Foundations of Physics, Vol. 18, No. 9, 1988
Local Observables, Nonlocality, and Asymptotically Separable Quantum Mechanics 1 K. Kong Wan 2 Received July 14, 1987 Quantum mechanics is troubled by the problem of nonlocality inherent in the theory. In a series of papers we explore the possibility of an algebraic formulation of quantum mechanics based on local observables which would incorporate nonlocality when small distances are involved but would be separable at large distances. This paper reviews some of the basic ideas and theories developed recently. These include a unified localization scheme, the introduction of local comoving evolution, local comoving observables, and related conservation laws. Technical considerations and mathematical jargon are kept to a minimum to avoid obscuring physical reasoning.
1. INTRODUCTION The nonlocality and measurement problems have attracted a great deal of attention in recent years alongside an increasing number of sophisticated experiments to probe the very foundation of quantum theory. There are many attempts to reinterpret quantum concepts and to model quantum measuring devices. We have studied the mathematical representations of physical observables from the point of view of geometric quantization to operator theory. (1 14) The question is whether operators such as -ihO/Ox used in the orthodox theory are the correct ones to describe observables. Some of the more extravagant claims of nonlocal correlations are based on the existing operator representation of observables. For example, the correlations between two wave functions ~ and ~ which are separated spatially by an arbitrarily large distance are due to the existence of 1 An invited paper in honour of David Bohm on the occasion of his 70th birthday. 2 Physics Department, St. Andrews University, St. Andrews, Fife KY16 9SS, Scotland, United Kingdom. 887 0015-9018/88/0900-0887506.00/0 © 1988 Plenum PublishingCorporation
888
Wan
operators A of the conventional kind with nonvanishing scalar product (~b[AO). Therefore a negative answer to the question posed above would have a bearing on the discussion on nonlocality and quantum measurement. This paper is a review of our work arising from a study of the above question in the context of nonrelativistic quantum mechanics. Emphasis will be on qualitative discussions of the basic thinking and ideas involved. When needed, relevant theorems and mathematical results are stated without proof. Detailed studies complete with relevant mathematics and proofs of theorems stated here can be found in the references cited. For simplicity, we shall confine our attention to a single quantum particle of mass m in one-dimension. An extension to three-dimensions or to a manyparticle system is either obvious or given elsewhere already. The states of such a system can generally be decomposed into bound states and scattering statesJ 15~16)Bound states are associated with the discrete eigenvalues of the Hamiltonian while scattering states correspond to the continuous part of the Hamiltonian. Scattering states describe the particle as it moves away to infinity. (15'16) We are primarily concerned with the asymptotic behaviour of the system at large times and at large distances away, u.e., our interests lie mainly in the scattering states. We know from scattering theory that, for a large class of potentials, the scattering states and their time evolution can be approximated by the states and evolution of a free quantum system. For simplicity and without loss of generality we shall confine ourselves from now on to the case of a free quantum particle of mass m in one-dimension.
2. PHYSICAL C O N S I D E R A T I O N S We shall begin with the following starting assumptions: ~7)
(S1) A measuring device is necessarily of finite spatial dimensions and a measurement is performed within a finite time interval; ($2)
No physical observables can correlate states infinitely separated
in space. Since a measurement process is an operation to ascertain the values of certain observables, we regard ($2) above to be equivalent to the statement that no measurement process can bring two infinitely separated states to interfere with each other. Our objective is to study observables compatible with above assumptions and then to formulate a quantum mechanics based on these observables. We shall attach a precise maning to the intuitive notions introduced above as we go along.
Local Observables and Nonlocality
889
3. N O T A T I O N S (1) Let N denote the real numbers as well as the one-dimensional configuration space, and let C denote the complex numbers. (2) Capital lambda A denotes a bounded interval, either open or closed, in the configuration space ~. For any real number t, tA denotes the interval tA = {tx: x ~ A } , e.g., tA = (ta, tb) if A = (a, b). (3) The Hilbert space L2(~) will be denote by ovf, and L2(A) by iF(A). Elements ~bin . ~ or in J r ( A ) always mean normalized elements and they are also referred to as wave functions or states. (4)
The
symbol
t5
denotes
the
usual
momentum
operator
15=-ihd/dx in J f while p itself is reserved for numerical momentum values. (5) The free Hamilton is H ° = ~ 2 / 2 m and the free time evolution operator is Ut° = e x p ( - ] H ° t ) , where } = i/h, i.e., given an initial state q~ the evoled state at time t is ~b,= U°~b. (6) XA denote the characteristic function of A on N, i.e., )~A(X) equals 1 for x in A and equals 0 for x outside A. (7) We shall have many occasions to take limits as time t tends to infinity. (47'17) For brevity we shall write lim for limt~ ~.
4. L O C A L OBSERVABLES 4.1. Physical Motivation
Following assumption ($1), we wilt take the view that a measuring device can only scan a finite region A in space. More specifically, we call a measuring device to be of size A if: (i) the device cannot detect a particle lying outside A, and (ii) it cannot correlate states outside A and states inside A. Such a measuring device can only measure observables AA which reflect properties (i) and (ii) above, namely if ~b(x) and t//(x) are wave functions with supports inside and outside A respectively we have (i) ( ¢ [ A ~ / ) = 0 and (ii) (~b I A ¢ ) = 0 . These conditions imply that A A must satisfy(I, 3,8)
AA=zAAAx •
(1)
We call this the localization condition, and an observable satisfying this condition is called a local observable. A local observable can be visualized as an observable which is zero outside a bounded region. An observable
890
Wan
refers to a selfadjoint operator in ~ here. We call an operator (selfadjoint or otherwise) a local operator if it satifies (1). Otherwise an observable or an operator is caled global. We can now state our starting assumption ($1) precisely as saying that a measuring device measures local observables only. In particular, a measuring device of size A measures local observables AA localized in A. Usual observables like the position x, the momentum /~= -ihd/dx, and the potential V(x) are global observables. Take the potential V(x). Some people may say that this can be measured since we can in principle construct an infinite array of detectors covering the whole space. We would argue the exact opposite, that it is in principle impossible to have an infinite array of detectors covering the entire universe. We can only have a finite set of detectors capable of ascertaining the values of V(x) in a bounded region A. In oder words, we can measure a local potential
V'A(X) = V(x), =0,
x ~A xCA
Clearly
V~(X)=zAVZA
(2)
The moral is that we need local versions of our usual global observables. The construction of local versions of a given global observable, a process known as localization, is quite simple for bounded observables. Theorem 1. (1'x8) Let A be a bounded selfadjoint operator on ~t~, and let ~b be a wave function of support inside a bounded region A. Define an operator by A~, = ZAAXA. Then (1)
A ~ satisfies localization condition (1).
