Fuzzy amplitude densities and stochastic quantum mechanics

Foundations of Physics, VoL 19, No. 3, 1989

Fuzzy Amplitude Densities and Stochastic Quantum Mechanics

Stanley Gudder 1

Received November 18, 1987

Fuzzy amplitude densities are employed to obtain probability distributions for measurements that are not perfeetly accurate. The resulting quantum probability theory is motivated by the path integral formalism for quantum mechanics. AIeasurements that are covariant relative to a symmetry group are considered. It is shown that the theory, includes traditional as ,,ell as stochastic quantum mechanics.

1. INTRODUCTION

In a previous work, ~5) we presented a mathematical framework for a quantum probability theory based on the concept of amplitude densities. We now extend this work to include fuzzy (or nonsharp) amplitude densitites. As we shall show, the resulting framework generalizes stochastic quantum mechanics. (~2"6~ In our opinion, the generalization we present has several important features. It displays an underlying objective reality for a physical system. It emphasizes the role of the measurement apparatus. Measurements (observables) are represented by functions and not by self- adjoint operators. It is based on guidelines motivated by the path-integral formalism for quantum mechanicsJ 3'4,7) Although the path integral formalism is not mathematically rigorous, except in certain special cases , ~7)

it has been highly successful for computational purposes,

Department of Mathematics and Computer Science, University of Denver, Denver, Colorado 80208.

293

0015-9018/89/0300-0293506.00/'0 © 1989 Plenum Publishmg Corporation

294 Gudder

The guidelines we follow are:

(1) An outcome of a measurement is the result of various interfering alternatives and each of these alternatives has an amplitude for occurring.

(2) The amplitude of an outcome is the "sum" of the amplitudes of the alternatives that result in that outcome.

(3) The probability of an outcome is the modulus squared of its amplitude.

We now amplify on the meaning of these guidelines. We first assume that a physical system can be in precisely one of a set of possible configurations (alternatives, potentialities) and that each configuration has a probability amplitude of occurring. When the system interacts with a measuring apparatus, an outcome results. In general, an outcome may result from many interfering configurations. By interfering we mean that the configurations can only be distinguished by disturbing the system; that is, by executing at least one different measurement. The amplitude of an outcome is found by summing (in the discrete case) or integrating (in the continuum case) the amplitudes of the configurations that result in that outcome upon executing the measurement. These guidelines form the heuristic axioms of the present approach to quantum probability theory.

2. MATHEMATICAL FRAMEWORK

We now present a rigorous mathematical framework based on the previous guidelines. This general framework can then be used to construct mathematical models for describing particular physical systems.

Let ~ be a nonempty set called a sample space and whose elements we call sample points. The sample points correspond to the possible configurations of a physical system S. In practice it would be impossible to describe the configurations delineating all the properties of S (for example, some properties might be unknown) so configurations are limited to those properties on which we wish to focus. A measurement is a map F from f2 onto its range XF = F(f2) satisfying the following conditions.

(M1) XF is the base space of a a-finite measure space (XF, SF, VF).

(M2) For every x ~ XF, F=~(x) is the base space of a a-finite measure space (F-a(x), Zx , #x).

We call F-~(x) the fiber over x, the elements of X r are called F-out- comes and the sets in ZF are called F-events. A measurement F corresponds

Fuzzy Amplitude Densities and Stochastic Quantum Mechanics 295

to a laboratory procedure or experiment that can be performed on 5'. For every co e £2, F(co) denotes the outcome resulting from executing F, using a perfectly accurate measuring apparatus, when S has configuration m. For x ~ XF, the fiber F-~(x) is the set of sample points that result in outcome x using a perfect apparatus. The measure VF is an a priori weight for the F-events that is independent of the state of S and the state of the measuring apparatus. In case of total ignorance, VF is a uniform measure such as Lebesgue measure, Haar measure, or the counting measure in the discrete case. Similarly, #x is an a priori weight on the fiber F-~(x).

Let F: O ~ XF be a measurement. A function f : XF X f2 ~ C is a fuzzy ampliude density for F(F-fad) if f satisfies the following conditions.

(F1) For every x', XeXF, f (x ' , ")[F-I(x)eLI(F-I(x) , X x, #x)"

(F2) As a function of x

(F3)

fx,(X) -~ fF_t(x ) f(x', CO) d]2x(O) ) ~ LI(XF, SF, •F)

Definiting H F = L2(XF, S F, re) we have

r(f)(x') = fx S, (x) dvAx) e H~.

(F4) t lF(f) t l .~= 1

An F-fad f represents a state for the combined system S + M where M is a measuring apparatus for F. Since a perfect (sharp point) measuring apparatus is impossible to achieve we must assume that M has an intrinsic inaccuracy which is included in the description of f and is responsible for its fuzzyness. If another measuring apparatus M ' is used to execute F then the corresponding F-fad f ' represents a state for S + M ' and in general f ' would be different than f. From another point of view, we may think of M as a probe of S (for example, a scattering experiment) involving a quantum test particle ~. In this case, f would contain a proper wave function or excition state of ~.(6) We denote the set of F-fads by fad(F).

For f ~ f a d ( F ) we interpret f (x ' , co) as the amplitude density that S has configuration co and a subsequent execution of F using a fixed measuring apparatus M results in outcome x'. Notice that the result using a perfect apparatus would be the outcome F(co) which, in general, differs from x'. Applying Guideline (2) we conclude that fx,(x) is the amplitude density that the result of measurement F using M is x' when the result using a perfect apparatus is x. Applying Guideline (2) again, F(f)(x') gives the amplitude density of the outcome x' when F is executed using M. We

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call F(f ) the (F, f ) wave function. Applying Guideline (3) we interpret tF(f)(x')l ~ as the probability density for F at the outcome x'. Condition (F4) ensures that IF(f)(x')I 2 is indeed a probability density. The function fx , (x) indicates the confidence in the outcome x' using the apparatus M. Since f_¥,ELI(XF, Z~F, VT), if ttfx,IILt~0, the function ~/;x,(x)= [ f ,c'( x )l / [I f x']l L ~ is a probability density called the ( F, f ) confidence function. If IIfx,tlL~ = 0, then f x ' = 0 almost everywhere so the outcome x' essentially never occurs and Z:,x' is not defined.

