IL NUOVO CIMENTO Vo]~. 22 B, N. 2 l l Agosto 1974

Foundations of Stochast ic E lec trodynamics .

I: General Formalism and the Free Radiation Field (*).

E. SANTOS

Departamento de Optica, Univers idad de Val ladol id - Val ladol id G I ~ T - Val ladol id

(rieevuto il 24 Settembre 1973; manoseritto revisionato rieevuto il 10 Aprile 1974)

Summary. - - Stochastic eleetrodynamies is classical eleetrodynamics with the hypothesis of a random background radiation in the whole space. The probabili ty distributions of the FoUrier components of this electro- magnetic radiation are assumed Gaussian. Lorentz invariance fixes the spectrum of the radiation except for a constant, measuring its intensity, which is identified with Planek's constant. A formalism is developed to deal with general stochastic problems, associating a Hilbert space with every set of random variables. Then, a mapping is defined of the algebra of continuous functions of the random variables onto the algebra of operators on the Hilbert space. A correspondence is found between probabil i ty distributions of the random variables and vectors in the t t i lbert space. The case of Gaussian random variables is considered in detail. The formalism is applied to the s tudy of the radiat ion field when some known amount of radiat ion is present besides the random back- ground. This closely resembles the usual quantizat ion of the free radiation field, although there are significant differences. In particular, not all vectors of the Hilbert space represent physical states in the present theory.

Introduction.

Q u a n t u m t h e o r y seems u n s a t i s f a c t o r y to m a n y people. I t has a n e legant

f o r m a l i s m and a v e r y good p red ic t ive power, b u t i t is no t concep tua l l y clear

(at least , for the a u t h o r of the p re sen t paper) . I n pa r t i cu la r , i t does no t p rov ide

an i n t u i t i v e p ic tu re of t he phys ica l wor]d. I n order to solve th is difficulty,

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction.

201

202 ~. sA~mos

it has been proposed the possibili ty of finding theories of hidden variables

which should describe a s u b q n a n t u m level, and should coincide with q u a n t u m theory in its predict ions (at ]east, in those which have been checked with ex- periments) . The subject of hidden variables in q u a n t u m theory has received great a t t en t ion in the last few years (1), and we do not enter into a discussion of it in this paper. Our purpose is to present a par t icular hidden-var iable theory which is, in principle, exper imenta l ly differentiable f rom q u a n t u m mechanics. There- fore, we ~re going to present a new theory, not merely a new in terpre ta t ion of q u a n t u m mechanics. (In fact , two theories which predict the same outcomes for MI possible exper iments are a single theory f rom a str ict ly physicM point of view.)

Actually, s tochastic eleetrodyn~mies is not a new theory, it is jus t el~ssieM eleetrodynamics unders tood wi thout undue restrictions, as we will show. I t has been considered b y severM authors (~), bu t a very l imited kind of problems h~s been solved till now. This seems to be due to two basic difficulties, one ma themat i eM and the other physical. The first difficulty is the lack of a ma th - em~tieM formal ism suitable to deal wi th general problems. I n fact , only the free radia t ion field, the free part icle and the harmonic oscillator h~ve been studied in some detail till now. A second difficulty seems to be the fear to de- pa r t too drast ical ly f rom quan t um mechanics. This has produced some con- fusion with the principles of the theory, which not always h~ve been established as fully classical. I n the present paper, the principles are clearly s tated, we believe. Also, a general formal ism is presented which m~y allow the solution of most problems of interest .

The outline of the paper is as follows. I n Sect. 1, the principles of the theory are presented. These ~re hard ly new, bu t t hey ~re s ta ted in a unified and precise form. I n Sect. 2, the m~themat icM formal ism is developed. Almost 911 is new here, we believe, a l though the idea of ~pplying a quantumlike formal ism to general stochastic systems has been considered presiously (3). Finally, in Sect. 3, the formal ism is applied to the s tudy of the free radia t ion field.

1. - Principles o f the theory.

Tile line of reasoning which leads to stochastic e leetrodynamics is as fol- lows. Let us assume tha t in space there are systems of charged particles (which will be called atoms) moving according to classical laws. Then, the a toms would

(1) See, for example, B. D'ESPAGNAT, Editor: ~oundations o] Quantum Mechanics (New York, N.Y., 1971). (2) A review of the field was given by M. SU]~DIN: Ann. Inst. Poincar6, 15, 203 (1971). New developments and further references can be seen in E. SANTOS: Lctt. Nuovo Cimento, 4, 497 (1972); Nuovo Cimento, 19B, 57 (]974). (3) It. G. HALL and R. E. COL~I~S: Journ. Math. Phys., 12, 100 (1971), and references therein.

