IC/67/24

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

CURRENT DEFINITION AND MASS RENORMALIZATION IN A MODEL FIELD THEORY

C. R. HAGEN

1967

PIAZZA OBERDAN TRIESTE

IC/67/24

INTERNATIONAL ATOMIC ENERGY AdENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

CURRENT DEFINITION AND MASS HEJTORMALIZATION BT A MODEL FIELD THEORY *

C.R. Hagen **

TRIESTE April 1967

*To be Bubmitted to "Nuovo Cimento" **On leave of absenoe from the Univereiiy of Rochester,NY, USA

ABSTRACT

It is demonstrated that a field theory of vector mesons inter- acting with massless fermions in a world of one spatial dimension has an infinite number of solutions in direot analogy to the case of the Thirring model. Each of these solutions corresponds to a different value of a certain parameter £ whioh enters into the definition of the ourrent operator of the theory. The physical mass of the veotor meson is shown to depend linearly upon this parameter attaining its bare value u, for the exceptional oase in which the pseudo-vector current is exactly conserved. The limits jA-o —> °o and JLLQ -» 0 are examined, the former yielding results previously obtained for the Thirring model while the latter implies a unique value for ^ and defines the radiation gauge formulation of Schwinger's two-dimensional model of electrodynamics.

-1- CURRENT DEFINITION AND MASS RENORMALIZATION IN A MODEL FIELD THEORY

1. VECTOR MSSOITS IN ONE SPATIAL DIMENSION Although the soluble relativistic models known at the present time are remarkably few in number, they have long been cherished by field theorists for their value as potential testing areas for new techniques in particle physics. The study of such models has however, been handioapped by the faot that the only theories which are both non-trivial and soluble in a four-dimensional space-time refer to non-renormalizable interactions, a circumstance which thus far has made it extremely difficult to attach much significance to some of the formal results which have been obtained from these models. On the other hand the soluble field theories in a world of one spatial dimension are generally much less singular than their three-dimensional counterparts and are therefore considerably more amenable to calcula- tion. In particular the four fermion interaction (THIRRING model ') 2 1 and electrodynamics with a massless fermion (SCHWINGER model ') are outstanding examples of soluble renormalizable theories in one dimension and as such have received a considerable amount of attention in the literature.

Somewhat less familiar perhaps is the case of a vector meson interacting with a massless fermion field in one dimension acoording to the Lagrangian

1/4 o>% fW ^

where B'(x) and ^(x) are hermitian operators which desoribe vector and spinor fields respectively. For the present, the ourrent is assumed to be defined by the formal operator expression

where the Birao matrices oCmay be taken to be the two-dimensional set

-2- and q is the anti-symmetrical matrix

which acts in the two-dimensional charge spaoe of the field Yi[x). For later convenience we have inoluded in (l.l) the coupling of o^fx) and B"(x) to the sources A^(x) and J^(x) respeotively. By using this external field device it is possible to generate all the vacuum expectation values of the theoryf thereby effecting a oojnplete solution.

The action principle oan "be formally applied to the Lagrangian (l.l) to derive the equations

^B (1.2.) and the equal-time commutation relations

-3- It is useful to note that of the above field eqs.,(l.2a), (l.2b) and

r , I are equations of motion for V(x) B^x) and Q0\x) respectively, while the constraint equation * *B =ei' serves merely to define B(x)in terras of the true dynamioal variables of the theory.

Before proceeding to the task of computing the general struoture of the Greens functions it is well to remark briefly upon the work of BROWN ^ and SOMMERFIELIr' in connection with this model. In particular it has been observed ' that the definition of the current operator used by Sommerfield is not unobjectionable inasmuch as it requires the application of a time-like limiting procedure. This, of course^ re- presents a significant departure from the usual concept of a canonical theory acoording to whioh all operators are required to be defined solely in terms of the fields on a given spaoe-like surface. Brown's definition of jA*(3c) as , -k with the limit x —•*• x'to be taken after setting x° = x clearly avoids this oriticism and leads to a consistent solution of the model. However, it is interesting to note that although it has been shown ' that in the case of the Thirring model the results of JOHNSON ' and Sommerfield can be derived from a temporally local definition of j^(x) the conclusions of Brown and Sommerfield differ with regard to the physioal mass ^u of the veotor meson. In particular,Brown has found

whereas, Sommerfield's mass renormalization is only half as great.