(2)
A~ is selfadjoint.
(3)
(AO)(x)= (A'Aqk)(X), x ~ A .
(4)
<~blA~b>= <¢blAA~b>.
Naturally, we would identify A~ with the localization of A in A. Unfortunately, such a simple prescription fails for unbounded observables in general because A] so constructed may not be selfadjoint if A is unbounded. The momentum operator p is a case in poivt. For p another simple prescription works. (2) Let ¢(x) be a smooth bump function ¢19) (Fig. 1). Then the operator @ defined by
@ = ½(@ + fie) = -ih(¢d/dx + ½d~/dx)
(3)
Local Observables and Nonloeality
891
/ a o
Ao A
Fig. 1. The bump function ~(x). ~(x)=l, x~Ao; ~(x)e(0,1), x~A-Ao; ~(x)=0, x¢A where A=(a, b), Ao= Eao,bo]. is selfadjoint, satisfies the localization condition, and behaves like/~ in A. Extension to three-dimensions is straight forward. A detailed study of local momenta can be found in Ref. 2. Protest voices may be raised against the emergence of apparently additional and artificial operators like ~/~ not present in orthodox quantum mechanics. However, the opposite is the case. Orthodox theory assumes that every selfadjoint operator in J f corresponds to some physical quantities. (2°) Hence, the operator ~/~, which is selfadjoint in ~f~, does describe a physical observable already in the orthodox theory. Given that all local observables are already include in orthodox theory, the question is what physical observables operators like V'a(x) and ~/~ describe. Let us suppose that we have a device, which uses a magnetic field, covering a finite region, that deflects particles, to try to measure the momentum of a charged particle in three-dimensional motion. (21) The question is whether such a device measures the global momentum observable represented by the operator -ihV. The analysis based on (S1) tells us that this device can only measure a local observable. We would therefore argue that the device measures a local momentum. The argument is that local observables like ~/~ are the directly measurable quantities and are therefore the "real" things, while global ones like 10 are not. In many practical cases, global operators may well be easier to handle, and they can be used to approximate their local counterparts in calculations. This is similar to the use of nonsquareintegrable functions like plane waves in practical calculations. The approximation is possible because of the mathematically well-defined
892
Wan
relationship between A~ and A, i.e., as A tends to ~ local operators Ah tend to A. (1) Intuitively, we can this link between ¢/5 and/5. In advocating local observables, we have introduced a conceptual departure from orthodox theory. A measurement of a local observable invariably involves a position measurement to the extent of ascertaining the presence of of the particle in A. Intuitively, one can visualize this as catching the particle in A and doing a measurement on it. The measurement of momentum by deflection in a magnetic field serves to illustrate this. Wave functions which are localized in A can correspond to a finite range (2, 22) of values of a local momentum ~/5; this is false for the global momentum/5.(2) In passing we should mention that the usual uncertainty relation (Ax)(A/5) >1lh arising from the commutator Ix,/5] = ih does not apply to x and 4/5. Instead (22) we have (Ax)(A~/5)>>. ½hl(~bl ~b)l. If A and Ao are sufficiently large so that (Climb)~ 1 for a class of states, we arrive at the expression (Ax)(A~/5)>~½h, and we can say that ~/5 behaves just like/5 in these states. However, if A and A0 are so small that the above expression is significantly violated, one may then say that the local observable ~/5 no longer resembles the global momentum/5, at least not for the class of states in question. What this means is that if the measuring device is too small it would measure a local observable which cannot be approximated by a global observable. If we want to consider local observables which can be approximated by global observables we must be careful about our choice of A. In the case of momentum we should impose a relation between A, Ao, and the range of momentum values. A reasonable demand is (bo - ao)(22 - -
2 1 ) >~ h
(4)
The idea of localization can be generalized to localization in the spectrum of any chosen observable using the spctral measure of that observable/3) The result is that any pair of global obervables can have local versions, albeit in the generalized sense, whose values can be simultaneously confined in exactly the same way as x and ~/5. We shall not pursue this further in this paper so as to avoid excessive abstract mathematics.~3) 4.2. Localization Procedure Our task is to devise a localization scheme applicable to all given (global) observables, both bounded and unbounded. The idea is to generalize the prescription for momentum /5. This boils down to the following.
Local Observables and Nonlocality
893
Mathematical Problem. Let A be a bounded open interval in N and Ao be a closed interval inside A (Ao could be arbitrarily close to A). Given a possibly unbounded (global) selfadjoint operator A in J(~ with domain D(A) construct a local selfadjoint operator A A localized in A with domain D(AA) such that for every wave function ~b(x) in D(A) there corresponds a wave function (~A(X)in D(AA) satisfying:
qJACx)=(J(X),x~A o (ii) (AAOA)(X)= (A(~)(X), xSAo (iii) (q$~lAA@a)=(fflA~b) (i)
If the given global operator is/O then the local momentum operator ~/~ happens to be a solution to the above mathematical problem. We call AA which solves the above problem a localization of A in A. This is further supported by the observation that if ~ is localized in Ao, i.e., (~ vanishes outside Ao, then ((~[AAqJ)= (@IA~). To solve the above problem, we should realize that a global operator A acts on the space LZ(N) while a local operator acts effectively on the space L2(A) of wave functions defined nontrivially only on A If we can establish a sensible map between L2(N) and L2(A), we would be able to relate operators in L2(R) to operators in LZ(A). It turns out that we can introduce a unitary transformation of L2(~) onto LZA) which relates operators in L2R) to operators in LZ(A) in a way which solves the mathematical problem posed above. The main results are summarized briefly without proofs in a few theorems below. Theorem 2. (8'9,23) Let A=(a,b) be a bounded open interval, Ao = [ao, bo] be a closed interval inside A, and A - Ao = (a, ao) 'J (bo, b). Let ~(x) be a smooth function on N which vanishes outside A, equals 1 in Ao and takes values ~ ( x ) e (0, 1) in A - Ao (Fig. 1). Define a function a from A to 11~by
iy dy'
x = if(y) = Jyo ~(Y') + Yo
(5)
where Yo is a chosen value in A0, y 6 A and x ~ N. Then: (1) The function a ( y ) i n c r e a s e s monotonically from - ~ to oo as y varies from a to b, and x---o-(y) = y for y e A0. Furthermore, o-(y) admits an inverse a l(x). (2) For every ~ ( x ) in egg the function g$(y) = ~-112(y) ~(y)) defined on A is square-integrable in A and hence belongs to ~ ( A ) .
