Hilbert space formalism of quantum mechanics without the Hilbert space axiom

Vol. 3 (1972)

REPORTS

HILBERT

ON MATHEMATICAL

No. 3

PHYSICS

SPACE FORMALISM OF QUANTUM MECHANICS WITHOUT THE HILBERT SPACE AXIOM

M. J. M&?Xr;rSKI Institute

of Mathematics, (Received

Warsaw

Technical

University,

Warsaw

November 15, 1971)

It is shown that the Hilbert space formalism of quantum mechanics can be derived from a set of seven axioms involving only the probability function P(A) a, E) (the probability that a measurement of an observable A in a state a will lead to a value in a Bore1 set E) and the complex field postulate. In particular, the existence of a Hilbert space corresponding to the given physical system needs not be postulated but follows from the axioms.

1. Introduction

In [4], G. Mackey formulated a set of axioms from which it is possible to deduce the usual Hilbert space formalism of quantum mechanics. The first six of his axioms are of more general nature. They imply that with each physical system we can associate a partially ordered o-orthocomplemented set 9 such that each observable can be identified with an 9-valued measure on the real Bore1 sets and each state can be identified with a probability measure on 9. (In [7] and [8] we have shown that the same conclusion can be obtained from a set of three axioms only). This system constitutes the usual framework for the so-called axiomatic quantum mechanics, which is a subject of extensive research, since it has turned out that some facts and theorems in quantum mechanics can be stated and proved already in this system without involving the Hilbert space. These six axioms are more or less physically plausible and natural. On the other hand, the next axiom of Mackey (the so-called Hilbert space axiom) is of entirely different character. It postulates that 3 is isomorphic to the lattice of all closed subspaces of a complex Hilbert space. It is easy to show that the acceptance of this axiom leads to the usual Hilbert space formalism of quantum mechanics. However, this axiom certainly cannot be termed as physically natural and plausible. The axioms would be perfect it they made up a set of possibly simple experimentally directly verifiable statements all expressed in the same language from which the Hilbert space formalism, in particular the Hilbert space axiom, would follow. In this paper we would like to show that this aim can be achieved almost in full extent if we properly inter-

M. J. M_@ZYr;rSKI

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prete the meaning of an experimentally verifiable statement. Such statement will be termed in the sequel as physically basic. The most direct link between each physical theory and the experimental framework is established by the probability function ~(4, tl, E). With each physical system we associate the set 0 of all observables and the set Y of all states. Let 9(R) be the family of all Bore1 sets on the real line R. Let p(A, a, E) denote the probability that a measurement of an observable A E 0 for the system in a state a E 9 will lead to a value in a Bore1 set E E B(R). For each A E 0, n E 9 and E E .92(R), the value of p(A, a, E) (a real number in [0, I]) can be determined, at least in principle, experimentally by a more or less complicated experimental arrangement. On the other hand, each physical theory allows to predict the values of p(A, a, E) theoretically. A theory is good if the calculated values of p(A, a, E) are in good agreement with the experimantal ones. We see that p(A, a, E) has a pure mathematical meaning: it is a function from 0 x Yx S?(R) into [0, 11. Physically, for each A E 8, IXE Y and E E a(R), p(A) a, E) is an elementary term of our theory which is interpreted as “the probability that a measurement of A in state a leads to a valuein E”. An elementary term is not yet a meaningful physical statement, it has no logical value. Nevertheless, from elementary terms we can built up meaningful statements applying first mathematical signs +, 1, =, <, according to the rules of algebra and then logical connectives and quantifiers according to the general rules of mathematical logic. For example, when we have determined experimentally the values of p(A, a, E), we can decide whether statements of the form ~(A~,a~,E,)=p(A,,a,,E,),p(A,,cc,,E,)~p(A~,cc~,E~) orp(AI,aI,EI)+p(Az, a,,&)
and quantifiers,

(such postulates

Accordingly,

we adopt

DEFINITION 1.

the following

A statement

if it can be expressed

signs and logical

B(R)

or to their subsets

of quantum

any

definition

about

connectives built

can be termed

which

may

of a physically system

function

and quantifiers

up according

aim is to express

mechanics

theory

a physical

in terms of the probability

braic

Our immediate

since only such statements

do not preassume

develop

is said to be physically

in the form of physically

basic

p(A) a, E) with the help of alge-

which refer to elements

leading

from them).

basic statement.

to the rules of mathematical

postulates

as physically

to the Hilbert

of 0, 9

and

logic. space formalism

basic statements.

