On the free scalar massless field in two-dimensional space-time

Vol. 15 (1979) REPORTS ON MATHEMATICAL PHYSICS No. 3

ON THE FREE SCALAR MASSLESS FIELD IN TWO-DIMENSIONAL SPACE-TIME

L. HADJIIVANOV* and D. Ts. STOYANOV”

Joint Institute for Nuclear Research, Lgboratory of Theoretical Physics, Dubna, U.S.S.R.

(Received September 28, 1977)

In the present paper the solutions of the quantum field problem for the free scalar massless field in two-dimensional space time are constructed. It is shown that the fields obtained cannot vanish at space-like infinity. The latter fact implies the existence of two conserved charge operators. The transformation properties of these solutions under the two-dimensional Lorentz group are examined.

1. Introduction

The scalar zero-mass quantum field in two space-time dimensions plays an essential role in the solution of the Thirring model. As it is known [l], such a field does not exist in the conventional sense. Non-trivial operators for such a field can be introduced only in an indefinite metric Hilbert space. However, as is seen in papers [2], [3], [4] this is not the only difference between the two-dimensional and the four-dimensional scalar field. The existence of a non-zero charge in two dimensions, whose meaning is close to that of the topological charge, is the most essential difference between the two- and four- dimensional model. The spontaneous breakdown of gauge invariance in the Thirring model is a corollary to the existence of such a charge. For the first time this breakdown was observed in papers [2], [3], [4]. However these articles contain an error which did not permit their author to obtain one more charge (i.e. roughly speaking the vacuum is charged by two charges, rather than one as it is in papers [2], [3], [4]). In order to under- stand the situation we first define the quantum scalar field in two dimensions. Such a field satisfies the two-dimensional wave equation

q F = 0 (0 = &-a:) (1.1) and the commutation relation

I+) 7 Y(Y)1 = wx--.J4 (1.2) where

D(x) = & s

dpz&(po) 6(p*) esipX = -~:_[e(x~+~~)--B(x~-x~)]. (1.3) _.

* Permanent address: INRNE, Bulgarian Academy of Sciences. bul. Lenin 72, Sofia, Bulgaria.

[3611

362 L. HADJIIVANOV and D. TS. STOYANOV

The special form of the commutator function implies the following statement (see remark

at the end of the paper). For fields satisfying (1.2) the quantity

Q = 1 dx’&,@“, x’) --Do

cannot be equal to zero. Indeed, ’ we differentiate first eq. (1.2) with respect

respect to f

in the region from --oo to co, we obtain

[Q, ~01 = -i.

(1.4)

to x0 and then integrate with

(1.5)

We now introduce the field (we call it conjugate for convenience):

?(x) = i’ dz’a,&o,z’). (1.6) -0-Z

If this integral exists it can easily be checked that the conjugate field satisfies eq. (1. I) and the commutation relation (1.2), and therefore the integral

Q = ~dU~ao~(x~,X’) (1.7) --co

should be non-zero too. It is the quantity (1.7) that is missing in papers [2], [3]. For this reason in the present paper we shall try to find (at first classical) such solu-

tions of equation (1 .l), for which there exist finite non-zero charges (1.4) and (1.7). We insist that these are the solutions which should be quantized. We do not impose the condition of finiteness on the integral (1.6) because it appears to restrict unnecessarily the class of possible solutions of (1 .l). We define the conjugate field by the following equation :

$(x) = 1 dz’a,fp(xo, zl)+R(xO) (1.8) -m

where R(x”) is a subtraction, necessary for the regularization of the integral on the r.h.s. of equation (1.8). We would suppose that R(x”) is such that the fields C&X) and G(x) satisfy the differential relation

a,+) +Epyay+(~) = 0 0.9)

where .sol = -.sIO = 1, eoO = alI = 0 (the metric tensor in two-dimensional space-time is go0 = -gll = 1, go, = g,, = 0). In quantum theory the new definition of e(x) (1.8) implies a change in the commutation relation between +(x) and v(x). That is why we now establish their form.

