scholarly journals Quantum Linear Scalar Fields with Time Dependent Potentials: Overview and Applications to Cosmology

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 115 ◽  
Author(s):  
Jerónimo Cortez ◽  
Guillermo A. Mena Marugán ◽  
José Velhinho
Keyword(s):  
Scalar Fields ◽  
Complex Structures ◽  
De Sitter ◽  
Quantum Theories ◽  
Bianchi I ◽  
Uniqueness Results ◽  
Klein Gordon ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes ◽  
Linear Scalar

In this work, we present an overview of uniqueness results derived in recent years for the quantization of Gowdy cosmological models and for (test) Klein-Gordon fields minimally coupled to Friedmann-Lemaître-Robertson-Walker, de Sitter, and Bianchi I spacetimes. These results are attained by imposing the criteria of symmetry invariance and of unitary implementability of the dynamics. This powerful combination of criteria allows not only to address the ambiguity in the representation of the canonical commutation relations, but also to single out a preferred set of fundamental variables. For the sake of clarity and completeness in the presentation (essentially as a background and complementary material), we first review the classical and quantum theories of a scalar field in globally hyperbolic spacetimes. Special emphasis is made on complex structures and the unitary implementability of symplectic transformations.

scholarly journals Complex Structures for Klein-Gordon theory on globally hyperbolic spacetimes

2021 ◽  
Author(s):  
Albert Much ◽  
Robert Oeckl
Keyword(s):  
Differential Equation ◽  
Complex Structure ◽  
Complex Structures ◽  
Main Ingredient ◽  
Globally Hyperbolic ◽  
Klein Gordon ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes ◽  
Friedmann Robertson Walker ◽  
Rigorous Method

Abstract We develop a rigorous method to parametrize complex structures for Klein-Gordon theory in globally hyperbolic spacetimes that satisfy a completeness condition. The complex structures are conserved under time-evolution and implement unitary quantizations. They can be interpreted as corresponding to global choices of vacuum. The main ingredient in our construction is a system of operator differential equations. We provide a number of theorems ensuring that all ingredients and steps in the construction are well-defined. We apply the method to exhibit natural quantizations for certain classes of globally hyperbolic spacetimes. In particular, we consider static, expanding and Friedmann-Robertson-Walker spacetimes. Moreover, for a huge class of spacetimes we prove that the differential equation for the complex structure is given by the Gelfand-Dikki equation.

scholarly journals THE QUANTUM THEORY OF SCALAR FIELDS ON THE DE SITTER EXPANDING UNIVERSE

2008 ◽  
Vol 23 (16n17) ◽  
pp. 2563-2577 ◽  
Author(s):  
ION I. COTĂESCU ◽  
COSMIN CRUCEAN ◽  
ADRIAN POP
Keyword(s):  
Scalar Field ◽  
Gordon Equation ◽  
Scalar Fields ◽  
Wave Functions ◽  
De Sitter ◽  
Scalar Quantum ◽  
Long Time ◽  
Klein Gordon ◽  
Free Scalar ◽  
Special Time

New quantum modes of the free scalar field are derived in a special time-evolution picture that may be introduced in moving charts of de Sitter backgrounds. The wave functions of these new modes are solutions of the Klein–Gordon equation and energy eigenfunctions, defining the energy basis. This completes the scalar quantum mechanics where the momentum basis is well known for long time. In this enlarged framework the quantization of the scalar field can be done in canonical way obtaining the principal conserved one-particle operators and the Green functions.

scholarly journals Dynamical gravastars from the interaction between scalar fields and matter

2021 ◽  
Vol 81 (10) ◽  
Author(s):  
Alejandro Cabo Montes de Oca ◽  
Duvier Suarez Fontanella ◽  
David Valls-Gabaud
Keyword(s):  
Scalar Field ◽  
Yukawa Potential ◽  
Direct Interaction ◽  
Scalar Fields ◽  
Classical Solutions ◽  
Internal Space ◽  
De Sitter ◽  
Matter Interaction ◽  
Klein Gordon ◽  
Trapping Forces

AbstractGravastars are configurations of compact singularity-free gravitational objects which are interesting alternatives to classical solutions in the strong gravitational field regime. Although there are no static star-like solutions of the Einstein–Klein–Gordon equations for real scalar fields, we show that dynamical gravastars solutions arise through the direct interaction of a scalar field with matter. Two configurations presented here show that, within the internal zone, the scalar field plays a role similar to a cosmological constant, while it decays at large distances as the Yukawa potential. Like classical gravastars, these solutions exhibit small values of the temporal metric component near a transitional radial value, although this behaviour is not determined by the de Sitter nature of the internal space-time, but rather by a slowly-varying scalar field. The scalar field-matter interaction is able to define trapping forces that rigorously confine the polytropic gases to the interior of a sphere. At the surface of these spheres, pressures generated by the field-matter interaction play the role of “walls” preventing the matter from flowing out. These solutions predict a stronger scattering of the accreting matter with respect to Schwarzschild black holes.