-4-

•f-- ' This leads one to anticipate that the ambiguity in the definition of j^(x) whioh has previously been noted by the author ' in the oase of the Thirring model might well be responsible for the conflicting results obtained by BROWN and SOMMERFIEID' '. It is in fact the intent of this paper to disouss in some detail the consequences of using the ourrent definition of Ref. 5 &&&• to demonstrate the existence of a more general olass of solutions of the theory described by (l.l) than has previously been known.

2. BOSON MATRIX ELEMENTS

Since the vacuum expectation values of all operators contain- ing j^(x) and B^(x) in any arbitrary combination can be obtained from the vacuum-to-vacuum transition amplitude \ 00*J 00^ (or more simply

from whioh one can immediately deduce the result

where

The effect of the interaction can thus be extracted entirely into an exponential factor containing derivatives with respect to the source terms. One consequently has only to solve the external field problem and evaluate the functional derivatives in order to complete the calculation of _ e There is one further

-5- Bimplifioation whioh follpwe from the removal of the interaction term, namely the factorization of <^0 1 0 ^> as indioated by

The calculation of <^0 | 0/_. can be carried out by straight- forward application of the field equation

In partioular one finds that

reduces with the aid of (2.1) to

J (2.2) where G^(x),the free field Green's funotion of the meson field,is given by

and satisfies the differential equation

subject to the usual causal boundary conditions. It is now trivial to integrate eq. (2.2) and obtain

0 1 0 \ \ J^U) G£*t*r*:) J,

-6- By contrast the,task of determining the structure of is oonsiderably more dlfficult inasmuoh as it requires a detailed consideration of the structure of the operator j/*(x). In particular, one must be oareful not to infer from the formal definition

(2.3)

and the apparent invariance of the theory under the infinitesimal gauge transformations

(2.4b)

1 (Ye being the pseudo-soalar matrix (X ) that j^x) and oj^x) = £**" jy(x) are oonserved. Since, however, the problem of calculating <^ 0 I 0 > from the oondition

and the field equation for i V has been disoussed in some detail in I, it is sufficient merely to quote the result. Thus we define the matrix element of j^x) to be - <«If

(2.5)

v 0 where A^ s €^* Ay and the lim x • x'is ^fco be taken after setting x° - x '. The parameters £ and M (which are required to Batisfy the oonstraint

in order to preserve Lorentz invarianoe ) apppear as a direct consequence of the fact that there is no unique way in which to lend meaning to the ill-defined operator expression (2«3). With the prescription (2.5) one obtains '

which immediately implies the result

where we have set

and D(-x) is the causal Green's function defined by

One has now to evaluate the expression

\

This can be accomplished by relatively straightforward techniques ^* and we shall consequently not reproduce here these somewhat tedious calculations. -The result one obtains can be- written in the form

-8- where C L tit

and ^(x) is the caueal Green's function

with the renormalized meson mass yu being given by the expression

(2.6)

It is, of course, necessary to require that both (l + VJT^) and >) •'be positive in order to ensure that the metric'of the Hilbert space be positive definite. The dependence of jx on the parameter ^ as expressed by eq. (2.6) forcefully illustrates the possibility of having physioally observable consequences associated with the freedom one has available in defining the current. Although the solutions of the Thirring model were also found to depend upon ^the appearance of this parameter in the mass renormalization of the vector meson is perhaps a somewhat more oonvinoing effeot.

It is of interest to note that in the case

-9- no-renormalization theorem for the meson mass • By inspection of the structure of the current-current correlation function in the source- free limit

(2.7) one can easily verify that £ = 0 corresponds to the case in whioh both j^x) and jA*) are conserved. This is a consequence of the fact that the one-meson intermediate state contribution to D^ (x) (which would oontradiot the equation d/j-i$ ** 0) vanishes in the limit £ => 0. On the basis of (2,5) one oan furthermore understand in an almost trivial way the origin of the different mass renormalizations obtained by Brown and Sommerfield. Since it has already been observed in I that the case ]*j =< f] » j? corresponds to the current operator used by Johnson and Sommerfield while Brown's definition is olearly based upon the choice £ = 1, the discrepancy of a factor of two between the mass renormalizations obtained in RefS. 3 and 4 is seen to be an immediate consequence of (2.5).