If both sides of eq. (1.2) are differentiated with respect to x0 and then integrated

THE FREE SCALAR MASSLESS FIELD 363

over x1 from - co to -t co, then having in mind (1 A), we obtain

X’-y’

G(x), d.Y)l- P(xO), V(Y)1 = i s dz’aoD(xo-yo, 2’). (1.10) -co

The integral on the r.h.s. of this equation is convergent. If we introduce the function

&x) = -& s

d2p&(p’) cQ2) e+x = -3 [e(x”+Xl)-e(xO-xl)] (1.11)

(following papers [2], [3]), then

%I

5 --m a,D(xO, z’)dz’ = 6(x)-+ (1.12)

It is easy to show that the commutator [R(xO), I&)] is a constant. Indeed, having in mind translational invariance, we can write

WO) 9 dJ91 = 4x0 -7”). (1.13)

(We certainly use the assumption that the commutators of two free fields are C-numbers.) Using eq. (1.10) it is easy to verify that

[@(x0, x1), d_md+-co = PwO), 9o)l (1.14)

and therefore, instead of (1.13) we can write

d(xO -v”) = @(x0, x1), $d.Y)llxI~-, . (1.15)

Now, using relations (1.9) and (1.3), we can write down the following chain of equalities:

%MXO--YO) = [~oC(xO, -co), F(Y)1

= a?[v(XO, x1>, pI(y)I]xI.+_03 = iafD(X”-YO, xl-yl)]xL+-oo = 0.

Therefore

[R(xO), v(y)] = d = const (1.16)

which proves our statement. Then it follows from (1.10) and (1.12) that

L+(x), y(y)] = i&x-y)-G+d. (1.17)

The constant d is arbitrary and depends on the choice of arbitrariness in the regularization of the integral on the r.h.s. of eq. (1.8).

Now we shall show that the field G(x) defined by (1.8) satisfies the commutation relation (1.2). Starting from eq. (1.17) and proceeding analogously to the way eq. (1.10) was obtained, we can derive

G(x), @(v)l- GJ(~, W”)l = iW--Y) (1.18)


where we have made use of the relation

X’

s dz’a&P, 2’) = D(x). (1.19) -CO

Now we shall show that

G(x), wJO)l = 0. (1.20)

Indeed, since we have already proved eq. (1.16), it is implicit that

G(x), NYO)l - [NxO), &Jo)1 = 0 * (1.21)

If we now use the equality

aoR = alP(xO, X1)l,l,-al (1.22)

(which is a necessary condition for R(x”) if eq. (1.9) is to hold), it follows easily that

[WO), WO)l = 0.

Then eqs. (1.21) and (1.19) imply

[+(x), @(HI = i%--Y), i.e. our statement is proved.

(1.23)

2. Classical solution

In this section we construct classical solutions of eq. (l.l), with the existence of the non-zero charges (1.4) and (1.7) being allowed. It is obvious from relations (1.9) that solutions which vanish in space-like infinity are excluded. Indeed, if for instance the field q(x) vanishes at x1 -+ f co then the quantity (1.7) should be equal to zero.

As it is known, the general solution of eq. (1.1) can be written in the form

1 W ___ -

A+(p’)CiPX+ 1,2n 2,p’, A-(pl)e’PX+R s

(2.1)

__- where p” = lp’l, while A+@‘) = A-($) are arbitrary functions (the line denotes complex conjugation) and R is a subtraction constant (the necessity of the latter will be seen in what follows). As well known, the decomposition into positive- and negative-frequency parts of the field v(x) can be produced using the formula:

v*(x) = -i~dz’D*(x-z)2{cp(z) (2.2)

where A’ao B = 8, A * B- Aa, B (see Appendix) and D* (x-z) are the frequency parts of the commutation functions D(x-z). The decomposition into D*(x-z), however, needs a regularization of integrals and therefore,

D*(x) = &- 1 g [eri~x-O(x- Ipll)]po,,pr,. (2.3)


Now we insert (2.1) into (2.2) (for definiteness we choose the + sign in formula (2.2)). After simple calculations we obtain

1 cp’(x)=- __

s dP1 A+(pl)e-iPX+

v/2x 2)p’l

+ -..& 41/2x s

dp’ -s(x-lp’l)[~-(O)-At(]+~~. IP’I

It is evident from this result that if A-(0)-A+(O) # 0’ the integrals in formula (2.1) are logarithmically divergent, and therefore a regularization is needed. In this case

R= -+&g e(x-IP’I)[A+(O)+A-(0)1

and instead of (2.4) we can write

v,‘(x) = A; s $- tA*(P’) e*ip"-A*(0)O(x-Jpl~)].