As a mathematical idealization, suppose we can construct a perfect measuring apparatus for executing F. Heuristically speaking, a corresponding f ~ fad(F) would have the form

X ! f ( , o~) = 6(x' - F(o~)) g(o)

for some function g: £2 ~ C. Then heuristically, we have

fx,(x) = fF-~(x) 6(X'-- F(o~) ) g(o~) d#x(o~) = 6 ( x ' - x) fe_llx ) g(m) d#x(O9 )

and

F(f) (x ' ) = ~r_l( ,) g(m) cl~x,(CO )

Since

IlL, It = fF-l~x')g(O) d#x,(c~ )

we have

= b ( x ' - x )

We define a sharp point limit to be a sequence jr, E fad(F) such that f , (x ' , o~) ~ 6(x' - F(o~)) g(o~) m an appropriate sense.

The previous discussion shows that in the case of an idealized perfect measurement, an f ~ fad(F) corresponds to a function g: g2 ~ C satisfying the following conditions.

(SI) For every XEXF, g l F - I ( x ) E L I ( F - I ( x ) , Zx , #x)

($2) F(g)(x)=-~F-l(x~gd#x~HF ($3) []F(g)]lt4r= 1

Fuzzy Ampfitude Densities and Stochastic Quantum Mechanics 297

A function g; Q-~ C satisfying ($1)-($3) is called a sharp amplitude density for F(F-sad) and the set of F-sads is denoted by sad(F). A study of F-sads is given in Ref. (5) where they were simply called amplitude densities.

We now give a general method for computing probabilities of events. Let F:f2--eXF be a measurement and let fe fad(F) . A set A _ t 2 is a generalized (F, f ) event if it satisfies the following conditions.

(El) For every XeXF, Ar~F-I(x)eZx. (E2) As a function of x

fF(A )(x', x) -- ~ f ( x ' , (D) d~tx.((D ) E L '1 (X;., ~F, VF) JA c~ F._l(x)

(E3) fF(A)(x') =-- ~XFfF(A)(x', X) dVF(X ) ~ Hr.

We denote the set of generalized (b, f ) events by g(F, f). Notice that ~ , O s g(F, f ) and fF(fg) = O, fF(f2) = F(f). The elements of g(F, f ) are the subsets of £2 for which a reasonable amplitude can be defined. In fact, fF(A)(x) is the "sum" of the amplitudes over the configurations in A that result in x upon execution of/7. We interpret fr(A)(x) as the amplitude of A given that x occurs. Since F(f)(x) is the amplitude that x occurs, we interpret F(f)(x)*fr(A)(x ) as the "probability" of A and x (where • denotes the complex conjugate). Notice that this is a generalization of Guideline (3) which states that

IF(f )(x)[ 2 = F(f)(x)* r( f)(x) = F(f)(x)* fF(t?)(x)

is the probability that x occurs. Motivated by these considerations, we define the (F, f ) pseudo-probability of A as

PF.f(A)= fxFfF(A ) F(f)* dVF= ( fF(A) , F ( f ) }

(Note that we are using the mathematician inner product that is linear in the first argument.) We interpret PF.f(A) as the pseudo-probability that A occurs when viewed by measurement F for a system described by fad f

In its present generality, P~:f(A) may be complex so it cannot always be interpreted as a probability. However, we shall later specialize to measurements for which P~,f(A)/>0. The reader may wonder why we do not define PF.u(A) as

fxF [J)'(A)[2 dye-= [Ifr(A)[[ 2

which is also a reasonable definition. In fact this is done by Oudder, ~) and compared with the present definition for the case f ~ sad(F). Although this

298 Gadder

latter definition gives a nonnegative "probability," our present definition PF, f (A) has the advantage of possessing additivity properties.

We can extend pseudo-probabilities to pseudo-expectations in a natural way. We call a function h: ~2 ~ N an LZ(F, f ) function if h satisfies the following conditions.

(El) Defining hf: .e~ F X O "'4" C by (hf)(x, co) = h(co) f ( x , co), for every Xt, XEXF, hf(x', . ) I F - I ( x ) e L I ( F l (x ) ,Sx , #x).

(L2) As a function of x

f r ( h )(x', X) =~ fF-qx) (hf)(x', co) d#~(co) ~ L I(XF, S F, Y F)

(L3) fF(h)(x') =- ~XFfF(h)(x', X) dVF(X ) ~ H E.

For h e L2(F, f ) , the (F, f ) pseudo-expectation of h is

EF.y(h) = (fF(h), F ( f ) )

Notice that for A e g(F, f ) , if ZA denotes the characteristic function of A, we have )~A eL2( F, f ) and EF, f()~A)=PF, i(A). It follows that EF, S is the unique linear extension of PF, S to L2(F, f ) .

We can also define (F, g) pseudo-probabilities and (F, g) pseudo- expectations for g ~ sad(F). In this case, A ~ g(F, g) if A satisfies (El) and

(E2') gr(A)(x) -- ~A ~ g-~Ix) g(CO) d#x(co) e H F

We then define P F, g( A ) = ( g v( A ), F( g ) >. Similarly, h ~ L 2( F, g) if h: f2 ~ R satisfies the following.

(LI ' ) For every X~XF, h g l F - l ( x ) ~ L l ( F ~(x) ,Sx ,#x) .

(L2') gv(h)(x) = ~r-l~x) (hg)(co) d#x(co ) s HF.

We then define EF, g(h) = <gr(h), F(g)>. We have studied Pr, g and Er, g in Ref. 5. Our main concern now is PF, U and Er, f for f e fad(F).

