Library, N.W. Bldg

FEB 1 1965

NBS MONOGRAPH 79

On the Statistical Tlieory of

Electromagnetic Waves in a

Fluctuating Medium (II)

Mathematical Basis of tlie

Analogies to Quantum Field Theory

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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UNITED STATES DEPARTMENT OF COMMERCE • Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Dmctor

On the Statistical Theory of

Electromagnetic Waves in a

Fluctuating Medium (II)

Mathematical Basis of the Analogies

to Quantum Field Theory

National Bureau of Standards Monograph 79

Issued December 7, 1964

K. Furutsu

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colo.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price 35 cents
Library of Congress Catalog Card Number: 64-60057
On the Statistical Theory of Electromagnetic Waves in a Fluctuating

Medium (II). Mathematical Basis of the Analogies to Quantum

Field Theory*

K. Furutsu

National Bureau of Standards, Boulder, Colo.

Many analogies to quantum field theory are inherent in the statistical theory of waves. This is due to the fact that basic equations exist in the latter theory which correspond closely to the funda- mental equations of the former theory; i.e., to the commutation relations and the Heisenberg equation

of motion. A probability density function of waves is introduced here which corresponds to the probability amplitude function in quantum mechanics. The boundary conditions at infinity foi this probability density function are then found to be expressed in the same form as the vacuum boundary

conditions in field theory. The theory of the statistical Green's functions and their relationships to the expectation values of the physical variables is also extensively developed, using auxiliary external sources of the wave and of the fluctuating medium. It is found that there exists a one-to-one corre- spondence between the formalism of Green's functions presented here and that used in field theory. The above correspondence may be important for a further development of the statistical theory of waves, just as the advanced techniques of field theory have greatly influenced the development of ther- modynamics (or statistical physics).

I. . Introduction

In recent years, important advances have been carried out in the field of statistical physics

by a successful apphcation of well-developed methods of quantum field theory to the study of sys- tems consisting of a large number of interacting particles which are in thermodynamic equilibrium

[Alekseev, 1961; Bardeen, Cooper, and Schrieffer, 1957; Matsubara, 1955; Martin and Schwinger,

1959]. Especially, the method of thermodynamic Green's functions has been extensively used in the study of important problems of statistical physics, and in principle, aU the necessary physical

observables seem to have been obtained in terms of the many particle Green's functions.

On the other hand, the situation is similar also in the statistical theory of ordinary waves in a

fluctuating medium, as is emphasized in part I of this series of papers [Furutsu, 1963].' In fact, all the necessary information about the statistical system of waves can be obtained in terms of the

manifold Green's functions. Let \))(xi, X2) be the single Green's function as^ defined by (1.2.14) and i//o(:x:i, X2) be the corresponding function in the unperturbed medium; then t/zUi, X2) is generally

the solution of an integral equation of the form

^(Xi, X2) = ^o(Xl, X2} + j{dx)^o{Xu X)8k{x)llj(x, X2), (1.1)

where 8k{x) = y4>{x) is the fluctuating part of k{x) representing the medium constant, and y and

4>ix) are a constant matrix and a scalar function having the vanishing average {({)(x)} =0, respectively. The expectation value {ipixi, X2)) ^G{xi, X2) is formaUy obtained by expanding (1.1) in a Born series,

Mxi, X2) = ijjo(xu X2) + j

{dx)\jjo{xi, x)y(t>{x)ilJo(x, X2)

+

J

^{dx){dx')\\)o{xi, x)y<^{x)^Q{x, x')y<i){x')^o{x' , X2)+ . . ., (1.2)

and taking the average value for each term of this series. Similarly, the dual Green's function,

G{x\, X2\x[, x'2) = {^{xi, x'j)\fj{x2, x'^), is obtained as the product of the series similar to (1.2), averaged

*The theory is developed in three dimensional space. Part I appeared in J. Res. NBS 67D (Radio Prop.) No. 3, 303-323 (May-June 1963).

' Reference to equations in the work will be written in the typical form (1.2.14). Added in proof: Recently, R. C. Bourret kindly notified the author of his recent paper [1962] in which he also suggested a strong correspondence with field theory by several examples; Nuovo Cimento, XXVI, 1.

1
over all possible values of (t){x). Thus the higher-order Green's functions can be obtained in the same way. Especially, when the fluctuation follows the gaussian multivariate normal distribution,

it is well known [Middleton, 1960] that

i<l){Xi)(f)iX2) . . . 4>{X2n)} ^'^p{<f>{Xi)(}){X2)) . . . {^{X2n - l)Mx2n)) , X2) . . . <^(X2n+l)) = 0,

(1.3)

where Xp stands for the sum over all the terms obtained from all possible combinations of different

pairs composing the set of variables xi, xz, ., X2n- On the other hand, the necessary and sufficient description of the whole statistical system of waves can be obtained only after obtaining the full set of these statistical Green's functions of

infinite number. Their knowledge is proven to be equivalent to that of the probability distribution

function of waves, defined such that the probability that the wave has particular values between

^x)' and ^{xY + dijjix) ' in the whole space S is given by

d^\XY{^[X]'\), (1.4)

where stands for the values \\){x)' at all points in 2: it means that, if all points in space are represented by a discrete set of points xi, X2, . . ., Xn of finite number, assiiming a scalar real value of i//{jc) for the moment, (1.4) represents (appendix 7) = #(:ci)'#(x2)' . . . diii{xNy{ijj{xiy,iij{x2y, . . ., <//(x^)'i). (i.s)

In the Fourier representation,

<.//(xi)',.//(jC2)', . . •''/'(^^)'l) = (^)'']^" d.^^\^ d\2 . . . £° dXN

Xexp[i{killliXiy+k2l}l(X2y+ . . . + \Nlli{XNy }]{kuX2, . . .,\n\), (1.6)

it is proven that all the coefficients of the Taylor series of {Xi, \2, . • ., ^n]) can be given in terms

of the statistical Green's functions. In fact, assuming the excitation of the wave by an external source jixy.

^(x) ^

j[

ilfix, x')j{x')idx') , (1.7)

the inverse Fourier transformation of (1.6) yields

. , |._ ^ (-tXi)"-(-tX2)'^' . (-iM"iv

{Ki,\2, . . .,Kn\)— > , . —

j

X {{^Xi)nxl,iX2)r^ . . . mXs)}"N), (1.8)

where ipixj) (without prime) is given by (1.7) and generally

mim2) . . . i|/(ym))=| {dy[) j

idy',) ... J

{dy;^

xG(yi,y2, . . .,jm\y[,y'2, • y'JjiydjiyD j^m)^ (1-9)

in terms of the Green's functions

Giyuyt ym\y\,y2, • • ,y'J= myuy[)^{y2,y2) ^[ym,y^)).
The above results are the same even if the wave function is a complex vector; the independent variables, corresponding to the above scalar ijjix), are to be prepared for each component and

also for each real and imaginary part.

When evaluating the Green's functions of (1.9), w^e may use the Born series expansion (1.2), together with (1.3), or other corresponding rule which purely depends on the prescribed distribution function of the medium fluctuation. Thus, especially when the fluctuation has a gaussian

multivariate normal distribution so that (1.3) is applicable, the diagram method can be used almost

in the same way as in the field theory (part I); i.e., in the integrand of each term of the series thus

obtained, every factor of tpoix, x') can be represented by a solid line on a sheet of paper connecting

the points x and x' , and every factor of {(^(x)(^(x')) can be represented by a dotted line connecting

the points x and x' , and the final result is obtained by summing up the contributions from all the

possible graphs in the given order of approximation. Thus, in the same way as in the field theory, we would have the self-energy parts of a graph, to each of which only two solid lines are connected,

and the vertex parts of a graph to each of which two solid lines and one dotted line are connected,

etc. Thus, the same techniques as used in the field theory [Dyson, 1949a, 1949b] are available to treat these parts of a graph (theory of renormahzation), as is done in part I.^ Here a question might arise: why are there such close analogies between the statistical theory

of waves and the quantum field theory? The Feynman diagram technique is a due consequence

of the quantization of field, the Heisenberg equation of motion, and the boundary (vacuum) condition at infinity. If the expectation values of all the physical observables in the field theory can

be expressed in terms of Green's functions, we ask whether, inversely, the commutation relations

and the equation of motion, which are the fundamental bases of the quantum field theory, could

be derived from the knowledge of the Green's functions. It may be possible [Schwinger, 1951—

1954], and the basic equations to be used for the proof will be those equations corresponding

(though not exactly) to (1.6) and (1.8) which give the connection between the Green's functions

and the probability density function of the waves. The latter function apparently corresponds to the p-obabiUty amplitude function in quantum mechanics, which satisfies the Schrodinger equation.

In this equation, all the physical variables are represented by matrices or operators which operate on the probability amplitude function, and the equation prescribes the time variation of the probability amplitude function. Thus, it implies the complete description of the quantum field theory.

The major purpose of this Monograph is to show the existence of fundamental equations in the statistical theory of waves that correspond to the commutation relations in quantum mechanics

between the physical variables involved, and also to the Schrodinger equation (or the Heisenberg

equation of motion in the Heisenberg picture). As a consequence, many analogies to the field theory are obtained. These fundamental relationships between the two theories will be worth

while since other useful techniques used in the field theory, such as unperturbational methods

not based on Born series, may then be available also for the statistical theory of waves.

In this Monograph, the distribution of the electromagnetic wave on a plane is represented in terms of a complete set of orthogonal modes of waves by the use of a method which is essentially

the same as the one Marcuvitz and Schwinger [1951] used for the representation of the electro- magnetic Green's function. A probability density function of waves is then introduced, which is somewhat different from that defined in (1.4) and (1.5), to describe the statistical system of the waves

in a fluctuating medium. A major effort is made to find the equation to be satisfied by the probability density function, followed by an inevitable introduction of matrix representations of the

physical variables involved; the boundary conditions at infinity are examined next. The Green's

functions are extensively discussed, together with their relationships to the expectation values of

physical observables.

The standard notations used in quantum mechanics are employed here unless otherwise

specially indicated.^

^ A similar diagram method is adopted in the theory of scattering by impurities in metals by Rickayzen [1961]. *For example, P. A. M. Dirac, The Principle of Quantum Mechanics, Oxford Press (1958). 3
2. Wave Equation and Representation of the Wave Function

In this Monograph, Maxwell's equations are used in a symbolic form, and the following notations are employed: Latin subscripts assume values ranging from 1 to 3, and a repeated index is to be

so summed. The coordinate vector in space is denoted by Xi = {x).

Using the conventional notations. Maxwell's equations are given in gaussian units (time factor

giiot) by

rotH — i{(iilc)eE = 4t7I/c, rotE + iicj/cj/xH = 0.

Introducing the matrices

Si= U U l\, 52= U U U , 53

the operator rot as a three-rowed matrix is expressed by

rot^Sidi = (sd), dj=i— (y=l,2, 3). aXj

Thus, Maxwell's equations can be expressed in the form

/O 1\ /iE\ /e 0^

which, on using the two-rowed Pauli's matrices,

\1 0/ \i OJ

takes the form

or

[piisd) — k]\jj = 7],

[yidi — k]i}j = T), y,=pi5,. (2.1)

Here

are one-column matrices of 6 elements, and

a 6 X 6 matrix, is Hermitian when the medium is lossless. This lossless assumption will be kept

throughout this paper, but we shall assume an infinitesimal anti-Hermitian part for k, having minus

pure imaginary eigenvalues, if necessary; the latter assumption corresponds to the outward

propagating wave condition at infinity.

4
The matrices s, and p, are all Hermitian and especially

where y* and y[ are the complex conjugate and transposed matrices of y,, respectively.

The matrices 5, and p, satisfy the following relations which are often very convenient to use:

SiSjSk + SkSjSi = 8ijSk + 8jkSi, SiSj — sjs, = i~^€ijkSk ,

52 + 52 + 52 = 2, p2 = p2 = p2 = l, p^p2=-p^p^ = ip^^ etc. (2.2a)

Here eyfc is the antisymmetrical tensor corresponding to €123 = 1; its elements are + 1 or — 1 as the

number of permutations necessary to change the order of subscripts i, j, k into the order 1, 2, 3

is even or odd. From (2.2a), it especially follows, for any two orthogonal vectors m and n in the

ordinary three-dimensional space, that

(ny)3 = n^iny), (ny){myf = {m} — {myf}{ny), (ray)(my)(ny) = 0,

(ray)2(my)2 = (my) 2( ny)2 = ^2^2 -{^jfiX n-yf. (2.2b)

The traces of these matrices are frequently used in actual computations, and the following

formulas hold:

Tr'(si) = Q, Tr'{siSj) = 28ij,Tr'{siSjSk) = i~^€ijk, Tr'{siSjSkSi) = Sij8ki + SjkSu. (2.2c)

Here Tr' stands for the trace in the subspace of 5i's matrices.

Besides the wave equation (2.1), it is convenient to introduce the adjoint wave function xjf defined by

i^[-yS-A:]= Tj, (2.3) — . — . . . where tj is an external source current for i}j which is independent of?}, and di operates on the coordi- nates X on the left side. At infinity, the medium constant k{x) is assumed to reduce adiabatically to a constant ko [Lippmann and Schwinger, 1950], while 'r)(x) = rj{x) should vanish there; this enables us to define

the outward and inward propagating waves explicitly there. Also, the field at any point in space

wiU be represented by a complete set of orthogonal modes of waves which can describe any field distribution on a plane perpendicular to the JCs-axis, passing through the point considered.

In free space or at infinity where k — ko, the wave equations (2.1) and (2.3) become

[(yd) -ko]ilf=4'[- (yd) - ko] = 0. (2.4)

which will have a complete set of solutions of plane waves as follows:

ilix(x) = e - + ^2.r2 + >^3-r3)^^, ;j;^(x) = e +''^i-^> +>^2^2+^3^2)

1;^, (2.5)

where Xi and \2 are arbitrary real values and As's are roots of

Det[{ky)-ko]=0, (2.6)4

and and |x are the solutions of

[{\y)-ko]^x=M(ky)-ko]=0. (2.7)

^For example, in the case of an isotropic medium, K3 is given by ± \/(oj/c)^e)Li — — X|.

5
There exists an orthogonal relation between the fx's and ^x's. In fact, if and ^v' are

solutions of (2.7) for the same \i and \2, it follows that

|x"[Aiyi + ^272 - ko]h'=-Kh"y3^y=->^h'y3h-- (2.8)

Hence, if k'3 9^ k'3,

|x"73^x' = 0, Xl^K, (2.9)

and, therefore, the following normalization is possible:

|x'73^x' = 6/^x.;, (2.10)

where the \i and X2 -components of X' and X" are to be the same, respectively. In the following, the subscript X of fx. wiU stand for not only the Xi, X2, and Xa-components but

also for a kind of polarization for the same X, if necessary, so that the orthogonahty similar to (2.10) holds between the f\'s of different polarizations. Thus, from (2.5) and (2.10), there results the surface integral over the plane xz = 0

.i/x"73»//x' d(Ts - (277)28(X'; - k[)8{\l - x;)8x'^i, dcr^ = dx^dx^. (2.11)

•(•3=0 Although the i|;x's thus defined constitute a continuous set of orthogonal functions, we may

obtain a discrete set of orthogonal functions by restricting the integration domain of the surface

integral on the left side of (2.11). Thus, if the whole system is assumed to be enclosed in a large

square column with the cross section of — Z, ^ ^ + L (i = 1, 2), while — <» < X3 < + <», the periodic

function condition leads to the discrete set of values 7rn/L (ra = 0, ±1, ±2, . . .) for Xi and Xz. The normalization condition (2.11) is then to be replaced by

i|/x"y3»|'x'cb"3 = 8x"X' = Sx^'x/Sa^'x^'Sx^'x^'. (2.12)

J J3 = 0

For L —* °o, the same results wiU be obtained as in the case of an unbounded space. On the other hand, the complex conjugate functions i//x's satisfy

</'t[-(y^)-M=0 (2.13)

which is the same equation as for the (|/x's in (2.4). Hence, it follows that (|/x can be expressed

in terms of the <//x's according to

i|;x' = /Vx'X"»//xt , (2.14)

where the summation convention for quantities having the same subscripts is used here and henceforth. From the definitions of i//x and \\s\ in (2.5), we infer that the matrix A^x'X" (with respect to the indices X' and X") has nonvanishing matrix elements only for such X' and X" having the same

Xi- and X2-components.

