Solutions Of Classical Field Equations Research Articles

Low-energy effective quantum field theoretic description of excitations about soliton configurations

Solitons are the classical field configurations connecting two trivial vacua. These are also the solutions of classical field equations of motion with particle-like properties. Moreover, they are localized in space, having finite energy, and are stable against decay into radiation. The coherent state description of kink-solitons is discussed in the present article. Further, the relation between topological solitons and occupation numbers corresponding to low momentum excitations are also discussed coherently. The description of the low energy excitations about solitons in quantum field theory is the main theme of this article. Further, a few physical observables, namely some low order correlation functions, are computed up to certain integral forms. Furthermore, we have shown that it is possible to detect the presence of soliton-like classical configurations in many-particle systems from the nature of the one-point function and non-conservation of momentum feature of one-point, two-point and three-point functions in this low-energy effective field theory of these excitations.

Early universe with modified scalar-tensor theory of gravity

Scalar-tensor theory of gravity with non-minimal coupling is a fairly good candidate for dark energy, required to explain late-time cosmic evolution. Here we study the very early stage of evolution of the universe with a modified version of the theory, which includes scalar curvature squared term. One of the key aspects of the present study is that, the quantum dynamics of the action under consideration ends up generically with de-Sitter expansion under semiclassical approximation, rather than power-law. This justifies the analysis of inflationary regime with de-Sitter expansion. The other key aspect is that, while studying gravitational perturbation, the perturbed generalized scalar field equation obtained from the perturbed action, when matched with the perturbed form of the background scalar field equation, relates the coupling parameter and the potential exactly in the same manner as the solution of classical field equations does, assuming de-Sitter expansion. The study also reveals that the quantum theory is well behaved, inflationary parameters fall well within the observational limit and quantum perturbation analysis shows that the power-spectrum does not deviate considerably from the standard one obtained from minimally coupled theory.

Soliton solutions of classical equations of motions in the modified formulation of the Yang–Mills theory

We show that soliton solutions of classical field equations exist in the modified formulation of the Yang–Mills theory, which produces the same formal perturbation theory as the standard formulation.

Nonabelian T-duals of the flat background

Classification of nonabelian T-duals of the flat matric in D=4 with respect to the four-dimensional continuous subgroups of the Poincare group is given. After dualizing the flat background we identify majority of dual models as conformal sigma models in plane- parallel wave backgrounds, most of them having torsion. We give their form in Brinkmann coordinates. Besides the pp-waves we find several diagonalizable curved metrics with nontrivial scalar curvature and torsion.Due to the nonabelian T-duality, we find general solution of classical field equations of all the sigma models in terms of d'Alembert solutions of wave equation.

Holographic interpretations of the renormalization group

Abstract In semiclassical holographic duality, the running couplings of a field theory are conventionally identified with the classical solutions of field equations in the dual gravitational theory. However, this identification is unclear when the bulk fields fluctuate. Recent work has used a Wilsonian framework to propose an alternative identification of the running couplings in terms of non-fluctuating data; in the classical limit, these new couplings do not satisfy the bulk equations of motion. We study renormalization scheme dependence in the latter formalism, and show that a scheme exists in which couplings to single trace operators realize particular solutions to the bulk equations of motion, in the semiclassical limit. This occurs for operators with dimension $ \varDelta \notin \frac{d}{2}+\mathbb{Z} $ , for sufficiently low momenta. We then clarify the relation between the saddle point approximation to the Wilsonian effective action (S W ) and boundary conditions at a cutoff surface in AdS space. In particular, we interpret non-local multi-trace operators in S W as arising in Lorentzian AdS space from the temporary passage of excitations through the UV region that has been integrated out. Coarse-graining these operators makes the action effectively local.

