ tions, and in quantum theory, in equations due to Schrodinger. In any case, the law of propagation may be imagined as inducing a steady fluid motion in the phase-space. It has proved to be a fruitful observation that in many important cases of classical dynamics, this flow conserves volumes. It may be noted that in quantum mechanics, the flow conserves distances (i.e., the equations are "uni­ tary"). 2 L. Pauling and E. B. Wilson, "An introduction to quantum mechanics," McGraw-Hill, 1935, p. 422. Dirac, "Quantum mechanics/' Oxford, 1930, §4. 3 For the existence of mathematical causation, cf. also p. 65 of Heisenberg's "The physical principles of the quantum theory," Chicago, 1929. 4 Cf. J. von Neumann, "Mathematische Grundlagen der Quanten-mechanik" nik," ]Berlin, 1931. p. 18.

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4. Propositions as subsets of phase-space. Now before a phase-space can become imbued with reality, its elements and subsets must be correlated in some way with "experimental propositions" (which are subsets of different observation-spaces). Moreover, this must be so done that set-theoretical inclu­ sion (which is the analogue of logical implication) is preserved. There is an obvious way to do this in dynamical systems of the classical type. 6 One can measure position and its first time-derivative velocity—and hence momentum—explicitly, and so establish a one-one correspondence which pre­ serves inclusion between subsets of phase-space and subsets of a suitable obser­ vation-space. In the cases of the kinetic theory of gases and of electromagnetic waves no such simple procedure is possible, but it was imagined for a long time that "demons" of small enough size could by tracing the motion of each particle, or by a dynamometer and infinitesimal point-charges and magnets, measure quan­ tities corresponding to every coordinate of the phase-space involved. In quantum theory not even this is imagined, and the possibility of predicting in general the readings from measurements on a physical system © from a knowl­ edge of its "state" is denied; only statistical predictions are always possible. This has been interpreted as a renunciation of the doctrine of pre-determination; a thoughtful analysis shows that another and more subtle idea is involved. The central idea is that physical quantities are related, but are not all computable from a number of independent basic quantities (such as position and velocity). 6 We shall show in §12 that this situation has an exact algebraic analogue in the calculus of propositions. 5. Propositional calculi in classical dynamics. Thus we see that an un­ critical acceptance of the ideas of classical dynamics (particularly as they involve n-body problems) leads one to identify each subset of phase-space with an experimental proposition (the proposition that the system considered has position and momentum coordinates satisfying certain conditions) and con­ versely. This is easily seen to be unrealistic; for example, how absurd it would be to call an "experimental proposition," the assertion that the angular momentum (in radians per second) of the earth around the sun was at a particular instant a rational number! Actually, at least in statistics, it seems best to assume that it is the Lebesguemeasurable subsets of a phase-space which correspond to experimental proposi­ tions, two subsets being identified, if their difference has Lebesgue-measure 0.7 * Like systems idealizing the solar system or projectile motion. * A similar situation arises when one tries to correlate polarizations in different planes of electromagnetic waves. 7 Cf. J. von Neumann, "Operatofenmethoden in der klassischen Mechanik," Annals of Math. 33 (1932), 595-8. The difference of two sets £ , , S% is the set (Si + S2) - Si-S, of those points, which belong to one of them, but not to both.

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But in either case, the set-theoretical sum and product of any two subsets, and the complement of any one subset of phase-space corresponding to experimental propositions, has the same property. That is, by definition8 The experimental propositions concerning any system in classical mechanics^ correspond to a "field" of subsets of its phase-space. More precisely: To the "quotient" of such a field by an ideal in it. At any rate they form a "Boolean Algebra"9 In the axiomatic discussion of propositional calculi which follows, it will be shown that this is inevitable when one is dealing with exclusively compatible measurements, and also that it is logically immaterial which particular field of sets is used. 6. A propositional calculus for quantum mechanics. The question of the connection in quantum mechanics between subsets of observation-spaces (or "experimental propositions") and subsets of the phase-space of a system ©, has not been touched. The present section will be devoted to defining such a connection, proving some facts about it, and obtaining from it heuristically by introducing a plausible postulate, a propositional calculus for quantum me­ chanics. Accordingly, let us observe that if «i, • • • , an are any compatible observations on a quantum-mechanical system © with phase-space 2, then10 there exists a set of mutually orthogonal closed linear subspaces 12, of 2 (which correspond to the families of proper functions satisfying ctif = Xlfi/, • • • , anf = X,-,n/) such that every point (or function) / e 2 can be uniquely written in the form / = cji + c 2 / ? + Czfz + * • • [ / * € ®i] Hence if we state the DEFINITION: By the "mathematical representative" of a subset S of any observation-space (determined by compatible observations a i , • • • , an) for a quantum-mechanical system ©, will be meant the set of all points / of the phasespace of ©, which are linearly determined by proper functions fk satisfying oLifk = Xi/fc, • • • , «n/jt = X„/*, where (Xi, • • • , Xn) e S. Then it follows immediately: (1) that the mathematical representative of any experimental proposition is a closed linear subspace of Hilbert space (2) since all operators of quantum mechanics are Hermitian, that the mathematical representative of the negative11 of any experimental proposition is the orthogonal 8

F. Hausdorff, "Mengenlehre," Berlin, 1927, p. 78. M. H. Stone, "Boolean Algebras and their application to topology," Proc. Nat. Acad. 20 (1934), p. 197. 10 Cf. von Neumann, op. cit., pp. 121, 90, or Dirac, op. cit., 17. We disregard complica­ tions due to the possibility of a continuous spectrum. They are inessential in the pres­ ent case. 11 By the "negative'' of an experimental proposition (or subset S of an observationspace) is meant the experimental proposition corresponding to the set-complement of S in the same observation-space. 9

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109

complement of the mathematical representative of the proposition itself (3) the following three conditions on two experimental propositions P and Q concerning a given type of physical system are equivalent: (3a) The mathematical representative of P is a subset of the mathematical representative of Q. (3b) P implies Q—that is, whenever one can predict P with certainty, one can predict Q with certainty. (3c) For any statistical ensemble* of systems, the probability of P is at most the probability of Q. The equivalence of (3a)-(3c) leads one to regard the aggregate of the mathe­ matical representatives of the experimental propositions concerning any physical system ©, as representing mathematically the propositional calculus for ©. We now introduce the POSTULATE : The set-theoretical product of any two mathematical representatives of experimental propositions concerning a quantum-mechanical system, is itself the mathematical representative of an experimental proposition. REMARKS: This postulate would clearly be implied by the not unnatural conjecture that all Hermitian-symmetric operators in Hilbert space (phasespace) correspond to observables;12 it would even be implied by the conjecture that those operators which correspond to observables coincide with the Hermi­ tian-symmetric elements of a suitable operator-ring M.u Now the closed linear sum Qi + Q2 of any two closed linear subspaces ft» of Hilbert space, is the orthogonal complement of the set-product Qi-ftj of the orthogonal complements 121 of the 12,; hence if one adds the above postulate to the usual postulates of quantum theory, then one can deduce that The set-product and closed linear sum of any two, and the orthogonal complement of any one closed linear subspace of Hilbert space representing mathematically an experimental proposition concerning a quantum-mechanical system ©, itself represents an experimental proposition concerning @. This defines the calculus of experimental propositions concerning ©, as a calculus with three operations and a relation of implication, which closely resembles the systems defined in §5. We shall now turn to the analysis and com­ parison of all three calculi from an axiomatic-algebraic standpoint. II. ALGEBRAIC ANALYSIS

7. Implication as partial ordering. It was suggested above that in any physical theory involving a phase-space, the experimental propositions concern12 I.e., that given such an operator a, one "could" find an observable for which the proper states were the proper functions of a. 13 F. J. Murray and J. v. Neumann, "On rings of operators," Annals of Math., 37 (1936), p. 120. It is shown on p. 141, loc. cit. (Definition 4.2.1 and Lenuna 4.2.1), that the closed linear sets of a ring M—that is those, the "projection operators" of which belong to M— coincide with the closed linear sets which are invariant under a certain group of rotations of Hilbert space. And the latter property is obviously conserved when a set-theoretical intersection is formed.

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ing a system <5 correspond to a family of subsets of its phase-space 2, in such a way that "x implies y" (x and y being any two experimental propositions) means that the subset of 2 corresponding to x is contained set-theoretically in the subset corresponding to y. This hypothesis clearly is important in propor­ tion as relationships of implication exist between experimental propositions corresponding to subsets of different observation-spaces. The present section will be devoted to corroborating this hypothesis by identi­ fying the algebraic-axiomatic properties of logical implication with those of setinclusion. It is customary to admit as relations of "implication," only relations satisfy­ ing S I : x implies x. S2: If a: implies y and y implies z, then x implies z. S3: If x implies y and y implies x, then x and y are logically equivalent. In fact, S3 need not be stated as a postulate at all, but can be regarded as a definition of logical equivalence. Pursuing this line of thought, one can interpret as a "physical quality," the set of all experimental propositions logically equiva­ lent to a given experimental proposition.14 Now if one regards the set Sx of propositions implying a given proposition x as a "mathematical representative" of x, then by S3 the correspondence between the x and the Sx is one-one, and x implies y if and only if Sx C Sy. While con­ versely, if L is any system of subsets X of a fixed class T, then there is an iso­ morphism which carries inclusion into logical implication between L and the system L* of propositions "x is a point of X , " X e L. Thus we see that the properties of logical implication are indistinguishable from those of set-inclusion, and that therefore it is algebraically reasonable to try to correlate physical qualities with subsets of phase-space. A system satisfying S1-S3, and in which the relation "x implies y" is written x C y, is usually15 called a "partially ordered system," and thus our first postu­ late concerning propositional calculi is that the physical qualities attributable to any physical system form a partially ordered system. It does not seem excessive to require that in addition any such calculus contain two special propositions: the proposition Q that the system considered exists, and the proposition (5) that it does not exist. Clearly S4: © C x C D for any x. @ is, from a logical standpoint, the "identically false" or "absurd" proposition ; D is the "identically true" or "self-evident" proposition. 8. Lattices. In any calculus of propositions, it is natural to imagine that there is a weakest proposition implying, and a strongest proposition implied by, 14 Thus in §6, closed linear subspaces of Hilbert space correspond one-many to experi­ mental propositions, but one-one to physical qualities in this sense. 16 F. Hausdorff, "Grundzuge der Merigenlehre," Leipzig, 1914, Chap. VI, §1.

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a given pair of propositions. In fact, investigations of partially ordered systems from different angles allindicate that the first property which they are likely to possess, is the existence of greatest lower bounds and least upper bounds to sub­ sets of their elements. Accordingly, we state DEFINITION: A partially ordered system L will be called a "lattice" if and only if to any pair x and y of its elements there correspond S5: A "meet" or "greatest lower bound" x f) y such that (5a) x fi y x, (6b) x U y 3 y f (6c) w 3 x a n d w Z)y imply t o D x U i / . The relation between meets and joins and abstract inclusion can be sum­ marized as follows,16 (8.1) In any lattice L, the following formal identities are true, Ll: L2: L3: L4:

a a a a

0 f) fl U

a = a and a U a = a. b = b 0 a and a U b = b U a. (b 0 c) = (a fl b) D c and a U (b U c) = (a U b) U c. (a fl 6) = a fl (a U 6) = a.

Moreover, the relations a 3 b, a 0 6 = 6, and a U 6 = a are equivalent—each implies both of the others. (8.2) Conversely, in any set of elements satisfying L2-L4 (Ll is redundant), a fl b = b and a U b = a are equivalent. And if one defines them to mean a 3 by then one reveals L as a lattice. Clearly L1-L4 are well-known formal properties of and and or in ordinary logic. This gives an algebraic reason for admitting as a postulate (if necessary) the statement that a given calculus of propositions is a lattice. There are other reasons17 which impel one to admit as a postulate the stronger statement that the set-product of any two subsets of a phase-space which correspond to physical qualities, itself represents a physical quality—this is, of course, the Postulate of §6. It is worth remarking that in classical mechanics, one can easily define the meet or join of any two experimental propositions as an experimental proposi­ tion—simply by having independent observers read off the measurements which either proposition involves, and combining the results logically. This is true in quantum mechanics' only exceptionally—only when all the measurements in­ volved commute (are compatible); in general, one can only express the join or 16 The final result was found independently by O. Ore, "The foundations of abstract algebra. I.," Annals of Math. 36 (1935), 406-37, and by H. MacNeille in his Harvard Doc­ toral Thesis, 1935. 17 The first reason is that this implies no restriction on the abstract nature of a lattice— any lattice can be realized as a system of its own subsets, in such a way that a fj b is the setproduct of a and b. The second reason is that if one regards a subset S of the phase-space of a system © as corresponding to the certainty of observing © in S, then it is natural to assume that the combined certainty of observing <& in S and T is the certainty of observing © in ST = S fl T,—and assumes quantum theory.