(2.5)

(2.6)

Formulae (2.6) can be used to obtain the asymptotic behaviour of v*(x) (and therefore the field v(x)) at space-like infinity. For this purpose in the first term on the r.h.s. of eq. (2.6) we insert

A*@‘) E Ai(A*(O)+A*(O)

and after rearrangement we obtain

(2.7)

1 Y’(X) = -$2x ,p’, s dp’- [Ai - A*(O)] eriPx+l/~A*(0)D*(x). (2.8)

It is easy to show that the integral on the r.h.s. of eq. (2.8) is finite at infinity, and therefore the leading asymptotic behaviour of the field q*(x) is given by the second term on the r.h.s. of these equations:

A*@) y*(~)(xr,*, - --~ lnp21x112

2 1/2T. (2.9)

(,u is a renormalization constant related to the constant x from eq. (2.3): p = e-r’(1)%). It is obvious that in order to obtain eq. (2.4) we have assumed that the quantities

A*(O) exist, thus choosing such solutions of eq. (1.1) which have the weakest growth at space-like infinity. We note that such a growth still admits formulae (2.2).

Having in mind eq. (2.6) it is easy to obtain the form of the corresponding conjugate field. For this purpose we use the differential relation (1.9) and obtain

g”,(x) = 2 P( -f$:- A*(pl)eripx+gf. (2.10)

’ The case A-(0)-A+(O) = 0 is excluded. Indeed, in the latter, R = 0, the integrals (2.1) are convergent. and it is easy to show that the quantity Q (see (1.4)) equals zero.


If A*(O) is finite (as we have assumed), then the integral (2.7) is convergent at p’ = 0

if A* (p’) is a smooth function at this point. If, however, A*($) jumps at p’ = 0 a

regularization is needed and g* is the regularization constant.

It can be shown that when A*($) is a smooth function at p1 = 0 the integral (1.7)

equals zero. Therefore, we set

A*($) = a*(p’>+&(p’)b*(p’) (2.11)

and shall assume

A*(O) = af(0). (2.12)

Here e(p’) = 1 if p1 > 0, c(p’) = - 1 if p’ i 0. At times it is convenient to assume that

e(0) = 0. Moreover, we shall assume that the functions a*(p’) and b*(p’) are independent

and smooth at p’ = 0. Inserting (2.11) in (2.10) we see that i* should have the form:

(2.13)

(implied by the equality ple(pl) = Ip’ I). Thus we finally obtain the following solutions of eq. (1.1)

q*(x) = T;zs dp’ ( u*~‘)e’i”“-a~(0)~(~-lP1l) +p b*(!‘) eTip.j IP I

(2.14) P

and analogously for the conjugate field

~(~~)er’pX-b*(0)e(x-Ip’I) +p .*cgl, IPll

eripx .

P (2.15)

Now it is easy to calculate the charges (1.4) and (1.7). Making the proper substitutions,

we obtain after simple calculations

Q=i [a-(0)-u+(O)], Cj = -i [b- (0) -b+ (0)] . (2.16)

3. Quantization

In this section we introduce the second quantization of the fields (2.14) and (2.15).

For this purpose it is convenient to use formulae (2.6). From the latter it is easy to write

down the inverse Fourier transform in the following form:

Ai = +i f 1/ s

a,~*(x)E*‘pVX’ (3.1)

and in view of eq. (1.2) we obtain the following commutator for A+($) and A-(q’)

[A+(P’), A-($)1 = 21~~1 Npl-$1. (3.2)

But it is easy to see that if we now calculate the mmmutator (1.2) again, we obtain under


the integral the quantity

[A+(@), A--W)1 IP’I 14’1 .

(3.2’)

It is obvious that we cannot use eq. (3.2) to determine (3.2’), since the equation

IP’I 1411fw, 49 = 0 (3.3)

has non-trivial solutions. Indeed, suppose that f(p’, ql) is a solution of (3.3). Then in general we have

[~+w)~A-w)l _ 2 q$_ql)+f(pl ql)

IP’ I w I IP’I , * (3.4)

So we see that we do not need formula (3.2) but a formula of type (3.4) and its determination is equivalent to the determination of the functionf(p’, ql). We first of all notice that an arbitrary solution of equation (3.3) has the form

f(Pl, 4l) = WPl) Nq’) +g,(q’) W) +gzW) d(ql). (3.5)

Therefore, from (3.4) we can uniquely determine the commutator of A+@‘) and A-(O) (or A+ (0) and A- (ql)). Namely,

& [A+@‘), A-(0)1 = we. (3.6)

In order to obtain the function f(p’, ql) from (3.5) we Iirst calculate the integral

s () ~1+ x eiPXdxl.

