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Finite temperature properties of the Dirac operator under local boundary conditions.pdf

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Finite temperature properties of the Dirac operator under local boundary conditions

a r X i v : h e p - t h / 0 4 0 4 1 1 5 v 2 2 6 A u g 2 0 0 4 Finite temperature properties of the Dirac operator under local boundary conditions C.G. Beneventano?, E.M. Santangelo? Departamento de F??sica, Universidad Nacional de La Plata C.C.67, 1900 La Plata, Argentina February 1, 2008 Abstract We study the finite temperature free energy and fermion number for Dirac fields in a one-dimensional spatial segment, under two different members of the family of local boundary conditions defining a self-adjoint Euclidean Dirac operator in two dimensions. For one of such bound- ary conditions, compatible with the presence of a spectral asymmetry, we discuss in detail the contribution of this part of the spectrum to the zeta-regularized determinant of the Dirac operator and, thus, to the finite temperature properties of the theory. Subject Classification PACS: 11.10.Wx, 02.30.Sa MSC: 58J55, 35P05 1 Introduction When the Euclidean Dirac operator is considered on even-dimensional com- pact manifolds with boundary, its domain can be determined through a family of local boundary conditions which define a self-adjoint boundary problem [1] (the particular case of two-dimensional manifolds was first studied in [2]). The whole family is characterized by a real parameter θ, which can be interpreted as an analytic continuation of the well known θ parameter in gauge theories. These boundary conditions can be considered to be the natural counterpart in Euclidean space of the well known chiral bag boundary conditions. Recently, it was shown [3] that the boundary problem so defined is not only self-adjoint, but also strongly elliptic [4, 5] in any even dimension. Also in reference [3], the meromorphic properties of the associated zeta function were ?Fellow of CONICET - E-mail: gabriela@obelix.fisica.unlp.edu.ar ?Member of CONICET - E-mail: mariel@obelix.fisica.unlp.edu.ar 1 determined for manifolds of the product type. For particular non-product man- ifolds, heat kernel coefficient