]?OUI~DATIO:NS OF S T O C H A S T I C ] ~ L E C T ~ O D Y N A ~ I I C S - I 2 0 3

be radiat ing, so that a certain amount of radia t ion mus t be always present in space. The rad ia t ion will act on the atoms, and it is inconsistent to app ly classical e leetrodynamics wi thout tak ing it into account. (Another point of view to present the same idea is to say t h a t the general solution of ~axwe] l equations is the sum of a par t icular solution plus the general solution of the homogeneous equations which represents a background radiation.) To as- sume f rom the ve ry begining t h a t there is no backgTound radia t ion is a too s t rong hypothesis. Stochast ic e lectrodynamics is the theory which results when hypotheses are made about the background radia t ion much weaker t h a n to assume t h a t it does not exist. I n fact , ve ry general principles lead to the con- clusion t h a t the amoun t of radia t ion is measured b y a single parameter . Ex- per iments mus t fix the value of this parameter , a zero value being equivalent to the nonexistence oR radiation.

I t is obvious t h a t we cannot describe the assumed background radia t ion in detail, so t ha t a stat ist ical theory is needed. The electric and magnet ic fields of the radia t ion mus t be considered stochast ic processes depending cont inuously on r and t (stochastic fields). The question is to characterize these stochast ic fields. I t is convenient to s tar t wi th the potent ia l vector A wr i t ten as a Fourier series in the form

(1) A ( r , t) = ~9 -~ ~ %.a e~ exp [ik. r -- i~ot] ~- complex conjuga te , k,A

where o4 ------ elk [ and a finite volume Y2 is considered with periodic bounda ry conditions. The Coulomb gauge is used, so t h a t the electric and magnet ic fields are given b y

1 ~A (2) E ( r , t) = B ( r , t) ~- rot A

e ~ t '

The pa ramete r s ek, a become r andom variables in this theory. The problem is to fix their probabi l i ty distributions.

I n the first place we will assume t h a t the %a are s tochast ical ly independent , t ha t is, their joint probabi l i ty distr ibution is just the produc t of the probabi l i ty distr ibutions for the individual r andom variables. This assumpt ion is suggested b y the fact t ha t Maxwell equations do not couple the cka. Some coupling is provided b y the in teract ion of the radia t ion with the atoms, bu t this coupling is weak and random, so t h a t the assumpt ion of stochastic independence seems sound. Hence, it results t h a t the problem is reduced to de termining the probabi l i ty dis tr ibut ion of every r a n d o m variable. The most plausible assump- t ion is t h a t the probabi l i ty distr ibution of every ok. ~ is a Gaussian. I n fact , the central-l imit t heorem of probabi l i ty theory shows t h a t the probabi l i ty distri- but ion of a r andom variable, which is the average of m a n y others wi th a rb i t r a ry distributions, is Gaussian when some ve ry general conditions are fulfilled.

204 ~ . SAN~0S

(Similar a rguments lead, for example, to the assumpt ion t h a t the accidental errors in measurements have Gaussian distributions.) For a real variable, a Gaussian distr ibutions is character ized b y two parameters , the mean and the s tandurd deviation. For a complex variable, four are needed. I n eq. (1) it seems clear t h a t the phases of the ok. ~ are all equally probable, so t h a t only one para- mete r is needed for every r a n d o m var iable : its s tandurd deviat ion (the mean is zero for r a n d o m phases). I n this case the square of the s tandard deviat ion (the variance) is jus t the square mean of the r a n d o m variable, which we will write E[]c~.~l ~] following the convent ional nota t ion of probabi l i ty theory. The problem is to de termine these quant i t ies for all ~ and k.

At this m o m e n t we pos tu la te t h a t the stochustic propert ies of the back- ground radia t ion are the same for all inert ial observers, which makes the theory Poincar5 inva r i an t . This pos tu la te gives rise to strong restrict ions on the quuntit ies E[I%,~I 2] as we show in the following. I n fact , a s t ra ight forward calculation gives the following expression ~or the expecta t ion value of the energy densi ty of the radia t ion field:

(3) E[~] = (~/~6~ ~) Z~co~ E[t~k ~12] d~ k

where the l imit f2-+ oo has been considered. Rota t iona l invar iance implies t ha t E[]%.~I ~] depends only on the frequency, so t h a t we have

E[u] = (1 /S~ , c~)fco 2/(co) d~ k , /(o~) = E[[ek,~l~.

