Methods of Weyl representation of the phase space and canonical transformations. I

The Weyl representation of an operator Â is a function A(x) in phase space. It is shown that a product Â 1 ... Â 2 n is represented by an integral over all (2 n +1)-sided polygons where the midpoint of one side is centred on x and the other midpoints take on the values A 1 ( x 1 ), ..., A 2 n ( x 2 n ). This leads to a new path integral for Û t = exp( - i ħ -1 Ĥt ) in the Weyl representation : U(x) is an integral over all the paths whose endpoints form a chord with x as its midpoint. No restriction is imposed on the form of the hamiltonian. Equivalence with previous path integrals generalizes these by substituting the Weyl hamiltonian for the classical hamiltonian when the latter does not have the simple form p 2 /2 + V(q) .

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CANONICAL TRANSFORMATIONS IN THREE-DIMENSIONAL PHASE-SPACE

International Journal of Modern Physics A ◽

10.1142/s0217751x09044760 ◽

2009 ◽

Vol 24 (25n26) ◽

pp. 4769-4788 ◽

Cited By ~ 1

Author(s):

TEKİN DERELİ ◽

ADNAN TEĞMEN ◽

TUĞRUL HAKİOĞLU

Keyword(s):

Phase Space ◽

Canonical Transformation ◽

Generating Functions ◽

General Framework ◽

Three Dimensional ◽

Two Dimensional ◽

Canonical Transformations ◽

Dimensional Phase Space ◽

Nambu Bracket ◽

Definition Of

Canonical transformation in a three-dimensional phase-space endowed with Nambu bracket is discussed in a general framework. Definition of the canonical transformations is constructed based on canonoid transformations. It is shown that generating functions, transformed Hamilton functions and the transformation itself for given generating functions can be determined by solving Pfaffian differential equations corresponding to that quantities. Types of the generating functions are introduced and all of them are listed. Infinitesimal canonical transformations are also discussed. Finally, we show that the decomposition of canonical transformations is also possible in three-dimensional phase space as in the usual two-dimensional one.

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QUANTUM-DEFORMED CANONICAL TRANSFORMATIONS, W∞ ALGEBRAS AND UNITARY TRANSFORMATIONS

International Journal of Modern Physics A ◽

10.1142/s0217751x94002405 ◽

1994 ◽

Vol 09 (32) ◽

pp. 5801-5820 ◽

Cited By ~ 7

Author(s):

E. GOZZI ◽

M. REUTER

Keyword(s):

Phase Space ◽

Pseudodifferential Operators ◽

Star Product ◽

Canonical Transformations ◽

Symbol Calculus ◽

Algebra Of Functions ◽

Unitary Transformations ◽

Symbols Of Operators ◽

Left And Right ◽

Gns Construction

We investigate the algebraic properties of the quantum counterpart of the classical canonical transformations using the symbol calculus approach to quantum mechanics. In this framework we construct a set of pseudodifferential operators which act on the symbols of operators, i.e. on functions defined over phase space. They act as operatorial left and right multiplication and form a W∞×W∞ algebra which contracts to its diagonal subalgebra in the classical limit. We also describe the Gel’fand-Naimark-Segal (GNS) construction in this language and show that the GNS representation space (a doubled Hilbert space) is closely related to the algebra of functions over phase space equipped with the star product of the symbol calculus.

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Canonical Transformations, Entropy and Quantization

Zeitschrift für Naturforschung A ◽

10.1515/zna-1987-0401 ◽

1987 ◽

Vol 42 (4) ◽

pp. 333-340 ◽

Cited By ~ 2

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.

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

International Journal of Modern Physics A ◽

10.1142/s0217751x09044620 ◽

2009 ◽

Vol 24 (24) ◽

pp. 4573-4587 ◽

Cited By ~ 2

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.

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