894
Wan
Remark. The effect of a is to leave the points in Ao unchanged while squeezing all the points in R outside Ao, i.e., in R - A o , into the small region A - A0. T h e o r e m 3. (8'9) The map L,¢ from 5¢f of ~ ( A ) defined by
qb(x) --. ~b(y)= (£,¢~)(y) = ~ 1/2(y)q~(a(y))
(6)
is a unitary transformation from Yf onto ,~(A) with the inverse transform L~-1 given by ~b(y) ~ q~(x) = (L~<~b)(x) = 41/2(o- l(x)) (~(a-l(x))
(7)
Remark. The map/~¢ leaves the portion of the wave function in the interval A0 unchanged while squeezing the remainder of the wave function in R - Ao into the small region A - A o. Theorem 4. (8'9) The map L e induces an isometric map (24) L e of 3eg into ~ by ~ ( ( x ) ~ O(x) = (L¢ 5U)(x) = ¢ -~/2(x) ~(a(x)),
xeA
x~A
=0,
(8)
with range #F(A). The adjoint L~ is given by
(L~O )(x) = ~I/2(cr-l(x ) ) •(a-l(x) ),
tpe~
and x e R
(9)
and we have the identity operator on Yf,
L~L¢=I
(10)
Remark. The map L¢ has the effect of localizing a widely spread-out wave function to the region A, while its adjoint L~ does the exact opposite. A wave function of support in A, when acted on by L~, becomes spread out all over E. In order words, L e localizes while L~ "globalizes" (Fig. 2, 3). T h e o r e m 5, (8'9) Given a map L e and any possibly unbounded operator A in ~¢g with domain D(A) define an operator AA in ~f by AA = L¢AL~ with domain D(A~)= LeD(A ) + Xao3/f, where AC= I ~ - A. Then:
(1) A3 is selfadjoint if and only if A is selfadjoint. (2) If A is selfadjoint the spctrum of AA is equal to that of A plus the additional value 0 (if not already in the spectrum of A).
Local Observables and Nonlocality
895
I ,~o
°
i') a
b
ao
Ao A
~"
Fig. 2. The operator L¢ localizes. ~(x) ( - - - - - - ) and (Le~)(x) (. . . . . . . . . ) differ outside A0, while in Ao we have ~ ( x ) = (Le Yt)(x)
(. . . . . . .
).
Remark. A~ is zero outside A, hence the term ZAc~cg in the additional eigenvalue 0 for A A.
D(AA) and
Theorem 6. (8,9) Let A be a selfadjoint and possibly unbounded operator in ~ with domain D(A). Let A~=L~AL~, ChaD(A) and q~= L¢ ~. Then:
f..--,..\ J
"'"'"i
°°
/
.
I. b
"" ao
o
b
Ao A
"
Fig. 3. The operator L~ globalizes. ~k(x) ( - - - - - - ) and (L~t~)(x) ( . . . . . . . . . ) differ outside Ao while in Ao we have (L*~/)(x)=¢(x) (. . . . . . . ). The values of ~(x) outside A do not affect L~ ~ at all.
896
Wan
(t)
~b(x)=Ci(x) and (AAq~)(x)=(Aq~)(x), x ~ A o.
(2)
(AIAAfb)=(cbIA~).
(3)
Ifq~(x)has its
support
in
A0then
~b(x)=q~(x),
x~
and
( ~ ] A A ~ ) = ( ~IAcIg ). There is no need to pursue the technicalities involved to appreciate what is going on. All we need is to recall the mathematical problem raised at the start of this section and then to realize that the problem is solved by an isometric map L¢. To visualize the map we observe that L~ does two things: (1) within A0, it acts like an identity operator leaving things unchanged, and (2) it transfers the effects A and q~ have outside A to the boundary region A - A o in tandem with the map a which maps ~ - A to A - Ao (Fig. 2).
Definitions: (D1) We call (i) ¢ a localizing function; (ii) L¢ the localizing isometry defined by 4; (iii) A0, A o, A respectively as the center and regio.~ of localization of ~ and of L¢; (iv) A - A o the boundary of ¢ and of L~.
(D2) We call AA = L¢AL~ the localization of A in A defined by 4We have now constructed a general localization scheme. The scheme may appear rather abstract and unrelated to previous schemes. In fact, the present scheme is really a generalization of the previous momentum localization scheme as we shall see presently when it is applied to /~. For brevity we shall write L and L t for L¢ and L~ from now on. 4.3. Three Examples (1) Local Momentum. It is readily verified that /~A = L1~Lt is equal to 4/3. A diagrammatic illustration of the effects of/~A and fi is available in Ref. (2).
(2) Local Potential.
VA(X) = LV(x)L* = ZA(X) V(a(x))
(11)
where )~A(X) is the characteristic function and a is the function defined in Theorem 2. Clearly, VA(X) = V(X) in Ao, VA(x) = 0 outside A. The values of V(x) outside A are embodied in VA(x) in the boundary. In other words, the values of V(x) in • - A are squeezed into the tiny A - Ao. (3) Local Hamiltonian. Given H = H ° + V, where H°=/~2/2m, the local Hamiltonian is H a = H A° + VA, H ° = D ] / 2 m . For the case of a
Local Observables and Nonlocality
897
harmonic oscillator, (21) the local Hamiltonian would have the usual eigenvalues (n _1)he) plus the value 0. The additional value 0 has no special significance as it is obtained by operating on a wave function lying outside A. 4.4. Nonuniqueness
Our localization scheme using the isometry L applies to both bounded and unbounded operators, and it is unique once the localizing function ~ is chosen. There are infinitely many different localizing functions with the same A and A o. Also, for bounded operators A a differ from A~ defined in Theorem 1. However, the nonuniqueness manifests itself only in the boundary A - A o which could be made arbitrarily small. In the center of localization A0, our localization A A is uniquely determined by A. The region A - A 0 , one can argue, corresponds to the physical boundary of a measuring device and the nonuniqueness of localization in / / - A 0 may correspond to all kinds of boundary effects in the physical boundary of a measuring device. These boundary effects will not be significant in most cases.
5. ASYMPTOTIC L O C A L I Z A T I O N AND SEPARATION OF STATES Having discussed the meaning and implications of our starting assumption (S1) we now turn to ($2). The notion of infinitely separated states and their existence must be established first. This section is devoted to this purpose.
Definitions." (4,5) (D1) A state described by ~b in . ~ is asymptotically localizable in tA if a proper interval A = Iv, w] in ~ exists such that lim!,A 1¢'12dx=1
(12)
(D2) Two states ~b and ~b' in ~ are asymptotically separable if they are asymptotically localizable in disjoint regions [tv, tw] and [tv', tw']. The corresponding evolved states ~bt and ~b't are said to be spatially separating. Clearly asymptotic localization in tA means that the probability of finding the particle in tA (outside tA) tends to 1 (to 0) as t tends to infinity. The definition of asymptotic separation is then self-explanatory.