The propert ies of /(co) in relat ion with Lorentz t ransformat ions can be best seen if the previous equal i ty is wr i t ten in the fo rm

E[u] = (1/4~r a c~)fco a/(co) 0(co) ~ (k 2 c 2 - - oJ ~) d 3 k dco.

I n the r ight -hand side all is Lorentz invar iant except r Therefore, this quan t i ty mus t t r ans form as an energy density. This gives

/(co) ~-- const/co.

I f this constant is wr i t ten ~r~c~ for convenience, it results

(4) E[Ick,~l ~ ] = =l~c2 /co .

F r o m eqs. (3) and (4) it follows tha t the mean energy associated to each degree of f reedom of the radia t ion field equals ~o/2. I t seems obvious tha t , in order t h a t the predictions of stochastic eleetrodynamics do not contradict exper imenta l results (predicted also b y quan t u m theory) , the constant ~ mus t

: F O U N D A T I O : N S O F ST 'OC] : [ASTIC E L E C T R O : D Y N A : M [ I C S - I 20~

be identified with P lanck ' s constant. The meaning of this constant in stochast ic e lectrodynamics is clear: it measures the in tens i ty of the background radiat ion. After tha t , we have fully character ized the background radia t ion b y eq. (1) and the following probabi l i ty dis tr ibut ion for the coefficients %.~:

(5)

where ~c is the a rgument of (the complex number) ck, a. This completes the prin- ciples of stochastic eleetrodynamies. After tha t , it remains only to find ma th - ematical techniques suitable to calculate the evolution of mater ia l sys tems in teract ing with a r andom background radia t ion character ized as above. I t is impor tan t to emphasize once more t h a t the theory is fully classical. The par- ticles (if t hey are pointlike) have a well-defined posit ion and veloci ty at any t ime, and the probabil i t ies which enter the theory are only measures of our ignorance about them.

As we have established it, s tochastic e lectrodynamies has two difficulties, which might be related. The first one is the infinite e lectromagnet ic mass of charged point particles, a p rob lem which is f~miliar f rom classical electrody- namics. We will not s tudy this p rob lem for the m o m e n t and, just as a caleu- lationM rule, we will assume t h a t the particles h~ve ~n infinite negat ive me- chanics] mass so t h a t the to ta l mass is finite. The second difficulty is the infinite energy densi ty of the background radiat ion, which is an obvious consequence of eqs. (3) and (4). I n quan t um eleetrodynamies ~ similar difficulty ~ppears: the infinite energy densi ty of the vacuum. I n our derivat ion, the divergent energy densi ty is a consequence of the assumpt ion of Poinear@ invarianee. Pos- sible solutions to this problem migh t be the inclusion of g rav i ty (s tudy of the background radia t ion in the f r amework of general re la t ivi ty) or the hypothes is of a sea of particles and antiparticles. For the m o m e n t we will app ly the rule of mak ing a cut in the frequencies whenever a difficulty appears and taking the l imit ~0m~ x --> C~ at the end of the calculations. E v e r y sensible predict ion of the theory mus t be independent of the cut.

2. - M a t h e m a t i c a l f o r m u l a t i o n .

We need a formal ism to s tudy the evolution of (classical) dynamica l sto- chastic systems. E v e r y dynamica l sys tem is determined b y a set of observables or dynamica l variables, which become r andom variables in a s tochast ic theory . As a consequence, the equations of mot ion are stochastic equations, i.e. func- t ional relations between several r andom variables. Therefore, we mus t deal with the set d of functions of some set ~ of r andom variables which describe the sys tem considered. I n order to define these sets more precisely, let us s tar t with a set ~ ~ {X1, ..., X~} of re~l (i,e. whose range is a subset of ~ ) r~ndom

2 0 6 ~ . s ~ N m o s

variables, which are assumed funct ional ly independent (i.e. there is no relat ion of the form ](X~, ..., X , ) = 0). We do not assume for the momen t t h a t they are s tochast ical ly independent (i.e. the probabi l i ty t ha t X~ takes the value xj and X~ the value x~ m a y not be equal to the produc t of the probabili t ies of the individual events). The set ~ / o f functions of X~, ..., X~ can be endowed with a convenient s t ructure if we restr ict d to be the set of continuous functions vanishing at infinity. I n fact , the set ~/ so defined has the s t ructure of an Abelian (commutat ive) C*-algebra. This is a par t icular case of the well-known result (4) tha t the set of continuous functions vanishing at infinity of a var iable defined in a locally compact space is an Abelian C*-algebra. (Note t ha t any set of n real r a n d o m variables is equivalent to ~ single r andom variable whose range is a subset of ~ . )

The r andom variables X~ ~ N are character ized b y their probabi l i ty distri- butions. These allow the calculation of the expecta t ion value of any r andom variable ] ~ J , which will be denoted b y

(6) ( / ) =_ E l i (X , , ..., x~ ) ] .