Inserting v+(x) from (2.6), one easily obtains

s v+(x)e +iPX&l = 1 (3.7)

A+ W> ____ =~$[y+(x)eipXdxl+A+(0)&$l){O(~-lql/)$-.

IP’I (3.8)

Using formulae (3.1) we can also calculate the commutator of A- (p’) and

q+ (x) eipxdxl. As a result we have - + v+(x)eiJ’xdxl, A-(ql) I

= 26(p’-ql). w

Now we are ready to find the explicit form of (3.2’). For that purpose we insert A+(p’)/lp’I

from eq. (3.8). Using relations (3.6) and (3.9) we finally obtain

[A+W), A-WI IP’I WI

= & d(p’ - ql) + 26(p1) 6(q1) s $$ WC-- WI). (3.10)

368 L. HADJIJYANOV and D. TS. STOYANOV

Eq. (3.8) also enables to calculate the commutator of A+(O) and A-(O). Indeed, the

following equality is readily verified :

P+(P’>, A-(O)1 _ IP’I 2i)(~‘)+[A+(O),A-(O)lr)(p’)S~~B(i?-lk’,). (3.11)

Comparing the latter with formula (3.6), we see that

[A+(O), A-(O)] = 0. (3.12)

In order to accomplish the quantization of the fields A*(p’) we must also implement

commutativity of the equal frequency fields. For the analogous quantization of the fields

G(x) we use the following formal procedure. If we write the field (2.15) in the form

(3.13)

then comparing (3.13), (2.11) and (2.12) we see that

B*(p’) = b*(p’)+~(p’)a*(p’) = s(p’)A*(p’). (3.14)

Since e(p’) enters into the integrals by means of the principal value (i.e. with the point

p1 = 0 dropped) we must formally set

&(Pl) 6(p’) = 0. (3.15)

On the other hand, as a generalized function ~~(p’) does not differ from unity, and

therefore the expression c(p) c(q) 6(p) S(q) must be formally defined as

F(P) d(P)&(q) d(q) = &(P)&(q) d(P) ‘VP- 4) = E’(P) d(P) 6(q) = 6(P) d(q). (3.16)

Now starting from eq. (3.10) and multiplying it by E($) and E(q’), and taking into account

relations (3.14), (3.15) and (3.16), we obtain the following commutators

1 --- P+(p’), B-(ql)] = & d(p’ -q1)+2d(p’)d(q1,S @- l/k,)&

IP’I lqll

1 1

IP’I 14’1 [A+W)Y B-W)1 = ,p’, ,q’, LB+@‘), A-(q’)] = P-$cyp’-q’). (3.17)

From the latter we can easily obtain the remaining commutators

& [A+@), B-W)1 = $- V+(p’), B-691 = 0, (3.18)

$7 [B+(pp’), E(O)] = 26($).

As in the previous case the equal frequency fields are commuting. So are the operators

A*(O) and B*(O), which in view of their definitions (2.12) and (3.14) (B*(O) = b*(O))

are independent. Relations (3.17) and (3.18), obtained above, show that on the r.h.s. of

eq. (1.17) one must take d = i/2, and therefore, instead of that equation we must have

[G(x), &)I = &x--Y). (3.19)


Using the relations (2.11) and (3.14) we can easily write down all the commutation relations in terms of the operators a*($) and b*(p’). In doing this we must take into account that all operators a*($) commute with all operators b*(pl). Since there are no principal difficulties in such a procedure, we omit it here.

4. One-particle states

Consider the “one-particle” states associated with the fields y(x) and F(x). The vacuum is defined as the state for which

v+(X)]o) = @+(X)]o) = 0, (O]O) = 1. (4.1)

According to (3.1) and analogous formulae for the creation and annihilation operators B*(ql) associated with the field g(y), eq. (4.1) means that

A+(p’)(O) = B’(q’)[O) = 0. (4.2)

When using test functions that temper the field operators, one must be very careful in view of the infrared divergency (see [l]). Suppose F belongs to the space of complex rapidly decreasing infinitely differentiable functions of two arguments S(R2). Consider its restriction to the light-cone (we denote the space of such functions by S(C+)):

_@I) = F(IP’LP’). (4.3)

In order that the single particle state have finite norm a regularization is needed:

If> =sg r~-wfw--A- mfme- IP’I>1l0> *

Indeed

<flf> = 4% rlf~P’~I’-If~~>l’~~~-I~‘l~1~

(4.4)

(4.5)

and obviously l(fJf)l < co. It is necessary to note that the second term on the r.h.s. of the commutator (3.10) guarantees convergence of the integral (4.5).