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meet of two given experimental propositions as a class of logically equivalent experimental propositions—i.e., as a physical quality }* \ 9. Complemented lattices. Besides the (binary) operations of meet- and join-formation, there is a third (unary) operation which may be defined in par­ tially ordered systems. This is the operation of complementation. In the case of lattices isomorphic with "fields" of sets, complementation corre­ sponds to passage to the set-complement. In the case of closed linear subspaces of Hilbert space (or of Cartesian n-space), it corresponds to passage to the orthogonal complement. In either case, denoting the "complement'' of an element a by a', one has the formal identities, L71: (a')' = a. L72: a fl a' = © and a U a! = Q. L73: a C b implies a! Z) &'. By definition, L71 and L73 amount to asserting that complementation is a "dual automorphism" of period two. I t is an immediate corollary of this and the duality between the definitions (in terms of inclusion) of meet and join, that L74: (a fl 6)' = a' U V and (a U &)' = a' fl 5' and another corollary that the second half of L72 is redundant. [Proof: by L71 and the first half of L74, (a U a') = (a" U a') = (a' fl a)' = (§)', while under inversion of inclusion (5) evidently becomes Q.] This permits one to deduce L72 from the even weaker assumption that a da! implies a = (5). Proof: for any x, (x fl x')' = (x' U x") = x' U x IDx fl x'. Hence if one admits as a postulate the assertion that passage from an experi­ mental proposition a to its complement a' is a dual automorphism of period two, and a implies a' is absurd, one has in effect admitted L71-L74. This postulate is independently suggested (and L71 proved) by the fact the "complement" of the proposition that the readings xi, • • • , xn from a series of compatible observations n\, • • • , \in lie in a subset S of (xi, • • • , x n )-space, is by definition the proposition that the readings lie in the set-complement of S. 10. The distributive identity. Up to now, we have only discussed formal features of logical structure which seem to be common to classical dynamics and the quantum theory. We now turn to the central difference between them—the distributive identity of the propositional calculus: L6: a U (6 fl c) = (a U b) fl (a U c) and a 0 (6 U c) = (a (1 b) U (a (1 c) which is a law in classical, but not in quantum mechanics. 18

The following point should be mentioned in order to avoid misunderstanding: If a, b are two physical qualities, then a U 6, a f\ b and a' (cf. below) are physical qualities too (and so are O and Q-f). But a C b is not a physical quality; it is a relation between physi­ cal qualities.

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From an axiomatic viewpoint, each half of L6 implies the other.19 Further, either half of L6, taken with L72, implies L71 and L73, and to assume L6 and L72 amounts to assuming the usual definition of a Boolean algebra.20 From a deeper mathematical viewpoint, L6 is the characteristic property of set-combination. More precisely, every "field'' of sets is isomorphic with a Boolean algebra, and conversely.21 This throws new light on the well-known fact that the propositional calculi of classical mechanics are Boolean algebras. It is interesting that L6 is also a logical consequence of the compatibility of the observables occurring in a, 6, and c. That is, if observations are made by inde­ pendent observers, and combined according to the usual rules of logic, one can prove L1-L4, L6, and L71-74. These facts suggest that the distributive law may break down in quantum mechanics. That it does break down is shown by the fact that if a denotes the experimental observation of a wave-packet \p on one side of a plane in ordinary space, a' correspondingly the observation of ^ on the other side, and b the obser­ vation of ^ in a state symmetric about the plane, then (as one can readily check): b fl (a U a') = b fl D = b > © = (b PI a) = (6 fl a')

= (b fl a) U (6 n a') REMARK: In connection with this, it is a salient fact that the generalized distributive law of logic:

L6*: ft(±*.*)= E ( f t «•.>(.)) breaks down in the quotient algebra of the field of Lebesgue measurable sets by the ideal of sets of Lebesgue measure 0, which is so fundamental in statistics and the formulation of the ergodic principle.22 11. The modular identity. Although closed linear subspaces of Hilbert space and Cartesian n-space need not satisfy L6 relative to set-products and closed linear sums, the formal properties of these operations are not confined to L1-L4 and L71-L73. In particular, set-products and straight linear sums are known23 to satisfy the so-called "modular identity." 19

R. Dedekind, 'Werfce," Braunschweig, 1931, vol. 2, p. 110. G. Birkhoff, uOn the combination of subalgebras," Proc. Camb. Phil. Soc. 29 (1933), 441-64, §§23-4. Also, in any lattice satisfying L6, isomorphism with respect to inclusion implies isomorphism with respect to complementation; this need not be true if L6 is not assumed, as the lattice of linear subspaces through the origin of Cartesian n-space shows. 21 M. H. Stone, "Boolean algebras and their application to topology," Proc. Nat. Acad. 20 (1934), 197-202. 22 A detailed explanation will be omitted, for brevity; one could refer to work of G. D . Birkhoff, J. von Neumann, and A. Tarski. 23 G. Birkhoff, op. cit., §28. The proof is easy. One first notes that since a C (a U b) f| c if a C c, and b f) c C (o U b) 0 c in any case, a U (6 f) c) C (a U b) f) c. Then one notes 20

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L5: I f a C c , then a U (6 fl c) = (a U b) D c. Therefore (since the linear sum of any two finite-dimensional linear subspaces of Hilbert space is itself finite-dimensional and consequently closed) set-products and closed linear sums of the finite dimensional subspaces of any topological linear space such as Cartesian n-space or Hilbert space satisfy L5, too. One can interpret L5 directly in various ways. First, it is evidently a re­ stricted associative law on mixed joins and meets. It can equally well be re­ garded as a weakened distributive law, since if a Cl c, then a U (b f] c) = (a fl c) U (6 fl c) and (a U b) fl c = (a U b) 0 (a U c). And it is self-dual: replacing C , fl, U by 3 , U, fl merely replaces a, 6, c, by c, 6, a. Also, speaking graphically, the assumption that a lattice L is "modular" (i.e., satisfies L5) is equivalent to24 saying that L contains no sublattice isomorphic with the lattice graphed in fig. 1 :

FIG.

1

Thus in Hilbert space, one can find a counterexample to L5 of this type. Denote by £i, £2, £3, • • • a basis of orthonormal vectors of the space, and by a, bf and c respectively the closed linear subspaces generated by the vectors (£2* + 10-"£i -f 10_,2n£2„+i), by the vectors £2n, and by a and the vector £ t . Then a, b, and c generate the lattice of Fig. 1. Finally, the modular identity can be proved to be a consequence of the assump­ tion that there exists a numerical dimension-function d(a)f with the properties D l : If a > b, then d(a) > d(b). D2: d^a) + d(b) = d(a f) b) + d(a U b). This theorem has a converse under the restriction to lattices in which there is a finite upper bound to the length n of chains25 (5) < aY < a2 < • • < an < Q of elements. Since conditions D1-D2 partially describe the formal properties of prob­ ability, the presence of condition L5 is closely related to the existence of an t h a t any vector in (a U b) f] c can be written £ = a + /3 [a e a, /3 e b, £ e c\. r But 0 = £ — a is in c (since £ e c and a e a C c); hence £ = « + 0 « a U (6 fl <•), and a U (6 f| c) Z> (a U 6) f| c, completing the proof. 24 R. Dedekind, "Werke," vol. 2, p. 2.r>:>. 25 The statements of this paragraph are corollaries of Theorem 10.2 of G. Birkhoff, op. cit.

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"a priori thenno-dynamic weight of states." But it would be desirable to interpret L5 by simpler phenomenological properties of quantum physics. 12. Relation to abstract projective geometries. We shall next investigate how the assumption of postulates asserting that the physical qualities attrib­ utable to any quantum-mechanical system © are a lattice satisfying L5 and L71-L73 characterizes the resulting propositional calculus. This question is evidently purely algebraic. We believe that the best way to find this out is to introduce an assumption limiting the length of chains of elements (assumption of finite dimensions) of the lattice, admitting frankly that the assumption is purely heuristic. It is known26 that any lattice of finite dimensions satisfying L5 and L72 is the direct product of a finite number of abstract projective geometries (in the sense of Ycblen and Young), and a finite Boolean algebra, and conversely. REMARK: It is a corollary that a lattice satisfying L5 and L71-L73 possesses independent basic elements of which any element is a union, if and only if it is a Boolean algebra. Again, such a lattice is a single projective geometry if and only if it is irre­ ducible—that is, if and only if it contains no "neutral" elements.27 XJ^ (5), Q such that a = (o fl x) U (a fl x') for all a. In actual quantum mechanics such an element would have a projection-operator, which commutes with all projec­ tion-operators of observables, and so with all operators of observables in general. This would violate the requirement of "irreducibility" in quantum mechanics.28 Hence we conclude that the propositional calculus of quantum mechanics has the same structure as an abstract projective geometry. Moreover, this conclusion has been obtained purely by analyzing internal properties of the calculus, in a way which involves Hilbert space only indirectly. 13. Abstract projective geometries and skew-fields. We shall now try to get a fresh picture of the propositional calculus of quantum mechanics, by recalling the well-known two-way correspondence between abstract projective geometries and (not necessarily commutative) fields. Namely, let F be any such field, and consider the following definitions and con­ structions : n elements Xi, • • • , xn of F, not all = Oy form a right-ratio [xi: • • •: xn]r, two right-ratios [x\: • • • :xn]r, and [£r. • • • :£ n ] r being called "equal," if and only if a z e F with £,: = x,z, i — 1, • • • , n, exists. Similarly, n elements yiy • • • , yn of F, not all = 0, form a left-ratio [yii • • • :yn]if two left-ratios [yr. • • • :yn]i and [ryi: • • • :rjn]i being called "equal," if and only if a z in F with rj{ = zyi) i = 1, . . . t n, exists. 26 G. Birkhoff ''Combinatorial relations in projective geometries" Annals of Math. 36 (1935), 743-8. 27 O. Ore, op. cit., p. 419. 28 Using the terminology of footnote,1* and of loc. cit. there: The ring MM' should con­ tain no other projection-operators than 0, 1, or: the ring M must be a "factor." Cf. loc. cit.13, p. 120.

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Now define an n — 1-dimensional projective geometry Pn-i(F) as follows: The "points" of Pu-i(F) are all right-ratios [xii • • • :x n ] r . The "linear subspaces" of Pm-i(F) are those sets of points, which are defined by systems of equations (XklXi +

• • • + OCknXn = 0,

k = 1, • • • , Til.

(m = 1, 2, • • • , the a** are fixed, but arbitrary elements of F). The proof, that this is an abstract projective geometry, amounts simply to restating the basic properties of linear dependence.29 The same considerations show, that the (n — 2-dimensional) hyperplanes in Pm-i(F) correspond to m = 1, not all a, = 0. Put an = y,-, then we have (*)

2/1*1 + • • • + VnXn = 0,

not all y{ = 0.

This proves, that the (n — 2-dimensional) hyperplanes in P m _i(P) are in a oneto-one correspondence with the left-ratios [yr: • • • :yn]i. So we can identify them with the left-ratios, as points are already identical with the right-ratios, and (*) becomes the definition of "incidence" (point C hyperplane). Reciprocally, any abstract n — 1-dimensional projective geometry Qn_i with n = 4, 5, • • belongs in this way to some (not necessarily commutative field F(QM_0, and Q„_i is isomorphic with Pn-i{F{Qn^))^ 14. Relation of abstract complementarity to involutory anti-isomorphisms in skew-fields. We have seen that the family of irreducible lattices satisfying L5 and L72 is precisely the family of projective geometries, provided we exclude the two-dimensional case. But what about L71 and L73? In other words, for which P„_i(P) can one define complements possessing all the known formal properties of orthogonal complements? The present section will be spent in answering this question.30* " Cf. §§103-105 of B. L. Van der Waerden's "Moderne Algebra," Berlin, 1931, Vol. 2. 30 n — 4, 5, • • • means of course n — 1 ^ 3, that is, that Qn-i is necessarily a "Desarguesian" geometry. (Cf. O. Veblen and J. W. Young, "Projective Geometry," New York, 1910, Vol. 1, page 41). Then F = F(Qn-i) can be constructed in the classical way. (Cf. Veblen and Young, Vol. 1, pages 141-150). The proof of the isomorphism between Q„_i and the P«_i(F) as constructed above, amounts to this: Introducing (not necessarily commutative) homogeneous coordinates X\, • • • , xn from F in Q n -i, and expressing the equations of hyper­ planes with their help. This can be done in the manner which is familiar in projective geometry, although most books consider the commutative ("Pascalian") case only. D. Hilbert, "Grundlagen der Geometrie" 7th edition, 1930, pages 96-103, considers the noncommutative case, but for afnne geometry, and n — 1 = 2, 3 only. Considering the lengthy although elementary character of the complete proof, we pro­ pose to publish it elsewhere. 30a R. Brauer, "A characterization of null systems in projective space," Bull. Am. Math. Soc. 42 (1936), 247-54, treats the analogous question in the opposite case that X C\ X' ?* 0 is postulated.

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First, we shall show that it is sufficient that F admit an involutory aniisomorphism W: x = W(x), that is: Ql. w(w(u)) = u, Q2. w(u -f v) = w(w) + w(v)f Q3. w(wi;) = w(v) w(u), with a definite diagonal Hermitianform w(x 1)71 £1 -f • • • + w(£n)7nfn, where Q4. w(xi)yixi

+ • • • + w(xn)ynxn

= 0 implies xx = .... = x n = 0,

the 7» being fixed elements of F, satisfying 10(7») = 7*. Proof: Consider ennuples (not right- or left-ratios!) x:(xi, • •• , x n ), £: (£i> • • • > £n) of elements of F. Define for them the vector-operations xz: (xxz, . . . , xnz)

(zin F),

* + £: (*i + $1, • • - , * » + fn), and an "inner product" (£ix) = w(£ 1)71X1 + . •. +

w(Zn)ynxn.

Then the following formulas are corollaries of Q1-Q4. IP1 IP2

(x, Q = w((it, x)), (£, xu) = (£, z)u, (€U, X) = w(u)U, x),

IP4

(x, x) = w((x, x)) = [x] is 5* 0 if x ^ 0 (that is, if any x» 7* 0).

iP3 (ff x' + x") = a, xo + a, x"), (r + r,*) = («', *) + a", *),

We can define x J_ £ (in words: "x is orthogonal to £") to mean that (£, x) = 0. This is evidently symmetric in x, £, and depends on the right-ratios [xi: • • • :x n ] r , [£1* • * • '£n]r only so it establishes the relation of "polarity," a i . 6 , between the points a: [xr. • • • :x n ] r ,

6: [ft: •. • :{J r of P n -i(F).