We assume wi thout proof t h a t the reciprocal is also true, i.e. t h a t the expecta t ion values of all functions ] ~ d fully determine the probabi l i ty distributions. I t is easy to see t ha t the expecta t ion value defines a linear, normalized, posi t ive funct ional (i.e. a (( s ta te ~)) over the C*-algebra ~/. I n part icular , normalizat ion is equivalent to the s t a t ement t h a t the stun of all probabili t ies is unity. As a conclusion we have the following result: to give a set ~ of r andom variables is equivalent to give a s ta te over an Abelian C*-algebra ~/.

After tha t , the existence of the quan t um formal ism guides us to search for representat ions of the C*-algebra ~4 in a Hi lber t space. Indeed, it is well known (5) t h a t the G.N.S. (Gelfand-Naimark-Segal) construct ion associates wi th every C*-algebra ~ / an isometrical ly *-isomorphic algebra of bounded operators in some t I i lber t space 54f. Fur thermore , each s ta te E[ ] over the al- gebra ~ is associated with a cyclic normalized vector ]~} ~ in such a way

t h a t

(7) E[]] = (~o]f lp) , Y]~ ~ ,

where f ~ d / / i s the representa t ive of ] ~ d , and J / / i s the set of all bounded op- erators on ~f'. Nevertheless, the correspondence given b y the G.N.S. construc- t ion is not the one interest ing for us. This is a guess mo t iva t ed b y the fact t ha t we are searching for a formal ism similar to the q u a n t u m one. Then, noncom-

(4) See, ~or example, O. E. LANFORD I I I : in Statistical Mechanics and Quantum Pield Theory, edited by C. D~WITT and R. STORA (New York, N.Y., 1971), p. 141. (5) See reL (~), p. 159.

F O U I ~ D A T I O : N S O F S T O C I : f A S T I C E L E G T I ~ O D u - I 207

mut ing operators m a y appear in it, whereas the G.~I.S. construct ion will as- sociate commut ing operators of ~ with the elements of the Abelian Ca-alge - bra ~r Consequently, we define another correspondence between zr and J / as follows.

We are searching for a suitable mapp ing of ~ onto J / fulfilling eq. (7). As the lef t-hand side of eq. (7) defines a linear funct ional over zr and the right- hand side a linear funct ional over ~//, we assume t h a t the mapp ing of d onto is linear. I t is also plausible to postula te t ha t the complex conjugation in zr corresponds to g e r m i t i a n conjugat ion in ~/ . After this, the algebraic corre- spondence would be fully defined if we s tate which operat ion in ~ / corresponds to the product in zr Obviously, it cannot be the product in J [ , which is not commuta t ive in general. (Incidental ly, we point out t h a t there is a similar p rob lem in q u a n t u m mechanics, which is unsolved. For example, if ~ is the opera tor associated with the co-ordinate z of a particle, and/3 the one associated with its m o m e n t u m p, quan t um mechanics does not say which is the op- era tor associated with the var iable x~p~.) I n order to solve the problem, let us assume t h a t 2~, ..., ~ e J / are the operators associated with the r a n d o m variables X~, ..., X , c ~ c ~r Then, we mus t determine the opera tor of d / which is the image of each monomia l like X~ ~ X~ ~ ~ ... X , . I n fact , once this

correspondence is given, the linear p rope r ty of the mapp ing d - + ~ ' would fix the opera tor associated with every polynomial of X~, ..., X~. Fur the rmore , we should assume wi thout proof t ha t this would fix complete ly the mapp ing of ~r onto ~ / .

