scholarly journals Isomorphic exponential Weyl algebras

1991 ◽  
Vol 33 (1) ◽  
pp. 7-10 ◽  
Author(s):  
P. L. Robinson
Keyword(s):  
Vector Space ◽  
Weyl Algebra ◽  
Associative Algebras ◽  
Exponential Form ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Weyl Algebras ◽  
The Subject ◽  
Symplectic Vector Space

Canonically associated to a real symplectic vector space are several associative algebras. The Weyl algebra (generated by the Heisenberg commutation relations) has been the subject of much study; see [1] for example. The exponential Weyl algebra (generated by the canonical commutation relations in exponential form) has been less well studied; see [8].

scholarly journals The structure of exponential Weyl algebras

1993 ◽  
Vol 55 (3) ◽  
pp. 302-310 ◽  
Author(s):  
P. L. Robinson
Keyword(s):  
Structural Properties ◽  
Associative Algebra ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Exponential Form ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Weyl Algebras ◽  
Zero Divisors ◽  
Central Simple

abstractWe present structural properties of the complex associative algebra generated by the canonical commutation relations in exponential form. In particular, we show it to be a central simple algebra that lacks zero divisors and is not Noetherian on either side; in addition, we determine explicitly its units and its automorphisms.

Some associative algebras related to $U(\mathfrak g)$ and twisted generalized Weyl algebras

2003 ◽  
Vol 92 (1) ◽  
pp. 5 ◽  
Author(s):  
V. Mazorchuk ◽  
M. Ponomarenko ◽  
L. Turowska
Keyword(s):  
Weyl Algebra ◽  
Associative Algebras ◽  
Graded Modules ◽  
Weyl Algebras ◽  
Generalized Weyl Algebra ◽  
Generalized Weyl Algebras ◽  
Weight Modules

We prove that both Mickelsson step algebras and Orthogonal Gelfand-Zetlin algebras are twisted generalized Weyl algebras. Using an analogue of the Shapovalov form we construct all weight simple graded modules and some classes of simple weight modules over a twisted generalized Weyl algebra, improving the results from [6], where a particular class of algebras was considered and only special modules were classified.

CLASSICAL LIE SUPERALGEBRAS OVER SIMPLE ASSOCIATIVE ALGEBRAS

2006 ◽  
Vol 92 (3) ◽  
pp. 581-600 ◽  
Author(s):  
GEORGIA BENKART ◽  
XIAOPING XU ◽  
KAIMING ZHAO
Keyword(s):  
Lie Algebras ◽  
Associative Algebra ◽  
Weyl Algebra ◽  
Identity Element ◽  
Lie Superalgebras ◽  
Associative Algebras ◽  
Matrix Algebras ◽  
Weyl Algebras ◽  
Conformal Algebras ◽  
The Matrix

Over arbitrary fields of characteristic not equal to 2, we construct three families of simple Lie algebras and six families of simple Lie superalgebras of matrices with entries chosen from different one-sided ideals of a simple associative algebra. These families correspond to the classical Lie algebras and superalgebras. Our constructions intermix the structure of the associative algebra and the structure of the matrix algebra in an essential, compatible way. Many examples of simple associative algebras without an identity element arise as a by-product. The study of conformal algebras and superalgebras often involves matrix algebras over associative algebras such as Weyl algebras, and for that reason, we illustrate our constructions by taking various one-sided ideals from a Weyl algebra or a quantum torus.

scholarly journals Singular Bogoliubov transformations and inequivalent representations of canonical commutation relations

2019 ◽  
Vol 31 (08) ◽  
pp. 1950026 ◽  
Author(s):  
Asao Arai
Keyword(s):  
Fock Space ◽  
Sufficient Conditions ◽  
Bogoliubov Transformation ◽  
Fock Representation ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Direct Sum ◽  
Boson Fock Space ◽  
Inequivalent Representations ◽  
Bogoliubov Transformations

We introduce a concept of singular Bogoliubov transformation on the abstract boson Fock space and construct a representation of canonical commutation relations (CCRs) which is inequivalent to any direct sum of the Fock representation. Sufficient conditions for the representation to be irreducible are formulated. Moreover, an example of such representations of CCRs is given.

scholarly journals Noisy effects in interferometric quantum gravity tests

2017 ◽  
Vol 15 (08) ◽  
pp. 1740014 ◽  
Author(s):  
F. Benatti ◽  
R. Floreanini ◽  
S. Olivares ◽  
E. Sindici
Keyword(s):  
Quantum Gravity ◽  
Weak Coupling ◽  
Scale Effects ◽  
External Environment ◽  
Planck Scale ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Photon Propagation ◽  
Gravity Theories ◽  
Planck Scale Effects

Quantum-enhanced metrology is boosting interferometer sensitivities to extraordinary levels, up to the point where table-top experiments have been proposed to measure Planck-scale effects predicted by quantum gravity theories. In setups involving multiple photon interferometers, as those for measuring the so-called holographic fluctuations, entanglement provides substantial improvements in sensitivity. Entanglement is however a fragile resource and may be endangered by decoherence phenomena. We analyze how noisy effects arising either from the weak coupling to an external environment or from the modification of the canonical commutation relations in photon propagation may affect this entanglement-enhanced gain in sensitivity.

scholarly journals Deformation of the Wheeler–DeWitt equation

2014 ◽  
Vol 29 (20) ◽  
pp. 1450106 ◽  
Author(s):  
Mir Faizal
Keyword(s):  
Commutation Relation ◽  
Canonical Commutation Relation ◽  
Big Bang ◽  
Minimum Length ◽  
Second Quantization ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
The Big Bang ◽  
Big Bang Singularity ◽  
Modified Theory

In this paper, we will analyze the consequences of deforming the canonical commutation relations consistent with the existence of a minimum length and a maximum momentum. We first generalize the deformation of first quantized canonical commutation relation to second quantized canonical commutation relation. Thus, we arrive at a modified version of second quantization. A modified Wheeler–DeWitt equation will be constructed by using this deformed second quantized canonical commutation relation. Finally, we demonstrate that in this modified theory the big bang singularity gets naturally avoided.

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