Commuting, translationary-invariant operational fields

COMMUTING, TRANSLATIONARY-INVARIANT OPERATIONAL FIELDS*

Khr. Ya. KHRISTOV

Sofia, Bulgaria

(Received 16 November 1963)

1. Statement of the problem

LET T(X) be a real operational field. The variable x is the radius vector of a point in ordinary three-dimensional space :‘?. (!Yithout any special changes all the discussion which follows can be generalized to the case where x is the radius vector in a space of any number of variables with a plane Ruclidean or pseudo-Euclidean metric). All the operators generally act in some infinite configurational space H.

The operators q(x) are real

(P+(x) = cpw, they commute with each other

cp(X)cp(Y) - cp(Y)cpb) = 0 (for any x and y)

(1)

(2)

and possess translational invariance

cp(a + x) = u(4cp(x)~-1(a) (for any a arxi x), (3)

where lita) is some unitary representation of the group of translations

U-l(a) = U+(a). (4)

In tnis paper we find the general form of the operational fields T(X), which satisfy (1) - (3). The conditions of non-equivalence and non- indivisibility of the required fields will not be fully used. We shall

. Zh. vjichisl. Mat. mat. Fiz. 5, 6, 991 - 1005, 1965.

18

Commuting, trans lat ionary-invariant operational fields 19

also not consider questions of the convergence of infinite sums and integrals, so that the proofs are to some extent formal.

In quantum field theory an exceptionally important problem is that of finding locally-commuting, Lorentz-invariant operational fields 9(y), depending not only on the space radius vector x, but on the time t: y =

(t, x) [I, p.7221. In the case of a real scalar field, obviously, the operators 9(y) will satisfy all the requirements (1) - (3) which we have imposed on q(x) for any fixed value of t. Consequently the operators p(y) in quantum field theory are one-parameter families s(t, x) of suit- ably chosen operators of the type we have considered

where T(t) is some presentation of the group of translations with respect to the time. Thus the problem of finding q(x) can be considered as a pre- liminary problem to that of finding Lore&z-invariant, locally-commuting fields. On the other hand if we replace the requirement (3) by tile more rigid requirement that the field T(X) is obtained from the Lorentz- invariant field q(y) for t = 0, condition (2) is equivalent to that of local conunutativity. Thus, our problem is also a model of the problem of finding the operational fields in quantum field tneory.

2. General equations for the elements of the matrix q(x)

It is well known that all irreducible unitary representations C’(a) of a group of translations are oue-dimensional, exp( - ip * a), where p is a real vector in E. Consequently we can choose a sasis 2 in i! so that the matrix ‘J(a) is diagonal. Let the basic vectors be characterized by the value of the variable m which has a finite or infinite number of components, running through the same set ‘J!. To each m tnere corresponds one value pm of the variable p, so that the elements of the matrix L’(a)

U(m, a, 12) = h(m - n) exp (-ipnI.a),

where n is a variable analogous to m arid running through the same values, and 6(m - n) is a generalized Kronecker symbol in the set ,i!, i.e. the product of S-symbols (of Kronecker matrices of Dirac functions) corresponding to all the components m. Then the elements rp(xJ will be gCm,x,n)

and (3) will take the form

cp(m, a + x, 72) = exp (-ip,.a)cp(m, x, n) exp (zpn.a).

20 Khr .Ya. Khristov

Let dm, 0, 4 = d m, n) and let x--+0 and a +x, tilencp(m, x, n) =

9(m, d exp[i(pm - pn) *xl.

Note that the operators of non-interacting fields are generally given in the form of Fourier integrals with respect to x [2, p. 301, i.e. in the form of linear combinations of exponential functions. Bearing in mind, however, the general representation of birth and annihilation operators c* [3, p. 461, it is easy to verify that for any fixed choice of m and n,

i.e. for each element p(m, x, n), the function under the integral differs from zero for no more than one value of the variable under the integral, so that the integral is reduced to one exponential function in accordance with the expression for q(m, x, n) found here.