As a simple example, consider the monomia l ] ~ X ~, X ~ ~ . I f 2 is the image of X, it seems plausible to assume tha t the image of ] is f ~ 2~. I f we write ] in te rms of the new variables {171, Y~}, whose sum is X, then we have

I t s associated operator is wri t ten, in te rms of {?~, ?)2} (which are the images of {Y~, Y~}), as follows:

f _ _ ~ 3 A 2 A A A A A A 2 A A S A A A A 2 ~ *4 3

y~ + (yly~ + y~y~y~ + y2yl) + (y~y~ + y~ylY~ + Y~Y~) + Y~ �9

A 2 ~ A ~ A ~ 4 2 Then, it is plausible to associate the opera tor (yly2~y~y2y~+y~yl)/3 with the var iable 37~ Y2, and so on. This leads to postula te the correspondence

(s) A~ 1 A/r n x ~ ... x ~ o - + ~(xl ... ~ ) ,

where S (symmetrizer) means the average of kl t imes ~1, ..., kn t imes ~. , re- a r ranged in all possible orderings. I t can be realized t ha t this is the only cor- respondenee which is conserved b y a linear change f rom the basic set ~

{X1, ..., Xn} to another one ~ ' ~ {Yl , . . . , Y~}.

208 ]~. SANTOS

Up to now, the Hi lber t space 5C has not been defined, and it can be chosen at will, provided t ha t eq. (7) is fulfilled. The choice clearly depends on the probabi l i ty distr ibutions of the r andom variables Xj ~ ~ . Actually, the Hi lber t - space formal ism which we are developing m a y be useful for some r andom variables and useless for some others. I n the following we s tudy the mos t im- po r t an t case, which arises when the r andom variables have independent Gaus- sian probabi l i ty distributions.

To begin with, let us consider a set ~ ~ {Q, P} of two Gaussian r andom variables, which are stochastical ly independent . Wi thou t loss of generality, we m a y assume tha t thei r probabi l i ty distr ibutions have zero mean and a var iance of one-half. ( I f this is not the case, we m a y change f rom Q to the new variable (Q - <Q>)/(2<Q ~> - 2<Q>~) ~, and similarly for P. ) I t is convenient to introduce a single complex r a n d o m variable A such t h a t

(9) A ---- (Q ~- i P ) / v ~ .

I t is easy to show tha t the new variable has a r andom phase and a var iance of one-half. I t s probabi l i ty dis tr ibut ion is therefore

(10) dP(A) = (2/~) exp [-- 21A]~ ] IAI diAl d~ , T ~ arg A.

I n order to find the Hi lber t space J~ which is suitable to represent the set ag of functions of Q and P, let us calculate f rom eq. (10) the expecta t ion value of the simple funct ion A ' A *~, which gives

(n) E[A~A*q =_fArA*~dP = (r!/2 ~) c~r,,

where 6,~ is the Kronecker symbol. At this m o m e n t we mus t point out tha t , s t r ict ly speaking, A * A *~ does not belong to d because it is a funct ion which does not vanish at infinity. Wi th some lack of rigour (which is familiar f rom Inost t r ea tcment s of q u a n t u m theory) we will include in ~r all continuous functions of Q and P , and in ~//[ all (even nonbounded) linear operators on ~ . Then, the postulates s ta ted till now (in par t icular eqs. (7) and (8)) imply t ha t

(12) <Ola(#r = (r!/20 a,o,

where / 0> e W is the vector associated to the given probabi l i ty distr ibution of the r a n d o m variables.

The suitable representa t ion space is found as follows. Consider first the case r = s; i t can be realized t h a t r! is just the number of pe rmuta t ions of r objects. For each t e r m in S(drdt*), r! is also %he number of ways in which the r op- erators d can be paired with the r operators d t. For instance, in a t e r m like

X~OIY:NDALTIONS OF S T O C H A S T I C E L E C T ] ~ O D Y N A M I C S - I 2 0 ~

d 2 d ~ the operators can be paired in the following 2 ! : 2 ways:

d d d * ~ , a a d a .

Let us give a numerical value to every paired term in such a way tha t the bracket <0]M]0> equals the sum of the values of the r! paired terms which can be obtained from it, M being one of the ordered products of r times d and r

times d t. Equat ion (12) will be fulfilled if the average value of all the paired terms is 1/2 r. I n fact, in this case the left-hand side of eq. (12) willbe an average

of ordered products, each one being the sum of r! paired terms with average value 1/2 r. A procedure to obtain an average value 1/2 r for the paired terms is

to give the value 1 to every paired term in which the operator d is to the left of the corresponding operator d t in every pair, and the value 0 to all other

paired terms. I t is easy to see tha t only one in every 2 ~ paired terms survives. l~or example, for r = 2, only three nonvanishing paired terms exist and we have

<O[S(d~d~)]O> = (d d d r -t- r ~ d d t a a~)/6 ,

]n agreement with eq. (12). These combinatorial rules have been inspired by the usual calculational rules of quan tum field theory, based on normal ordering,

Wick 's theorem and so on. This stresses our idea tha t quan tum field theory m a y be just a theory of stochastic (classical) fields, whose deep physical meaning

has been hidden up to now. In order to simplify the rules established above, we realize tha t the value of

any bracket tha t begins with d ~ or ends with d is zero because in all paired terms

which can be obtained from it there is at least a pair with the wrong order

between d and d +. So, we will write

(~3) dlO> = o, < o [ ~ = o,

where the 0 of the r ight-hand side means the null vector of the Hilbert space.