The polars to any point b: [ft: • • • :£»]r of P n _i(F) constitute a linear subspace of points of P„_i(F), which by Q4 does not contain b itself, and yet with b gener­ ates whole projective space P n _i(F), since for any ennuple x: (x^ • • - , x„)

x = x' + Mfl-K*,*) where by Q4, [£] -^ 0, and by IP (£, x') == 0. This linear subspace is, therefore, an n-2-dimensional hyperplane. Hence if c is any fc-dimensional element of P n _i(F)i one can set up inductively k mutually polar points 6 (l) , • • • , ¥k) in c. Then it is easy to show that the set c' of points polar to every 6(1), — • , ¥k)—or equivalently to every point in c— constitute an n-fc-1-dimensional element, satisfying c f) cf = (3) and c U c' = Q. Moreover, by symmetry (c')' 3 c, whence by dimensional considerations c" = c. Finally, c D d implies c' C d', and so the correspondence c—*d defines an involutory dual automorphism of P n -i(F) completing the proof.

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In the Appendix it will be shown that this condition is also necessary. Thus the above class of systems is exactly the class of irreducible lattices of finite dimensions > 3 satisfying L5 and L71-L73. III.

CONCLUSIONS

15. Mathematical models for propositional calculi. One conclusion which can be drawn from the preceding algebraic considerations, is that one can construct many different models for a propositional calculus in quantum me­ chanics, which cannot be differentiated by known criteria. More precisely, one can take any field F having an involutory anti-isomorphism satisfying Q4 (such fields include the real, complex, and quaternion number systems 31 ), introduce suitable notions of linear dependence and complementarity, and then construct for every dimension-number n a model Pn(F), having all of the properties of the propositional calculus suggested by quantum-mechanics. One can also construct infinite-dimensional models P^F) whose elements consist of all closed linear subspaces of normed infinite-dimensional spaces. But philosophically, HankeFs principle of the "perseverance of formal laws" (which leads one to try to preserve L5)32 and mathematically, technical analysis of spectral theory in Hilbert space, lead one to prefer a continuous-dimensional model PC(F), which will be described by one of us in another paper.33 PC(F) is very analogous with the model furnished by the measurable subsets of phase-space in classical dynamics. 34 16. The logical coherence of quantum mechanics. The above heuristic considerations suggest in particular that the physically significant statements in quantum mechanics actually constitute a sort of projective geometry, while the physically significant statements concerning a given system in classical dynamics constitute a Boolean algebra. They suggest even more strongly that whereas in classical mechanics any propositional calculus involving more than two propositions can be decomposed into independent constituents (direct sums in the sense of modern algebra), quantum theory involves irreducible propositional calculi of unbounded com­ plexity. This indicates that quantum mechanics has a greater logical coherence 31 In the real case, w(x) — x; in the complex case, w(x + iy) = x — iy; in the quaternionic case, w{u + ix + jy + kz) = u — ix — jy — kz; in all cases, the X» are 1. Conversely, A. Kolmogoroff, "Zur Begriindung der projektiven Geometrie," Annals of Math. 33 (1932), 175-6 has shown that any projective geometry whose A:-dimensional elements have a locally compact topology relative to which the lattice operations are continuous, must be over the real, the complex, or the quaternion field. 32 L5 can also be preserved by the artifice of considering in P^F) only elements which cither are or have complements which are of finite dimensions. 33 J. von Neumann, "Continuous geometries/* Proc. Nat. Acad., 22 (1936), 92-100 and 101-109. These may be a more suitable frame for quantum theory, than Hilbert space. 34 In quantum mechanics, dimensions but not complements are uniquely determined by the inclusion relation; in classical mechanics, the reverse is true!

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than classical mechanics—a conclusion corroborated by the impossibility in general of measuring different quantities independently. 17. Relation to pure logic. The models for propositional calculi which have been considered in the preceding sections are also interesting from the standpoint of pure logic. Their nature is determined by quasi-physical and technical reasoning, different from the introspective and philosophical considera­ tions which have had to guide logicians hitherto. Hence it is interesting to com­ pare the modifications which they introduce into Boolean algebra, with those which logicians on "intuitionist" and related grounds have tried introducing. The main difference seems to be that whereas logicians have usually assumed that properties L71-L73 of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities L6 as the weakest link in the algebra of logic. Cf. the last two paragraphs of §10. Our conclusion agrees perhaps more with those critiques of logic, which find most objectionable the assumption that a' U b = Q implies a C b (or dually, the assumption that a f) b' = (§) implies b 3 a—the assumption that to deduce an absurdity from the conjunction of a and not b, justifies one in inferring that a implies b).zb 18. Suggested questions. The same heuristic reasoning suggests the follow­ ing as fruitful questions. What experimental meaning can one attach to the meet and join of two given experimental propositions? What simple and plausible physical motivation is there for condition Lo? APPENDIX

1. Consider a projective geometry Qn_i as described in §13. F is a (not neces­ sarily commutative, but associative) field, n = 4, 5, • • • , Q n -i = Pn-i(F) the projective geometry of all right-ratios [xii • • • :xn]ry which are the points of Qn-i- The (n — 2-dimensional) hyperplanes are represented by the left-ratios [y\m-' • • Vn]iy incidence of a point [xii • • • x„]r and of a hyperplane [yx\ • • • :yn]i being defined by (1)

t , Vi Xi = 0 i = 1

All linear subspaces of Q„_i form the lattice L, with the elements a, 6, c, • • • . Assume now that an operation a' with the properties L71-L73 in §9 exists: L71 (a')' = a L72 a fl a' = ® and a U a' = Q, L73 a db implies a' Z> br. 35

It is not difficult to show, that assuming our axioms LI-ft and 7. the distributive law L(i is equivalent to this postulate: a' \J b = [] implies a C b.

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They imply (cf. §9) L74

(a fl b)f = a' U 6' and (a U 6)' = a' H 6'.

Observe, that the relation a C 6' is symmetric in a, 6, owing to L73 and L71. 2. If a: [xii • • • :# n ] r is a point, then a' is an foi: • • • ?/„]/. So we may write: • • • :vn]i, [[xii * i : -•■ • :-* :x • • ]n ,]^== for. for----:ifn]i,

(2)

and define an operation which connects right- and left-ratios. We know from §14, that a general characterization of a' (a any element of L) is obtained, as soon as we derive an algebraic characterization of the above [xii • • • 'xn]r • We will now find such a characterization of [xii • • • :x„] r , and show, that it justifies the description given in §14. In order to do this, we will have to make a rather free use of collineations in Qn-i. A collineation is, by definition, a coordinate-transformation, which replaces [xx: • • • :xn]r by [x t : • • • :£„] r , n

w Xj »v z» Z; = 2 S3 w»7 #»

(3)

for; = 1, • • • , n.

i-2

Here the o># are fixed elements of F, and such, that (3) has an inverse. n

(4)

Xi •W = X / -. v»y 0«v 37/, *».

for t = 1, • • •, n,

==

f - l

the On being fixed elements of F, too. S*( *« ==

(3), (4) clearly mean

(l if fc k = 1 (1 (00 if if kk ** 11

n

(5)

:

n

/_j 6,A, 2Lf ft; ft; «*; «*; == &A,

W»/ftifcOik= =5/Jfc. 221/ ^ <*>%} djk-

j/ = 1

t = 1

Considering (1) and (5) they imply the contravariant coordinate-transformation for hyperplanes: [yii • • • :y n ]/ becomes [t/r. • • • :£„]/, where n

(6)

& V< hi, ft = £ ViOijf

for j == 1,1,. .... . , ,n,n, forj

i = 1

n

(7)

3/< = £

&■«&•»

;= i

fort = 1,

n.

(Observe, that the position of the coefficients on the left side of the variables in (4), (5), and on their right side in (6), (7), is essential!) 3. We will bring about W

:5,„] [5 tl :.• •••• :$,„], [5,-i: • • • :5 :6in]i t n ] r = [Sal

for for ii == 1, 1, •••• ,, n, n,

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by choosing a suitable system of coordinates, that is, by applying suitable collineations. We proceed by induction: Assume that (8) holds for i = 1, . -. , m — l(ra = 1, • • • , n), then we shall find a collineation which makes (8) true for i = 1, • • • , ra. Denote the point [d{i:. - • :8in]r by p*, and the hyperplane [fc:. •. :5<»]| by h* our assumption on (8) is: p*' = h* for i = 1, . . - , m — 1. Consider now a point a: [x\\ • • • :x n ] r , and the hyperplane a': [i/ii • • • :i/n]j. Now a ^ p*' = h* means (use (1)) xt- = 0, and p? ^ a' means (use (8)) y{ = 0. But these two statements are equivalent. So we see: If i = 1, • • • , m — 1, then x,- = 0 and y% = 0 are equivalent. Consider now p*: [5mi: • •. :5 mn ] r . Put p i : [*/!:••• :*/*]*• As 8™i = 0 f or i = 1, • • • , m — 1, so we have t/< = 0 for i = 1, • • • , m — 1. Furthermore, P* A p*' = 0, p* ^ 0, so p* not ^ p*'. By (1) this means yl ^ 0. Form the collineation (3), (4), (6), (7), with Ba = an = 1,

6mi = cotm = t/*"1 y*

for t = m + 1, • • • , n,

all other 0„, «# = 0. One verifies immediately, that this collineation leaves the coordinates of the pt: [da: • •• :5tn]r, i = 1, • • • , n, invariant, and similarly those of the pV: [&n:' • • '!>im]h i = 1» * * * m ~~ 1J while it transforms those of

pi': lifT:---:!*]! into [5mi:- • -:5 mn ]/. So after this collineation (8) holds for i = 1, • • • , m. Thus we may assume, by induction over ra = 1, • • • , n, that (8) holds for all i = 1, • • • , n. This we will do. The above argument now shows, that for a: [xx: • • • :x n ] r , a':fe/i*.• • • :t/n]*, (9)

Xi = 0 is equivalent to y* = 0,

for t = 1, • • • , n.

4. Put a: [xi:...:xn]r,a': [yi:---:y n ]i, and 6: [£r. • • • :£n]r,&': [i?i: • • • :*?n]z. Assume first in = 1, T?2 = *?, 773 = • • • = i?n = 0. Then (9) gives {1 ^ 0, so we can normalize £ 1 = 1, and £3 = . . . = £n = 0. £2 can depend on 772 = v only, so £2 = ft(rj). Assume further Xi = 1. Then (9) gives yx ?± 0, so we can normalize t/i = 1, Now a ^ V means by (i) 1 + TJX2 = 0, and b ^ a! means 1 + 2/2/2(1?) = 0. These two statements must, therefore, be equivalent. So if x 2 5* 0, we may put T; = - x i 1 , and obtain y2 = - (/«(if))_1 = - foC- si"1))"1- If x2 = 0, then 1/2 = 0 by (9). Thus, x 2 determines at any rate y2 (independently of Xz, - - yXn):y2 = (p2(x2). Permuting the i = 2, • • • , n gives, therefore: There exists for each i = 2, • • • , n a function ^?,(x), such that y* = ^»(x,). Or: (10)

If a: [l:x 2 : • • • :x„]r,

then a': [l:^ 2 (x 2 ): • • • :^n(xn)L.

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Applying this to a: [lix*: • • • :xn]r and c: [l:uii • • • :un]r shows: A s a ^ c ' and c ^ a' are equivalent,.so n

(11)

n

]T) ^t(tZt) x» = — 1 is eauivalent to J 3 ^fo) w* = — 1. i - 2

t=2

Observe, that (9) becomes: (12)


5. (11) with x 3 = • • • = xn = uz = • • • = un = 0 shows: ^2(^2)22 = — 1 is equivalent to ^2(^2)^2 = — 1. If £2 F^ 0, W2 = (•—¥>2(z2))-1, then the second equation holds, and so both do. Choose x2, u2 in this way, but leave x*, • • • , xnj Uz, • • • un arbitrary. Then (11) becomes: n

(13)

n

]C ^t(^t) #* = 0 is equivalent to ]T^
% - 3

Now put x6 = • • • = £n = ^6 = • • • = un = 0.

Then (13) becomes:

^3(^3)23 + ^4(^4)0:4 = 0 is equivalent to ^3(^3)^3 +


= 0,

that is (for x 4 , u± 5^ 0): (a)

xzx~ll =
(b)

usulx

(14) is equivalent to =
Let a:4, x 3 be given. Choose uZi uA so as to satisfy (b). Then (a) is true, too. Now (a) remains true, if we leave uZj u4 unchanged, but change x 3 , x4 without changing x3x^1. So (b) remains too true under these conditions, that is, the value of
ipzixz) =
This was derived for x, z 9^ 0, but it will hold for x or z = 0, too, if we define ^34(0) = 0. (Use (12).) (15), with z = 1 gives

where a t ; 9^ 0.

(For i = j put an = 1.) Now (15) becomes

(17)


=

Of42^34(2)«23.

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123

Put x = 1 in (17), write x for z, and use (16) with j = 2:
(18)

where 0, 7,- 5* 0.

(j8 =
w(tm) = 10(^)10(1;)

results (12) and (18) give (20)

w(u) = 0 if and only if u = 0.

Now write w(z)y 7* for pw{z)p~l, fa. Then (18), (19), (20) remain true, (18) is simplified in so far, as we have ft = 1 there. So (11) becomes (21)

£

ti
» «2

(21) is equivalent to n

£

w(x,-)7,-tti = - 1

t =2

£2 = x} Ui = w and all other Xi = u\• = 0 give: w(w)72X = — 1 is equivalent to w(x)72w = — 1. If x ?* 0, w = — 77 x ty(x) -1 , then the second equation holds, and so the first one gives: x = — 77 1 w(u)~l = — y'2~1(w( — y~2~1w(x)-1))-1. But (19), (20) imply w(l) = 1, w(w~l) = w(w) -1 , so the above relation becomes:

x = - Y 7 > ( - Y 7 V * 7 1 ) ) - 1 = -7l 1 ^((-7l 1 ^(x)- 1 )- 1 ) = -y^lw(w(x)(-y2))

=

-yllw(-yt)w(w(x)).