In the present generality, $ ( F , f ) need not be a a-algebra and PF, S need not be countably additive. However, weaker regularity conditions hold. A nonempty collection 5 e of subsets of g2 is an additive class if 5P is closed under the formation of complements and finite disjoint unions.

Theorem 2.1. If F: ~"2-"i'Y F is a measurement and f e f a d ( F ) , then g(F, f ) is an additive class and PF.S is an additive complex-valued set function on g(F, f ) with PF, S(O) = 1.

Proof. Clearly, g2 ~ g(F, f ) and

PF, f(g2) = (fF(12), F( f ) ) = IIF(f)jl2 = 1


If A ~ ,g(F, f ) , then for every x ~ X F the complement A c of A satisfies

A C r~ F-l(x) = F l(x)\A r~ F-~(x) ~ Xx

Hence, (El) holds and as a function of x we have

fr(AC)(x ', x) = [ f (x ' , co) d#x(co) oA co., F-lfx)

-l(x) 'dA ~F-I(X )

= fx, (x)-- fF(A)(x ' , X)~ LI(XF, i~F, ~'F)

Thus, (E2) holds and moreover,

fg(A<)(J() = fxrfx,(X ) dVv(X ) -- fxFfF(A)(x' , X)dVF(X )

= F(f)(x') --fF(A)(x') E H g

Hence, (E3) holds for A C so g(F, f ) is closed under complementation. Now let A i~ d(F, f ) , i = 1 ..... n, be mutually disjoint. Then for every x ~ Xr we have

( U Ai)r~F-J(x)= U (Air~F- l (x ) )~Sx

so (El) holds. Since the A,- are mutually disjoint, as a function of x we have

= ~ fF(Ai)(x', x)~ LI(XF, SF, VF)

Hence, (E2) holds and moreover,

fF (U Ai) (x'): fx F fF (U Ai) (xt, X) d F(X,--~ 2 .fF(Ai)(X '~ HF

Thus, (E3) holds for U Ai so £(F, f ) is closed under the formation of finite disjoint unions. We conclude that ¢(F, f ) is an additive class. Finally,

SO PF, f is finitely additive, i

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Suppose that f2 is the set of configurations of S for a fixed time. Then F: 12~XF and f E f a d ( F ) are also fixed in time, We now briefly consider dynamics. In this case we let O ' = f2 x ~ where co '= (co, t ) s O' is a configuration of S at time t. Corresponding to the measurement F: 12 ~ Xv we define measurements Ft: D(Ft) ~ XF, te ~, as follows. The domain F t is

D(Ft) = {(~o, z )~ f2': z = t} _ 12'

and for (09, t) ~ D(F,) we define F,(co, t) = F(co). We then have XF, = XF for all t e N. Define ZF, = 27 F and VF, = VF for all t e N. Hence, HF, = Hr for all t e R . Moreover, for any XeXF=XF, Ft - t (x )=F-l (x)x t . Define ~ ' x = S x X t , #'x(Axt)=l~x(A) for all A e S x. We have thus defined a collection of measurements { F t : t e R } which we call the dynamic measurement corresponding to the static measurement F. We interpret F t as executing the measurement F at time t. The domain D(F,) is the set of all configurations that are relevant to the measurement F t .

Now suppose f : XF x t2' --* C satisfies

f t[XFXD(Ft)]sfad(Ft) for all t e R (2.1)

Then f can be considered to be a fad for all the Ft, t s ~. For A c__ ~2' and t e R we define

A t = {(¢o, "c) ~ A: "c = t} = A ~ D(Ft)

I f A t ¢ g ( F t , f ) for all t ~ , then

PF,,f(A') = <JF,(At), F t ( f ) >

is the pseudo-probability that A occurs at time t when viewed by the measurement F. If PF, f(A t) is integrable with respect to Lebesgue measure dt, we define

['F,s(A)= f~ P~;.f(A') dt (2.2)

Notice that Pe. f is additive. However, fi~.f(A) cannot be interpreted as a probability even if it is nonnegative. For example, it follows from the above that Pv.f(g2')=~. One can define conditional pseudo-probabilities PF, f (A [B) = PF.f(A)/PF.f(B ) when A ___ B a n d PF, f(B) 5 ¢: 0. (6)

The previous treatment can be thought of as a Heisenberg picture since the measurements F t were functions of time and the fad f was fixed. We can also use the Schr6dinger picture in which F is fixed and f varies


with time. If f : X F × Q' -~ C satisfies (2.1), define f t e fad(F) by f t (x, co) = f (x , (co, t)). For A _~ Q' define

A t= {oJegT: (oJ, t ) eA} ~f2

Then A t = A t x t for all t eN. We now show that the two pictures are equivalent.

Theorem 2.2. Suppose f:X~-xf2'--+C satisfies (2.1) and A_~f2' satisfies A ' ~ N ( F t , f ) for all t ~ . (a) fF,(At)=f~(At), Ft(f)=F(f~), Pr,.f(A t) = Pg, i,(At) for all t ~ N. (b) If PF, f (A') is integrable with respect to t, then

PF, u(A) = PF.f,(A,) dt= (ftF(A,), F( f t ) ) dt - - e~o - - o 0

Proof. (a) for x', XSXF,=XF we have

' J A t ~ F t (x) t~ F-I(x)

: f~-(At)(x', x)

Hence,

= ftF(At)(X')

It follows that F t ( f ) = F ( f t) and

PF,,f(A t) = (fF,(At), r , ( f ) ) = (fry(At) , F(U <) ) = PF.u,(At)

(b) This follows from (a) and Equation (2.2). I

3. REGULAR AND COVARIANT MEASUREMENTS

We now specialize the treatment of Section 2 to measurements with certain regularity properties. If F: f2 --+ X F is a measurement, then the wave function mapf~-->F(f) is a transformation from fad(F) into the unit sphere SF of H F. If there exists a set W _~ fad(F) such that F: ~ --+ Se is bijective, then F is .f-regular. For an ~-regular F we use the notation

g(F, . ~ ) = (~ {8(F, f ): f E.~}

If ~ __. N(F, ~ ) is a a-algebra and PF.f(A) >~ 0 for every A ~ ~ and f ~ ~ ,

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then F is (~, i f ) positive. If F is (~, i f ) positive, it follows from the additivity of PF, f that

O~PF, f ( A ) ~ P F , f(£2 ) = 1

for every A e ~ and f ~ f f . Hence, PF, f(A) can be interpreted as a probability and PF.f is a finitely additive probability measure on M.

