]L NUOVO CIMENTO VOL. 9B, N. 1 II Maggio 1972

A Noncommutative Representation of Classical Dynamics. Connections with Field Quantization {')

J. A. CRAWFORD (**)

B e r k e l e y , Cal .

(ricevuto il 31 Agosto 1970; manoscritto rcvisionato ricevuto il 25 Maggio 1971)

S u m m a r y . - - Einstein's program of classical field theory for physics has not been proven unrealizable, lIowever, Bell has shown that a theory of this type could not bc consistent with all of the predictions of quantum mechanics. I t is reasonable to assume that fields in a theory of the Einstein type could be described as classical dynamical systems. We begin by formulating conditions for an operator representation in Hilbert space of the phase functions of a finite classical system. I t is proved that such a representation cannot be commutative. The Weyl representation ful- fills the required conditions: in Appendix A it is shown to map the quadratically integrable phase functions isometrically into the Hilbert- Schmidt operators. I t follows that the image U of the phase probability density is a statistical operator in the sense of von Neumann, though not generally nonncgative definite. The Weyl representation is then formally extended to infinite classical systems and applied to interacting relativistic tensor wave fields (spinor fields, not being single valued, present difficulties). With the choice t~ ~ u ~, this representation sat- isfies the requirements of canonical quantization. The Hamiltonian operator and operators for the usual field quantities are the same as in quantum field theory, ltowever, an additional field interaction term appears, modifying von Neumann's equation for the time dependence of U. Essential divergence difficulties should not arise, since no diver- gcnccs occur in classical field theory. This representation of classical field theory is amenable, in just the same way as quantum field theory, to a configuration space representation and a corresponding particle interpretation. These aspects will be discussed in a later publication.

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (**) Postal address: 1033 Kecler Ave., Berkeley, Cal. 94708.

1 - I1 Nuovo Cimenio B. 1

2 J . A . CRAW~ORD

1 . - I n t r o d u ~ i o n .

1'1. Quantum mechanics and the hypothesis o/ field quantization. - Dirac's quantization of the electromagnetic field (l) marked the birth of the paradigm of (( quantum theory ~), through which the laws of quantum mechanics, verified in detail for aonrelativistic systems of particles, were assumed valid quite generally for all physical systems. The extension of these laws to fields, giving rise to the quantum theory of fields, disclosed a heuristic circularity as this theory was found to yield the very systems of particles whose laws had been borrowed to construct it.

Quantum field theory became, with the theory of relativity, one of the pillars upon which modern physics was erected. Yet these two theories are mutually contradictory. The geometrical nature of the metric tensor g~ in Einstein's equation

(1) R~ ,--�89 ,R = --zT~,

implies that the energy-momentum tensor of matter T~, is a classical field. This is impossible when matter is described by quantum field theory. The blame for this incompatibility between relativity and quantuIa field theory can, we suggest, probably be fixed upon the quantum-theoretical paradigm, partly in view of the latter's heuristic circularity and partly because of the inconsistencies (divergences) which it develops when applied to fields.

On the other hand, quantum field theory, handled with art, has been suf- ficiently successful to support the belief that it is not far from the truth, and that a correct physical theory must bear it a mathematical resemblance. These considerations combine to suggest that a valid physical field theory, if one exists, may be classical even though physical quantities in such a theory must somehow be represented by noncommuting operators. A theory of this type, implementing Einstein's program of <( classical theory ~ (5), would imply a rejec- tion of the quantum-theoretical paradigm.

BELL has shown that this type of theory could not be consistent with ~ll of the predictions of quantum mechanics (a). An experimental search for one possible violation of these predictions is now in progress (4).

(1) P. A. M. DIRAC: Proc. Roy. Soc., 114, 243a, 710 (1927). (2) A. EINSTEIN: Albert Einstein: Philosopher-Scientist, edited by P. A. SCHILPP (New York, 1951), p. 675. (3) J.S. BELL: Physics, 1, 195 (1964); Proc. S.I.F., Course IL (New York, 1972), p. 170. (4) J . F . CLAUSER, M. A. HORNE, A. SHIMONY and R. A. HOLT: Phys. Rev. Lett., 23, 880 (1969). Other possible violations are discussed by P. M. PEARLE: Phys. Rev. D, 2, 1418 (1970).

A NONCOMMUTATIVE R E P R E S E N T A T I O N OF CLASSICAL DYNAMICS ETC.

1"2. The physical interpretation o] a classical field theory. - We must expect that a valid classical field theory should be expressible in the language of par- ticle quantum mechanics, since it should allow a description of physical experi- ments in familiar terms. This implies that such a theory must be capable of a configuration space representation.

A complete discussion of the physical interpretation of a classical field theory thus requires the development of its configuration space representation. Only in that context could such things as quantum-mechanical observables and the related uncertainty principle be given any meaning.

From the formal Weyl representation of fields to be developed later, a configuration space representation can indeed be constructed, essentially in the same way as from quantum field theory. Space limitations prevent us from discussing this in the present paper. We limit ourselves to giving the field operator represent~ttion, and exhibiting its mathematical similarity to quantum field theory.

2. - The operator representation of classical physical quantifies.

We have noted that the success of quantum field theory suggests that physical quantities occurring in a valid classical field theory would have to be represented by noncommuting operators.

I t is sometimes asserted that the existence in quantum theory of mutually incompatible physical quantities is a direct consequence of the noncommuta- t ivity of the operator representation. Thus the notion of a noncommutative representation for classical fields seems paradoxical: these systems do not possess incompatible physical quantities. Actually, the quantum-theoretical origin of incompatibility is more involved: it contains the additional ingredient of the nonnegativity of yon Neumann's statistical operator. This nonnegativity is deduced from Dirac's rule for' constructing the operator representing the function /(9t) of a real physical quantity ~R from the operator associated with ~R:/(~R)o~ ----/(~o~) (5).

I t is reasonable to assume that fields in a theory of the Einstein t~pe could be described as classical dynamical systems. We shall formulate condi- tions for an operator represelltation in Hilbert space for the physical quantities associ~ted with a finitc clas.~ical dynamical system, in which Dirac's rule is not assumed to be valid. Instead, we shall require the quadratically integrable phase functions to be mapped isometrically into the space of Hilbert-Schmidt operators. On this hasis it will be shown that commutative representations do not exist.

(5) J. VON :NEUMANN: Mathematical Foundations o/ Quantum Mechanics (Princeton1, 1955), p. 313.

4 J . A. CRAWFORD

A representat ion fulfilling the required conditions is constructed. This is made possible by the proof of a theorem (Theorem A.6) concerning Weyl oper- ators. A representat ion for a sys tem of interact ing relativistic classical tensor fields is then defined in Sect. 4 as a formal extension of the representa t ion for a finite classical system. The la t ter representa t ion forms the subject of the present Section and of Sect. 3.

2"1. Fin i t e classical dynamical systems. - Consider a classical dynamica l system with ] degrees of freedom. In this sys tem physical quanti t ies are defined as complex functions a(pl , . . . , ql) of 2] canonical variables pl , ..., p~,

q l , . . . , qr, whose range is the set of real numbers . These variables define a 2/-dimensional Euclidean phase space. Physical quanti t ies are then complex functions on the phase space, or (( phase functions ~>.

We shall write

Pi = x~, qi = xt+i, i = 1, . . . , / .

The funct ion a(xl , ...7 x2~) will be denoted compact ly as a(x) and the phase volume e lement dxl . . , dx2r as dx. Al ternate ly , we shall occasionally write a(x) in the form a(p, q) and dx as d p . d q .

Then there is a probabi l i ty densi ty funct ion on phase space u(x) such

tha t the expecta t ion value is

(2) exp [2] =fdxu(x)a(x) for every physical quant i ty 9 / = a(x).

We shall assume u(x) to be bounded. Since i t is nonnegative and integrable, i t belongs to Z1; its boundedness then ensures tha t it is also an element of L2, the space of quadrat ical ly integrable phase functions.

For a physical quant i ty 71 in Z2, exp [~] may be expressed according to (2) as the scalar product

(3) exp [9/] = (~, a ) .

The Hi lber t -Schmidt operators (the bounded linear operators of domain s and finite norm) of a Hi lber t space @, themselves form a l:filbert space ~:. Since L2 is a Hi lber t space, one may then map it l inearly and isometrically onto any I t i lber t subspace !~9 or ~E. Such a mapping preserves the conjuga-

t ion in L~ if

a(x) ~ L~ -+ A* ~ ~ ,

when

a(x) e L~---> A e ' ~ .

A NONCOMMUTATIVE REPRESENTATION OF CLASSICAL I)YNA~IICS ETC. 5

The scalar product in ~ may be wri t ten , ~s in ~ , in the form

(A, B) = Tr ( A ' B ) ,

and the isometry of the mapping requires

(3) now becomes

(a, b) = (A, B ) .

exp [91] = Tr ( U A ) ,

rec~lling yon Neumann 's expression for the expecta t ion value of a physical quan t i ty in quan tum mechanics (6).

