B 2 0 2 G R I B O V , P O M E R A N C H U K , A N D T E R - M A R T I R O S Y A N

in Eq. (B3). The contours CV and C2' (Figs. 16 and 17) differ from those represented in Figs. 2 and 3 in that they are continued to oo.

The function F(t,h,fa) has singularities at tll2—t^12

+fa1/2, h=0, fa=0. If Re/21/2>Re*1/2, the singularity

ilm==tii2_l2i/2 0f th e function F{t,thfa) is absent on the physical sheet of the plane fa represented in Fig. 16 since the point h for which Re/i1/2<0 lies below the cut made in Fig. 16 left of the singular point h=0.

Therefore at Re/1/2 < Re/21/2 the singularity of the

integral over fa:

1 1 y F(t;thfa) <l>3\tM = / dh

2 2iJCl>j+l-a(t1)-a(t2)

arises only for such j , t, and fa for which the zero of the denominator appearing across the cut h>^n2 and de­forming the contour C/ of integration coincides with the

I. INTRODUCTION

USING the methods of quantum field theory, we shall compute the characteristic functional for an

electromagnetic field in thermal equilibrium within an enclosure of arbitrary size and shape. From this func­tional, we shall compute the moments or correlation functions and the probability distributions for any number of field components at the same or different points in space-time.1 We shall see that the probability distribution is a multivariate Gaussian function. There­fore, all correlation functions are expressible in terms of the two point correlation function. To exemplify the result, we shall explicitly calculate this correlation function for an unbounded domain and for a semi-infinite domain bounded by a perfectly conducting plane. For the unbounded domain our results agree with

* This research was supported by the U. S. Office of Naval Re­search, under Contract No. NONR 285-(48).

1 Of course, the distribution functions are physically meaningful only when they refer to points at which the field components commute. For the electric and magnetic field components, this means that no two points lie on the same light cone.

points h= 0. This singularity, given by the condition

j+l = a(fa)+a(0),

i.e., j=a(fa), appears on the cut of the plane fa, deforms the contour (as is indicated in Fig. 17) and reaching the line Re/2

1/2=Re/1/2 does not lead to the singularity of integral (B2)

//(*) = - f ^(t,fa)dfa. (B4) 2tJc2>

This means that the singularity of this integral arises only from the region of small (or complex) fa, fa<t- Since / is small the quantity h1/2=tll2—fa112 is also small (or if it is not, it is complex). In either case the particle masses cannot enter into the expression giving the loca­tion of the singularity. Actually the singularity of inte­gral (B2), (B4) arises from the point h=fa=t/^.

those of Sarfatt,2 Bourret,3 and Mehta and Wolf.4

The correlation functions for a semi-infinite domain do not seem to have been calculated previously.

The deduction of the Gaussian distribution functions for black-body radiation in an unbounded domain has already been given by Glauber5,6 and Holliday.7 These distribution functions were used implicitly by Purcell8

and explicitly by Mandel and Wolf9 in order to analyze the intensity interferometry experiments of Hanbury-

2 J. Sarfatt, Nuovo Cimento 27, 1119 (1963). 3 R. C. Bourret, Nuovo Cimento 18, 347 (1960). 4 C . L. Mehta and E. Wolf, Phys. Rev. 134, A1143, A1149

(1964). 5 R. J. Glauber, Phys. Rev. Letters 13, 84 (1963); Phys. Rev.

130, 2529 (1963); Quantum Optics and Electronics: The 1964 Les H ouches Lectures, edited by C. DeWitt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, Science Publishers, Inc., New York, 1965).

6 R. J. Glauber, Phys. Rev. 131, 2766 (1963). 7 D. Holliday, Phys. Letters 8, 250 (1964); see also D. Holliday

and M. L. Sage, Ann. Phys. (N. Y.) 29, 125 (1964). 8 E. M. Purcell, Nature 178, 1449 (1956). 9 L. Mandel and E. Wolf, Phys. Rev. 124, 1696 (1961).

P H Y S I C A L R E V I E W V O L U M E 1 3 9 , N U M B E R I B 12 J U L Y 1 9 6 5

Statistics of the Thermal Radiation Field*

EVELYN FOX KELLER

Courant Institute of Mathematical Sciences, New York University, New York, New York (Received 8 December 1964; revised manuscript received 27 January 1965)

The characteristic functional is calculated for a system of bosons obeying linear field equations. The system is assumed to be in equilibrium, and the density matrix is taken to be of the form {{n} \p\{m}) = UK dnKmK(l —zK)zK

nK, where K labels the individual modes. From the characteristic functional, the moments and distribution functions of an arbitrary number of field components are derived. In addition, it is shown how to obtain the density matrix from the characteristic functional, and, for the system in question, the original density matrix is recovered. Explicit calculations are performed for the electromagnetic field in an unbounded domain and in a semi-infinite domain bounded by a perfectly conducting plane.

S T A T I S T I C S O F R A D I A T I O N F I E L D B203

Brown and Twiss.10 Glauber5 has also obtained the probability distribution for the number of photons counted by a detector.

