Lie Series and Canonical Transformations in Complex Phase Space

1988 ◽  
Vol 43 (5) ◽  
pp. 411-418 ◽  
Author(s):  
B. Bruhn
Keyword(s):  
Phase Space ◽  
Generating Function ◽  
Series Representation ◽  
Canonical Transformations ◽  
Lie Series ◽  
Complex Phase ◽  
Lie Transformations ◽  
Complex Valued

This paper considers the Lie series representation of the canonical transformations in a complex phase space. A condition is given which selects the canonical mappings from the Lie transformations associated with a complex-valued generating function. Some special types of mappings and some simple algebraic tools are discussed.

Canonical Transformations, Entropy and Quantization

1987 ◽  
Vol 42 (4) ◽  
pp. 333-340 ◽  
Author(s):  
B. Bruhn
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Arbitrary Function ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Integrable Hamiltonian Systems ◽  
Eigenvalue Spectrum ◽  
Complex Phase ◽  
Algebraic Treatment ◽  
Canonical Coordinate

This paper considers various aspects of the canonical coordinate transformations in a complex phase space. The main result is given by two theorems which describe two special families of mappings between integrable Hamiltonian systems. The generating function of these transformations is determined by the entropy and a second arbitrary function which we take to be the energy function. For simple integrable systems an algebraic treatment based on the group properties of the canonical transformations is given to calculate the eigenvalue spectrum of the energy.

scholarly journals QUANTUM CANONICAL TRANSFORMATIONS IN WEYL–WIGNER–GROENEWOLD–MOYAL FORMALISM

2009 ◽  
Vol 24 (24) ◽  
pp. 4573-4587 ◽  
Author(s):  
TEKİN DERELİ ◽  
TUĞRUL HAKİOĞLU ◽  
ADNAN TEĞMEN
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Generating Function ◽  
Canonical Transformation ◽  
Star Product ◽  
Product Form ◽  
Canonical Transformations ◽  
Exponential Form ◽  
Alternative Approach ◽  
Point Transformations

A conjecture in quantum mechanics states that any quantum canonical transformation can decompose into a sequence of three basic canonical transformations; gauge, point and interchange of coordinates and momenta. It is shown that if one attempts to construct the three basic transformations in star-product form, while gauge and point transformations are immediate in star-exponential form, interchange has no correspondent, but it is possible in an ordinary exponential form. As an alternative approach, it is shown that all three basic transformations can be constructed in the ordinary exponential form and that in some cases this approach provides more useful tools than the star-exponential form in finding the generating function for given canonical transformation or vice versa. It is also shown that transforms of c-number phase space functions under linear–nonlinear canonical transformations and intertwining method can be treated within this argument.

scholarly journals Complex Covariance

10.14311/1809 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Frieder Kleefeld
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
Special Relativity ◽  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Space Time ◽  
Complex Phase ◽  
Dirac Equations ◽  
Klein Gordon ◽  
Complex Valued

According to some generalized correspondence principle the classical limit of a non-Hermitian quantum theory describing quantum degrees of freedom is expected to be the well known classical mechanics of classical degrees of freedom in the complex phase space, i.e., some phase space spanned by complex-valued space and momentum coordinates. As special relativity was developed by Einstein merely for real-valued space-time and four-momentum, we will try to understand how special relativity and covariance can be extended to complex-valued space-time and four-momentum. Our considerations will lead us not only to some unconventional derivation of Lorentz transformations for complex-valued velocities, but also to the non-Hermitian Klein-Gordon and Dirac equations, which are to lay the foundations of a non-Hermitian quantum theory.

scholarly journals Tunneling mechanism due to chaos in a complex phase space

2001 ◽  
Vol 64 (2) ◽  
Author(s):  
T. Onishi ◽  
A. Shudo ◽  
K. S. Ikeda ◽  
K. Takahashi
Keyword(s):  
Phase Space ◽  
Complex Phase ◽  
Tunneling Mechanism

scholarly journals Coherent representation of fields and deformation quantization

2020 ◽  
Vol 17 (11) ◽  
pp. 2050166 ◽  
Author(s):  
Jasel Berra-Montiel ◽  
Alberto Molgado
Keyword(s):  
Phase Space ◽  
Deformation Quantization ◽  
Probability Distributions ◽  
Series Representation ◽  
Quantum Field ◽  
Husimi Distribution ◽  
Normal Star ◽  
Holomorphic Representation ◽  
Functional Distribution ◽  
Space Description

Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows to establish a [Formula: see text]-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution, which may be convenient in order to calculate probability distributions of quantum field observables without performing formal phase space integrals at all.

On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

scholarly journals A probablistic proof of the series representation of the MacDonald function with applications

1998 ◽  
Vol 21 (3) ◽  
pp. 463-466
Author(s):  
M. Aslam Chaudhry ◽  
Munir Ahmad
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Generating Function ◽  
Open Problem ◽  
Series Representation ◽  
Moment Generating Function ◽  
Macdonald Function

A series representation of the Macdonald function is obtained using the properties of a probability density function and its moment generating function. Some applications of the result are discussed and an open problem is posed.

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