On the other hand, in view of (l), condition (2) shows that with the help of one unitary transformation V(r, m) we can reduce all the matrices q(x) to the diagonal form

q(m,x, n)= 2 v-i(m,r)~(r,X)V(~,II). t

Here r is a new variable, analogous to m, which characterizes the basic vectors in a new canonical basis Be. Let R be the set which it runs through. The function q~(r, x1 gives the diagonal elements of the matrix q(x). By virtue of (1) it is real. The matrix V(r, m) is unitary and does not depend on x

CV(r, m)v+(m, s)= 6(r- s). m

Here the variable s is analogous to F and runs through the same values, and 6(2 - s) are Kronecker symbols in Be. Equating both of the expres-

sions found for cQ(m, x, n) we obtain

dm, n) = 2 V-i( m, +44r, x)v(r, n>exp[i(Pm-Pp,)~4.

By differentiating with respect to x (assuming that (assuming that this is possible) we eliminate Ip(m, n)

i 2 V-‘(m, r) “!Jix) V(r, n)- zpmF(m, r)q(r, x)V(r, n)+ 7 +

+ 2 Wm, +W: x)W, 4~~ = 0.

We multiply on the left by V(r, m) and on the right by V-‘(n, s), first replacing r by t in the formula. We obtain

Cammut ing, translationary-invariant operational fields 21

W(r, s) = CT+, m)pT?Y-‘(m, s). m

Then by virtue of (6) we find that

(7)

(8)

3. Conditions of integrabifity for V and q~

Since y is a one-component function of the variable r and x, and (8) is a three-component (vector) equation, which must be valid for any r,

s and x, it is clear that the function W must satisfy the same conditions of integrability in order that we may obtain non-trivial solutions (depending on r and x) for $r, x1. To find these conditions we note that all the equations are invariant with respect to the substitution

F = F(r), (9)

if

V(r m) _ -!I!! ‘j2- t - I I ar V(T, m) t *u(r,x)= g+w, (10)

where Ia?/ al-1 is the Jacobian of the transfofmation (9). We can con-

sider (9) as the transition to the new basis & which differs from 2,

only in the numbering of the basic vectors. Therefore instead of r we introduce three new and also multicomponent variables p ‘, p and r ’ so that the following conditions are satisfied.

1. For any fixed choice of the variable p’ the variable r ’ assumes only a denumerable (or finite) set of the variables p’ and F-’ all the of one another and continuously in these components, where the number and r’.

2. The equality

of values, and for any fixed choice c~ponents of p vary independently some (open) region in the space of of components of P aay depend on p’

21, (pi’, ri’, Pi, x> = 9 (Pz’, r2'7 Pz, x)

holds for any x when and only when rl’ = r-2’ and pr = p2.

i-11)

22 Khr . Ya. Khr is t ov

Let 9’ be the set wnich p’ runs through, f?‘(p’) the set which r ’ runs through for any given p. o(p’, r ‘1 the region which p runs through for

given p’ and I-’ and n’(r’, p) the set which p’ runs through for given r’ and P.

Rearing in mind that the first term in (81 is zero if r # s and that the first factor of the second term is zero if r’ # s’ or p # u (from (111 it is obvious that if r ’ # s’ or p # a it differs from zero) we find tilat the matrix v(r, s) must be of the form

W(r,s) = iuk(r)h(p’- o’)&(p - 0)6(r’- s’) +

+ v(r, cq6(p - a)h(r’- s’); (12)

where r’E I?‘, p E !)(r’) and p’ E :)‘(r’, p). (Summation by repeated

indices is implied). Here k characterizes the components pk of the variable p and 6k(p - a1 represents the derivative of the function S(p - o) with respect to pk. Yaturally the dimensionality of the function S(o-al may depend on the values of r ’ and p ‘. Since the varianle p is real and

tue matrix 1’ is unitary we find from (7) that the matrix \V(r, s) is self-

conjugate. Consequently

uh (r) = Uk* (r), VW, r’r P, 0’) = v* (0’9 r’, P, P’),

wnere the index * means the complex conjugate.

