The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a
Dirac field in (1+1) dimensions.
Definition
The Thirring model is given by the
Lagrangian density
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
:
where
is the field, ''g'' is the
coupling constant, ''m'' is the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
, and
, for
, are the two-dimensional
gamma matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
.
This is the unique model of (1+1)-dimensional, Dirac fermions with a local (self-)interaction. Indeed, since there are only 4 independent fields, because of the
Pauli principle, all the quartic, local interactions are equivalent; and all higher power, local interactions vanish. (Interactions containing derivatives, such as
, are not considered because they are non-renormalizable.)
The correlation functions of the Thirring model (massive or massless) verify the Osterwalder–Schrader axioms, and hence the theory makes sense as a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
.
Massless case
The massless Thirring model is exactly solvable in the sense that a formula for the
-points field correlation is known.
Exact solution
After it was introduced by
Walter Thirring
Walter Thirring (29 April 1927 – 19 August 2014) was an Austrian physicist after whom the Thirring model in quantum field theory is named. He was the son of the physicist Hans Thirring.Thirring, H. Über die Wirkung rotierender ferner Massen ...
,
[
] many authors tried to solve the massless case, with confusing outcomes. The correct formula for the two and four point correlation was finally found by K. Johnson;
[
] then C. R. Hagen
[
] and B. Klaiber
[
] extended the explicit solution to any multipoint correlation function of the fields.
Massive Thirring model, or MTM
The ''mass spectrum'' of the model and the ''scattering matrix'' was explicitly evaluated by
Bethe Ansatz. An explicit formula for the correlations is ''not'' known. J. I. Cirac, P. Maraner and J. K. Pachos applied the massive Thirring model to the description of optical lattices.
[
]
Exact solution
In one space dimension and one time dimension the model can be solved by the
Bethe Ansatz. This helps one calculate exactly the mass spectrum
and
scattering matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
. Calculation of the scattering matrix reproduces the results published earlier by
Alexander Zamolodchikov
Alexander Borisovich Zamolodchikov (russian: Алекса́ндр Бори́сович Замоло́дчиков; born September 18, 1952) is a Russian physicist, known for his contributions to condensed matter physics, two-dimensional conformal ...
. The paper with the exact solution of Massive Thirring model by Bethe Ansatz was first published in Russian.
[
Translated in ] Ultraviolet
renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
was done in the frame of the Bethe Ansatz. The fractional charge appears in the model during renormalization as a repulsion beyond the cutoff.
Multi-particle production cancels on mass shell.
The exact solution shows once again the equivalence of the Thirring model and the quantum
sine-Gordon model
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of sur ...
. The Thirring model is
S-dual to the
sine-Gordon model
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of sur ...
. The fundamental fermions of the Thirring model correspond to the
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
s of the
sine-Gordon model
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of sur ...
.
Bosonization
S. Coleman
[
] discovered an equivalence between the Thirring and the
sine-Gordon model
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of sur ...
s. Despite the fact that the latter is a pure boson model, massless Thirring fermions are equivalent to free bosons; besides massive fermions are equivalent to the sine-Gordon bosons. This phenomenon is more general in two dimensions and is called
bosonization
In theoretical condensed matter physics and quantum field theory, bosonization is a mathematical procedure by which a system of interacting fermions in (1+1) dimensions can be transformed to a system of massless, non-interacting bosons.
The metho ...
.
See also
*
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
*
Gross–Neveu model
The Gross–Neveu (GN) 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 ...
*
Nonlinear Dirac equation
:''See Ricci calculus and Van der Waerden notation for the notation.''
In quantum field theory, the nonlinear Dirac equation is a model of self-interacting Dirac fermions. This model is widely considered in quantum physics as a toy model of self-i ...
*
Soler model
The soler model is a quantum field theory model of Dirac fermions interacting via four fermion interactions in 3 spatial and 1 time dimension. It was introduced in 1938 by Dmitri Ivanenko
and re-introduced and investigated in 1970 by Mario S ...
References
External links
On the equivalence between sine-Gordon Model and Thirring Model in the chirally broken phase
{{DEFAULTSORT:Thirring Model
Quantum field theory
Exactly solvable models