Canonical Transformations in a Higher-Derivative Field Theory

It has been suggested that the chiral symmetry can be implemented only in classical Lagrangians containing higher covariant derivatives of odd order. Contrary to this belief, it is shown that one can construct an exactly soluble two-dimensional higher-derivative fermionic quantum field theory containing only derivatives of even order whose classical Lagrangian exhibits chiral-gauge invariance. The original field solution is expressed in terms of usual Dirac spinors through a canonical transformation, whose generating function allows the determination of the new Hamiltonian. It is emphasized that the original and transformed Hamiltonians are different because the mapping from the old to the new canonical variables depends explicitly on time. The violation of cluster decomposition is discussed and the general Wightman functions satisfying the positive-definiteness condition are obtained. PACS numbers: 11.10.Ef, 11.15.Tk ∗On leave of absence from Departamento de F́ısica, Universidade Federal Fluminense, Outeiro de São João Batista s/n, 24020-005 Centro, Niterói, RJ, Brazil. 1 2 Introduction In the last few years several authors have turned their attention to the study of higher-derivative field theories. The reader is referred to [1-3] for the motivation of such investigations, as well as for a brief sketch of the historical development of the subject. It is the purpose of this paper to show that certain statements concerning higher-derivative field theories with chiral symmetry are based on hasty arguments, and to make the discussion concrete a two-dimensional higher-derivative free model of even order and its local gauge invariant generalization (higher-derivative Schwinger model of even-order) are considered. We also address ourselves to some unexplored aspects of two-dimensional quantum field theories with higher-derivative couplings. One occasionally finds in the literature statements to the effect 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 chiralgauge transformations [3]. Here we show that this is not the case by explicitly constructing a higher-derivative model of even order whose classical Lagrangian exhibits chiral-gauge invariance. The local operator solution of this higher-derivative model is obtained in terms of a product of local spacetime functions and usual “bona fide” Dirac spinors, thus defining a mapping between two sets of field operators. It is pointed out that, being explicitly time dependent, such a mapping leads to a new Hamiltonian H ′ that differs from the original Hamiltonian H . The canonical character of the transformation is stressed and the generating function involving old and new variables is obtained. The same analysis can be extended to higher-derivative models of odd order. The theory involves an indefinite-metric “Hilbert space” and cluster decomposition is violated. It is shown, however, that the general Wightman functions for the canonical free and massless Dirac fields satisfying the positive-definiteness condition can be recovered by considering correlations between the original field variables and their conjugate momenta. 1 Free Model Attention has been called recently [2,3] to Lorentz invariant field models in two-dimensional spacetime defined by the Lagrangian density Lo = i ξ̄ ( ∂/ ∂/ † )∂/ ξ , (1.1) 3 whose local gauge-invariant generalization can be exactly solved. With the purpose of introducing the extension to models with an even number of field derivatives, let us consider the Lorentz transformation (LT) properties in two dimensions. In this case, the behavior of fields under a LT is better analyzed using the light-cone variables : x = x 0 ± x . Under a LT these variables transform according with x → λ x , x → λ x , such that ∂+ → λ∂+ and ∂− → λ∂− , where λ ∈ (0,∞). In terms of spinor components ξ(α), the kinetic term (1.1) can be written as Lo = i ξ ∗ (1) ∂ 2N+1 − ξ(1) + i ξ ∗ (2) ∂ 2N+1 + ξ(2) . (1.2) Thus, under a LT the upper and lower spinor components obey the transformation law ξ(α) → λ 1 2 )γ 5 αα ξ(α). Taking these considerations into account, the extension to the case with an even number of Fermi fields derivatives can be introduced through the following (Hermitian) Lagrangian density Lo = ξ̄ γ 0 ( i / ∂ † i / ∂ ) ξ , (1.3) where the imaginary factor i was inserted for future convenience. In this case, the Lorentz transformation properties of the spinor components are ξ(α) → λ 5 αα ξ(α). Let us now perform the quantization of the free model defined by the Lagrangian (1.3). For the sake of simplicity we consider the upper component only. Proceeding in the spirit of Ref. [2], we consider the configuration space generated by ξ n (1) = (∂−) n ξ(1) , n = 0, 1, .., 2N − 1 , (1.4) so that the associated momenta are given by π n ξ = (−1) (N−1−n) (∂−) (2N−1−n) ξ ∗ (1) . (1.5) Since the system under consideration exhibits constraints, the canonical quantization must be perThe conventions used are : ξ = ( ξ(1) , ξ(2)) T , ǫ 0 1 = g 0 0 = − g 1 1 = 1 , γ 5 = − γ γ 1 , γ γ 5 = ǫ ν γν γ 0 = ( 0 1 1 0 ) , γ 1 = ( 0 1 −1 0 ) , γ 5 = ( 1 0 0 −1 ) 4 formed using the Dirac method, which leads to the equal-time anticommutators { ∂ p − ξ(1)(x) , ∂ q − ξ ∗ (1)(y) } ET = i (−1) δq+p , 2N−1 δ(x − y) . (1.6) In momentum space, the solution of the equation of motion for ξ(1) is a linear combination of derivatives of δ( k 2 ) 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+) ξ(1)(x) = m 1 2 −N √ 2 π ∫ +∞ −∞ d k e ik ( x 0 + x 1 ) 2N−1 ∑ p=0 (− imx) p p ! ξ̃ p (1)(k) . (1.7) In the above expression the finite arbitrary mass scale m is introduced in order to ensure the usual dimension for the spinor component fields ξ̃ p (1)(k), which satisfy { ξ̃ p (1) (k) , ξ̃ q ∗ (1) (k ) } = δq+p , 2N−1 δ(k − k ) . (1.8) The anticommutation relations of the mode expansion operators can be diagonalized via the linear transformation ψ̃ p (1)(k) = 1 √ 2 ( ξ̃ p−1 (1) (k) + ξ̃ 2N−p (1) (k) ) , (1.9a) ψ̃ (1) (k) = 1 √ 2 ( ξ̃ p−1 (1) (k) − ξ̃ 2N−p (1) (k) ) , (1.9b) with p = 1, 2, .., N . Now a set of 2N usual free Dirac spinors in coordinate space ψ j can be introduced. Defining their upper components by ψ p (1)(x) = 1 √ 2π ∫ d k e k · (x 0 + x 1 ) ψ̃ p (1) (k) , (1.10) we can write ξ(1)(x) = 2N ∑ j=1 fj(x )ψ j (1) (x ) . (1.11) 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 ξ(α) are implemented if we perform the LT combined with the redefinition m → λm. 5 The Dirac fields ψ j (1)(x ) are quantized with positive (negative) metric for j ≤ N ( j > N ). The factors fj(x ) are given by fj(x ) = m 1 2 −N √ 2 [ (− imx) j− 1 (j − 1) ! + (− imx) 2N − j (2N − j) ! ]