This defines the vector ]0} c J((. I n the second place, it is easy to see tha t

(14)

and _~ being arbi t rary ordered products of d and d t. I n fact, the nonzero

paired terms on the left-hand side are the same which appear in the first bracket

on the right plus those in which the product dd+ is paired. But these are as m a n y as the nonzero paired terms of the~last bracket. As eq. (14) holds for

any 2~ and ~ , we can write

(15) [d, d t] =_ dd t - - d t d = 1 .

210 ~. sA~mos

After that , we realize tha t eqs. (13) ~nd (114) imply tha t eq. (12) holds also for r ~= s. In fact, in this case an operator d can be carried to the right or an op- erator d t to the left by repeated use of eq. (15) and every braneket vanishes. Therefore, eqs. (13) and (15) fully characterize the random variable A as h~ving the probabil i ty distribution eq. (10). ( I t is obvious that , instead of eqs. (113) and (15), we might have postulated the following ones:

(16) d*[o>= o, <old= o, [a +, d]= 1.

This change corresponds to the charge conjugation of quan tum field theory. ) I f we consider the algebra generated by the operators d and d t, which fulfil

eq. (15), as the algebra of linear operators on a t t i lber t space 3/z, this space is isomorphic with the space of complex square-integrable functions of a single real variable, a result well known from quan tum mechanics. (Remember tha t , if we label ~ and p the t t e rmi t ian operators associated with the random variables Q and P, eq. (15) implies [q, t3] = i.) As a summary we have shown tha t our postulates about the mapping ~ r (in particular, l inearity and eq. (8)) plus eqs. (9), (13) and (15) characterize the r~ndom variables Q and P as having a joint probabil i ty distribution which is the product of two independent Gaussian distributions. In quan tum mechanics it is well known tha t eqs. (9), (13) and (15) imply tha t either Q or P have separate Gaussian probabil i ty distributions, bu t a joint distribution does not exist for observables represented b y noncommuting operators. I t can be realized tha t a subtle bu t significant difference exists between quan tum theory and the present formalism.

The generalization to more than two basic variables is s traightforward. Let us assume tha t we have the set ~ ~_ {Q1, P~, ..., Q,, P ,} , where all Qj and P~. are stochastically independent Gaussian random variables with zero mean and variance one-half. We introduce the complex variables Aj related to the previous ones by equations similar to eq. (9). Then, it is not difficult to see tha t the stochastic independence implies tha t

(17) E[f~(A~, A,)/j(A~, Aj)] = E[],(A,, A,)]E[/~(Aj, A~)],

where ]~ is an arb i t rary continuous funct ion of A~ and A~ (or Q~ and P,) . This suggests to postulate tha t the Hilber t space J/f, associated with the set ~ of random variables, is the tensor product of the spaces ~ , each one associated with a complex random variable Aj. This means tha t the representat ion of the algebra d of continuous functions of Q1, P1, ..., Qn, P . is determined b y the representations of the algebras ~/j c ~u/of functions of Qj, JPj. As a conse- quence of the postulate, the operators (on 5r ~) d~ and d*j commute with dj and d*s. Then, all ~he results obtained up to now can be summarized in the equations

~ d r ~ (18) [ai, dj] [ , , d~] = O, [d,, aj] = a . ,

F O U N D A T I O N S O1 ~ S T O C I I A S T I Q : E L E C T R O ] ) u - I 2 1 1

which fix the s tructure of the algebra Jff (or the Hilber t space g/f) and

(19) d~lO> = O, <O]d~ = O,

which fix the ket 10> and the bra <0[. These equations, plus the eorrespondence aft-+ J r , fully characterize the set of random variables A; as a set of stocha- stically independent random variables with probabi l i ty distributions given

by eq. (10).

3. - Appl icat ion to the free radiat ion field.