Although the analysis described so far refers to the electromagnetic field, the same considerations apply to any boson field governed by linear equations. To illustrate this, we shall calculate also the characteristic functional and the two point correlation function for a scalar meson field in thermal equilibrium.

We next solve the inverse problem of determining the density matrix for a field when its characteristic functional is known. For this purpose, we make use of the representation developed by Glauber.6 In the case of a characteristic functional which leads to Gaussian distribution functions, we shall show that the density matrix has the form of that for black-body radiation.

Our results shed some light on a question which has been raised concerning the applicability of classical physics to random electromagnetic fields. We know that all the statistical information concerning a field is contained in the characteristic functional. From this functional, both the quantum-mechanical density matrix and the probability distributions of the field components can be determined. Therefore, two com­pletely . equivalent descriptions of a random field can be given—one in terms of a quantum-mechanical density matrix, and the other in terms of a set of probability distributions for field components which may be regarded as classical quantities.11 The two descriptions will give identical results for all quantities provided that they both correspond to the same charac­teristic functional. Of course, the correct probability distributions cannot be determined classically, but once they are determined, they can be used without any further reference to the quantum-mechanical nature of the fields.

II. DEFINITION OF THE CHARACTERISTIC FUNCTIONAL

The electromagnetic field in a domain V devoid of sources can be described in terms of a Hermitian vector potential operator A(r,£). Within V, A satisfies the wave equation

( V 2 - c - W ) A = 0 . (la)

In addition, some appropriate subsidiary condition is required to fix the gauge of A, such as

V-A=0 , (lb)

the transversality condition. On the boundary of V, we assume that A satisfies a real linear homogeneous boundary condition which makes the Laplacian operator Hermitian. From A the electric and magnetic field operators E(r,0, and B(r,£) can be obtained via the

10 R. Hanbury-Brown and R. Q. Twiss, Nature 177, 27 (1956); Proc. Roy. Soc. (London) A242, 300 (1957); A243, 291 (1957).

11 See footnote 1.

relations E=- i r 1 d«A, (2)

B = V X A . (3)

I t is convenient to introduce the product solutions of (1) of the form VK(x) = uK(r)eiat. From (1) it follows that uK (r) satisfies the reduced wave equation

(V 2+c-W)uK ( r ) = 0. (4a)

The functions uK must also satisfy the transversality condition

V-u,(r) = 0. (4b)

If uK satisfies the same boundary condition as A, then uK is an eigenfunction of the domain and coK, the corresponding eigenvalue, is real. The index K labels the various eigenfunctions and corresponding eigen­values. When V is bounded the eigenvalues are discrete and the index K may be restricted to discrete values. Unbounded domains are covered by our treatment if we interpret K as a continuous index, and understand that summation over K means integration. Alternatively, we shall consider V to be bounded and pass to the limit of an unbounded domain in our final result. Since the equations and boundary conditions for uK are real, it follows that uK*, the complex conjugate of uK, is also an eigenfunction corresponding to the same eigenvalue coK. We assume that the uK have been chosen to satisfy the orthonormality conditions

J u ,*(r) -u , , ( r )dr=5„, . (5) J v

By utilizing a complete set of uK(r) we can express A in the form

X O J I , (r)er-™«'+ aJuK* (r)e*«««]. (6)

The coefficient aK and its Hermitian adjoint a J are the annihilation and creation operators for photons of the /rth mode of the field. We assume that they satisfy the familiar commutation relations for independent harmonic oscillators,

[aK ,<v]= [_aK\a<K>f] = 0 , (7)

[_aK,aK^~\=8KKr. (8)

From (6)-(8) the commutation relations of the com­ponents of A at two space-time points can be found.

The statistical description of any quantum-mechani­cal system, such as the electromagnetic field in V, is expressible in terms of a density operator p. In terms of p the expectation value (0) of any operator 0 is given by

<0)=tr(pO). (9)

To find the expectation of A or of any function of A it is convenient to introduce the characteristic functional

B204 EVELYN FOX KELLER

F M defined by

Fl\\ = /exJi ^x)-A(x)dx)\. (10)

Here x= (t,t) is a four vector, ^ is an arbitrary real vector function and the integration extends over the domain r in V and — <*></< oo.