We s?iall assume that the functions qy and L’ are different respect to o. !Fe substitute (12) in (Sl and find tnat

iab le with

a+ (r7 4 uk(r) -- = %(r’,x) ah ax * (14)

We also substitute (12) in (7). We interchange m and n in the equation ootained and multiply on the right by v(s, a). Implying SUmmitiOn with

respect to s we obtain

iuk (r) avA;km’ + 2 v(r, d) V(d, p, r’, m) = V(r, m)pm. (15) a’

In place of % w: introduce two varianles P and p, where 9 is an addi- tional variable which is characterized by the various values of m which correspond to the given value of p. Then (15) takes tie form

iuh (r) aV(r, CL, P)

dPh + l$?r, 4 W, P, f, CL, p) = VP, CL, P)P. (19

a

Commuting, translationary-invariant operational fields

Note that the substitution

W, x1 = ,Zx(r, Hex+ Wx) P

transforms (4) into the equation

juk (r) dx Cry P>

hh = x(r, PIP,

which is similar to (141, and conversely if

U(r, jk,p,x) = V(r, P,P) exp (-iP*x),

then this function will satisfy the equation

23

(17)

(18)

of type (14).

If we consider V(F, ~1, p) as a family of functions of the argument F

with parameters p and ~1, they are orthonormal

2 V(r, c1, W’(r, y, q)=~h-%W--q? (19)

and form a complete set of functions in 8.

For given U& and Y we can consider (14) and (16) as equations for 1~ and V. Since there is no differentiation, nor integration, nor summation with respect to F' and s’ in (14), it can be split up into separate sub-

systems for each component corresponding to given p’ and F’. In (16) there is no differentiation nor summation with respect to F’ so that it can also be split up. In order that the solutions may exist some conditions of integrability must be satisfied, viz. functions Uk and v, which do not depend on ~1 and p, must be chosen so that, first, the sequence of all equations (lS), which are characterized by values of the parameter

I F , may permit of some sequence of SOlUtiOUS of V(F, u, p), which are characterized by the variables u and p, which form a complete set of functions with respect to all functions F(F) in the space of variables F and, second, to different values of p and F‘ there correspond different solutions y(r, x) of equation (14) which are independent of p’,

4. The conditions of integrability for uk and v

According to (17), to each solution of equation (16) there corresponds

24 Khr.Ya. Khristov

one solution of equation (18), while both these functions coincide if x = 0. Then the condition of completeness, which states that any function F(F) can be presented in the form of a linear combination of solutions of equation (16), with suitable p, shows that equation (18) must also have a solution which reduces to an arbitrarily preassigned function F(F) if x = 0. This means that equation (18) must be not -only integrable, i.e. have a non-zero solution but also be completely integrable, i.e. has a solution equal to any preassigned function F(F) if x = 0. In order to find this condition of complete integrability we first of all write (18) in the form

Here for brevity the arguments are not written out, F = 6(p’ - u’), and summation with respect to u ’ is implied (6 and v are matrices related to the arguments p’ and CT’). We take the curl of this equation and obtain

a duk------

dPk

where x denotes the vector product in the three-dimensional space fi. (If the space E has more than three dimensions, instead of the vector product the antisymmetric part of the direct product appears). Eliminating the derivatives with respect to x from these two equations we find that

In view of the fact that the product of the adiSDMK?triC matrix Uk x Ul

by the symmetric matrix a20/apk+, is zero, this equation is transformed into

- 6uk x 2+i(ulXv+vXw)]$ +[iukx~-t-VxV]~=~

(21)

The condition of complete integrability of equation (18) gives the following equations for Uk and v:

8uk x au1 aphfi(ul xv+v x lll)Z 0, (22)

!k x $+ivXv=O. (23)

Equation (14) must also be integrable for any value of F’ (this argu-

ment will be omitted in future) and must have a solution ~(p, x), which

Commuting, translationary-invariant operational fields 25

does not depend on p’ and is such that, according to fIl1, if pl # 13,

Q (Pi! x) = $3 fP2,X) (24)

for at least one x. If we take t;le curl with respect to (14), with

P ’ = Pl’, we obtain

a uip k apk

a* - x ax = 0,

Where ul, k = u(pl’, p). In addition, from (141 with p’ = pz ‘, we have

so that, on eliminating the derivative witn respect to x, we find that

= 0,

or

ui, k x u2, I -+u,,kx~$=o. aPk%X

125)

This relation expresses the condition of integrability, because if y(p) is some function, which satisfies these equations, we can find a function y(p, x) which satisfies (141, and which reduces to y(p) if x = xg. In particular, if pl’ = pz ’ = p’ the first term in this equation will be 9

and we obtain first-order equations for v(p):

