A theory of classical limit for quantum theories which are defined by real Lie algebras

1978 ◽  
Vol 19 (7) ◽  
pp. 1600-1606 ◽  
Author(s):  
Kai Drühl
Keyword(s):  
Lie Algebras ◽  
Classical Limit ◽  
Quantum Theories

scholarly journals COSET REALIZATION OF UNIFYING ${\mathcal W}$ ALGEBRAS

1995 ◽  
Vol 10 (16) ◽  
pp. 2367-2430 ◽  
Author(s):  
R. BLUMENHAGEN ◽  
W. EHOLZER ◽  
A. HONECKER ◽  
R. HÜBEL ◽  
K. HORNFECK
Keyword(s):  
Lie Algebras ◽  
Classical Limit ◽  
Classical Case ◽  
Quantum Level ◽  
Minimal Models ◽  
Classical Formula

We construct several quantum coset [Formula: see text] algebras, e.g. [Formula: see text] and [Formula: see text] and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying [Formula: see text] algebras of Casimir [Formula: see text] algebras. We show that it is possible to give coset realizations of various types of unifying [Formula: see text] algebras; for example, the diagonal cosets based on the symplectic Lie algebras sp (2n) realize the unifying [Formula: see text] algebras which have previously been introduced as [Formula: see text]. In addition, minimal models of [Formula: see text] are studied. The coset realizations provide a generalization of level-rank duality of dual coset pairs. As further examples of finitely nonfreely generated quantum [Formula: see text] algebras, we discuss orbifolding of [Formula: see text] algebras which on the quantum level has different properties than in the classical case. We demonstrate through some examples that the classical limit — according to Bowcock and Watts — of these finitely nonfreely generated quantum [Formula: see text] algebras probably yields infinitely nonfreely generated classical [Formula: see text] algebras.

NEW SOLUTIONS OF THE YANG-BAXTER EQUATION AND THEIR YANG-BAXTERIZATION

1991 ◽  
Vol 06 (04) ◽  
pp. 559-576 ◽  
Author(s):  
M. COUTURE ◽  
Y. CHENG ◽  
M.L. GE ◽  
K. XUE
Keyword(s):  
Lie Algebras ◽  
Braid Group ◽  
Classical Limit ◽  
Usual Sense ◽  
Baxter Equation ◽  
Group Representations ◽  
Simple Lie Algebras

Through the examples associated with simple Lie algebras B2, C2 and D2, an approach of constructing new solutions of the spectral-independent Yang-Baxter equation (i.e. the braid group representations) was shown. These new solutions possess some particular features, such as that they do not have the classical limit in the usual sense of Refs. 2, 5, etc., and give rise to the new solutions of the x-dependent Yang-Baxter equation by using the Yang-Baxterization prescription.

Classical limit for Lie algebras

1990 ◽  
Vol 199 (1) ◽  
pp. 187-224 ◽  
Author(s):  
Aurel Bulgac ◽  
Dimitri Kusnezov
Keyword(s):  
Lie Algebras ◽  
Classical Limit

EXPLICIT TRIGONOMETRIC YANG-BAXTERIZATION

1991 ◽  
Vol 06 (21) ◽  
pp. 3735-3779 ◽  
Author(s):  
MO-LIN GE ◽  
YONG-SHI WU ◽  
KANG XUE
Keyword(s):  
Quantum Group ◽  
Lie Algebras ◽  
Braid Group ◽  
Classical Limit ◽  
Group Representation ◽  
Direct Method ◽  
Explicit Solutions ◽  
Simple Lie Algebras ◽  
Higher Dimensional

We present an explicit prescription for trigonometric Yang-Baxterization. Given a braid group representation (BGR) in appropriate form, our prescription generates explicit solutions to the quantum Yang-Baxter equations (QYBE). We have proved the correctness of this prescription, together with a variation of it, for BGR’s having two or three unequal eigenvalues. All the known explicit results obtained by Jimbo and Bazhanov are reproduced. More YB solutions are obtained from “standard” BGR’s associated with fundamental and higher dimensional representations of simple Lie algebras. Our prescription also applies to the new “nonstandard” BGR’s, which are beyond the reach of present quantum group techniques but obtainable by our previous direct method. New explicit examples include the standard ones with 6 of SU(3), 10 of SU(5) and exotic ones with C2 and D2, etc. The classical limit of our YB solutions is also studied. It turns out that some of our exotic QYB solutions have an unusual classical limit. Our results suggest that the QYBE embodies a much richer structure than we thought.

scholarly journals DEGREE CONES AND MONOMIAL BASES OF LIE ALGEBRAS AND QUANTUM GROUPS

2017 ◽  
Vol 59 (3) ◽  
pp. 595-621 ◽  
Author(s):  
TEODOR BACKHAUS ◽  
XIN FANG ◽  
GHISLAIN FOURIER
Keyword(s):  
Lie Algebras ◽  
Classical Limit ◽  
Polynomial Algebra ◽  
Monomial Ideals ◽  
Lattice Points ◽  
Graded Algebra ◽  
Negative Part ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Skew Polynomial

AbstractWe provide ℕ-filtrations on the negative part Uq($\mathfrak{n}$−) of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}$−. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an ℕ-filtration on any finite-dimensional simple $\mathfrak{g}$-module. We prove for type An, Cn, B3, D4 and G2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$.

Coherent States and Classical Limit of Quantum Electrodynamics

2017 ◽  
Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor
Keyword(s):  
Electromagnetic Field ◽  
Quantum Optics ◽  
Coherent State ◽  
Quantum Electrodynamics ◽  
Classical Limit ◽  
Coherent States ◽  
Quantum Fluctuations ◽  
Squeezed States ◽  
Quantum Theories

The coherent state formalism is used in order to study the classical limit of quantum theories and, in particular, quantum electrodynamics. It is shown that we can construct states for which the quantum fluctuations are negligible. Interesting cases include the electromagnetic field in a cavity, or ‘squeezed’ states used in quantum optics.

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