Monogenic functions and representations of nilpotent Lie groups in quantum mechanics

Cnops, Jan; Kisil, Vladimir V.

doi:10.1002/(sici)1099-1476(19990310)22:4<353::aid-mma44>3.0.co;2-#

Cited by 21 publications

(12 citation statements)

References 19 publications

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“…polynomials expansions and their applications are well-known in harmonic analysis, Fourier analysis and wavelet analysis (cf [6,11,12,31]) they have not received that much attention in quantum mechanics after the approach of Cnops and Kisil [10] in which the representation of nilpotent Lie groups like the Heisenberg group H n arises in the hypercomplex formulation. Let us get some motivation from the representation of nilpotent Lie groups (see [29], chapter 10).…”

Section: Motivation and Main Resultsmentioning

confidence: 99%

Fock spaces, Landau operators and the time-harmonic Maxwell equations

Constales¹,

Faustino²,

Kraußhar³

2011

J. Phys. A: Math. Theor.

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We investigate the representations of the solutions to Maxwell's equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the solutions to the time-harmonic Maxwell system in terms of series expansions involving spherical harmonics resp. spherical monogenics. Also, a thorough investigation for the series representation of the solutions in terms of eigenfunctions of Landau operators that encode n-dimensional spinless electrons is given. This new insight should lead to important investigations in the study of regularity and hypo-ellipticity of the solutions to Schrödinger equations with natural applications in relativistic quantum mechanics concerning massive spinor fields.

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Section: Motivation and Main Resultsmentioning

confidence: 99%

Fock spaces, Landau operators and the time-harmonic Maxwell equations

Constales¹,

Faustino²,

Kraußhar³

2011

J. Phys. A: Math. Theor.

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“…In a similar way the De Donder-Weyl formalism for classical field theories [17] requires Clifford numbers [14] for their quantisation. It was recently realised [9] that the appearance of Clifford algebras is induced by the Galilean group-a nilpotent step two Lie group with multidimensional centre. In the one-dimensional case an element of the Galilean group is (s 1 , .…”

Section: Discussionmentioning

confidence: 99%

“…e n are imaginary units spanning the Clifford algebra. The associated Fock spaces were described in [9]. In [28] we quantise the De Donder-Weyl field equations (similarly to our consideration in Subsection 4.1) with the help of composed antiderivative operator A = n 1 e i A i , where S i A i = 4π 2 I.…”

Section: Discussionmentioning

confidence: 99%

p-mechanics as a physical theory: an introduction

Kisil¹

2003

J. Phys. A: Math. Gen.

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The paper provides an introduction into p-mechanics, which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics. p-Mechanics naturally provides a common ground for several different approaches to quantisation (geometric, Weyl, coherent states, Berezin, deformation, Moyal, etc.) and has a potential for expansions into field and string theories. The backbone of p-mechanics is solely the representation theory of the Heisenberg group.

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“…We present an abstract scheme, which also could be applied to other analytic function theories [10,27]. We start from a dry construction followed in the next Section by classic examples, which will justify our usage of personal names.…”

Section: Wavelet Transform and Coherent Statesmentioning

confidence: 99%

“…One could apply it to analytic function theory of several variables. This gives a variety of different function theories both known (see below) and new [10,27]. For example, the biholomorphic automorphisms [38, Chap.…”

Section: Example 63mentioning

confidence: 99%