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arXiv:hep-th/0105290

Notes about equivalence between Sine-Gordon theory (free fermion point) and the free fermion theory

S. V. Kryukov

Published May 29, 2001• 0 citations
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Abstract

The space of local integrals of motion for the Sine-Gordon theory (the free fermion point) and the theory of free fermions in the light cone coordinates is investigated. Some important differences between the spaces of local integrals of motion of these theories are obtained. The equivalence is broken on the level of the integrals of motion between bosonic and fermionic theories (in the free fermion point). The integrals of motion are constracted without Quantum Inverse Scattering Method (QISM)and the additional quantum integrals of motion are obtaned. So the QISM is not absolutely complete.

DOI
10.48550/arxiv.hep-th/0105290

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The space of local integrals of motion for the Sine-Gordon theory (the free fermion point) and the theory of free fermions in the light cone coordinates is investigated. Some important differences between the spaces of local integrals of motion of these theories are obtained. The equivalence is broken on the level of the integrals of motion between bosonic and fermionic theories (in the free fermion point). The integrals of motion are constracted without Quantum Inverse Scattering Method (QISM)and the additional quantum integrals of motion are obtaned. So the QISM is not absolutely complete.
This PhD thesis addresses the problem of the calculation of the one point functions (1PF) in integrable two-dimensional quantum field theories. A method for their calculation has been developed in the context of the sine-Gordon Theory. The integrability of the model was used to build a basis of local operators to describe the six-vertex model. This basis, called fermionic, is interesting because the vacuum expectation values of its operators are expressed in terms of determinants and the fermionic structure can be extended to the continuous limit in order to characterize local operators in the CFT. In this thesis, we continue to work on this approach, aiming to generalize the fermionic basis to the supersymmetric sine-Gordon model (ssG). We derived scaling equations governing the thermodynamics of the ssG theory, reproducing the BLZ generating function. Then, we described the integrable structure of the ssG model using the fermion-current basis. We focused on the fermionic part and calculated its one point functions. These results were verified with a different approach based on the reflection relations.
We extend the notion of space shifts introduced in [FV3] for certain quantum light cone lattice equations of sine-Gordon type at root of unity (e.g. [FV1][FV2][BKP][BBR]). As a result we obtain a compatibility equation for the roots of central elements within the algebra of observables (also called current algebra). The equation which is obtained by exponentiating these roots is exactly the evolution equation for the "classical background" as described in [BBR]. As an application for the introduced constructions, a one to one correspondence between a special case of the quantum light cone lattice equations of sine-Gordon type and free massive fermions on a lattice as constructed in [DV] is derived. Supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 288 e-mail: nadja@math.tu-berlin.de 1 Introduction Let us consider a class of integrable lattice systems defined by an evolution equation of the following type: g u = V 0 (g l \\Gamma g r ) + g d = 0; (1) where...
This work didactically presents the mathematical procedures required for the construction of the creation and annihilation operators for a free quantum particle considering the coordinates of the light cone. For that, the relationships between the aforementioned coordinates and the coordinates (ct, x, y, z) are listed, in addition to the use of the Klein-Gordon-Fock equation in the formalism of the light cone coordinates. Finally, the temporal evolution operator and the quantum operators of creation and annihilation of the integral type of motion are obtained.
We study a massive Thirring-like model in 2-dimensional space-time, which contains two different species of fermions. This model is a field theoretical version of the quantum mechanical model originally proposed by Glöckle, Nogami and Fukui, where different fermions interact with each other through δ-function potentials. We derive a corresponding boson model by the bosonization technique in the path integral formulation. This is a simple but non-trivial extension of the freedom of the bosonization technique. Operator correspondences between fermion and boson fields are given. One of these could not be realistically expected from the naive correspondence of the original single-species models. It is essential for this point that in our model fermions of the same kind do not interact with each other directly. We find that for a specific value of the coupling constant, one boson field becomes free while the other is a Sine-Gordon field. For this case, therefore, our two-species model is equivalent to the ordinary Sine-Gordon model of a single boson field.