From 3/(X) it is possible to infer the structure of the equal- time commutator of the current and oharge densities. In particular one finds for x - x

7^ V**" _1_ (2.8)

One can also determine from D' (x) the structure of the time-ordered product

\

-10- vhioh upon comparison with {2.1) clearly Bhows that

As in the oase of the Thirring model one can conclude from this result that it is not necessarily possible to infer that *-J— - 0 merely heoause the ourrent happens to be conserved for vanishing external field. Although it will subsequently he shown that in this model the current is conserved in the source-free limit, it is only for the oase Y\ - 0 (i.e., f6r the situation in whioh j^(x) is strictly oonserved in the presenoe of an external souroe) that one oan consistently take j (x) to be independent of A°(x).

3. PEEMION MATRIX ELEMENTS

Since operation on < 0 | ° ^"AT by suitable combinations of -4 -rj-^and — j^-v.yields the vacuum expectation value of all possible operator functions of B/*(x) and j^(x), there remains only the problem of calculat- ing the fermion Green's funotions in order to complete the solution of the model. These are the time-ordered funotions

\ o>A Ae

where £(x^t x^, ... xin) is the completely anti-symmetric function of its argument which is equal to unity when xL, x^, ... x^ form a decreasing time sequence. Following the procedure of the preoeding seotion one can extraot the effect of the interaction into an exponential of functional derivatives to write (3.1) as This can be further simplified "by assuming G.(x|j to have the form

where Q (x1, x., ... xitt) is the free field Green's function defined hy

Prom the equation of motion of the fermion field there follows

so that one obtains U, t- (3.2)

The remaining calculations required to evaluate explicitly X are G, T (xo 2_t ••• X^H) somewhat tedious and we consequently are oontent once again merely to quote the result. This oan be put into the form

vhere N. r V

-12- and

Eq. (3.3) evidently implies that in the absence of source terms the two point function has the structure '

It is to be noted that as in the Thirring model all the results derived using our generalized definition of the ourrent operator reduce to those obtained by SOMMERPIELD ''for the oaBe

Prom the structure of G (xL, i^, ... x£H) one can immediately evaluate the equal-time commutator of j^x) with V(x)* This is a consequence of the form of the discontinuity relation

- '% ^ from whioh there follows

(3.4)

Al «)J f <*')> -^. ^ ^ 0*) SO*-*'). (3.5)

One has thus encountered a situation which is reminisoent of that found 5} in the case of the Thirring model ' where the commutator of the oharge density j (x) with V(x) also showed a significant departure from the canonical form. Although one might be tempted to conclude that the situation is somewhat simpler here by virtue of the faot that the commu- tator of 3j(x) with V^x) assumes the canonical value independent of the value of Jf , it is easy to show that the other equal-time commutation relations involving $5(x) fail to assume the usual form unless ? *» 0.

-13- To demonstrate this result* we note that from the preceding seotion one has in the absence of source terms

which implies the equal-time commutators of j°(x) with the dynamioal variables of the meson field

as well as the corresponding results for j£(x)

O (3.7a)

It will be seen in the subsequent section that these commutation relations can also be derived direotly from the operator formalism of the theory.

Eqs. (3.4) and (3.6) display the remarkable fact that the commutators of the charge density operator jo(x) with the fundamental field variables assume the canonioal form only for the limiting case J1 = 0, i.e., when the current j/^x) is oonstruoted to be strictly invariant under the gauge transformation (2.4a). Similarly (3.7) illustrates that in the case <• / 0 there is a discrepancy between the actual structure of the j^(x) commutator with G° (x) and its usual oanonioal form. One thus has an exact analogue of the situation encountered in the Thirring model where it was observed'*'' that the slightest departure from strict gauge invariance implied a dependence of the equal-time commutation relations on the dynamics of the theory* It is clear from this result that extreme caution must "be exercised in attempting to infer the commutation relations of any current operator in the event that the relevant symmetry does not describe a strict invariance operation of the theory.