Using (2.14) and (2.12), we find

iVx"X'" <^^"r3<^x'^/o-3=8x"x', . (2.15)

Jx3 = 0

SO that the matrix A'^ is defined by the relation

Afl!'yzWda3 = {N-'\:y, (2.16) /J •1-3 = 0

6
which is a Hermitian matrix, as is seen from the left side. For later convenience, we shall normalize

the i//\'s such as to have

\Det{N)\ = l. (2.17)

From the orthogonality condition (2.15), we may expect the possibihty to expand any (//-function

of the form yli// in a Unear combination of yli/zx. Since 73 = 73 according to (2.2b), the eigenvalues

of 73 are found to be either 0 or ± 1, and thus the components of ipy and for which 73 has zero eigenvalue do not contribute to the normalization (2.15). Also, since 7|(7|</') = (731//), yl^ is the

component of (// corresponding to the eigenvalue of 73=1, for which there is no component with

the eigenvalue of 73 = 0. Thus, we may decompose an arbitrary \jj(x) on any plane, say of X3 = 0, into two parts,

y}j(x) = ijj{cT{x)) + i(,{n{x)),

with

ylijjix) = i}j{aix)), (1 - yi)i)j{x) - i|/(n(x)), (2. 18a)

where i//(cr(jc)) represents the tangential components of the electromagnetic field, and \fj{n{x)) the

normal components with respect to the surface a of :j:3 = 0. The component ijjinix)) is not independent of ijjicrix)), but can be derived from the latter with

the aid of the wave equation (2.1) (appendix 1):

4,(n{x)) = A3-3HI - ym{ydT)-k}il,{aix)) - 7^(x)]

,

;jjinix)) = [Mo-ix)){ - {yh -k}- r){x)]a- yDJ^z\ (2.18b)

where n is the unit vector in the direction of the X3-axis,

kz3 = Tr'[(\-yl)k}, ar=Oi, ^2, 0),

and Tr' is the trace of the referred matrix in the subspace of the 5j's matrices (see (2.2c)). Now any \\i((t(x)) and i|/(o-(x)) may be expanded as ^

i//(o-(jc)) = i//x'(o-(Ai))A^vx"t/'v{o-], ^{(t{x)) = i,Ki(TW\"\'Ki(^(x)), (2. 19a)

where, in view of (2.16),

l//*.<(T(x))73l//X'(c7(jc))6?0-3 = (TV"') V'V, f2.19b) JJ3=0

and thus

»/'x[o-] = J^>/'*y3»/'c?o-3, t|ix'[o-] =

J

^ysipydaa, (2.19c)

and cr stands for the surface X3 = 0 in this case. Therefore, we dso infer, from the completeness

of the orthogonal functions, that

{i}jy{a-(x))}aNy,'{il,ti(Tix'))}0 = (ysU'biaix) - (xix')), (2.19d)

where the 8-function on the right side is the two dimensional 8-function on the plane cr.

^Generally, when rjix) 9^ 0 and/or k{x) ko, t}i(n(x)) 9^ i))y{n{x))Nyy \liy'[cr] .

7J4-501 0-64-2
So far we have been considering the field on the special plane of X3 = 0 and its representation

by the complete set of plane wave solutions of (2.5), which, on the plane :t3 = 0, become

M<^{x)) = e-'^^i^i"'2-^2^yl^K, l^x(o-U))-=e+'^^l^l+^2^2^^;,y2_ (2.20)

Since we could do the same on other planes for which x^t^ 0, we shall redefine \lt\{a{x)) and [jj\{(T{x))

by (2.20), which thus become independent of the coordinate X3, in the following. Then the rela- tions (2.19a-d) are Vcdid on any plane cr perpendicular to the JCs-axis, and in view of (2.18) and

(2.19), the whole field distribution on cr can be obtained in terms of i/;x [cr]. Besides the plane wave solutions (2.20), any other sets of wave functions, e.g., those based on

the cylindrical coordinate system, might be used as well to represent the field on the plane cr. Therefore, in the following, we shall exclusively use the notation if/aicrix)) (having Latin subscripts),

instead of ilj\{cr(x)), to mark a complete set of orthogonal functions for representing the field in cr. Correspondingly, t/zxCcr] will be replaced by t//a[cr], and the Greek subscripts will exclusively be

used for representing the (six) components of the wave function.

Now i//o[o"i] and i//b[cr2] on different surfaces cri and cr-z are linearly connected as follows:

let Ua{x, [(72 ]) be the solution of the wave equation satisfying the boundary condition that yl^aix, [0-2]) = Ua{(r{x), [cr-z]) tends to ipa{cr{x)) as cr^ 0-2. Thus,

[(yd)-k]Ua{x,[cr2])-0, Ua{a-{x),[a2]) \^^^ = i}ja((rix)). (2.21)

Here, according to (2.19a-c),

U„(o-i(jc), [0-2]) = t//c(o-i(x))7Vc-6U6„ [0-1,0-2] , (2.22)

where, also in the following, cr(x) is replaced by cri{x) when the point is on a particular surface

0-1, and

U6„[o-i,o-2]= [.//,*y3U„(o-(x), [o-2])</o-3. (2.23)

The boundary condition in (2.21) corresponds to the condition

U ha [cr, , 0-2]

! ^, _ {N-%a . (2.24)

On the other hand, the complex conjugation of (2.21) and (2.22) yields

U*(;c, [o-2])[-(y3)-^]=0,

U*(o-i(x), [0-2])= U+ft[o-2,o-i]yV6c<|'*(o-i(m (2.25)

where

U„+ft[o-2, o-,] = U*„[o-i, 0-2] . (2.26)

Therefore, the matrix U +[0-2 , 0-1], thus defined, is the Hermitian conjugate of U [cri, 0-2]. Now, since

di[Ut {x,[(ri])yiUb(x,[o-2])] = 0 (2.27)

because of the wave equations in (2.21) and (2.25), it follows from the gaussian theory that

I

U * (x, [cri])ysUb(x, [crzDdcTa = U„+ [en, (T]NcdU db[(r, 0-2] = const, (2.28)

8
independent of the position of the surface cr. Therefore, setting (t = u^ and cr = cri in (2.28) and

using the boundary condition (2.24), we see that, since N^h—^ab is Hermitian from (2.16), U lb [o-i, 0-2] = U afiki, 0-2] • (2.29)

Thus U [cTi, (72] turns out to be the Hermitian conjugate of U [0^2, cti] . In the same way, we also see from (2.28) that, by replacing 0-2 by cti and cr by 0-2,

U acki, 0-2] Ned U d6[o-2, 0-1] = A^- lab, (2.30)

which gives

^'al [O-I, (Tl\ = Nac U cdK, (Ti]Ndb, (2.31a)

or, symboUcally,

U-'[cTuO-2] = NU[(T2,(Ti]N, (2.31b)

where U~*[o-i,cr2] is the reciprocal matrix of U[(Ti,(r2] defined by U "^[cti, (T2] U [o-i, cr2] = 1

.

From the definition of U[o"i,cr2], it is now evident that >|/[cri] and i//[cr2] are connected by

the relation

ilJa[(Ti] = Uab[(ru(T2]Nbc^c[a-2], 17 = 0 (2.32a)

or, simply,

i//[cr,]= U[o-i,o-2]A^<//[o-2], 17 = 0, (2.32b)

which is obtained, on using (2.19), from (2.22) with the boundary condition in (2.21). We observe that the six-rowed matrix with respect to the Greek subscripts, U (cri(xi), cr2(x2)), defined by

U(o-i(Xi), (r2iX2)) = \Ja((TliXi), [cr2])Nab^bi(r2iX2)) = Ma-i{xi))NabUbc[(Ti, a2]Ncd4idi(J-2ix2)) (2.33)

generally diverges and thus does not exist except for the case of (Ti — a2 where, because of the

boundary condition (2.24) and (2.19d),

Uia(xi),cr{x2))=y38((T{xi)—cr(x2)), cri^cr2 = CT. (2.34)

This situation is recognized as the special case of an isotropic homogeneous medium of k = ko by using the complete set of plane wave solutions of (2.5) for ^aix); as observed in the footnote

of (2.6), UA.\[cri, (T2] would then contain the factors e-'^3(-r3-j-3') where A3—>±i<» as \i, A.2^±oo.^

In the general case of rj 7^ 0, the field at a point Xi on cti is determined by the boundary dis- tribution i/'[(r2] of the field at (T2 and the external source 17 between cti and (T2, and is given by

(appendix 2).

»/'to) = 2a,b L UaiXi, [(T2])Nab^b[o-2]

-i\ ^ dx3[(T]Ua(Xu[(T])Nabi Vb^MvMdoS

J (T2 J (T -a-yDkasixiy^-qixi), (2.35)

*The general divergence of (2.33) involves the major difference from the case where Xs is the time coordinate and \jj

is a wave function depending on the time explicitly; then \3 = ± V^^ + A^ + \| ^ ± » as \i, \2^±°° so that U(cri(a:i), C2to)) would exist generally. 9
where

iljl*ix) = <//6*(o-(x))[l + {- iydr) -k}(l- yDh^ix)-'] (2.36)

is the complex conjugate field expected from the tangential components i|/*((t(a:)) according to (2.18b) in the medium of k (but not of ko). Now, all the integrals involved in (2.35) are convergent

for any physically prescribed value of i|/[t72], but the order of the integrations and the series summation with respect to the Latin subscripts are not interchangeable in the series (2.35). The equation (2.35) gives, using the matrix notation,

#ar,] = U [o-i, a-iW^M + \ ' dxaicr] U [o-j, (t]Nv[o-1

'^a[o-]=-i[ »//*((T(x))[l + {(7X)-^}(l-ri)Ai3W*)c?o-3, (2.37)

which reduces to (2.32b) in the case of 17 = 0. In the same way, we have the solution of (2.3) in terms of the boundary distribution i|/[cr2] as

follows:

Va[o-]=i j^ri{x)[l + k33'{l-yl){-{y%)-k}]iPa{cT(x))d<T3. (2.38)

The complete set of wave functions ilfa{(r{x)) can be divided into two groups, say i//|(cr(a;)),

according to their directions of propagation. In the case of the plane wave functions iff\s of

(2.5), A.3 is an analytic function of ki and \2 in the whole range and tends to'± i °o for |\i |, IX2I ~^

if an infinitesimal negative imaginary part of the medium constants are assumed, corresponding

to an infinitesimal loss in the medium. Therefore, the set of wave functions j//x's may be mathematically divided into the two groups whose Xs's tend to — I'^and respectively, as \k\=(\^ + \iyl^—> <x>. As may be seen from the definitions in (2.5), the former group represents

the waves propagating in the positive direction of the JCa-axis for the range of |\| where X3 assumes

a real value, while the latter represents the waves propagating in the negative direction. Also,

according to (2.5), the former group of solutions are not bounded in the domain X3 < 0 for |X| ^ «,

while the latter are not in the domain X3 > 0. Thus, generaUy the two sets of functions ipfix\

may be mathematically defined by those wave functions which are bounded in the domains X3 > 0

and X3 < 0, respectively, when |A.| ^ «>. This situation would be the same for the i/;^'s in the case

of the most general orthogonal functions.

From (2.9), on noticing the different assignment of the phase terms for i// and we deduce

I

i//^y3</'^G?o-3 = 0, (2.39)

where 's are to be defined in the same way as for the (//^'s.

3. Introduction of a Probability Density Function for Waves

As we have seen in the preceding section, the distribution of the wave function ^{xi) on a

surface, say cti, is completely described by its tangential components i/;(cri(^i)), which can be rep- resented by «|/a[cri]'s (or simply ^[(Ti]) as defined by (2.19). On the other hand, (//[cti] is completely determined if the boundary value ip[(T2] is given on another surface o"2, provided that the

medium between cti and (72 is prescribed. However, when the medium is fluctuating, j//[cri] would assume various values, and we could expect only a probability of having some value i//[cri]' for a given boundary value <//[cr2]'. We define the probability density function {^[cti]', o-i|i/;[cr2]', 0-2) in such a way that, for the given boundary value ^[az] ' at 0-2, the probability that i//[o-i] at cti takes

10
a value between i/»[cri]' and ijj[a-i]' + dijj[cri]' is given by

(#[o-i]'){<//[o-i]', o-i|i//[o-2]', 0-2). (3.1)

Here, (<fi//[cri]') is defined by

(#[0-1]') = UadMo-iYRdipaWi]',, (3.2)

where the subscripts R and / stand for the real and imaginary parts of the referring quantity,

respectively.

Although (//[cTi]' and ^[0-2]' are quite independent of each other in most cases of a fluctuating

medium, they are stiU bounded by the associated theory of reciprocity; from (2.37), we have, for a variation c?j/;[o-2]' of the boundary value on 0-2,

d#o-i]'= U[o-i, o-2F#[o-2]'. (3.3)

Therefore, according to appendix 3,

(#[o-i]') =|Z)ei(U[o-,, o-2]Af)|Wo-2]')- (3.4)

Here, in view of the normalization (2.17) and (2.30), \Detm = '^,

DetiU[(Ti, (T2]NU [0-2, o-i]) = Z)ef (U [o-i, (72]) Det (U [0-2, cri]) = \Det{U[(Tucr2W, (3.5)

where (2.26) and (2.29) are taken into account. Thus (3.4) reduces to

(c?i//[o-i]') = (#[(T2]'), (3.6)

which is true, independent of the (lossless) medium between cti and (T2. By the substitution of

(3.6) in (3.1), we infer that (3.1) also gives the probability that »//[cr2] has a value between t//[(r2]' and iIj[(J2]' + dxjj[<J2]' for the given value of t/'[cri]'. Thus, according to the definition of (3.1),

<l//[0-,]', 0-i|l//[0-2]', 0-2) = <(A[0-2]', 0-2|»//[0-i]', 0-1). (3.7)7

We shall see a more direct proof of (3.7) in section 5.