Superconductivity in Restricted Chromo-Dynamics (RCD) in SU(2) and SU(3) Gauge Theories

Characterizing the dyonically condensed vacuum by the presence of two massive modes (one determining how fast the perturbative vacuum around a colour source reaches the condensation and the other giving the penetration length of colored flux) in SU(2) theory, it has been shown that due to the dynamical breaking of magnetic symmetry the vacuum of RCD acquires the properties similar to those of relativistic superconductor. Analysing the behaviour of dyons around RCD string, the solutions of classical field equations have been obtained and it has been shown that magnetic constituent of dyonic current is zero at centre of the string and also at the points far away from the string. Extending RCD in the realistic color gauge group SU(3), it has been shown that the resulting Lagrangian leads to dyonic condensation, color confinement and the superconductivity with the presence of two scalar modes and two vector modes.

Dual Superconductivity in Restricted Chromo-Dynamics (RCD)

Characterizing the dyonically condensed vacuum by the presence of two massive modes (one determining how fast the perturbative vacuum around a colour source reaches the condensation and the other giving the penetration length of colored flux), it has been shown that due to the dynamical breaking of magnetic symmetry the vacuum of RCD acquires the properties similar to those of relativistic superconductor. Originally present global SU(2) symmetry in RCD has been broken to U(1) reducing the four dimensional action to two dimensional one by using an Ansatz which incorporates a non-trivial coordinate dependent phase between the components of SU(2) doublet. Analyzing the behaviour of dyons around RCD string, the solutions of classical field equations have been obtained and it has been shown that magnetic constituent of dyonic current is zero at centre of the string and also at the points far away from the string. The conditions for this current to be maximum at a transverse distance from the string have also been obtained.

Towards chaos criterion in quantum field theory

Chaos criterion for quantum field theory is proposed. Its correspondence with classical chaos criterion in semi-classical regime is shown. It is demonstrated for real scalar field that proposed chaos criterion can be used to investigate stability of classical solutions of field equations.

One-loop quantum energy densities of domain wall field configurations

We discuss a simple procedure for computing one-loop quantum energies of any static field configuration that depends non-trivially on only a single spatial coordinate. We specifically focus on domain wall-type field configurations that connect two distinct minima of the effective potential, and may or may not be the solutions of classical field equations. We avoid the conventional summation of zero-point energies, and instead exploit the relation between functional determinants and solutions of associated differential equations. This approach allows ultraviolet divergences to be easily isolated and extracted using any convenient regularization scheme. Two examples are considered: two-dimensional $\phi^4$ theory, and three-dimensional scalar electrodynamics with spontaneous symmetry breaking at the one-loop level.

Sphalerons and large order behaviour of perturbation theory in lower dimension

Sphalerons - unstable static solutions of classical field equations in ( d + 1)-dimensional spacetime - may be viewed as euclidean solutions in d dimensions. We discuss their role in the large order asymptotics of perturbation theory. Specifically, we calculate their contribution to the large order behaviour of the ground state energy in a quantum mechanical model. When the number of negative modes is odd, the single sphaleron contribution dominates, while this contribution vanishes when the number of negative modes is even. These results are confirmed by numerical calculations.