4. OPERATOR FORMALISM

It is useful at this point to examine "briefly the operator struoture of the model in the source-free limit in order to understand more fully some of the results derived in the preceding sections. In particular we note that (2.5) is entirely equivalent to the operator definition r so that upon using

and verifying that in the limit 3C-»x'the product Vt1) V(x0 which occurs in the term linear in e can "be replaced by its vaouum expectation value, one obtains

or

(4.2)

This result has the important oonsequence that since 3^x) is a veotor in a two—dimensional space time, the operator

-15- cannot transform as a vector under Lorentz transformations. On the basis of (4.1) it is now possible to derive in a trivial way the commutation relations (3.4) and (3.6) while (4*2) similarly implies (3*5) and. (3.7a). One can furthermore write

which with the aid of (2.S) readily yields (3.7b).

Having verified that the commutation relations of emerge as an immediate consequence of the definition of the current, we next turn our attention to the energy momentum tensor of the ' theory. From the conditions

P'f 0(x) ] - id,O(x)

p°, O(x)] -*idQ0(x) where 0(x) is any operator funotion not explioitly dependent on the co-ordinates, we infer "that a suitable set of momentum operators are

where — - t. (4.3)

-16- and

(4.4)

with all boson (fermion) operator products implicitly assumed to be appropriately symmetrized (antisymmetrized). Sinoe the presence of the terms £ 4jp- B1 and -f-^p B* in T°°(3c) cannot be readily deduced from the energy momentum tensor as obtained by straightforward application of the action principle, we shall not attempt its con- struction from first principles. It is sufficient merely to note that (4.3) and (4.4) conform to SCHWINGER's commutator condition

r_oo/ v -_oo / * "I r™ oi/\ «.O1/\1#KC/ **^\ / \

on the energy density and that the operators P^ and

satisfy the struoture relations

o1 [P1 , J ].

[p°, J01]- appropriate to the two-dimensional Lorentz group. It is of interest to observe that in the limit

ev

-17- one obtains the Thirring model with

. X

- o

in complete agreement with I.

Finally we note that for k^ - J*1 - 0 the Hamiltonian P° given above implies the conservation lav

d (4.6) despite the fact that j^t*) is not generally conserved in the presence of source terms. This» however, appears to "be nothing more than a dynamical accident arising from the extremely simple oharacter of the exoitation spectrum of jA'(x). In faot it follows immediately from

or

that if j^(x) were coupled to a second vector meson field one would find (4.6) to be no longer valid. One thus has in the veotor meson case what is perhaps a somewhat more convincing argument than exists in the Thirring model concerning the connection between the conservation laws and the form of the equal-time commutation relations. In particular, since in the Thirring model the current operator has no massive excitation, both 0^(x) and j£(x) may be said to be "accidentally" oonserved for A^ » 0 and it was therefore essential in I to use the external source device to infer the relation between current conservation and the canonical commutation relations« However, in the present case even in the limit A/* « J^1 = 0 one sees that the oanonioal form of the commu-

-18- tation relations of jj-fx) obtains only for the case £ • 0, i.e.,when the pseudo-vector current is exactly conserved. This result appears to lend considerable strength to the argument advanced in I concerning the requirement of strict gauge invariance aa a necessary condition for assuming the canonical form for the equal-time commu- tation relations of the ourrent operator.

5. THE ZERO MASS LIMIT

In the oourse of the preceding sections considerable stress has been plaoed upon'the non-uniqueness which characterizes the solutions of a theory of vector mesons in two dimensions . However, it is worth noting that there is one speoial oase of this model particularly deserving of attention in whioh such an ambiguity is not expected to persist.. He refer here to the JA0=> 0 limit for which the field equation (l.2c) in the absence of the souroe terra J^" reduces to

This result implies, by virtue of the anti-symmetry of Gr (x) that ^(x) is conserved and that one must consequently impose the condition f| a 0 if a oonsistent solution is to be obtained.

In the more familiar example of a four-dimensional spaoe-time it is well known ' that this case requires the decoupling of the longitudinal mode of the meson field if one is to retain the correct number of independent dynamical variables in the limit of vanishing meson mass. In a two-dimensional space—time> however, the entire meson field must clearly decouple from the charge field, thereby leaving only a possible residual interaction of the spinor field with itself. It is a matter of some interest to explicitly illustrate this decoupling and we shall therefore display here the precise manner in which the energy density (4.4)reduces to the corresponding (radiation gauge) result.

