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<ul><li><p>COMMUTING, TRANSLATIONARY-INVARIANT OPERATIONAL FIELDS* </p><p>Khr. Ya. KHRISTOV </p><p>Sofia, Bulgaria </p><p>(Received 16 November 1963) </p><p>1. Statement of the problem </p><p>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 gener- ally act in some infinite configurational space H. </p><p>The operators q(x) are real </p><p>(P+(x) = cpw, they commute with each other </p><p>cp(X)cp(Y) - cp(Y)cpb) = 0 (for any x and y) </p><p>(1) </p><p>(2) </p><p>and possess translational invariance </p><p>cp(a + x) = u(4cp(x)~-1(a) (for any a arxi x), (3) </p><p>where lita) is some unitary representation of the group of translations </p><p>U-l(a) = U+(a). (4) </p><p>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 </p><p>. Zh. vjichisl. Mat. mat. Fiz. 5, 6, 991 - 1005, 1965. </p><p>18 </p></li><li><p>Commuting, trans lat ionary-invariant operational fields 19 </p><p>also not consider questions of the convergence of infinite sums and inte- grals, so that the proofs are to some extent formal. </p><p>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 </p><p>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. </p><p>2. General equations for the elements of the matrix q(x) </p><p>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 compo- nents, 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) </p><p>U(m, a, 12) = h(m - n) exp (-ipnI.a), </p><p>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) corre- sponding to all the components m. Then the elements rp(xJ will be gCm,x,n) and (3) will take the form </p><p>cp(m, a + x, 72) = exp (-ip,.a)cp(m, x, n) exp (zpn.a). </p></li><li><p>20 Khr .Ya. Khristov </p><p>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. </p><p>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, </p><p>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 accord- ance with the expression for q(m, x, n) found here. </p><p>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 </p><p>q(m,x, n)= 2 v-i(m,r)~(r,X)V(~,II). t </p><p>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 </p><p>CV(r, m)v+(m, s)= 6(r- s). m </p><p>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- </p><p>sions found for cQ(m, x, n) we obtain </p><p>dm, n) = 2 V-i( m, +44r, x)v(r, n>exp[i(Pm-Pp,)~4. </p><p>By differentiating with respect to x (assuming that (assuming that this is possible) we eliminate Ip(m, n) </p><p>i 2 V-(m, r) !Jix) V(r, n)- zpmF(m, r)q(r, x)V(r, n)+ 7 + </p><p>+ 2 Wm, +W: x)W, 4~~ = 0. </p><p>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 </p></li><li><p>Cammut ing, translationary-invariant operational fields 21 </p><p>W(r, s) = CT+, m)pT?Y-(m, s). m </p><p>Then by virtue of (6) we find that </p><p>(7) </p><p>(8) </p><p>3. Conditions of integrabifity for V and q~ </p><p>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 (de- pending on r and x) for $r, x1. To find these conditions we note that all the equations are invariant with respect to the substitution </p><p>F = F(r), (9) </p><p>if </p><p>V(r m) _ -!I!! j2- t - I I ar V(T, m) t *u(r,x)= g+w, (10) </p><p>where Ia?/ al-1 is the Jacobian of the transfofmation (9). We can con- </p><p>sider (9) as the transition to the new basis & which differs from 2, </p><p>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. </p><p>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. </p><p>2. The equality </p><p>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 </p><p>21, (pi, ri, Pi, x> = 9 (Pz, r2'7 Pz, x) </p><p>holds for any x when and only when rl = r-2 and pr = p2. </p><p>i-11) </p></li><li><p>22 Khr . Ya. Khr is t ov </p><p>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 </p><p>given p and I- and n(r, p) the set which p runs through for given r and P. </p><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 </p><p>W(r,s) = iuk(r)h(p- o)&(p - 0)6(r- s) + </p><p>+ v(r, cq6(p - a)h(r- s); (12) </p><p>where rE I?, p E !)