4. Lagrangian Variational Principle and Action Function of Waves

As in quantum mechanics, the action function is fundamentally important also here in the

statistical treatment of the waves. We define the Lagrangian density function L by

L = ^[_yidi-k]ijj-^r)- Vijj, (4.1)

and assume that the wave equations for \}j and i|/ are to be derived from the following variational

principle for xfj and i.e.,

8^ = 6 jL(dx) = 0. (4.2)

^ Since the probability density function thus defined is a resJ function of both i/zalcilj,, i/(a[o-i]/, and ^a[o'2]^ , ijjalcri]',

,

we should write it, strictly speaking, in the form

<t//[o-i]', i|»*[o-i]', cri|i|/[cr2]', iIj*[<T2]', 0-2)
Here, the space integration is to be performed over all the space between the surfaces cti and 0-2, which are assumed as planes perpendicular to the A^s-axis; further, the variations 8{}j{x) and 8i|/(x) are to vanish at the boundary surfaces.

On the other hand, adding the variations of rjix), rjix), and k(x) for later convenience, we find with the aids of partial integration

8A = f [8i//{ [jidi -k]ilj-v} + {ilj[-yidi-k]- v}8iIj]

J (72

+ i f ^y38ilid(T3-

I ^y38ifjd(T3 + \ 8L{dx), (4.3)

where

8L(jc) = - {4,{x)8Tq(x) + 8rj(x)\lf(x)} - ^ix)8k{x)^{x). (4.4)

According to the variational principle, the integrand of the first integral in (4.3) should vanish

because it depends on the variations Sip and 8i/; between ai and (T2, and thus the wave equations

(2.1) and (2.3) are derived. The t7(jc) may be interpreted as a source current density for \]j(x). However, i|; has no physical meaning as yet and is introduced only for convenience.

Next we consider the variation 8A due to arbitrary displacements 8xj{x) of the whole system,

keeping the values of and \Jj at every point, so that ip{x — 8x) at the point x — 8x becomes the new wave function i|/(x) at the point x after the displacement. Hence, the corresponding variation

84f will be given by 81// = — {dldxj)^8xj, which are to be substituted in (4.3) for the variations of the wave function at the boundary surfaces.

Now the total variation 8xA due to the displacement in space is obtained by adding the contribution due to the change of the integration domain: thus, assuming uniform and independent

displacements 8jCi[(7i] and 8A:i[(T2] on C7i and (T2, respectively, we obtain

8j,A = Pj[(T,'\8xj[(T,] - Pj[(T2]8xj{cr2l (4.5) where

^ Picr] = ^j3id<rz,

Tij = -^yidjilj + L8ij. (4.6)

With the aid of the wave equation (2.1) and the relation (2.18), all the Ti/s can be expressed in terms of the tangential components ifjicrix)), 4j{(t{x)) and the tangential derivatives of t//(cr(x)); for instance, in view of the identity yidi —y^ds^yrdr, we obtain from (4.6)

7^33 = i// [(y3r) — k]^ — ^jrj—rj^ = M(r}[{iydT) - k}k^i{l - yl) {{ydr) - k} - A:].//(o-) - r}[l + A^'d " ylHiydT) - kmcr) - Mo-)[l + {{ydr) -k}{l- yl)k^3']v + ^^I'd " yf)^- (4.7a)

On the other hand, using the wave equations (2.1) and (2.3), Psicr] is given in a simple form as

P3[(t] = [

[4/yl{iiydT) -k)iij--n}- v^jdo-^

f _ ^ _

(4.7b) = [{M- (y^r) -k)- T))yly\) - i},v]dcr3

with the supplementary conditions

12
{l-yl){{{ydr)-k)ilj-'n}=0,

{U-iydr) -k)-r}} a -y^) = o,

which define ijj{n(x)) and ^in{x)) in terms of i|/(<t(jc)), dr^icrix)) and i}j(ct{x)), 3ri|/(cr(*)), respectively.

Thus, the total variation 8^, amounting to the sum of (4.3) and (4.5), becomes

8A = dA{ilj[(Ti],(ri\\})[(T2],cr2) = i ijj{a-}y38\}f(o-)da3 + Pj[cri]8xj[(Ti]

J o-i -if Mo-)y38iljicr)da3-Pj[<T2]8xj[(T2] + \'8L(dx), (4. J a2 J '^2 8)

which formerly gives the differential equation to be solved in order to obtain A itself. Now we conveniently introduce a surface functional derivative defined in such a way that we

have from (4.8)

8AI8i}ja(o-i{x)) ^i{^iai{x))y3}a,

(4.9) — 8AI8ipa{cr2{x)) =i{4j(cr2{x))y3}a,

where x is on the referred surface cti or cr2. Thus, from the completeness of the differential equation (4.8), i.e., {dldx3[a-]){8l8ijja{(T(x)))A = {8l8^a{o-{x))}(dldx3[(T]}A, we have

dj{Mx)y3}a-{8l8Mo-{x)})Pj[(r], ;=1, 2, 3. (4.10)

In the same way, if we introduce A' defined by

v4'=^()//[o-i],cri|(//[o-2],o-2)-i|

J -

j

J^y34fdo-3, (4.11a) we find, using (4.8),

8A'^-i\ I -( I 84jy3Ho-3 + Pj[(Ti]8xj[cTi] -Pj[a2]8xj[(T2] + 8L{dx), (4.11b)

which gives the equation corresponding to (4.10): - 5j(73«//(jc))a = ( 8l8Mo-(x)))Pj [a] . (4. 12)

In the special case of j= 'i, (4.12) together with (4.6) and (4.7) gives the wave equation in terms of ^{(t(x)). Evidently (4.10) and (4.12) correspond to the Hamiltonian canonical equations

in dynamics.

The explicit solution of (4.8) is obtained as follows: for arbitrary variations of ip{(Ti{x)), t//(o-2(x)) and r]{x),

8^(t//Lo-i],o-i|i//[o-2],o-2) = t j I -J \^y384id(T3- ^8r){dx),

1 J(Tl J<T2 J J<T2

which gives, since (|/[cr] is given by (2.38) and thus the ^ is perfectly independent of ifj and -q,

A{ilj[(Ti],o-i\4i[cr2],o-2) = il \ -I \^y3^d(T3-\ ^ 4ividx), (4.13)

L Joj Jo'2 J Jo'2

where the integration constants are chosen such that A vanishes for i//[cri] =i/;lo-2] =tj = 0. The action A, given by (4.13), can be expressed in terms of (//[(Ti], i//[cr2], iqlcr], and the corre- sponding quantities for j|/ and rj which are defined by (2.19), (2.37), and (2.38), and thus

13
^(i//[o-i],o-i|t//[o-2], o-2) = i{^[o-i]yVi//[o-i]-i|/[o-2]iVi//[cr2]}

-i\ ^[(T]Nri[(T]dxs[a] + V(x)il-yl)k3s(x)-''nix){dx). (4.14)

Here, although i^[cri] and »//[cr2] are independent variables oiA, t|/[cri] and i^[cr2] are not independent but are to be understood as related by (2.38); this is required from the "equation of motion"

(4.10) for y = 3 which is equivalent to the wave equation for In the special case of Tj = 0, (4.14) takes the following form, when using (2.38) for i|;[cri] with

the boundary value (|/[cr2]' and replacing i/;[cri] by »//[o"i]' and i/'[o-2] by (//[era]',

^(i/;[o-i] ', o-i|./;[o-2] ', 0-2) = ii^Lo-z] W{i/;[o-2] - <|/[c72] '} = ji^;[o-i]7V{.|/[o-i]'-.//[o-x]}. (4.15)

Here, >|/[cri], i//[cr2], and i|/[cri], having no primes, are the values determined from the boundary

values '/'[era]', t//[cri]', and «|/[o-2]', respectively; according to (2.37) and (2.38),

r 0-2 i/zM - U[o-2,o-i]yV.//[o-,]'+ U[o-2,o-]A^i7[o-]c?;c3[o-],

^[o-i] = .i;[o-2]'yVU [0-2,0-1], etc. (4.16)

The proof of (4.15) is given in appendix 4.

5. Equation Formulation of the Probability Density Function of Waves

We first assume a given boundary value i/'[o-i]' at 0-1 and ask the probability that i//[o-2] at 0-2 wiU have a value in the range between »|/[o"2]' and ^\(T'i\' +c?i//[o-2]'. Then, according to the defini- tion (3.1) of the probabihty density function, it foUows that

(#o-2]',o-2|i//[o-i]',o-i)=Av[8((//[o-2]'-./;[o-2])]. (5.1)

Here, t/'[o-2] is defined by (4.16) and is thus the value at 0-2 determined by the boundary vedue >//[o-i]'; Av stands for the average value of referred variable over all the possible values of the medium

constant k, and the complex 8-function is defined by

S(i/;[o-]'-.//[o-]") = n„S(.//„[o-];j-.//„[o-]'^)8(.|/„[o-];-./;a[a-]'p. (5.2)

Indeed, if the medium does not fluctuate, the probabihty density is to be the S-function of (5.1). By the use of the integral representation

8(x) = ^

J"

e'^^rfX, (5.3)

and also the first expression of (4.15) for the action function A, we find the following fundamental

relation between the action and the probability density function:

{i//[o-2]', 0-2! #0-1]', 0-1) =Av

I

{d^\(J1^;Nl2Tt) exp [Hm U(i//[o-,] ', o-:|.//[o-2] ', 0-2)] ] . (5.4)

Here, in view of (4.16) with (A.3.10) and (2.17),

[0-2 ] 'AO - ( [0-2] ') = {d^\sJx'\)^{d^Wx\N). (5.5)

14
Therefore, using the second expression of (4.15) for A, it follows that (5.4) can be also expressed in the form

<i/'[o-2] ', o-2|i//[o-, ]

', o-i) = Av [8(./;[(ri] ' ~^[<ti])] , (5.6)

whose right side is equal to {(//[cti] ', o-i|(/;[o-2] ', 0-2) according to the definition (5.1). Thus, we

have a direct proof of (3.7). Now (5.4) can be expressed in the form

{^[cTi]', o-i|i/;[a-2]', 0-2) =Av

J -co /277) exp[^(^-^*)] (5.7)

where

^=^(i//[o-,]',o-i|./;[o-2]',o-2).

and A* is the complex conjugate of A. Although (5.7) gives the probability density function only when ^ = 0, we sheill keep 17?^ 0 in the

following for mathematical convenience, with the understanding that ^ is to vanish in the final result.

5.1. Matrix Representation of Fluctuating Medium and Representation of the Expectation Value as an Inner Product of Hilbert Vectors

For a further development, we need assumptions concerning the distribution function of the

medium constant k in the wave equations (2.1) and (2.3). The k is here assumed to be fluctuating

around a mean value ka and thus will generally take the form (compare with (1.4.1) and (1.4.6a))

k{x)^ko+y^{x), (5.8)

where both k^ and y are constant Hermitian matrices (having the same physical dimension) operating on the (//-vector, and <^{x) is a scalar real function. The y may be defined such as to have

Av[0(x)] = O, Av[0(x)2]= 1. (5.9)

Here the medium fluctuation is assumed to be homogeneous over aU the space except at infinity

where the fluctuation is to vanish adiabaticaUy.

We now introduce the distribution function (</)|0[S]') which is defined so that the probability

that (^[S], i.e., the distribution of 0 over all the points of the space 2 have some values between

<^[2]' and <^)[2]' + rf</)[2]', is given by (0|(^[S]')c^<A[S]'i the space 2 being the whole space here

(appendix 7). The Fourier representation of {</)|<^[2]') is usually called "characteristic function" and is very convenient in most cases. It is defined by the relations

J —00

<<;,k[S]')=J°" <0|(/,[2]')rf0[X]'<0[X]'|At[2]'), (5.10)«

where [compare (A.7.20)]

*The definition of <f/i.[2]' and d(\)[%]' should be given properly (appendix 7). Also, the upper and lower limits of

integrations with respect to </)[X]' and are always to be ±00 and will not be assigned explicitly in the following.

734-501 0-64-J 15
(/i[S]'|<^[X]')-exp ij^ ix{xycf>{xy(dx)^,

<(/)[X]V[S]')=(/x[2]'|(/)[S]')* = exp[-iJ^ fi(x)'cl>{xy(dx)'^, (5.11a)

I

((/,[X]V[X]')^//x[S]'(/x[2]'|(/>[S]")=8((^,[S]'-(A[2]"),

|(A.[S]'|(A[S]')^f(Am'<0[2]'|M[S]")=8(/.[X]'-/.[2]"), (5.11b)

and, from the normalization condition of the distribution function,

<</'Ia[2]')U' = o =

I

{ct>\<t>[iy)dct>[lV^l. (5.12)

When the fluctuation follows the gaussian multivariate distribution with the mean value zero

as in (5.9), (<^|)Lt[2]') is assumed to take the form

(</>|/i[2]') =exp ,x{x)'D{x-y)Myyidx){dy) . (5.13)

Here the function D(x — y) is easily identified as the expectation value of the product ^{x)^{y); i.e.,

Ax [(f>ix)(f>iy)]= j{(l>\ct> [X]')d(t>[iycl>{xy(l>{yy = D{x-y), (5.14)

which is derived by the use of (5.10), (5.12), (5.13), and the partial integration together with

(8/8/>t(jc)')((^>|/^[X]')=-j[ D{x-y),ji{yy(dy){cj>\fJL[Xy), (5.15)

where the left side is the functional derivative as ordinarily defined.

There exists an operational method for getting the average value Av [Q] of any functional

^(</)[X]) over all possible distributions of the medium. Thus, using the Fourier transformation

(5.10) with (5.11),

Av [Qm=j ((M(/)[X]')(?(<^W[X]' =

jj

(</,|)a[X]')^/At[X]'(/x[X]'|C(^)l/>t[X]")rf)u[2]"| (^[S]"|0[2]')<f<^>[2]', (5.16)

where

{,ji[xy\Q{ct>)\fx[xr)-j ()Lii2]'ic/,ii:]')W)(0i2]vi2]")rf0i2]'. (5.1?)

Hence, when Q= (f){x) or 0(x)", we have on reference to (5.11b)

()Lt[2]-l</)(x)|/Lt[2]") = (- i8l8ix(xy)8{ti[l] ' - MX]"),

( M [X] 'I ct>{x)„ 1 /Li [X] ") = (- i8l8fi{xyr8ifi[ X]' - ^[X]"). (5. 18)

Here we conveniently introduce the "matrices" (/)(x) and fx-ix) whose matrix elements are defined

by {/LttX] '|(/)(x)|AtiX]") in (5.18) and

{ti[xy\ti{x)\ix[xr)=ix{xy8{fji[xy-ix[ir), (5.19)

the latter being a diagonal matrix. Then, using the ordinary matrix multiplication rule

{lJi'\AB\fji") = j{fjL'\A\fjL"')diJi"'{iJL"'\B\ix"}, (5.20)

16
()Lt[2]'|<^(^)"|/u.[S]") in (5.18) can be obtained as the matrix element of the matrix (/)(x)". More

generally, if Q{(l>) is a functional expandible with respect to </>, (5.17) can be regarded as the matrix

element of the corresponding matrix Q{<p).

