Higher-Derivative Schwinger Model II
L. V. Belvedere, R. L. P. G. Amaral and N. A. Lemos Instituto de Física - Universidade Federal Fluminense
Outeiro de São João Batista s/n, 24020 Centro, Niterói, Rio de Janeiro, Brasil
Abstract Contrary to what is sometimes suggested in the literature, it is shown that it is possible to construct a two-dimensional higher-derivative fermionic field theory of even order whose classical Lagrangian exhibits chiral-gauge invariance. By means of both canonical and functional methods the anomalous divergence of the axial current is derived and the mass acquired by the gauge field is found. The original field variables are expressed in terms of usual Dirac spinors through a canonical transformation, whose generating function allows the determination of the new Hamiltonian.
In the last few years several authors have turned their attention to the study of higher- derivative quantum field theories [1]. We have discussed a higher-derivative generalization of two-dimensional quantum electrodynamics (Schwinger model) in a previous article [2], to which the reader is referred for the motivation of such investigations as well as for a brief sketch of the historical development of the subject. In the present work we address ourselves to additional aspects of the model considered in the paper just quoted. It is occasionally suggested that the chiral symmetry restricts "the number and the kind" of covariant derivatives in the fermionic Lagrangian, so that, for instance, only the appearence of an odd number of covariant derivatives of the fermion fields is compatible with invariance (at the classical level) of the fermionic Lagrangian under chiral-gauge transformations [3]. Here we show that this is not the case by explicitly constructing a higher-derivative Schwinger model of even order whose classical Lagrangian exhibits chiral-gauge invariance.
1 Free Model
The extension to the case with an even number of Fermi fields derivatives can be introduced N through the following (Hermitian) Lagrangian density Co = £"f° {i$*ifi) Ç, where the imagi- nary factor i was inserted for future convenience. In two-dimensions, the behavior of fields under a Lorentz transformation is better analyzed using the light-cone variables : xi = x°-±. x1. Under a LT these variables transform according with i+ -t k+ , i" -• A"1 x~ , such that d+ —• A"19+ and cL —> AtL , where A € (0, oo). In this case, the Lorentz transformation properties of the spinor components are £(oj —» A^»» £(o).
406 Let us now perform the quantization of the free model. For the sake of simplicity we consider the upper component only. Following our previous paper, the configuration space is generated by Çfa = (d-)n £(i) , n = 0,1,..,2N - 1 and the associated momenta are given (is 1 n (2N 1 n by jfj" = (-l) '~ ~ )(9_) ~ ~ 'f("1). The canonical quantization must be performed using the Dirac method, which leads to the equal-time anticommutators { 9f £(i)(a:), dl f (i)(y) } = ,-(-1)"-P-I X 1 1 6q+P,2N-\ ^(i — y )- In momentum space, the solution of the equation of motion for ^j is a sum of derivatives of S(k2) up to the order (N — 1). The Fourier representation for the operator solution of the model which leads to a local field operator is given by (k = k+)
In the above expression the finite arbitrary mass scale m is introduced1 in order to ensure the usual dimension for the spinor component fields |^j(i), which satisfy {£(*)(&)> £(')(&')} =
The anticommutation relations of the mode expansion operators can be diagonalized via #o(*) = Tstfii'i*) + Í™~'W) ' C"W = TstttfW ~ fw""(*))• with P = 1,2, ..,Ar. Now a set of 2A' usual free Dirac spinors in coordinate space rji3 can be introduced. Thus, we can write t,(i){x) = T%=i fj{x~) i>n) (2+)- The Dirac fields i/>i.(x+) are quantized with positive (negative) metric for j < N {j > N). The expression for the lower component + f(2)(z) is obtained interchanging the light-cone variables x <-» x~. The anticommutator between i(i) fields are given by { f(i)(x), £(i)(y) } = (x~ — y~ )2N-X x 5 (a;+ — y+ ), where the function S(x) is the anticommutator of two usual Dirac fields. Although the £ field does not represent a genuine spinor field operator, the corresponding spinorial nature is carried by the infrafermions if)' field operators, which ensure the correct micro-causality requi- rements. In this sense, the general quantum field features are implemented by the infrafermion operators. The bosonization of the free theory is straightforward and can be carried out using the stardard bosonization formulae, leading to JV independent
1 It should be remarked that there is another solution, in which the arbitrary dimensional parameter m is not introduced, but it leads to a non-local field operator. In order to circumvent this problem and obtain a solution in terms of usual fermionic field operators, the parameter m must be introduced. In this case, the previously referred Lorentz transformation properties of the spinor components £(o) are implemented if we perform the LT combined with the redefinition m —» Am.