Upon utilizing (6) for A the integral in (10) can be written in the form

[l(x)-A(x)dx=cz(—) [ X A + X . W ] . (11)

The coefficient A* in (11) is defined by

XK= / ^ ( ^ ) - u K ( r ) g - ^ ^ . (12)

When (11) is used in (10) and the commutation relation (7) is recalled, (10) can be written as

^ M = ( I I exp[^(V2coK)1/2(X.aK+XK*a«t)]). (13) K

To proceed further we must specify the density operator. For thermal equilibrium, the density operator takes

the form

p = e - ^ / t n r ^ , (14a)

where the Hamiltonian H is given by

# = Z i%uK(aKaKf+aK

1[aK) = Y, ^ K ( w K + i ) . (14b) K K

Here nK=aJaK is the number operator for the /cth mode and (3=l/kT, where k is Boltzmann's constant and T is absolute temperature. I t follows from (14) that

into (18) we obtain

^ M = I Iexp-{ ( ; 2 (V2co K ) |X , | 2 K+i)} K

= e x p { - i : c2(V2ay)|XK|2<»«+!>}. (19) K

From the expansion of A(x) according to Eq. (6), we notice that

<il<(^Mi(y)>=E^(V2«0{^(*)vy*(y)<n1c+l>

By using (20) and the definition (12) of A K we have, in dyadic notation,

\fo-•)X(y):(A(x)A.(y))dxdy

=E|x,|v. 0-+i }. (21)

P=HPK, (15)

with

p K =exp[—^co K (^ K +| ) ] / t rexp[—^co, (w K +| ) ] . (16)

More generally, let us consider operators with matrix elements of the form

{pt)nKmK=^nKmK{^~Z>)ZKVK y (17)

where zK is a scalar function of K. I t is clear that Z^K,pK

fl=0 and [aKt,pK/] = 0 for H^K'. For density operators with this property, (13) becomes

FLX^mexp{ic(fi/2^K)^(XKaK+\^a;)}). (18)

The expectation value in (18) is just the characteristic function for a single mode K and this is given by Bloch's theorem.12 When we insert this characteristic function

12 F. Bloch, Z. Physik 74, 295 (1932).

Upon making use of (21), we can rewrite equation (19) in the form

^ [ X ] = e x p J — U(x)My):(A(x)A(y))dxdy\ . (22)

This is our general result for the characteristic func­tional of an electromagnetic field described by a density matrix of the form (15) with the pK satisfying (17). In particular it applies to a field in thermal equilibrium for which p is given by (14).

III. MOMENTS OF THE RADIATION FIELD

Once the characteristic functional is known, the calculation of all moments and distribution functions is straightforward. The only difficulty arises from the noncommutativity of the field operators at different points in space-time. From the commutation relations imposed on the operators aK and aK\ it is possible to compute the commutator [Ai(x),Aj(y)~]. When the field satisfies the transversality condition (1) it is found that the commutator vanishes only for time-like pairs of points x and y. (See, e.g., Heitler.13) The electric-and magnetic-field components commute more gener­ally, but still do not commute for pairs of points lying on a light cone.

Nevertheless, we can define an nth-order moment as follows:

Inil'"'tin{xh'"Xn)

1 = - E (Ail(xd-~Ai.(xn)). (23)

The summation in (23) extends over the set P (x\ • • • xn) of all permutations of X\ * Xn* For a set of points

13 W. Heitler, Quantum Theory of Radiation (Clarendon Press, Oxford, England, 1954), p. 405.

S T A T I S T I C S O F R A D I A T I O N F I E L D B205

{%i} such that [A(xi),A(%j)~] = 0 for all points in the set, (27) reduces to the usual ^th-order moment

For an arbitrary set of points {xi}, it follows from the definition of F[\\ that In can be obtained by taking the wth-order functional derivative of F[\]- That is

1 n ' v^lj * * * %n) ^

SWF[X]

Skiiixi)" -d\in(xn) (24)

X(z)=0

When F[\] is given by (22), we find by using (24) that all moments for n> 2 can be expressed in terms of

h'^KiM^Ajiy^+iA^Aiix))-]. (25)

All odd moments vanish and the even moments (i.e., n even) are found to be given by

= 2^ 1 1 2\Aia\Xa)Aiy[Xy) partitions pairs

+Aiy(xy)Aia(xa)). (26)

The summation in (26) extends over all partitions of the integers 1, • • •, n into pairs, and the product extends over all pairs (01,7) in each partition.

Let us write the symmetrized product AB-\-BA as {A,B}. Then we have, in particular, from (26) and the vanishing of odd moments,

7i ' (*0 = <4<(*i)>=0, l2il>h(%h%2)=h({Ah(x1),Ai2(x2)}),

Izil>i*>is(xhX2,xz) = 0,

lAiuii^iA(xhX29Xz9XA)

= \{{A n (xi),A h (x2) } ){{A h (xz),A k (xA)} ) +l({Ai1(x1),Ah(xd)}}({Ai2(x2),Aii(xi)}) +i{{Ail(x1),Ak(xA)})({Ai2(x2)1Ah(xz)})

+l2il'H^h^2i2'h(X2,XZ) . (27)

The basic quantity l2ilii(xi,X2) can be obtained from (6). For a cubical domain of volume V with periodic boundary conditions

U k ( x ) = y - l / 2 e ( X ) ^ r . (28)

The vectors e ( X ) ( \ = l , 2) are unit polarization vectors orthogonal to k, and k is a vector such that V1!sk/2w has non-negative integers as components. Here and hereafter the index K is replaced by a double label consisting of the vector k and the polarization index X. We shall write cojc instead of coK since o)K=kc is in­dependent of the polarization index X and of the direc­tion of k.