A function f : XF X £2 ~ C is a fuzzy amplitude function for F(F- fa f ) if f satisfies conditions (F1), (F2) and (F3). Thus, an F-fad is a normalized F-faf. Denoting the set of F-fafs by faf(F) we see that faf(F) is a complex linear space. If F is ~--regular, then the map F: f f ~ SF extends to a linear bijection from the linear hull g _= far(F) of ~- onto H g which we also denote by F. For f ~ f f , A ~g(F, Y), it is easy to check that A satisfies (El), (2) and (E3) so fe(A)~ Hr is well-defined. For A ~ i f (F , ~ ) , define F(A): HF---,HF by P(A)tp=(F-t~,)F(A). Then F(A) becomes a linear operator satisfying F(A ) F(f) = fF(A) for all f e f t . Hence, for every f E o~- we have

PF, u(A) = ( fr(A ), F(U) } = ( F(A ) F(f), F(U) } (3.1)

Moreover, if F is (~8, i f ) positive, then (P(A)t~, 0} I>0 for every 0 E HF, A ~ 2 , and we conclude that/~(A) is a positive bounded linear operator on HF. Since F(£2)= L it follows that /~ is a finitely additive normalized POV (positive operator-valued) measure on 2 .

Let G be a group of bijections from £2 onto ~2. If F: £2 ~ XF is a measurement, G is an F-symmetry group if there exists an irreducible, projective, unitary representation g ~ Ug of G on H r. We may think of F(f) ~ UgF(f) as the change of the wave function due to the symmetry transformation g on £2. Moreover, if F is Y-regular, we can transfer the action of Ug to ~ by defining 0 ~ ( f ) = F-1UgF(f) for every f ~ ~-, g ~ G.

Then f~-* 0 g f gives the change in the f a d f due to g on £2. We say that a (~3, ,~-) positive measurement F is covariant with respect to an F-symmetry group G if gA ==- { gco: ~o E A } ~ ~ for every A ~ ~ , g ~ G and

P F . f ( g - I A ) = PF, O j ( A ) (3.2)

for every f ~ i f , A ~ ~ , g ~ G. Both sides of (3.2) given the probability of A after the symmetry transformation g. Applying (3.1) and (3.2), we have for all f e f f , A ~ )

( F(g-~A) F(f), F(f) } = PF, f(g-~A ) = PF.Oj(A)

= (F(A) F(Ogf), r ( u g f ) )

= <_F(A) UgF(f), UgF(f)}

= (U*F(A) U,F( f ) , F ( f ) }


It follows that

F(gA) = Ug/~(A) U* (3.3)

for all geG, AeM. In a similar way, Eq. (3.3) implies Eq. (32) and are therefore equivalent.

We now consider dynamics for the present situation. Let F: (2--+ XF be a (D~, ~ ) positive, static measurement with corresponding dynamic measurement F t: £2' ~ XF, t e ~. Assume that the free dynamics is described by a dynamical group V t on HF. This is, V t is a weakly continuous one- parameter, unitary group on HF such that VtF(f) is the wave function at time t if the wave function at time t = 0 is F(f), f ~ . As before, we transfer the action of V t to g by defining f t=F-~VtF(f) , t~ ~, f e Y . For f e ~ , define f ' :XFX(2 ' by f'(x,(cn, t))=ft(x,o~). Then f ' l [XF × D(Ft)] ~ fad(Ft) for all t ~ ~, f ~ ~ . Now suppose A e £2' satisfies A s ~ ~ for all t e ~, and PF,,u,(A t) is measurable with respct to t. It follows from Theorem 2.2(b) that for every f ~ . ~ we have

PF, T,(A) = PF, r(AO dt= (F(At) F(f'), F ( f t ) ) dt - - o o . . - ~

= (V*F(At) VtF(f), F(f) ) dt (3.4) - - o o

If we define the Bochner integral

then Eq. (3.4) becomes

P~f.(A) = ( F(A) g(f), F(f) )

for every fe~,~. In this case, F is an unnormalized finitely additive POV m e a s u r e o n H F .

Now let G be a group of bijections from f2' to s'2' that possesses a projective, unitary representation g v-+ Ug on Hr. We say that the dynamic measurement {Ft: t ~ ~ } is covariant with respect to G if

PF.f,(g-IA) = Pr,(Ogfy(A)

for every f e ~ , g~ G, and A ~ 2 ' satisfying the conditions of the previous paragraph. As in Eq.(3.3) it follows that {Ft: t ~ } is covariant with respect to G if and only if for every g ~ G and A as above we have

F( gA ) = Ug F(A ) U*

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304 Gudder

We now consider an important class of (~, o~) positive measurements. Let F: £2 ~ XF be a measurement and let { ~o: o) e £2 } be a set of vectors in H F satisfying ll~,olI=[l~o,,![ for all co, o)'e£2. We call {~o,: o)eQ} an F-resolution of the identity if for every ~, ~ e HF, the function (¢, 4)~(o)) = (¢ , ~ } ( ~ , ¢} satisfies the following conditions.

(R1) (¢ ,O)¢tF- l (x )eL~(r- l (x ) ,Sx , l~x) for x e X r . (R2) (0, ~b)~(x) =- ~r-~(~) (O~b)¢(e)) dgx(CO ) e LI(XF, ZF, VF). (R3) ~xv(¢, ~)~(x)dvr(X)= (0, 0}.