On the other hand, using the representations of (/)(x) and ijl{x) in (5.18) and (5.19), we observe

that their commutation relations are

[(li(x),tJL(x')] =-i8{x-x'),

[<l>{x), 4>(x')] = lixix), fjiix')] = 0 (5.21) 9

with the ordinary bracket convention for commutator

[A,B]^AB-BA. (5.22)

Now, on the right side of (5.16), ((^|/x[X]') is naturally interpreted as the representation of a

left-hand Hilbert vector ((/)| on which the matrix Qicj)) operates from the right side, and the term

</x[S]"|«^[S]')ci0[2]' = 8(;a[S]")- <iu-[2]"|0^) (5.23)

as the representation of a right-hand Hilbert vector |0^) on which (^(0) operates from the left side. Thus (5.16) may be symbolically expressed as

Av [(3(0)] = (</>!(?(«/.) 1 0^) (5.24)

with the normalization

(0 K)=j <0lM[S]')fl?M[2]'<)Lt[2]'|O^) = ((/)|At[S]')U-=o = l, (5.25)

which follows from (5.12). Here, the Hilbert vectors ((/)| and |0^) are specialized by (5.15) and

(5.23), respectively, and thus

<0l 0( x)+ij D{x-x')fJL{x'){dx') a, (5.26a)

lx{x)\Q^)=Q, (5.26b)

where both <^{x) and ix{x') are now the matrices defined by (5.18) and (5.19). The equations (5.24-26) give a complete description of the expectation value of any functional

Q{4>)i which is given as the matrix element of an equivalent matrix Q{^) between the prescribed

Hilbert vectors (0| and |0^).

The representations of 0(%) and ix{x) in (5.18) and (5.19) are the special representations in which

)U,[2] is diagonal. However, we could use various representations by suitable transformations,

for which the commutation relations (5.21) and the equations (5.24—26) will remain the same. For instance, the representations in which 0[S] is diagonal are obtained by the use of the trans- formation function given by (5.11), and then (5.24) becomes

Av[(3(0)]=| I (c/,|0[S]')^i0[X]'(0[S]'|W)|c/>[S]")c?(/,[2]"(0[S]"|O^) =

1

(0|(/)[X]') (0[X]"|O^) = l, (5.27)

'Equation (5.21) is derived from the identity

ix[x) = I b{x — x') iJi,{x'){dx').

17
which turns out to be the original definition of the expectation value. The equation (5.26a), for {(f) \ can be replaced by a simple condition corresponding to (5.26b) for |0^) as foUows. We put

<</)| = (0,i|e-«(^] (5.28)

with

GP] =

| 1 1^ ^{x)D{x-x')fx(x')(dx){dx'), (5.29)

and substitute in (5.26a), and then operate e

+ ^[^] from the right side. Here, using the lemma

e-^Qe^o = Q-^G,Q]+^[G,[G,Q]]-j^[G,[G,[G,Qm + . . ., (5.30)

and also the commutation relations (5.21), we see that

e-«[^l0(x) e«[^] - <i>{x) - i joix -x') /i(x') {dx') = <t>{x),

e-Gt^V(^)e''f^^ = /^W, (5.31)

since the terms of the orders higher than the second in (5.30) vanish for Q = <t>ix). Thus, the condition for {O^tl, corresponding to (5.26b), turns out to be

(0^\(l>(x) = 0, <Orf,|0^) = l, (5.32)

which means that (0,^,1 is a left-hand eigenvector of 0 having the eigenvalue <f)' = 0. On the other hand, using (5.28) in (5.24) and taking into account the condition (5.26b) and thus

G[2]|0^)=0,

Av [Q{<t>)] = <0,^|e-«(^](?((/))e«Pl|0^) (5.33) = {OM<t>)K), (0*10^) = !,

where <f>{x) is defined in (5.31) and

e-«ra</)(x)«e«[^] = 0(;c)", e-«[^W)e«[^] = (?(0) (5.34)

for any functional Q{^) expandable with respect to 0. Equation (5.33) gives the expectation

value of any Q as the matrix element of the corresponding matrix representation between the

Hilbert vectors (0<i| and |0;x)- It is very convenient to replace fj.{x) by iJi(x) + q{x), where q{x) is an arbitrary ordinary real function and wiU be called "the external source" of the medium in the following. Thus,

of (5.29) and <fiix) of (5.31) are respectively replaced by

G[S] =

| 1

1 ifx{x)+g{x))D{x-x'){fxix') + qix')){dx){dx'l

Equation (5.30) is derived from the expansion

a being an ordinary number, where ^'"''s are obtained by successive differentiation of both sides with respect to a at a=l.

Added in proof: This transformation is to be understood as a similarity transformation.

18
<f>ix) = e-G[S]0(^)eG[S]^ - i j^Dix - x')ifx(x') + q{x')){dx'),

{if>ix))o ^ {OMx)K)=-i jD{x-x')q{x')idx') {5.35a)

in terms of the notation

<F)o= (0*|F|0;.). (5.35b)

Now, for the arbitrary functional Q{<f>), we consider, on using the conditions for (0,^,1 and |0^),

i.e., (5.32) and (5.26b),

{4,{x)Q{<l,))o-- {Q{<l,)<t>{x))o = {Q{4p){<l>{x)+{<f,{x))o})o = {[Qi4>),(f>{x)])o+ (0(x))o<(?(«A))o, (5.36)

where, using the commutation relation (5.21),

[(?(0), (i,ix)]=-[(i>{x), Qm= mstiix))Q{4>) = i{8l8q{x))Q(if,), (5.37)

since the dependence of fj.(x) on <f>{x) is the same as that of q(x). Thus, (5.36) becomes

{<f,{x)Q(4>))o=={{<l>ix)}o + ii8l8qix))}{Q{<^))o. (5.38)

On using (5.35), (5.38) yields for q = 0

(0(x)(?(0))o = £D{x-x'H{8ld<f>{x'))Qmo{dx'), (5.39)

which turns out to be the equation equivalent to (1.5.18). From (5.38), we observe that

{4,{xr)o-{{<Hx))o + i{8l8q{x))}\ <0<^|0^)=1,

or, more generally,

{Qmo-Qi{<Hx))o + mi8qix))), (5.40)

in which we readily find, e.g., {<f,{x)4,{x'))oU = o-D{x-x'). (5.41)

On the other hand, we observe from (5.38) that, on reference to (5.35),

i(8/8g(x)){e -c«ra<(2(0))o} = e'^'^H<Kx)Oiit»))o,'

(5.42)

Go[S]=G[S]L=o,

and thus, generally,

(i(8/8g(x)))«{e-««l^l((?(0))o} = e-«o[i)<0(x)n(?(0))o. (5.43)

19
Therefore, according to Taylor expansion,

9=0 = <exp -i q{x)'il>{x){dx)

9 = 0 ,Go[2]U=o=0, (5.44)

in view of (5.43). Hence, we finally have, in terms of the convention Go[S]' = Go[S]|g = g',

(W))oU = ,'=e+Gom{exp — i J

<l>ix)q{x) '(dx) (?(0))o

9 = 0 (5.45)

which gives the explicit dependence of (^(0))o on the external source qix)'. In the special case

of (3 = 1, (5.45) shows

(exp ; <l>{x)q{x)'{dx) )o = e-Gom', (5.46)

which is the generating function giving

(<b(x^)<l>{x2) . . . <fi(xn))o = {i8l8q{xi})(i8l8q(x2)) . . . (i6/8g(*„)) exp [- Go[S] ] |9 = o. (5.47)

On the other hand, differentiating both sides of (5.45) with respect to qix)' , we observe

i(8l8qixy){Qi<f>))oU = g'= (0(x)(?(0))o- (0(^))o((?(0))o|9 = 9S (5.48)

reproducing (5.38) again. As the special case,

i{8l8q(x)){<l>{x'))o={<t>(x)4>{x'))o- {<f,{x))o(<l>ix')}o, (5.49) on dropping the prime of q{x)'.

5.2. Mathematical Preparation for Matrix Representation of Wave Variables

According to the expression (5.7) of the probability density function of waves, this can be

given, on using (5.33), in the form

Here

(#0-1]', 0-i|l//[0-2]', 0-2) = {04,\m(Tr]', ar\tf»[(T2]', (T2)K).

mo-iY, |^[o-2]', 0-2) =

I

{d^[a2]'l27r) exp {A -A*)

A=A{iIj[(Ti]' , C7i|l/;[o-2]', 0-2)U[I]-^«f2],

id^ia-Y I2tt) = na(d4,a[crY l2TT)R{d;jja[(rY I2tt),,

(5.50a)

(5.50b)

and *lf[(r] stands for the implicit dependence of ^[cr] on the operator 0[2], so that the bracket

(•/'[ci]', cri|i/»[cr2]', 0-2) is stiU an operator due to the involved. Thus, for instance, the representation, in which 0[2] is diagonal, (5.50a) gives

<#0-,]',0-i|.//[0-2]",0-2)=|J {0M^Y)d<t>[lV{C[(T^Y,(Tl\aO-2]\o-2} C?0[S]"<0[2]"|O,

(5.51a) "

" Indeed, when ^ = 0, (5.51a) leads to the original definition of the average value, since (<^[2]"|0^) = 1, while (0,j,\<t>['L]') turns out to be the given probability density function, yielding the probability (0*|<^[2]' that has a value between

<f>[l]' and <f,[1.]' + d>f,[X]'. 20
Here

(^[o-i]', o-ilM', 0-2) = ma,]', (Tr\i}t[(T2]", 0-2) U = *'8(<^P]' (5.51b)

and ^[0"!]' stands for both i//[cri]' and 0[X]', and ^[0-2]" for both (//[o-2]" and 0[2]". In the special case of (Ti = (72 = cr,

{a<rV, o-\a<T]", or) = di^ia] ' - ./;[o-]")8(0[2] ' - 0[S]"), (5.51c)

which is evident from (5.6). The bracket term defined by (5.51b) wiU be called the Transformation

Function in the following.

For later convenience, we also introduce the following notation for an arbitrary functional

Q o{ilj[(T'\',4,[(Ty, and 0[S]'(x3[o-i] >;«;3[o-] ^:<;3[o-2]);

<C[ori]',o-i|(?(i//, .J, 0)1^0-2]", 0-2) = [ (c?,|;[o-2]7277)W,^',0')exp[^(^-^*)l8(0[2]'-0[X]").

J J (552)

Here, i/;' and are the solutions of the wave equations with the boundary values of t//[cri]' (or i//[o-2]") and i|;[o'2]', respectively, according to (2.37) and (2.38); therefore, they can be defined also by the use of iffix)' =— Sv4/8tj(x) and {jjix)' =— 8AI8iq{x) according to (4.8) with (4.4).

It is noticed that there is no prime for the arguments of Q on the left side. Also, in order to be

self-consistent, the order of the variables in Q on the left side is very important and must be "well- ordered," as wiU be seen later. In the special cases of ^=1, 0(jc), or i//[c7i] (or ^[az]), we find, from (5.52), that

(^[O-,] ', O-ii l|^[o-2]", 0-2) = (Cki] ', O-il ^[0-2]", 0-2),

(ao-i]', o-Mo-m(T2]", 0-2) =.//ki]'(acri]', o-iiao-2]", 0-2),

(aa-iY, o-i|0(;c)|ao-2]", 0-2) =<P(xy{a<ri]', (TiICM", 0-2), (5.53)

and a similar equation result for «//[(r2], instead of t//[cri]. Now we investigate the change of (^[cti]', cri|^[o-2] ',0-2) due to the variations of ^[cri] ', {[0-2] ',

a:3[o'i], X3[cr2], and also due to those of 17 and rj; on reference to the first equation of (5.53) with (5.52),

Haa-iY, o-ilCM', 0-2) = {Cio-iY, aS{8A-8A*)\a(r2Y, 0-2), (5.54)

where, from (4.8) with (4.4) and (5.8), 8A^il ijjy38iljd(T3 + P3[cTi]8x3[(Ti]-i I i//738i//c?a-3 — P3[o-2]8x3[o-2] + ( 8L{dx),

8L{x) =- {^{x)87){x) + 8r](xmx)} - {Mx)yMx)}8(t>(x), (5.55)

and the variation due to 80 is added for later convenience.

6. Introduction of Matrix Representation of Wave Variables and

Commutation Relations

In (5.54) combined with (5.55), we first consider the variation of (^[cri]', o"i|^[o-2]' 0-2) due to

8(//a[o"i]' (where the Latin subscript is attached to express the components expUcitly) or the

corresponding variation 8i|/a(cri(x))'. According to the definition of the { )-notation of (5.52), the 8\\ta{(T i(x)) contained in 8A on the right side of (5.54) can be taken out of { ) and primed, as

in (5.53). Hence

21
8<C[o-i] ', o-i |C[o-2] ', 0-2) = i <to-3(x)<C [o-i] ', o-il I

{il/Wyalal^o-z] ', (Tz )8i/;„(o-i(x))'

= i< ^[o-i] o-il I

{^[a;]N}a\a(T2] ', o-2)8.//„[o-i] ', (6.1a)

the latter of which is obtained on reference to (4.14). Since (6.1a) is true for arbitrary Bifjaicriix))',

it foUows that

-i(8/8i//a(cri(:t))0<C[o-i]\o-i|4[o-2]\o-2)-(ao-i]\o-i||{,i;((Ti^^^^^ (6.1b)

1 -iX8/8»|;aki]')(C[o-i]\o-i|C[o-2]\o-2) = (C[o-,]',o-i|-{,|,[o-i]7V}a|ao-2]',o-2). (6.2a)

In the same way, we obtain

+ i{8l8ilJa{cT2ix)y){C[(Ti]\ar\ao-2]\a-2}=a[a-,]\ai\^{M^^^^^

+ m8ilJa[(T2]'Ha<ri]\a,\a<r2]\(T2)^{a<ri]\(Tr\^{Ua2]N}a\a^^ (6.2b)

and similar equations for the complex conjugate variables.

For the variation due to the displacements 8j!;3[cri] and 8x3[(72], (5.54) gives, with reference

to (5.55),

iidldXs[(T,]){aCTiy, 0-,|^[0-2]' 0-2) = <^[0-,]', CTi | ^[o",] | C M' , (T2) ,

-i0/aX3[0-2]KC[0-l]', O-llCM', 0-2) = (C[0-l]', 0-,|jr[0-2]|^[0-2]', 0-2). (6.3)

Here, from (4.7b),

^[0-]^i{P3[0-]-P3*[0-]}

=

1

jjcTsUiiydT) -k)rp-n}{;}jy!)-{i{ydr)-k*)ijj*-in*}{;ij*yi)- ri^+rj*ri (6.4)

where the order of ^ and i// is interchanged for later convenience. The superscript T stands for the transposition of the matrix referred to, y* = y^i=~yu and k* = k'''. In the special case of o-i = o-2 = cr, (6.1b) and (6.2a) give, on referring to (5.51c),

a{(T\\hm(T{x))y,]a\^[cTr)=-imu(r{x))'m{o-]'-

(C[o-]1H^J[o-]A^}a|a<r]'') = -j(8/8.|/„[(T]O8(i/,[o-]'-,/,[(T]W[S]'-0[S]'^ (6.5)

where cr is omitted in the above angle brackets on the left side since there is no explicit dependence

on it.Now we introduce the "matrices" \{\lf{(T{x))y3\a and H«l'[o'] A'^ja whose matrix elements

(CM'IH<|i(o-(x))r3}ak[o-]") and {IWV Wi^fWWMWY) are defined by those in (6.5), re- spectively; these constitute the ordinary matrix representations of the differential operators — i{8l8*l}a{(T{x))') and — i(8/8»//a[cr]')5 respectively, and therefore, for instance, (6.2a) and (6.2b) for

CTi ¥= (T2 can be represented as

I

(ao-i]'IH^(o-i(A:))y3}a|C[o-i]"),((;^[cr,]")(^l[o-,]", o-i|^[o-2]', 0-2) = {iW{\',(Tr\ H'?(o-i(AJ))y3}a|ao-2]',o-2), (<f^ [o"] ') = (^M V</> [2] ', (6.6a)

22
I

(C[0-:]',0-,|C[0-2]",0-2)(rfC[0-2]")(^[0-2]"| H^i(T2{x))ys}a\aCT2V)

= <C[o-:]',o-,| H<|i(o-2W)r3}a|C[o-2]',o-2), etc. (6.6b)

Thus, we infer that (^[cti]', cti

|

^

[0*2

]

' , 0-2) constitutes a right-hand Hilbert vector for the operator

H<l'(o'i)73}a, and a left-hand vector for ^{i|/(cr2)73}a.