2. The axiom system Let 0 be the set of all observables (two non-empty family

abstract

and 9’ the set of all states of a given physical

sets from a mathematical

point

of Bore1 sets on the real line R (which is a Boolean

of view), and let 9(R) a-algebra).

system be the

With each quantum

HILBERT SPACE FORMALISM

mechanical into

system there is associated

[0, I], denoted

by ~(4,

OF QUANTUM

a function

MECHANICS

from the Cartesian

c(, E). We postulate

product

that this function

211 0 x Y x .4?(R)

satisfies the following

axioms. AXIOM 1.

For every .4 in Loand every cz in 9,

the map E HP(A)

a, E) is a probability

measure on 9?(R). Before stating further axioms, we need some abbreviations. Let b= 8 x B(R). For each a E Y and a=(A, E) E 8, let p,(a) =p(A, cc,E). Two elements a, b in d are said to be equivalent, a-b, iff for all CIin 9, p,(a)=p,(b). For each a=(A, E)E b, let a’= (A, R-E). Let 0 denote any element in d of the form (A, 0) where AE 0. Let 1 denote any element in & of the form (A, R). AXIOM 2. +p,(aj)<

For every sequence

1 jbr all tl E 9,

a,, a,, . . . of elements

there exists

of 8 where, .for if j, p,(aJ+

an a in d such that p,(a) = fp=(aj)

for all a~ 9.

j=l

AXIOM 3. For every subset Q of d there exists an a in 8 such that p,(x)p,(e) for some a E Y. AXIOM 4. for all CIE Y, some CIE 9.

For every two non-equivalent elements a, b in 6 satisfying p,(a)~,(a) for

AXIOM 5. For every two atoms e, , e, and every a in d such that pJe,)>p.(a) some CIE Y, p,(el)
For every a not equivalent

for

to 0 and to 1, there exists b E d not equivalent

to a’ such that 1 E {a, b)” and 1 E {a’, b’}“. AXIOM 7. There exist at least four non-equivalent elements ai, i= 1, 2, 3,4 a, not equivalent to 0, such that p,(ai)
in 6,

Observe that all these axioms are physically basic statements, according to our definition. Although in some axioms there appear such terms as “e is an atom” or “c E (a, b}“” which are not primitive notions of our language, they can easily be expressed in terms ofp, only and then substituted into the axioms to obtain physically basic statements containing only p,, algebraic signs, and logical signs and quantifiers, as required in Definition 1. For example, the definition of an atom can be written as e is an atom=(e*O)r\{

A [(ame)r\(amO)]* OS8

v(p,(aj>pJe)}, ae.v

212

M. J. M4CZYfiSKI

where e-O=

V p,(e)#O, (IEY

aWe= V p,(a)fp.(e) ; L7E.Y and c E {a, b)” means

A {

A [ A p,(a) < p,(d)] A [ dsb

Of course, the statement but it satisfies the formal

a~9

A9db) 4 p,(d)]*

obtained after substituting the above requirement of the definition.

cz?5p~.(c) G p=(d)] > -

definitions

may be long,

3. The representation theorem

We can now state and prove our basic theorem. THEOREM 1.

Let p: 0x9~

B(R)+[O,

l] be a function satisfying Axioms l-7.