Put herein a: = 1, as w(w(l)) = w(l) = 1, so — y^lw( — 72) = 1, w( — 72) = —72 results. Thus the above equation becomes (22)

w(w(x)) = x,

and w( —72) = — 72 gives, if we permute the i = 2, • • • , n, (23)

w(-7») = - 7 . - . 1

Put Ui = - 7 7 in (21).

Then considering (22) and (19)

n

(24)

]C x» t = 2

n

=

1 *s equivalent to ^

w(x,) = 1

i =2

obtains. Put x2 = x, £3 = y, x4 = 1 — x — 2/, x5 = *' * = x n = 0. Then (24) gives w{x) + w(i/) = 1 — t^(l — x — y). So w(x) + w(y) depends on x + y only. Replacing x, y by x + 2/, 0 shows, that it is equal to w(x + y) + w(0) = w(x + y) (use 20). So we have: (25)

w(x) + w(y) = w[x + y)

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(25), (19) and (22) give together: w(x) is an involutory antisomorphism of F. Observe, that (25) implies w(—1) = —t^(l) = — 1, and so (23) becomes w(yi)

(26)

= 7*.

7. Consider a: [x\\ • • • :xn]r, a': [yii • • • :yn]i. If xi ?± 0, we may write a: [l:x2x~[l: • • • :xnx1l]r, and so a': \\:w{x2x1l)y2\ • • • :^(xnrc71)7n]/. But w{xiXll)yi

= w{xil)w(xi)y% =

w(xi)~lw(xi)yi,

and so we can write a':[w(xi):w{x2)y2: too.

• • • m.w(xn)yn]i

So we have

(27)

y{ = w(Xi)yi

where the 7,- for i = so (26) holds for all i obeying (26), if x{ j± Permutation of the

for i = 1, ••. • , n,

2, • • • , n are those from 6., and 71 = 1. And w(l) = 1> = 1, • • • , n. So we have the representation (27) with 7* 0. i = 1, • • • , n shows, that a similar relation holds if x2 7* 0:

(27+)

Vi

(26+)

=

iff+ixtht

w+(7t) = 7t,

w+(x) being an involutory antisomorphism of F. (w+(x), 7 I may differ from w(x), 7,!) Instead of 71 = 1 we have now yt — 1, but we will not use this. Put all Xi = 1. Then a': [yii • • • :yn]i can be expressed by both formulae (27) and (27+). As w(x)\W+(x) are both antisomorphism, so w(l) = w+(l) = 1, and therefore [t/ii. • • :yn]i = [71-• • • ^TJI = frl": • • • :7nl* obtains. Thus ( T D ^ T T = (Ti)"^* = 7.', yt = 7l7» for t = 1, • • • , n. Assume now x2 5^ 0 only. Then (27+) gives 2/» = w+{x%)y\1 but as we are dealing with left ratios, we may as well put Vi = (7t)~ 1 ^ + (^»)7T = (7i+)~~1w+(x)7+7». Put 0+ = 7 + 5^ 0, then we have: (27++)

Vi

=

P+^w+MP+yi.

Put now xi = x 2 = 1, x 3 = x, all other a:,- = 0. Again a': [yii • • • :i/n]z can be expressed by both formulae (27) and (27++), again w(l) = w+(l). Therefore [2/1:2/2:2/3:2/4: • • • :2/n]z = [71:72:^(^)73:0: • • • :0]i = [7i:7 2 :/3+- 1 w;+(a:)^+73:0:.. • :0]i obtains. (27).

This implies w(x) = /3+_1w(x)/3+ for all x, and so (27++) coincides with

Quantum Mechanics

123

Logic of Quantum Mechanics

125

In other words: (27) holds for x% j* 0 too. Permuting i = 2, • • • , n (only i = 1 has an exceptional role in (27)), we see: (27) holds if Xi ^ 0 for i = 2 , ' . . . , n. For xi ^ 0 (27) held anyhow, and for some i = 1, • • • , n we must have Xi 9* 0. Therefore: (27) holds for all points a: [xii • • • :x n ] r . 8. Consider now two points a: [xx: • • • :xn]r and 6: [£r. • • • :£n]r. Put a'- [yi- • • • -2/n]*, then b ^ a' means, considering (1) and (27) (cf. the end of 7.):

(28)

£ wifrhih = 0.

a ^ a' can never hold (a 0 a' = 0, a 5^ 0), so (28) can only hold for x,- = &, ifalla;i = 0. Thus, n

(29)

X) w(xi)jiXi = 0 implies £1 = • • • = xn = 0. i= l

Summing up the last result of 6., and formulae (26), (29) and (28), we obtain: There exists an involutory antisomorphism w(x) of F (cf. (22), (25), (19)) and a n

definite diagonal Hermitian form ^2 w(z»)7*£*' in F (cf. (26), (29)), such that for a: [xii • • • :xn]r, b: [fa • • •: £„]r b ^ a' is defined by polarity with respect to it: (28)

2

wfo)Y<& = 0.

This is exactly the result of §14, which is thus justified.

The Neumann

124

Compendium

QUANTUM LOGICS (STRICT- AND PROBABILITY-LOGICS) Reviewed by A. H. TAUB

IN AN unfinished manuscript, written about 1937, von Neumann proposes to deal with the question: How does the system of logics apply to those mathematical models, which various current physical theories use to describe the physical world, i.e. which they substitute in its place? The system of logics is to be understood . . . to include the propositional calculus only. "Let S be the physical system, or rather the mathematical model of a physical system, to which we wish to apply logics. The system L of logics is then the set of all statements a, b, c, . . . which can be made concerning S. Such a statement is always one concerning the outcome of a certain measurement, which is to be performed on S. For a more detailed discussion, cf. 1, §§ 1-4. The fundamental relations and operations for elements of L are these: (I) The relation of "implication": a ^ b. a ^ b means this: If a measurement of a on 5 has shown a to be true, then an immediately subsequent measurement of b on 5 will certainly show b to be true. (II) The operation of "negation": —a. — a obtains as follows: The same measurement which is used to decide about the validity of a is also used to decide about the validity of —a, but when the result concerning the validity of a is "yes", then the one concerning —a is "no", and conversely. a ^ b has clearly the following properties: (A) a ^ b and b ^ a together are equivalent to a = b. (B) a ^ b and b ^ c together imply a ^ c. We also define: (III) a ^ b means b ^ a. a < b (a > b)-means a ^ b (a ^ b) but a ^ b. Now (A), (B) and (III) permit us to infer immediately: (C) If © stands for any one of the four relations ^ , ^ , < , > , then a ® b and b ® c together imply a ® c. (C) shows, that a < b (that is a ^ b) defines a "partial ordering" of L. We also infer from (A) to (C): (D) Given a, b, a c is called a "greatest lower bound" of a, b, if for every u (in L) u ^ a and u ^ b together are equivalent to u rg c. If such a c exists at all, then it is unique. (E) Given a, b, a b is called a "least upper bound" of a, b if for every u (ill L) u ^ a and u ^ b together are equivalent to u ^ b. If such a b exists at all, then it is unique. The existence of the logical operations of "conjunction" ("a and b") and "dis­ junction" ("a or b") induces us to postulate: (IV) Any two a, b possess a [unique, cf. (D)] greatest lower bound c, to be denoted by ab—this is "a and b". 195 Reprinted from John von Neumann Collected Works, ed. A. Taub, Vol. IV, pp. 195-197.

Quantum 196

Mechanics

125 REVIEW BY A. H. TAUB

(V) Any two a, b possess a (unique, cf. (E)) least upper bound b, to be denoted by a + b—this is "a or b". "These considerations lay the foundations for an axiomatic treatment of L. But we are not prepared yet to take this up systematically. We must first discuss the influence of another basic constituent of logics—as we wish to see it—and also discuss several examples. "So far the only structure L possesses is defined by the "primitive notions" a g b, — a, and the "derived notions" a + b, ab. To this extent we will call L the system of "strict logics". In applications of L to actual physical reality, how­ ever, a further structure of L appears, which can only be expressed in terms of "probability. In other words: For any well defined state of our knowledge con­ cerning the mathematical description of physical reality, that is for any reasonable mathematical model 5, a probability function exists. So we have in L: (VI) The {real number-valued) function called "probability function": P(a, b). P(a, b) = 9(9 a real number) means this: If a measurement of a on S has shown a to be true, then the probability of an immediately subsequent measurement of b on S showing b to be true, is equal to 9. If we consider the structure which L acquires by making use of the function P(a, b)—that is of all relations P(a, b) = 9 (of course necessarily 0^9^ 1)— then L appears as a new system, which we will call the system of "probability logics". "It is easy to see that the system of strict logics is part of thesystem of probability logics, since a S b, —a can be defined in terms of P(a9 b) = 9 (0 :§ 0 ?g 1). Indeed (VI) makes the following statements obvious: (F) P(a, b) = 1 is equivalent to a g b. (G) P(a, b) = 0 is equivalent to a g — b. Hence a !g b is directly defined by (F), that is by P(a, b) = 1; and - a is in­ directly defined by (G), as the (unique) c, for which u ^ c is equivalent to u S — <* that is, for which P(u c) = 1 is equivalent to P(u, a) = 0. Conversely (F), (G) define P(a, b) = 9 for 9 = 0, 1 in terms of strict logics. For a 9 with 0 < 9 < 1, however, no such "reduction" of P(a, b) = 9 to strict logics seems possible. It is of course feasible to perform the procedure described in (VI) above on a large number N of specimens 5*, . . . , S% of St and then to interpret P(a, b) = 9 as a frequency statement, i.e. if we measure on each 5 * , . . . , S% first a, and then in immediate succession b, and if then the number of those among 5*, . . . , S% where a is found to be true is M, and the number of those where a, b are both found to be true is M\ then: (H) P(a, b) = 9 means that M'\M - 9 for N -> oo. This view, the so-called "frequency theory of probability" has been very brilliantly upheld and expounded by R. V. Mises. This view, however, is not acceptable to us, at least not in the present "logical" context. We do not think that (H) really expresses a convergence-statement in the strict mathematical sense of the word—at least not without extending the physical terminology and ideology to infinite systems (namely, to the entirety of an infinite sequence 5 J, S*, • • • )—and we are not prepared to carry out such an extension at this stage. The approximative

The Neumann

126

QUANTUM LOGICS

Compendium

197

forms of (H), on the other hand, are mere probability-statements, e.g. "Bernouilli's law of great numbers": (Happr.) For any two e, S > 0 there exists an N0 = N0(e9 <5), such that for N ^ N0 the probability of \M'/M - 0\ < e is > 1 - S. (We assume, that N -> oo, implies M -> oo, that is we exclude "absurd" a's.) And such probability-statements are again of the same nature as the relation P(a9 b) = 0, which they should interpret. We prefer, therefore, to disclaim any intention to interpret the relations P(a, b) = 6 (0 < 6 < 1) in terms of strict logics. In other words, we admit: Probability logics cannot be reduced to strict logics, but constitute an essentially wider system than the latter, and statements of the form P(a, b) = 6 (0 < 9 < 1) are perfectly new and sui generis aspects of physical reality. So probability logics appear as an essential extension of strict logics. This view, the so-called "logical theory of probability" is the foundation of J. N. Keynes's work on this subject. Von Neumann had intended to discuss four examples: 1. A system "which behaves in the sense of classical physics (in particular mechanics) and which possesses only a finite number of possible different states". 2. A system similar to 1 in which the number of states is discretely infinite. 3. A system similar to 1 in which the number of states is continuously infinite. 4. A system where the finiteness of the number of states remains essentially untouched by the "classical" way of looking at things is replaced by a "quantum mechanical" one. He also intended to give a final synthesis by combining these two kinds of extensions. .Unfortunately the manuscript contains only a discussion of example I.