825/18/9-3
898
Wan
A classical particle moving freely goes to spatial infinity. A free quantum particle will also move to spatial infinity as time tends to infinity in the sense that the probability of finding the particle in any bounded region in space tends to zero as time tends to infinity. (~5~ An ensemble of noninteracting classical particles of mass m with momentum values in the range [my, mw] will freely evolve into the region [tv, tw] for sufficiently large time t, when the initial positions of the particles become insignificant. A similar situation prevails for quantum particles as seen in the following
Theorem 7. (4,5,25_27) Let OF(p) be the Fourier transform (28) of 0(x). Then
lim f, A lOtl2 dX= fmA lOFl2dp
(13)
Observe that the right-hand integral is the probability of the particle in state 0 having a momentum value in the range mA. The theorem implies that (t) a state 0 is asymptotically localizable in tA = [tv, tw] if and only if 0 corresponds with certainty to the momentum values in mA = [my, mw], and (ii) two states 0 and 0' are asymptotically separable if and only if they correspond to disjoint ranges of momentum values. Any wave function 0(x) in ~ whose Fourier transform OF(p) is of compact support [ P l , P2], i.e., OF(p)=0 for all pC [pl, p~], will localize asymptotically into [tv, tw] where v = p~/m, w = pJm. An example is
OF(p) = (p2--pl) -1/2, = 0,
pE [Pl, P2] P ¢ [Pl, P2]'
~ elp2x -- elp,x ] 0(x) = (2nh(p2 - p l ) ) - 1 / 2 [_
~x
J
where J = i/h. Two wave functions whose Fourier transforms have disjoint supports will separate asymptotically. Since Ot does not converge in a meaningful way to any nontrivial element in ~ as t tends to infinity, we cannot introduce "states at infinity" that are "infinitely" separated within the rigid orthodox Hilbert space formulation of quantum mechanics. We do have states which are separated in time by an arbitrarily large distance, as measured by Itv'-tw[. Later we shall introduce an algebraic formulation in which we can introduce "states at infinity" in a meaningful way. An alternative time evolution will also be considered.
Local Observables and Nonlocality
899
6. Q U A S I - L O C A L O P E R A T O R A L G E B R A
Knowing that states do go to infinity and can also separate spatially, we want to consider again what kind of observables would satisfy both our starting assumptions ($I) and ($2). We have given a meaning to ($1) earlier with the consequence that ($1) admits local observables only. We can now interpret assumption (5;2) as requiring a physical observable A to satisfy tim(~b,lA~b',)=0 whenever ~b, and ~b't are spatially separating. Clearly a local observable satisfies this requirement. We are naturally led to the tentative conclusion that ($1) and ($2) require that we restrict to local observables only. Let dL denote the set of all bounded local operators on egg. We shall confine ourselves to bounded operators because: (1) they are mathematically easier to handle, (2) there are well-known quantum theories based on bounded operators, e.g., the quantum logic approach (29-31~ and the C*-algebra approach, (32-35) and (3) an unbounded observable can be approximated with arbitrary accuracy by bounded observables via its spectral representation. (35-37) The set d r can be extended without violating the spirit of (S1) and ($2) in different ways. (6'7'38'39) The smallest extension is to take the completion s ~ of dL in the operator norm in the following way. Let An, n = 1, 2,..., be a sequence of local observables in dL converging uniformly to an observable which is not in de. It follows from uniform convergence that for any given small positive number s there exists an integer N such that for every A, with n >i N, we have I(~tA~b) - (q~l A,q~)l
(14)
for all states ~b in Jzt~. Physically this means that we can obtain the value (~blA~), with any predetermined accuracy and without prior knowledge of the state from the measured value of (~blAn~b) ifn is large enough. We are justified to claim that A is also measurable in a way consistent with the spirit of (SI) and ($2). The completion s ~ of dL consists of dL and the uniform limits of all uniformly convergent sequences of local operators in alL. We shall call s ~ the quasi-local operator algebra and a selfadjoint member of s ~ a quasi-local observable following the standard terminology in algebraic quantum theories. (32 35) Apart from the physical justification for including quasi-local observables, the extended set ~ has the added advantage of having the mathematical structure of a C*-algebra, namely: (1) all operators in s ~ are bounded, (2) for any A, B in s ~ and e in C, the operators A*, cA, A + B and AB are again in 4 , and (3) s ~ contains the uniform limits of all uniformly convergent sequences in alL. Familiar observables like fi and H ° are excluded not so much because they are unboun-
900
Wan
ded (bounded functions of /~ are also excluded) global. The identity operator I is not in ~ either. It of normalization to include the identity operator technical point can be achieved by annexing the set to form a new C*-algebra (3) s~, i.e., s~ = sgL + CI.
but because they are is desirable for reasons in a C*-algebra. This C I = {c~I: ~ ~ C} to s ~
7. AN ASYMPTOTICALLY SEPARABLE Q U A N T U M MECHANICS 7.1. Preliminaries on C*-Algebra Formulation
Closely associated with a given C*-algebra are the positive linear functionals. To be specific, let us focus our attention to the C*-algebra on J~f.
Definitions:
(34, 40)
(D1) A linear functional on ~ is a map co of ~ into the complex numbers, i.e., co: ~ ~ C with the property that for all ~,/~ in C and for all A, B in ~ / w e have co(~A + fiB) = ~co(A) +/~co(B)
(D2) A linear functional is positive if co(A'A) is real and nonnegative for all A in ~ , and in normalized if co(I) = 1. (D3) A normalized positive linear functional (NPLF) co, is normal if there exists a density operator p on ~ such that co(A)= Trace(pA) for all A in ~/. Remark. Definition (D3) is possible because every density operator p on ~ generates an N P L F on ~ by co(A)= Tr(pA). We shall denote this N P L F by cop. In particular, when p is a one-dimensional projector p = I~b)(~b[ we see that cop(A)= (~blA~b). We shall use the symbol co~ to highlight this. The basic idea of the C*-algebra approach is to associate with a physical system a C*-algebra with the interpretation that the selfadjoint elements of the algebra correspond to the observables of the system. States are then introduced as normalized positive linear functionals on the algebra, with the interpretation that the value of such a functional acting on an observable gives the expectation value of that observable in the state described by that functional. For a one-particle quantum mechanical system, the standard procedure is (1) to associate the system with the C*algebra B(~tf) which consists of all bounded operators on ~ and (2) to
Local Observables and Nonlocality
901
assume that states correspond to normal NPFLs on B(o'eg). It is not hard to see that these are equivalent to the corresponding postulates in the orthodox theory. There is no problem incorporating time evolution in quantum there in this algebraic set-up. The resulting theory is equivalent to orthodox quantum mechanics formulated in Hilbert space. (35) This algebraic approach has found many physical applications. (32-35'4°43) The choice of the C*-algebra is crucial. We shall propose an algebra different from B ( ~ ) to produce an asymptotically separable theory.