On the basis of (6), (23), and (28), we have

h I^Hxhx2) = c'(2V)-1 E —<«kx+J>

Here e^X) denotes the ith component of e (x), and (#1— #2)= CM)- l n the limit V—•> 00, the sum ^ k becomes (2T)~zJ*dk, and (29) becomes

l2hi2(xhx2) = ic2H^yz jdk EKx+J)(2co f c)~1

X (g.-(k.r-«*« + 6r»(k.r-«tO)gf.i(X)^.j|(X) # (30)

The two point correlation function of the electric-and magnetic-field components can be obtained from (30) by means of the defining relations (2) and (3). If v^kx)=(wk), as is the case for thermal equilibrium, we can make use of the relation

7 > C% Cj — 0{j K%KjK ,

X=l,2

where &2=X^ k?. We then find

ft r <{£,-(*i),£y(*2)}>= a>kdk(nk+±)

(2TT)3 J

(31)

Xeik'T coso)kt\ (32)

(33) ({BiixtXB^^cHiEiixdM^)}), ch r

Xe*,rcos&)fc*(eyjfcj). (34) (2x)»

For thermal equilibrium it follows from (14) that

<Wk>= («**•*-1)"1. (35)

In this case the above formulas can be explicitly evalu­ated. For 19* j , we find from (32) that <{£;,£,})=(). For i=j and Ei the component of E parallel to r, we shall call the moment defined by the left side of (32) 5iong(r,0. Then (32) yields

«iong(r,0=({JS<(*i),£<(*0»

ch

ir2r2

/•OO

Jo kdk-

coskct

e<*k—l

[ sinkr ~| coskr + £ ' ( * ) .

Here a=h(3c, and

ctr D'{x)^

27T2f2 Jo [ sinkr

coskr kr

(36)

(37)

is a singular contribution from the vacuum fluctua­tions. Equation (36) can be rewritten as

8: long =

ch

4Tar

> <^(k.r -W fcO_J_ e -*(k. r -co&0)g. i (X)^ 2 (X) (29)

' \ dr/

x[Lfar+ct)\+L(^(r-ct)Y\+D'(x). (38)

B206 E V E L Y N FOX K E L L E R

Here L(T) is the Langevin function

2a r00 sin ( T T - ^ T ) 1 L(r) = —\ <tt = cothr . (39)

W o (eah-\) T

For i='j and E{ the component of E perpendicular to r, let us call the moment defined by the left side of (32) (§iat(r,/). Then (32) yields

«iat(r,0=-<{^i(*i),^(*2)})

Apart from the singular terms D'(%) and D"(x)y our formulas are the same, to within a numerical coefficient, as those of Sarfatt1 and Bourett.2

A simple form can also be obtained for ({E»-(#i), Bj{%?)}) for Ei and Bj components of E and B perpen­dicular to r. In that case, Eq. (34) yields

{{Eiix^B^)})—

he Wdk-

coskct

c2h

4 7 r 2 f

r°° /sinkr \ / k2dk coskctl coskr 1

Jo \ kr J

X C ^ - l ^ + i ] . (43) 2TT

X

2Jo eah—\

r sinkr 1 /sinkr \~[ f coskr )

_ kr k2r2\ kr /_ +D"(x). (40)

Here fie r

D"(x) = — / kzdk 47T27o

X coskct rsin&r 1 /sinkr \~|

_ kr k2r2\ kr coskr 1 (41)

is another singular contribution arising from the same vacuum fluctuations. Equation (40) can be rewritten as

r d ' )§long(r,t)

r d \ { fie /

\ 2drJ

f r d\( tic / d\

\ 2dr/\^rard\ drj

x | V - ( r + c o Y b z / - ( r - 6 f l ) +D"(x).

(42)

From (34) we see that if i=j, then{{Ei(xi),Bi(x2)}) = 0. Another case of interest is that of a semi-infinite

domain bounded by a perfectly conducting wall at x=0. We will consider first a finite cubical domain of volume V with two perfectly conducting walls at x—0 and x=V1/d, and later let V—> oo. We assume that the field is periodic with period Vllz in the directions parallel to the conducting walls. The normalized eigenfunctions corresponding to the eigenvalue o)k==kc are then

uk X= (2F)-^ 2 {n(n.e^)) (^ k ' r +^ ' ' k / - r )

+ n X ( e ^ X n ) ( e * , r » « * , , f ) } . (44)

Here n is the unit vector normal to the conducting plane, and k' = k— 2(n-k)n. The vector k is restricted to those of the previous values for which (n«k)<0. Since nX (e (X)Xn) = e (x) — (n«e(x))n, Uk\ may be re­written as

uk X= (2F)-1 / 2{n(n.e ( x>)2e i k ' - r+e ( x )(^k* r-^k * r)}. (45)

From (6) and (27) we have

({Ai(x1),Aj(x2)}) = ftc2 £ [^kx,^ri)#\x>y(r2)e-*^+^\x ) 4(ri)#kxj(r2)^G = 2 ^ ^ ^ i ^ 2 ) . (46) k 2o)fc

We have chosen a coordinate system such that n lies along the x axis. Let us assume, as in blackbody radiation, (wkx)=(wk) and then consider separately the two possibilities i= j and i^j.