‘ilk x (26)

Let ytn(p) be the complete set of f~ln~tiona~ly-independent solutions

of equations (251, such that any solution of (25) can be put in the form

*((P) = E’(V(P)). (27)

Tine function F is generally not arbitrary, but nevertheless exists, since the general solution of any linear equation can be presented as a linear homogeneous function of some of its solutions. Since (25) expresses the condition of integrability, any solution y~(p, x) of equation (14) can be put in the form

If the solutions ly”(p) are such that at least one pair of values of p1 and pz exists such that I” = v”(p2) for all n, then according to (28) we shall also have y(p1, x) = +~(pz, x) for any x, however we choose

26 Khr .Ya. Khristou

1’. But according to (24) this is impossible. Consequently there must be sufficient solutions v”(p) so that if pl f p2 we have yn(pl) # y”(pz) for at least one II. From this it follows that the set y”(p) is complete, i.e. not only any solution of (25) but also any function yl(p) can be put in the form (27) and belong to F (of course we do not also require the function F(p) to satisfy (25)). In particular if the number of components of the variable p is i<, the system (2.5) must also have at least K functionally independent solutions (if their number is less than K the equation y”(p) = c” for a given choice of c”, has more than one solution

for p, which on;y contradicts what we have proved). Then bearing in mind that any linear combination of solutions of equations (25) is also a solution, on taking p = p” and 1 = lo, we can find a solution u/O of equations (25) such that if p = p” all the derivatives 89” ,‘a~, are zero except that aQO / dplo = 1. Then from (26) we find that uk satisfies the equation

uk, x _ = 0 @k

(29)

if p = p” and 1 = lo, i.e. for all p and 1, since p” and 2’ can be chosen arbitrarily. We now show that the function uk(r) does not depend on p’. Let pl’ and ~2’ be such values of p’ that if r’ = r-‘O, p = p”, k = k” and a = a0

(Ilka, va, xa* ti = 1, 2, 3 are the components of Uk, v, x).

Equation (14), with a = a0 and p’ = pl’ or p’ = pz’, gives

all, 0 - = qO(pl’, P)-- 7 w alCl

a32 a6’k - = qy(P2’, P)-. axa0 apk

Substituting p = p” and y = v” we obtain

a4J0 ___ = q,“(Pz’, P”) 7 axd

which is impossible in view of (30). This proves that uk(r) does not depend on p ‘. Then the condition of integrability (25) is reduced to (26) and is satisfied in view of (29). Consequently equation (14) is completely integrable. So long as Uk does not depend on p’ we obtain Ukxv+vxuk = 0 and so (22) also reduces to (29). Thus the conditions of integrability of equations (14) and (16) reduce to (29) and (23).


5. Finding the functions uk

Roth equations (29) and (23) are not standard in the sense that, in

view of the presence of vector products, the determinants of the coeffi-

cients of the derivatives of the unknowns Uk and Y with respect to any

of the independent variables pk are 0. Therefore these equations cannot

be solved for the derivatives of the unknowns with respect to any pk and consequently we cannot apply Cauchy’s theorem, which defines the set of their solutions. We reduce the systems (29) and (23) to simpler form by a suitable choice of the variables pk. We make a substitution of the

type (9)

Then

Consequently, in order to preserve equations (5), (5), (7) and (12) we

must assume that

(321

Then equations (29). (23), (16) and (14) are also preserved for Uk, Y, if and v. Consequently the theory is invariant with the change of variables (31). Using this, for any fixed value of r’ we can carry out a

change of variables of the type (31) such that the new functions Uk assume the simplest possible form.

SUppoSQ that we need one solution uk(p) of equations (29). We first of all reduce to canonical form the components Uk’. We shall consider the general case where there is among them at least one which is non- zero in some region of the variables pk. According to (32)

(33)

28 Khr.Ya. Khristou

If we wish to make all tne iI1 equal to zero then the pl(p,,,) must be functionally-independent solutions of the equation

In particular, if o has a finite number of components K, then this equation must have I! functionally-independent integrals. But, as is well known, a linear homogeneous equation whose coefficients are not all zero, has no such number of functionally-independent solutions. As a consequence

one of the components “kl must be different from zero, and so we assume

that

iii’ = 1, Ux’ = 0 tx = 2, 3, . . .j. (34)