Let

B=OxB(R) and let _Y={(laj: UE~}, where lal={x~&: p,(x)=p,(u) for all KEY). Suppose that in 9 1~1
we must state some facts from lattice theory involved adopted here is consistent with [6], where the proofs

(1) A partially ordered set 9 is a complete lattice if every subset of L has its join in 9, since then, for every subset S, the set of all lower bounds has its join which is evidently the meet of S. (2) We say that b covers (I in a lattice and write a
e of a lattice

with the least element

0 is called

(i.e., a< b and a#b) an atom if O
(4) A lattice 9 is called atomistic when every non-zero element a of 9 is the join of atoms contained in a. It can be shown that a lattice 8 with 0 is atomistic iff Y satisfies the following condition: ac b implies the existence of an atom e such that non e,
HILBERT SPACE FORMALISM

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MECHANICS

213

(5) We say that a lattice 2 with 0 has the covering property if for every atom e E Y and any a E S’ such that a n e =0 we have a
(i) al n a=0 and a’- ua=l (ii) a
(i.e. al is a complement of a).

implies al>bl. = a for every a.

When a
we say that a and b are orthogonal and write a I b.

(7) An orthocomplemented lattice .& is called orthomodular when 9 _Lb implies (c u a) n b = c u (a n b) for every c< b. It can be shown that an orthocomplemented lattice is orthomodular iff for every a< b there exists c E B such that a I c and au c =b (see [6], 29.13). (8) In an orthocomplemented lattice _Y two elements a and b are said to split, acrb, if there exist a,, bl, cl, a, I bl, a, I c, bl _L c such that a=al LJc and b=bl u c. An element z E Y is said to be a central element if z-a for all a E 2. The set of all central elements is called the center of 2. It can be shown that z is a central element in an orthomodular lattice iff z has a unique complement, namely zL (i.e., if z n z1 =0 and zu z1 = 1 imply z,=z’) (see [6], 36.9). An orthocomplemented lattice is said to be irreducible if

the center of _Y consists only of 0 and 1. (9) A lattice _Y with the covering property . .
is said to be of length n if there exist

a,, a2, . . . , a, such that O
(10) If P is an orthocomplemented to be an Z-valued measure if (i) for E, FELB((R), En (ii) if Ei n Ej=O

F=0

complete lattice, then a map ,u: B(R)-+2

is said

implies ,u(E) I p(F);

for i#j, then ~(E~uE~u...)=~(E~)u~((E2)u...;

(iii) ,~(0)=0,

p(R)=l.

A family {,u~}, -4 EO, of P-valued measures is said to be surjective if for every a E LZ there exist A E 0 and E E S?(R) such that ~~@)=a. (11) If 2 is an orthocomplemented

complete lattice, a map m: _Y+ [0, l] is said to be a

probability measure on L? if

(i) m(O)=O, m(l)=l; (ii) if a, I aj for i#j, aiE 5?, then m(al uazu

...)=j$lT?l(c.lj).

M. J. Mz&ZYrjSKI

214

A family {m,}, CIE Y, of probability measures m,(a)
on .!Z is said to be jiiZZ if for a, b E 9,

iff p,(u) Proof of the representation theorem: It is evident that the relation “u-b =p,(b) for all c( in 9”’ is an equivalence relation in 8. 9’ is then the set of all equivalence classes of the relation -, i.e. _Y = G/-. Let 0= 101 and 1= / 1 I. It is also evident that the relation < in 9 defined as “la/ d lb1 iff p,(u)
f p,(uj)

for all a E 9’.

j=l

We shall show that Ial is the join of the set {]a,, /a,], . ..}. i.e. lul=lu,l u lazl u . . . Let lujl
f

p,(uj)

for all

aEY.

j=l

This

implies p,(b’) + f p,(aj> < 1

,

i.e.

jrlPa(aj)C

1 -Px(b')=Pz(bj

j=l

for all a E 9. We conclude that p,(a)dp,(b) for all ME 9, i.e., Ial d Ibl, and Ial is the least upper bound of Iail, j= 1,2, . . . Since lulp,(e) for some CIE 9, we see by (2) and (3) that e E E is an atom iff lel is an atom in the lattice 8. Now Axiom 4 implies that for every two elements ICI\, Ibl in 9 such that lal” is equivalent to la,1 = Ial u lel, A xiom 5 means that for every two le,I,
HILBERT SPACE FORMALISM