The matrix representations (6.5) for |{t|/(cr)'y3}o and ^{4f[<^]^}a can be regarded as the repre- sentations in which the "matrices" i//a(cr) or t/zaLo"] (and i/'*(cr) or »/{j*[cr]) and i.e., ^[o-],

is diagonal:

(^o-] '

I U<t{x)) U[o-]") = U<rix)y ' -^[ct]") 8(<^[2] ' -

"U[o-]'|.A„[o-]|ao-]'')=.//a[o-]'8(.//[o-]'-./,[o-]'')6(0[X]'-0[2]''), (6.7)

these being the special case of cri =o"2 =o" in (5.53). Also, the matrix i/;* turns out to be the Hermitian conjugate of i/;, as is directly seen from the representations similar to (6.7). Therefore, using the ordinary bracket convention (5.22) for the commutator of two matrices

and the matrix multiplication rule

{ilj[ay\AB\ilj[ar}^j{iP[ay\A\rljiay''m[ay'^ (6.8)

we obtain, in this representation, the following commutation relations:

[ Hi/'(o-(*))y3}a, {ys^icrix'))}^] =— iiy3)paSiaix) — (Tix')),

[HU(^]N}a,llJb[(T]'\=-i8ab, (6.9a)

[the 8-function being defined in (2.19d)] and also the similar Hermitian conjugate equations

[ My3./;*(o-(x))}a, {«//*(o-(x))73}0] =- i (y^USicTix) -aix')),

[HN4>n(r'\}a,^IJ*[(r]]=-i8ab, (6.9b)

corresponding to the complex conjugate wave variables. AU the other commutators vanish:

[ilJa[(T],il,i>[<T]] = [M<r], M<r]] = [C M' ^ftM] = [M<r],ilJ*[(T]]^0, etc. (6.9c)

We cilso observe directly from (6.5) and (6.7) that the wave variables and 0 commute:

[i//[o-], (^[S] ] = ] = 0, etc. (6.10)

On the other hand, since 0 does not commute with 0 and /x, separately, the wave variables

^, etc., are found not to commute with 0 and /x (refer to the later equation (6.22)).

The two equations in (6.9a) are dependent and derived from each other by the use of the relation

(2.19a-d); i.e.,

t//(o-( x)) = M(r{x)) Nab ^b [o-] , Mo-{x)) = 4Jb [a-] Nba C M^))'

{iPa {o-{x))}a Nab {^t i(r{x'))}p = {y3)aP 8(cr{x) - o-ix')). (6.11)

23
Since t/'[cr] and are commutative, the matrix representations of all relevant variables

involved are possible, sich that <//[cr], »|/*[o-], and i.e., ^[cr], become simultanpously diagonal.

6.1. Equation of Motion of Transformation Function (Schrodinger Equation)

The equation (6.3) prescribes the spatial change of the transformation function {^[cti]',

o^i I {[(T2]", 0-2) with respect to the coordinates X3[cri]and JC3[cr2], together with the boundary condition (5.51c):

<C[o-i]',o-i|^[o-2]",o-2)U,^. = 8(.A[o-i]'-.//[o-2]")8(0[2]'-0[S]'O. (6.12)

According to the original definition (5.52) of the bracket ( ) notation, the (//[o-i], »/'*[(Ti], and <ft contained m^[cri ] can be taken out of the brackets and primed as in (5.53), and thus the remaining

terms in the brackets contain only i|/[cri] and i|>*[cri] in linear form according to (6.4). On the other hand, for the brackets containing (//[cti] and ii/*[o"i], we can use the rule (6.6) with (6.5), and the corresponding one for (//*[cri]. Hence, in (6.3), if we substitute the expression

(6.4) ioT ^[cTi] with <//[cri], (|/[cri], i//*[cri], and t//*[cri], which are now aU matrices, in the given

order, »//[cri] and i/'*[o-i] being defined by (6.7), and if we apply the ordinary matrix multiplication

rule (6.8), the correct equivalence is obtained for the right side of (6.3) between the one originally

defined by (5.52) and the one given in the matrix representation. Thus, (6.3) is equivalent to

iO/a;c3[o-i])<C[o-,]',o-,|^[o-2]',o-2)=| {a(TlV\^[o^]\acT,]")ida(r^]") <^ [o-ij", o-^ICM ', 0-2),

-i0/a^3[0-2])<^[0-i] ',0-11^0-2]', 0-2) =j (a0-l]^0-,|^[0-2]^0-2)(c^^L(T2]'')<C[02]''^^[02]|^[0-2]'),

(6.13)

(<fC[o-])-(#[o-])</0[X].

Thus, the order of the variables m^[(r] is very important because they do not generally

commute. However, by the use of the commutation relations (6.9), ^[cr] in (6.4) can be replaced

by

g^[(T] =|| do-^l^ yl { [(ydr) - k]il, - r)} - ^* yU [(ydr) - k*^* - r,*} -7)4'+

=

1 1 das[m-(yX)-k]-r,}yl^-{^'^[-{y%)-k*]-V*}yir-^V + ^*V^^

(6.14)

with the supplementary conditions (4.7c), where k* = k''' is the complex conjugate matrix. The

proof is not difficult by the use of (4.7a) in (4.6). On the other hand, the right side of (6.14) is just the Hermitian conjugation of that of (6.4), and thus^lcr] is found to be Hermitian in the representation used in which ^[tr] is diagonal.

Therefore, we see from (6.13), together with the boundary condition (6.12), that

(^o-i]', o-i|C[o-2]', 0-2)* = (^0-2]', o-2|C[o-i]', (7,) (6.15)

provided that (^[0-2]', cr2|^[cri]', ai) on the right side is defined by the solution of (6.13) with the

boundary condition (6.12) even for JC3 [0-2 ] <JC3 [(Xi ] . From (6.15), we see that, when A:3[cr2]<X3[o"i],

(C[cr2] ', cr2|{[(7i] ', CTi) can be defined by the complex conjugation of (CCcil 'i o'ilt[o'2]', o"2). Further, if t)=t)* = 0, the both sides of (6.15) wiU be real, as may be seen from the result (5.6) in connection with the relation (3.7). 24
The equation (6.13) evidently corresponds to the Schrodinger equation in quantum mechanics;

the time in the latter is simply replaced by X3[o"] in the former.

6.2. A Few Relations Between Transformation Functions

There is the following relation between the transformation functions: for any plane (T2 which

is perpendicular to the JC3-axis, it holds that

iCla-iV, c7i|C[o-3]', 0-3) =1 (Cio-xV, 0-^0-2]', o-2)(c?^[o-2]')(ao-2]', o-^I^M', 0-3). (6.16)

The proof of (6.16) is given first by showing that the right side is independent of the coordinate

*3[cr2] and then, by putting X3[cr2] XsIcts], both sides are found to be equal by the use of (6.12); the first step of the proof is very simple by the use of (6.13). Applying the functional differentiation —28l8ria{x) to both sides of (6.16), where the point x

is between the surfaces (T2 and 0-3, we have, on account of (5.54) and (5.55) or the later equation

(6.25),

(6.17)

If the surface 0-2 is chosen so that the point x is on it, the rule (6.6a) can be used on the right side

of (6.17), and thus

<C[0-l]',0-l|.i;.(x)|^[0-3]',0-3)=| I

(^[O-,]', 0-i|C[0-2]',0-2)

X(<^C[0-2]')(C[0-2]'|l|>a(^)k[0-2]")(^i4[0-2]")(C[0-2]", CT2\aCTsV , CT3) . (6.18)

We could do the same things also for i//a(x), i|/*(jc), and )//*(%) by applying — 2(8/8 T)a(x)), + 2(8/8i7*(x)), and + 2(8/8 t)*(x)), respectively. Thus, h generally follows that, for any function

F(x) of the wave variables \\t{x), (//(jc), \li*(x), and ^*(x) on a2,

(C[o-i]', CT^\F(x)\i[a3\' , o-3>=|| (^[o-i]', o-,|C[o-2]', <T2)(dl,[_a2Y)

X <a0-2]'|TO|C[0-2]")(rfC[0-2]")<^[0-2]", 0-2|C[0-3]', 0-3). (6.19)

6.3. Wave Equation in the Heisenberg Representation

The matrix representations (6.5) and (6.7) for \1){<t{x)) and \lf{cr(x)), respectively, can be interpreted

to be the representations in which i//[cr] is diagonal, the point x being on cr. However, i//[cr] itself changes with the coordinate ;«;3[cr], and thus, if their matrix representations were based on the

variable »|/[(t] at a fixed surface, say (t = c72, they would change exphcitly with Ai3[o-]. As in quantum mechanics, the former representation will be called the Schrodinger representation and the

latter the Heisenberg representation. Let F{x\) be an arbitrary function of wave variables \\f{xi) and ^{xi) on (Xi; then, according to

(6.19), its representation in which ^[0-2] is diagonal can be expressed, on referring to (6.15), as

<^[o-2]'|F(:c,)|C[o-2]") =

J

|(C[o-2]',o-2|C[c7i]',o-:)(rfC[o-i]')

X <^[o-,]'|F(;c.)|^[o-,]") (di[cx,^") <^[o-,]",o-i|^[o-2]",(T2). (6.20)

Applying i(a/ajC3[a-i ]) to both sides of (6.20) and using(6.3), we observe, since (C[o-i] ' |F(xi)|^[o-i]")

is independent of jc3[o-i], that

25
i0/a;C3[0-i])<C[0-2]'|F(xi) 1^0-2]") =-j <C[o-2]',cr2|:^[o-,]|C[o-i]',o-.) ida^T^V) (C[o-i]',o-,|FUi) 1^0-2]", 0-2) + j

U[<T2y, (T,\F{Xr)\aO-lV,0;){dC[(TrV) (a0-l]',0-,|/[0-i]|C[0-2]", 0-2) = <C[o-2] ', 0-2

1 -X[o-i] Fix^)+Fixi)^[a,] |C[o-2]", 0-2) (6.21)

or, in the matrix representation with the convention (5.22) for the commutators,

i{^|^X3[(T^])Fix^)=[F{x,),^[(T^]], (6.22)

which corresponds to the Heisenberg equation of motion.

For instance, if we put F(xi) = y3'/'(xi) = y3<|'(o-(xi)) and use the commutation relations (6.9a)

and<^[o-] given by (6.14), the equation (6.22) becomes

i {dldx3)y3iljixi) =-yl[{{ydT)-k} ifiixi) -7](xi)], orylHiyd)- k} if/ixi) —qixi)] = 0, (6.23a)

or, using the supplementary condition (4.7c),

[(yd) - k] ifjixi) = 17(^1). (6.23b)

In the same way, putting F{xi) = ijj{xi) ya,

iMxi) {- (y^) -k}- rj(xi)] yl = 0, (6.24a) or Mxi)[-(y%-k] = r)(Xi). (6.24b)

Equations (6.23b) and (6.24b) are exactly the same as the original wave equations (2.1) and

(2.3), respectively. The same results are also obtained for i//* and

In the same way as the coordinate derivatives of the wave variables (but not of <f>{x)) with re- spect to X3 are obtained by (6.22), the derivatives with respect to other external variables such as

r]{x), Vix), and q{x) are also important, and are obtained in a similar form: using (5.54) with (5.55) we observe

{8l8riaix)){C[(Ti]' , o-i|^[o-2]', 0-2) ^-haia-iV, a-ilMxMa-^V, 0-2),

i8l8'n*{x)){a<riV, o-i|C[o-2]', o-2)-+HC[o-i]', (r^l^jjtixMfr-zV , 0-2), (6.25)

where the point x is to be in the space between (Ti and cr2. On the other hand, differentiating both sides of (6.19) with respect to y]{x') and using (6.25), we have, as (6.22) is derived,

i8l8'na(x')m<TiV, o-i|F(^)|4[cr2]', 0-2) =- 1 <C[o-i]', <J,\T[Fixmx')]\a(r2]', 0-2) + {a(TiV,cri\8'F{x)l8'r)„{x'}\ao-2V,(T2), xaCo-i] > X3 [0-2]. (6.26)

Here, T stands for the coordinate ordered product

(A{x)B(x'), X3>x^,

T[A{x)Bix')]=\ (6.27)

[B(x')A(x), X3 < x's.

26
and 8'F{x)l8'riaix') is the direct functional derivative when Fix) is an expHcit functional of 17. In the same way, for :r3[<Ti] > JC3[cr2],

imv*ix')){a<riV, o-i|FW|C[o-2]', 0-2)=+ i (^[o-,]', o-^\T[F(x)^:(x')]\a(T2V, 0-2)

+ (a^riV, a-^\8'FixWv^ix')\aa2]', (T2). (6.28)

Generally, we observe that, for J[;3[cri] >:r3[cr2],

8(a<Tiy,(Ti\F{x)\a(T2V,<T2)

(^[o-i]',o-i|r Fix) f'^l{8Lix')-8L*ix')}idx')

Jt2 ^ + 8'F(;c)|C[o-2]', 0-2), (6.29)

where, according to (5.55) with (5.8),

\{8Lix) - 8L*ix) } = - ijix)S4>{.x) - ^{^ix)8'nix) + Sr)ix)iljix) - ^*ix)8'n*ix) - 8ri*ix)ip*ix) }, (6.30)

with

jix) = -^ {iijix)yiljix) - ^*ix)y*ilj*ix)}. (6.31)

Equation (6.29) gives the variation of 8(^[o-i]', cri|F(jc)|^[o-2] ', (72) for any wave variable Fix)

involved in the space between cri and 0-2.*^ Since ipinix)) and i//(/i(x)) given by (2.18b) are not independent variables, but functions of

ijjicrix)), T]ix), and ijj'ia-ix)), vix), respectively, special care is necessary when using (6.29) for these

variables. For instance, using (2.18b),

i8l8'naix'))a[(TiV, o-i|i/;M«|^[o-2]', 0-2) =-{C[o-iV, cri\^T[iP0inix))Hfaix')] + {k33ix)-m-yl)}0a8ix-x')\ao-2V, 0-2), (6.32)

where the term corresponding to 8'Fix) in (6.29) does not vanish.

