Gordon Field Theory Research Articles

Strong/weak duality symmetries for Jacobi–Gordon field theory through elliptic functions

Abstract It is shown that the standard sin/sinh Gordon field theory with the strong/weak duality symmetry of its quantum S-matrix, can be formulated in terms of elliptic functions with their duality symmetries, which will correspond to the classical realization of that quantum symmetry. Specifically we show that the so called self-dual point that divides the strong and the weak coupling regimes, corresponds only to one point of a set of fixed points under the duality transformations for the elliptic functions. Furthermore, the equations of motion can be solved in exact form in terms of the inverse elliptic functions; in spontaneous symmetry breaking scenarios, these solutions show that 
kink-like solitons can decay to cusp-like solitons

Entanglement entropy after a partial projective measurement in 1 + 1 dimensional conformal field theories: exact results

We calculate analytically the Rényi bipartite entanglement entropy of the ground state of 1 + 1 dimensional conformal field theories (CFT) after performing a projective measurement in part of the system. We show that the entanglement entropy in this setup is dependent on the central charge and the operator content of the system. When the measurement region A separates the two parts B and , the entanglement entropy between B and decreases like a power-law with respect to the characteristic distance between the two regions with an exponent which is dependent on the rank α of the Rényi entanglement entropy and the smallest scaling dimension present in the system. We check our findings by making numerical calculations on the Klein–Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators. We also comment on the post-measurement entanglement entropy in the massive quantum field theories.

How instanton combinatorics solves Painlevé VI, V and IIIs

We elaborate on a recently conjectured relation of Painlevé transcendents and 2D conformal field theory. General solutions of Painlevé VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGT related to instanton partition functions in supersymmetric gauge theories with Nf = 0, 1, 2, 3, 4. The resulting combinatorial series representations of Painlevé functions provide an efficient tool for their numerical computation at finite values of the argument. The series involves sums over bipartitions which, in the simplest cases, coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the Gaussian Unitary Ensemble, and all-order conformal perturbation theory expansions of correlation functions in the sine–Gordon field theory at the free-fermion point.

Soliton–potential interaction in the nonlinear Klein–Gordon model

The interaction of solitons with external potentials in the nonlinear Klein–Gordon field theory is investigated using an improved model. The presented model has been constructed with a better approximation for adding the potential to the Lagrangian through the metric of background space–time. The results of the model are compared with another model and the differences are discussed.

The field theory of symmetrical layered electrolytic systems and the thermal Casimireffect

We present a general extension of a field-theoretic approach developed in earlierpapers to the calculation of the free energy of symmetrically layered electrolyticsystems which is based on the sine–Gordon field theory for the Coulomb gas. Themethod is to construct the partition function in terms of the Feynman evolutionkernel in the Euclidean time variable associated with the coordinate normal to thesurfaces defining the layered structure. The theory is applicable to cylindricalsystems and its development is motivated by the possibility that a static van derWaals or thermal Casimir force could provide an attractive force stabilizing adielectric tube formed from a lipid bilayer, an example of which is provided by thet-tubules occurring in certain muscle cells. In this context, we apply the theoryto the calculation of the thermal Casimir effect for a dielectric tube of radiusR andthickness δ formed from such a membrane in water. In a grand canonical approachwe find that the leading contribution to the Casimir energy behaves like−kBTLκC/R which gives rise to an attractive force which tends to contract the tube radius. We find thatκC∼0.3 for the case of typical lipid membrane t-tubules. We conclude that except in the case of avery soft membrane this force is insufficient to stabilize such tubes against thebending stress which tends to increase the radius. We briefly discuss the role ofthe lipid membrane reservoir implicit in the approach and whether its nature inbiological systems may possibly lead to a stabilizing mechanism for such lipid tubes.

Evolution of the energy and momentum distribution of electron beam in the gamma-ray laser synchrotron sources

Evolution of the relativistic electron beam in the laser-beam Compton scattering was investigated. The quantum kinetic equation for the electron beam distribution function is derived in the frame of the Klein–Gordon field theory. When the beam energy spread exceeds the energy of scattered photons, this equation is reduced to the diffusion-like kinetic equation which describes the laser cooling of electron beam and includes quantum fluctuations limiting the cooling process. In the opposite limit of high beam energies, we describe the evolution of transverse beam emittance in the laser-beam Compton scattering.

HIDDEN SYMMETRIES IN QUANTUM FIELD THEORIES FROM EXTENDED COMPLEX NUMBERS

The two-dimensional space–time sine–Gordon field theory is extended algebraically within the n-dimensional space of extended complex numbers. This field theory is constructed in terms of an adapted extension of standard vertex operators. A whole set of nonlocal conserved charges is constructed and studied in this framework. Thereby, an algebraic nonperturbative description is possible for this n-1 parameters family of quantum field theories. Known results are obtained for specific values of the parameters, especially in relation to affine Toda field theories. Different (dual)-models can then be described in this formalism.