The conditions

OF QUANTUM

(ii) and (iii) in (6) are evidently

satisfied

la] ~+\a] l. We also have Ial u la/‘- =l, since Ial a, E)+p(A, CI,R-E)=p(A, = IA, El, P,(~)+P,(a’)=P(A? lbl= lJ (1~1: IuI E S} implies lb\‘= n {]u]k Ial E S} ( i.e. in L). Hence la/ u Ial i =l implies Ial n Iall =0 and (i) the map

MECHANICS

215

in view of the definition

of

I Ial’- and by Axiom 1 for a cc, R)= 1 =p,(l). Observe that de Morgan’s laws are satisfied in (6) is also satisfied. Thus _?Z’

is an orthocomplemented lattice. The lattice 2 is orthomodular. In fact, let Ial
= P,G> + P,(C> = Pa(a) + [I - A(41

= PAa> + I? - (P&4 + 1 -Pm)]

= P,(b)

for all u (here instead of p,(c’) we have substituted p,(u) + (1 --p,(b)), since ICI_L= Ial u lb/ 1 for all CIE 9, i.e. Id/ = Ibl and Ial u ICI=Ibl where and Ial _L lbll). H ence p,(d)=p,(b) IuI I ICI. Thus, by (7), 2 is orthomodular. Axiom 6 implies that for every Ial ~0, 1 there exists Ibl #lull- such that Ial u lb] =l and Ial n Ibl=O (1 ~{a’, 6’}“, i.e. 1-1~1~ u /bll, is equivalent to Ial n Ibl =0 by de Morgan’s law). Thus every element Ial of 3 different from 0 and 1 has at least two complements, are 0 and 1, namely Ial i and Ibl. H ence the only central elements with unique complements and by (8) the lattice 9 is irreducible. Axiom 7 shows that there are at least four elements la,/, la,/, la,/, Ia,] of S? such that O< Iul/ < Ia,] < Ia,] < 1~1. By (9) it is evident that the lattice 3 is of length 24. Hence 2 is an irreducible complete orthomodular AC-lattice of length 24. For each AEU, the map Pi: EHI(A, E)] is an Y-valued measure. In fact, we have /(A, E, u E2. ..)I i#j, since Ei n Ej=O implies /(A, Ei)l l_ = IL4 &)I u I(4 Edl u . . . for E, n Ej=O, _L I(A, Ej)l for i#j, and by Axiom I, P(A,a,E,utlu...)=j~~Pia,z,Ej)

We have

for all CI.

already proved that such an equality for the orthogonal sequence /(A, E,I), I(4 &)I >. . . means that I(A, E, u E2 u . ..)I = I(A, E,)l u I(A, E2)l u . . . it is now evident that all the conditions in (10) are satisfied. Since every element of 2 is of the form I(A, E)j, the family {puA},A E 0, is surjective. Finally, it is evident from Axiom 2 that for each CIEY, the map m,: l(A, E)I t+p(A, CI,E) is a probability measure on 2. The family, {m,}, C(E9, is full by the very definition of the order < in 9. We clearly have p(A, CI,E)=ma o pA(E). Thus we have shown that each function p(A, M, E) satisfying the Axioms l-7 generates an irreducible complete orthomodular AC-lattice _Y of length 24 with a surjective family (~1~1, A E 0, of Y-valued measure and a full family of probability measures {m,}, 01E Y, such that p(A, CI,E)=m, okra. The proof of the converse statement (the last part of Theorem 1) is straightforward and we shall omit it. Note only that not every orthomodular lattice admits a full set of probability measures on it.

216

M. J. M.&ZY%KI

It is interesting to observe that the fact that the logic _Y of p is a a-orthocomplemented partially ordered set can be derived from Axiom 1 and Axiom 2 (see 171). Consequently, in axiomatic quantum mechanics we can base our theory on these two axioms only (see [S]). 4. The Hilbert space formalism of quantum mechanics To show that _%’can now be associated with a Hilbert space, we have to appeal theorem proved by M. D. MacLaren [5] and Piron [9] (see also [6], 34.5).

to a

THEOREM 2 (MacLaren). Let 9 be an irreducible complete orthocomplemented AC-lattice of lenght 24. There exists a division ring K with an involutorial anti-automorphism l-t,%* and there exists a vector space H over K with a definite Hermitian form f such that 9 is ortho-isomorphic to the lattice 9n(H) of H-closed subspaces of H.