The application of the precedent formalism to the free electromagnetic radiat ion is straightforward. (Provided tha t we assume tha t the results ob- tained up to now can be extended to the ease in which the set ~ of basic random variables is not finite.) All postulates of Sect. 1 about the random background radiat ion can be summarized associating to every random variable ek. x of eq. (1) the operator

(20) dk.~ = (2~c~I~ ~ dk,~,

and assuming tha t the operators dk. ~ fulfil eqs. (18) and (19) (with j ~ {k, ~}). I t is also easy to s tudy the free radiat ion field when some amount of ra-

diation is present in space besides the random background. For example, let us assume tha t the additional radiat ion is a plane wave, determined by the potent ial vector

(21) A(r , t) = (2~e~/Y2~oo)�89 ~~ cos (ko" r - foot@ ~),

where the real numbers b and ~ measure the ampli tude and the phase of the wave, and the constant before b is a convenient dimensional factor. The coef- ficients ck. ~ of the Fourier expansion of the radiat ion field are Gaussian random variables with variance given by eq. (4), Also, the mean of every random va- riable is zero except for %0.;~o, which has a mean equal to

(22) E[%0.~o] = (2z~c~/(Oo)�89 exp [ib].

This means tha t the random variable %~ is the sum of a random variable with probabil i ty distr ibution given by eq. (5) plus the constant given by eq. (22). Then, we associate a set of operators {dka } (fulfilling eqs. (18) and (19)) with the random variables %.x by means o f eqs. (20), exept for %~ whose associated operator will be

(23) Cko.~o= (27d~e2/e~189 @ b exp [i~]).

212 m. s ~ T o s

This fully characterizes the radia t ion field as a r a n d o m background radia t ion (defined in Sect. 1) plus a plane wave (given b y eq. (21)).

An a l te rnat ive way to character ize this radia t ion field is to change f rom the set of operators {dk,a} to the new set {d'~.a} such tha t

A? A A I

(24) ak,~= a~,~ , %o,~o= ak.,~~ @ b exp [id].

All p r imed operators fulfil eqs. (18) and (20), bu t instead of eqs. (19) we have now

(25) ~' ~' %,~]0} = 0 if k r ko and ~ # 2o, a~..~.lr ---- b exp [i~] ]0>,

and the conjugated of these equations. The vector ]r is the same one defined in eqs. (19), relabelled for convenience. I t is also possible to define a new vector ]0'} such tha t , for all k and X,

(19') - la~,~ = ak,~[O > = O .

:Now, we m a y say t h a t eqs. (18) and (20) hold for any s ta te of the radia t ion field, whilst the vector 10> (fulfilling eqs. (19)) represents the pure background ra- diat ion and the vector [r (sat isfying eqs. (25)) represents the r andom back- ground plus a plane wave. I n this way the states of the radia t ion field are character ized b y vectors of the t I i lber t space, as in q u a n t u m theory. Equa- tions (25) are formal ly identical to those defining the coherent states of the radia t ion field in q u a n t u m theory (6).

I n order to generalize, we define the pure states of the radia t ion field in stochastic e lectrodynamics as those in which there is a r andom background r ad i a t i on - - a s s ta ted in Sect. 1 - -p lus a per/ectly known amoun t of radiat ion. Then, this given radia t ion can be wr i t ten as a superposi t ion of p lane waves such as

(26) A(r , t) = 2(2~c2/~2) ~ ~ bk, z E x cos ( k . r -- cot + ~,~).

The propert ies of the field when there is a r andom background plus a known radiat ion, represented b y eq. (26), are s ta ted b y associating an operator dk, ~ with each r andom var iable ck. ~ b y means of eq. (20), assuming t h a t these op- erators fulfil eqs. (18) and postula t ing t h a t the vector ]0} ~ jig, which represents the state, fulfils the equations

(27) dk,~lr = bk. ~ exp [ia~,~] ]r

{~) See, ior example, S. ST]~HOLM: Phys..Lett., 60, 1 (1973), for a st~ldy o~ coherent states.