First suppose i=j. Then Eq. (46) becomes

<»*+*>/ h2\ l2ii(%h%2) = c2fi(2V)-1 Re £ ( 1 [V k - r +^ k ' - r + (25 i n - l ) (^ k - r i -* k , - r 2 +e i k ' - r i - i k - r 2 )>-^^. (47)

k 2a>k \ k2J In the limit V —» oo the sum F_1X)k becomes (2ir)~zfdk where integration is performed over all k such that

(n*k) < 0 . From (47) we see that iV*(#i,#2) can be considered as a sum of two terms Pu and Qu;

J2H= Pii(xhX2) + Qii(xhX2) . (48)

pa corresponds to what we have previously found in the absence of conducting walls and Qil results from the presence of the conducting walls. Pu is given by

) / W \ J 1 J(eik.r_[_gtk'.r)gr-»«fce

c2h r {nh+h)i pn(Xhx2)==— Re(27r)-3 / dk

2 J (n»k)<0 2c0fc

ch

2(2TT)2

r[A-\-B) sinkr B /sinkr y\ kdk cosco^(wfc+i)| — ( — coskr J .

kr k2r2\ kr -(49)

Here A = [\+ (fi/r)22 and B= P-—3(/yr)2] , where r is the magnitude of r.

S T A T I S T I C S OF R A D I A T I O N FIEL W D B207

To obtain an explicit expression for Qii(xi,x2) it is convenient to specify the orientation of r. Let us first suppose r is parallel to n. Then we find from (47)

<2(rlln)™(>l,#2) = ch

2(2TT)2 (2din~ 1) / kdk cosco^(w&+2)

A+B\$mk(r-2s) B fsmk(r-2s) X \ ( ) 1 — c o s f c ( r - 2 * ) l | , (50)

i(r-2s) k2(r-2s)2L k(r-2s) J J

{ /A+J5\smk(

i \ 2 / k(r

where s is the perpendicular distance between r± and the wall. A and B are defined as above. For r perpendicular to n, (47) yields:

(a) (*||n)

r / a2 \

<?»"(*i,*») = (27r)-2fe / kdk(l+k^—)D(k,r,s){nh+i) ,

(b) 0||r)

Q"(xux2)=(2Tr)-ific kdk(-\-k~2— )D(k,r,s)(nk+%),

(c) (i±n±r)

r"> dk/ d2 d2 \ Qu(x1,**)= (2x)-2fe —( — + — )D(ft,r,j)<n*+i>.

(51)

Here I is a unit vector in the direction of the i axis. The superscripts fin, rr, and JLJL indicate that both the i and j components of the field have been chosen in the directions parallel to n and r in the first two cases, and perpendicular to both n and r in the third. D(k,r}s) is given by

D(k,r,s)^coskd I d/ji coskrfx Jo(ks(l—fi2)112). (52) l Considering now the case i^j, we find that

I^(po\,X2) = 0 unless i is parallel to n and j parallel to r. In that case we find

[ (rln) r(xhx2)

d d he

dr ds (2TT)2

r°°dk

Jo & (nk+h)D{k,r,s). (53)

I t is interesting to note that in computing the charac­teristic functional we have made no use of the specific nature of the electromagnetic field. In fact, the same functional will describe any linear boson field whose density matrix has the required form. The only modifi­cation which must be made is that the eigenfunctions uK(r) denned by (4) must be replaced by the eigen-functions of the equation which describes the field in question. For example, let us consider the thermal equilibrium of a scalar meson field <p satisfying the Klein-Gordon equation (V2— c~2dt

2—m2) <p (r,£) = 0.

The characteristic functional for this field is [cf., (22)]

^[X] = e x p { — f\(x)\(yX<p(x)cp(y))dxdy\ , (54)

where X (x) is now an arbitrary real scalar function. As in the case of the electromagnetic field, all moments can be expressed in terms of the second-order moment, as in (26). I t is therefore useful to compute the second-order moment explicitly. For a cubical domain of volume V with periodic boundary conditions, the Klein-Gordon equation has the plane wave solutions e~ia}ktUk(r) = V-ll2eik'T~ia,kK By using them, and letting V become infinite, we find the two point correlation function h=h({<p(x),<p(y)})tobt

l2(x7y) = hc(^jrr)' /»0O

Jo

kdk

o (k2+m2yi2

XZsin(kr+a)kt)+$in(kr—a)kf)~]' (55)

When tn—0 and {tik)= (e~Phck— l ) - 1 we can evaluate this integral and express the result in terms of Langevin functions. If we ignore singular contributions on the light cone due to vacuum fluctuations, we obtain from (55)

h(x,y) = —^L(-(r+ct))+L(-(r-ct)X\. (56) 16irarL

B208 EVELYN FOX KELLER

IV. DISTRIBUTION FUNCTIONS transform of (26). We have

If [ A - a ( ^ , ^ f e ) ] = 0 for all a, 0 = 1 , • • - , » , then pn{Ai f{xx) • • • A{ '(xn)) it is possible to introduce the distribution functions n ' ' %n