We thus have for every ~k, ~b e H F

=fx .IF (~, ,~)(~o,O)d#,~(co)dvr(x) F

For q~ e HF, define

(3.5)

and

F A x

(W~¢)(o))-- (0, ~ ) =,}(co)

and define the positive operators P~ o n H F by

P~(0) = (~, ~,> ~. = ~;(o))¢,,

Equation (3.5) shows that the following Bochner type integral satisfies

fa P~, do) = I

For t) e Hr, define fo: XF × £2 --* C by f~(x, o)) = ~o~(X) ~(o)) and let ~ -= {f~: ¢ e St}. The next result shows that F is ~-regular.

Theorem 3.1. Let {¢o,: o)e£2} be an F-resolution of the identity. (a) For any C e S r , f~efad(F) and F is ~-regular. For any a-algebra ~___g(F, ~ ) , F is (~, ,~) positive. (b) For any q* e SF, F(f6)= ~. (c) If A e g(F, ~ ) and Ax = A ~ F-l(x), then for any ~ e SF we have

P~;T¢(A ) = ~ X "A[F~ [~(o))[2 d#x(o) ) dV F(X )


Proof. Since f~,(x', co)= (P~/s)Cx'), by the properties of a Bochner integral, the following integrals exist.

;x f~ f* (x', co) d,u:~(co) dvFCx) >~ - l ( x )

It follows that for ¢ 6 S F , f~, 6fad(F) and F(f~,) = ¢. Hence, F: ~ - ~ S e is bijective and F is ~¢-regular. If A 6 d'(F, ~ ) and ¢ 6 S F we have

F ,c

=[fv, iA< P~ dlx~Cco)dvFCX)] +(x') C3.6)

Hence,

f f <p o, o > :VF .,ix

We conclude that F is (~, Y¢) positive for any a-algebra 1 ~_ d~(F, ~¢). Moreover, since

~'CA)~=-~(A) F(fo)= f~,F(A)

the second part of (c) follows from Eq. (3.6). I

Corollary 3.2. Suppose an F-resolution of the identity {~, : co e f2} exists and M and is a a-algebra in ¢(F, ~ ) . Ca) Then P~:f is a (a-additive) probability measure on ~ for every f ~ and _F' is a normalized Ca-additive) POV measure on ~ . (b) If h: f2 ~ ~ is an LZ(F, J~¢) function that is measurable with respect to ~ , then for any f = f , ~ we have

3 0 6 Gudder

Proof (a) This follows from Theorem 3.1 and the aaddit ivi ty of the Bochner integral. (b) Since h is measurable with respect to ~ and from (a), PF, S restricted to ~ is a probability measure, (O, ~ , PF, f) is a probability space so the integral ~oh(co)PF, T(dco) is defined as the usual Lebesgue integral. Now suppose h=S,,ZA ,, a i ~ , A i E ~ ] , i= 1 ..... n, is a simple function. We then have

EF, f(h) = S a E F , f()~A,) = XaPF, u(Ai) = f h(co) PF, f(dco)

A standard limit argument now gives the result. |

We thus see that in this case, Pr.f and EF.f have the desirable properties of a probability measure and an expectation.

Let {~,o: cocO} be an F-resolution of the identity. We define K,: f2 x f2--+ C by K~(co', co) = (~o,,, ~ ) . Then

and

K~'(CO', co)= (~o~, {~o') = K¢(CO, co')

IX IF Ke(co', co") K¢(CO", co) dt~(co") dVF(X) r - l ( x ) -

: fXFfF_,(x)(~,o', ~o)")(~o~", ~ )d~x(~" )dVF(X)

= (~o,', {o,) = K,(co', co)

It follows that K, is a reproducing kernel. (6) Now suppose we have a dynamic measurement F, on £2 '=£2x N corresponding to the static measurement F and assume the system evolves in accordance with a dynamical group V(t): H F ~ H F. Define the F-propagator

We interpret

K¢((CO', t'), (co, t ) )= (~o,,, V(t'-t)~o~)

K~((CO', t'), (co, t ) ) / l l~ol l 2

as the transition amplitude from (co, t) to (co', t') as viewed by the measurement F. The following calculations show that K¢((co', t'), (o, t)) has the usual properties of a propagator.

K*((CO', t'), (co, t ) )= ( V(t'-- t)~,, ~o2' ) = ( ~,o, V ( t - t ' )~ , )

= K¢((CO, t), (co', t'))


M oreover,

fX>fF K¢((co', t'), (o)", t")) K¢((co",)t"), (co, t)) d#~(co") dVF(X ) -1(x~

F -t{x)

r -~(x)

= { V ( t " - t ' ) ~ , , V( t"- t )~,~)= (~o', V( t ' - t " ) V(t"-t)~,o )

= (~o', V( t ' - t ' )~ ,o)= K~((CO', t')(co, t))

We now briefly discuss quantum field theory in the present context. Let F: £2 ~ XF be a measurement that possesses a resolution of the identity { ~o~ : co e Y2 }. For f g e He = W¢ H F define

Then ( f , g ) is an inner product on H~ and H e becomes a Hilbert space. Moreover, We: HF-~ H~ is an isometry. If we have n identical systems, each described by £2, the resulting combined system has sample space £2"=£2 x -.. x £2 (n times). We define the n-particle boson Hilbert space F,((2) to be the symmetric tensor product

F.(£2)=H}"=H~s...sH¢

In a similar way, one can define the n-particle fermion Hitbert space, but for illustrative purposes, we shall confine ourselves to the boson case. The corresponding boson Fock space is

F(£2) = r . (£2 ) , ro( ) = c n ~ O

An element ~b e F.(£2) can be considered to be a symmetric function of n variables ~b(col,..., c%), coie£2. For an f ~ o ~ , x'~XF, we define the local annihilation operator Af(x'): F (£2)~ 1"(£2) as follows. For ~b ~ F . (Q)

[ Au(x')(~ ](COl,'", co~- l )

~ n - 1 / 2 f ~ f * ( x t , (.On)~((DI,.., , (.On)d~:(O.)n)dVF(X ) Jx r OF~I( x )

308 Gudder

It can be shown that Af(x' ) is a closed densely defined operator on F(12) whose adjoint (the local creation operator) satisfies

[A~(x')cb](col,..., co,+ 1)

n+l = ( n + l ) -1/2 ~ f(x',coj)(~(cot ..... coj_l,coj+t ..... o9,+1)

j = l

One can now define mathematically rigorous field operators. For example, the number operator Nf is given by

Nr= fx A~(x') Af(x') dVF(X' )

where the Bochner integral is well-defined on a dense domain.