EXTENSION OF SINE–GORDON FIELD THEORY FROM GENERALIZED CLIFFORD ALGEBRAS

Linearization of homogeneous polynomials of degree n and k variables leads to generalized Clifford algebras. Multicomplex numbers are then introduced in analogy to complex numbers with respect to the usual Clifford algebra. In turn multi-complex extensions of trigonometric functions are constructed in terms of "compact" and "non-compact" variables. It gives rise to the natural extension of the d-dimensional sine–Gordon field theory in the n-dimensional multicomplex space. In two dimensions, the cases n = 1, 2, 3, 4 are identified as the quantum integrable Liouville, sine–Gordon and known deformed Toda models. The general case is discussed.

M-Theory from Worldsheet Defects in Liouville String

We have argued previously that black holes may be represented in a D-brane approach by monopole and vortex defects in a sine–Gordon field theory model of Liouville dynamics on the worldsheet. Supersymmetrizing this sine–Gordon system, we find critical behavior in 11 dimensions, due to defect condensation which we interpret as the worldsheet analog of D-brane condensation around an extra space–time dimension in M-theory. This supersymmetric description of Liouville dynamics has a natural embedding within a 12-dimensional framework suggestive of F-theory.

Algebraic approach to quantum field theory on non-globally-hyperbolic spacetimes

The mathematical formalism for linear quantum field theory on curved spacetime depends in an essential way on the assumption of global hyperbolicity. Physically, what lie at the foundation of any formalism for quantization in curved spacetime are the canonical commutation relations, imposed on the field operators evaluated at a global Cauchy surface. In the algebraic formulation of linear quantum field theory, the canonical commutation relations are restated in terms of a well-defined symplectic structure on the space of smooth solutions, and the local field algebra is constructed as the Weyl algebra associated to this symplectic vector space. When spacetime is not globally hyperbolic, e.g. when it contains naked singularities or closed time-like curves, a global Cauchy surface does not exist, and there is no obvious way to formulate the canonical commutation relations, hence no obvious way to construct the field algebra. In a paper which appears elsewhere in this journal, we report on a generalization of the algebraic framework for quantum field theory to arbitrary topological spaces which do not necessarily have a spacetime metric defined on them at the outset. Taking this generalization as a starting point, in this paper we give a prescription for constructing the field algebra of a (massless or massive) Klein--Gordon field on an arbitrary background spacetime. When spacetime is globally hyperbolic, the theory defined by our construction coincides with the ordinary Klein--Gordon field theory on a globally hyperbolic background. We explore some basic features of our generalized Klein--Gordon theory on arbitrary spacetimes, and study its specific properties on simple examples of non-globally-hyperbolic backgrounds that contain closed time-like curves or naked singularities.

ON THE KOSTERLITZ–THOULESS TRANSITION ON COMPACT RIEMANN SURFACES

A Lagrangian for the two-dimensional vortex gas is derived from a general microscopic Lagrangian for 4 He atoms on an arbitrary compact Riemann Surface without boundary. In the constant density limit the vortex Hamiltonian obtained from this Lagrangian is found to be the same as the Kosterlitz and Thouless Coulombic interaction Hamiltonian. The partition function for the Kosterlitz–Thouless ensemble on the general compact is formulated and mapped into the sine–Gordon field theory.

QUANTIZED BREATHERS IN A DOUBLE SINE-GORDON SYSTEM

We examine the excitations of a quasi-one dimensional easy plane antiferromagnet, in a symmetry breaking applied magnetic field. The antiferromagnetic exchange includes a small Dzyaloshinski Moriya component that produces the easy plane. In the classical continuum limit, this system is described by a double-sine-Gordon field theory, which supports soliton excitations and the multi spin wave bound states/resonances of the discrete spin system. The latter correspond to the quantized breather excitations of the double sine Gordon field theory.

An alternate proof of ultraviolet stability of the two-dimensional massive sine–Gordon field theory in all regions of collapse

The upper bound for the ultraviolet stability of the two-dimensional cosine interaction ∫Λ:cos αφξ:dξ, Λ⊆R2, in finite volume Λ is proven for α2 ∈ [4π,8π[, where the theory has been shown to be superrenormalizable [see, e.g., G. Gallavotti, Rev. Mod. Phys. 57, 471 (1985)]. Ultraviolet stability in this interval was proven previously (F. Nicolò, J. Renn, and A.Steinmann, ‘‘On the massive sine–Gordon equation in all regions of collapse,’’ preprint II Università di Roma, 1985). Here we give a second proof using renormalization group methods based on a multiscale decomposition of the field by showing that the large fluctuations may be controlled by their small probability. The method essentially follows the one given by Nicolò [F. Nicolò, Commun. Math. Phys. 88, 681 (1983)] for α2 ∈ [4π, (32)/(5) π[.