7.2. The Theory An obvious choice of an alternative algebra in keeping with our starting assumptions ($1) and ($2) is ~ . Some people may feel uneasy in the disappearance of any trace of the familiar momentum 6. It is possible to incorporate 6 in an enlarged algebra if we so wish. It turns out that such an enlargement would help us to introduce "states at infinity" which are not available in orthodox theory. We shall therefore incorporate -6 in the form of bounded functions of/~. Let L ~ ( ~ ) denote the set of all essentially bounded Lebesgue measurable complex-valued functions f(x) on ~, and let L~(-6) denote the set of operators of the from f(/~), fEL~(~). Functions of a selfadjoint operator can be defined through its spectral representation. (37) Here it is quite sufficient, if one wishes, to regard L~(-6) as the set of bounded functions of p on an intuitive basis. If f(x) is a real function of x, the operator f(-6) is selfadjoint. The question now is how we can interpret such a selfadjoint operator f(-6) as an observable without violating ($1) and ($2). The answer is to consider f(/~) as an observable at infinity, an interpretation based on the following. Theorem 8. (27'39) Let g(x) be any function of compact support in L ~ ( ~ ) . Then (15) s-lim U°*g(mx/t) U° = g(-6)
Corollary 1. (1)
lira II(g(mx/t)-g(P))(~t]l = 0
(16)
(2)
lim(<~b,I g(p)~b,>- <~btl g(mx/t)~>)=O
(17)
Observe that g(x) is a hump-like function in that it vanishes outside a bounded region in ~. Hence g(mx/t) represents a hump-like function spreading and moving to spatial infinity as time t increases. The above corollary suggests the interpretation of g(/~), when it is selfadjoint, as an "observable at infinity" in that the expectation value of g(-6) is obtained as
902
Wan
the limiting value arising from the local observable g ( m x / t ) and state ~, as they both spread and move to spatial infinity. The observable g(/~) is relevant, if at all, only at large time when the particle is far away, hence the name "observable at infinity." Note that every f(/~) in L~(/3) also satisfies our assumption ($2). Now we can annex L~(/~) to ~ to form a final C*-algebra .52~.(6'7'39) One way of doing this C39)is to take the direct sum ~ ' = , ~ ® L°°(/3)
(18)
An N P L F on sJ consists of a sum of two terms, 3 i.e., (2 = co~ ~ c o [ where co~ is an N P L F on s~ and a)~ is an N P L F on L~(/J), and we have O(A • f ( P ) ) = co~(A) + c0~(f(/5))
(19)
How do we interpret co~ and c)~? For (o~ let us restrict ourselves to normal N P L F s defined earlier. In orthodox theory a density operator p describes a state. At time t the state will be described by another density operator which we shall denote by p t . (22'30) A state in algebraic theory corresponds to a normal N P L F cop and we expect the evolved state to correspond to cop. The question now is what happens to coo, as t tends to infinity. For a wave function ~b, the limit lira ~b~ does not make much sense. The same is true for p,. Despite this we can still define the limit lira cop, in a physically meaningful way based on the following. T h e o r e m 9. (39) Given a normal N P L F coo on ~ N P L F co~ on L°°(p) defined by P
co;(g(/~)) = lim cop,(g(mx/t))
there is a unique (20)
for all g ( x ) introduce in Theorem 8. Remark. Expression (20) is a generalization of (17) in Corollary t: (20) coincides with (17) if p t = I~b,)(~btl. We can let co; act on ~ by co;(A) = lim cop,(A). Then we will have c o ; ( A ) = 0 . A natural interpretation of co; emerges, i.e., it is a state at infinity, o9; acts nontrivially only on operators at infinity. 3 Proof:~44,45~ Let cg1 and ~2 be two C*-algebras of operators respectively. Then their direct sum cg1~ cg2 is the C*-algebra form Ca (~ Q , where C 1 ~ ('~1 and Ca s ~q2, on the direct sum defined by HCI®C211=sup{ItClll, tlC~ll}. Let 091 and oJ2 be tively. We call the map f2:~1 q~ cg2 ~ C defined by
0(C1 q~ C~) = oh(C1) + o~2(C2) the direct sum of ~o1 and a) 2 with the notation I2 = 091 ~9 ~o2.
on Hilbert spaces ~ and of direct sum operators of the space 3'g ~ out, with the norm N P L F ' s on ~1 and cg2 respec-
Local Observables and Noniocality
903
Definition. An N P L F on L~(/~) generated by a density operator according to Theorem 9 is called a normal N P L F at infinity. We can now write down the formal statements of our algebraic formulation in the following two basic postulates. Postulate 1. A free quantum particle in ~ has associated with it the C*-algebra s~l=s~OL°~(/}) of operators on ~ O ~ . Bounded observables correspond to selfadjoint members of sJ. Postulate 2. A state 12 of the system is represented by a convex combination of a normal N P L F £2o and a normal N P L F at infinity [2p~, i.e., £2 = c[2p+c[2 ~ (c, c' >~O, c + c ' = 1). Conversely, every such combination describes a state. If C in a¢ is selfadjoint then 12(C) is the expectation value of the observable represented by C for the system in state/2.
Remark. [2p = cop ® 0, or cop, may now be called a normal state while fay = 0 • cop,, or cop,, a normal state at infinity or simply a state at infinity. Time evolution can also be set up formally in an algebraic theory. (7'34} Our present qualitative exposition does not require such formality. It is quite sufficient to consider that in the Schr6dinger picture the time evolution of a state £2 is given by [2, = c[2 p' + c'12;. Note that only normal states evolve. A normal state 12p, evolves into a state at infinity [2p which then ceases to evolve. oo
co
¢o
7.3. Asymptotic Separation A fundamental problem facing quantum theory is whether a pure state can evolve into a mixed state. The answer is emphatically no in the orthodox theory which is then said to be nonseparable. Nonseparability is one of the main sources of conceptual difficulties manifested in, for example, the de Broglie paradox346) In the present algebraic set-up the notions of pure and mixed states can be quantified in the following definitions.
Definitions: (D1) A state 12 is pure if it cannot be decomposed into a nontriviat convex combination of two different states, (33) i.e., if 12=c1121+c2122, where cl, c 2 > 0 and cl + c 2 = 1 ~ 1 2 = [21 =122 Otherwise the state is called mixed and is said to be a mixture of states 12~ and [22.