In view of Theorem

1 and Theorem

2, we obtain

the following

corollary.

COROLLARY 1. Zf the function p(A) a, E) satisfies the Axioms 1-7, then there exists a vector space H over a division ring K with a Hermitian form f such that _Y is orthoisomorphic to the lattice _Yn(H) of H-closed subspaces of H.

To proceed further we have to decide what division ring should be associated with the lattice 2. Our axioms so far give no clue of this matter. The most natural examples of a division ring are fields, in -particular the field of real numbers, the field of complex numbers, and the field of quaternions. All these fields were exploited in quantum mechanics, but the field of complex numbers is undoubtedly the most frequently used. At this moment it seems rather improbable that a new physically basic statement similar to the Axioms 1-7, and implying that K is the field of complex numbers will be found. We are rather inclined to accept the fact that K should be the field of complex numbers on the basis that it is the one most frequently used in physics, and no division ring which is not a field of numbers has shown to be of any practical meaning. The following postulate is therefore most natural in this situation. THE COMLEX FIELD POSTULATE.

The division ring K determined by 9

is the field of

complex numbers. This is the only postulate of our system which is not expressible in the form of a physically basic statement and whose validity has to be confirmed by the fact that the theory developed from it gives predictions which are in agreement with reality. If K is the field of complex numbers (where I* =$, then a vector space H over K with a definite Hermitian form f is called an inner product space (a pre-Hilbert space). The hermitian form f (x, y) is then denoted by (x, y> and is called an inner productl. A Hilbert space is a complete inner product space. We now apply the following theorem. 1 Since f is definite, it may be assumed positive definite when K is the field of complex numbers.

HILBERT SPACE FORMALISM

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MECHANICS

217

[l], see [6], 34.9). Let H be an inner product space. of H-closed subspaces of H is orthomodular if and only if H is a Hilbert

THEOREM 3 (Amemiya-Araki

The lattice Z&H) space.

Since by Theorem 1 and Corollary 1 we know that .Yn(H) is orthomodular, we conclude that His a Hilbert space. Moreover, since in a Hilbert space Ha subspace is H-closed if and only if it is closed (see Halmos [3], p. 24), we have Z#)=L?(H), is the lattice of all closed subspaces of H. Hence, we obtain the following

where 8(H) corollary.

COROLLARY 2. If p(A, u, E) satisfies the Axioms 1-7 and we accept .the complex field postulate, the logic 8 of p is orthoisomorphic to the lattice Z(H) of all closed subspaces of a complex Hilbert space H. Corollary 2 is in fact the Hilbert space axiom of Mackey. We see that we obtain the Hilbert space as a consequence of the Axioms 1-7 and the complex field postulate. In the sequel we shall identify _Y with 9(H). We can now easily pass to the usual Hilbert space formalism of quantum mechanics. Using the spectral theorem and Theorem 1, we see that each observable A corresponds to an .F-valued measure pA which is now a spectral measure and can be identified with a self-adjoint operator A acting in H. Each state a corresponds by Theorem 1 to a probability measure m, on 8(H). A probability measure m, on L?(H) is said to be pure if it cannot be expressed as a non-trivial convex combination of other probability measures (i.e. if ma= c timi with c ti=l, ti>O, mi probability measures on Z(H), implies ti=l i=l

for some i). A state tc is pure if its probability measure m, is pure. A. M. Gleason showed in [2] that every pure probability measure on the lattice of closed subspaces of a Hilbert space arises from a unit vector in H, that is, if m, is a pure probability measure, then there exists a vector u, E H, Ilurrll = 1, such that m,(M) = (PMu,, u,) for every ME .5?(H), where P, is the projection on the subspace M. We now have by the spectral theorem and Theorem 1: p(A,