Nonlinear field equations and twistor theory

In 1967 Roger Penrose introduced what is now called twistor geometry (Penrose 1967). This geometry is an amplification of the geometry of lines in projective space studied extensively by Felix Klein and others in the late 19th century. Penrose made the fundamental observation that this classical geometry could be used in conjunction with a Radon transform style integral geometry to provide rather complete sets of solutions of numerous partial differential equations, both linear and nonlinear, which arise naturally in theoretical physics. The Radon transform, which involves integration over straight lines in R 3, was used by F. John to generate solutions of hyperbolic partial differential equations in 1942. Penrose integrates over complex projective lines, real 2-spheres, and the data of the integration is complex-analytic functions of 3 complex variables. This is analogous to (and indeed related to) Weierstrass' representation of solutions of the minimal surface equation in terms of triples of holomorphic functions of a single complex variable which satisfy an algebraic relation. In this paper we discuss the origins of the differential equations which are amiable to twistor-geometric study from the realms of theoretical physics. Twistor geometry is most suitable for physical theories which are relativistically invariant, including Maxwell's description of electromagnetic phenomena from the 19th century (Maxwell's equations), Einstein's theory of gravitation (the equations of general relativity), and the contemporary descriptions of elementary particles (quantum field theory). Twistor geometry interacts in a nontrivial manner with the mathematical models involved in all of the above physical theories. In particular, it has led to new solutions of some of the nonlinear partial differential equations involved (Einstein's equations, Yang-Mills equations, equations of a magnetic monopole). We will describe some recent progress on twistorgeometric representations of solutions of classical field equations. There are two aspects to this. First, there is the problem of translating a given system of differential equations into twistor-geometric language. This has been done successfully in many different cases, and will be described in Section 3. Once a given problem has been transferred to a twistor-geometric context, it becomes, in general, a specific problem involving algebraic geometry, several complex variables, and algebraic topology involving holomorphic data on complex manifolds. The only partial differential equations remaining in the twistor context are the CauchyRiemann equations characterizing holomorphic functions. In effect, the partial differential equation has been transformed into a geometric problem. This is analogous to the Fourier transform transforming a partial differential equation with constant coefficients into an algebraic problem, where differentiation is replaced by the (algebraic) symbol of the differential operator in question. The second aspect of the recent progress on the twistor-geometric representation of the solutions of the classical field equations concerns itself with solving the geometric problem in question and transforming back to space-time (or Euclidean space), generating (sometimes new) solutions of the classical field equations.

New meron-type solutions

New meron-type solutions are constructed to the Thirring model. During the last few years a considerable amount of work has been devoted to the study of the classical solutions of field equations within various fields of theoretical physics. An interpretation of classical solutions was given in the form of vacuum expectation values of quantum fields. This has represented an interesting ground of investigation for a deeper understanding of the formal aspects of the theory (1 ). Among the classical solutions, instanton and meron solutions had a particular place. Since the first instanton solutions (2), the one and multiple instanton solutions were found (3) and the determinants, the one loop contribution in the presence of the background field of the classical solutions, were calculated (4). Self-coupled spinor models were also studied in this line of research, and meron- and instanton- type solutions of pure Thirring (5) and G~rsey (6) models were found (7). In this paper, we construct new meron-type solutions for the original Thirring model. We start with tile Lagrangian in the Euclidean domain

Supersymmetry and classical solutions

We obtain classical solutions to the field equations of the massless supersymmetric Wess-Zumino model and to the field equations of the interacting SU(2) gauge supermultiplet. This is done by applying finite supersymmetry transformations to the known solutions of the scalar field equation with ϕ 4 interaction and the Yang-Mills field equations. The relevance of supersymmetry to the solution of classical field equations involving anticommuting fermion fields is discussed.

Cancellation of the zero-mode singularities in soliton quantization theory

We give a proof of an exact cancellation of the zero-mode singularities in each order of the loop expansion determining the quantum corrections to soliton (particle-like) solutions of classical field equations. The cancellation restores an underlying symmetry broken by a specific classical solution, and is owing to the stability of the soliton with respect to small perturbation generated by the symmetry group transformations.

Renarks on the classical limit of quantum field theories

Recently, there has been an increasing interest in computing quantum mechanical corrections to solutions of classical field equations. In this note, we want to proceed in the opposite way and we summarize theorems about the classical limit of relativistic quantum field models. These results are a byproduct of the so called ‘constructive’ approach to quantum field theory.

Gravitation in the light cone gauge

The formulation of gravitation theory in the light cone gauge is studied. After a brief discussion of Yang- Mills theory for purposes of illustration, tensor and scalartensor gravitation are investigated. We show that if the gauge conditions are properly chosen the constrained components of the metric tensor can be explicitly solved for by quadrature, so that the field theory can be reformulated entirely in terms of the physical transverse fields. It is also shown that the light cone gauge is useful for finding wave solutions of classical field equations. Occasional reference is made to dual models, primarily to explain our motivation, but familiarity with them is not required for an understanding of this paper.

Solutions of classical field equations with locally finite kinetic energy

Classical non-linear field equations are discussed in the space of functions having locally finite kinetic energy. An existence and uniqueness theorem for the Cauchy problem is presented.