The transformation (31), which reduces IZ~’ to such a form, is given by

the formulae

After this transformation equations (29) can be written in the form

au13 dUi" di.Q ~+---u~3-=0, ___=o,

aup apk dPh aE?i

- 0. dpi-

From the last two equations we find that

uz3 = Fz3 (px) (x = 2,3, . . .) ) (35)

and the substitrltion of these expressions in the first equation gives

&2Ff _ F

dP% saF12 = 0

%&ii- (x=2,3 ,,.., l=1,2,...) (36)

(summation with respect to the repeated index K is implied). From now on these variables 6~ and functions & Will be denoted Simply by Pk and Uk. We make use of the freedom which remains so as to change nk2 into

canonical form, while retaining (34). This second transformation is brought about again by (31). In view of (34) and the first equation (29) we must have

from which we obtain

if = pi + J’t(p& ;x = Fx(pn) (x, h = 2,3,. . .). (37)


In view of (35) the second equation (29) reduces to

~FI aFi ~2 + ux2 -- = Fi2 + F,c2 7 I= 1, dPx dpx if

dF ux&F1 = F,2h

(38)

I dPx dPx 1‘ 1=2,3,... .

It is best to assume that all u12 are zero, but this is impossible, because in the general case, where not all &2 are zero, the equation F,2(@‘/ ap,) = 0 has an insufficient number of functionally-independ-

ent solutions for the transformation (31) to be unique and reversible. In particular if )a runs through J? values and K through h’ - 1 values, then it is clear that the equation F,2(dF / b’p,) = 0 will have only K - 2 functionally-independent solutions, and the conditions of non- degeneracy (31) requires !I - I such solutions.

We therefore assume that

ui - 2 = Pi2 = 0, U22 = Fz2 = 1, iI&2 = F,2 = 0 (P = 3,4, . . .) . . (39)

According to our assumption not all qh2 are zero and consequently the functions Fh2 can be defined so that all these equalities hold. In view of (3.5) it is clear that the requirement of reversibility of the transformation (31) does not impose any restrictions on F12 and we can assume that n12 = Fr2 = 0. Substituting (39) in (36) we obtain dFF/dp2 = 0

and consequently u13(ph) = F13(px) = G?(p,) (P = 3, 4, . . .).

We make use of the remaining freedom in the choice of variables ok to simplify u13(p ) = G13(p,).

In view of (34) the second equation (31) gives

a- ap2Pi = 0, _dpz=o,

aP2 -q& = 0. dP2

Hence from (37) we find tb t

~~=pi+Fi=pi-tGi(p,), G = F2 = ~2 + G2tpp.),

ii% = Fv = G,(p,) (I.&v = 3,4,. . .).

Then the third equation (32) reduces to

30 Khr.Ya.Khristou

;‘a = uks I U13-f up

3 ~GI qq- = ,‘&3 +G 3aG1

IL %P

if I_= 1,

,!&={ u~+~,~$$=C.~+G$$ if 1=2, (40)

dG”

t UP3 dp, = Gp3 f$ if l=V=3,4,....

P

Proceeding from the same considerations as in the investigation of (38) we obtain

ui3 = 7&3 = 0 7 u33 = 1 7 uv3=o (v=4, 5, . ..). (41)

by which a simplification of the vectors uk is accomplished.

6. Finding the function v

We have found that without loss of generality we can use the values Of (34), (39) and (41) for uk:

ui= (1,0,0), uz= (OAO), u3= (O,O,l), uu=o (v=4,5, . ..).

Then (23, reduces to

or in expanded form

dv3 au* --+ i(vW- v3v2) = 0, g-ap3

as ad --- aP1 am

+

ad d”3+ i(v3v1

a&G -api i(W-- v”v’) = 0,

(421

- VW) = 0,

(43)

or even

~~ww,P,~‘)+ Q v(p’,p,z’) x v(r’,p, o’) = 0,

where rot,, is the curl in the space of variables ~1. p2 and p3.

The remaining arguments pV do not enter into (43) in explicit form, but v may depend on them, so that all the arbitrary elements (constants and functions), which appear in the general solution of this equation, will depend arbitrarils on pv.