If f(0) = 0 (we denote by S,(C+) the subspace of S(C+) where this holds) then

(flf> z 0. (4.6)

The completion in the norm

Ilfll” = <flf> = 2sg lfW>I’ (4.7)

of the pre-Hilbert space &(C+) is in fact the Hilbert space Lz of functions

f(p’) square-integrable with respect to the invariant measure dpl/lp’l. We note that the condition f(0) = 0 is equivalent to the condition k+(O)lf) = 0, since (see [2])


A+ (Wf> = 2 s 49 w’l_f~P’)lo) = vv9lo>~ (4.8)

It is evident that for functions from S,(C+) the second term on the r.h.s. of (3.10) is zero and the commutators obtain their usual form.

5. Lore&z transformation properties

Here we study the transformation properties of the fields p(x) and F(x) under transformations belonging to the Lore& group O(1, l), which in two-dimensional space time, where no space rotations exist, are just hyperbolic rotations in the (x0, x1) plane:

cha shcc A, =

I 1 sha cha ’ (5.1)

Note that (il,x)“_+(II,x)’ = e*;a(xo+xl) (5.2)

and for the case p” = lpll

(5.3)

We first of all note that in view of the commutators (3.17) and (3.18) we have

G*(x), f(Y)1 = o”‘(w% (5.4)

i,“(x) = TfS dp’ 1 x0-X’fiO _eTiPx = T_ln _____.. P’ 4r; xO+x’TiO (5.5)

From the latter formulae it is easy to obtain the following identity:

L7i(A,x) = &x)T$-. (5.6)

Taking into account eq. (5.4), this implies that the fields F*(x) and q*(x) are not scalars. Moreover, the Lorentz transformations of these fields cannot be homogeneous. There- fore, eq. (2.2) defines the frequency parts of the field operators only up to certain additive constant operators. The reason for this ambiguity is the principal value term on the r.h.s. of eqs. (2.11) and (2.12). Indeed, if we introduce the notation

(5.7)

&(x) = L 21/2x c dp’ [&(p’)

l WI eTipx--u*(O)6(x-I$[)],

ii(X) = ---L=- J& [b*(pl)e”PX-_*(0)~(~-lpll)], 21/2x s

f?+(x) = &P 21/2x s

dp’ 7 b*(pl)e*ipx,


then it is easy to verify that for instance the integrals

s dZ’D’(X-z)&?,,‘(z), ~drlD*(X-z)ta&;(z)

as well as analogous integrals for +f (x) are not determined. This arbitrariness can be settled by means of the following identities:

-i~rlzlD*(*-z)~&$(Z) = -iS~~‘~‘(x-~)~~b(=)+~*(~)-~*(- co),

where A*(z) = iD*(x-z) [@$(z)+l/2xD*(z)b*(O)] (see Appendix). Now instead of eq. (2.2) we can write analogous but unambiguous equalities

q*(x) = -i~~z’Di(.u-z)~~~~(z)-~~dz1~*(x-z)~~~~(z)+Ai(~)--A~(-~) (5.9)

as well as

G*(x) = +z’D’( x-z)~~l?~(z)-iSdz’~*(~-z)Labpl,‘(z)+~*~~)-A*(-03). (5.10)

Having in mind the above relations it is easy to obtain the transformation laws for the fields v*(x) and e*(x) with respect to the Lorentz group.

We note first that CJ+$(X) and @i(x) are scalars, i.e.

v;‘&(‘4,‘X) u, = &(x),

u,-‘~;(rl,‘x) u, = @i(x)

where iJ, is the representation of the two-dimensional Lorentz group in the carrier space of the field operators. Then using eq. (5.6), we obtain the following transformation law for the fields under consideration

U,-‘~f(A,‘X) u, = qJ”(x)--=l-b*(O), 2 ]/2X

,yl* (&‘x) u, = F’(x) + u

-u*(o). 2i2x

So we see that neither of the fields q*(x) and +*(x) is a true Zorentz scalar. The ap- pearence of inhomogeneous terms is due to the fact that the Lorentz group in two- dimensions is abelian.

