Soliton operators in the quantum equivalence of the CP1 and O(3) - σmodels

We discuss some interesting aspects of the well known quantum equivalence between the O(3) - σ and CP1 models in 3D, working inthe canonical and in the path integral formulations. We show first that the canonical quantization in the hamiltonian formulation is freeof ordering ambiguities for both models. We use the canonical map between the fields and momenta of the two models and compute therelevant functional determinant to verify the equivalence between the phase-space partition functions and the quantum equivalence in allthe topological sectors. We also use the explicit form of the map to construct the soliton operator of the O(3) - σ model starting from therepresentation of the operator in the CP1 model, and discuss their properties.

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Shape Dynamics and the Linking Theory

10.1093/oso/9780198789475.003.0012 ◽

2018 ◽

Author(s):

Flavio Mercati

Keyword(s):

Phase Space ◽

Degrees Of Freedom ◽

Line Element ◽

Hamiltonian Formulation ◽

Operational Definition ◽

Extended Phase Space ◽

Extended Phase ◽

Problem Of Time ◽

Definition Of ◽

Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

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Hamilton-Jacobi Theory

10.1093/oso/9780198822370.003.0019 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Phase Space ◽

Bbgky Hierarchy ◽

Distribution Functions ◽

Symplectic Integrators ◽

Partition Functions ◽

Von Neumann ◽

Equipartition Theorem ◽

Recurrence Theorem ◽

Phase Amplitude ◽

Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

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Phase space path integral for modified pöschl-teller potential

Journal of Physics Conference Series ◽

10.1088/1742-6596/1766/1/012031 ◽

2021 ◽

Vol 1766 (1) ◽

pp. 012031

Author(s):

M S Allal ◽

B Bentag

Keyword(s):

Phase Space ◽

Path Integral

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Unifying Theory for Casimir Forces: Bulk and Surface Formulations

Universe ◽

10.3390/universe7070225 ◽

2021 ◽

Vol 7 (7) ◽

pp. 225

Author(s):

Giuseppe Bimonte ◽

Thorsten Emig

Keyword(s):

Free Energy ◽

Path Integral ◽

Casimir Force ◽

Hamiltonian Formulation ◽

Current Fluctuations ◽

Practical Application ◽

Casimir Forces ◽

Surface Approach ◽

Maxwell Stress Tensor ◽

Electromagnetic Fluctuation

The principles of the electromagnetic fluctuation-induced phenomena such as Casimir forces are well understood. However, recent experimental advances require universal and efficient methods to compute these forces. While several approaches have been proposed in the literature, their connection is often not entirely clear, and some of them have been introduced as purely numerical techniques. Here we present a unifying approach for the Casimir force and free energy that builds on both the Maxwell stress tensor and path integral quantization. The result is presented in terms of either bulk or surface operators that describe corresponding current fluctuations. Our surface approach yields a novel formula for the Casimir free energy. The path integral is presented both within a Lagrange and Hamiltonian formulation yielding different surface operators and expressions for the free energy that are equivalent. We compare our approaches to previously developed numerical methods and the scattering approach. The practical application of our methods is exemplified by the derivation of the Lifshitz formula.

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ON CANONICAL QUANTIZATION OF THE GAUGED WZW MODEL WITH PERMUTATION BRANES

International Journal of Modern Physics A ◽

10.1142/s0217751x11054668 ◽

2011 ◽

Vol 26 (26) ◽

pp. 4647-4660

Author(s):

GOR SARKISSIAN

Keyword(s):

Riemann Surface ◽

Phase Space ◽

Boundary Conditions ◽

Canonical Quantization ◽

Simons Theory ◽

Gauge Fields ◽

Time Line ◽

Chern Simons ◽

Special Case ◽

Chern Simons Theory

In this paper we perform canonical quantization of the product of the gauged WZW models on a strip with boundary conditions specified by permutation branes. We show that the phase space of the N-fold product of the gauged WZW model G/H on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of the double Chern–Simons theory on a sphere with N holes times the time-line with G and H gauge fields both coupled to two Wilson lines. For the special case of the topological coset G/G we arrive at the conclusion that the phase space of the N-fold product of the topological coset G/G on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of Chern–Simons theory on a Riemann surface of the genus N-1 times the time-line with four Wilson lines.

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TI/PIMC METHOD WITH THE TAKAHASHI–IMADA APPROXIMATION FOR THE EQUILIBRIUM CONSTANT OF THE O + HCl ⇌ OH + Cl REACTION

Journal of Theoretical and Computational Chemistry ◽

10.1142/s0219633613500260 ◽

2013 ◽

Vol 12 (04) ◽

pp. 1350026 ◽

Cited By ~ 1

Author(s):

MARCIN BUCHOWIECKI

Keyword(s):

Monte Carlo ◽

Temperature Dependence ◽

Equilibrium Constant ◽

Path Integral ◽

Thermodynamic Integration ◽

Partition Functions ◽

Integration Path ◽

Path Integral Monte Carlo

The thermodynamic integration/path integral Monte Carlo (TI/PIMC) method of calculating the temperature dependence of the equilibrium constant quantum mechanically is applied to O + HCl ⇌ OH + Cl reaction. The method is based upon PIMC simulations for energies of the reactants and the products and subsequently on thermodynamic integration for the ratios of partition functions. PIMC calculations are performed with the primitive approximation (PA) and the Takahashi–Imada approximation (TIA).

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PATH INTEGRAL QUANTIZATION OF SUPERSTRING

Modern Physics Letters A ◽

10.1142/s0217732310032287 ◽

2010 ◽

Vol 25 (02) ◽

pp. 135-141

Author(s):

H. A. ELEGLA ◽

N. I. FARAHAT

Keyword(s):

Phase Space ◽

Differential Equations ◽

Path Integral ◽

Equations Of Motion ◽

Singular System ◽

Constrained Systems ◽

Classical Structure ◽

Path Integral Quantization ◽

Total Differential

Motivated by the Hamilton–Jacobi approach of constrained systems, we analyze the classical structure of a four-dimensional superstring. The equations of motion for a singular system are obtained as total differential equations in many variables. The path integral quantization based on Hamilton–Jacobi approach is applied to quantize the system, and the integration is taken over the canonical phase space coordinates.

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Dirac’s Method for the Two-Dimensional Damped Harmonic Oscillator in the Extended Phase Space

Mathematics ◽

10.3390/math6100180 ◽

2018 ◽

Vol 6 (10) ◽

pp. 180 ◽

Cited By ~ 1

Author(s):

Laure Gouba

Keyword(s):

Phase Space ◽

Harmonic Oscillator ◽

Canonical Quantization ◽

Extended Phase Space ◽

Two Dimensional ◽

Class Constraint ◽

Extended Phase ◽

Damped Harmonic Oscillator ◽

Points Of View ◽

Space Description

The system of a two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem that has already been addressed by many authors that we present here with some fresh points of view and carry on a whole discussion. We show that the system is singular. The classical Hamiltonian is proportional to the first-class constraint. We pursue with the Dirac’s canonical quantization procedure by fixing the gauge and provide a reduced phase space description of the system. As a result, the quantum system is simply modeled by the original quantum Hamiltonian.

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