407 of the Lagrangian density. The Hamiltonian density is then given by
2N-1( «(*)= E (-l)W-1-n(^-)2'V-1 n=0 I ' and the corresponding Hainiltouian // evolves the field £(ij(-r)- Since the expression of the •original configuration space variables in terms of the 2A' usual Dirac fields ti>J is explicitly time dependent, the above Hamiltonian docs not describe the time evolution of the latter fields. Nevertheless the Hamiltonian H' that describes the time evolution of the Dirac fields can be obtained from H by canonical methods.
To this end, we consider mapping from the set of old variables £('Jj, to the set of new variables, i.e., {£(")(£)} —» Wm(i+)} , and take into account the expressions for all the ori- ginal phase space variables. Then, we have the set of transformation equations: £("j(a:) = J2f=i \d2fj(x~)]il>^(x). After this point transformation, the new momenta are obtained com- puting the variation of the action around a solution of the equation of motion and we can identify i i/>^ as the canonical conjugate momenta of the ibfo fields. The anticommutators for the Dirac fields are thus reobtained without resort to the anticommutation relations of the original field solution. In order to obtain the expression of the new Hamiltonian, let us consider the generating function for the point transformation. In classical mechanics the generating function of a point transformation from the set of conjugate variables {pi,(n} to the set {Pi,Qi} can be written as Sl{P,q,t) = H, Z5! Çi(<7,<)- The transformation equations can be recovered by means of Qi = |jr , pi = f^, while the expression for the new Hamiltonian function is given by
H' = H + dofl. The generalization of this formalism to field theory can be performed considering the "generating functional" of the transformation from the new ipi variables to the old £n variables as the sum of the old momenta multiplied by the corresponding old variables written in terms of the new ones, that is,
2N-1 ( 2N •)
This generating function allows one to obtain the transformation through
where c = 0(1), for j
I which is the correct Hamiltonian density associated with the free Dirac spinor components *A(Ji)(z)> and the one to be bosonized.
408 2 Local Gauge Invariance
Following Ref.[2], the Schwinger model with derivatives of order 2 Ar is denned by the Lagran- gian density £ = - j (J>.. )2 + * $?N *, where $2Ai is the covariant derivative of order 2N 2N 0 definedby$ = 7 (ifUty)", where the usual covariant derivative is#> = 'y"(dll-ieAli). The operator solution is a straightforward adaptation of the odd-order case considered in our previous paper. Following the same steps outlined there, we obtain the physical fermion field operator as
where a is a spurious operator with scale dimension of zero value. The gauge field is given by À{x) = -\yf^e^dut(x), with S satisfying the equation of motion [O + lKsL]t(x) = 0. M li The anomalous divergence of the axial current is then given by ô 0^ = — ^elill3- ''. Since the operator solution represents a chiral operator gauge transformation acting on the free field, the model quantization can be performed following the same steps outlined in the free- field case. The configuration space is generated by *
J. J. Giambiagi, Nuovo Cimento 104 A, 1841 (1991);
C. G. Bolini and J. J. Giambiagi, J. Math. Phys. 34, 610 (1993)
2 - R. L. P. G. do Amaral, L. V. Belvedere, N. A. Lemos and C. P. Natividade, Phys. Rev.
D47, 3443(1993)
3 - J. Barcelos-Neto and C. P. Natividade, Z. Phys. C 49, 511 (1991)
4 - L. V. Belvedere, J. A. Swieca, K. D. Rothe and B. Schroer, Nucl. Phys. B 153,112 (1979);
L. V. Belvedere, Nucl. Phys. B 276, 197 (1986).
5 - M. B. Halpern, Phys. Rev. D 12, 1684 (1975); 13, 337 (1976).
6 - K. Fujikawa, Phys. Rev. Lett. 42, 1195 (1979); Phys. Rev. D21, 2848(1980); ibid. D22, 1499 (1980) (Erratum) 409