4. STOCHASTIC Q U A N T U M MECHANICS

This section shows that our previous work generalizes stochastic quantum mechanics for a single spinless particle in three space dimensionsJ 1'2"6~ We take for our sample space the six-dimensional phase space

= ~6 = {(_q, _p). q, _p~ ~3}

Define the measurements Q: (2 ~ R 3, P: f2 ~ ~3 by Q(q, p) = _q, P(_q, _p) = p

and let S Q = X p = B ( ~ 3 ) , dvQ=dq, dvp=dp. On the fiber Q l ( q ) = q × ~3 we let L'q = _q × B(~3), dl~q = dp -and on -p-l(_p) = ~3 × p, we let- S e =

B(~ 3) ×_p, dl~ e = d q. Of course, Q, P correspond to position and momentum measurements, respectively.

We first consider traditional quantum mechanics; that is, the case of sharp amplitude densities corresponding to perfect measuring apparata. If f ~ sad(Q) we have

Q(f)(q_) = f f (q , p) d p e Lz(N 3, d q) = HQ

and if f ' ~ sad(P) we have

P(f')(_p) = f f'(q, p) dq ~ Lz(~ 3, d p) = Hp

where ItQ(f)lt = ]lP(f)ll = 1.


We now construct a set of sads that correspond to the traditional quantum states. For

Ip e L2(~ 3, dq) c~ L I ( ~ 3, d q)

denote the Fourier transform by

# (W~)(p) = ~(p) = (2nh)-3/2 j ~(q) e l , ~ "~/h dq

and the inverse transform by

(W*g,)(q) = ~(q) = (2nh) -3/2 f ¢,(_p) e,_q-e/h dp

Define f e , f ~ : O -~ C by

fq,(_q, _P) = (2gh)- 3/21~(£0_ ) ei~e/h

f'o(q, P) = (2nh)- 3/2O(q) e-iq-'e/h

For A ffB(N6)--B(~) we have

f~o(A )(q_)= ..f, f*(q' p) ap = (2~h)-'/2 ..f,q ¢(p_ )eiq-e/h d p

= ( z < ~ b ) ~ ( q )

f;.(A)_(n)=f~ f;(q, e )~=(2=h) - ' /~ I~ lis(q)e-'q-'-;l~dq _P P

= 0c~eO) ~ (_p) It follows that

and

Q(fo)(_ = Q = q) to (re(q_) O(q)

P(f'~)~) = f~e(sg)~) = ~(p)

Hence, if 11~'1t L2 = 1, then f~ e sad(Q), f ~ e sad(P). Using the notation ~.~ = {f¢,: OeLlc~L2}, o ~ ' = { f ; : OeL~nL2}, fov-+Q(fo) and f'~-'~P(f'o) are linear bijections from .N and ~ ' onto L 1 r~ L 2, respectively. Moreover, for any A ~B(N6), if we define Q.(A)O=feQ(A), P(A)(J = f l e ( A ) , then 0(A) and P(A) are densely defined linear operators on H e and He, respectively, which satisfy

PQ,f,(A) = (O(A)~, ~ ), Pe, f~(A)= (P(A)~, (9)

310 Gudder

It is clear that PQ.Z~, PP..f'~ are complex-valued measures and Q,/3 are

operator-valued measures on B(~6). In general, the first two are not nonnegative and the second two are not positive operators. For a set of the form A x B ~ B(R 3) × B(~ 3) we have

and

Hence,

Similarly,

and

Hence,

f~Q(A x B)(q) = ZA (_q) j" f ,(q, P) @ = Z~ (q) W*Z. WO(q_)

Po~,f+(A x B ) = (ZA W*zsW$, ~h )

O.(A x B) = ZA W*zs W

f'~e(A × B)(p) = XB(_P) f A f'~(q' p) dq = Zs(_P) WZA W*6(_p)

P p,f~ (A x B) = (Zt~ PIZZA W*~, ~ )

P(A x B) = ZB WZA W*

In particular, Q(A x ~3) = •A, /5(N3 x B) = Zs, 0( N3 x B) = W*ZB W, P(A x ~3 )= WZA W*, so these are projection-valued (PV) measures on B(~3). Moreover, we have the usual quantum mechanical formulas

PQ, f~(A x ~3) = Pp.d,(A x ~3) = f A [~(q)[2 d q

PQV.~(~ 3 X B)=ep.f~(~3 x B) = fB t~(P)I2 @

It is shown in Ref. 5 that other results of traditional quantum mechanics, including dynamics, can be obtained in this way.