This suggests t ha t we might look for a linear operator representa t ion in for physical quanti t ies, coinciding with the above mapping for all quanti t ies in L2. We assume tha t this representa t ion has the proper ty tha t , if a(x)--+A,

then a(x)--> A*. The operators corresponding to the x~ are therefore self-adjoint, and we shall assume their spectrum to be the set of reals. For convenience we say tha t the 2/-dimensional vector x is represented by the vector operator X. Alternately, we shall sometimes write the [-dimension,~l vector (Xx, ..., X~) as P, and (Xs+~, ..., X~) as Q. We ussume fur ther t h a t every physical quant i ty is represented by some funct ion of the X~.

2"2. The nonexistence o/ commutative representations. - Clearly a necessary and sufficient condition for a representa t ion to be commuta t ive is tha t the X, commute with one another . Le t this be the case, and let ~R = r ( x ) ~ 0 be a real physical quant i ty in L2 represented by the self-adjoint operator Q(X). We require q(X) to be ;~ t t i lber t -Schmidt operator. I t mus t then possess a finite norm. I ts norm must therefore be finite in the L2 representat ion, Q(X) being there represented by the mult ipl icat ive operator ~(x). Let {~(x)} be an or thonormal basis in Z2. We then require

(4) fdxle(x) < co.

Since ~(X) is a self-adjoint Hi lber t -Schmidt operator, its eigenvectors form an or thonormal basis in 5)(7). We may then choose the y)~(x) to be eigen-

funct ions of ~(x), so t ha t

e (x )~ , (x ) = e,W,(x).

(s) g. VON NEUMANN: Mathematical Foundations o] Quantum Mechanics (Princeton, 1955), p. 316. (7) N. DUNI~ORD and J. T. ScnWhRTZ: Linear Operators (New York, 1963), p. 905.

6 ft. A. CRA~VFORD

The requirement (4) may then be wri t ten

[e,l~< o o . f

Whenever ~ (x )~ ~ , ~i(x) must vanish. Since, however, ~(x) has a non- vanishing norm, the measure of the domain D,, where ~(x) ---- ~i, must be finite. Any function F(x)eL~ vanishing outside D, is then an eigenfunction of e(x) belonging to the eigenvalue ei (with the exception of y~(x)~ 0). I t is easy to show tha t there exist an infinity of l inearly independent functions with this property. Therefore, every eigenvalue of ~(x) is infinitely degenerate.

Since (~(X), 0(X)) = (r, r ) > 0, o(X) cannot vanish. Therefore not every O, vanishes. Le t then ~- ~ 0. We have

Hence Q(X) has infinite norm, and so cannot be a tI i lbert-Schmidt operator us required.

Commutat ive representations therefore do not exist. The necessity for a noneommutat ive representation, which in quantum theory implies the existence of mutual ly incompatible physical quantit ies, is here implied by their absence.

2"3. The Weyl representation. - Since no commutat ive representation exists, we look for a noncommutat ive one. To this end we introduce the commutat ion relations

(5) [P,, Pj] = O, [Q,, Q~] = o , [P,, Qj] -- (i~r) -1 (~.,

where I is the uni t operator. The self-adjoint operators

N, = �89 [~(P~ + Q~)- 1], i = 1 , . . . , / ,

commute with each other and possess a pure discrete spectrum of nonnegative integers. A set of simultaneous normalized eigenvectors {r of the Ni, there- fore consti tutes a basis in gj. We can and shall assume tha t Y3 is so constructed tha t it contains a unique (to within an arbi t rary phase factor) normalized element r such tha t

3"ir = 0 , i = 1, ..., t"

Each set of simultaneous eigenvalues of the Ni then determines ~ corresponding r uniquely, to within an arbi t rary factor exp [iOn].

Let now

a(x) = f d u ~ ( u ) exp [ixx]

A NONCOMMUTATIVE REPRESENTATION OF CLASSICAL DYNAMICS ETC. 7

be any element of L~ (x being a 2j-dimensional re:~l vector variable). I t fol- lows that ~(x)e Z~, the space of quadratically integrable complex-valued func- tions of x. We then call the linear operator

T[a(x)] =fdx exp [ixX]

a Weyl operator, and denote it by A, the capital letter corresponding to a:

/~[a(x)] = A .

Since nx, xX are dimensionless, x, and X, must have the same dimensions. I t then follows from (5) that action cannot possess dimensionality in this rep- resentation.

We can thus define a linear mapping of L~ onto the space of Weyl operators by the correspondence a(x)--~A. I t will be proved in Appendix A that this space is a proper Hilbert subspace ~ of 2, and that the mapping a(x)-->A is a conjugation-preserving isometric mapping of J5~ onto !~.

In particular, if A, B are the images in ~ of the L~ elements a(x), b(x), we have

(a, b) = (A, B).

This mapping thus fulfills precisely the requirements of Subsect. 2"1 for an operator representation of the elements of Z~. We call it the Weyl repre- sentation for elements of Z~.

We have seen that the probability density in phase space u(x), assumed to be bounded, is an element of L2. I t therefore possesses an image U in and, for any a(x)eL2, we have

Thus

(6)

(~, a) = ( U*, A ) .

exp [a(x)] = Tr (UA).

Furthermore, since u(x) is real, U is self-adjoint. However, in contrast with quantum theory, U is not necessarily a nonnegative definite operator. From Theorem B.1 we have nevertheless

Tr U =fdx (x) = 1

Finally, since U e929, U is a Hilbert-Schmidt oper,~tor, and since it is also self-adjoint, its eigenvectors form an orthonormal basis in ~, and no eigen- value of U (except possibly zero) is infinitely degenerate (7).

S J. A. CRAWFORD

2"4. Expression o] u(x) as a Wigner density. - For purposes of comparison with the Weyl.Wigner correspondence in quantum mechanics (8.1o), it is of some interest to point out tha t the probabili ty density u(x) is the Wigner density corresponding to the operator U. The following proof is formal. Let

exp [iuX] A

be a Weyl operator. According to Theorem A.6 we have

( U*, A) = (2~z)2' f dxv( - - x)a(x)

Since ~(x) is an arbi t rary element of Z'~, i t follows tha t

o r

(2~)21v(--x) = Tr (U exp [ixX]) ,

v(x) ~- (2ze) -~1 Tr ( U exp [-- ixX]) .

Taking the Fourier transform, we obtain

u ( x ) : (2 )-2,fdx Tr ( U exp [-- ixX]) exp [ixx],

Wigner's expression for the quasi-probability density corresponding to the stat ist ical operator U. u(x) is, of course, an authent ic probabili ty density, and remains so as t ime proceeds, according to the Liouville equation.

By contrast, in Moyal's representation of quantum theory in classical phase space (8), the Wigner quasi-probability density, even if it is originally non- negative definite, does not in general remain so in the course of time.

2"5. Extension o] the Weyl representation to ]unctions not in L2. - We may ask in general whether the Weyl representation for elements of Z2 can be extended to functions b(x) not in 3~2, in such a way as to satisfy relations anal- ogous to (6), i.e.

(7) exp [b(x)] = Tr ( U B ) ,

where b(x) --~ B. The linear properties of expectation values require tha t such an extension

be linear. Since, in general, the operator B can no longer be expected to have

(8) J. E. ~OYAL: Proc. Cambridge Phil. Soc., 45, 99 (1949). (9) G. A. BAKER: Phys. Rev., 109, 2198 (1958). (10) j . C. T. POOL: Jour. Math. Phys., 7, 66 (1966).

A NONCOMMUTATIVE REPRESENTA'[ ' I ( )N OF CLASSICAL DYNAMICS ETC.

domain ~, the definition of Tr (UB) presents complications. We shall in this Section develop the extcnsion oi' the Weyl represent~t ion in ~ formal manner , neglecting all consideration of thc dom~ins of operators. Traces of operators will be assumed without proof to satisfy the rules

(8) Tr (cC + dD) = c Tr C + d T r D ,

(9) Tr (CD) = Tr (DC),

C, D being operators in ~, and c, d constants. Formal computat ions with traces, following the rules (8) and (9), will enable

us to write expressions of the type Tr (UB) as traces of Weyl operators, thus allowing us to utilize Theorem B.1 ~md so express expecta t ion values of func- tions b(x) not in L2 in the form (7).