REFERENCE

1. GARRETT BIRKHOFF and J. VON NEUMANN, The logics of quantum mechanics, Ann. Math., vol. 37 (1936) pp. 823-843.

Ergodic Theory

127

J O H N V O N N E U M A N N A N D ERGODIC THEORY J. FRITZ

The history of ergodic theory as a branch of mathematics dates back to the early 1930s when, motivated by problems related to the mathemati­ cal foundation of classical statistical mechanics, B. O. Koopman and J. von Neumann initiated a systematic study of dynamical systems. In view of the pioneering work of L. Boltzmann 1 on gas theory, thermodynamics should be understood based on the 'ergodic' behavior of the underlying molecu­ lar dynamics. Ergodicity has been mathematical interpretated and justified, and about 60 years later an extremely powerful theory with many fruitful applications was developed in many other fields. A dynamical system with discrete time is a finite (cr-finite) measure space (X,X,\) equipped with a measure preserving transfomation T : X — i ► X. This means that T is measurable, and X(T~1A) = X(A) whenever A G X, where T~l A — {x G X : Tx G A) and Tx is the image oi x e X under the action of T. If t is a natural number, Tl : X »—> X is defined by iterating the map T t times, i.e. T° is the identity map, T 1 = T, and we have Tt+Sx = TlTsx for all integers t, s > 0 and x G X. The problem of ergodicity is related to the convergence of the arithmetic means, St, of the iterates of T, 1 l~l s

tf = 7J2 f(T*x) -> / a s * -> +°°

U)

for / : X t-» IR and x G X. Of course, the notion of convergence and the class of allowed functions should be specified here. Roughly speaking, T is ergodic if the limit / does not depend on x. In the case of dynamical systems with continuous time we are given a measure preserving transformation Tl for all t > 0 such that T° is the identity and Tt+S = TlTs. Then the sum in (1) is replaced by a time integral, and some conditions on continuity of Tl as a function of time t are also needed. In statistical mechanics, X is chosen as an energy shell, that is a surface consisting of points with constant energy in the phase space JR6N of a system of N particles, A is the surface measure, and Tl is the flow generated by Newton's equations of motion. In view of the Liouville theorem, this flow is a group of measure preserving transformations, it is defined even for negative times. As was shown by H. Poincare, 9 the tra­ jectory Tlx starting from x G X returns infinitely often to any neighborhood of x, which is a consequence of ergodicity. For simplicity we discuss basically

The Neumann

128

Compendium

the case of discrete time. In 1931, B. O. Koopman 7 observed that if T is in­ vert ible and its inverse T _ 1 is also measurable then the operator Uf , defined by Uf(x) = f(Tx) is unitary in the Hilbert space H = TL2(X,X) of square integrable functions. Von Neumann, who knew everything on operators in Hilbert space, immediately made a decisive step by proving his celebrated mean ergodic theorem [41].* If U is a unitary operator in a Hilbert space, then 1 '-1 ^ i r n ^ - £ t / 5 / = Pf for all / € « , (2) 5=0

where P is the orthogonal projection onto the invariant subspace Hi of H with respect to U\ Hi is characterized by / = £//, i.e. by f(x) = f(Tx) A-a.s. in the original formulation of the problem. Therefore the ergodicity of T means that it is metrically transitive in the sense that every invariant func­ tion is A-a.s. which is a constant. Actually von Neumann proved this result for systems with continuous time, his argument was based on the spectral representation of unitary operators. Shortly after this work, G. D. Birkhoff3 managed to extend the mean ergodic theorem to integrable functions and proved a.s. convergence of the arithmetic means Stf as t —> +oo. Although this paper appeared before [41], [41] was acknowledged. A simplified proof of the mean ergodic theorem was given by F. Riesz10; and various extensions were obtained later by S. Kakutani, 5 E. Hopf4 and others. In a next paper [43] with B. O. Koopman another basic notion of ergodic theory was discussed. A dynamical system is mixing if

t^HAT-.B)=mm

(3)

for any pair i , B G ^ . Mixing implies ergodicity, and it is characterized by spectral properties of the associated unitary operator U in the case of invertible transformations as follows: Koopman and von Neumann proved in [43] that T is mixing exactly if the only eigenvalue of U is 1 and its multiplicity is 1. Further basic results were obtained in [46] in the opposite case when the unitary operator U associated to an ergodic T has a pure point spectrum. The eigenvalues of U form a subgroup of the unit circle on the complex plane, and every such group consists of the eigenvalues of a unitary operator U with pure point spectrum corresponding to an ergodic dynamical system. It is interesting to note that the problem of ergodicity of irrational rotations of the unit circle was also investigated in this paper, establishing connections with number theory. The first result on the measure theoretical classification of dynamical systems was also proven in [46], see also [97]. In fact, a criterion of equiva­ lence (isomorphism) for ergodic systems was obtained in terms of the spectral * Numbers in square brackets correspond to the Bibliography listed on pp. 677-689.

Ergodic Theory

129

properties of the associated unitary operators. Let T\ and T^ be measure pre­ serving transformations on the measure spaces ( X i , X\, \\) and (X2, ^ 2 , ^ 2 ) , respectively. They are isomorphic if there is an invertible and measure pre­ serving transformation M between X\ and X2 such t h a t T2M = MT\. T h e above-mentioned result of von Neumann states t h a t if T\ and T2 are in­ vertible and ergodic, and the associated unitary operators, U\ and U2 have pure point spectra, then T\ and T2 are isomorphic if and only if U\ and V2 are unitary equivalent. 30 years later A. N. Kolmogorov 6 and Ya. G. Sinai 1 1 found a new invariant of dynamical systems called entropy, and D. Ornstein 8 proved in 1970 t h a t Bernoulli shifts with the same entropy are isomorphic. References 1. L. Boltzmann, Einige allgemeine Satze iiber das Warmegleichgewicht unter Gasmolekiilen, Wien. Ber. 66(1871)679-711. 2. G. D. Birkhoff, Proof of the Ergodic Theorem, Proc. Nat. Acad. Sci. 17(1931) 656-660. 3. G. D. Birkhoff and B. O. Koopman, Recent Contributions to Ergodic Theory, Proc. Nat. Acad. Sci. 18(1932)279-282. 4. E. Hopf, The General Temporally Discrete Markov Process, J. Ratl. Mech. Anal. 3(1954)13-45. 5. S. Kakutani, Concrete Representation of Abstract L-Spaces and Mean Ergodic Theorem, Ann. Math. 42(1941)523-537. 6. A. N. Kolmogorov, On the Entropy per Unit Time as a Metric Invariant of Automorphisms, Dokl. Akad. Nauk 124(1959)754-755. 7. B. O. Koopman, Hamiltonian Systems and Transformations in Hilbert Space, Proc. Nat. Acad. Sci. 17(1931)315-318. 8. D. Ornstein, Bernoulli Shifts with the Same Entropy are Isomorphic, Adv. Math. 4(1970)337-352. 9. H. Poincare, Methodes nouvelles de la mecanique celeste, Paris 1899. 10. F. Riesz, Sur la theorie ergodique, Comm. Math. Helv. 17(1945)221-239. 11. Ya. G. Sinai, Dynamical Systems with Countably Multiple Lebesgue Spectrum, Izv. Acad. Nauk SSSR Mat. 25(1961)899-924.

130

The Neumann

Compendium

ERRATA for "Proof of the Quasi-Ergodic Hypothesis" All footnotes (in the text) 15-28 are incorrect. Their numbers should be increased by one: 16 for 15, 17 for 16, . . ., 29 for 28. The footnotes 1-14 are correct, immediately after footnote 14, footnote 15 should follow. p. 263, line 5 (from bottom). Formula should read: Ux for U2. p. 265, line line line line

6: read £ for f> 8: read t — s —> oo for / — s —► 0 18: read J n |S(/)| 9R

p. 266, line 6: read c for C line 14: read F(X) = 1 for F(A) = A line 28: insert footnote 20 at the end of the first sentence. p. 268, line line line line line

9 (formula): read j M for Jm 20 (formula): read in the suffix A = x.v(P) f° r ^ = Zn(P) 22 (formula): read x°N(p) f o r W 2 (from bottom): read %M /LIM for Xm /um 1 (from bottom) (formula): read ;&(/>) for X°(P)

p. 269, line 1: read rx, sx for tlsl line 3: read tKp(P9 N) ^ X%(P) for t^P, line line line line line line

N) - **(/>)

4: read v —> oo for v —► oo 5: read JM for J w 14 (formula): read ;&(/>) = C^ for X%(P) = C„ 15: read Cv for Cn 19 (formula): read CN for Cn 25 (formula): read J n for J

p. 270, line 1: insert of finite measure between every measurable set and A remaining invariant. p. 271, line 13 (from bottom) (formula): read fi for m,,in the numerator and in the denominator, line 5 (from bottom): read . . . orders . . . for . . . order. p. 272, line 8 (formula): read ]T for 2 *n the denominator. line line p. 273, line line line

21: read Z<w) for Z<w) 22: read 2 ? m ) for Yi{n) 10 (footnote): read (E(y)ff) for {E(y\f) 21 (footnote): read %A f° r <*A 22 (footnote): read % for X or x (in 12 places).

Ergodic Theory

131

PROOF OF THE QUASI-ERGODIC HYPOTHESIS

1. The purpose of this note is to prove and to generalize the quasi ergodic hypothesis of classical Hamiltonian dynamics1 (or "ergodic hy­ pothesis/' as we shall say for brevity) with the aid of the reduction, recently discovered by Koopman,2 of Hamiltonian systems to Hilbert space, and with the use of certain methods of ours closely connected with recent in­ vestigations of our own of the algebra of linear transformations in this space.3 A precise statement of our results appears on page 79. We shall employ the notation of Koopman's paper, with which we assume the reader to be familiar. The Hamiltonian system of k degrees of freedom corresponding with the Hamiltonian function H(qi, . . . , qk, Pu • - •> Pk) defines a steady incompressible flow P —>- Pt = StP in the space 3> of the variables (qu . .., qk, pu . . ., pk) or ''phase-space/' and a corresponding steady conservative flow of positive density p in any in­ variant sub-space 12 d 3> (12 being, e.g., the set of points in <£ of equal energy). The Hilbert space £ consists of the class of measurable functions f(P) having the finite Lebesgue integral J\i |/|2pda>, the "inner product" 4 of any two of them (/, g) and "length" ||/|j being defined by the equations

(f,g) -fQfgpd»; 11/11 = VUJ).

(1)

The transformation Ut is defined as follows: W )

- f(S,P) = f(Pt);

obviously it has the group property 261 Published in N.A.S. Proc, Vol. 18, pp. 70-82 (1932). Reprinted from John von Neumann Collected Works, ed. A. Taub, Vol. II, pp. 261-273.

(2)

132 262

The Neumann

Compendium

Mathematics

UtUs = Ul+S, Uo = I;

(3)

and in virtue of the conservative character of the flow, and the resulting invariance of (Utf, Utg), it is unitary. The spectral reduction of Ut in terms of its "canonical resolution of the identity" JE(X)5 is furnished by a theorem due to Stone,6 and gives us

Ut = J[J eitxdE(k),

(4)

this being the symbolic expression for the fact that, for all /, g of £ , we have, in terms of Stieltjes integrals,

(U,f, g) - f _ + J eitx d(E(\)f, g).

(4')

The pith of the idea in Koopman's method resides in the conception of the spectrum E(\) reflecting, in its structure, the properties of the dynamical system—more precisely, those properties of the system which are true "almost everywhere/' in the sense of Lebesgue sets. The possibility of applying Koopman's work to the proof of theorems like the ergodic theorem was suggested to me in a conversation with that author in the spring of 1930 In a conversation with A. Weil in the summer of 1931, a similar application was suggested, and I take this opportunity of thanking both mathematicians for the incentive which they furnished me for undertaking the investigations of this paper. 2: For the sake of brevity, we shall introduce the following notation: We shall replace pdu by dv, writing J^ — pdu — JQ — dv. By the "weight pQ of the Lebesgue-measurable set 0 ( d & ) with respect to the density p" will be meant the quantity /x0 = Jl pdu = J^ dv. By a "zero set" we shall mean a set of zero weight, and hence, since throughout tt, 0 < pi < p < P2, a set of zero Lebesgue measure. If 0 is a set of points P of SI or $>, we shall denote its. characteristic function by %e = Xe(P); i c , Xe(P) = 1 or 0 according as 0 does or does not contain P . If /(P) is any measurable function, the set of points for which/(P) > X, etc., will be denoted as usual by [/(P) > X], etc. We have the identity [x[/>\] = 1] = [f > X], etc. By the strong convergence of a sequence /i, / 2 , . . . in ^ (/n —> f) will be meant that ||/w — / | | — > 0 as n —> » . By weak convergence (fn —'-*- f) we mean, on the other hand, that for an arbitrarily chosen g of ^>, {fn* g) — > (/» g) as n —>■ oo. It is shown that fn —> / implies fn—*~f, but not conversely.7 In general, expressions depending for their precise meaning on the nature of the convergence considered will be suffixed by the corresponding convergence symbol, thus we shall write

Ergodic Theory

133 Mathematics

263

"separable (—>)," "everywhere dense (—>)," etc. All these notions subsist if n is replaced by one or more continuously varying parameters. By /i-convergence of a sequence of point sets 0i, 0 2 , . . . ( CZ12 or $>) will be meant the strong convergence of the corresponding measure functions: 0„ —> 0 if xen —> Xe> or, what is the same thing, n[Qn + 0 — 0 n 0] —> 0 as n —>- oo. Clearly 0, lim 0 n , and lim 0W, will all differ by at most zero sets. The greatest lower bound and least upper bound of a set [ ] will be denoted, as usual, by inf [ ] and sup [ ]. 3. The starting point of our investigations is the construction of the operator

/} /), ' '.* =7^~sf!^Urdr(s<(S< U

attS =JZTsJs

(5)

this being, as before, but the symbolic expression of the fact that for all /, g in § ,

(*'.'f'S> =TZrsf (UTf,g)dr;

(50

the existence of atiS is easily proved.8 We will show that, for each / of §i ■) as t — s —>■ <*>, irrespectively of the mode of variation of s, t. We have from (5'):

K*/ll2 = M «i.J) = jzsf'

Wrf> <".*f)dT 17

^^fsfs^ ^^''

since U0 is unitary and Ul = U^1 = £/-ff,9

IUM/I! \M\* 2 = * JfZ-tf£ JT^f

f W (Ur-,f,f)dTd
which reduces, on making the change of variables r — a — x, r + Jis + \x\ * =

Tt—W* / \ll ~~ 5s ) J ~(t-s) \

~~ ) J

(t-s-\*\KU,fj)dx.

-(t-s)

This may be calculated with the aid of (4'), inasmuch as the various changes in the order of integration are permissible, on account of the uniform convergence of the Stieltjes integral in (4') 10 (for all values of t—at present, x). Thus:

The Neumann Compendium

134 264

Mathematics

1

K*/H2 = jfZT&J

/•+(<-*)

, , r r+"

(t-s-

\*\)\J_w

.

