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The Gross-Neveu model is a quantum field theory model of Dirac fermions interacting via four fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 by David Gross and André Neveu as a toy model for quantum chromodynamics, the theory of strong interactions. It consists of N Dirac fermions, ψ1, ..., ψN. The Lagrangian density is : using the Einstein summation notation where g is the coupling constant. If the mass m is nonzero, the model is massive classically, otherwise it enjoys a chiral symmetry. This model has an U(N) global internal symmetry. Note that it does not reduce to the massive Thirring model (which is completely integrable). It is a 2-dimensional version of the 4-dimensional Nambu–Jona-Lasinio model (NJL), which was introduced 14 years earlier as a model of dynamical chiral symmetry breaking (but no quark confinement) modeled upon the BCS theory of superconductivity. The 2-dimensional version has the advantage that the 4-fermi interaction is renormalizable, which it is not in any higher number of dimensions. ==Features of the theory== Gross and Neveu studied this model in the large N limit, expanding the relevant parameters in a 1/N expansion. After demonstrating that this and related models are asymptotically free, they found that, in the subleading order, for small fermion masses the bifermion condensate acquires a vacuum expectation value (VEV) and as a result the fundamental fermions become massive. They find that the mass is not analytic in the coupling constant g. The vacuum expectation value spontaneously breaks the chiral symmetry of the theory. More precisely, expanding about the vacuum with no vacuum expectation value for the bilinear condensate they found a tachyon. To do this they solve the renormalization group equations for the propagator of the bifermion field, using the fact that the only renormalization of the coupling constant comes from the wave function renormalization of the composite field. They then calculated, at leading order in a 1/N expansion but to all orders in the coupling constant, the dependence of the potential energy on the condensate using the effective action techniques introduced the previous year by Sidney Coleman at the Erice International Summer School of Physics. They found that this potential is minimized at a nonzero value of the condensate, indicating that this is the true value of the condensate. Expanding the theory about the new vacuum, the tachyon was found to be no longer present and in fact, like the BCS theory of superconductivity, there is a mass gap. They then made a number of general arguments about dynamical mass generation in quantum field theories. For example, they demonstrated that not all masses may be dynamically generated in theories which are infrared-stable, using this to argue that, at least to leading order in 1/N, the 4-dimensional theory does not exist. They also argued that in asymptotically free theories the dynamically generated masses never depend analytically on the coupling constants. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gross–Neveu model」の詳細全文を読む スポンサード リンク
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