Pn(Ai1f(x1),Ai2

f(x2),' -,Ain'(xn)). These are defined to _ 1 ^ , f° , " give the joint probability of the operators A^fa), ~(2w)nJ^ J^ *^«Aia{xa)_] • • -,Ain(xn) taking on the values A^'fa), • • • ^ i / W , respectively. From them, all moments can be obtained v , _ , * , N . . , N . , NV_. ,^.N

in the usual way. That is, X e x p [ - i ^\MAMAM)1 • (62)

(f(Ai1'(xi),AiS'(xa)9--,Ain'(xn))) T y s integral can be evaluated, being simply the

/

r Fourier transform of a multivariate Gaussian. The

• • • / Pn(Aii(xi),' • -,Ain'(xn)) answer can be written as X/04n'(*i),. • ^Ain/M)dAn

f(x1)- • . ^ ^ ( ^ ) . (57) ^ » / W » ' • '^* '(*»)) (2x)~w/2

Here / is an arbitrary function of the components Aia\ = Because of the commutativity of all the operators (detGa^)112

in question, it is possible to choose a representation in n

which all Aia(xa) are diagonal. In that case, the expec- Xexp{ — ̂ zL \G~ )apAia (xa)Ai0 (xp)}, (63) tation value of a product of operators A ̂ (xi) • • -Ain (xn) obtained by taking the trace, as in (9), is the same as w h e r e Q JS a m a t r i x with components the expectation value obtained from (23). The distri­bution functions P n ^ i / W ^ ' - ^ t / W ) can be ob- Gap=(Aia(xa)Aip(xp)), tained from the characteristic functional F\X\ as a n d ( G _i )^ i s t h e fe/5) e l e m e n t of t h e i n v e r s e m a t r i x # follows. Let F o r n= 1? G i s a s c a l a r

*(*)=E A«<$(*-*«K«, (58) G=(Af(x)). (64)

For a uniform, isotropic system in equilibrium, where n^ is the unit vector corresponding to ia. Then ( N\_i / / J 2/n\\ (CX\

(Ai (x))—^{A (0)). (65) F\\1 = {exp{i[\1Ail(x1)+' • -+\nAin(*n)']}) T h e n G i s independent of the choice of component i

= tr{p expft £ KAivixa)]}. (59) and space-time vector x, and (63)-(65) yield P [ ^ / W ] = ( M ^ 2 > ) - 1 / 2 e x p [ - M / , ( * ) / < ^ > ] . (66)

Using the formula for F[X] given by (22), we find -p 9

( e x p ^ O i ^ ^ ^ O + ' - ' + X n ^ ^ J ] } ) / (AHx)) (Ai(x)Aj(y))\ G=( (67)

, , » , W J , , , , „ x fm. A \(Mx)A3iy)) (A?(y)) J = exp{ — i 22 XaXfiiAiaiXajAieixfi))}. (60) and

G~ But (exp{i[\iAil(x1)+'->+\nAin(xn)'l}) can also be {Ai

2(x)Aj2(y))-(Ai(x)Aj(y))2

obtained from (23): ^ , w ) - ^ ( ^ ( y ) ^

<exp{i[Xiili(*i)+-' - + X ^ ^ ( ^ ) ] } ) X-iAi^Ajiy)) (A*(y)) J '

= / • • • / dAii(xi)' - -dAin'(xn) Again, for a uniform isotropic system in equilibrium G J J denends onlv on (i— i). and (x—v). That is

depends only on (i—j), and (x—y). That is XPniA^ixx),- -,Air!{xn)) expp £ \aAic!(xa)~]. 1 g(p) g(x-.y)d..

(61) W-y)** g(0) / g(0) *<*- ,)*<A

V (*-?)«« e(0) /

Therefore, the distribution function Pn(Ain'(xi),- • -, where g(x) — {Ai(x)Ai(0)). The distribution function is ^.^'(xn)) can be obtained by taking the Fourier then

1 f lg(0)lAm+Af(y)l-28ilg(x-y)Ai(x)AJ(y)] P^Aiix)^^)^ exp . (70)