-]

e^d(E(\)f,f)^dx

eixx eixx {t(t s s

+

=(^/_? [fZ7 ~~- -HH^^ ^^ =(^/_ ; if-Z7 c s ( x) (

=(T^y.i [j[ 2

'jr^L

° * - * -* ~ *)-^j^w/./)

s)X /r+*m+ " 1 — cos (/ — s)A

—»—' ( £ ( x ) / , / ) —^Td(£(X)/./).

J-~ L 'Ac—

This integral has a non-negative integrand and a non-decreasing expres­ sion after the d-sign; hence we may obtain an upper bound for it as follows: First, break it up into / J —e

and / J — co

+

/

; (« > 0, to be considered

Je

later).

Then replace the integrand in the first part by 1, and t h a t in the T 1 12 4 second part by -rrr r = — — 2 . Finally, replace the field L hv "" s)eJ V "~~ s) € +» We shall then have of integration in the second by / / . We shall then have

KJII* * f_*'d{E{\)f,f) + ( f _ 4 i ) v ^%(£(X)/,/) KJll* = {(£(<)/,/) {(£(«)/,/) - (£(-«)/,/)} + ((f / _ 4,J)) VV (/■/)• (/»/)• Hence, as * - s —*■ + » , ffii || 0, so t h a t atJ — > 0. We now introduce the projection operator JE 0 defined as follows: (£(c) — E( — e))f—> Eof as e—>* 0. The existence and projective nature of E0 is easily deduced from the fact that E(e) — E( — e) is a nonincreasing "function' 1 of e.n Thus, we are able to express the condition t h a t 0 in the form: E0f = 0. Suppose t h a t E0f = / ; then, since E(e) — E( — e) ^ E0t we have E(e)f = / , J E ( - e ) / = 0, 12 i.e., £(X)/ = / f o r X > 0, = 0 for X < 0. [Hence, for all g, d E (U = X-" f*'**C " X d(£(X)/ ( Wf>' g) i) == (/(/■ tf, g) W/. g) = ' f)'g)>

so that, for all values of /, Utf = / . Let 9tt be the linear manifold in & corresponding with EQ.

For every

Ergodic Theory

135

Mathematics

265

/ of 2ft, Eof = / ; hence UJ = /, and atsf = /, so that as / — s —> » we have *tj—+f = E0f. (6) For all / orthogonal to 2ft on the other hand, we have 'uf—>0

= £o/.

(6')

Now l e t / be an arbitrary point of f>; we can write/ = /i + /2, where /i is in 2ft, and/ 2 , orthogonal to 2ft. Then it will follow from (6), (6'), that we still have, as t — s —>* 0, *iiSf—+Eof.

(6")

Throughout 2ft, UJ = / . Conversely, if UJ = / , it will follow that and hence by (6"), / = E0f, i.e., / belongs to 2ft. Thus, 2ft is the class of all solutions of the equation UJ = / (i.e., the identity in /). 4. Let us examine 2ft more closely. Its elements / are characterized by UJ = /, and hence, in virtue of (2), by

f(P) =f(P,),

(7)

the == sign holding for all / but with the possible exception of a zero set of points P (in general, dependent on /). Hence, if/ is in 2ft, $ ( / ) will be also, provided J* | g(\) 12d\ is finite. Now / can be expressed as the limit ( —>- ) of functions of the form 5(/) where g is susceptible of but a finite number of values, and these, in their turn, are linear combinations of similar functions susceptible only of the values 0 and l.13 The latter, being of the form g ( / ) , belong to 2ft. If we denote by $ the class of all functions belonging to 2ft and taking on only the values 0 and 1, we may say that $ spans ( —> ) the closed ( —> ) linear manifold 2ft. If / belongs to « f we shall write f/(P) = 1] = A, and / = / A ( = X A ) , (cf. § 2). Since /xA = | | / 1 | 2, a n d / is in £), /xA must be finite; and since / is in $£ Z) 2ft, it follows from (7) that the transformation P —> Pt changes A by at most a zero set. These two properties, its finite weight and invariance under P —> Ph characterize A. We shalj call any set having these properties a A-set. Evidently fi will be a A-set if and only if pQ is finite. The class of all A-sets, being a subclass of the class of all measur­ able sub-sets of Q, is separable;14 that is, there exists a sequence Ai, A2, . . . of A-sets such that any A-set can be expressed as the limit (in the sense of /i-convergence) of a subsequence of Ai, A2, By methods which we have used in another connection,15 we are able to replace the sequence Aj, A2, . . . by another sequence of A-sets, A/, A2', . . . ,

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such that, firstly, any member of one sequence may be expressed in terms of elements of the other with the finite repetition of the operations of taking the logical sum ( + ) , the logical product ( X ) and the logical difference ( —); secondly, A / , A2', . . . may be set into one to one corre­ spondence with certain rational numbers p„ = p O O , A^ = A(prt), 0 < p w < 1 , in such a manner that p m < pn implies A(pm)CA(p„); and, thirdly, inf p„ = 0 and sup p n = 1. We now define the function G(P) as follows: (= inf G(P) < = mi^pn f ° r w h i c h F i s i n A ( P » ) ] » 1• =— i1> when no such pw exists. By its construction, G{P) is invariant under P — > Pt (remaining un­ changed apart from zero sets); and inf G{P) = 0, sup G(P) = 1. Since [G(P) < pw] = A( Prt ), it follows t h a t / , , ; = fAipn) = F(G) (for we may set F(\) = A for X < p n , = 0 for X > pn); and therefore this property remains true for every/ A w , 1 6 and a n y / A of St (cf. definition of —>* for sets). And since every / of 90? is the limit ( —>■ ) of a sequence of linear combinations of functions of $ , it follows t h a t every such / is a function of G.17 Finally if X < A, tx\G{P) ^ X] is finite. For a Pn > X may be found, whereupon [G(P) ^ X ] d [G(P) ^ Pn] = A( Pn ) = A'M, and »\n = ||/ A ;||*, which is finite for any/ A ' w of £ ( d £ ) . Any function G(P) like the above, such that G{Pt) = G(P) for all t except perhaps at zero-sets, such that X' = inf G{P) and X" = sup G{P) exist, and that, if X < X;/, JJL[G(P) ^ X] is finite, and which possesses the property t h a t every / of 90? may be expressed as $(G), shall be called a universal integral. We have shown that one universal integral always exists; obviously there are infinitely many. 1 8 The class 2ft coincides with the totality of expressions g(G) (with Jn | i$(G(P)) \2dv finite). This makes it possible to express Eof in terms of G. We shall give below, instead of our original method of computation, an abbreviated method for which we are grateful to Mr. M . H. Stone. 5. For an arbitrary / of ^>, E0f belongs to 90?, and is, accordingly, of the form g(G)._ Let X < X" ( = sup G), and define f (X) to be 1 for X < X and 0 for X ^ X. Let us set A(X) = [G(P) < X] (we have seen t h a t /*A(X) is finite). Then we have, on the one hand,

foE 7 ^ o0/f(P)UG(.P))dv ( P ) r ( G ( P ) ) ^ = f%(G(P)MG(P))dv"> f$(G(P)MG(P))dv">

and, on the other hand,

Ergodic Theory

137 Mathematics

267

f»E«KP)!(G{P))dv = (Eof, KO) = (/, £0r(G))2> = (/, f(G)) = fnf(P)!(G(P))dv faf(P)t(G(P))dv =

fMAl»f(P)dv. iJ(P)dv.

Thus:

fX^(\)d„\(\) /\(X)d = f_f(P)dv. f_f(P)dv. MA(X) = J X'

(8)

yA(Xj

As X increases from X' t o X", u\(\) goes in a non-decreasing fashion from 0 to /xfi; and juA(X + 0) = MA(X). In intervals Xx < X < X2 where /xA(X) is constant, A(X) changes a t most by points P of a zero set, i.e., Xi < G(P) < X2 can be true a t most on a zero set; thus the behavior of g(X) in such intervals does not affect the relation E0f = g(G)—we may take g(X) constant upon them. I t follows t h a t the familiar theorems on the differentiation of Lebesgue integrals may be applied, the independent variable being here x = /xA(X). From such considerations it follows t h a t d\ ^ M X ) / ( ^ W d {MA(X)|

exists f o rall

x in X' < X < X", except for a set of values

of X for which the corresponding set of values x = MA(X) is of zero measure, 22 and this derivative is equal t o \$(\). The correspondence P—>• x, obtained by setting X = G(P)f x = M(X), carries a set 9 (Z ft into a set 0 on the x-axis so t h a t fxQ = measure of 0.23 THus we have, except for at most a zero set on ft,

mP)_

Rif,.»/;yi] L

a{M(X)|

.

Jx=/(io

(9)

This is naturally true for X = X" only when x = JJL\(\") exists, i.e., when Mft is finite. In the case X = G(P) = X;/, we carry out the above process with f (X) = 1 for X = X'7, = 0 for X ^ X". On setting A = [G(P) = X" j , the following becomes clear: If juA = 0, g(X") does not affect £ 0 / ( P ) . If /zA = <»> E0f(P) must be zero; for it is constant on A, and belongs to ^p. If juA is > 0 and finite, the considerations which lead t o (8) show t h a t 5(X")-MA = fJ{P)dv.

(8')

Since A = A(X') - A(X" - 0) = Q - A(X" - 0), it follows t h a t £

/ ( p ) = = =

JK

'

fa-A«'-O)f(P)d S^a--Aw-o)f(P)dv A ft'-O)/^)^

jM u [nn-- A A (( \\ f"f -- o) juin-ACx"o)] o)l

(( ff oo rr G{p) „ G ( P ) =

"). XV)

(90

This formula subsists when /x[ft — A(X" -■ 0)] = 0 or ®, provided we agree t o replace the right-hand member by 0 in the case where the de­ nominator vanishes (and hence the numerator also) or is infinite.

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Mathematics

When fiQ is finite, (9') leads t o (9) (cf. 2 2 ); otherwise, it forms its natural generalization. 6. Let M and N be any measurable sub sets of Q, with fiM, JJLN finite. Then XM(P) and x.vCP) are in $r>, and we may apply our results to them. I t follows from (5') t h a t (s xxt X.u), which equals J]i(xt>5 xx{P)dv, is equal to

Thf{fau***}d* = i~rs£ ^Setc-) = fJlnZs,, (P,N)dv, N)dv, (P, mZs,t where we have set ZSft (P, N) equal to —-— times the linear measure of the set of r-values for which S _ T P ( = P _ T ) is on N.2A T h a t is, Zs>l(P, N) is the mean time of sojourn of P in N between the times s and / (actually, with the sign of r changed, b u t this is immaterial). Since the above is true for all M9 we have er,.,x.v(P) = = 2 Zs>t N),
(10)

and in virtue of (6")> ZS,(P, N) — > EoXx(P) = X%(P) as / - 5 — ^ co,

(11)

in the sense of strong convergence in § . On applying (9), (9') t o / = XNt we have 0

_ p ( M [ A ( X ) X N]\l L

d AlA(X)| d{ixA(\)\

JX-Xn(P)

(12)

when G(P) < X', and N X A(X 0)] M [[ Q XHF) ° ~ ^ x A(x" "~ x*(P) -= M ~ 0)] [« - A ( \ " - 0)] M M [« - A ( \ " - 0)]

(12')

when G{P) = X". T h e right-hand members have a meaning except possibly for P on a zero set: in (12), cf.;22 in (12'), we take 0 when the denominator is infinite, or when it (and hence, the numerator) vanishes. Let us express the content of (11) in the following three ways: (A) explicitly as strong convergence in ^>; (B) as point convergence of a sub­ sequence; 17 (C) as weak convergence, or rather as the implication of the latter regarding the inner product of (11) with an arbitrary Xm (/*m, finite). A.

f

\Z5A ( P , N) - X°n(P)]*dv — > 0 as / - s —> + » .

Ergodic Theory

139

Mathematics

B.

269

For every sequence / lfJl ; k, s2; . . . with tn — sn — > + oo, there is a sub sequence / v s„y (v = 1, 2, . . . ) such that for all P of ft with the possible exception of a zero set, Zs , tn (P, AT) —►• X* (P) as V

>* a > ,

C. For each subset Af of ft of finite fxM, fmZ5tl{Pt N)dv —>J\iX%(P)dv as / - 5 —>- + o>. We observe that although x?v is expressed in (12), (12') in terms of the non-uniquely determined universal integral Gt its dependence upon G is only apparent: each of (11), A, B or C, determines X°N uniquely. 7. The existence, for each point P, of the limit of the mean sojourn Zst(Py M) is a consequence of A, B or C, and applies to any Hamiltonian system. Our system is ergodic if and only if this limit, x%(P), is inde­ pendent of P, i.e., when X°N(P) = CH, a constant.

(13)

When this is true, we must have in the case /*ft = oo that Cn = 0; for otherwise, || x!v II = °°> whereas x°v is in £ . But when ytt is finite, we have: fx°x(P)dv = (x%, 1) = (E0x.v, 1) = (x.v, £ol) = (x.v, 1) = fXx(P)dv = nN. Hence, by (13), 7 MA Cn = - .

(130

This is obviously true, from what was said earlier, when /& ==■ oo. It is now a simple matter to tell whether the system is ergodic or not, and we do not even need the more complete results of § § 4 and 5. First, suppose that (13) (and consequently (13')) is true for arbitrary N. L e t / be an element of 90?. Then, on the one hand, we have Sx°N(P)f(P)dv

= !^fQf(P)dv ij.il

=

Cn-N,

and, on the other hand, Ax%{P)]{P)dv

= (x%,f) = (£ox.v,/) = U.v,E0f) = fxN(P)f(P)dv

=

= ( x .v,/)

fNf{P)dv.