Le t

a( x) = f d x g ( x ) exp [ixx] ~ L.2

and assume x~a(x)EL.~. Then

= f d x ~ ( x ) " ~ (--'~-~ exp [ixx])=f d. (i ~--~(x)) exp [ixx] xia(x)

The Weyl operator associated with x~a(x) is therefore

o) ] =fax ~a(x)

' ~ - i exp [i•

We may write this form~dly as (~)

[ixX]) 01) T[x~a(x)]=fdx:~(x)(--,,~exp =

Let r(x) be a polynomia,1, and assume

r(x)u(x) ~ L~.

I t is then clear from the foregoing how the Wcyl operator represent ing r(x)u(x)

(~t) X~.A is nol, iden~icM to T[xia(x) 1 as given in (I0). The letter's domain is g); the domain of Xj.A is the intersection of the domain of X~ and the range of A, and so does not extend t,) the whole of .r We shall, as stated earlier, ignore all considera- tion of the domains of operators.

10 a.A. CRAWFORD

may be expressed in terms of U. We need consider only the case when r(x) is a monomial:

r ( x ) = r e ( x ) = e ~ , x , , . . . x , . .

Assuming that x~u(x), x~._x~u(x), ... all belong to L~, (11) yields

r E m / x ) , ( x ) l = ox, v)... )).

I t then follows from (8) and (9) tha t

From the commutation relations (5) it is easily shown that

x, .(x,._

is invariant to arbitrary permutations of its (~ factors ~) Xj (see also the proof of Theorem A.2). Thus we find

Tr (T[m(x)u(x)])= Tr ( U c [ X ~ . ( X i . ( . . . . ( X , . _ . X ~ ) . . . ))]).

If we extend the Weyl representation by associating with m(x) the operator

T[m(x)] = M = cXq. (X, . ( . . . . (X~,_. .X, .) . . . ) ) ,

we then obtain, from Theorem B.1,

f dxu(x) re(x) = exp [m(x)] ---- (UM) . Tr

For an arbitrary polynomial r(x) we define the associated operator R = = T[r(x)] in the extended Weyl representation as the sum of the operators M associated with the monomials in r(x).

I t is readily verified that this extended representation fulfills the require- ments set forth in Subsect. 2"1. I t then follows that

exp [r(x)] = Tr (UR).

Theorem 1. If r(x)----rl(x)r2(x), r~(x) and r2(x) being polynomials on dis- joint sets of degrees of freedom, then

/ ~ = R I R2 = R2171 �9

A N O N C O M M U T A T I V E R E P R E S E N T A T I O N O F C L A S S I C A L D Y N A M I C S E T C , 11

Proo]. I t is sufficient to assume tha t rl(x) is the monomial m(x) discussed above. Then

(12) R - -cX, . . (X, . ( . . . . (X,n .R~ ) . . .)) .

Since R~ commutes with each of the X~ in this expression, it is readily shown, as before, tha t all (( factors ~) on the right side of (12) are arbitrarily permutable. Hence

We may extend the Weyl representation also to functions b(x) which are not polynomials. Let b(x) be ~ function on a subset of {X~} ~nd ~ssume tha t it has a Fourier t ransform in the space S determined by this subset:

b(x)--fdxff(x) exp [ixx] =fdxff(x) o ~o ~.vi" (xx)",

where xx is now a scalar product in the space S. Since our representation is required to be linear, if it is to satisfy (7), we

require only the operator representing the polynomial

r(x) = (xx)~ : (~ ~:ix,) ~ �9

From our adopted rule, this is

R= [ ~ , X d . ( [ ~ i X , ] . ( . . . .([~. ~iX d . [ ~ X d ) . . . ) ) , p ((factors,,.

Since all of these (( factors ~) are identical, they mutual ly commute, and

= .

i

The linearity of the representation then implies

T[b(x)] ---- B ---- xff(x) = duff(x)

8 Z

exp [ixX]

(in agreement with the Weyl representation for elements of L2 when the subset of {Xr is identical with {Xd).

We may then write

exp [b(x)] = duff(x) p~ exp [(xxF] = duff(x) | z~ Tr [U(xX)~] = Tr (UB). 8 8

12 J.A. CRAWFORD

Theorem 2. If c (x )=a(x )b (x ) , a(x) and b(x) being functions on disjoint sets of degrees of freedom, then

C = A B = B A .

The proof proceeds as in Theorem 1, making use of the expansion

~ p [ i ~ x ] = 5 ( ~ x ) , . V.'

2"6. The Weyl representation /or Poisson brackets. - In order to derive the operator representation of the Liouville equation, we must first obtain the Weyl representation for Poisson brackets. Let a(x) ~ Z2, s(x) be a polynomial. We wish to ascertain the operator corresponding to the Poisson bracket

~s ~a ~s ~a (s(x), a(x)}

= ~ ~"q~ ~p, ~P, ~q~

in the Weyl representation. Let 2 be a real parameter, and consider the following linear transformations

in Z2:

e,(~) = q , - G ~p--~ ' i = 1, . . . , 1 ,

0,(4) = iv , + ~ ~ , i = / + 1, . . . , 2 / .

We may write these transformations in the form

(13) Q,(2) = x, + ).u,, i ---- 1, ..., 2],

where

(14)

g~---- - - , i = 1 , . . . , ] ,

i = t + 1 , . . . , 2 i . g i ~ i ~ ~Xi_ 1

Consider now the operator S = s(X) representing the polynomial s(x) in the

Weyl representation. We then construct the transformation r -- s(p) in L~

by substituting p for X in the expression s(X).

Theorem 3.

d q~ 31 ~ ~ (15) - - = ~ ~ , - - .

A N O N C O M M U T A T I V E REPRESENTATION OF C L A S S I C A L D Y N A M I C S E T C . 1 3

Proo]:

Lemma 1. If (]5) holds for q~,, ~b~, i t holds for cr ~b~ + q~, c ~ con- stant. The proo[ is trivial.

Lemma 2. If (15) holds for q~, it holds for ~'q~l, ? = 1, ..., 2].

To demonstrate this we may write, with the help of (13) and (14),

= + [q;, m ] ~ "

Suppose first tha t )< ] . Then [~, z~] vanishes unless i = ) § then

1 8 1

Hence

We have

1 b 1 ~ _8~b 1 1[ 1 8 ] 1 8q~

The same result can be derived in the ease j > ], completing the proof of this Lemma.

We now observe tha t q~ = ~j satisfies Theorem 3. Indeed

d ~ 2t 8~ - - ~ --= ~ : r , ~ = 1 , . . . , 2 1 .

d2 i=l ~i

Theorem 3 is thereby proved. I t is easily verified tha t the operator ~s(X)/SXi represents the polynomial

8s(x)/Sx~ in the Weyl representation. Theorem 3 is therefore satisfied also by 8r We may then write

q ~ " ( 2 ) = ~ ,=1 d2 8q, ,=I j=l ]=1

By iteration it is readily proved tha t

(16) ~.=1 ~,~,~ . . . ~i~" dn=l

14 J . A . CRAWFORD

From the MacLaurin expansion

= s

we derive the iden t i ty

(17) 5 b r ~b(r) __~b(r) ~ (1/2)~ - = 2 5 r=O L ~ odd P ~ r=O

Choosing the b, in such a way tha t the coefficient of ~(m)(0) on the right-

hand side of (17) is ~m.1, we obtain b0----1, b 2 = - - 1 / 2 4 , b4 = 7/5760, b6 = = - - 31/967680, ..., br vanishing when r is odd. (17) may then be wri t ten

(18) ~ '(0) = [~(~) - - r �89 - - (1/24)[r - - r �89 +

+ (7/5760)[~5,4,(1) - - ~b(4,(_ �89 _ (31/967680)[r _ ~b(e)(_ I)] -~ . . . .

We remark that , since ~b(2) is a polynomial in the ~,(2), the expansion (18) possesses a finite number of terms.

We observe tha t ~b(0)= s(x). Hence from Theorem 3

~s(x) �9 ' ( o ) = L zv, ~ - .

~=1 f~X~

Therefore

2/ ~s(x) 1 ~ ~s ~a ~s a a 1 (19) ~b'(0)a(x) = ,=15 ~' Vx, a(x) = - - ~ , ~ ~q~ ~p, ap, ~q, in (s, a} .

Thus our problem is the constr~ction of the Weyl operator corresponding to qS'(O)a(x). I f we utilize the expression (18) for ~'(0), we must find the operators corresponding to r and q)(~)(--1)a(x). We note tha t

I t can be shown tha t (12)

OA/~Q~ = i~[P~, A ] ,

These two expressions may be wri t ten

~A/~X~_, = i~[Xi, A] ,

~A/~Xi+I = - iT~[X,, A] ,

8AIS_P ~ = - - i~[Q~, A] .

i > / ,

i < / ,

(12) H. WF.YL: The Theory o] Groups and Quantum Mechanics (New York, 1931), p. 275.