Wan
904
(D2) Two normal states G2p and f2p, are said to be asymptotically disjoint (v) if lim(~bt [A~b~) = 0 for all A in 4 , whenever ~b is an eigenvector of p and ¢' is an eigenvector of p'. The corresponding states at infinity, f2~ and f2~, are said to be disjoint. The concept of disjointness (41) is related to the absence of correlations. This is seen clearly in (D2) above when p = [~b)(~b[ and p ' = I~b')(~b'l. Then f2~ and £2~, are asymptotically disjoint if lim(~btlA~b;)=0, A s s ~ , i.e., if the correlations (~b,lA~b~) become vanishingly small at large times. If asymptotically disjoint states exist, and moreover, if asymptotic disjointness turns out to be caused by spatial separation in the configuration space we would have achieved our objective, i.e., a quantum mechanics based on a C*-algebra consisting of observables which are measurable by finite measuring devices and which do not correlate infinitely separated states. The theorems and corollaries quoted below confirm our expectation. Theorem 10. (7) Every wave function ~b in ~ normal state f2~ on d . The converse is true.
corresponds to a pure
Theorem 11. (7) Two pure normal states g2o and asymptotically disjoint if ~b and ~b' are asymptotically separable.
f2~, are
Corollary 2. (7) Two normal states £2p and (2p, are asymptotically disjoint if every eigenvector of p is asymptotically separable from every eigenvector of p'. Theorem 12. (7) If ¢, 0 are asymptotically separable and c/, = cqb + fl0 (e, f l e C ) then £2eco = lc~[2(22 + [fl]2 £2~co and f2eco is a mixed state at infinity. Corollary
3 . (7)
(1)
Every normal state at infinity is mixed.
(2)
Every pure normal state evolves into a mixed state at infinity.
We have seen that, by restricting the C*-algebra to ~' and by introducing observables and states at infinity, the theory is asymptotically separable in the sense that a pure state evolves in time into a mixed state and a coherent superposition of two pure states can evolve into a mixture. The present theory also provides a characterization of a system at infinity. The system behaves like a classical system at infinity in that all observables at infinity commute. The theory presented can be readily extended to simple scattering systems and to many-particle systems. The mathematical structure can also be extended or modified. Spin can be incorporated by taking the tensor product algebra of ~ and the algebra of spin operators. (39)
Local Observables and Nonlocality
905
7.4. Some General Remarks
Many of the ideas and mathematical objects presented are well-known as the references cited show. Quasi-local observable, their relationship with finite measuring devices and the C*-algebra approach were put forward by Haag and Kastler in the context of quantum field theory. (3z) Local observables approach was used to construct models of measuring devices. (4~43~ There are also different constructions of local obsevables. (47) Terms like observables at infinity, classical observables and macroscopic observables have been used by various authors in different contexts, (4°'41'48'49) The idea that nonoverlapping wave functions should have vanishing correlations has also been discussed by a number of authors. (5°-53) We have utilized many of these ideas to establish la theory specifically for a quantum mechanics of a finite number of particles. 8. LOCAL C O M O V I N G E V O L U T I O N AND LOCAL C O M O V I N G OBSERVABLES Up to now we have employed the orthodox time evolution operator U~ which is a function of/~ and is a global operator. The prominent role played by U ° runs entirely contrary to our whole thinking and to the explicit relegation of functions of/~ to the minor status of observables at infinity. This apparent inconsistency can be tackled by introducing what we call a local comoving evolution. 8.1. Local Comoving Evolution Operators
Suppose a wave function ~b(x) is localized in a region A(0)= (a(0), b(0)) at time t= O, and that at time t > 0 the wave function evolves to the region A(t)= (a(t), b(t)). It is well-known that at time t > 0 the support A(t) of the evolved wave function ~bt will necessarily be the entire space under the usual evolution operator U°. ~s4'55) This is known as the instant infinite spreading of the wave packet, and it is partly a result of the global nature of/~ and U °. To appreciate how such an infinite spreading arises we have only to realize that the Fourier transform ~F(p) of a compact-supported wave function b(x) does not vanish over any finite range of values of p. It follows that in the orthodox theory which associates the global operator /~ with the momentum the particle could have arbitrarily large momentum values. Intuitively, this suggests that the particle would have arbitrarily large velocity values v =p/rn which would result in an instant infinite spreading of the wave packet. In our local theory the global operator/~ has no role except as an observable at infinity. It follows that we will not be
906
Wan
able to obtain a complete knowledge about the particle's velocity from ~bF(p), since OF(p) now refers to the probability amplitude of an observable relevant only to states at infinity. Having cast aside /~ and the Fourier transform, we must turn to local momentum observables. What we want is a velocity probability distribution relevant to the given wave function in A(0) nearby, i.e., not at infinity. In our local theory the relevant observables are then the local momentum observables in A(0). The recognition of this immediately reminds us that a compact-supported wave function can correspond to a finite range of local momentum values. (2) If we then relate the intuitive notion of velocity to local momentum values, we would arrive at a finite range of velocities for the wave packet which in turn must entail a finite spreading of the wave packet. Having spelled out our basic thinking we shall now. proceed to formulate a theory of time evolution within our local theory. Suppose an initial state ~k is prepared which is localized in the region A(0)= (a(0), b(0)) and which corresponds to a certain range (21,22) of values of a local momentum ~0/~, whose localizing function ~0 has A(0) and Ao(0)= (ao(0), b0(0)) as its region and center of localization respectively, i.e., corresponds to zero probability of having a value of ~0/~ outside the range (21, 22) and nonzero probability of having a value of ~0/~ within any small interval inside (21,22). (2,56) At time t > 0 the state 0 will propagate to a region A(t)= (a(t), b(t)). Our task is to decide what a(t), b(t) should be, and to set up an evolution of ~ which would lead to such a propagation. Interpreting the velocity range corresponding to the local momentum range (21,22) to be (vt, v2), where vl=21/m, v2=22/rn, we take the obvious choice a ( t ) = a ( O ) + v l t and b ( t ) = b(0)+ v2t. Let Ao(t ) be the closed interval [ao(t ), b0(t)], where ao(t)=ao(O)+vlt, bo(t)=bo(O)+vzt. Starting from 4o we shall introduce a one-parameter family of localizing functions ¢~ with region A(t) and center Ao(t) of localization propagating in time in accordance with the velocity range (vl, v2) by ¢,(x) = C o ( X - v~ t),
x E (O(t),
ao(t))
= 1,
x~Ao(t)
= ~o(x - v2t),
x ~ (bo(t), b(t))
=0,
xCA(t)
Let L, be the one-parameter family of localizing isometries defined by the above family of localizing functions ~t. Define a two-parameter family of operators U(t, s) on ~ as follows.
Definition.
t t,s~ U ( t , s ) = L t U , _os L s,
and t,s>~O.
