Generalized Phase Space Formulation of the Hamiltonian Dynamics

1974 ◽  
Vol 52 (16) ◽  
pp. 1532-1546 ◽  
Author(s):  
M. Razavy ◽  
Frederick James Kennedy
Keyword(s):  
Phase Space ◽  
Hamiltonian Dynamics ◽  
Equations Of Motion ◽  
Hamiltonian Function ◽  
Hamiltonian Operator ◽  
Classical Dynamics ◽  
Time Translation ◽  
Number Of Generators ◽  
Space Formulation ◽  
Generalized Phase Space

In an n dimensional phase space, the generator of the time translation can be written in terms of a Hamiltonian and a set of Poisson brackets for the phase space variables. When the velocity vector in this phase space is divergenceless, then the equations of motion reduce to those obtained by Nambu. The extension of the Hamiltonian dynamics to the phase space of arbitrary dimensions enables one to find a generalized Hamiltonian function for equations of motion involving time derivatives of any order (even or odd) of the coordinates. The problem of quantization of Nambu's generalized dynamics is studied, and it is shown that in certain cases, for a system moving under a set of constraints, it is possible to replace the Hamiltonian operator by an infinite number of generators of time translation functions. Some examples from classical dynamics and quantum mechanics are given to show the range of applicability of the generalized phase space formulation.

scholarly journals Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 518 ◽  
Author(s):  
Alessandro Sergi ◽  
Gabriel Hanna ◽  
Roberto Grimaudo ◽  
Antonino Messina
Keyword(s):  
Constant Temperature ◽  
Degrees Of Freedom ◽  
Open Quantum Systems ◽  
Hamiltonian Dynamics ◽  
Quantum Systems ◽  
Classical Dynamics ◽  
Open Quantum ◽  
Time Translation ◽  
Translation Symmetry ◽  
Lie Brackets

Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nosé–Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics/Monte Carlo algorithms, which we outline herein.

Statistical dynamics of multiply-periodic systems

1956 ◽  
Vol 52 (2) ◽  
pp. 287-300 ◽  
Author(s):  
M. Born ◽  
D. J. Hooton ◽  
N. F. Mott
Keyword(s):  
Phase Space ◽  
Probability Density ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Universal Constant ◽  
Physical Reality ◽  
Statistical Interpretation ◽  
Essential Difference ◽  
Classical Dynamics ◽  
Deterministic Theory

ABSTRACTClassical dynamics is generally considered as the prototype of a deterministic theory; the equations of motion determine the coordinates q and the momenta p at any time provided they are given at an initial instant. It is pointed out that this is an unrealistic assumption, for it is inevitable that there should be small uncertainties Δq, Δp; a point of the mathematical continuum has no physical significance. The uncertainties can be taken into account without violating the deterministic equations by introducing a probability density P in phase space p, q. The function P satisfies the partial differential equation which expresses Liouville's theorem. Thus it can be shown that an initial uncertainty spreads out in phase space, so that finally all states of the system consistent with the mean initial values of the constants of the motion are equally probable. This property holds for one degree of freedom just as well as for many, and is not a consequence of our ignorance concerning large numbers of particles. The essential difference between classical mechanics and quantum mechanics consists not in the physical necessity of a statistical interpretation, but in the further introduction of a probability amplitude, the square of which is the probability density; thisimplies the restriction of the initial uncertainties as expressed by the uncertainty laws, and the phenomenon of interference of probabilities which makes necessary a revision of our concepts of physical reality. It is remarkable that the product of the uncertainties of a properly chosen pair of variables is constant also in classical mechanics, although of course its value is not a universal constant as in the quantum case.

scholarly journals Covariant phase space with boundaries

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Daniel Harlow ◽  
Jie-qiang Wu
Keyword(s):  
Phase Space ◽  
Boundary Conditions ◽  
Hamiltonian Dynamics ◽  
Equations Of Motion ◽  
Field Theories ◽  
Boundary Terms ◽  
Space Formalism ◽  
Spatial Boundary ◽  
Original Construction ◽  
Space Method