a, E)=mcroCLA(E)=ma(M)=(P~uUdl,

ac)=
a,>,

where M=pA(E) is the subspace of H determined by the Z(H)-valued measure pA and the Bore1 set E, PM= Pi is the projection on this subspace. The map EHP~ is the spectral measure corresponding to the self-adjoint operator A. We can now sum up the obtained results in the following theorem. THEOREM 4. Let p(A) CI,E) be a function satisfying the Axioms l-7 and the complex field postulate. There exists a complex Hilbert space H such that each observable A corresponds to a self-adjoint operator A in H with the spectral measure PA and each pure state M corresponds to a unit vector u, E H. For every A E 0, everypure state tl E Y and every E E W(R) we have . p(A, a,E)=. (*)

218

M. J. M.&ZYfiSKI

If we are interested theorem

only in the mean value of A in the state CI, we have by the spectral

5. Concluding remarks We have seen that our axiom system allows to replace the Hilbert space axiom by an apparently weaker and more natural complex field postulate. This postulate is the only axiom of our system which is not expressed in the “experimental language” of the Axioms 1-7. It remains an open question whether it is possible to deduce the complex field postulate from axioms expressed in terms of the probability function p. It seems, how,ever, that such axioms, even if possible, would be extremely complicated. Our axiom system does not allow to specify what kind of a complex Hilbert space should be associated with a quantum system. We know that most frequently an infinite dimensional, separable, complex Hilbeit space is used. It would be possible to add new axioms lo deduce the separability and infinite dimensionality of our Hilbert space, but since we are interested only in the most general axiom system leading to a Hilbert space, we shall not pursue this possibility. From the physical point of view, the only way of distinguishing between two observables and two states is by means of measurement. Consequently, we could add to our system the following axiom:

Ifp(A, LX,E)=plA’ =p(A,

a, E) for all CI and a11 E, LX’,E) .for all A and all E, then LX=a’.

then A= A’.

Similarly,

if p(A, a, E)

An assumption of this type is usually made in axiom systems for quantum mechanics (e.g. Axiom 2 of Mackey [4]). However, by accepting this axiom we do not change the logic 3. Even if this axiom does not hold, we could make it hold by identifying those observables and states which are indistinguishable by measurement (i.e., by introducing an equivalence relation “A-A’ iff p(A, CI,E)=p(A’, a, E) for all CI and all E” in the set of all observables 0, and similarly in the set of all states 9, and considering instead of 0 and 9 the sets 01~ and Y/of equivalence classes of these relations). Such an identification would influence neither the logic 8 nor the representation theorem. The only gain would be that the maps A I-+I(~~and CIH~, of 0 into the set of all _Y-valued measures and of 9’ into the set of all probability measures, respectively, would be one-one (but not necessarily onto!). The basic equality(*) in Theorem 4 would still hold, and since we are interested only in recovering the values of p(A, u, E) from pA and m,, this additional assumption is not necessary and we do not make it. Acknowledgement The author is indebted to Professor K. Maurin for encouragement discussion at his seminar during the preparation of this work.

and a valuable

HILBERT

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REFERENCES [l] Amemiya, I., and H. Araki, Publications Research Inst. Math. Sci., Kyoto Univ. Ser. A 2 (1966), 423. [2] Gleason, A. M., J. Math. Mechanics 6 (1957), 885. [3] Halmos, P. R., Introduction to Hilbert space and theory of spectral multiplicity, Van Nostrand, New York, 1957. [4] Mackey, G., The mathematical foundations of quantum mechanics, W. A. Benjamin Inc., New York, 1963. [5] MacLaren, M. D., Pacific J. Math. 14 (1964), 597. [6] Maeda, F., and S. Maeda, Theory of symmetric lattices, Springer Verlag, New York, 1970. [7] Mqczyriski, M. J., Bull. Acad. Polon. Sci., Ser. Sci. math. astr. et phys. 15 (1967), 583. [8] -, Reports Math. Phys. 2 (1971), 135. [9] Piron, C., Helv. Phys. Acta 37 (1964), 439.