From the equality of the final expressions for all N9 we conclude that / ( P ) = C. Secondly, suppose that, conversely, every function of 93? is a constant. Then \N> belonging to 2)?, is a constant, and (13) is true. Thus the system will be ergodic if and only if 2)? consists exclusively of constants. This will be true if and only if $ consists exclusively of con­ stants, 13 i.e., that the A-sets all differ from 0 or from ft at most by zero sets. Thus we have proved the theorem:

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Mathematics

E.

The system is ergodic if and only if every measurable set A remaining invariant under St {except for points of a zero set) reduces to 0 or to ft {except for points of a zero set).2b Since in E the ergodic condition is the non-existence of any measurable A-sets (4^0, ft), one might be tempted to suppose that the ergodic condition as stated in (13) would have to hold for a correspondingly broad class of sets. This, however, is not the case: If (13) is true for all open sets N of finite nN, it wrill be true, by continuity, for all ^-limits of such sets (cf. §2),—i.e., for all measurable sets N of finite /xiV.26 Indeed, it is only necessary to require its truth for sets N which are the sums of a finite number of the neighborhoods of an arbitrary "topologically equivalent system of neighborhoods" in ft,26 for instance, for sums of finite numbers of spheres. S. From a purely mathematical standpoint, the question as to the validity and most appropriate generalization of the ergodic hypothesis has been fully answered: these special problems have been reduced to the general problem of the integrals of the system—the structure of G{P). Thus, the system is either ergodic, or else* there is a non-constant G{P), in which case ft is decomposable into subsets like [G{P) = Xi], [Xi < G{P) < X2], etc., upon which the flow has a sort of ergodic character, as is easily shown by means of (12), (12'). But from the point of view of physics, there remains the difficult question as to the existence and nature of G{P) in each particular case. I t might happen t h a t there are integrals of the system in the classical sense, i.e., analytic, or at least continuously differentiate, as would be true, for example, if G{P) were of this character; in which case they could be used for the reduction of the dimensionality of ft (cf.,2 p. 315, last line). Or it might happen t h a t no such integrals exist, in spite of the fact that G{P) is non-constant. Conceivably this last situation is impossible when the Hamiltonian II is analytic (or even continuously differentiable); if so, the proof of this fact would be most useful. But it appears that the proof could not be obtained alone from the general formal considerations in Koopman's method, i.e., from (3) and from

UtF(MP),MP), ...) = FiUMP), U,MP), ...),

(14)

(cf.,2 p 318). For (4), (14) remain true in the case that the one to one map P — > Pt of ft upon itself is any one-parameter group of the following properties: a. Pt is a measurable function of /, b. P — > Pt maps every measurable P-set in a measurable P r s e t with the same measure. Since a, b permit of discontinuities of Ph it is easy to give examples with only discontinuous integrals.

Ergodic Theory

141 Mathematics

271

Indeed, even when P — > Pt is defined by means of equations of the type — xi = onixu . . . ,.T/),. . ., — xi = ai(xlt

. . ., .v/),

(16)

(P:(xi, . . . , xi))t and when b is valid—when P — > Pt is indeed an "incompressible continuous flow"—there are examples where all the integrals are discontinuous, and yet are not constants. (In an example, / = 2, ai/oi2 is continuous in P , but ah a 3 themselves, discontinuous.) We shall not pursue this question further. We may observe, in conclusion, how remarkable it is t h a t the concept of Lebesgue measure should play so important a role in a so essentially physical a question as the validity of the ergodic hypothesis, or, more generally, in the value of the limit of the mean sojourn, lim Z(P, N). Even in the case where N is an open set or, indeed, the sum of a finite number of spheres—which has an immediate physical significance— the function X°N(P) given by the above limit does only need to be measur­ able! In the last analysis, one is always brought to the cardinal question: "Does P belong to 0 or not?" where the set 0 is merely assumed to be measurable. The opinion is generally prevalent that from the point of view of empiricism such questions are meaningless, for example, when 0 is everywhere dense—for every measurement is of limited accuracy. The author believes, however, that this attitude must be abandoned, and gives the following reason as an argument: Suppose that 12, in which P varies, have a finite measure, m9.. Since 0 is measurable, it follows from a familiar theorem of Lebesgue that limm[e

™

X K(P, e)) mK(P, e)

(where K(P, e) is a sphere of center P and radius e), exists at each point of 0, and = 1 with the exception of a zero set. 27 Similarly for all points of 12 — 0, where it is zero, with the same exception. The same is true when the spheres are replaced by many other sorts of figures, e.g., cubes. 27 Consider a sequence of partitions of 12 into systems of disjoint cells, Z ( " , . . ., Zi? (n = 1,2, . . . ) , such t h a t the maximum diameter en of Zi (w) , . . ., Z(kn approaches zero as n — > <». The limited accuracy of measurements finds its expression in the fact t h a t we have to consider different order of accuracy (viz., 1, 2, . . . ) ; where, by an experiment of order of accuracy w, shall be meant the mere process of distinguishing in which Z nv (v = 1, . . ., kn) P lies. Suppose t h a t a measurement of order n has established that, for in-

142

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Mathematics

stance, P lies in Z ( " ; ; then the (geometric) probability that P belong to . . w[Z ( : } X e ] e is —-—^—TT^-—~. So that if W [ Z

^(*

6]

< « or > 1 - 5

(5 > 0),

(15)

we know with a probability > 1 — 5 the answer to the question, " I s P i n © or not?" The fact that we will be able to answer this question with a probability > 1 — 8 of being right has the a priori probability (i.e., before the observation is made) of

(i) = w («) w =

B) EE/( -v" rn « Z™ z™ == £>Z? E^I^T;

E «^("i

mn

where 2 ^ ( w ) represents the summation over all values of v satisfying (15). If we could prove t h a t w\?] —>- 1 as n —>■ oo f it would become clear that, granted a sufficiently high accuracy of experiment, the above question could be answered with an arbitrarily great degree of certainty—i.e., the question has physical meaning. (This is seen by taking, e.g., w^ > 1 -8). Suppose t h a t w^ —>- 1 as w —>■ o° is untrue. Then for infinitely many values of n, w ( ^ < 1 — rj (for a certain rj > 0); so that if XT(»>n; implies summation over all values of v for which (15) is violated, we shall have n) XT;n) Z(:} ^ ^ i? rj «o. ma. «m E'v ^ The set S of all points P which belong to infinitely many such sets J]*(J,) Z ( " } will then also have a measure > rjmQ > 0, in virtue of a theorem of Arzela's. 28 If P is on S, it lies on infinitely many sets S ^ Z ^ ; suppose it to be, for example, on Z ^ . Since vn belongs, for infinitely many values of n, to X"in)> (15) is violated for these values, the ratio

(WJ

Vn

mZ vn determines neither the limit 0 nor 1. But this is in contradiction with the theorem of Lebesgue, in the case where the Z ( ? } , s are such that its hy­ pothesis applies (e.g., when Z ( ? }, s are cubes). 1

For the formulation and critique of this theorem, cf., e.g., Entykl. d. Math. Wiss., 4, Art. 32 on Statistical Mechanics, by P. and T. Ehrenfest, specially 30-36. The original formulations are to be found in Wien. Ber., 63, [2] 679 (1871) (Boltzmann), and Cambr. Phil. Soc. Trans., 12, 547 (1879) (Maxwell). 2 These PROCEEDINGS, 17, [5] 315-318 (May, 1931). 3 Cf., e.g., the discussion in the author's paper, "Allgemein Eigenwerttheorie Hermitescher Funktionaloperatoren" {Math. Ann., 102, [1] 108-111 (1929)). This paper, as well as the author's paper, "Zur Algebra der Funktionaloperatoren und Theorie der normalen Operatoren" {Math. Ann., 102, 3 (1929)), will be referred to in the present paper under the abbreviations E and A, respectively.

Ergodic

Theory

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Mathemaiics

273

♦Cf. E, 54-55, 109. Cf. E, 91-92. 6 These PROCEEDINGS, 16, [2] 172-175 (Feb., 1930); also, cf. a paper soon to appear in the Ann. Math. 7 Cf., regarding these concepts, the article of Hellinger and Toeplitz in the Math. Encyklopadie, 2, c. 13, 1435 (1928); further, cf. A, 378-381. 8 Cf., e.g., the similar proof in E, 112, top. 9 R* is the adjoint of R in the terminology of matrices: the con jugate-transposed matrix. Cf., e.g., 112. 10 The integrand, e , / x , is uniformly bounded, the expression after the d-sign, (E(y),f), of bounded variation: / l + J 0 d(E(y)f, f) = (f,f). 11 E, 91, 77-78. 12 Cf. the theory of projection operators outlined in E, 74-78; similarly for the discussion to follow. 13 Cf. E, 110. 14 Cf. Hansdorflf, Mengenlehre, 127 (1927), line 6; or E, 110. 15 Cf. E, 110. 16 Cf. the corresponding construction in the proof of theorem 10; in A, 401-402. There, the permutable projections £ l f E2, . . . and Fif F2, . . . took the place of the A-sets Ai, A2, . . . and A / , A2', . . . ; but this distinction can be abolished by replacing each A-set by the operator, £ x : J(P) >- x\(P)f(P)17 For clearly, XA + M = XX. + XM — X\XM, X\xM = XAX'A/. XA-M = XA — XAXJl/18 If the sequency / l f / 2 , . . . converges ( >-), a subsequency will converge in any point, excepted a 0-set (cf., f.i. E, 111). Therefore a limit of functions of G is a function of G. 19 Thus, e.g., each T(G) is one, if T(y) is a monotonically increasing function. 20 On the other hand, our construction shows that it suffices to confine $ to the second Baire class (the convergence is pointwise convergence except for zero sets). 21 This transformation from Lebesgue to Stieltj^s integrals goes back to Lebesgue. Cf. Ann. de Vtcole Normale, 3, 27 (1910), p. 407. It is sufficient to establish it for a real variable, for it is then easily extended to an arbitrary Q, which may always be mapped in a measure-preserving manner upon the real axis. Maps of this sort are given by Lebesgue (loc. cit.) for n-dimensional space, and may easily be extended to il. 22 Since £(G) belongs to 2ft, it is left unchanged by E0. 23 At points of discontinuity of x = MA(X), where x experiences a jump of a whole interval, the differential quotient has a meaning, and is equal to the difference quotient between X -f- 0 and X — 0: 8

Jl (x+0) - A (x - 0) f(P)dv + 0) - A ( X - 0 ) ] *

M[A(X 24

Cf. the author's paper, "Uber Funktionen von Funktionaloperatoren," Ann. Math., 32, [2] 196 (1931), (Satz 3), as well as the reference (20). 25 This transformation consists in a change in the order of integration; since in the Lebesgue integrals that appear, everything is bounded, it is permissible. 26 For nil = 001 naturally only the former will come into question. 27 Cf. E, 110. 28 The generalization to other figures is to be found in its broadest form in Caratheodory, Vorlesungen fiber reelle Funktionen (Leipzig-Berlin, 1918), 492-494, in particular, Theorem 3. The f u n c t i o n / ( P ) appearing there is to be defined a s / ( P ) = A'G(P). 29 Cf., e.g., de la Vallee Poussin, Cours d'Analyse infinitesimale, 1, 2 (Louvain-Paris, 1909)„ pp. 68-69.

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OPERATOR METHODS IN CLASSICAL MECHANICS, II BY PAUL R. HALMOS AND JOHN VON NEUMANN

Introduction The purpose of this paper is two-fold: to map all measure spaces for which this is possible on the unit interval, and to apply such mapping theorems to the study of ergodic measure preserving transformations with a pure point spectrum. "Mappings" between two measure spaces may be interpreted in two ways, as set mappings and as point mappings, and accordingly we give below two sets of necessary and suflBcient conditions for the existence of a mapping from a given space to the interval. The first of these, the set mapping or algebraic iso­ morphism theorem, seems to be known, and although it has never been explicitly stated in the literature there are many proofs of special cases of it on record. We give an explicit proof of it and use a construction of the proof in proving the second, point mapping or geometric isomorphism, theorem. This second theo­ rem depends on the new concept of normal measure space: a seemingly artificial concept which is, however, useful for two reasons. First, it is purely measure theoretic (and not topological), in character, and hence is applicable to the measure spaces usually discussed in probabiUty theory; second it is hereditary under all the usual operations on measure spaces (such as the formation of direct products, decomposition into direct sums, etc.). Using the concepts and results of the mapping theorems just described, and of the Pontrjagin duality theorem concerning compact and discrete abelian groups, we are able to show that every ergodic measure preserving transforma­ tion with a pure point spectrum is isomorphic to a rotation on a compact abelian group. This is a "normal form" theorem for a certain class of measure pre­ serving transformations and can be used to answer many questions, such as the existence of square roots, commutative transformations, etc., concerning such transformations. Although this paper is a continuation of an earlier work of one of us1 it is to a large extent independent of this earlier work. The proofs of the main theorems mentioned above are logically complete here; only in some of the applications, as for example in discussing the relation between point mappings and set map­ pings, do we make use of the results of (I). 1. General measure spaces; the algebraic isomorphism theorem Let X be any set, and 9C any Borel field of subsets of X; let m be a non negative, contably additive, finite measure defined on 9C. The system [X, 9C, m), which we shall usually denote by X, or, if necessary to indicate its 1

See John von Neuraan, Zur Operatorenmethode in der klassischen Mechanih, Annals of Mathematics, vol. 33, (1932), pp. 587-642. In the sequel we shall refer to this paper as (I). 251 Published in Ann. Math., Vol. 43, pp. 332-350, © 1942 Princeton University Press. Reprinted from John von Neumann Collected Works, ed. A. Taub, Vol. IV, pp. 251-269.