Abstract The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically treat total derivatives and boundary terms, which has led to some confusion about how exactly to apply the formalism in the presence of boundaries. In particular the original construction of the canonical Hamiltonian relies on the assumed existence of a certain boundary quantity “B”, whose physical interpretation has not been clear. We here give an algorithmic procedure for applying the covariant phase space formalism to field theories with spatial boundaries, from which the term in the Hamiltonian involving B emerges naturally. Our procedure also produces an additional boundary term, which was not present in the original literature and which so far has only appeared implicitly in specific examples, and which is already nonvanishing even in general relativity with sufficiently permissive boundary conditions. The only requirement we impose is that at solutions of the equations of motion the action is stationary modulo future/past boundary terms under arbitrary variations obeying the spatial boundary conditions; from this the symplectic structure and the Hamiltonian for any diffeomorphism that preserves the theory are unambiguously constructed. We show in examples that the Hamiltonian so constructed agrees with previous results. We also show that the Poisson bracket on covariant phase space directly coincides with the Peierls bracket, without any need for non-covariant intermediate steps, and we discuss possible implications for the entropy of dynamical black hole horizons.

scholarly journals On the Dirac eigenvalues as observables of the on-shell N = 2D = 4 Euclidean supergravity

Open Physics ◽  
2008 ◽  
Vol 6 (4) ◽  
Author(s):  
Ion Vancea
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Gauge Transformations

AbstractWe generalize previous works on the Dirac eigenvalues as dynamical variables of Euclidean gravity and N =1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their value for the Dirac eigenvalues.

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.

PATH INTEGRAL QUANTIZATION OF SUPERSTRING

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.

scholarly journals Quantum gravity, dynamical phase-space and string theory

2014 ◽  
Vol 23 (12) ◽  
pp. 1442006 ◽  
Author(s):  
Laurent Freidel ◽  
Robert G. Leigh ◽  
Djordje Minic
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
String Theory ◽  
Quantum Gravity ◽  
Momentum Space ◽  
Symplectic Geometry ◽  
Complex Geometry ◽  
Hamiltonian Dynamics ◽  
Natural Extension ◽  
Dynamical Phase

In a natural extension of the relativity principle, we speculate that a quantum theory of gravity involves two fundamental scales associated with both dynamical spacetime as well as dynamical momentum space. This view of quantum gravity is explicitly realized in a new formulation of string theory which involves dynamical phase-space and in which spacetime is a derived concept. This formulation naturally unifies symplectic geometry of Hamiltonian dynamics, complex geometry of quantum theory and real geometry of general relativity. The spacetime and momentum space dynamics, and thus dynamical phase-space, is governed by a new version of the renormalization group (RG).

scholarly journals NONCOMMUTATIVE METAFLUID DYNAMICS

2006 ◽  
Vol 21 (03) ◽  
pp. 505-516 ◽  
Author(s):  
A. C. R. MENDES ◽  
C. NEVES ◽  
W. OLIVEIRA ◽  
F. I. TAKAKURA
Keyword(s):  
Phase Space ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Additional Term ◽  
Dissipative Force ◽  
Covariant Form ◽  
Covariant Quantization ◽  
Noncommutative Spaces ◽  
Classical Phase Space

In this paper we define a noncommutative (NC) metafluid dynamics.1,2 We applied the Dirac's quantization to the metafluid dynamics on NC spaces. First class constraints were found which are the same obtained in Ref. 4. The gauge covariant quantization of the nonlinear equations of fields on noncommutative spaces were studied. We have found the extended Hamiltonian which leads to equations of motion in the gauge covariant form. In addition, we show that a particular transformation3 on the usual classical phase space (CPS) leads to the same results as of the ⋆-deformation with ν = 0. Besides, we have shown that an additional term is introduced into the dissipative force due to the NC geometry. This is an interesting feature due to the NC nature induced into model.

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