7. Analogies to Quantum Electrodynamics

The important equations obtained in the preceding sections will be as follows: from (5.26b),

(5.32), (5.35a), (5.50a), and (6.22),

(i//[o-i]',o-i|i/;[o-2]',o-2) = {0,i,|(./»[o-i]',o-i|^[(T2]',o-2)|0^), (7.1a)

{OMx) = 0, M(^)|0^)=0, (0^|0^) = 1, (7.1b)

0(x) = (kix) - i Dix-x'){fiix') + qix')}idx'), (7.1c)

iidldxs[(T])Fix) = [Fix), g^[cT]\. (7.1d)

Here Fix) in (7. Id) stands for the wave variables ^ix), ijfix), i//*(x), and ijj*ix), but not the medium

variables 0(j;), (t>ix), and ^t(jc). The total Hamiltonian^[cr] is given by (6.14) together with the conditions (4.7c) which define i//(n(jc)) and iljinix)) in terms of ipiaix)), (/;(cr(*)), and their tangential derivatives on the surface cr, respectively. Thus, with the conditions (4.7c), S^W] may be given by

At this stage, 80(x) in (6.30) is to be an arbitrary variation and thus independent of the (f>{x) whose eigenvectors are used as the basis for the representation of the transformation function.

27

i
^[o-] = <^o[o-]+^i[o-],

^o[o-]-| j^d(rs[^{{ydT)-ko}rli-4f*{(ydT)-k*}n

^A(r]^j^dcT:i j(x}4>{x)--^{;jjr) + v^-4,*7)*-ri*i}j*}^, (7.2)

where j{x) is defined by (6.31), and the commutation relations of the wave variables and the medium

variables involved are given by (6.9a-e) and (5.21), respectively. In (7.1) and (7.2), we observe an important analogy to the field theory: Equation (7. Id) for the wave

variables is exactly the same as the Heisenberg equation of motion in quantum mechanics due to the Hamiltonian;^/ [cr], the time being replaced by the coordinate JCaM, whUe <l>{x) contained in ^ [cr] plays the role of another field interacting with the wave field. Equation (7.1c) gives the«^(A:) in terms of the canonical conjugate variables (t>{x) and /xix), and (7.1b) may be interpreted as the vacuum condition for the <^-field (the corresponding condition for the wave field wiU be described

in the next section). Equation (7.1a) will, of course, correspond to the probability amplitude function in the vacuum state of the </^-field. On the other hand, the major difference from the field theory will be the nonexistence of the equation of motion for the medium variables <p, jx, and 0

given in the form of (7. Id), which would determine their behavior along the coordinate x^. In the case of quantum electrodynamics, the longitudinal field can be removed (on account of

the Lorentz condition) to give rise to the Coulomb interaction terms between the charges involved

in the Hamiltonian.'^ Therefore, it will be interesting to check a few results that would come from

the corresponding use of the Coulomb gauge in electrodynamics. On referring to (7.1a) or (5.51a), we observe from (6.3) that

i{dldx^Wi]){x\,W,]' , o-j|i//[o-2]', 0-2) = (#0-1]', o-,|/[o-i]|#o-2]', 0-2), (7.3a)

-i{dldxi(Tt]){^{(Jx\', 0-i|t//[cr2]', 0-2) = (<|/[C71]', (JxY/[(J-M[(T2\ , (T2} , (7.3b)

where, according to (7.2),

^[o-] =^o[o-] +^i[o-], ^,[o-] - 1^ d(T3j{x)<f>{x) , (7.4)

when 7] = r} = ri* = rj* = 0.

On the other hand, according to the lemma (5.38),

<#o-i]', o-i|y(^)</)(x)|#o-2]', 0-2) = {{4>{x))o + i{8l8qix))}{il,[(Ti]', (rMxMcrzY , (T2), (7.5)

where, according to (5.35a),

<0(x))o = - i D(x - x')q(x')(dx'}. (7.6)

In order to evaluate the functional derivative with respect to q(x) in (7.5), we notice the lemma (6.29)

with (6.30) and thus have

jix) l'''j{x'){84>(x')l8q{x)}{dx') |i//[o-2]', 0-2),

Jo-2 J

i{dl8q{x)){rlj[<Ti]', o-x\j{xMcT2Y , 0-2) = (#o-i]', o-.|r

L J(r2 -I (7.7) where, according to (7.1c),

8(f>{x')l8q{x) = -iD{x' -x). (7.8)

Coulomb potential is, of course, an ordinary potential not quantized.

28
Therefore, we finally observe from (7.4), (7.5), (7.7) and (7.8) that, for q = 0.

(./;M',o-i|^i[o-i]|i//[o-2]',cr2)

=-j^i//[o-i]',o-i|r j

dasix) j'^' {dx')j\x)D{x-x')j{x')^\Ha2\\a})^ 9 = 0. (7.9)

On the right side of (7.9), the (ft variable is not present, and the term — ij{x)D(x — x')j{x') may

be interpreted as a two-body interaction between the "charges"' j(;t) and j(x') with the interaction

potential — iD(x — x'). However, a large difference from the electrodynamics is that it is an inter- action of the charges on the surface cti with all the charges distributed over the space between cri and (T2, and consequently, it follows that'^ [cti] in (7.3) depends not only on the wave variables on

(Ti but also on the variables on other surfaces involved between cri and era. It implies that the

relation corresponding to (6.16) does not hold, and thus a new mathematical problem arises when

trying to use (7.9).

8. Expectation Values of Physical Variables and Their Relations With

Green's Functions

As mentioned in section 2, the medium fluctuation is assumed to vanish adiabatically at in- finity, and also the external sources r} = ri = r}* — rj* = 0 there. Hence, we can explicitly define

the incoming waves and the outgoing waves there such that

iIj[(t] = i//+[o-] + <|/-[o-], = ^+[(7] + {dip[a]) = ( a?^+[o-])(#-[o-]),

where Xsia] = ± <», and i//-[cr] and ii;-[cr] are the components of <//[cr] and t//[cr] belonging to the waves

propagating in the positive and negative directions, respectively. According to (2.19), (8.1) is equivalent to

ipix) = iIj+(x) + (|/-(x), 4,{x) = ^+{x) + 4j-(x), (8.2) where

i)j^(x) = ^\)a(x)Nab^t [o-] , ^^(x) = [ar]Nba^\tt{x) , (8.3)

the explicit subscripts being attached.

When evaluating the expectation value of any physical variable, we notice that the excitation

of the waves is given by the external current source, rj and/or by possible incoming waves pre- scribed at infinity. First, we shall consider Av [^(x)fi(y)] which is the average value of the product

of two physical variables A at x and B at y. Here the incoming wave components (//""[cti]' at a;3[cri] — + <» and '/'^[cTa]' at X3[cr2] =— <» are assumed to be given, and the outgoing waves may have various

possible values due to the fluctuation of the medium. Then, tracing the same course of evaluation

as we did when deriving (5.51a), we have, if the point x is on the surface (Tj- and the point y on CTy and if XsIctjc] ^ Xzlay],

Av [A(x)B{y) ] = ]im XsicTi]-^ + X3 [0-2] - 00

////// <O<il0[2]')<^0P]'(#+[o-i]')(aa-i]',o-x|ao-x]',o-xM(x)'(</ao-.]')

//// <^M^]')d<l>[^]'(dr[(riV) {ao-i]',o-i|fe]',o-2)

idr[<T2Y)d<t>m"{<f>[X]"K)

where

ao-i]'-^(r[o-i]',r[o-,]',0[2]'), ao-2]'=(r[o-2]',r[or2]',«^[2]"). (8.4) 29
Here, it is very convenient to introduce the right- and left-hand Hilbert vectors lO"*") and {0~|

defined by

«//+[o-2]|o+) = .i;+[o-2]|o+) = (//+*[o-2]|o+) = .i;+*[o-2]|o+) = 0,

(0-|r[o-i]= (Oir[cri]= (0-|r*[o-i]= (Oir*[o-i] = 0, (8.5)

or, in the representation in which (//^[(t] and t|/+[<T] or i//"[cr] and «|/"[cr] are simuhaneously diagonal,

their matrix elements may be given by

<Mo-2]', Mo-2]'\0+)=d{il,+[o:z]')8{H>^[cT2]'l2n),

<0i.//-[o-x]',,|;-[o-i]')=8(r[o-i]')8(t|;-M727r), (8.6)

where the 8-function in this case is defined as in (5.2). The above representation is possible

since (//-[cr] and t/;-[o"] commute each other and thus they can be represented simultaneously in diagonal; using the orthogonality (2.39) and the commutation relation (6.9a-c),

[H4/Ho-]N}a,x}if[o-]]=-i8ab. (8.7)

In order to get the representations of |0+) and^(0~| in which ^[0-2] and t//[o-i] are diagonal,

respectively, the transformation function ( i/;[(j]'|i/;[cr]') between the two variables i/;[o-] and

i|/[tr] is necessary and is given by

{^[(T]'\U(TV)=eXp [i| IJo-3{,|;W + r'y3^*'}

^exp [iHM(TVNab4'b[<TV + iljt[o-yNt,a^*[aV}] = (r[(rV\4j-[cry){r[<Ty\r[<ry),

(«/'*M'lr[o-]') = exp [iH^n<rVNat>4f^[o-y + 4>r[<^VNba^r[(^y}], (8.8)

where, by the use of (6.5), it is easy to see that

j{^[(Ty\lmcT]N}a\Harm[ar){il,[ar\4,[a^^ (8.9)

and a similar equation holds for 2{^</'*[c] }a- Also, it is normaUzed in such a way that

|(i/;[o-]'|t^;[o-]0(c?.|;[o-]7277)(^;[o-]'|.//[o-]'')=8(.//[o-]'-i/,[o-]''). (8.10)

Thus, using (8.8),

(.//[o-2]'|0+)-(i//+[o-2]', .A-[o-2]'|0+)=| (.//-[cr2]'i^+[o-2]')

X(rf.|;+[o-2]727r)(i//+[o-2]', .i;+[o-2]'|0+)=8((//+[o-2]'). (8.11a)

In the same way,

(0-|»//[c7i]')=8(i//-[o-,]'). (8.11b)

30
We observe from (8.11) that |0+) has a uniform distribution with respect to ^'[ctz]' and (0~|

with respect to i//'''[cri]'. It impHes that any eigenvector of the eigenvalue (|/+[cr]'=0 represents

the uniform distribution with respect to i//"[o"]', and any vector of i|;~[o-] ' = 0 with respect to if/^la]'. Therefore, when there are no incoming waves at the infinity and thus i//~[cri] ' = (/;+ [0-2]' =0

in (8.4), the right side of (8.4) can be entirely expressed in terms of (0" | and |0 + ) as follows: using (8.11a, b).

<O„|0[S]')^f«^[2]'(c/.|/ + [o-i]')(4[o-i]',o-,|C[o-]',o-) =

I

(0^|«^[2]') (Oi.//[o-.]')(rf^[o-:]')(C[o-x]',o-,|^[o-]', 0-) - (0*|(0i^[o-]',o-),

J

|(^[o-]',o-|^[o-2]',o-2) (#-[o-2]')#[2]'<0[2]'|O^) = (ao-]',o-|O+)|O^). (8.12)

Thus, in the case of no incoming waves, (8.4) can be given by

Av[A{x)B{y)] = {nA{x)Biy)]), ^-[o-,] ' = ,/,+[o-2] ' = 0, (8.13)

in terms of the notations

(F) -<0|F|0)/(0|0),

with

<0| = <0,tKOi, |0)=-|0+)|0^); (8.14)

this new notation is to be distinguished from the similar one {F}o introduced in (5.35b).

(F) is generally a functional of the external sources 17, 17*, rj, rj*, and q, and it will usually be

possible to expand it into the power series, e.g.,

(F) =1;^ [f (dx) {7}a{xy (8/87,„(x)) + r,*(x)' (8/87,*(;c))}l" (F) U=r,*=, = exp j^idx) {UxY i8l8'naix)) + -nt (;c)' (8/8T,*(^))}j (F) | t,=„*=o (8.15)

which is the expansion with respect to 17 = 17' and 17* = 17*'. The above series, however, terminates

after a finite number of term in the case where F is a polynomial of the wave variables if) and

being the case of most interest, as will be seen later (Green's functions).

On the other hand, from (6.26) and (6.28), we see, if T[F] — F, which does not contain 17 and

17* explicitly, that

(8/8t,„(;c))" (0|F|0) - {0\T[F{ - \ |0),

(8/8r,>))" (0|F|0) - {^\T\_F{ i .|;>)}"] |0). (8.16)

Therefore, on referring to (8.15), we have, if r[F] =F,

<0 |F |0) 1,^,, = ( 0 |r[exp ( - iu;) F] |0) |.=.*=o, (8.17)

with ^ =

\\^ {-.ii(z)T7(Aj)' + r,*(A;)' Ai^'^x)} {dx). (8.18)

Thus, using (8.17) and (8.18), we observe from (8.13), dropping the primes from 17' and 17*', that Av [F]-(rm)-<0|r[exp(-iM;)F]|0)/<0|nexp(-iu;)]|0), (8.19)

where 17 = 17* = 0 on the right side except for those in the'term w.

31
In the same way, we can get the similar explicit dependence on 17 and V* by the use of (6.29)

whh (6.30). The equation (8.19) gives the explicit dependence of the external sources on the expectation

value of any physical variable involved when there are no incoming waves at the infinity. Here

it is remarked that the same equation (8.19) can be used even when the excitation of the system is provided by incoming waves <//~[o"i]' and (/'"'"[era]' prescribed at the infinities xslcri] — + °° and

^sLcTa] =~°°, respectively, by introducing the following effective values T7^of tj and tj* of tj*:

Tj^ix) =iy3[ — \p-{cri{x)y8{x3[(T] — X3[(Ti]) + i//+(o-2(:r))'8(x3[cr] —xslaz])],

7}|(X) = 1 [llj-*(0-My 8{x3[cr] -Xs[o;])-i}j^*((T2{x)y8{Xs[(T] -X3[(T2])] Js- (8.20)

Here i//-(jc) is defined by (8.3) and cr is the surface on which the point x exists. The proof of (8.20)

is not difficult if one notes that |t/''^[o-2] ') and {i}j~[(Ti]' \ defined by

|i//+[o-2]') =exp

2J 0-2 |o-).

{^-[criVl = <0i exp \ da-3{

' J cri (8.21)

turn out to be the eigenvectors of the eigenvalues ^'*'[(T2]' and (//"[cti]', respectively (appendix 5). In the same way as in (8.19), we could get the expUcit dependence of q on (F) according to

(5.45). Thus, in the most general case where aU the external sources rj, tj*, rj, rj*, and q are not

zero, we find, on referring to (8.17) and (8.18), that the equation (8.19) still gives their exphcit de- pendence, if w in (8.18) is replaced by

^ — j

^dx) <t>{x) q (x) + {- M^) -nix) - "nix) i\i(x) + y)*(x) i}j*(x) + i//*(x) y)*{x)} (8.22)

As a few direct results of (8.19) with (8.22), we have, for T[F] =F which does not contain

17 and tj* explicitly,

(dlS'nMKF) ^-UT[FMx)] ) + hiniMx)),

(dl8r,*Jx)){F)= UnF^:ix)])- h{F){^*a{x)),

i{8l8q(x)){F) ^{T[F<f,{x)] ) - {F){<t>{x)),

and also the similar equations with respect to Vaix) and rja*ix).

On the other hand, in the same way as (7.5) is obtained, we have

{<f>{x)} = {<f»{x))o + i{8l8q{x)){0\0)l{0\0),

while, tracing the same course of evaluation as in (7.7) with (7.8),

i{8l8q(x)){0\0)=-i j

D(x-x')(0\j{x'}\0){dx').