scholarly journals Klein–Gordon field from the XXZ Heisenberg model

2019 ◽  
Vol 28 (01) ◽  
pp. 1950020 ◽  
Author(s):  
Jakub Bilski ◽  
Suddhasattwa Brahma ◽  
Antonino Marcianò ◽  
Jakub Mielczarek
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Heisenberg Model ◽  
Anisotropy Parameter ◽  
Scalar Fields ◽  
Scalar Field Theory ◽  
Interaction Terms ◽  
Klein Gordon ◽  
Xxz Heisenberg Model ◽  
Linear Scalar

We examine the recently introduced idea of Spin-Field Correspondence (SFC) focusing on the example of the spin system described by the XXZ Heisenberg model with external magnetic field. The Hamiltonian of the resulting nonlinear scalar field theory is derived for arbitrary value of the anisotropy parameter [Formula: see text]. We show that the linear scalar field theory is reconstructed in the large spin limit. For [Formula: see text], a nonrelativistic scalar field theory satisfying the Born reciprocity principle is recovered. As expected, for the vanishing anisotropy parameter [Formula: see text], the standard relativistic Klein–Gordon field is obtained. Various aspects of the obtained class of the scalar fields are studied, including the fate of the relativistic symmetries and the properties of the emerging interaction terms. We show that, in a certain limit, the so-called polymer quantization of the field variables is recovered. This and other discussed properties suggest a possible relevance of the considered framework in the context of quantum gravity.

scholarly journals Self-Adjointness in Klein-Gordon Theory on Globally Hyperbolic Spacetimes

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Albert Much ◽  
Robert Oeckl
Keyword(s):  
Large Class ◽  
Hilbert Spaces ◽  
Hyperbolic Manifolds ◽  
External Potential ◽  
Globally Hyperbolic ◽  
Spatial Part ◽  
Klein Gordon ◽  
Globally Hyperbolic Spacetimes ◽  
Hyperbolic Spacetimes

AbstractWe prove essential self-adjointness of the spatial part of the linear Klein-Gordon operator with external potential for a large class of globally hyperbolic manifolds. The proof is conducted by a fusion of new results concerning globally hyperbolic manifolds, the theory of weighted Hilbert spaces and related functional analytic advances.

scholarly journals BIANCHI I SOLUTIONS OF EFFECTIVE QUADRATIC GRAVITY

2012 ◽  
Vol 21 (04) ◽  
pp. 1250037 ◽  
Author(s):  
DANIEL MÜLLER ◽  
JULIANO A. DE DEUS
Keyword(s):  
Numerical Solutions ◽  
Relativity Theory ◽  
De Sitter ◽  
General Relativity Theory ◽  
Spatial Metrics ◽  
Effective Gravity ◽  
Bianchi I ◽  
Quadratic Theory ◽  
Quadratic Gravity ◽  
Planck Time

It is believed that soon after the Planck time, Einstein's general relativity theory should be corrected to an effective quadratic theory. Numerical solutions for the anisotropic generalization of the Friedmann "flat" model E3 for this effective gravity are given. It must be emphasized that although numeric, these solutions are exact in the sense that they depend only on the precision of the machine. The solutions are identified asymptotically in a certain sense. It is found solutions which asymptote de Sitter space, Minkowski space and a singularity. This work is a generalization for nondiagonal spatial metrics of a previous result obtained by one of us and a collaborator for Bianchi I spaces.

scholarly journals FINITE ACTION KLEIN–GORDON SOLUTIONS ON LORENTZIAN MANIFOLDS

2006 ◽  
Vol 03 (07) ◽  
pp. 1349-1357 ◽  
Author(s):  
V. V. KOZLOV ◽  
I. V. VOLOVICH
Keyword(s):  
Mass Spectrum ◽  
Elliptic Equations ◽  
Gordon Equation ◽  
Infinite Family ◽  
De Sitter ◽  
Lorentzian Manifolds ◽  
Klein Gordon ◽  
Finite Action ◽  
Discrete Mass ◽  
Klein Gordon Equation

The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this paper we consider such a problem for the hyperbolic Klein–Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An infinite family of square integrable solutions for the Klein–Gordon equation on the Friedman type manifolds is constructed. These solutions have a discrete mass spectrum and a finite action. In particular the solutions on de Sitter space are investigated.

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