A N O N C O M M U T A T I V E R : E P R E S E N T A T I O N OF C L A S S I C A L D Y N A M I C S E T C .

and are subsumed in the formula

(20) z,a(x) -+ [X,, A ] ,

Thus

and

Q,(�89 X~.A + �89 A] = X~A

o2(-�89 ~ X ~ . A - - � 8 9 A] = A X , .

and

I t is then readily verified that, for the polynomial r s(p),

r189 ~ s(X)A

In particular

~(-- �89 --> As(X) .

a(")r189 a(x) ~ ~(")s(X) ~ i . . . . ~Qi, ~Xi ... ~Xi, A

~(,)q5(__1) a(x) ---> A ~(")s(X) ae,~ ae,, axT~.7. ~ : r ,

and

Thus, according to (16) and (20),

15

i --~ l, . . . ,2] .

in=l

From (18) and (19) we then obtMn

1 ( 21 ) i z

I X,. , ~(")s(X)

[ I [ i ' 7 ~ Xi , X~, X~, X ~ , [ ~ X z ~ X ~ X ~ X i , A . , . .

3. - The Liouvi l le equation.

The Liouville equation for the probability density in phase space ut(x), considered as a function of the time t as well as the phase point x, is

(22) ~u,(x)/~t = {h(x, t), u , (x ) } .

16 g. A, CRAWFORD

In the Weyl representa t ion we have

(23) at(x) ---> Us.

We consider the case when the Hami l ton ian function h(x, t) is a polynomial

in the xi. F rom (21), (22) and (23) we then obtain the equat ion of mot ion

for Us

(24) ~ d V,=at [~' u~]-- ~, x,, x,, ~x~-x,' u, + i . i=1

+ 5760,.,.~.t=ll [ toxtexkex,ox,' . . . .

3"1. Constants o/ motion. - Let c(x) be a polynomial cons tant of motion

not depending explicit ly on the t ime. We have

and

{c, h} = 0

~{~, ~,} ={~, (h, u,})= -{h, (.,, ~}}-{~,,, {~, h})={h, {~, ~,}}.

Thus {c, ut} obeys the Liouville equation. I t follows tha t the initial condition

(e, us} = 0

is conserved in t ime. In the Weyl representa t ion this may be s ta ted as follows: The ini t ial condition

is conserved in t ime. In part icular , if e(x) is a quadrat ic constant of motion,

then the condition

(25) IV, Vii = 0

is conserved. The relat ion {c, h} ~ 0 may also be expressed in the Weyl representat ion.

To accomplish this we note first tha t the linear t ransformations in L2 already discussed are also t ransformat ions of phase functions b(x) not in L~, and pos- sessing an image in the Weyl representat ion. I t is easily verified tha t also in

this case

g,b(x) -~ [X,, B] ,

x~b(x) --> X~ " B .

A NONCOMMUTATIWE REPRESENTATION OF CLASSICAL DYNAMICS ETC. 17

I t then follows t h a t the expression (21) is also val id when a(x) is replaced b y b(x). Wri t ing s(x) ~ c, b(x) = h, (21) yields

0 ~ [ C , H I - - ~ X~,X~, tt + . . . .

In par t icular , if c(x) is a. quadra t ic cons tant , we have

(26) [c, H] _= o .

4. - The statistical theory of interacting relativistic wave fields.

We now a t t e m p t to ex tend the Weyl represen ta t ion to in terac t ing relat ivis t ic

wave fields. A sys tem of this sort can be expressed wi thout difficulty as a classical dynamica l sys tem wi th a denumerab le set of canonical variables.

For our purposes i t is essent ial t h a t the canonical var iables should be real . Le t the set (%(r)}, i - 1 ,2 , ..., be a real o r thonormal ba.sis in 3-space, and con-

sider a pa i r of canonically conjugate fields ~(r) , n(r). We m a y express these in

t e rms of denumerable canonical var iables p~, q~ through the canonicM trans- fo rmat ion

c o c o

(27) yJ(r) ---- ~ q~%(r) , n(r) = ~ p , ~ ( r ) . ~=I i = l

I f ~(r) , z ( r ) are real , so are pl, q~, and the extension of the Weyl representa-

t ion can be carr ied through in the s t ra ight forward manner outl ined below. The same is t rue if we are dealing with l inear ly independent complex conjugate pairs of canonically conjugate fields y;(r), z ( r ) and ~(r) , ~r(r). These can be ex-

pressed in t e rms of independent real canonically conjugate fields yd r ) , z d r ) and ~2(r), s~(r) through the canonical t r ans fo rma t ion (~a)

(28) { ~ = 2-~(~'~ + iw2),

= 2-�89 - - i ~ ) ,

~fi 2-�89 - - i~f2) ,

= 2-�89 § in2) �9

One or the other of these s i tuat ions occurs when we deal wi th tensor wave fields. Spinor wave fields, being double valued, p resen t a special problem

and will not be deal t with in this paper . We proceed, therefore , to ex tend the

Weyl representa t ion formal ly to sys tems of in terac t ing rela t ivis t ic tensor wave

fields. The p rob lem here is to go to the l imit ] = c~. We have seen tha t , if

a(xi~, . . . , xi.) is a funct ion of n canonical var iables , i ts image in the Weyl rep-

resenta t ion is uniquely de te rmined independent ly of ]. We note also t h a t the

(13) G. WENTZEL: Q u a n t u m Theory o/ F ie lds ( N e w Y o r k , 1949), p . 12.

2 - II Nuovo Cimento B.

18 J. A. CRAWFORD

stat ist ical operator Ut is a self-adjoint t t i lber t -Schmidt operator of uni t trace for all values of ].

We are thus led to postulate tha t , for any stat ist ical s tate of the infinite system, there exists a self-adjoint Hi lber t -Schmidt operator Ut of uni t trace, such tha t , for any funct ion a(x~,, ..., x~.),

cxp [a] = Tr (UtA) ,

A being the image of a in the Weyl representat ion. I t is well known that , for relat ivist ic classical canonical fields, whether

or not they are placed in interact ion, several impor tan t quadrat ic constants of mot ion arise: the components of the to ta l m o m en tu m vector, of the to ta l angular mom e n tum and, if the case arises, the to ta l electric charge (14). I f c(x) is a quadrat ic constant of motion, we may then conclude tha t the init ial con- dit ion

(29) [C, Ut] = 0

is conserved, and t ha t

(30) [C, HI = 0 ,

according to (25) and (26). We remark nex t tha t the t t ami l ton ian funct ion for interact ing relativistic

tensor fields is, in general, of degree no higher than the four th in the canonical variables (and no higher than the second in any one pair of these variables). In these cases the equat ion of mot ion (24) for U~ becomes

i7~1 dUt_dt [H, U t ] - X~, ~ X i ' Ut (31) ~,~1 Xi, �9

We have seen (Subsect. 2"3) tha t action must be dimensionless in the Weyl representat ion. Sett ing ~ z,-1 it may be verified tha t the Hamil tonian oper- ator H in this expression is identical with the corresponding operator in quantum field theory (15). The eq. (31), therefore, differs from yon :Neumann's equation

for the stat ist ical operator of quan tum field theory only by the presence of the

second t e rm on the r ight-hand side. I t is easily seen tha t this addit ional t e rm arises purely from the field interactions. On the other hand, the relations (29)

and (30) are identical with the corresponding relations in quan tum field theory (16).

(14) G. WENTZEL: Quantum Theory o] Fields (New York, 1949), p. 51, 123. (15) G. WENTZEL: Quantum Theory o/ Fields (New York, 1949), p. 4. This statement is true also for all of the usual field quantities. (16) We note that, if a constant of motion c(x) is represented by the operator C in the Weyl representation, [c(x)] 2 is not in general represented by C 2. If C has a pure discrete

A NONCOMMUT ATIVE REPP~ESENTA'[ ' ION OF CLASSICAL DYNAMICS ETC. 19

The formal cl~ssic~d theory we have outl ined bears thus a m a r k e d mathe-

mat ica l resemblance to q u a n t u m field theory. I t is indeed, l ike q u a n t u m field theory, susceptible of a configuration space represen ta t ion through which i t

m a y be in te rp re ted in the language of par t ic le q u a n t u m mechanics (it can be seen t h a t creat ion and unnihil~tion operators curt be defined in the usual way

f rom the operators .P~, Q3. This aspect of the theory will be developed in a la ter publ icat ion.

Final ly, i t should be recallcd tha t , since el~ssica.l fieId theory is free f rom

divergences, we should expect the fo rmal r ep resen ta t ion of classical fields

presented here to be devoid of essent ia l divcrgence difficulties such as charac-

ter ize the q u a n t u m theory of fields.

5. - R e m a r k s o n t h e M a x w e l l f ie ld .

