scholarly journals Classical Analog of Extended Phase Space SUSY and Its Breaking

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Gagik Ter-Kazarian
Keyword(s):  
Phase Space ◽  
Ground State ◽  
Degrees Of Freedom ◽  
Susy Breaking ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Shape Invariance ◽  
Classical Analog ◽  
Algebraic Properties ◽  
Practical Measure

We derive the classical analog of the extended phase space quantum mechanics of the particle with odd degrees of freedom which gives rise to ()-realization of supersymmetry (SUSY) algebra. By means of an iterative procedure, we find the approximate ground state solutions to the extended Schrödinger-like equation and use these solutions further to calculate the parameters which measure the breaking of extended SUSY such as the ground state energy. Consequently, we calculate a more practical measure for the SUSY breaking which is the expectation value of an auxiliary field. We analyze nonperturbative mechanism for extended phase space SUSY breaking in the instanton picture and show that this has resulted from tunneling between the classical vacua of the theory. Particular attention is given to the algebraic properties of shape invariance and spectrum generating algebra.

Shape Dynamics and the Linking Theory

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.

scholarly journals HAMILTONIAN ANALYSIS FOR ABELIAN AND NON-ABELIAN MASSIVE THEORIES IN THREE DIMENSIONS

2013 ◽  
Vol 28 (08) ◽  
pp. 1350019
Author(s):  
ALBERTO ESCALANTE ◽  
JOSÉ L. OSIO
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Three Dimensions ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Extended Action ◽  
Full Structure ◽  
Hamiltonian Analysis

A pure Dirac's method for Abelian and non-Abelian massive theories in three dimensions is performed. Our analysis is developed on the extended phase space, reporting the relevant structure of the theories, namely, the extended action, the extended Hamiltonian, the full structure of the constraints and the counting of degrees of freedom. In addition, we compare our results with those found in the literature.

scholarly journals Path Integrals for (Complex) Classical and Quantum Mechanics

10.14311/1414 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
R. J. Rivers
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Degrees Of Freedom ◽  
Path Integrals ◽  
Classical Mechanics ◽  
Classical Particle ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Complex Extension ◽  
Classical And Quantum Mechanics

An analysis of classical mechanics in a complex extension of phase space shows that a particle in such a space can behave in a way redolent of quantum mechanics; additional dimensions permit ‘tunnelling’ without recourse to instantons and time/energy uncertainties exist. In practice, ‘classical’ particle trajectories with additional degrees of freedom arise in several different formulations of quantum mechanics. In this talk we compare the extended phase space of the closed time-path formalism with that of complex classical mechanics, to suggest that ℏ has a role in our understanding of the latter. However, differences in the way that trajectories are used make a deeper comparison problematical. We conclude with some thoughts on quantisation as dimensional reduction.

TRANSITION STATE THEORY IN EXTENDED PHASE SPACE

1993 ◽  
Vol 07 (19) ◽  
pp. 1263-1268
Author(s):  
H. DEKKER ◽  
A. MAASSEN VAN DEN BRINK
Keyword(s):  
Phase Space ◽  
Transition State Theory ◽  
Unstable Mode ◽  
Planck Equation ◽  
State Theory ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Theoretical Concepts ◽  
Rate Processes ◽  
Mode Energy

Turnover theory (of the escape Γ) à la Grabert will be based solely on Kramers' Fokker–Planck equation for activated rate processes. No recourse to a microscope model or Langevin dynamics will be made. Apart from the unstable mode energy E, the analysis requires new theoretical concepts such as a constrained Gaussian transformation (CGT) and dynamically extended phase space (EPS).

scholarly journals Reality of the Wigner Functions and Quantization

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Sadollah Nasiri ◽  
Samira Bahrami
Keyword(s):  
Phase Space ◽  
Quantum Statistical Mechanics ◽  
Direct Consequence ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Reality Condition ◽  
Energy Quantization ◽  
Extended Action ◽  
Space Formulation ◽  
Quantum Statistical

Here we use the extended phase space formulation of quantum statistical mechanics proposed in an earlier work to define an extended lagrangian for Wigner's functions (WFs). The extended action defined by this lagrangian is a function of ordinary phase space variables. The reality condition of WFs is employed to quantize the extended action. The energy quantization is obtained as a direct consequence of the quantized action. The technique is applied to find the energy states of harmonic oscillator, particle in the box, and hydrogen atom as the illustrative examples.

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

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 180 ◽  
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.

scholarly journals COVARIANT DESCRIPTION OF PARAMETRIZED NONRELATIVISTIC HAMILTONIAN SYSTEMS

2004 ◽  
Vol 19 (15) ◽  
pp. 2473-2493 ◽  
Author(s):  
MAURICIO MONDRAGÓN ◽  
MERCED MONTESINOS
Keyword(s):  
Phase Space ◽  
Gauge Transformation ◽  
Hamiltonian Systems ◽  
Symplectic Structure ◽  
Canonical Formalism ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Algebra Of Observables ◽  
Covariant Description ◽  
Constraint Surface

The various phase spaces involved in the dynamics of parametrized nonrelativistic Hamiltonian systems are displayed by using Crnkovic and Witten's covariant canonical formalism. It is also pointed out that in Dirac's canonical formalism there exists a freedom in the choice of the symplectic structure on the extended phase space and in the choice of the equations that define the constraint surface with the only restriction that these two choices combine in such a way that any pair (of these two choices) generates the same gauge transformation. The consequence of this freedom on the algebra of observables is also discussed.

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