2Tr<j?(0)-8ng*(x-y)yi* M 2 g2(0)-^g2(*) J

S T A T I S T I C S OF R A D I A T I O N F I E L D B209

I t is possible to extend the definition of the distri­bution functions P n [ ^ / ( # i ) , - • - , , 4 ; / ( # „ ) ] to include those points where ZAia(xa),Ai^(x^)']9^0. This is done simply by replacing the product (Aia{xa)Ai&{x^)) by the corresponding symmetrized product ^{{Aia(xa), Aip(xp)}) wherever it appears. Then Eq. (63) remains unchanged provided that the matrix Ga$ is appropriately symmetrized. That is, Ga^={^{Aia(xa),Ai0(x^)}). I t is then possible to compute all the symmetrized moments from these distribution functions at all points in space-time. Caution must be taken, however, in the interpreta­tion of the distribution function at those points where noncommutativity occurs. Since it does not make sense to speak of joint probability distributions for non-commuting operators, the usual interpretation of Pn

must be abandoned at these points. Nevertheless, since the above extension of the

definition of Pn permits the calculation of the sym­metrized moments everywhere, the full set of functions {Pn} provides sufficient information to re-obtain the characteristic functional. This is because the functional Taylor expansion of F[\] involves only symmetrized moments. That is

30 %n

= E - £ n=0 fl\ u, • • • ,in

Xn(#l)* ' *Xin(#n)

8nF\X]

x-<5AnOi)- -5\in(xn)

A n ( # l ) ' • '\in(%n) •

Xln(il'"-in)(xh'",Xn)J (71)

where Inil'"'in(xh- • -xn) are defined by Eq. (27).

Consequently, the full set of distribution functions {Pn}, defined everywhere, provides a complete descrip­tion of the system.

For comparison with experiment, it is convenient to obtain the distribution functions of the electric- and magnetic-field components. For the electric field, Eq. (58) is replaced by

1 d AT

\(x) = X) \J(x—xa)nia, cdt<x=i

while for the magnetic field we would take

X(#) = — VX Z) Aad(#— oca)niL

(72)

(73)

For any mixture of the two, the appropriate combination of (72) and (73) would be employed. In the same way as above it is found that the distribution functions of the electric- and magnetic-field components are the same multivariate Gaussian functions with the ap­propriate replacement of Aia by Eia or Hi<x. Because of

the wider domain of commutativity of the electric- and magnetic-field operators, the distribution functions of these fields have a correspondingly wider range over which they can be physically interpreted.

V. OBTAINING THE DENSITY MATRIX FROM THE CHARACTERISTIC FUNCTIONAL

In Sec. II we have derived the characteristic func­tional for a system described by a density matrix of the form (17). In this section, we will solve the inverse problem for an arbitrary system. We will show that, given the characteristic functional (^[X]) = (exp[ifX(x)'A(x)dx']), it is possible, by judicious choice of ^(x), to obtain the density matrix p.

I t will be convenient for this purpose to use the basis employed by Glauber.5 In particular, we will obtain the matrix elements (a|p|j8), where | a) = 11* I a*) and \/3) = IL | / 3K) - The kets \aK) and |/3K) are eigenstates of the annihilation operators aK. That is,

aK\aK) = aK\aK), aK\(3K) = l3K\(3K),

(74)

where aK and j3K are complex numbers. I t follows that (aK| and' (j3K\ are eigenstates of the creation operator

(aK\aJ={aK\aK*, 0«\aJ=((3K\p*.

(75)

Glauber has shown that these states, although not orthogonal, do provide a complete basis in terms of which any state of the system can be expressed. We will call this representation the "a" representation. In order to simplify the calculations, we will consider a single mode of oscillation. That is, we will show how to find (aK\p\pK). The results for a full set of modes are ob­tained by straightforward generalization of the results for a single mode.

The matrix elements of p in any other representation can be obtained from the appropriate transformation formula. In particular, the matrix elements pmKnK

= (mK\p\nK), (where \mK) and \nK) are eigenstates of the photon-number operators a^aK)y are obtained as follows:

KnK=~ jd*aKjd*l3K(aK\p\PK)

X e x p [ - | W - i | / 3 « | 2 ] - — • (76)

By Sd2a we understand the double integration

/

OO /»00

d(Rea) d(Ima); -00 J —00

that is, integration is over the entire complex plane.

= - / d2aK(aK\p expfaaf—yfadloLK) IT J

B210 E V E L Y N F O X K E L L E R

In the "a" representation where

( e x p ( * / ^ ) - A ( * y * ) ) Z ( r i ) T 2 ) = exp(|C(7 2)2+i(T1T2+T1*7S*)]}.

Making a cyclic permutation of the operators under the r / r \ -i trace, we have

= tr p expf i J X(x) • A(x)dx 1

/ (71,72) = - / d2aK{aK I exp (iy2*af)p

= — I d2a(a\p exp (i l(x)-A(x)dx )\a). (77) IT J \J I Xexp(7iaK

+—71*^) exp(iy2al()\aK)Z (84)

Let us take ^(x) to be

*(*) = i(yi*^K+-y^K-)+ (72*.++72***-) n . = **(*) , ( j Xexp[i(72aK+72*aK*)])Z.

where 71 and y2 are arbitrary complex numbers, and But exp(7iaK t-7 l*aK) is just the displacement operator \K± are functions defined by £(71), introduced by Glauber,6 which has the property

