scholarly journals A PROPOSAL FOR A DIFFERENTIAL CALCULUS IN QUANTUM MECHANICS

1994 ◽  
Vol 09 (13) ◽  
pp. 2191-2227 ◽  
Author(s):  
E. GOZZI ◽  
M. REUTER
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Hamiltonian System ◽  
Differential Calculus ◽  
Cotangent Bundle ◽  
Differential Forms ◽  
Lie Derivative ◽  
Exterior Derivative ◽  
Tensor Calculus ◽  
Exterior Calculus

In this paper, using, the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a quantum-deformed exterior calculus on the phase space of an arbitrary Hamiltonian system. Introducing additional bosonic and fermionic coordinates, we construct a supermanifold which is closely related to the tangent and cotangent bundle over phase space. Scalar functions on the supermanifold become equivalent to differential forms on the standard phase space. The algebra of these functions is equipped with a Moyal superstar product which deforms the pointwise product of the classical tensor calculus. We use the Moyal bracket algebra to derive a set of quantum-deformed rules for the exterior derivative, Lie derivative, contraction, and similar operations of the Cartan calculus.

scholarly journals A primer on exterior differential calculus

2003 ◽  
pp. 85-162 ◽  
Author(s):  
D.A. Burton
Keyword(s):  
Continuum Mechanics ◽  
Differential Calculus ◽  
Differential Forms ◽  
Lie Derivative ◽  
Exterior Differential ◽  
Exterior Calculus ◽  
Vector Calculus ◽  
Exterior Forms ◽  
Pedagogical Application

A pedagogical application-oriented introduction to the cal?culus of exterior differential forms on differential manifolds is presented. Stokes' theorem, the Lie derivative, linear con?nections and their curvature, torsion and non-metricity are discussed. Numerous examples using differential calculus are given and some detailed comparisons are made with their tradi?tional vector counterparts. In particular, vector calculus on R3 is cast in terms of exterior calculus and the traditional Stokes' and divergence theorems replaced by the more powerful exterior expression of Stokes' theorem. Examples from classical continuum mechanics and spacetime physics are discussed and worked through using the language of exterior forms. The numerous advantages of this calculus, over more traditional ma?chinery, are stressed throughout the article. .

QUANTUM-DEFORMED GEOMETRY ON PHASE-SPACE

1993 ◽  
Vol 08 (15) ◽  
pp. 1433-1442 ◽  
Author(s):  
E. GOZZI ◽  
M. REUTER
Keyword(s):  
Vector Field ◽  
Phase Space ◽  
Mechanical System ◽  
Time Evolution ◽  
Cotangent Bundle ◽  
Symplectic Manifolds ◽  
Hamiltonian Vector ◽  
Lie Derivative ◽  
Hamiltonian Vector Field ◽  
Quantum Analog

In this paper we extend the standard Moyal formalism to the tangent and cotangent bundle of the phase-space of any Hamiltonian mechanical system. In this manner we build the quantum analog of the classical Hamiltonian vector-field of time evolution and its associated Lie-derivative. We also use this extended Moyal formalism to develop a quantum analog of the Cartan calculus on symplectic manifolds.

scholarly journals On the Geometry of Relativistic Anyon

1997 ◽  
Vol 12 (24) ◽  
pp. 1783-1789 ◽  
Author(s):  
A. Nersessian
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Irreducible Representations ◽  
Hamiltonian Reduction ◽  
Poincare Group ◽  
Covariant Model ◽  
Lobachevsky Plane ◽  
Spin Generator

A twistor model is proposed for the free relativistic anyon. The Hamiltonian reduction of this model by the action of the spin generator leads to the minimal covariant model; whereas that by the action of spin and mass generators leads to the anyon model with free phase space which is a cotangent bundle of the Lobachevsky plane with twisted symplectic structure. Quantum mechanics of that model is described by irreducible representations of the (2+1)-dimensional Poincaré group.

scholarly journals Poisson Algebra of Differential Forms

1997 ◽  
Vol 12 (31) ◽  
pp. 5573-5587 ◽  
Author(s):  
Chong-Sun Chu ◽  
Pei-Ming Ho
Keyword(s):  
Differential Calculus ◽  
Symplectic Geometry ◽  
Differential Algebra ◽  
Differential Forms ◽  
Poisson Algebra ◽  
Exterior Derivative ◽  
Complex Dimension ◽  
Definition Of ◽  
Darboux's Theorem ◽  
Made In

We give a natural definition of a Poisson differential algebra. Consistency conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on the differential calculus in a simple canonical form by a coordinate trans-formation. This is in analogy with the standard Darboux's theorem for symplectic geometry. For certain cases there exists a realization of the exterior derivative through a certain canonical one-form. All the above are carried out similarly for the case of a complex Poisson differential algebra. The case of one complex dimension is treated in detail and interesting features are noted. Conclusions are made in the last section.

scholarly journals Non-Smooth Geodesic Flows and Classical Mechanics

1969 ◽  
Vol 12 (2) ◽  
pp. 209-212 ◽  
Author(s):  
J. E. Marsden
Keyword(s):  
Phase Space ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Configuration Space ◽  
Smooth Function ◽  
Cotangent Bundle ◽  
Classical Mechanics ◽  
Riemannian Metric ◽  
Geodesic Flows ◽  
Image Position

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In fact, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q-space or the configuration space), we may define a smooth function Tg on the cotangent bundle T*M (q-p-space, or the phase space). This function is the kinetic energy of q, and locally is given by

Basic Forms

2020 ◽  
pp. 97-102
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Total Space ◽  
Lie Derivative ◽  
Connected Lie Group

This chapter describes basic forms. On a principal bundle π‎: P → M, the differential forms on P that are pullbacks of forms ω‎ on the base M are called basic forms. The chapter characterizes basic forms in terms of the Lie derivative and interior multiplication. It shows that basic forms on a principal bundle are invariant and horizontal. To understand basic forms better, the chapter considers a simple example. The plane ℝ2 may be viewed as the total space of a principal ℝ-bundle. A connected Lie group is generated by any neighborhood of the identity. This example shows the necessity of the connectedness hypothesis.

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