1. Introduction

The Gross-Neveu (GN) model describes a system of massless fermions with self-interacting fermions which generates a dynamical mass. The GN model was born as a toy model of Quantum Chromodynamics (QCD) ¹. At first, it was studied using semiclassical functional methods, which is rigorously justified in the large N limit. In the traditional context, the GN model has shown a rich phase structure, in particular, there we have chiral symmetry breaking. Which, at first sight, it could not be allowed in 1+1 dimensions due to infrared fluctuations, unless it is invoked the large N limit ². Despite its simplicity, it keeps many exciting features, such as discrete chiral symmetry, dynamical mass generation, and asymptotic freedom.

Curiously, this model has also application in condensed matter physics, where it describes the conductivity in some polymers. In particular, it can be mentioned the case of trans-polyacetylene, which, in a simplified continuous model, is described by the symmetric GN model ³. Besides, the massive GN model has a condensed matter analog in the modelization of polymers with non-degenerate ground states ⁴.

The standard approach to the GN model leads to an homogeneous condensate solution, i.e., independent of the space coordinates. After several years, it was realized that there are crystal solutions of the model, giving a rich interpretation in the realm of condensed matter physics ⁵. The spatial dimension could be constrained to a finite size to consider Casimir type forces. Without having in mind any particular model, we asked ourselves to consider many boundary conditions (BC’s) to simulate a broad variety of possible physical scenarios. Those are periodic, antiperiodic and confining BC’s.

Having a limited spatial extension leads us to consider the Casimir energy associated with such spaces. We found that the Casimir energies and forces are sensitive to the BC’s and parameters involved. Using the generic name of “universe" for the spatial dimension we identified the stable, metastable, attractive or repulsive regimes, depending on the parameters and BC’s we use

There is previous work in the direction we propose. In particular, in ⁶, it was computed the Casimir energy with MIT Bag Model boundary condition, obtaining the behavior of the Casimir energy as a function of the distance between the points and the mass of the fermionic field. On the other side, the renormalization issue of the GN model was addressed in ⁷, using the worldline Montecarlo approach.

In our study, we shall concentrate on the homogeneous solutions of the GN model for a finite space of fixed size L. We are interested in the behaviour of physical parameters for different boundary conditions (BC’s). The boundary conditions considered are periodic, antiperiodic and two types of confining conditions.

The HF approximation implies the use of a large momentum cutoff. Since we shall deal with systems of finite spatial size, the momentum integrals must be replaced by summation on discrete modes, meaning that the natural regularization to be used is the zeta regularization technique ⁸.

In this work, we first ask about the ultraviolet dependence of the physical parameters on the BC’s, considering the GN model at zero bare mass (m0=0) where temperature and chemical potential are not considered. We assume that the spatial length L is a fixed parameter, so, if the physical mass is independent of the cutoff, it implies that the beta function does not depend on the BC’s. There appears an arbitrary mass scale and the functional dependency of the dynamical mass depends on the BC’s.

The second step in our work is to study the Casimir energy and force due to the quantum fluctuation of the effective free system that arises from the HF approximation. We consider the non-dimensional parameter μ=mL, since the value of m is fixed by ultraviolet considerations, the variation of μ is equivalent to the variation of L. We find that the value of energy and Force are sensitive to the BC’s. In particular, the signature of the energy clearly differs in the small size limit, but it is universally positive for infinite size limit. On the other hand, the force is also sensitive to the BC’s, implying situations where the forces are such that they compress or expand our space depending on the BC’s used. There is also a universal metastable point where the force becomes zero independently of the BC’s used. For the large L limit, the force becomes negative for any BC’s considered.

2. The Gross-Neveu Model

The Gross-Neveu Lagrangian is given by

where i runs from 1 to N. For the sake of simplicity, from now, we suppress the index i. The Euler Lagrange equation from (1) is given by

In the framework of Hartree-Fock relativistic approximation, it is assumed the expectation value ⟨ψ¯γ5ψ⟩=0 and ⟨ψ¯ψ⟩=Nρ.

It was introduced a finite mass in order to consider a general expression and using the convention

we end up with the expression

where m=m0-g2Nρ and ρ=⟨ψ¯ψ⟩/N.

From (4) we obtain a free Dirac equation

In order to obtain a stationary solution, we use the usual decomposition

Making the redefinition of the fields

we obtain a general solution

where Ω=λn2-m2 and the constants α and β are not independent since they are determined by the boundary conditions.

3. Hartree-Fock for different boundary conditions

Following the standard procedure ⁹, it is possible to compute the negative energy in an infinite space taking the value of m as a parameter to be determined

where G=Ng2 and Λ is a momentum cut off.