^(. ) - T l im^(W*)«»--^^-«p( ± ^) °M^= k + ^ e s p i ^ * - ^ * ) . (85) Hence

= *<*•(*)• (79) 1

Here ur{r) = uK{r) and u^r) = ut*{r). Because '(71,7*) = - / <P«.<««|p|a«+7i> exp[t(7ja«+T«*«.*)] 7T ./

J _ f f ^ = l i a > 0 XexpCi(7i««*-a«7i*)]Z(7i,72). (86)

2 7 T W - 0 0 T — ^ € . , • / I i « \ i i •

Letting yi~PK—CLK, we can obtain {aK\p\pK) by taking = 0, a < 0 , the Fourier transform of

it follows from the plane wave expansion of A(x) (6) [/(71,72) exp(-§7ia ,*+§a/yi*)Z- 1 (71,72)]. that

r That is: (a) I ^K+(x)'A(x)dx=aK,

(b) / 1K- (x) - A (x)dx= a^.

(80) (aK\p\ pK) = 7T-1 fd2y2(I(yhy2)

Xexp[ - | (7 i a / -« K 7 i* ) ]Z~ 1 (7 i ,72 ) )

From our choice of *(*) [Eq. (27)], we then have Xexp[-i(72*a«*+72a«)] . (87)

f ( ^ ) - A ( * ) i f a = ( 7 i a / - 7 i * a O + f ( T A + 7 . % t ) . (81) ^ t t i n g i ^ + J ^ i + f t , , 7 i f J « - « < = * i + « i , and / • • 72=^i+W2, Eq. (87) can be rewritten as

We now use the fact that, if A andJB are operators such (aK\p\pK) = Tc~1 exp[—\(fiKa*—a A * ) ] that £ 4 , 5 ] is a "c" number, then

eA+B==eAeBe-ii2[A,B]t /g2) X / dsids2e~i(hlsl~h2S2)

Then we can write <exp(fy^(^)-A(«)^)> (nowafunc-tion of 71 and 72) as follows: X e x p ( - ^ 1

2 - | 5 22 ) / ( g 1 + ^ 2 , si+ts*). (88)

/ / f \ \ Using (83) to define 1(71,72), we now have an explicit I(71,72) = ^ e x p U / ^(ff)-A(a;)<foj\ formula for obtaining the density matrix p from the

^ ^ ^ ' ' characteristic functional F[X]. To actually compute

= tr[p e x p ( T i ^ - 7 i * < 0 exp(*y A ) ^ ^ k i s ° f C 0 U r S e n e c e s s a r y t o k n o w ™ f o r t h e

system in question. Xexp{iy2*aJ)~]Z(71,72), (83) In particular, for a system described by a charac-

S T A T I S T I C S OF R A D I A T I O N F I E L D B211

teristic functional of the form Equation (86) becomes

X e x p p £ (T2KO;K+72K*Q;K*)]

F[\l = exJ— [l(x)l(y):(\(x)A(y))\ (89) * (hurt*)) = 11* *J ^ « < a « | p k + 7 i

we can show that the most general equilibrium density matrix is

p „ „ = a * » ( l - * ) s - . (90) X e x p { | E C(T2«)2+i(7i«T2«+7i«*72«*)

This corresponds to black-body radiation if we take + (7i«a« — a«7i« )]} • (95) 2=exp(—8ho). In order to show this, we substitute r™. . , • a • . • ~ / \ i r- i , /TON • . ,„«s i r i Then, taking yu—pt—aK, we obtain a.(») as denned by (78) into (89), and find ' 5 ' y '

/(7i,72) = exp(-(«1I+|)|72+i7i*i2) (a|p|^)^({a«}|p|{/3»})=n- I ffiaj({yu,yu}) = exp[-<«,+i){*1

2+*2H-g12+e-2

2 "wJ

+2(g#t+glsJ}l. (91) Xexp[-.X(7i.W4-Y»«a.)]

Inserting this into (88) and performing the integration x ^ *\J_I 12 indicated, we find, after some algebraic manipulation, Xexp{~2 1, L(7i«a« -aKyu )+|72*|

(aK\p\t3K)= (nK+iy* ewla*pK(n.)/(l+(n.))] +i(7i«7s«+7i«*T2«*)]} • (96) Xexp[ - JI aK 1 2 - J10K12]. (92) F o r ^ [ X ] g i v e n b y (89)? w e find

Using (76) to obtain p w we find /({Ti«,7»«» = exp(-J L<*«44>|7*«+*7i«*[2) K

Pnxm,= 5 n A ( l -2)2 w S (93) = n/(7i«,7.«) (97)

where 2= <»«)/(l+(»«)). The procedure outlined above can be immediately and

generalized to obtain the matrix elements of p corre- <{<*.} |p|{/J.»=II<a.|p|/S.>. (98) sponding to all modes of oscillation. It is simply necessary to consider an infinite set of complex numbers Consequently, the full density matrix, in the occupation {7I*,Y2K} in place of the numbers yx and 72. Then (78) number representation, is given by is replaced by

^(*) = EC*(7u*^+-7i«V)+(72A«++72«*V)]- (94) X [ < 0 / ( H > , » > . (99)