We now consider fad(Q) and fad(P) which are more in accordance with stochastic quantum mechanics. Let ~ L 2 ( R 3 , dx) with ]14]]= (2~h) -3/2 and define

~_q, p (_x) = e"e (x- e)/h ~(_x -- q)


Then it can be shown [in Ref. 6] that {{_q.p: (q, _p)~g?} is a Q-resolution of identity. For tp eL2({R 3, d_x), t!t//{l = I, we define, as in Section 3,

i~(_q, _p)= (w~g,)(q, p)= <g,, ~,e) and

fv(x, (q, p)) = {~,e(~ ) ~(_q, p)

It follows from Theorem 3.1 and Corollary 3.2 that f~, ~ fad(Q), Q ( f v ) = and Q is (B(£2), ~,~) positive. Moreover, Po,I~ is a probability measure and

is a normalized POV measure on B(~) and for all A e B(f2) we have

P°'Y~(A) = Jfa l~(_q, p)[2 dq dp

Q.(A) = f f a P~'p dq dp

To treat the momentum measurement, let ~(_k), q~(k) be the Fourier- Plancherel transform of ~(k), O(k), respectively, and define

(q. p(k) = e i-q' (k- p/h ((_k _ p)

~'(_q, _p)= <~, C_~,e > f~(k, (_q, p)) = ~.e(k) ~'(q, _p)

Then {(v,e: (q, _P) ~ f2} is a P-resolution of the identity and for all A e B(Q) we have

PP're(A) = ffa [~l'(q, p)[2 dq dp

We now compute the (Q, f ) confidence function for f = f ~ .

Theorem 4.1. For ~b ~ S O and f = f e,, we have

fx,(X_) = (27rh) 3 t¢(_x'- ~) j20(x ' )

f1:~.(_x) = (2=h)3 I¢(x ' - x)t 2

312

Proof Applying the Fourier-Plancherel theorem that is rigorized in the usual manner) we obtain

Hence,

Gudder

(in a formal way

=f e'~"x'-<'lh~(x'-x_) f I/s(_y)e ~(-Y--x>/~*(y-_x)d_yd_p

= f tp(_),') {(x_'-- x)~*(_y-- x_) f e @''~'- ~,)if, dp d y

= (2r~h) 3 ~ ( x ' - ~ ) f ds(y) ~ * ( y - x) 6(~ ' -_y) d y

= (2~ / i ) ~ l ~ ( _ x ' - _x)l 2 q,(_x')

¢ I1~,tl L,---J I L,(_x)l a_x= lq,(_x')l

If tp(_x') ¢-0, we have

Zf, x ' ( - x ) = lf_~'(S)l/ilfv'llL~ = (2/~h) 3 t~(_ x ' -_x) t 2 |

In a similar way, the (P, f ) confidence function becomes

2z_k (_k) = (2~h)~ I ~(_k ' - _k)l 2

Moreover, it is shown in Ref. 6 that for every A e B(~ 3)

PQ.f(A x ~3) = Pp.f(A x IR 3) = fA is3 Zf. _~'(~)l~s(-x)[ 2 d~x e x'

.Pe, i(R3 x A)= Pe,](~3 x A ) = f A ;n 3 X]@'(-k ) )~(-k ))2 d k d_k'

We now discuss the covariance of Q and P. Let g be the Euclidean group

g = {g=(a_,v,R):a, v6R3, R s S O ( 3 ) }

For g = (q, v, R), _a corresponds to a space translation, v a velocity boost and R a rotation. The action of g on £2 = N6 is given by

g(q, _p) = (Rq~ + a_, Rp + mr,)

Fuzzy Amplitude Densities and StGchastic Quantum Mechanics 313

where m is the mass of a particle being measured and the product law in d ~ is

gl g2 = (_a~, _v~, R1)" (_a2, _v2, Re)

= (_a I "-[- R 1 _a2, _Vl + R 1 _v2, RI R2)

For ~h(x) 6 LZ([~ 3, d_x) = HQ define

U(g, v, R) $(_x) = exp[im _v. (_x - a_)/h] Ip(R -~(_x -- _a))

It can be shown that U is an irreducible, unitary, projective representation of # on HQ and hence e2 is a Q-symmetry group. The proof of the following theorem is indicated in Ref. 6.

Theorem 4.2. If ~ is rotationally invariant (~(R_x)=~(x) for all x ~ ~3, R ~ SO(3)) then the (B(O), ~ ) positive measurement Q is covariant with respect to #.

Proof Since ~ is rotationally invariant we have

~e(q.e)(~) = 4(Rq+~.Rp+,.~)(_X)

= exp [i(R_p + m y ) . ( x - Rq_ - a ) / h ] ~(x_ - Rq_ - a )

= exp( --im v -Rq/h) exp(im v- (x - _a)/h) = ~q, e(R-l(x_ - a_))

= exp( - i m v- Rq_/h)[ U(g) ~q. p] (x)

Hence, for every g E # we have

U(g) ~.q. p = exp(im _v- Rq_/h) ~g(q, e)

For • ~ HQ, we obtain

[U(g)Tp, e U(g)*¢](_x)

= exp Jim t2. (y - _a)/h] [Tq.e U(g)*$](R -I(5 - _a))

= exp[im v- (_x - a)/h] ( U(g)*th, ¢q, e ) ~q_,e (R -J(x - _a))

= (~k, U(g)~q,_p)[U(g)~q_.e](x)

= (~t, ~g(q, pl ) ~gfq._p)(X) -- (Tg(q.e)~)(x)

Hence,

U(g)Tq_,pU(g)* = Tg(q_.p)

314 Gudder

and we have for every A e B(sC2)

U(g) Q_.(A) U(g)* = II A U(g) Tq.eU(g)* dq dp

= If z~(q_,_P)Te(~.e~ dq dp

= ff xgA(q_, _p) r~, p ~ @

=fig L,aqaP=O(gA) ! , 4 - ' - - -

In a similar way, we can show that if ~ is rotationally invariant, then the (B(f2), ~ ) positive measurement P is covariant with respect to &

We next consider the dynamic measurement ( Q , : t ~ N } on D' = ~ x N. The appropriate symmetry group is the Galilei group

N = { g = (b,_a, v, R): beN, a. ve N3, R e S O ( 3 ) }

where a, v, R have the same meanings as in the Euclidean group and b corresponds to time translation. The action of g = (b, a, v, R) on f2 '= N 6 x E is given by

g(_q,_p, t )= (Rq+_vt+_a, Rp+mv, t + b )

and the product law in N is

g l g2----- ( b l , _a l , _vl, R1) , (b 2, a 2, _v 2, R2)

= (bl + b2, a_l + Rla_~ + b,.v_l, v_l + Rlv_2, R~R2)

For 0(_x) E L2(N 3, d x) = HQ, define

;U U(b, a_, v, R) O(x)=exp b + m v . ( x - a ) tp (R-~(x- g - b_v))

Then U is an irreducible, unitary, projective representation of N on H 0 and hence ~¢ is a {Q,: t e N}-symmetry group.