907
Local Observables and Nonlocality
U(t, s) is a partial isometry <24) with initial domain Jt°(A(s)) and final domain ~£(A(t)). At first sight U(t,s) seems to be a formidable and abstract mathematical object. A closerlook reveals that U(t, s) performs a triple of Theorem 14. (8'9)
clear actions. Recall that we are looking for an evolution operator which will map the initial state in A(0) onto the evolved state in A(t). Now set s = 0 and let U(t,O) acts on ~. Then, (1) L*o globalizes ~,, (2) the global evolution U ° then acts on the globalized function ~ = Lo~, + and finally (3) the evolved globalized function 5u, = U ° ~u is localized in the stipulated region A(t) by L,. Clearly, U(t, 0) has the desired effect of evolving the initial wave function from A(0) to A(t). This alone will not qualify U(t, 0) as our desired evolution operator. Two more requirements should be demanded. Firstly, U(t, s) should possess mathematical properties similar to the standard properties of an orthodox evolution operator. (36'57) This is satisfied as seen in Theorem 15. Theorem 15. (8'9)
(1) (2) (3)
U(t, t ) = ~m,), the characteristic function of A(t). U(t, r) U(r, s)= U(t, s). U(t, s) is strongly continuous in t and s.
Secondly, U(t, 0) and U ° should approximate each other at least in the range of time duration of interest, i.e., at large times, so that U(t, 0) does not deviate from U ° drastically. This turns out to be the case provided the natural condition given by equation (4) is satisfied; then H(U(t, 0 ) - U°)lp II becomes negligibly small for sufficiently large times. (8'9) Large times here correspond to times when asymptotic localization effectively takes place; then both operators U ° and U(t, 0) lead to evolution into the region A(t). Numerical examples are available in Ref. (58). The distinctive features of U(t, 0) are (1) it is not a fixed global operator like U ° but is a statedependent operator, and (2) it acts effectively only in the moving support of the wave function, i.e., the evolution from time 0 to S and then to t is given by ~ ~ ~(s) = U(s, 0)~ ~ ~(t) = U(t, s) ~b(s). We therefore call U(t, s) a local comoving evolution operator. To achieve a consistent local theory we can now replace the conventional global evolution operator U ° by the local comoving evolution operator U(t, 0), i.e., instead of Or= U°~ we have the evolved state, denoted by ~,(t), given by
~,(0= u(t, 0)~
(21)
At large times, there is agreement between U ° and U(t, 0). At other times
908
Wan
U(t, 0) may deviate from U ° significantly. We may have to modify U(t, 0), e.g., by adopting different expressions for Ao(t ) and A(t). More work has still to be done.
8.2. Local Comoving Obervables Under a local comoving evolution, ~b(t) moves along as a compactsupported wave packet in A(t). A class of observables immediately presents itself as being special to the wave function ~b(t) at time t, namely those localized in A(t).
Definition. (8'9) Let A be a (global) selfadjoint operator in H and let ~b be a given wave function which evolves according to ~ ( t ) = U(t, O)tp. Then the localization A t in A(t) of A, i.e., At = LtAL~, is called a local observable comoving with the state ~b(t). An obvious example is the local momentum /~t= ~t/5 comoving with ~(t). At time t = 0 , the relevant local momentum is/~o = ~o/~ in A(0) where the initial wave function ~b is located, and at time t > 0 the relevant local momentum is/~t in A(t) in which the evolved wave packet ~b(t) is situated. To measure /50 we require a measuring device of size A(0) while /~, demands a different measuring device of size A(t). This is consistent with the present thinking that [3o and /~t are two different observables at different locations, and hence require two different measuring devices placed in two appropriate locations to perform the two measurements at respective times. The intuitive picture is that the wave packet (or the particle) carries its observables along with it. This makes sense also when we consider conservation laws later. Our second example is the local Hamiltonian comoving with ~(t), H°=L,H°L*,=~/2m. This is then the local energy observable carried along with the wave packet. We can write down the differential form of the comoving evolution equation ~b(t) = U(t, 0)~p in terms of H ° as i h - - ' ~ = [ H ° + i h \ Ot JL*t] ~b(t)
(22)
This is a drastically modified Schr6dinger-type equation.
8.3. Conservation Laws in Local Comoving Evolution Conventional conservation laws are to do with global observables. A bounded global observable A is conserved under the evolution U ° if (~ktlA~Pt) = (~jA~k) for all ~ in ~ . Obviously this does not apply a local comoving evolution. We reformulate the notion of conservation as follows.
Local Observables and Nonlocality
909
Definition. (8'9) A bounded local observable At comoving with a state qi(t) is conserved in state tp(t) if (O(t)IA,O(t)) is time-independent. This definition is in keeping with our intuition that ~b(t) carries its observables along with it, with (O(t)]A~b(t)) being measured in A(t) at time t. This new definition is closely related to the conventional one as seen in Theorem 16. Theorem 16. (8'9) Let A be a bounded (global) observable and let At be the corresponding local observable comoving with state ~b(t). If A is conserved under U ° in the conventional theory then At is conserved under local comoving evolution in state O(t). Consequently, for a free particle, the local comoving momentum /~t and the local comoving Hamiltonian H ° are conserved in state O(t) as expected. 9. CONCLUDING REMARKS We have reviewed an asymptotically separable theory based on local observables for a one-particle system. The theory is extendable to, say, twoparticle systems. For such systems, one's attention is attracted to the EPR problem on correlations between two particles when they are far apart. (59-61) It seems to us that the case for an asymptotically separable theory is even stronger for two-particle systems. This is due to a new feature which we call a process of chronological disordering. (sS) In a typical experiment to measure correlations a laboratory assistant at the origin sends out a large number of identical pairs of particles consecutively, at regular intervals, with one of the particles in each pair moving to the right and the other going to the left. An observer OR on the far right is ready to measure an observable of the particles reaching him and similarly an observer O L on the far left is prepared to act. It is not difficult to arrive at the conclusion that, when OR and OL are sufficiently far apart, the chronological order in which they detect the particles reaching them will be random so that it would no longer correspond to the order the particles were sent out by the laboratory assistant. The first particle detected by OR, and on which he obtained his first datum, may be from the nth pair of particles sent out by the assistant while the first particle observed by Oc may be from the lth pair. The breakdown of the original pairing of the particles means that O R and OL cannot correlate their respective experimental data. As a result, only one-particle observables are measurable and two-particle observables capable of correlating two infinitely separated particles must therefore be excluded from a two-particle theory. Hence, we should again have an asymptotically separable theory. Such a theory violates the conservation laws of some familiar global two-particle observables, e.g., the
910
Wan
square of the total (global) angular momentum(46,53); some people may regard this as an insurmountable dilIiculty, However, the process of chronological disordering tells us that as the observers move far apart, they become unable to measure those two-particle observables so that it becomes meaningless to talk about these two-particle observables, let alone their conservation. The basic idea of state-dependent evolution embodied in (20) was rendered most in Bohm's quantum potential interpretation of quantum mechanics, where he introduced an equation of motion involving the quantum potential which was itself determined by the wave function. (6~,63) We are very much influenced by the works of Professor David Bohm. It gives me great pleasure to have this opportunity to contribute to the celebration of his 70th birthday.