I n the last Section i t w~s implici t ly suggested t h a t re la t ivis t ic classical cunonicul fields m:~y tu rn out, a f t e r all, to yield a val id physical theory of the

corresponding boson part ic les . I f this view is correct , we should expect the physical e lec t romagnet ic field to be none other t han Maxwell 's cl~ssical field. Yet one h~s tod~y the impress ion thut this possibil i ty has been disproved.

Our purpose in this Section is to po in t out briefly theft this impress ion has

no solid basis in fact . A centrul a rgumen t against the physicul va l id i ty of

classical e lectromugnet ic theor): is its fai lure in predic t ing the spontaneous

emission of radia t ion f rom exci ted a toms (JT). Only the q u a n t u m theory of radia t ion will yield correct ly this spontaneous emissio~.

Bu t this ,~rgument against M~xwcll 's theory is not decisive. A w~lid theory of the in te rac t ion of the cl~ssical e lect romagnet ic field with a toms would quite possibly have to t re~t the electrons themselves also f rom the s tandpoin t of classical field theory. Thus the hybr id model usual ly discussed c~nnot logic~lly serve as a b~sis for a decision for or ag~inst the classical Ma,xwell field.

The f~ilure of the semi-clussicul theory of a tomic radiution to predic t spon-

taneous emission is due fundamen ta l l y to the impossibi l i ty of considering

quantum-mecb~niea.1 electrol~s :~s the source of ~ cl:~ssic:~l fieJd, a.s required

by Maxwcll 's equations. The difficulty here is ent i re ly analogous to the one

spectrum (for instance if C is the field electric-charge operator), and if U is some steady state, it may be that U corresponds to a single eigenvalue c of C, i.e., for every eigen- vector r of C, (r Ur 0 except when Cr162 I t follows that Tr(UC 2) = = [Tr (UC)] 2. This does not imply, however, that the dispersion of e(x) vanishes: a sharp ~ value ~) of C does not imply a sharp wdue of c(x), and there is no inconsistency in the fact that C has a discrete spectrum, while c(x) has a continuous one. (17) G. WENTZEL: Quantum Theory o/ l~ields (until 1947). TheoreticM Physics in the Tweatieth Century, edited by FIERZ, MARCUS and WEISSKOPF (New York, 1960), p. 48.

20 J. A.. CRAW~ORD

already mentioned of interpreting the quantized energy-momentum tensor of matter as a source of the classical gravitational field in Einstein's equa- tion (1). Thus it illustrates rather the incompatibility of quantum and classical theories than the failure of classical theory itself.

Indeed, with a classical field theory of electrons the difficulty would not arise, and no a priori reasou would exist for supposing that spontaneous emis- sion could not be correctly accounted for in the context of Maxwell's theory.

A second argument against the physical validity of classical electromagnetic theory is Gibbs's statistical theory of classical canonical systems which predicts equipartition of energy in thermal equilibrium, in contradiction with what occurs for black-body radiation. However, Gibb's theory is a hypothesis which has never been deduced from the classical canonical equations of motion, except in certain special or limiting cases unrelated to black-body radiation. BOLTZl~ANN has derived the H-theorem in the limiting case of a dilute gas. SINAI (18) has proved the validity of the ergodie theorem for a gas of hard

spheres. But ordinary classical dynamical systems without constraints fail conspicuously (19) to satisfy Birkhoff's ergodic theorem (20), and there exists no reason for supposing that Gibb's hypothesis is valid for the interacting harmonic oscillators of classical cavity radiation. Brout and Prigogine's deriva- tion of a Gaussian distribution for a harmonic-oscillator amplitude in thermal equilibrium (21) is consistent with other hypotheses besides that of Grm3s.

Thus, 5Iaxwell's theory, taken together with a classical field theory of electrons, has not been proved inconsistent with Planck's radiation law, and the alleged differences between the predictions of classical theory and quantum theory in this instance may reveal themselves as insubstantial as the emperor's clothes. A similar suggestion, based on different arguments, has already been advanced by BocomE~I and LOlNa]~R (2~).

6. - A representation preserving the dimensionality of action.

We have seen in Subsect. 2"3 that the Weyl representation requires action to be dimensionless. I t follows that the use of this representation in a physical

theory implies the existence of a conversion factor between, for instance, energy

(is) j . G. S~NAI: Dokl. Akad. Nauk SSSR, 153, 1261 (1963). (19) A. I. KHINCHIN: Mathematical .Foundations el Statistical Mechanics (New York, 1949), p. 56. (2o) A. BLANC-LAPIERRE, P. CASAL and A. TORTRAT: M~thods math&~atiques de la mdcanique statistique (Paris, 1959), p. 61. (~1) R. BROUT and I. PRmOGINE: Physica, 22, 35 (1956). (2~) p. BOCCmERI and A. LOINGER: .~ett. Nuovo Cimento, 4, 310 (1970).

A NONCOMMUTATIVE REPRESENTATION OF CLASSICAL DYNAMICS ETC. 21

and frequency units. The value of this factor would be determined from the exper imenta l value of /~ and the relat ion h = ~-~ postula ted in Sect .4.

We may ask, however, if .r representa t ion similar to the Weyl representa- t ion could be constructed which would allow the dimensional i ty of action to be preserved. There is no difficulty in doing this. Such a representa t ion is, however, mathemat ica l ly less simple than the one we have chosen.

Le t P~, Q~ ( i - 1, ..., 1) be operators satisfying the commuta t ion relations

[P'~, q~] = ~ . ,

where /z is an a rb i t ra ry parameter . Le t now

q) = fda. d'r162162162 ~) exp [i(ap + 'vq) a ( p ,

be some phase funct ion in Z2, and associate with i t the new operator

=fdr d~a(a, ~) exp A ' [r

We may write P: = (/2z)�89 Q: (#z)~Q~, and P] = ~, Q~ satisfy the commuta- t ion relations (5). F rom (6) it is then easily deduced tha t

exp [a(p, q)] = ( /~)1Tr (U 'A ' ) .

The equat ion of motion for U I is der ived in exact ly the same manner as (24), # taking the place of z-1 at every step of the derivation. We thus find, in a self-explanatory notat ion,

A mathemat ica l resemblance to quan tum field theory can then be obtained following the argument of Sect. 4, if we s e t / z - - h and ident i fy (/~:r)tU: as the stat ist ical operator before going to the limit ] =- c~. The identification of the universal constant ~ with a pa ramete r whose value is a rb i t ra ry is precisely what

permits a scale t ransformat ion and defines action as a dimensional quant i ty .

The author is indebted to Dr. H. EKSTEIN for a valuable correspondence, to Dr. J. S. BELL for a s t imulat ing exchange of let ters on local hidden-variable

theories, to Dr. H. P. STAI~I " for extensive and useful discussions and, finally, to Prof. G. W~.~ZEL for a few very valuable observations concerning the present work. He also wishes to t hank a referee for some construct ive suggestions.

22 J . A . C R A W F O R D

APPENDIX A

The Hilbert space ~ o f W e y l operators.

A'I. Introductory lemma. - With a view to carrying out certain l imiting processes, we introduce the Hilbert-Schmidt operator in ~ ~ ( 0 ~ ~ 1), with

I

hr---- ~Sr , , N,---- �89 [z(P~ + Q~)- - I ] . i = l

These l imiting processes will, in each case, involve let t ing $ approach uni ty, so t ha t ~a formally approaches the uni t operator I .

We verify first tha t ~N is a Hilbert-Schmid* operator. We note t ha t ~ is a linear operator. To show tha t it is also bounded, let r = - ~ c , r c SJ; then

r

T

2Vr = n~r

Since ~ is self-adjoint, i t then follows tha t its domain is ~3 (a3). ~ therefore possesses a norm ][~II, ~nd we need only show tha t this norm is finite. We have

r r v t=O v2=O v r=O i = 0 n = O

L e m m a A.1. If Z is a bounded linear operator of domain ~, its norm IIL[[ satisfies the following relation:

I!LI[~ = l im IIL~I] ~ . ~--+1

Proo/. Since L is a bounded linear operator of domuin g2, the operutor L~ ~ shares this property. I t therefore possesses a norm. Hence we have

r r

I f L has a finite norm, ~ (Lr Let) converges absolutely. Thus for every

e /2> 0 there exists a M, such tha t

8 (Let , Let) < ~ if M > M6.

r = M

(~a) M. H. STONE: Linear Trans/ormations in Hilbert space (New York, 1964), p. 58.

A NONCOMMUTATIVE REPRESENTATION OF CLASSICAL DYNAMICS ]~TC. 23

Therefore

r=M r=M

5Tow

M--1 M--1

Z (ir Lr - Z 6~'(Lr Lr =/(6) [r=l r=l

is continuous at 6 = 1. Hence the re exists a ~ > 0 such tha t

]<6) : I/<6)-f(m)l<~

whenever 16 - - 1 ] < 5. I t follows tha t , for every assigned e > 0, there is a ~ > 0 such tha t

(A.~) ~ ( ~ r 1 6 2 LOr) 62n*(LCr, ~ r r~l r=l

whenever 16--11< b. Since the lef t -hand side of (A.1) is easily shown to be nonnegative, it is equal to its ~bsolute value. I t then follows immediate ly t ha t

l im [IL6~[I ~ = Imlf~- ~'-1

If, on the other h~md, I]L][ ~ , then, for any v > 0, there exists a M~ such tha t

if M > M ~ . Now

M

r=l

M M

~-+1 ~ 1 r=l r=l

Thus l im ]IL6xll ~ is greater t han any assigned v, and henee is infinite. We ~->1

therefore have, in :my ease,

IILJI ~ = ~ rlL6~l[ ~

A'2. Proper Weyl operators. - A Weyl operator

A = f d x g ( x ) exp [iuX]

is said to be a proper Weyl operator when its kernel a(x) is bounded and belongs to L~, the space of ~tbsolutely integrable functions of x. For convenience we shall often write the /-dimensional vector (zl, . . . , zs) as a, and the vector

24

With this notation we may write

xX = ~P + 'vQ, dx ---- do" d'v .

J . A . C R A W F O R D

(A.2)

~ow

Ia(x) ~",(r exp [ixX]r < a ~ ' , ,

where a~ is the bound of a(x). The sum

r n l . . . .wn~ i = 1 i=O n=O

converges. Hence, the series

Z a(u)~",(r exp [ixX]r

is uniformly convergent in x, by the Weierstrass test, and (A.2) may therefore

Also, ~(u) will be sometimes written in the form a(a, ~).

Theorem A'I. Proper Weyl operators are linear, bounded and of domain ~.

Proof. The linearity property is obvious. Let r be ~n element of ~, A a proper Weyl operator. A t is defined if and only if

+,

converges. I t is then equal to (At, At) . We have

Hence A has domain ~ and is boanded.

Theorem A'2. If

A =fax=(,,) exp [ixX]

is a proper Weyl operator, then

lim ~ (r A~r ----- (2~)~fa(0). ~-->1

Proo]. I t follows from Theorem A.1 tha t A~ ~ has domain ~), so that A ~ r exists. We have

X (r ~ r = X ~nr(r a c t ) = -- X fax~(x)~",(r e~p C~xX1r �9 $' r

A N O N C O M M U T A T I V E R E P R E S E N T A T I O N O F C L A S S I C A L D Y N A M I C S E T C , 25

be wr i t ten

lqow

I

exp [ iuX] ~ = H exp [ i z , ] ~ ' , , i=2

Z i ~- (~iPi -~ T i Q i .

I t follows from the e lementary propert ies of the r t ha t we can write

(r ~ ) ' ,, l -[ exp [iZ,]$~'r --~ l ~ (r exp [iZ~]~N,r i=1 i = l

and

(A.4)

with

r r i ~ l

= H (r cxp [~z,]~-,r = H s, t = l n~=0

co

n~=0

and

(A.5) ~ r = n~r i = 1 , . . . , f .

We now fix our a t t en t ion on a par t icular value j of the index i , and denote t r by the symbol Znj, so t ha t (A.5) may be wri t ten

~ ' ) ~ n j : ni~nj , i = 1, . . . , ] .

With this nota t ion we may drop the subscript j a l together and convenient ly write

(A.6) s, = ~ ~,(r ex~ [;zj]r = ~ ~(z~, e~p [iZ]zo). n j--0 n--0

Our first task is to compute this expression. We have

(zo,

Because of the completeness of the t I i lber t space ~ (~4), i t is readi ly shown tha t we ma y wri te this in the form

co i p (A.~) (z~, exp [iz]z~) = Z : ~ (z~, z~z.).

~=oP:

(24) M. H. STON]~: Linear Trans]ormations in Hilbert Space (New York, 1964), p. 3.

26 J.A. C~AWFO~D

TO evaluate the expression (Z., Z~Z~) we introduce the destruction and crea- tion operators C~, C* defined as follows:

Ci-.~ (~/2)�89 *-- c, - (~ /~)~ (.g, + iO,) , i = 1 , . . . , I .

These operators satisfy the commutat ion relations

[c~, c~] = o , [ c , , c~] = , ~ z . (A.8)

We obtain

Sett ing i----j and dropping the subscript j, we have

Z = yC + ~C*, y = (2zr)-~ (a + iT) .

Recalling the definition of the dot product of two operators Y-Y' to be

Y. Y ' = � 8 9 Y 'Y) ,

we consider the expression

y,.

where each Ym is either yC or ~7C*. Consider the ident i ty

Ym" (~Tm~-l" k~)- Ym+l" (Yva" L~) : 1 [[Yra, ~/-m+l], S]

for any operator S. In view of (A.8), the r ight-hand side vanishes. Since the dot product is commutat ive, it is then clear tha t E is invariant to every per- muta t ion of the << factors >> Ym.

Defining

r ; - T c . ( . . . . . . . . ) ) ,

in which there are r <~ factors ,> yC and p- - r <~factors ~> ~7C*, we may there- fore write

) \ \ r / denotes the binomial coef f ic ien t .

P v a l u a t i n g T~, we obtain

1 ~o (~) T r ~ ~p~ . ( yC) s ( fC*)p - r ( yC) r-s~

with (Z~, C~C*~-~C . . . . ~,,~ obta in ing

A N O N C O M M U T A T I V E R E P R E S E N T A T I O N O F C L A S S I C A L D Y N A M I C S E T C .

Hence 1(;) (:) F r o m the famili~r proper t ies of destruct ion and creution operators we

obta in

c' ,z, . = [ ( ~ : - - t ) : l z , . - , ,

C*tzm = [ m! Z,,,+t,

the useful convent ion thn t l! = ~ if l ~ O. We ma,y now evuluate

This yields

27

0 , p o d d ,

(A.9) (Z, , Z"Z, ) =

: p12 ( n + ~ • p e v e n .

F r o m (A.6), (A.7) and (A.9) we m a y wri te

.=o - n~o .=o ( ~ ) ~ ( z . , Z ' z . ) .

We begin by evnluat ing the :mxiliary expression

i ~ , , ~ ~ ~ io. S t ( 0 ) = - - -: - ~ ( x ~ , z ~ r z ~ ) �9 ~:o ,~o (2r) ! (Z~' Z~Z~) r=o(~r) !

From (A.9) we have

~ ; ( ) E~n(zn, Z2rZn)= E ~ n r r (~ +,~)! n=O n=0 2r ] y ] 2 r = ( n - t - s - - r ) !

2 - 2,. 171"2 2~" . : o ~=o ( n + s - - r ) !

I t is easily verified, by the r,~tio test , t ha t the series

~ o .~o (n + , ~ z r ) !

0 , p o d d ,

(n + s) ! (n + s ~ p / 2 ) ! ' p e v e n .

2 8 J . A. C R A W F O R D

converges absolutely. We can therefore reverse the order of summation and write

2o(;) = 2 '---~ 2r I~' ]*" ~._~.~.(t-Fr)! i r! = ,:o t! 2~, 2~ ] r l ' ( 1 + ~ ) ' h - ~ ) ' + ' "

Therefore

O" 1 ( r ) [~,[2r [I+~\'T_ 1 ~ 1 (01~,1'(1+ #)]" S'(0)=.=~o(~-)!2" 2r i ~ - - ~ 1 - - - - ~ ) ' " 1 - - ~ , ~ 0 H \ 4~- -~) ] "

This converges for every real value of 0 to

We note that

Thus the series

x [OIkP(x + ~)]

Sj(1)= i i ~"(--1)" Z2,X.)

(--1)" ~: ~" (2r i~ (Z., Z"Z.)

r=O n ~ o

is absolutely convergent, and, therefore, equal to the series obtained by re- versing the order of summation, i.e. to El. Hence S~----S~(--1), or

where we have restored the subscript j, so that

7~ = (2n)-.~(aj + ivy) .

From (A.4) we see that !

(A.10) ~ (r exp [ixX] ~ r = 1-[ S, = r i = I

= ( l - - ~ ) - ' e x p [ 4(1--~)~=~

since

= (1 -- $)-J exp [

!

i=2

s ~ ( 1 - ~)J '

A N O N C O ~ M U T A T I V E REPRESENTATION OF CLASSICAL DYNAMICS ETC. 2 9

Writ ing ~ = 1 - - } , we find f rom (A.3) and (A.10) tha t

lim~}"r(r ACr) : l im~(r lim (dxzt(x) ~-Jexp [ (2=~)x2]--

= lim exp cxp [-- : ,~-+o d

= (2~) 2I [~(x) exp [~/8Jr]]~=o = (2~)21a(0) .