(r) and p E :)(r, p). (Summation by repeated </p><p>indices is implied). Here k characterizes the components pk of the vari- able 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- </p><p>conjugate. Consequently </p><p>uh (r) = Uk* (r), VW, rr P, 0) = v* (09 r, P, P), wnere the index * means the complex conjugate. </p><p>We s?iall assume that the functions qy and L are different respect to o. !Fe substitute (12) in (Sl and find tnat </p><p>iab le with </p><p>a+ (r7 4 uk(r) -- = %(r,x) ah ax * </p><p>(14) </p><p>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 </p><p>respect to s we obtain </p><p>iuk (r) avA;km + 2 v(r, d) V(d, p, r, m) = V(r, m)pm. (15) a </p><p>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 </p><p>iuh (r) aV(r, CL, P) </p><p>dPh + l$?r, 4 W, P, f, CL, p) = VP, CL, P)P. (19 </p><p>a </p></li><li><p>Commuting, translationary-invariant operational fields </p><p>Note that the substitution </p><p>W, x1 = ,Zx(r, Hex+ Wx) P </p><p>transforms (4) into the equation </p><p>juk (r) dx Cry P> </p><p>hh = x(r, PIP, </p><p>which is similar to (141, and conversely if </p><p>U(r, jk,p,x) = V(r, P,P) exp (-iP*x), </p><p>then this function will satisfy the equation </p><p>23 </p><p>(17) </p><p>(18) </p><p>of type (14). </p><p>If we consider V(F, ~1, p) as a family of functions of the argument F with parameters p and ~1, they are orthonormal </p><p>2 V(r, c1, W(r, y, q)=~h-%W--q? (19) </p><p>and form a complete set of functions in 8. </p><p>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 condi- tions 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 </p><p>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 differ- ent solutions y(r, x) of equation (14) which are independent of p, </p><p>4. The conditions of integrability for uk and v </p><p>According to (17), to each solution of equation (16) there corresponds </p></li><li><p>24 Khr.Ya. Khristov </p><p>one solution of equation (18), while both these functions coincide if x = 0. Then the condition of completeness, which states that any func- tion F(F) can be presented in the form of a linear combination of solu- tions of equation (16), with suitable p, shows that equation (18) must also have a solution which reduces to an arbitrarily preassigned func- tion F(F) if x = 0. This means that equation (18) must be not -only inte- grable, 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 </p><p>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 </p><p>a duk------ </p><p>dPk </p><p>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 </p><p>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 </p><p>- 6uk x 2+i(ulXv+vXw)]$ +[iukx~-t-VxV]~=~ (21) </p><p>The condition of complete integrability of equation (18) gives the following equations for Uk and v: </p><p>8uk x au1 aphfi(ul xv+v x lll)Z 0, (22) </p><p>!k x $+ivXv=O. (23) </p><p>Equation (14) must also be integrable for any value of F (this argu- </p><p>ment will be omitted in future) and must have a solution ~(p, x), which </p></li><li><p>Commuting, translationary-invariant operational fields 25 </p><p>does not depend on p and is such that, according to fIl1, if pl # 13, </p><p>Q (Pi! x) = $3 fP2,X) (24) </p><p>for at least one x. If we take t;le curl with respect to (14), with </p><p>P = Pl, we obtain </p><p>a uip k apk a* - x ax = 0, </p><p>Where ul, k = u(pl, p). In addition, from (141 with p = pz , we have </p><p>so that, on eliminating the derivative witn respect to x, we find that </p><p>= 0, </p><p>or </p><p>ui, k x u2, I -+u,,kx~$=o. aPk%X </p><p>125) </p><p>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 </p><p>and we obtain first-order equations for v(p): </p><p>ilk x (26) </p><p>Let ytn(p) be the complete set of f~ln~tiona~ly-independent solutions </p><p>of equations (251, such that any solution of (25) can be put in the form </p><p>*((P) = E(V(P)). (27) </p><p>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 </p><p>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 </p></li><li><p>26 Khr .Ya. Khristou </p><p>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</p></li><li><p>Commuting, translationary-invariant operational fields 27 </p><p>5. Finding the functions uk </p><p>Roth equations (29) and (23) are not standard in the sense that, in </p><p>view of the presence of vector products, the determinants of the coeffi- </p><p>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 </p><p>be solved for the derivatives of the unknowns with respect to any pk and consequently we cannot apply Cauchys 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 </p><p>type (9) </p><p>Then </p><p>Consequently, in order to preserve equations (5), (5), (7) and (12) we </p><p>must assume that </p><p>(321 </p><p>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 vari- ables (31). Using this, for any fixed value of r we can carry out a </p><p>change of variables of the type (31) such that the new functions Uk assume the simplest possible form. </p><p>SUppo...</p></li></ul>