Physica A 387 (2008) 5101–5109 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Fock space for fermion-like lattices and the linear Glauber model Érica M. Silva a,c , Paulo T. Muzy b , Ademir E. Santana c,∗ a Núcleo de Física, Universidade Federal do Tocantins, 77020-210, Palmas, TO, Brazil b Instituto de Física, Universidade de São Paulo, Cidade Universitária, Caixa Postal 66318, 05315-970, São Paulo, SP, Brazil c Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF, Brazil article info Article history: Received 31 December 2007 Received in revised form 15 March 2008 Available online 8 April 2008 PACS: 05.30.Fk 05.50.+q 75.10.Pq Keywords: Linear Glauber model Fock space Spin lattice abstract The concept of Fock space representation is developed to deal with stochastic spin lattices written in terms of fermion operators. A density operator is introduced in order to follow in parallel the developments of the case of bosons in the literature. Some general conceptual quantities for spin lattices are then derived, including the notion of generating function and path integral via Grassmann variables. The formalism is used to derive the Liouvillian of the d-dimensional Linear Glauber dynamics in the Fock-space representation. Then the time evolution equations for the magnetization and the two-point correlation function are derived in terms of the number operator. © 2008 Elsevier B.V. All rights reserved. 1. Introduction The notion of Fock space, or number representation, was introduced in classical physics by Schönberg [1] to describe classes of linear equations, such as the Liouville equation in phase space. Later, Doi used this procedure to deal with reaction- diffusion processes [2], pointing a way to treat stochastic equations with the methods of quantum field theory. In this realm, the creation and annihilation field operators describe, for instance, the reagents taking place in a chemical reaction. This theory has been developed in several directions [3], including the work by Martin, Siggia, and Rose [4] deriving a functional formalism for the nonequilibrium statistical mechanics. A version of path integrals for bosons was proposed by Peliti [5], and it has been extended and applied to different systems [6,7]. In the trend of developments over the last decades, approaches for stochastic models based on number representation seem, at first sight, disconnected from each other, thus raising practical obstructions which have been addressed, for instance, by Grassberger and Scheunert [8] and Andersen [9] (for a review, see Ref. [3] and references therein). One of these difficulties is that number representation has been used in quantum theories because of the indistinguishability of subatomic particles, an aspect described by probability amplitudes. For classical systems, however, probability amplitudes have been defined in association with a generalization of the Liouville theorem [1,10–25]. The physical and mathematical nature of such a formalism have also been analyzed with representations of Lie groups [26], where the Fock space is taken as a representative vector space of symmetry. In this sense there is no “ ¯ h” in the method, resulting in full consistency with classical physics and no ambiguity with quantum theory. Many of these developments were carried out for bosons [27], leaving the case of fermion-like lattices to be studied in a broader sense. ∗ Corresponding author. Tel.: +55 61 3307 2900; fax: +55 61 3307 2363. E-mail address: asantana@fis.unb.br (A.E. Santana). 0378-4371/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.04.015