F O V N D A T I O N S O F S T O C H A S T I C E L E C T R O I ) u - I 213

a n d i t s c o n j u g a t e d ones. Hence , we see t h a t p u r e s t a t e s of t h e f ree r a d i a t i o n

f ield in s t o c h a s t i c e l e e t r o d y n a m i c s a re q u i t e s imi la r to t h e cohe ren t s t a t e s of

q u a n t u m e l e e t r o d y n a m i c s . W e see t h a t in s t oc ha s t i c e l e e t r o d y n a m i c s t h e r e is

no r o o m for p u r e s t a t e s wh ich are n o n e o h e r e n t . I n p a r t i c u l a r , t h e v e c t o r s of

w h i c h in q u a n t u m e l e c t r o d y n a m i c s a re a s s o c i a t e d w i t h one p h o t o n , t w o p h o t o n s ,

etc. do n o t r e p r e s e n t p h y s i c a l s t a t e s in s t ochas t i c e l e c t r o d y n a m i e s . I n o t h e r

words , each pu re s t a t e is r e p r e s e n t e d b y a v e c t o r in 2 , b u t n o t al l v e c t o r s in 24 ~

do r e p r e s e n t p h y s i c a l s t a tes . Never theless~ t h o s e wh ich r e p r e s e n t s t a t e s ( the

c o h e r e n t s t a t e vec to r s ) f o r m an o v e r - c o m p l e t e set~ as is wel l k n o w n f r o m q u a n t u m

t h e o r y . I n s t o c h a s t i c e l e c t r o d y n a m i c s i t is also poss ib le to def ine m i x e d s t a t e s

as t h o s e in wh ich t h e r e is some i m p e r ] e c t l y k n o w n a m o u n t of r a d i a t i o n bes ides

t h e r a n d o m b a c k g r o u n d . These s t a t e s m a y b e a s s o c i a t e d to d e n s i t y m a t r i c e s

as in q u a n t u m t h e o r y .

I t is n o t e a sy to see w h e t h e r t h e p re sen t , p u r e l y class ical , t h e o r y m a y a]low

t h e i n t e r p r e t a t i o n of e x p e r i m e n t a l f ac t s u s u a l l y a s s o c i a t e d to p h o t o n s . I n

s t oc has t i c e l e e t r o d y n a m i c s , p h o t o n s are p u r e l y m a t h e m a t i c a l o b j e c t s u sed for

t h e c lass i f i ca t ion of vec to r s in t h e H i l b e r t space a s s o c i a t e d w i t h t h e r a d i a t i o n

field. I n fac t , t h e H i l b e r t space can be c o n s t r u c t e d as a F o e k space , w h i c h is

t h e s u m of t h e H i l b e r t spaces of t h e s t a t e s w i t h zero p h o t o n s ( the v a c u u m ) ,

one p h o t o n , etc. As q u a n t u m t h e o r y has a c c u s t o m e d us to s p e a k a b o u t p h o t o n s ,

i t seems no t e a sy to i n t e r p r e t t h e e x p e r i m e n t s w i t h a p u r e l y c lass ica l t h e o r y

wh ich exc ludes t h e m . This p o i n t wil l be d i scussed in d e t a i l in s u b s e q u e n t

pape r s . Also, t h e e v o l u t i o n of s y s t e m s of c h a r g e d pa r t i c l e s a c c o r d i n g to sto-

chas t i c e l e c t r o d y n a m i e s wil l be cons ide r ed e lsewhere .

�9 R I A S S U N T 0 (*)

L 'e le t t rodinamiea stoeastica ~ l ' e le t t rodinamiea elassiea con l ' aggiunta dell ' ipotesi di una radiazione di fondo easuale in tn t to lo spazio. Si suppone ehe le distr ibuzioni di probabil i ts delle eomponenti di Fourier di questa radiazione elet tromagnetiea siano ganssiane. L ' invar ianza di Lorentz determina lo spettro della radiazione a meno di una eostante, ehe ne misnra l'in~ensit&, ehe si identifiea con la eostante di Plank. Si svi lappa nn proeedimen~o formale per t r a t t a r e problemi stoeast iei generali, assoeiando uno spazio di Hilbert a ogni insieme di var iabi l i easuMi. Poi si istitnisee nna maiopa dell 'algebra delle funzioni continue delle var iabi l i eausali sull 'algebra degli operator i hello spazio di Hilbert. Si t rova una eorrispondenza, fra le dis tr ibuzioni di probabil i t~ ed i ve t tor i nello spazio di Hilbert . Si analizza de t tagl ia tamente it easo di var iabi l i ganssiane. Si appliea il proeedimento allo studio del eampo radia t ive quando sia pre- sente una quantit& nora di radiazione altre al sottofondo cansale. Cib somiglia s tret- t amente alla quantizzazione solita del campo delle radiazioni libere, benehg ei siano differenze signifieative. In part ieolare, in quests teoria non tu t t i i ve t tor i dello spazio di Hilbert rappresentano s ta t i fisiei.

(*) Traduz ione a cura della Redazione .

14 - II Nuovo Cimento B.

214 ~. s ~ o s

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