Theorem A.3. The set ~* of proper Weyl operators is a s tar algebra.

Le t A, B e ~ * , a constant . We need only prove 1) eA ~ * , 2) A + B ~gg*, 3) A*~9~*,

4) A*B ~ * .

Pro@ 1), 2): The proofs follow from the fact t h a t eo:(u), g(x) + fi(x) are bounded and elements of L~.

3): The existence of A* is a consequence of Theorem A.1 (25). We have

* = f d ~ ( ~ ) exp ~- ~,,,:~ = f d ~ ( - - ~) exp ~ X ~ ,

a ( - -u ) is bounded and an element of L~. Hence A*69~*.

4):

A * ~ = (.(d,,. ~.~. d,,'. d ~ ' ~ ( - - ~ , - ~)~(~', ~'). (A.11)

-exp [ i (aP + ,Q) ] exp [i(a'P + w'Q)] .

To t ransform this expression, we make use of a theorem (~) which 's ta tes t ha t

exp [i(~P + ~Q)] = exp i ~ exp [i~Q] exp [ieP],

if

[P, Q] -- (i7~)-1/.

F r o m the commuta t iv i t y of P~, Qj operators wi th different subscripts, i t follows t ha t

(A.12) exp [i(aP + ,~Q)] = exp ~ ~'~ exp [i'~Q] exp [i~P].

Taking the adjoint of this expression and changing the signs of a, % we easily

(25) ~. H. STONE: Linear Trans]ormations in Hilbert Space (New York, 1964), p. 64. (2~) W. 0. KERMACK and W. H. ~r Proc. Edinburg Math. Soc., 2, 224 (1931).

30

show that

(A.13)

J . A. C R A W F O R D

exp [iaP] exp [i'rQ] = exp [ / a x ] exp [i~rQ] exp [iaP].

Making use of (A.12) and (A.13), we then obtain

exp [i(aP + ~Q)] exp [i(a'P § ~'Q)] =

= exp ~ (ax + a 'x ' ) exp [i'cQ] exp [i~P] exp [ix'Q] exp [ia'P] =

= e x p ~ ( a x + a " ~ ' ) exp a'~' e x p [ i ( ' c + x ' ) O ] e x p [ i ( a + a ' ) P ] =

, , 1 [ i ,1 ":')1 " _1

�9 exp [i[(a + a') P -5 ('~ -5 "~') Q]] = exp ~ (a'~' - a 'x) �9

�9 exp [i[(a -5 a')P -5 (x -5 x ' )Q]] .

Subst i tu t ing this expression in the integrand of (A.11) and effecting a change of variables, we obtain

A*B =fd,,. d~,(,,, ,~) exp [i(~P -5 ,~Q)]

with

7<o,-) =fd~' d~'~(,'--~, ~'--,)~(~', ~ ' ) e x p [ ~ (a, '-- a',)] .

To show tha t A*B ~9~* we need, therefore, only prove tha t V(~, ~) is bounded and an element of L~ (this implies tha t y(a, .r)eJ~, asserting tha t A*B is a Weyl operator). We have

�9 )1 < fdo' d~'l~(,,'--,~, ~'--':)1 I~(o', ~ ') l< a~fdo' d~"l~(~', I:"(r "12' ) I ~

where al is the bound of ~(a, x). Hence y(a, x) is bounded. We have

--fd -) Ifd ' -')1< oo.

Therefore },(r Ir) s/]~. Hence A*B~gA*.

Theorem A.4. If

B =fdxfl(x) exp [ixX]

is a proper Weyl operator, it is a Hilbert-Sehmidt operator whose norm is equal

A N O N C O M M U T A T I V E R E P R E S E N T A T I O N OF CLASSICAL D Y N A M I C S E T C . 3 1

to the norm of the funct ion

= fdx f l ( • exp [ i x x ] . b(x)

Proo]. To show tha t B 6 ~ we need only show tha t its norm [IB[I is finite. F rom Le m ma A.I and Theorem A.] i t follows tha t

(A.14) [IB[] ~ = lira iIB~"lls.

We m a y wri te

r r

I t follows from Theorem A.3 t ha t C ~ B*B is a proper Weyl operator :

- - f d x 7 ( • ) exp [ixX]. C

Thus

Um [] ~ [is = lira ~ (r c v ~ r ~---~1 ~ ' 1 r

From (A.14) and Theorem A.2 it then follows tha t

lIB II ~ = ( 2 ~ ) - r ( o ) = ( ' 2 ~ ) . f d ~ l ~ ( ~ ) l s = f d ~ r b ( x ) l s = lib(*)1Is

Since b(x) cL2, []BI] is finite. The theorem is therefore proved.

Theorem A.5. If

A ----j 'd~(x) e~p [i..x]. . j'd~fl(~) exp [.;~x]

~re proper Weyl operators , then

.(A, B) -- (a, b)

with

a(x) = fdx~(x) exp [ixx], b(x) =fdxf l ( • exp [ixx] .

Proo]. This theorem is a direct consequence of the fact t ha t the mapping

a(x) -> A , b(x) --~ B

is l inear and isometric.

(A.15)

(A.16)

F rom Theorem A.4

IIA + B I12 = lla + ~ rl s ,

IIA + i~FJ 2 = fla + ibll 2.

3 2 J . A. CRAWFORD

(A.15) yields

[[A[[ ~ -k ][B[I~ ~- (A, B) -k (B, A) : Hail ~ + [[bH2 -[- (a, b) ~- (b, a).

]tence, from Theorem A.4,

(A, B) ~- (B, A) : (a, b) ~- (b, a) .

From (A.16) we deduce similarly that

(A, B)- - (B, A) : (a, b)--(b, a) .

I t follows that

(A, B) = (a, b).

A.3. The Hilbert space o] Wef t operators.

Theorem A.6. Weyl operators form a ttilbert subspace ~ of ~:. The mapping a(x) ~ L2 --> A ~ ~ , where

A : f d x ~ ( x ) exp [ixX], a(x) ----fdxa(x) exp [ixx] ,

is linear, isometric and conjugation-preserving.

Proof. Let {h.(t)} be the set of t termite functions. We have

r

[dth,(t)h ( t )= O m n m �9

We can define an orthonormal basis {,/,(x)} in s by writing

2I

• r (X) = 1 - [ hn,(~r , r : ( n l , n2, ..., n~i) .

Since Kermite functions are botmded and absolutely integrable, the func- tions ~,(x) are bounded and elements of JS[. Consequently the operators

H~ = fdxn~(x ) exp [ixX]

are proper Weyl operators. According to Theorem A.5 we then have

(H,, H,) = (2=)-j 'dxn,(x)ns(x) =

The set {H,} is therefore an orthogonal set in ~:.

A NONCOMMUTATIVE REPRESENTATION OI,' CLASSICAL DYNAMICS ETC. 3 3

Consider now operators of the form

r r

These operators must then be elements of ~;, and obviously form a Hilbert subspaee ~ or 3;. We now show tha t ~ is the space of Weyl operators. Let fl(x) be some element of L~. We can then write

r r

B ~_j dxfl(x) exp [ixX]

is then a Weyl operator. Let 95, ~p be two elements of g~. We have

(A.17)

The series

(qS, Bye) = fdx/~(x)(95, exp [ixX] y~) = f d x ~ b~(x)(r exp [ixX]y)) .

b~(x)(95, exp [ixX]~0) r

converges uniformly to 8(• exp [i• a simple consequence of the uni- form convergence of ~br~,.(x). We m a y therefore interchange the integra-

r

t ion and summation signs iu (A.17), obtaining

(r BW)= ~ brfdxvr(X)(r exp [ixX]w): ~r br(r HrW)= (95, ~ b~H~W).

Hence

B = ~ b~H~. r

This is an e lement of ![B, since ~ [br[ 2 < o o . Conversely, the operator

A = ~arRre~ r

is easily shown to be equal to the Weyl operator

f d x ( ~ a ~ , ( x ) ) e x p [ ixX].

Thus the space of Weyl operators is the Hilbert subspace ![9 of 3;. Let then

A =fdx (x)exp [ xxl, B =fd ( )exp [ixX]

3 - I I N u o v o C i m e n t o B .

34

be two Weyl operators.

where

Hence

(A.18)

where

We have then

A = ~ a ~ H ~ , B = ~ b ~ H ~ ,

r r

(A, B) --~ (2~)~s~ ~b~---- (2z)~1fdxa~)fl(x) = (a, b), r

a(x) =fdx (x) exp [ixx] e L,2 ,

b(x) =fdxfl( ) exp [ixx] e L2 .

J . A. ORAWFORD

The mapping a(x)e Z2--> A e ~9 of Z2 onto ~ is obviously linear. (A.18) then implies that this mapping is also isometric. To show further that it is conjuga- tion-preserving, we note that if

a(x) =fdx~(x) exp [ixx] --> A =fdxg(x) exp [iuX] ,

then

Theorem A.7. !~9 is a proper subspace of ~:.