2 We now show that we can choose vl, uplEcl and $l=cl pzzc2 arbitrarily

for any values of the remaining arguments and by this t;le solution of (43) (or (44)) is uniquely determined. We take the divergence of (44) and obtain

v.rol, v - (rot, v) .v = 0. ($5)

If instead of rotp v in accordance witn (44) we here substitute -iv x v, we obtain the matrix identity

v. (v x v) - (v x v) .v = 0. (46)

This shows that the condition of complete integrability is satisfied. Conversely, if the second and third equations (43) are valid for anv pk. and the first for some given values of ~1, the derivative of its left- hand term will be zero for the same value of ~1.

Since vl, 2

vpl=cl and &=cl p2=c2 are given for all values of the re-

maining arguments, then from the first equation (43) with p1 = cl we can

determine viQzCQ so that this equation is valid if pl=cl . The question

reduces to the solution of one arbitrary linear first-order differential equation, or, to be more precise, to one system of integro-differential equations, in view of the summation with respect to T ’ in (44). After we have shown that this equation holds if p1 = cl it will also hold for any

pl. Knowing vl, 2 3

upl=cl and vpl=cl, from the last two equations (43) we

can uniquely determine u2 and vu3 for aRy ~1.

Paragraphs 5 and 6 exhaust the general case when I( 3 3 and among the components Uk’, UK2 and up3 there is at least one which is different from zero (case A). If f? < 3 or in some of the equations (33), VU?), (40) all the uk’, uK2 or uV3 are zero, various degenerate cases are obtained

7. ‘he case of degeneracy

which can be reduced to the following three.

Case B. U1 = (1, 0, O), u2 = (0, 1, O), ucI = 0. Then (42) reduces to

al73 _+ i(vW- uW)= 0, afJz (47)

The sum of the derivatives with respect to pl and p2 of the first two of these equations is identically zero in view of these equations themselves.

32 Khr.Ya. Khristou

Consequently the condition of integrability is satisfied (as in case A

we can take ul, 2

91=c1 and v&=c2 arbitrarily).

The third equation (17) determines II’ uniquely and the first two determine v3 for all values of p.

Case C. U1 = (1, 0, oj, uK = 0. Equation (42) gives

2 Here we can take ti’, upl=cl and viIXc, arbitrarily, only if ~zpl=~~ and

viI=cl commute. Then it is nor, difficult to verify that the condition of

integrability is satisfied. (The derivative of the left-hand term of the

first equation is zero, so that it is satisfied for any ~1). Consequently the second and third equations determine v2 and v3 uniquely for any ~1.

Case D. All the Uk are zero. Then (42) reduces to

vxv=o. (49)

The components of the vector v are given by three arbitrary communting matrices ua(p’, a’).

8. The solution of the equations obtained

All the equations (45), (47), (48) and (49) are linear homogeneous ordinary differential equations with variable coefficients and free terms and are of the form

it!E--SH dt -

-HHS+HlJ (50)

(except for the first equation (48) and equation (49), which simply express the commutativity of certain matrices u’, u2 and u3, #here ‘3 is the required. and /I and :fu given, Hermitian matrices which depend on t.

In our case t can be p1 or ~2, .T will be u2 or v3, !f the various already-

known components of IJ~, v’, v3, and Ii0 their derivatives. The basis vectors in the configuration space of the operators are characterized by the values of the variable p’. 9s is known from quantum field theory, the general solution of equation (501 is given in the form

s(t) = E-‘(t) [Ho + SE(t)Ho(t)E-l(t)dt]E(t), 1”

(51)


where 80 is an arbitrary constant matrix and

E(t)= Texp[ ii H(t)&], t4

where T indicates that we must satisfy the exponential expression following after T with respect to t.

The variables pp (0 < /3 < a, a = 3, 2, 1, 0 in cases A, 8, C and D respectively) are chosen in relation to the remaining components p< of the variable p. These last come in everywhere in exactly the same way as I“, the difference lying only in the fact that r’ assumes discrete values and pt varies continuously. Therefore it is advisable to change the notation; the new variable p will only join pp; then the number of its components will be no more than three, and the variable r ’ will join the old variable r’ and all the values of pt.