To this end it is convenient to introduoe the operator

-19- which has the property that

where for the present we ignore certain problems associated with the definition of the inverse Laplaoian. In terms of these new oanonioal variables one oan rewrite the energy density as

(5.1)

where the ocourrenoe of the last term is a oonsequence of the equation

which follows from a careful evaluation of the limit. Upon making the change of variables

(5.1) becomes T" ' " ' "" "''•'

')

One oan now take the limit /*, -> 0 provided that (in accord with our earlier remarks) n is simultaneously required to vanish. Since this oondition does not preolude the possibility of n//*1" approaching a finite limit, we set f*

with f

m In the fi-o 0 limit the terms describing the interaction between the meson field and the spinor field obviously vanish and one obtains T -^

where for simplicity we drop the prime notation and write

Although this result clearly displays the decoupling of the meson and Bpinor fields as well as the neoessity of requiring the vanishing of the parameter n , there yet remains an ambiguity in

-21- the final form of the Green's functions of the theory which is associated with the definition of the inverse Laplacian. In particular we note that the most general form of the funotion 3L defined by

is

where a and b are arbitrary constants. Since, however, the only combination which appears in (5*2) is ^(^ ) the parameter a is not relevant and can consequently be taken to be zero. This implies the momentum space representation o

where

and the free parameter b is related to OCand (i by

Although BROW asserts ' that both a and b must be zero, this result is readily seen to be merely a consequence of his use of an energy density which differs from (5.2) by a spatial derivative. On the other hand, since the energy density employed by Brown (unlike (5.2)) does not satisfy Sohwingerls commutator condition on T (x), it appears that (5*2) may be a more satisfactory form for this operator. However, it is to be noted that despite the occurence of this ambiguity in the definition of the inverse Laplacian (and consequently also in the Green's functions of the theory) the mass of the boson excitation remains unaffected by these considerations,

6. CONCLUDING REMARKS

By the extension of the techniques of I to the case of a veotor meson as carried out in this paper, a number of results have been obtained which could not be illustrated within the somewhat

-22-

.,.;:~-;lfe;;lg simpler structure of the .Thirring model. Most striking of these is perhaps the remarkable dependence of the mass renormalization on the detailed definition of the current operator as described by r -/* ^

It is particularly worth noting that since for non-zero /^ the parameter ^ was not constrained to be positive, the physical meson can be either more or less massive than the bare partiole. This illustrates the considerable care which must be exercised in attempting to infer the sign of a mass renormalization on the basis of general arguments.

In the oontext of current applications, a particularly suggestive result of this work*arises from the marked dependence of the equal- time commutation relations of the oharge operator upon the detailed definition of the current. The fact that the canonical result was found to obtain only in the limit in whioh the relevant oharge operator generated an exact symmetry of the system might well be taken to suggest the existence of significant problems in oonnection with reoent applications of current algebras. Certainly much work remains to be done in this area of research before its basio assumptions can be considered to be reasonably well founded within the framework of

Lagrangian field theory.

ACKNOWLEDGMENT

The author is indebted to Professor Abdus Salam and the IAEA for making possible his stay at the International Centre for Theoretioal Physics, Trieste.

-23- REFERENCES AND FOOTNOTES

1. W. THIRRI1JG, Ann. Phys. (HY) 3.» 91 (1958).

2. J. SCHWIFGER, Phys. Rev. 128, 2425 (1962).

3. L.S. BROW, Kuovo Cimen-fco 2£, 617 (1963).

4. C. SOMMERFIEID, Ann. Phys. (HY) 26_, 1 (1964).

5. C.R. HAGEN, ICTP, Trieste, preprint IC/67/23 (1967), hereafter referred to as I. 9

6. K. JOimSOK, Nuovo Gimento 2jO, 773 (I96l).

7. More pertinent perhaps is the fact that since the Thirring model is merely the limit of the vector meson theory (l.l)

in which e andyu.o "both beoome infinite, the free parameter which was found- to ooour in the current definition of the former case must clearly be relevant in the present context as well.

8. Although this result for G(x - x') dearly indicates the existence of an ultraviolet divergence in the wave funotion renormalization of the fermion field, we consider this to be merely an indica- tion of a defect in the model itself and shall consequently not dwell on this point in further detail.

9. J. SCHWItfGER, Phys. Rev. 121t 324 (1962).

10. D.G. BOULWARE and W. GILBERT, Phys. Rev. 126, 1563 (1962).

-24-

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