Appendix

Here we shall prove the identity (5.8) and explain its sense. We first introduce the following notation :

L,‘(x) = g$(x)fJ2x Ijf(x)bf(O), (A.1)

Q(x) = g(x)Q&- b)f(x)af(O), (~4.2)


-G(x) = ~p(x)T~‘2x-D*(x)af(O), (A-3)

Z;(x) = ~;(X)f1/5a*(x)bf(O) 64.4)

where @Xx>, $,“(x), vi( x and @B(x) are defined by eq. (5.7). It is easy to prove that ) arbitrary L* from (A.l)-(A.4) vanishes when x1 + _t co. As an example we prove this

statement for L;(x). Inserting y:(x) and D”*(x) into the r.h.s. of (A.l), we get

L;(x) = ‘-.P{F [b*(p’)-b*(O)]e-. 2 Ij;“ri

(A.51

From the latter our statement follows immediately if we keep in mind that

eiipx P--T_

P - WI.

bx:-t*m c4.6)

The quantities L introduced by (A.l)-(A.6) satisfy the following differential equations

a, L; (x) + EaY& 2; (x) = 0 ) (A-7)

ap L; (x) + Epy aYE; (x) = 0.

Now we can pass to the proof of identity (5.8). Consider the integral

J = -iSdz’D”(x--)~~L~(z). (A.8)

Writing its explicit form by using the first eq. (A.7) and the analogous identity

a,D*(x)+E;&+(X) = 0 (A.9)

we can bring the integral (A.8) to the form

J = -iSdZ’a;ij*(x-,)L~(*)+iSdzD~(x-z)a;~b(z).

Integrating by parts and subsequently making use once more of the identities (A.7) and (A.9), we obtain

-i~d.zlDi(x--z)~~L,i(z) = -iS~~‘5*(x-~)~iL(z>+A’(co)-n*(-cu, (A.10)

where we have introduced the notation

Af(z’) = iD’(x-z&z). (A.1 1)

Now in order to obtain the identity (5.8) we must insert L;(z) and L;(z) from (A.l) and (A.4), respectively, into (A.lO), having in mind the obvious equality

-i~dzlo’(x-z)~~D*(z) = +z’~f(x-z)~~D*(z) = G(x). (A.12)

It follows from (A.1 1) that the constants A* (co) and A* (- co) are determined by the

asymptotic behaviour of ii(z) and are nonzero if and only if the following asymptotic

THE! FREE SCALAR MASSLESS FIELD 373

condition holds :

~P(4lp+m - A. In lzl 1 (A.13)

If we now refer to the definition of ii (equality (A.4)), remembering that the quantities

T1/2x D’(x)bi(O) are leading terms in the asymptotic behaviour of plft(x), we find that the constants A(co) and A( - co) are determined by the next terms in the asymptotic expansion of the given fields. The additional arbitrariness, which was discussed in Section 5, is due to these constant operators.

That is why by fixing these constant operators together with eqs. (5.9) and (5.10) we determine completely the fields y*(x) and p1* (x), after which their transformation properties with respect to the two-dimensional Lorentz group take the form (5.12).

Finally we note that the results of this appendix do not contradict those of the second paragraph where in fact the fundamental role in obtaining eq. (2.6) was played by the components C&(X) of the field v*(x), for which formula (2.2) remains valid.

Remark. In fact eq. (1.5) follows directly from the equal times canonical commutation relations

PIJVJ(x), dYNx~=y~ = -w--3 not only in the case of two-dimensional space-time. Eq. (1.5) is obtained by integrating the upper equality over the space-like surface x0 = const.

REFERENCES

[l] A. Wightman: Problems in the relativistic dynamics of quantized fields, Nauka, Moscow 1968. [2] N. Nakanishi: Progr. Theor. Phys. 57 (1977), 269. [3] -: ibidem 57 (1977), 580. [4] -: ibidem 57 (1977), 1025. [.5] B. Klaiber: Boulder lectures in theoreticaI physics, vol. XA, Gordon and Breach, 1967, p. 141.

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On the free scalar massless field in two-dimensional space-time