After Uk and v have been found we must substitute the expressions obtained in (14) and (16) in order to find y and V. Equation (14) reduces to

a* (PI r’, x> y% B %‘(P, f, x> az, = a

i ,whereYa aB _ 1, if cz= fl <a,

af+ -

0, if a=f3>a or a#fi

(summation with respect to the repeated index p is implied). Hence we directly obtain the general solution

\c, (PI f, x) = Pa(pa + “6; r’) (0 < p < a < 3). (52)

According to (24), Pa(xb, r) cannot have periods T in .x3 which are con-

tained in D(r ‘J, but if ri’ # rzi, then Fa(xb, ri’) # ~o(rg, n’) for at

least one xp. Consequently in cases A, B and C the function w depends, respectively, on the 3rd, 2nd or 1st components of xi3 and p+ and in case D it is independent of x and p. Equation (15) reduces to

where vu are chosen so that the condition of integrability of (42) is satisfied. The solution of this integro-differential equation can be reduced to a purely integral equation. For this purpose we introduce the Fourier transform Q of tne function V:

34 Khr.Ya. Khristov

where y has components Y? (0 < ? < a). In view of the fact that p varies only in some region D(p ‘, r ‘), this definition is not single-valued. To

eliminate this non-uniqueness without loss of generality we can take V = 0 everywhere outside the region $p’, r ‘):

Substituting (54) in (53) we obtain

2 @(p’, Y, r’, z, u’) Q ( (I', 2, J, P, P) + (VP, 3s - P”> Q (P', Y, f-', PL, P) = z. 0' (55)

= Qb', Y, r’, P, P)P”,

where

w(p’, y, r’, z, o’) = 1 dp exp{i(y - z) . p}v(p', p, r’, 2). (56)

Here the variable z is analogous to y and runs Thus the question of finding 3 reduces to that functions and the corresponding eigenvalues of

U,a (p’, y, r/, z, (3’) - 6 (Y - z) yaa’ q/p

After we have found the functions Q and the

through the same values. of finding joint eigen- the three operators

(a = 1,2,3). (57)

corresponding eigenvalues P of the operators (57) we can find V from (54). The operators (57) are selfconjugate and commute among themselves so that they have just the necessary set of solutions for constructing the unitary matrix Q, and consequently also V, for any r ‘. For any r’ these solutions are characterized by the parameter P which runs through the spectrum of the eigenvalues of the operators (57). and by one more parameter ct’, which, in the case of degeneracy, characterizes the various mutually orthogonal eigenfunctions, corresponding to the given p. This also determines the sets P and :W’(P), on which P and p’ are given. Note that in case D (55) is reduced to

Cv(p’, r’, o’) Qd, J, FL’, P) = J’(P’, J, P’, P)P. a’

(58)

We shall denote by Q+(p', y, p', p) the solutions of equation (55) for any given r‘. They (and according to (54) the functions V,#(p', p, p,, p) also) form orthogonal systems of vectors in the spaces II, ‘,“corresponding to the quantities r ‘. All these spaces are orthogonal to each other. Consequently, to obtain the unitary matrix I’(r, m) it is sufficient to assume that


w, m) = v(p’, p, F’, m’, CL’, P) = a(F’ - m’) vrr (p’, p, $, P) ,‘ (59)

where the variable m’ is analogous to r ’ and runs through the same values, so that m = (m’, u’, p).

We note in conclusion that the basis 3, in which we have found the elements of the matrix q(x), is not arbitrary: in it the representations of the group of translations U(a) are diagonal, but the basis is not uniquely determined. Consequently, among the fields q(x) we have found there may be some which are equivalent. The solutions found are probably decomposable, since the operators T(X) commute and consequently all of them can be reduced to diagonal form. but the solutions in the general case are not diagonal. Non-decomposable solutions, however in this sense do not interest us because they are not invariant with respect to (3), since with translations they will be transformed one into the other. Therefore we introduce the idea of complete decomposability, which requires decompositions not only of all the matrices q(x), but also of all L’(a), or otherwise the solution p(x) will be completely decomposable, if a matrix exists, not a multiple of the unit matrix, which commutes not only for all $x1 but also for every u(a). In view of the fact that the matrices 6(r - slv(r, x1 and (I(a) are diagonal, the presence of the factor 6(r’ - m’) in (59) shows that if m’ takes on more than one value the field q(x) is completely decomposable, since the various cells of the matrix p(x), and also of the matrix u(a), which are characterized by the quantities m’ and n’ (n’ is a parameter analogous to m’ and running through the same values) are zero if m’ # n’. This does not of course exhaust the question of the complete decomposability and equivalence because we cannot assert that all the remaining solutions are not equivalent and not completely decomposable.

