The Zeta Function Method and the Harmonic Oscillator Propagator

We show how the pre-exponential factor of the Feynman propagator for the harmonic oscillator can be computed by the generalized ζ-function method. Besides, we establish a direct equivalence between this method and Schwinger’s propertime method. ⋆ e-mail: [email protected] † e-mail: [email protected] 1 In a recent paper that appeared in this journal [1] the harmonic oscillator propagator was evaluated by a variety of ways, all of them based on path integrals. In fact, some of them did not involve any explicit computation of the Feynman path integral, but their common starting point was actually an expression for the harmonic oscillator propagator which was explicitly derived by path integral means, namely (we are using as much as possible the notation of reference [1]): DF (zf , tf ; zi, ti) = ( detO detO(o) )−1/2 √ m 2πih̄(tf − ti) exp { i h̄ S[zcl] } , (1) where O = ω + d 2 dt2 , O = d 2 dt2 , (2) and the determinants must be computed with Dirichlet boundary conditions. In the previous equation S[zcl] means the classical action, that is, the functional action evaluated at the classical solution satisfying the Feynman conditions z(ti) = zi and z(tf ) = zf and the factor before the exponential is the so called pre-exponential factor, which we shall denote by F (tf − ti). In reference [1] three distinct methods were presented for the computation of F (tf − ti): (i) it was computed directly by the products of the corresponding eigenvalues of O and O (some care must be taken here since both products are infinite, but their ratio is finite); (ii) it was computed with the aid of Schwinger’s propertime method [2] (an introductory presentation of this method with simple applications can be found in reference [3]); (iii) it was computed by the Green function approach (a variety of simple examples worked out with this approach can be found in references [4, 5]). In this note we just add to the previous list one more method for computing the preexponential factor of the harmonic oscillator propagator, namely, the generalized ζ function 2 method, so that this note can be considered as a small complement of Holstein’s paper [1]. In fact, every time we make a semiclassical approximation, no matter it is in the context of quantum mechanics or quantum field theory, we will get involved with the computation of a determinant of a differential operator with some boundary conditions. If we try naively to compute these determinants as the products of the corresponding eigenvalues we will get ill defined expressions. Hence, it is imperative to give a finite prescription for computing determinants for these cases. The generalized ζ-function method is precisely one possible way of doing that. It was introduced in physics in the middle seventies [6] and it is in fact a very powerful regularization prescription which has applications in several branches of physics (a detailed discussion can be found in reference [7]). This method, as we will see, is based on an analytical extension in the complex plane. We think that the harmonic oscillator propagator is the perfect scenario for introducing such an important method, because undergraduate students are all familiarized with the quantum harmonic oscillators and besides, it is the first non-trivial example after the free particle. In what follows, we shall first introduce briefly the ζ-function method, then we shall apply it to compute F (tf −ti) for the harmonic oscillator propagator and finally, we shall establish a direct equivalence between this method and Schwinger’s propertime method. Consider an operator A and let us assume, without loss of generality, that it has a discrete set of non-degenerate eigenvalues {λn}. When there is only a finite number of eigenvalues detA is just given by the product of these eigenvalues and we can write: