Quantization Methods: A Guide for Physicists and Analysts

Ali, S. Twareque; Engliš, Miroslav

doi:10.1142/s0129055x05002376

Cited by 213 publications

(226 citation statements)

References 192 publications

(223 reference statements)

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“…[12], [1]), it is known that for f, g, say, smooth with compact support, one has the asymptotic expansion…”

Section: Toeplitz Operators and Semiclassical Limitsmentioning

confidence: 99%

Noncommutative coherent states and related aspects of Berezin–Toeplitz quantization

Chowdhury¹,

Ali²,

Engliš³

2017

J. Phys. A: Math. Theor.

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Dedicated by the first and the third authors to the memory of the second author, with gratitude for his friendship and for all they learnt from him AbstractIn this paper, we construct noncommutative coherent states using various families of unitary irreducible representations (UIRs) of G nc , a connected, simply connected nilpotent Lie group, that was identified as the kinematical symmetry group of noncommutative quantum mechanics for a system of 2-degrees of freedom in an earlier paper. Likewise described are the degenerate noncommutative coherent states arising from the degenerate UIRs of G nc . We then compute the reproducing kernels associated with both these families of coherent states and study Berezin-Toeplitz quantization of the observables on the underlying 4-dimensional phase space, analyzing in particular the semi-classical asymptotics for both these cases. I IntroductionNoncommutative quantum mechanics (NCQM) is an active field of research these days. The starting point here is to alter the canonical commutation relations (CCR) among the * shhchowdhury@gmail.com † twareque.ali@concordia.ca ‡ englis@math.cas.cz 1 respective positions and momenta coordinates and hence introduce a new noncommutative Lie algebra structure. Consult [13,8] for a detailed account on this approach. There is another approach available where one starts with noncommutative field theory (NCFT) studying quantum field theory on noncommutative space-time. An excellent pedagogical treatment to noncommutative quantum field theory can be found in [15]. Refer to the excellent review [9] to delve further into the studies of NCFT. A noncommutative quantum field theory is the one where the fields are functions of space-time coordinates with spatial coordinates failing to commute with each other. Among others, Snyder and Yang were the leading proponents to introduce the concept of noncommutative structure of space-time (see [14,16]). Introduction of such assumption of spatial noncommutativity eliminates ultraviolet (UV) divergences of quantum field theory and runs parallel to the technique of renormalization as a cure to such annoying divergence issues in quantum field theory. NCQM can then be seen as nonrelativistic approximation of NCFT (see [10, 3]).In yet another approach (see [4], [5]), the authors start from a certain nilpotent Lie group G nc as the defining group of NCQM for a system of 2 degrees of freedom and compute its unitary dual using the method of orbits introduced by Kirillov (see [11]). Although G nc does not contain the Weyl-Heisenberg group G WH as its subgroup, the unitary dual of G WH is found to be sitting inside that of G nc . Various gauges arising in NCQM are also found to be related to a certain family of unitary irreducible representations (UIRs) of G nc . The Lie group G nc was later identified as the kinematical symmetry group of NCQM in [6] where various Wigner functions for such model of NCQM were computed that were found to be supported on relevant families of coadjoint orbits associated with G nc . The purp...

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“…[12], [1]), it is known that for f, g, say, smooth with compact support, one has the asymptotic expansion…”

Section: Toeplitz Operators and Semiclassical Limitsmentioning

confidence: 99%

Noncommutative coherent states and related aspects of Berezin–Toeplitz quantization

Chowdhury¹,

Ali²,

Engliš³

2017

J. Phys. A: Math. Theor.

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“…We refer to [22,23] for many details on this topic. What we want to do here is to show that bicoherent states can also be used for this purpose.…”

Section: Quantization Via Bicoherent Statesmentioning

confidence: 99%

Intertwining operators for non-self-adjoint Hamiltonians and bicoherent states

Багарелло

2016

Journal of Mathematical Physics

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This paper is devoted to the construction of what we will call exactly solvable models, i.e. of quantum mechanical systems described by an Hamiltonian H whose eigenvalues and eigenvectors can be explicitly constructed out of some minimal ingredients. In particular, motivated by PT-quantum mechanics, we will not insist on any self-adjointness feature of the Hamiltonians considered in our construction. We also introduce the so-called bicoherent states, we analyze some of their properties and we show how they can be used for quantizing a system. Some examples, both in finite and in infinite-dimensional Hilbert spaces, are discussed.1 This paper is dedicated to the memory of Syed Twareque Ali, much more than just a colleague!

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“…This is the main result of this letter. The proof is easy, here it is: matrix elements of operators associated to A * p , A q and A * p A q are calculated using (2). It follows that…”

Section: Ordering Algebramentioning

confidence: 99%

“…Now, when two classical observables are quantized an additional ordering rule is needed in order to quantize their product. As a matter of fact the usual methods of quantization, including the most advanced ones like, for instance, geometric quantization [2], allow to quantize only a restricted set of classical observables and do not provide any ordering rule. For instance, the generators of some Lie algebra may be quantized, but even the simplest functions of them like polynomials must be considered separately.…”

Section: Introductionmentioning

confidence: 99%