Now the free Hamiltonian is given by Ho = (-h2/2rn)V 2 and the free dynamics is described by

0(_x, t)= v,~,(s)= e-im'/h ~,(_x)


For ~/eS0, define ~(q, p, t)= W¢~(x_, t) and the corresponding fad is given by

f'~(x, q, p, t) = ~q_.p(x) ~(q, p, t)

As is Section 3, it follows that for all A ~ B(¢2')

? Q(A)= Vff(2(A')V, dt= V* Tq pdqdp Vtdt

-- fff A e"° ' /~ Tq, p e - iHo t/h dq d p dt

is an unnormalized POV measure. As in Theorem 4.1, it can be shown (6) that if ~ is rotationaly invariant, then

U(g) Q(A) U(g)* = Q_(gA)

and hence the dynamic measurement {Q,: t e N} is covariant with respect to (¢. In a similar way, {Pt: t~ N} is covariant with respect to (¢.

We close this section with a consideration of the relativistic situation. The sample space is the relativistic phase space

(2 = {(q, p): qe R 4, p e f~,}

where ~ , = ~ + u qr;~ is the mass hyperboloid

$/'~ = {p: p==p-p=rn2c ~, p ° X 0 }

p . p = (pO)Z_ (pl)2_ ( p 2 ) 2 (p3)Z. The appropriate symmetry group is the Poincar6 group

= {(a, A) : a t [R 4, A e Y }

where 5o is the proper Lorentz group and a e ~4 corresponds to spacetime translation. The action of g = (a, A) on f2 is given by

g(q, p) = (Aq + a, Ap)

and the product law in ,G ~ is

gl g2 = (al, A1) . (a2, A2) = (Ala2 + a~, AIA2)

A relativistically covariant position or spacetime measurement with desirable properties does not existJ 6) However, we can define a dynamic

316 Gudder

energy-momentum measurement {Pt: t ~ ~ } as follows. The domain of P~ is

D(P~) = {(q, p): qO= ct} ~

and for (q, p)eD(P~) we define P,(q, p ) = p . Then for every t~ II~, Xp~= ~ and we let 27p, = B(~m) and vp, is the Lorentz invariant measure

re, = dv m( p ) = cS( p 2 - m2 c 2) d4p

Notice that

~,(_p)

where ~O(p)= ~((p2 +m2c2)t/2, p) and a similar result holds on ¢ /~ with ~(_p) = ( _ (pZ + m2cZ)l/z, p). On the fiber PFl(p) = o" = ct x ~3, we define Sp = ct x B(~ 3) and #p is the Lorentz invariant measure

d#p = 2e(p °) p~ day(q)

where e(pO)= _+ t for pO~ 0 and day points in the direction of the normal to the spacelike hyperplane a. (~6) It is not hard to show (6) that d#p dv m = dq dp on a x ~U + . An irreducible, unitary representation of ,~ on H p =

Lz(~um, Vm) is given by

U(a, A) ()(k)=ei~k/hqS(A tk)

which shows that ~ is a symmetry group for Pt: t ~ ~. For (q, p)Eg2, let A~ be a pure Lorentz boost by the 4-velocity

v = p / m and let t1~L2(~ "+, Vm) be rotationally invariant with 111711 =2mc(2z~h) ~-3. Now define tlq, p~ L2(V + , v~) by

11q.p(k )[ U(q, A,)r/](k) = eiq~/~11( A~-~k )

Then

{qq.p: (q, p) ~ £2, qO = ct}

is a Pt-resolution of the identity. (6~ Hence, if Pq, p is the positive operator Pq, p~ = (~b, ~/q,p)r/q,p we have

f qo = ~ P ~. p dq dp = I+


where I_+ are the respective identity operators o n L2('C/'m~ , Vm). For ~b s L2(~P,~, Vm) define ~(q, p ) = (4, ~/q,p)¢+. Then by Theorem 3.1

f ( ~ , (q, p)) = ~.,p(IC) ~(q, p)qO = ct

is a fad for P, and for every A e B(a x ~.~) we have

P,(A ) = ff A P*p dq dp

Moreover, Pt is (B(a × ~m), ~ ) positive. As before, for A E B(t2) we define

P(A)=f~ P,(A')dt

and it can be shown that {Pt: t,G ~} is covariant. That is,

U(a, A) P(A) U*(a, A) = P((a, A) A)

for all (a, A) ~ ~ and A ~ B((2).

REFERENCES

1. S. T. Ali and E. Prugove~ki, "Extended Harmonic Analysis of Phase Space Representation of the Galilei Group," Acta. AppL Math. 6, 19-45 (1986).

2. S. T. Ati and E. Prugove6ki, "Harmonic Analysis and Systems of Covariance for Space Representations of the Poincar~ Group," Acta. Appl. Math. 6, 47-62 (1986).

3. R. Feynman, "Space-time Approach to Non-relativistic Quantum Mechanics," Rev. Mod. Phys. 20, 367-398 (1948).

4. R. Feyman and A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

5. S. Gudder, "A Theory of Amplitudes," J. Math. Phys, 29, 2020-2035 (1988). 6. E. Prugove~ki, Stochastic Quantum Mechanics and Quantum Spacetime (Reidel, Dordrecht,

1984). 7. L. Schulman, Techniques and Applications of Path Integration (Wiley-Interscience, New

York. 1981).

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Fuzzy amplitude densities and stochastic quantum mechanics