9. The construction of the operational field T(X)

In conclusion we show what operations are necessary and in what order they must be carried out in practice to construct the required field q(x). In the same way it will be shown that the equations found are not

,only necessary but also the sufficient conditions, which characterize the field Q(X). Here we shall omit the variables m’, n’, r ‘, s’, because there are completely decomposable solutions corresponding to them. The Einding of Q(X) reduces to the following operations.

1. We choose the variables p’ and p, i.e. we choose notations for their components and indicate the set D’ in which p’ varies and the

36 Khr.Ya. Khristov

region D(p’) in which p varies for a given p’. Here a(p’J (the number of components of p) is not more than 3. We also find the region D and set D’tpj which p and p’ run through for a given p.

2. We choose arbitrary functions which contain the general solutions Of equations (43), (47), (48) or (49) depending on the number of components, viz:

(a) a=3; the following are arbitrary ui = z9(p’,pi, p2, p3, o’) : u”,,_,, =

02(p’, p2, p3, a') xf &-c,, p:=cz =I u3(p', pp (5');

(b) a = 2; the following are arbitrary ~1 = ui(p’, pi, p2, o’), u~,,~, = n

uZ(p’, p2, 0') H Q,=c,,p,=c* = u3(p', 0');

(c) a = 1; the following are arbitrary ~9 = vi(,p’, pi, (T’), z&_ = $(p’, 0

and T+,,=~, = u3 (p’, 6’) , while u&, and v;,=c, commute;

(d) a = 0; all va = va(p ‘, a’) are arbitrary and they all commute.

Here all ua are selfconjugate with respect to p’ and u’,

3. We find all the remaining components of Y (for all the remaining values of the ,variable p) using:

(a) The first equation (43) with pr = cl, to determine v:~=~,, and

the remaining two equations (43) to determine v2 and II~ for any ~1;

(b) the third equation (4’7) to determine v2, and the first two to determine v3;

(c) the second and third equations (48) to determine v2 and v3.

Here p varies in D and p ’ in D’(p).

4. We choose the function F,(xp), which satisfies the requirements,

given for the derivation of (52) and with the help of this equation we determine w(p, x) = F,(pp + xp) (0 < i3 < a).

5. From (56) we determine w. After this from (55) we find compatible orthonormal functions ? and the corresponding eigenvalues of the p operators (57). The set P of possible values of the variable p is similarly determined as the spectrum of the operators (5’7) and the set h!‘(p) of possible values of the variable ~1’ for a given p.


6. We substitute y and V in (5) and find the elements ~(cl’, p, x, g,

v ‘) of the required field q(x).

Each of these solutions is characterized by the parameter F’, i.e.

Q(X) = q+‘(x). To find the general solution we must still choose a unitary matrix Ttm’, F') and assume that p(x) = ~(m’, p’, p, x, q, v’, n’) =

~)((m’,r’)~cp,/(~~,p,x,q,y’)T-i(F’, n’). 7’

Acknowledgements. I wish to express my indebtedness and thanks to N.N. Bogolyubov and A.A. Logunov for their interest in my work and also to I.T. Todorov for useful advice and critical remarks on the progress of the work and to N.A. Chernikov for discussing and helping with the

planning of the paper.

Translated by H.F. Cleaves

REFERENCES

1. SCHWEBER, S.S. An introduction to relativistic quantum field theory.

Peterson and Co., New York, 1961.

2. BOGOLYUBOV, N.N. and SHXRKOV, D.V. Introduction to quantum field

theory (Vvedenie v teoriyu kvantovannykh polei). Gostekhizdat

Moscow, 1957.

3. VENTTSEL. T. Introduction to the quantum theory of wave fields

(Vvedenie v kvantovuyu teoriyu volnovykh polei). Gostekhizdat, Moscow, 1947.

4. SCHWARTZ, L. Theorie des distributions. T.I. Hermann, Paris, 1950.

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Commuting, translationary-invariant operational fields