Effect of boundary conditions on dimensionally reduced field-theoretical models at finite temperature

E Cavalcanti,A P C Malbouisson,C A Linhares,J A Lourenço

doi:10.1103/physrevd.100.025008

Abstract

Here we understand \textit{dimensional reduction} as a procedure to obtain an effective model in $D-1$ dimensions that is related to the original model in $D$ dimensions. To explore this concept we use both a self-interacting fermionic model and self-interacting bosonic model. Furthermore, in both cases, we consider different boundary conditions in space: periodic, antiperiodic, Dirichlet and Neumann. For bosonic fields, we get the so defined dimensional reduction. Taking the simple example of a quartic interaction, we obtain that the boundary condition (periodic, Dirichlet, Neumann) influence the new coupling of the reduced model. For fermionic fields, we get the curious result that the model obtained reducing from $D$ dimensions to $D-1$ dimensions is distinguishable from taking into account a fermionic field originally in $D-1$ dimensions. Moreover, when one considers antiperiodic boundary condition in space (both for bosons or fermions) it is found that the dimensional reduction is not allowed.

Highlights

The construction and use of quantum field-theoretical models at dimensions different from the usual space-time in D 1⁄4 3 þ 1 are usual in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
We call a dimensional reduction the possibility that the model of the system (A) becomes or brings information about a planar model—like the one of case (B)—if we consider the limiting process to take the length to zero: L → 0. If we generalize this problem to an arbitrary number of dimensions, we can ask ourselves whether there is a relationship between a model in D dimensions and a model in D − 1 dimensions; this is the major objective in the present study
III C the procedure of dimensional reduction is ill defined. This result shows that the use of antiperiodic boundary conditions in space forbids the procedure of dimensional reduction both for bosonic and fermionic fields

Summary

INTRODUCTION

The construction and use of quantum field-theoretical models at dimensions different from the usual space-time in D 1⁄4 3 þ 1 are usual in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. We call a dimensional reduction the possibility that the model of the system (A) becomes or brings information about a planar model—like the one of case (B)—if we consider the limiting process to take the length to zero: L → 0 If we generalize this problem to an arbitrary number of dimensions, we can ask ourselves whether there is a relationship between a model in D dimensions and a model in D − 1 dimensions; this is the major objective in the present study. The logic is that at high temperatures there occurs a decoupling between static (a zero mode) and nonstatic contributions (nonzero modes) This reasoning occurs when dealing with periodic boundary condition, when we refer to antiperiodic boundary condition—as is the case of fermions in the thermal dimensional reduction—we do not have static modes [22]. It seems that a fermionic model in D dimensions is not related to a model originally built in D − 1 dimensions [35,36]

GENERIC MODEL AND BOUNDARY CONDITIONS

DIMENSIONAL REDUCTION

Bosonic field

Fermionic field