Since we have a finite spatial size, the wave number k is discretized

implying

where r is a number which depends on boundary conditions. So, we have

Dealing with the summation implies a treatment of Epstein zeta function

using properties of gamma function and by means of Jacobi inverse summation formulae , we have

From the above integral, we can recognize a gamma function and a second kind Bessel function ¹⁰, leading us to the general expression

We are interested in the value s=-1/2, but there is a singularity in such point, so we can isolate that term by meaning of a series expasion in term of s=-1/2+ϵ, when ϵ→0, giving us the expression for the Epstein zeta function

where we use tha fact that Kn(x)=K-n(x). Note that there is a diverging term, which can be seen as an ultraviolet cutoff. Since the power 1/2 in the summation is replaced by a term 1/2-ϵ, it appears a mass scale η by means of the regularization procedure ⁸.

We have the following momentum decomposition for the BC’s to be considered:

We note that the imposed boundary conditions over kn have the general structure

Introducing the parameter of mass η and rewriting the energy density we have

As we see, the summation term can be expressed as an Epstein zeta function, so we have

Minimizing the energy density respect to μ we can obtain a dimensionless expression

as Xμ=0, the coupling parameter G is given by

For the four considered BC’s, we obtained the following expressions for the energy density:

where it was introduced the non-dimensional variables μ=mL and η~=ηL.

In the following step, we minimize the energy densities with respect to μ. Then, we use (21) and obtain for each BC an expression for G , which can be resumed to the form

where ϵ=s+1/2 goes to zero and must be considered as the ultraviolet cut-off. On the other side, the finite terms depends on the BC’s.

For finite ϵ, it gives the impression that the running of G should depend on the BC’s. But, if we fix the value of G in an arbitrary scale, the limit ϵ→0 is universal, independent of any BC.

We observe from the general relation (21), that there is dependency of the constants μ and η~ for each BC through the transcendental equation:

being C an arbitrary constant.

Considering the traditional point of view where the physical μ must be independent of the cutt-off ϵ we have a renormalization group equation

For any BC, it is computed the beta function

meaning an universal behaviour of G(ϵ) as it is shown in Fig. 1.

4. Casimir Energy for global boundary conditions

Imposing BC’s of the form

In Eqs. (8) and (9) we have solutions of the form

where the values of α,β depend on the imposed BC on the problem. If we define the invertible matrix

the constants are given by

meaning that

The BC can be expressed as

Then, the eigenvalue condition is given by

We can include the periodic and anti periodic case by the parametrization

From (8) and (9), we have

Eq. (35) leads to the condition

Since Ω=λ2-m2, we have

Where μ=mL and r a parameter which depends of the boundary conditions, so the general expression for the Casimir energy for spinor field is given by

We can recognize that the summation term in (37) an Epstein zeta function, so we use the representation in (15) in order to obtain the Casimir energy. We redefine the Casimir energy given by

The Casimir Force can be obtained by the usual way FCas=-dECas/dL, but considering the redefinition given before and η~≡ηL, we have

where FCas=F/m2.

5. Specific boundary conditions

6. Conclusions and discussion

The first part of this article focused on the study of the ultraviolet behavior of the GN model for different BC’s, using zeta function regularization and assuming a homogeneous solution. We found that the beta function is independent of the type of boundary condition used, and that there appears a mass scale of arbitrary value. The generated dynamical mass should depend on the BC’s, if we have no prescription on the arbitrary mass scale.

Later, assuming a homogeneous solution, we studied the Casimir energy and forces for different BC’s, if we concentrate on the behavior of ξ/μ from Figs. 2 and 3, we notice the following features:

For the Casimir forces, from Figs. 4 and 6, we conclude that:

It is shown in Fig. 6 that for η~≥4, there is a common point μ where the Casimir force becomes zero for any boundary condition.

From the above considerations, we conclude that for BC’s periodic and confining i, there are two regimes of forces, being positive for “small” μ, representing a universe that has an expanding tendency. On the other hand, when μ is “big", our universe is a shrinking one.

For the antiperiodic and confining ii, there is a more complex situation, since its behavior depends on the value of η~. For η~≤η~*, the force is always negative, hence there is an shrinking universe. For η~≥4, there is mixed case as it is shown in Fig. 6, there are the points A,A' and the universal point B. Between A(A') and B, the force becomes positive. It is also clear that B is an unstable point and the points A(A') are repelling points.

This study suggests that the further natural step is to consider a general relativity study where the spatial dynamics are affected by the quantum fluctuations of the Casimir energy and confirm if the BC’s determine the existence of shrinking or expanding low dimensional universes.