{,l>{x)) = {*t>{x))o-i j

D{x-x')(j{x'))idx')

=-ij D{x-x'){q{x')+{j{x'))}(dx').

(8.23a)

(8.23b)

(8.23c)

(8.24)

Hence,

(8.25)

"Using the field theoretical statement, (/W) corresponds to the vacuum fluctuation current, e.g., of the electron

field.

32
However, when the nonphysical external sources t)=t7* = 0, (<f>(x)) should be independent of

the wave variables since it is the expectation value of the fluctuating part of the medium, and thus

(0U)) = («/»U))oor (/U)) = 0. The equation for (<//U)), corresponding to (8.25), is obtained by applying the brackets ( and ) to the left and right sides of (6.23b) with (5.8) to give

[{yd) - ko] (^Ij(x) ) - 7(0(x) iltix)} = iqix),

which, using the lemma (8.23c) for the second term on the left side, gives

[{yd)-ko-y(4>{x))-iy(8l8q{x))] <i//(;c)) ^iqix). (8.26a)

Here, according to the boundary condition (8.5), {ijj-(x)) are to vanish at X3 = ^^, respectively,

and thus (i//(x)) satisfies the outward propagating wave condition at infinity. In the same way, for (i/;(jc)), we have

(<^(^)> [- iy% -ko-y(<f>(x}}- iy ihldqix))] = rjix), (8.26b)

and also the similar equations for (i//*(z)) and ({}j*{x)).

The Green's functions Ganix, x') and G*^ (x, x') for the original wave function and the complex

conjugate wave function are respectively defined, on referring to (8.23) and also (6.32), by

Ga$ {X, x') = {8ldr]0{x')) (^aix)) =-HnUx)Mx')])-{(k3s {x)-^)il-yl)},0 8(x-x'),

G^{x,x') = {8l87j;{x')){i}j* (x)) = HT[^*M^;{x')])-{{k33 {x)-')il-yl)}a^8{x-x'), (8.27)

where r), rj*, t), and t)* are to vanish after the functional differentiations (but q9^ 0 for a while).

Also (\}jix)} = {iif*(x)) =0 for T7 = rj* = 0, according to (8.26). In the same way, we can define the

Green's function g(x, x') of the medium by

gix, x') = i(8l8q(x')) (<l>(x)) = (r[0(;c)0(*')]) - {<f>ix')) = D(x-x') + j^{dx")D{x- x"){8l8q{x')){j{x")} = D{x- x'), (8.28)

where (8.23c) and (8.25) are used and 17 = 17* = t)= tj* = 0. On the other hand, operating 818'qix) on both sides of (8.26a), and putting 7] = iq* — rj=r}* = 0

after it, we have, according to the definition (8.27) of the Green's functions,

[{yd) -ko- y{<f>{x)} - iy{8l8q{x))]G{x, x') = 8{x - x'). (8.29)

The equation (8.29) is also derived directly from the explicit expression in (8.27) on using the wave

equation (6.23b) for 17 = 0 and the commutation relation (6.9a—c). In the same way,

[(yd) -k*- y*(<l>(x)) - iy*{8l8q{x))]G*{x, x') = 8{x-x'). (8.30)

33
Here, in (8.29) and (8.30) is given by (8.25) with 0(*)) =0.^^ Thus, the simultaneous equations (8.29) and (8.30) with (8.25) give a rigorous description of Gix, x') and G*{x, x'). In a similar way, we may define a double Green's function by

Gixi; X2\x[;x^) = {8l8ri(xmi8'n*{x^)){T[ilj{x,)ijj*{x2}] ) | ,=,. = ^=^.=0 (8.31)

where the subscripts for the wave components are included in the coordinates Xi and x'i. Using

(8.27), this function is given by

Gixi; X2 \x[; x^)^- \ {T[ilj{x,)ilj*{x2)Mxm*ixi)] > - i8l8'n*{x'2) ) {i}jHx2)kM-'}a-yl)id{xi-xl)-{8l8'nix[) ) {Mxi)ksAx2)-'){l-yi) 28(^2-^^ - < A;33 (^1

)

'ks3 (:c2

) -

1 ) (1 - - yl)2S(^i

x[)8(x2 -xi),

where (1 —yDi, i = 1, 2, operate on the coordinates xu On the other hand, in the same way as (8.26a) is derived,

3^{xMT[ijj{x^)xjj*{x2)]}=vixi){V(x2)), (8.32)

where 3^{x) is the functional differential operator on the left side of (8.26a) or (8.29). The above

equation further reads, using the complex conjugate of (8.26a),

^*ix2mxi){niij{xi)rix2)] >vixiWxz), (8.33)

where ^*{x) is the functional operator on the left side of (8.30). Applying 8l8r)ixl) and 8l8Tq*{xi) on both sides of (8.33) and putting 'q = r)* = rj =rj * — 0 after it,

^*ix2)^(xi)G{xx; X2\x[; x'^) = Six^- x'Mx2- x'^). (8.34)

Here, the two differential operators J^(jci) andj^ *(aj2) are commutative on account of the symmetry

g{x,x')^gix',x) from (8.28). Generally, the Green's functions of waves are defined by

Gixi,X2, . . ., x„;yuy2, . . .,ym\x[,xi, . . . , x^; y{ , y!, , . . .,y^)

^{(8l8'n(x[)){8l8'n{x',)) . . . (8/8i7(V))}{(8/8i7*(y;))(8/8rj*(yy) .... i8l87)*(y'm))}

X{il,{ximx2) . . . ilj{xn)i}>*iyi)i(f*{y2) . . . «|/*(ym))U=^-=^i=^.=o, (8.35)

and aU the other Green's functions which might be possible vanish identically; for instance,

G{x,\x2,X,) = {8|87}{X2))i8|87^{x,)){i|J{x^))

According to the definitions (6.31) and (8.27) for j(x) and the Green's functions, respectively,

iJ{x})=iTr{yG{x,x') + y*G*(x,x')]^

= iTi{y{G{x, x') + G-'{x', x)}]

where G^J^x' , x) = G*J^x, x'). Now it takes the same form as given for the corresponding vacuum current in field theory. See, for instance, Schwinger [1951]. 34
should vanish because, otherwise, it would imply a quadratic dependence of (tpixi)} on the external source 17 according to the expansion (8.15). According to (8.35), the expectation value of any functional F of ^ and i//* proves to be expressible in terms of the Green's functions in view of the series expansion (8.15). In the next section, the theory of the Green's functions will be developed, starting from (8.25),

(8.29), and (8.30), by using the unperturbational method of Schwinger for quantum electrodynamics.

9. Theory of Green's Functions of Waves

The functional differential equation (8.29) is equivalent, as wiU be shown later, to the integrodifferential equation

[{yd)-ko-y(<l>{x))]G{x,x') -j Ak{x,x"){dx")G{x",.x') =8{x-x') . (9.1)

Here (<f>ix)) is given by (8.25) with 0(^)) —0 and the operator AA: is defined later. According to the boundary conditions (8.5), G{x, x') is to be the outward propagating wave at infinity, as stated

below (8.26a).

Following Schwinger [1951], it is convenient to regard G{x, x') and y as the following matrices

with respect to the coordinates x and x' including the subscripts:

{x\G\x')=Gix,x'), {x\y{^)\x')=y8{x-mx-x'), {x\p\x' ) =iidldx)8(x- x') . (9.2)

Then, y{<f>ix)) and y(8l8q{x)) in (8.29) are represented by matrices as

y(<l>{x)) j yiemmda y{8l8q{x))^ j

yimSqimdi), (9.3)

Thus, on comparing with (9.1), (8.29) can be given in the form

iyp)-ko- j y{Om))id^)-Ak G = l, (9.4)

where the operator is defined by

AkG = ij (d^)ym8IHm- (9.5)

In the same way, from (8.30),

iyp)-k*-j y*(a(0(^))(<i^)-AA:*jG* = l, Ak*G* = ij idi)y*(mi8qim* (9.6)

In order to evaluate the functional derivative in (9.5) and (9.6), we introduce the matrices

Yii) and r*(^) defined by —(8/8(0(^)))G-' = G-H8G/8(0(a)}G-\

r*{0 = -{8l8{<l>{0))G*-' = G*-'{8G*l8(ff>{^))}G*-K

We then have the relation

(9.7)

i8l8{4>(mG-GmG, {8l8(4>iO))G*=G*r*{OG*, (9.8)

which give, using (9.4) and (9.6), m= y{e + (8l8{4>{mAk, r*(0 = y*i^)+(8l8(4>(mAk*. (9.9)

35
On the other hand, using (9.8) with (8.28),

i{8i8q(n)G= j(di)GmGD{^-n, m8q{n)G*= j{doG*T*m*D{^-i'). (9.10)

Therefore, substituting (9.10) in (9.5) and (9.6), we find that

Ak=

jj {d^m')y(e)Gnm^-n aa*=JJ

{di)(dny*inG*T*(m^-^'). (9.11)

Further, using (9.11) in (9.9) together with (9.8), we get

m-y(e+j j (dnidnyinGmGne')D{e-n

+11 idnide)y{nG{8nniB{<t»imDie-n (9.12)

and a similar equation resuh for r*(^).

The operators AA;, AA;*, F, F*, G, and G* can be constructed by successive approximation.

Thus, to the first approximation, we find, using (9.11) and (9.12),

{x\^k\x') ~ yGo{x-x')yD(x-x'),

{x\r{i)\x') ~y8(x-mix-x') + yGo{x-^)yGo{^-x')yD{x-x'), (9.13)

where Go refers to the 0th order approximation.

The Fourier transformation of the above equations in the limit ^ = 0 are also important; in that case, because of the translation invariance, G{x, x') = G{x — x'), {x\Ak\x') = {x — x"\Ak\x' —x"),

etc., and also {^{x)) — Q. Thus, putting

G(x) =-^ {dt)e-i^'^m, D{x)^-^ r {dt)e-^^^-m\

{x\Ak\x'}^j^^ (<ii)e-«'<-^-^'"AM?),

(ZTT)"' J-oc

(^|F(ak')=7^6 f f"" (c?«i)(<^«2)e-*'(^-f+'^'^-^'')F(ti,?2) (9.14)

and the similar equation results for G*, AA:*, and F*; then, from (9.4), (9.6), and (9.11),

[{yt)-ko-m)]G(t)^l, [-{yt)-k^-Ak*{t)]G*{t) = l, (9.15a)

or

G{t) = Go{t) + GomkG(t), Go{t)=[{yt)-ko]-\ (9.15b)

and

Mit) = -^J^ {dt')yGit + t')r{t',t)D{t'^). (9.15c)

The spectrum of possible waves in the fluctuating medium is given by the poles of G{t), each

pole of which gives the possible propagation constant.

In the actual situation, the correlation function between the waves at different points in space

is very important, and it is thoroughly described by the Green's function introduced by (8.31). Thus, in view of (8.15),

Ay [i}ja(xi)ip^{x2)] =

jj

idxi)idx^)Ga0;ys{xi;x2\xl;x^)r)y{x'i)rjUx2). (9.I6)

36
Here, the Green's function

G{xuX2\xl;x^) = {xi,X2\G{l,2)\x^,x^) (9.17)

satisfies the functional differential equation (8.34), which can also be represented by the following

integro-differential equation corresponding to (9.4) and (9.6) (appendix 6), i.e.,

F(1)F*(2) G(l , 2) - /(1 , 2)G{ 1 , 2) -1

,

where

F(l) = F*(2) =

(9.18)

(9.19)

and the subscripts 1 and 2 refer to the coordinates Xi and X2, respectively. On comparison with

(8.34), the interaction term 7(1, 2) is found to be (appendix 6)

/(i,2)G{i,2)=

[

[

{deidi')yi{m^-ar2*inGih2)+i

[ {dOyiimm/8q{mia,2)Ga,2)},

^ (9.20)

which, to the first approximation, gives

{xuX2\I{\,2)\x[,x'2) ~y,ytD{x,-X2)8{xr-x[)b(x2-x'^). (9.21) i«

The perturbation theory, as apphed in the above, is not to be confused with the Born series

expansion with respect to the couphng constant y. As stated in part I, it is equivalent to the se- lective summation of all the Feynman diagrams of the ladder type.

10. Summary and Discussion

The electromagnetic field distribution on an infinite plane cr is completely described by the

i//a[cr] 's (or simply (//[o"]) which are its components of a complete set of orthogonal modes of

electromagnetic waves defined on the plane a. When two infinite planes cti and (72 are parallel

to each other, t//[cri] is uniquely determined in terms of i//[cr2] if the medium involved between

CTi and (72 is given, while it may assume various values corresponding to a probability distribution

function if the medium is fluctuating. Thus, the probabiUty density function of waves is introduced

(sec. 3). Here, the surfaces C7i and (T2 may be concentric spherical surfaces having the same

origin, or cyhndrical surfaces having the same axis, since the existence of a complete set of ortho- gonal modes of waves on such surfaces is known [Stratton, 1941].

Also, an adjoint wave variable i//[(t] is inevitably introduced to describe the statistical system,

and both t|/[cr] and (^/[cr] are represented by matrices or operators which operate on the probability

density function of waves or, more generally, on the transformation function associated with it (sec. 5). These operators correspond to the canonical variables in field theory and satisfy the

commutation relation of Bose statistics. Also, their spatial changes are prescribed by the same

equation as the Heisenberg equation of motion in quantum mechanics. Further, the boundary

condition of the wave (i.e., the outward propagating wave condition at infinity) is given in the same

form as the vacuum condition in field theory. In this connection, the i^[o-] seems to be more fundamental than the wave itself on the surface

cr, since the former makes it possible to develop the whole theory as divergence-free, while it is not always true for the latter (sec. 2); not only can the matrix representations of <//[cr] and i//[cr] be

well defined on the surface of reference, but also the commutator exists between >/'[cri] and <i/[cr2] on different surfaces cr\ and cr2, while the matrix representation of the field \\f{x) at a point x is not determined uniquely unless otherwise the surface (on which the point x exists) of reference is specified. Also, the commutator generally does not exist between \\)(xi) on tri and Mx2) on ctz.

"'Taking the limit of g = 0, (9.18) with (9.21) provides the equation (6.7) in part I. 37
although there exist some special matrix elements of the commutator. Excluding the above points,

the analogies of the basic equations to those in field theory are rather remarkable, and many powerful methods known in the latter may be available, based on the fundamental relationships developed

in this paper. For instance, the Feynman diagram technique in field theory is a consequence of

the quantization, the Heisenberg equation of motion, and the boundary (vacuum) condition. Therefore, we could develop the corresponding technique, as discussed in part I, by adopting the inter- action representation for the wave variables, and also by using the boundary conditions specified by

<0| and jo) of (8.14), i.e. (5.26b), (5.32), and (8.5). Also, the theory of statistical Green's functions can be developed almost in the same form as in the field theory. For instance, it may be noticed

that the equation (9.18) takes the same form as the two particle Green's function in field theory,

and therefore the methods used for solving the latter may be available to get the present Green's

function in an appropriate approximation.

In this respect, the situation will be the same as in the recent theory of thermodynamic system

consisting of many particles, where the theory has been rapidly developed after introducing the

advanced techniques in field theory.