The question arises: can it be that ~ , which is a Hilbert subspace of ~, is in fact identical with ~?

I t can be proved that ![9 is a proper subspace of ~. Since this fact is not utilized in this paper, a proof obtained by the author is omitted. I t consists in showing that the Hilbert-Schmidt operator M, whose only nonvanishing matrix element in the {r basis is

C ~ r M r ~ - 1 ,

and whose norm is therefore unity, has a projection onto !~t), whose norm is less than unity, equal to 2-m. Hence M is not an element of ![9.

APPENDIX B

The trace of a Weyl operator.

Let L be an operator of domain gJ whose adjoint exists, and {Z~}, (v/~} two orthonormal bases in ~3. We define the trace of Z, denoted by (Z, Z, ~),

A N O N C O M M U T A T I V E R E P R E S E N T A T I O N O F C L A S S I C A L D Y N A M I C S E T C .

to be

(Z, Z, ~) = ~ ( ~ , Z,)(X~, Z ~ ) r s

whenever this double sum converges absolutely.

Lemma B.1. I f (L, Z, YJ) is defined, then

(L, Z, ~) = (L, Z (', ~(~))

if the r ight-hand side is defined. We therefore write

Proo]. Since

35

r r

This expression was evaluated in (A.10), giving us

~ (~r exp [ixX]r ~ (1 --~)-~ exp[

If we write

<1+ 1 8~(1

8 ~ ( 1 - ~)J '

r

must converge absolutely, we have

(~s, Zr)(Z~, Z ~ ) = (~s, Z ~ ) = ~ ( ~ , Z;1))(Z; 1), Z ~ ) r r

for any orthonorinal basis {Z~I)}. Hence

(z , z, ~) = (L, Z ,1), ~ ) .

Likewise, since L* exists, if we choose the basis {~} in the domain of L* i

(yzs, Or)(~)r, L ~ ) : ~ (Z*er , Y)~)(~s, Qr) : r s r s

- = ~ ( C >, F))(z; '>, L~(?>). r s r s r s

Lemma B.2. ~', defined in Subsect. A.] of Appendix A, is a Weyl operator.

Proo]. We have

3 6 J . & . C I C A W F O R D

it is clear that ~(x)e Z~. We may therefore define the Weyt operator

R(~) ~ f d x e ( x ) exp [ixX].

Let

A exp [ixX]

be some Weyl operator. According to Theorem A.5 we have

(B.1) (R(~), A) ---- (2~)~'fdxq(x)a(x).

On the other hand~

~ow the series

Z I(~Nr exp [ixx]r r

converges uniformly in x by the Weierstrass test: indeed,

and

Hence

co f ~ n o

r n t , . . . , n I i = l i = o n=O

(~NCr, exp [ixX]r r

converges uniformly in x, and we can therefore write

Comparing with (B.1), we see that

(~N--R(~), A) = 0

---- (2z)2Jfdx~(x) ~(x) .

for every Weyl operator A: hence R(~) is the projection of ~ onto ![9. prove now that ~--~R(~)~ we need only verify that

To

(~, ~ ) ---- (R(~), R(~)).

A N O N C O M M U T A T I V E I ~ E P I C E S F N ' [ ' A T [ O N O F C L A S S I C A L D Y N A M I C S E T C . 3 7

We have

(~, ~ ) = ~ ~-~ = (~ --$~)-, , r

On the other hand,

(R(~), R(~)) = ( 2 ~ ) - dxle(x)] ~ = ( 2 ~ ) - ~ q l - ~)-~lfdu exp[- 4~(1.- ~=)J

(<~7 + ~)] =

), e x p [ - - ( x ~ + y~)] =

= (1 --~")-~ = (~, ~ ) .

Thus ~ is ~ Weyl operator and it may be written in the form

~ i i =~-)j exp L~xAj.

.Lemma B.3. If M is a ttilbert-Schmidt operator,

lim (~-", M) = Tr M

whenever Tr M is defined.

Proo]. We note that

(~, M) = ~ ~,~r(r Me, ) . r

The existence of Tr M implies that

rs rs r

converges absolutely. Hence

converges absolutely. Now

c(r Mr r

F"r(r M r Ic(r Mr c ~ 1 , 0 ~ < 1 .

By the Weierstrass test, ~ [~,(r Mr therefore converges uniformly in $ r

38

for 0 < ~ < 1 . The same is therefore t rue of

~o,(r Me,). r

Consequently,

Theorem B.1.

~:---;"1 -7 ~ ~ 1

I f

A =~d~t~(x) exp [ixX] 3

is a Weyl operator , and

then

~(x) =fdx~(~) exp [ixx],

J . A. CRAWFORD

obtaining $ = 1 - - ~ , we may express this as follows:

i a Tr A = lira $- x exp - - exp [u2/8~] a(x) ---- ~-~-o

We ma y write this relat ion formally as

Tr A = f d x ~(x) Tr exp [ixX]----- (2~)~I~(0)

obtaining the convenient formal iden t i ty

(2u) -2' Tr exp [ixX] = 5(x) ,

Tr A = f d x a ( x )

whenever Tr A is defined.

Proo]. F rom Lemma B.2, Lemma B.3 and Theorem A.5 i t follows t h a t

Tr A = lira,_,1 ( ~ ' A ) = limr ( 2 ~ ) ' f d x o (~) ~(~) =

A NONCOMMUT&TIVE Ir OF CLASSICAL DYNAMICS ]ETC. 39

t h e r i g h t - h a n d s ide b e i n g D i r a e ' s & f u n c t i o n fo r t h e v e c t o r • T h e r e l a t i o n

e x p [i(r + xQ)] e x p [i(a'P + x'Q)] =

= e x p [ i (,~x ' - a'~)]e:,p [,i[(a + a ' ) P + (-c + x') Q]] ,

d e r i v e d i n c o n n e c t i o n w i t h t h e p r o o f of T h e o r e m A . 3 , e n a b l e s us t h e n t o d e d u c e t h e m o r e g e n e r a l f o r m u b ~

(2x) -2t T r ( e x p [ i x X J e x p [ - - i x ' X J) = 6 ( x - x ' ) .

�9 R I A S S U N T O (*)

Non si 6 d imos t ra to t he il p r o g r a m m a einsteiniano di una teorin dei enmpi clnsicn in fisica 5 irrcnlizzabih' . Per6 Bell ha d imost ra to che u n a t e o r i a di quest() t ipo non po- t rebbe essere in nccordo con t u t t e 1(3 prcdizioni della mecc~nica qunntist icn. E ragionevole supporre che, in una teor'i~ del t ipo di quelle di FAnstein i cnmpi possano essere descri t t i come sistemi dinamici classici. Si comincin formubmdo le condizioni per unn r~ppre- sentnzione hello spnzio hilbert:iano delle funzioni di fase di un sistemn classico finito. Si d imost ra che unn tale rappresentazion(~ non pu6 essere commutn t iva . La rnppre- sentazione di Weyl soddisfa le condizioni r iehieste: ne l l 'Appendice A si mostrn che rappresentn lc funzioni di fase integrabil i quadra t i camen te i somet r icamente negli ope- ra tor i di Hi lber t -Schmidt . Ne setZuc the l ' immagine U dclln densi ts di probabi l i ts di fase b u n operntore st~tistieo nel senso di yon Neumnim, qunn tunque in generale esso non sin non negat ivo definito. Si cs tende allora fo rmnhnen te in rnppresentnzione di Weyl a sistemi elassici intiniti (, in si apl)liea a campi d 'ondn tensorial i rclnt ivist ici in teragent i (i e~n,pi spinoriali, non essendo a valorc unieo, presentnno d(qle diffieolt'~). Con In sceltn h --- :r ~, ques ta rapprcsentnzione soddisfn le condizioni per In quant iz- zazione cnnonicn. Idoperalove hamil toniano e gli operator i delle usuali grandezzc del cnmpo sono uguali a quelli della teor ia quant i s t ica dei campi. Per6 coinparc un ulte- riore termin(~ dell'interazion(~ dci cnmpi, t he modifica l 'equnzione di von Neumnnn per in d ipendenza di U dal tempo. Non dovrebbero sorgere esseuziali diflieolt'~ di diver- genza, poich5 non vi sono divergenze nella teoria classica dei campi. Questa rappre- sentnzione delln teorin classica dei campi ~, r ieonducibile, proprio nelh) stesso modo della teor ia quant is t iea dei cnmpi, ;~d una rappresentaz ione hello spazio delle confi- gurazioni ed alla corrispo)~dente in te rpre taz ione con particelle. Si discuteranno qucsti aspe.tti in una pubblienzionc suceessiw~..

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