ACKNOWLEDGMENTS The author expresses his thanks to M. J. Hodkin and P. J. Sumner for a critical reading of the manuscript and for many suggestions which led to a much improved paper.
REFERENCES 1, K. K. Wan and T. D. Jackson, Phys. Lett. 106A, 219 (1984). 2. K. K. Wan, T. D. Jackson, and I. H. McKenna, Nuovo Cimento 81B, 165 (1984). 3. K. K. Wan and R. G. D. McLean, J. Math. Phys. 26, 2540 (1985). 4. K. K. Wan and R. G. D. McLean, Phys. Left. 94A, 198 (1983). 5. K. K. Wan and R. G. D. McLean, Phys. Lett. 95A, 76 (1983). 6. K. K. Wan and R. G. D. McLean, J. Phys. 17A, 825 (1984), Corrigenda 17A, 2367 (t984). 7. K. K. Wan and R. G. D. McLean, J. Phys. 17A, 837 (1984), Corrigenda 17A, 2367 (1984). 8. D. R, E. Timson and K. K. Wan, preprint, 1987. 9. D. R. E. Timson, PhD Thesis, St. Andrews University, 1986. 10. K. K. Wan and K. McFarlane, J. Phys. 13A, 2673 (1980). t l . K . K. Wan and K. McFarlane, J. Phys. 14A, 2595 (1981). 12. K. McFlarlane and K. K. Wan, J. Phys. 14A, L1 (1981). 13. K. McFarlane and K. K. Wan, Int. J. Theor. Phys. 22, 55 (1983). 14. K. K. Wan and I. H. McKenna, Algebras, Groups, and Geometries 1, 154 (1984). 15. W. O. Amrein, Nonrelativistic Quantum Dynamics (Reidel, Dordrecht, 1981). 16. W. O. Amrein, J. M. Jauch, and K. B. Sinha, Scattering Theory in Quantum Mechanics (Benjamin, Reading, Mass., 1971). 17. W. Thirring, Quantum Mechanics of Atoms and Molecules (Springer, New York, 1979). 18. P. Pfeifer, Helv. Phys. Acta. 53, 410 (1980). 19. R. Abraham and J. E. Marsden, Foundations of Mechanics (Benjamin/Cummings, Reading, Mass., 1978). 20. E. G. Beltrametti and G. Cassinelli, The Logic of Quantum Mechanics (Addison-Wesley, Reading, Mass., 1981), p. 3.
Local Observables and Noniocality 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 6t. 62. 63.
911
A. Messiah, Quantum Mechanics, Vol. I (North-Holland, Amsterdam, 1967), p. 145. P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), p. 2& K. K. Wan and C. Viasminsky, Prog. Theor. Phys. 58, 1030 (1977). J. Weidmann, Linear Operators in Hilbert Spaces (Springer, New York, 1980), p. 85. J. D. Dollard, Comm. Math. Phys. 12, 193 (1969). R. R. Strichartz, 3. Func. Anal. 40, 341 (1981). V. Enss, Comm. Math. Phys. 89, 245 (1983). E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), p. 140. G. W. Mackey, Mathematical Foundations of Quantum Mechanics (Benjamin, New York, 1963). J. M. Jauch, Foundations of Quantum Mechanics (Addison-Wesley, Reading, Mass., 1968). P. Mittelstaedt, Quantum Logic (Reidel, Drodrecht, 1978). R. Haag and D. Kastler, J. Math. Phys. 5, 848 (1964). G. G. Emch, Algebraic Methods in Statistical Mechanics' and Quantum Field Theory (Wiley, New York, 1972). O. Bratteli and D. W. Robinson, Operator Alhebras and Quantum Statistical Mechanics', VoL I (Springer, New York, 1979). N. N. Bogolubov, A. A. Logunov, and I. T. Todorov, Introduction to Axiomatic Quantum Field Theory (Benjamin, Reading, Mass., 1975), Chap. 23. E. Prugovecki, Quantum Mechanics in Hilbert Space (Academic Press, New York, 1981). P. Roman, Some Modern Mathematics for Physicists and other Outsiders (Pergamon, New York, 1975). K. K. Wan and R. G. D. McLean, Phys. Lett. 102A, 163 (1984). K. K. Wan and T. D. Jackson, Phys. Lett. l l l A , 223 (t985). H. Primas and U. Muller-Herold, Adv. Chem. Phys. 38, (1978), Chap. 3.5. K. Hepp, Helv. Phys..Acta. 45, 237 (1972). G. G. Emch, Helv. Phys. Acta. 45, 1049 (1972). B. Whitten-Wolfe and G. G. Emch, Helv. Phys. Acta. 49, 45 (1976). M. A. Naimark, Linear Differential Operators, Vol. II (Harrap, London, 1968), p. 209. W. Arveson, An Invitation to C*-Algebra (Springer, New York, 1976), p. 2t. F. Selleri and G. Tarozzi, Riv. Nuovo Cimento 4(2), (1981). W. M. de Muynck, Found. Phys. 14, 199 (1984). C. E. Lanford and D. Ruelle, Comm. Math. Phys. 13, 194 (1969). K. K. Wan, Can. J. Phys. 58, 976 (1980). W. H. Furry, Phys. Rev. 49, 393 (1936). E. Schr6dinger, Proc. Camb. Phil. Soc. 31, 555 (1939). M. Mugur-Schachter, in Quantum Mechanics a Half Century Later, J. Leite Lopes and M. Paty, eds. (Reidel, Dordrecht, 1977). G. C. Ghirardi, A. Rimini, and T. Weber, Nuovo Cimento 36B, 97 (1976). G. C. Hegerfeldt, Phys. Rev. D10, 3320 (1974). G. C. Hegerfeldt and S. N. M. Ruijsenaas, Phys. Rev. D22, 377 (1980). F. W. Byron and R. W. Fuler, Mathematics of Classical and Quantum Physics, Vol. I (Addison-Wesley, Reading, Mass., 1-969), Chap. 5.1 t. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II (Academic Press, New York, 1975), p. 282. K. K. Wan and D. R. E. Timson, Phys. Lett. IliA, 165 (1985). A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1951). M. Vujicic and F. Herbut, J. Math. Phys. 25, 2253 (1984). D. Bohm, Phys. Rev. 85, 166 (t952). C. Philippidis, C. Dewdney, and B. J. Hiley, Nuovo Cimento 52B, 15 (1979).