Appendix 1

Operating 1 — yi on both sides of the wave equation (2.1), i.e.,

iyd)ilj = kilj + ri, (A.1.1)

and using the algebraic relations (2.2b) and also (2.18a), the left side of (A.1.1) becomes

(1 - yl){yd)ilj(x) - (1 - yl){ydT)ymx) = (1 - y|)(yar)»//(o-(x)), (A. 1.2)

while the right side becomes

{l-yl){ki}j{x) + 7]{x)}=a-yl)[ka-yl)i}j(x) + kylip(x) + 'n{x)] = k3silj(n{x)) + (1 - yl ){kilj{a-(x)) + rj{x)}, (A.1.3)

where

(l-yi)Ml-yl) =(1-5P(1-5|)-^33(1-5|), (A.1.4)

since 53 has the eigenvalues 0, ±1 and thus the only nonvanishing matrix element is the diagonal

element corresponding to the eigenvector of S3 = 0. Applying Tr' on both sides of <A.1.4),

fe3 = rr'[(l-s|)A;]. (A. 1.5)

Thus, by equating (A. 1.2) and (A. 1.3),

iltin{x))^k33ix)-m-yl)[{iydT)-k}iljicr{x))-r,ix)]. (A.1.6)

Appendix 2

Using the wave equation (2.21) for \Ja{x, [0-2]), (2.35) gives, on dropping the subscript 1 from

Xl,

[(yd) - k] i},{x) = - [(yd) -k]il-yl) kM'

+ ya U aix, {a^)Nab \

i<rix'))\\ + { - (y^^) - k{x')}{\ - yl )h:,(x')

' ]t,(a;')c?o-^, J cr (A.2.1)

d' being the differentiation with respect to x' and cr the surface containing the point x. Here, on 38
referring to (2.33) and (2.34), the second term on the right side of (A. 2.1) becomes

y| [ 8{a{x)-a(x'))[l + {-{yd-;,)-k}a-yl)ksz{x')-']rj{x')da',

J a-

^ylvix)+ yUiydT)-k}a- yl)kM-'vM. (A.2.2)

Hence, since (y3)(l — y5) = (y3r)(l — y§) in the first term on the right side of (A.2.1), the right side

of (A.2.1) becomes

ylriix) - (1 - y|){(ya7-) - A:} (1 - yl)kM - > r,{x) = -qix), (A.2.3)

where, using (A. 1.4), the following relations are used:

(i-y|)(rar)(i-y|)-o,

(l-yl)ka- yl) = (l- yl)^33. (A.2.4)

Thus the ijjixi) given by (2.35) satisfies the given wave equation together with the boundary values on (72. Appendix 3

Let da and db be complex numbers connected by db = e'^da, then

dbn = cos 6 dan — sin 6 dai,

dbi —sin d daR + cos 6 dai , (A. 3.1)

or, in matrix form,

/c?6fi\ _ /cos 6 — sin d\ /daR\

\db, I ~\smd cos d)\da,r {^:i.2)

Hence,

dbndbi = duRdai, or (db) = {da). (A.3.3)

In the general case in which db = wda, it follows that

(db)=\w\''(da). (A.3.4)

When db is a complex vector having two components (dbi, dbz) and connected with da = (da\,

/dbi\_/wu wnX/daiX

\db2) U21 W22)\da2r ^^-^-^^

each matrix element Wij should be considered as a two-rowed matrix as in (A. 3. 2). Thus, intro- ducing the two-rowed matrices 1 and I defined by

it follows that any complex variable w can be represented by

w^wr\-\-wi\, Det{w)^wi + w^=\wY. (A.3.7)

Hence, the whole set of elements wij can be represented in the space resulting from the direct

product of the two independent spaces, one of which is spanned by 1 and I. Therefore the total39
determinant of (A. 3.5) can be obtained independently from the product of the determinants con- nected with the subspaces. Thus

{dbi) (dbz) = \Det {wij) \ \dai) {daz).

Here Det{wij) is the ordinary determinant with respect to the indices i and j. In the general case where

(A.3.8)

it foUows that

dbi dbt dbn '

idb,)idb2). . .{dbn) =

d {bi, 62, • • bn) d {ai, 02, . . ., an)

d{bub2,. ,6n) '

dai da2 . . . dan,

J(dai) {da^) . . . (dan).

(A.3.9)

(A.3.10)

(A.4.1)

d (ai, az, . ., an)

Appendix 4

In the first equation on the right side of (4.15), the first term becomes, using (4.16),

t(i;[o-2]'iVi/;[o-2] =f>[o-2] w| U [0-2, o-i]7V(|/[o-i]' +

J''" U [0-2, (T]N'q[_(j]dx^[(T]

= i4j[cri]Nilj[ai]' -i \ iJj[(r]N7)[(T]dx3[(T],

J(T2

which gives the first and third terms of (4.14). In the same way, in the second equation on the right side of (4.15),

-ii|i[o-i]yVi//[cri]=-iij;[o-i]7v| U[cri,o-2]7Vi//[o-2]'+|'" U[(Ti, (T]Nri[(T]dx3[(r]^

_ fcTl _ =-i>[o-2]Wi//[o-2]'-i il,[o-]Nri[(T]dx3[(T], (A.4.2)

which gives the second and third terms of (4.14).

Appendix 5 ^ Using the definition of ifj-ix) of (8.3) together with (2.19a-c) and (2.39), (8.21) gives

lr[o-2]')=exp {t|/6 [o-2]/Vftal//J [0-2] ' + [0-2] 'Nab^l* [0-2] } |0+),

(»//-[o-i]'|-(0-| exp (A.5.1)

Hence, using the commutation relation (8.7) and the boundary condition (8.5), they are found to satisfy

«/'+[0-2] I 0+[o-2]')=.//:[o-2]' I l//nO-2]'),

l'/'^[o-2]')=o,

(•//-[o-i]' |'/';[o-,]=(i//-[o-i]'|./'-[o-i]',

<r[o-i]'|^;^[o-i]=0. (A.5.2)

40
In view of (8.32), Appendix 6

.i^(l)G(l,2)-G*(2). (A.6.1)

Here G*(2) stands for the conjugate Green's function of the coordinates and x'^. Hence, using

F(l) and F*(2) defined by (9.19), and also (9.4) and (9.6), i.e.,

(' F(1)G(1)-1, F*(2)G*(2)-1, (A.6.2)

it follows that F*(2)^11)G(1,2) = 1, F(l),y~*(2)G(l,2) = l. (A.6.3)

On the other hand, from (8.34) and (9.18),

^~(1)^~*(2)G(1,2) = 1,

/(1, 2)G(1, 2) =F(1)F*(2)G(1, 2) - 1, (A.6.4)

which gives, together with (A.6.3),

/(I, 2)G(1, 2) = F(1)(F*(2)-.5^*(2))G(1, 2) = (F(l)-jn:i)) (F*(2)-J^*(2))G(1, 2)+r{l) (F*(2)-J^*(2)) G(1, 2), (A.6.5)

whose last term becomes

CSTd), F*(2)] G(l, 2)+ (F*(2) -T*{2)).9~{1) G(l, 2), (A.6.6)

since J^(l) and.^ *(2) commute, as noted below (8.34). The last term of (A.6.6) vanishes, as is seen using (A.6.3) and the first equation of (A.6.4).

Thus (A.6.5) becomes

7(1, 2)G(1, 2) = (F(1)-^11))(F*(2) - 9''*(2))G(1, 2) + [yil), F*(2)]G(1, 2). (A.6.7)

Here, F*(2) = G*(2)-» from (A.6.2), and thus using (9.7),

[ .y-(l), F*(2)] f I

{d^)yM){mq(0)F*{2)]

=ijj {do{my,{^)Yunmqm<f>{n) = 11 idim')y,{^)Y*{nDi^-n (A.6.8)

On the other hand, using the second equation of (A.6.5), the first term on the right side of

(A.6.7) becomes, since F(l)-i = G(l) from (A.6.2),

(F(l) -y-(l))G(l)/(l, 2)G(1, 2) = |- A^i + i I

{d^)yi{mi8q{^))] {G(l)/(1, 2)G(1, 2)}, (A.6.9)

which, on account of (9.5), gives

+ i j {d^)yi{mmimmi, 2}Ga, 2)}. . (A.6.10)

41
Thus, (A.6.7) finally becomes

/(1,2)G(1,2)- 1 j

{md^')yiiOTnnD{^-nG{l,2) + i ! {d^)y^imm8|8qimIih2)Gil,2)}.

(A.6.11)

In the same way, another expression is obtained in which the numbers 1 and 2 are interchanged.

Appendix 7

Let be a complex function of a real variable x, defined in a finite range of L ^ x ^ — L,

and be an integer, N'^ 1, and

Ax = LIN, Xn = nAx, re = 0,±l,±2, . . .,±N. (A.7.1)

Then, the set of values J{xn) (N— 1 ^ — N) can be represented by

f{xn)='''^ {2L)-"Vme-'-'"^l\ f(X-N)=Axs), (A.7.2)

where

fm^ {2L)-'i^^xei''"^nli'J{xn) . (A.7.3) n=-N

The proof can be obtained by substituting (A.7.3) into the right side of (A.7.2) and by using the

identity

A'-l N-1 (p-in(n-l) ^^i7r(n-l)\ fl n = I X (2A0-V-<-n-.)/^= 2 (2A0-^e^-'"'"-"/^=^" _

' (A.7.4)

m=-N m=-N ^ ) K

•

In the hmit for A^-^<»,

/m = (2L)-i/2

J'^

dx e---/^^) . (A.7.5)

Thus, (A.7.3) corresponds to the Fourier transformation.

On the other hand, putting

7r(m, n) = TT{n, m) = (2iV)-i/V'''"-^n/^==(2A0-'V'"««/^,

7T*{m,n) — Tr{ — m,n); (A.7.6)

TT*{m, n) being the complex conjugate of trim, n), (A.7.3) can also be represented by the alternative

form

fm= TT{m,n)f{Xn){^xyl\ {k.l.1)

Here, in view of (A.7.4),

m=-Af I2 Tr*{n,m)TT{m,t) = Ki. (A. 7.8)

Hence, according to (A.3.10), (A.7.7) gives

n (rf/m)=|Z)ef(7r)|2n {dAxn)m^l\ (A.7.9)

m=~N n=-N

where, for a complex differential having the real and imaginary parts dfR and df/, respectively,

42
{df) is defined as

{df) - dfndfj. (A.7.10)

while Detin) is the determinant of the matrix 7r(m, n) with respect to the indices m and n. On the other hand, it follows from (A. 7.8) that

\Det{TT)\^ = l. (A.7.11)

Thus, (A.7.9) can be reduced to

fi {dfm)= n idjixn)m"^). (A.7.12)

m=-N n=-N

In the hmit A^—* «, Aa;—* 0 and thus the discrete set of vedues /(jc„)'s tend to a continuous set of values when f(x) is a continuous function of x, while the coefficients /m's still remain as a dis- crete set of values in so far as the range x^—L is bounded.

The result (A.7.12) is obtained for the general case of a complex function f{x). When f{x) is a

real function, (A.7.12) is modified as follows: from (A. 7.2), with the reference to {A.7.1),

where, since/{xnYs are real values.

m=l

f-m —fZi

(A.7.13)

being the complex conjugate of fm, and also the boundary values /o and f-N prove to be real values. Therefore, the real part of (A.7.13) becomes

where

and

/(x„)(A;c)i/2 =^^' \J{xn\m,v)fm,., (A.7.14)

m=0 ^=1

\V2fmR, V=l f/o, V=\

[V2/,„;, v = 2, [/_^, v^2, (A.7.15)

{N-^I^ cos {irmxnlL), v = \

\J(xn\m, v) =VJ'''(m, v\xn) —\ {m^O)

\N-'I^ sin (TTmXnlL), v^2.

U{Xn\0,v)=UnO, P\xn) = - [(2IV)''i'(-)\ v = 2. (A.7.16)

On the other hand, the real part of (A.7.4) can be given in the form

N-\ 2 ^ ^ \J(xT\m.,v)\J'''{m.,v\xi) = hn (A.7.17)

Hence, the determinant of the matrix U defined by (A.7.16) with respect to the indices Xn and

(/ra, v) satisfies the relation

Det{U)Det{lF) = \ox {Det{U)Y = \. (A.7.18)

Thus, from (A.7.14) and (A.7.15),

n {#^n)(A;c)'/n - n n df-^' " = n ^^c^^/-)} • d/odf-^

n=-N m=0 v=l m=l (A. 7. 19)

43
If a bracket (/>t[Z,] ) is defined by

(tJi[L]\4>[L])=exp

as in (5.11a), it may be considered as the limit

i j

ix(x)(f){x)dx^ ,

lim exp

I i{tJL(xn)Ax'l^}{(l){xn)^x'l /V— cc \-n=-N

(A.7.20)

(A.7.21)

Here, applying the representation (A. 7. 14) to ijL(xn) and (}){xn) with (A. 7. 17), the expression (A.7.21) can be reduced to

(/x[L]|(/,[L])=lim exp (A.7.22)

Thus, <i(^[L] can be defined by

A'-l 2

A'->" m=0

nn^=1

^i(/>[L] = Hm f] fl {(27r)-i/2<f</,,,

which can also be represented as follows in view of (A.7.19):

lim n {d<i>{xn){l^l2TTyl^]. (A.7.23)

.A'-»oo n=-N A similar equation holds for c?/u,[L]. Here, as is already noted, although the discrete sets of

values <i>{xn) and /u,(x„) generally tend to continuous sets of values as A^—»oo, the (/)m, i^'s and /Hm, c's remain discrete in so far as the space — L ^ % ^ L is bounded.

References

Alekseev, A. I. (1961), The application of the methods of quantum field theory in statistical physics, Soviet Phys. Uspekhi

4, 23. Bardeen, J., L. N. Cooper, and J. R. Schrieffer (1957), Theory of superconductivity, Phys. Rev. 108, 1175. Dyson, F. (1949a), The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev. 75 , 486. Dyson, F. (1949b), The S matrix in quantum electrodynamics, Phys. Rev. 75, 1736. Furutsu, K. (1963), On the statistical theory of electromagnetic waves in a fluctuating medium (I), J. Res. NBS 67D (Radio

Prop.), No. 3, 303-323. Lippmann, B. A., and J. Schwinger (1950), Variational principles for scattering processes (I), Phys. Rev. 79, 469. Marcuvitz, N., and J. Schwinger (1951), On the representation of the electric and magnetic fields produced by currents and discontinuities in wave guides (I), J. Appl. Phys. 22, 806. Matsubara, T. (1955), A new approach to quantum-statistical mechanics, Prog. Theoret. Phys. 14, 351. Middleton (1960), Introduction to statistical communication theory, ch. 7 — sec. 7.3 (7.28), McGraw-Hill Book Co.

Martin, P. C, and J. Schwinger (1959), Theory of many-particle systems (I), Phys. Rev. 115, 1342. Rickayzen, G. (1961), Scattering by impurities in metals, "Many-body problem," W. A. Benjamin, Inc., New York. Schwinger, J. (1951), On the Green's functions of quantized fields, I, II, Proc. Natl. Acad. Sci. 37, 452; 455. Schwinger, J. (1951-1954), Theory of quantized fields I-VI, Phys. Rev. 82, 914; 91, 713; 91, 728; 92, 1283; 93, 615; 94, 1362.

Stratton, J. A. (1941), Electromagnetic theory, McGraw-HiU Book Co., New York.

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