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Ergodic Theory

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dependence on 9C and m, by Z(9C, m) is called a measure space. Sets E e 9C are called measurable; we shall use also the usual terminology of the Lebesgue theory in describing functions as measurable, integrable, etc. A measure space is complete if every subset of a measurable set of measure zero is itself measurable (and has, of course, measure zero). Since it is always possible to extend the definition of m to a Borel field 9C' 3 9C so that X(9C', m) is complete, we shall lose no generality, and gain somewhat in simplicity, by assuming completeness. In any measure space X we shall write 93 = 93(9C) for the Boolean algebra of measurable sets modulo sets of measure zero. We shall make use of the notations of set theory, ( C , + , etc.) in 33, and of the fact that we may consider m as defined on S3. We discuss now the concept of separability in measure spaces. A Borel field 9C (or a measure space X(9C, m)) is strictly separable if it contains a count­ able collection of sets such that the smallest Borel field containing all of them, (the Borel field spanned by them), is 9C itself. Two sub Borel fields, (5 and SB, of the Borel field 9C of measurable sets in a measure space X(9C, m) are equivalent if to every set E in either one of them there corresponds a set F in the other such that the symmetric difference (E — F) + (F — E) has measure zero. A measure space is separable if there exists a strictly separable Borel field Q con­ tained in and equivalent to 9C.2 A concept, which lies logically between separa­ bility and strict separability, more useful than either of these, is proper separa­ bility. A measure space X(9C, m) is properly separable if there exists a strictly separable Borel field (3 CZ 9C, such that to every E e^Xl there corresponds an F € G with E C F and m(F - E) = 0.3 We observe that this definition is self dual: by applying the condition to X — E we readily obtain a set Fed with F C E and m(E — F) = 0. We shall make use of the fact that if X is separable (or properly separable) and Q is the strictly separable Borel field described in the definitions above then 93 (9C) = S3 ((3). In the case of (properly) separable measure spaces it will be necessary to indicate in the notation the strictly separable Borel field used; we shall write X = X(9C, Q, m). We shall call sets of Q Borel setsf and functions measurable (Q) Baire Junctions. (A real valued function f(x) is measurable (G) if the inverse image under / of every real Borel set S, i.e. the set {x \f(x) e S], belongs to (3.) 2 This is not the usual form in which this definition is given. Cf., for example, J. L. Dobb, One—parameter families of transformation, Duke Mathematical Journal, vol. 4, (1938), p. 753. That our definition is, however, equivalent to the usual one is proved by Paul R. Halmos, The decomposition of measures. Duke Mathematical Journal, vol. 8, (1941), p. 387. We observe X is separable if and only if the Boolean algebra S3(9Q has a countable number of generators. 3 The concept of proper separability, first introduced by W. Ambrose and S. Kakutani, Structure and continuity of measurable flows, Duke Mathematical Journal, vol. 9, (1942), pp. 25-42, is fundamental in measure theory. Although it is possible to give examples of separable but not properly separable measure spaces, these examples are all of a more or less pathological kind. One such example is the unit interval, with the Borel field of all sets of Lebesgue measure zero and their complements in the role of 9C.

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A measurable set E in the measure space X(9C, m) is indecomposable if it contains no proper measurable subsets other than the empty set; an element E e S3(SQ is an atom if it contains no proper subelements, other than 0, in S8(SQ. A measure space is non atomic if 33 (9C) has no atoms: in other words if every measurable set of positive measure contains measurable subsets of smaller posi­ tive measure. From the point of view of a study of the structure of measure spaces indecomposable sets and atoms are uninteresting: we shall generally assume that the former consist of exactly one point and the latter are absent. More specifically our assumption will be described in the following terms. A countable sequence, A\, A2, • • • , of subsets of X is a separating sequence if to every pair of points, x 9^ y, we may find an integer n with x e An, y eX — An . If there exists in X a separating sequence of measurable sets, an indecomposable set contains exactly one point. We shall now show that the assumption of the existence of a separating sequence of measurable sets has a similar effect on atoms. Let E be a set of positive measure which contains no measurable subsets of smaller positive measure. I t follows that for each n one of the two sets, EAn , and E(X — An) has measure zero and the other one has measure m{E). By a slight change of notation we may assume m(EAn) = m{E) for n = 1, 2, • • • . If we write IJn=»i An = A, then we have m(EA) = m(E); since, however, A can contain at most one point, this implies that for some point x e E we have m(E — x) = 0. In other words the existence of a measur­ able separating sequence implies that the weight of an atom is concentrated at one point; if, for example, we assume that the measure of a point is always zero, we may infer that the space is non atomic. Since in a measure space, which has by definition finite measure, there can be at most a countable set of points of positive measure, and since their measure theoretic structure is clear, we shall generally assume non-atomicity explicitly. If Xi(9Ci, mi) and X2(9C2, m?) are measure spaces, a set isomorphism between Xi and X2 is a measure preserving isomorphism between the Boolean algebras S3(9Ci) and 93(9C2). More specifically a set isomorphism is a one to one mapping T from SB(9Ci) on 33(9C2) which is such that T(Xi - E) = X2 -

TE,

T{2_,n=l En) = Z^n=l TEn, mi(E) =

m2(TE).

If such a mapping T exists, Xi and X2 are set isomorphic. After one more comment on notation we shall be ready to state and prove our first result. Since the unit interval plays a fundamental role in our investiga­ tions and is used as a yardstick with which to compare other measure spaces, we find it convenient to introduce a special notation for it. We shall denote the unit interval by X, the collection of Lebesgue and Borel measurable sets by

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9C and Q respectively, and Lebesgue measure by m. In our terminology X = XiVC, 5 , m) is a properly separable measure space.4 THEOREM 1. A necessary and sufficient condition that a measure space of total measure one be set isomorphic to the unit interval is that it be separable and nonatomic. PROOF. Since the unit interval is separable and non-atomic and since these properties are evidently invariant under set isomorphisms, the necessity of our conditions is clear. To prove their sufficiency, let X(9C, (2, m) be the given measure space, m(X) = 1, and let At, Az, • • • be a countable sequence of Borel sets which span (3. We may assume (by adding a superfluous set to the {An\ if necessary) that ^ n - i 4 „ = X. Then we may make correspond to every rational number r, 0 g r ^ 1, a set B f such that (i) {An} and {Br} span the same field; (ii) r < s implies Br C B$ ; (iii) Ur>.Br = £ . ;

(iv) IL5 r = 0;i; r B r = X.6

We now define, for every real number a, 0 ^ a ^ 1, a set Ba by Ba = H»->a Br. It is clear that this definition of Ba is consistent with its previous definition in case a is rational, and that the family of sets [Ba] satisfies the conditions (ii), (iii), (iv), (where in (iii) and (iv) we extend the products and sums over an arbitrary countable set of real numbers r for which inf r = s in (iii), inf r = 0 and sup r = 1, respectively, in (iv)). Moreover, condition (i) implies that Ba eQ for all a and that the Borel field spanned by the Ba is Q itself. Given now the family Ba we may find a (uniquely determined) function/(x), defined for x e X, 0 ^ f(x) ^ 1, for which {x \f(x) ^ a] = Ba ; we may, for example, define (1)

f(x) = inf [a\x

eBa}.

The class of all sets of the form (2)

r\E)

=

[x\f(x)eS],

where E is an arbitrary Borel set in the unit interval, is a Borel field contained in (3; since it contains all Ba , and therefore all An , it coincides with Q. Let F(a) = m{x\f(x) ^ a] = m(Ba) be the distribution function of f(x): F(a) is monotone non-decreasing from 0 to 1 as a ranges between 0 and 1, and is continuous from the right. (This much is always true, of an arbitrary distri­ bution function.) In our special case we assert that F(a) is continuous. For 4 In the sequel we shall sometimes use the notation X(9C, (J, m) for the perimeter of the unit circle in the complex plane: it is clear that this space has the same measure theoretic structure as the unit interval. We shall always make it clear whether the symbol X has its real or its complex meaning. 5 Cf. (I), p. 602; see also J. L. Doob, Stochastic processes with an integral valued parameter, Transactions of the American Mathematical Society, vol. 44, (1938), p. 91.

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if a = do is a discontinuity of F(a), then {x \f(x) = OQ] is a set of positive measure which therefore, (non-atomicity), has Borel subsets of smaller positive measure. Such a subset cannot be put in the f o r m / - 1 (E), contrary to what we have already proved. For any x, 0 ^ x ^ 1, we define/(x) = inf [a \ F(a) ^ x}. I t is well known (and easily verified) that J(x) is a strictly monotone increasing (not necessarily continuous) function of x, which increases from 0 to 1 as x does, and which is continuous on the left. Moreover the distribution function of/(£) is again F(a). For any Borel set E C X, consider the setf~x(E): we assert that the collection of all sets of this form, (which clearly forms a Borel field), coincides with Q. This is true since the increasing character of ]{x) implies that every interval (0, x) has the f o r m / - 1 (2?), where E can even be chosen as an interval. Suppose that it ever happens t h a t / " 1 ^ ) = f~l(E2). (We shall now make use of the fact that for an arbitrary Baire function g(x), 0 g g(x) ^ 1, the correspondence E —> g~l{E) = {x \ g(x) e E], is a homomorphism of Q into (2, i.e. that g~\X - E) = X - g~\E), and g"\Ei + & + • • • ) = g~\Ex) + g~\E2) + • • • ) • Ifjra write E' = (Si - E2) + (E2 - Ex) for the symmetric difference between Ex and E2, then it follows from the equality of the distributions of f(x) a n d / ( x ) , that m{r\E')} = m{J~\E')} = 0. Con­ versely, of course, JT\SI) = r\E2) imphes the same result. Consequently the correspondence f~l(E) +± J~l(E) is one to one, not neces­ sarily between Q and 5 , but certainly between 35 = 53(9C) = 99(G) and $ = 93(9C) = 93(5). It is clear that this correspondence preserves measure, and the homomorphic nature of the mappings E —> f~l(E) and £ —> J~l{E) shows that it is also an algebraic isomorphism. This concludes the proof of Theorem 1. 2. Normal spaces; the geometric isomorphism theorem If Xi(9Ci, mi) and X2(QC2, ra2) are measure spaces, a point isomorphism between X\ and X2 is a one to one mapping from almost all of Xi on almost all of X2 such that Ex e 9 k if and only if E2 = TEX t SC2, and then mi(Ei) = m2(E2). If such a mapping T exists, Xi and X2 are point isomorphic. Our problem in this section is to find necessary and sufficient conditions in order that a measure space be point isomorphic to the unit interval. The funda­ mental concept in this connection is that of a normal space. DEFINITION 1. A measure space is proper if it is complete, properly separable, and non-atomic, and if it contains a separating sequence of Borel sets. DEFINITION 2. A proper measure space is normal if to each real valued univalent Baire function f(x) there corresponds a set X0 of measure zero such that the range, f(X — Xo), is a Borel set. The following lemmas concerning proper and normal spaces will be useful in the sequel. LEMMA 1. On every proper measure space X(9C, Q, m) there exist real valued bounded univalent Baire functions.

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PROOF. Since X is certainly separable and non-atomic the construction of the proof of Theorem 1 applies. We assert that the real valued bounded Baire function/(x) defined by (1) is univalent. For if the set [x | f(x) = a] contained more than one point, then the intersection of this set with a Borel set separating two of its points could not be expressed in the form {x | f(x) elS}. Since, how­ ever, the proof of Theorem 1 establishes that every Borel set has this form, f(x) must be univalent. LEMMA 2. If X(9C, fl, m) is a proper measure space with the property that the condition of Definition 2 is satisfied by every bounded function then X is normal. PROOF. Let f(x) be any univalent Baire function, and let G(y) be any con­ tinuous function which maps the infinite interval, —oo < ?/ < + °°, in a one to one way on a finite interval. Then g(x) = G(f(x)) is a Baire function which is univalent and bounded, hence, by hypothesis, there is a set X0 of measure zero such that g(X — X0) is a Borel set. The image of this Borel set under the one to one continuous mapping ^(y) is the range/(X — X0) which is therefore also a Borel set. LEMMA 3. 2/ X(9C, Q, m) is a normal space, B C X is a Borel set, and f(x) is a real valued univalent Baire function, then there is a set B0 C B of measure zero such thatf(B — B0) is a Borel set B0 can even be chosen in the form BXQ , where Xo is a Borel set of measure zero, depending on f but not on B. PROOF. We shall carry out the proof in three steps, first establishing the existence of a suitable Bo corresponding to a fixed B, then showing that Bo may even be chosen as a Borel set, and, finally, proving on the basis of our separability hypotheses, that we may choose B0 in the form described in the statement of the lemma. (i) We observe that the first statement asserts, essentially, that a Borel set in a normal space is itself a normal space. Accordingly, using Lemma 2, we may assume that f(x) is bounded. Let/'(x) be a bounded univalent Baire func­ tion on X, (Lemma 1); by appropriate linear transformations of f(x) and of f'(x) we can secure

0 ^ f{x) ^ 1 < /'(*) throughout X. Then the function f*(x), defined to be equal to f(x) on B and to/'(x) on B' = X — B is a univalent Baire function on X, hence for a suitable set Xo of measure zero, f*(X — X0) is a Borel set. The intersection of this Borel set with the closed interval (0, 1) is also a Borel set: this intersection is, however, precisely f(B — B0), where B0 = BX0. (ii) Let Bi be a Borel set of measure zero, B\ Z> B0. Applying the result of (i) to X — Biwe may find a set B2 3 B\ of measure zero such that f(X — B2) is a Borel set. We proceed similarly by induction, choosing B3 IP B2 to be a Borel set of measure zero, choosing B4 3 Bz so that f(X — B*) is a Borel set, and so on. We have B0 CZ B\ C B2 C B3
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character of the sequence {B n }, B* = ]C»-o#2n+i, so that B* is a Borel set. Similarly X - B0* = X - £ ; .