Quaternion wave equations in curved space-time

Relativistic dynamics with energy and momentum restricted to an anti-de Sitter space is presented. Coordinate operators conjugate to such momenta are introduced. Definition of functions of these operators, their differentiation and integration, all necessary for the development of dynamics is presented. The resulting algebra differs from the standard Heisenberg one, notably in that the space–time coordinates do not commute among each other. The resulting time variable is discrete and the limit to continuous time presents difficulties. A parallel approach, in which an overlap function, between position and momentum states, is obtained from solutions of wave equations on this curved space are also investigated. This approach, likewise, has problems in the that high energy behavior of these overlap functions precludes a space–time definition of action functionals.

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Quantum wave equations in curved space-time from wave mechanics

Symmetry and Perturbation Theory ◽

10.1142/9789812776174_0031 ◽

2007 ◽

Author(s):

Mayeul Arminjon

Keyword(s):

Wave Equations ◽

Space Time ◽

Wave Mechanics ◽

Curved Space ◽

Quantum Wave

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Path integrals and the solution of the Schwinger model in curved space-time

Physical Review D ◽

10.1103/physrevd.33.2262 ◽

1986 ◽

Vol 33 (8) ◽

pp. 2262-2266 ◽

Cited By ~ 12

Author(s):

J. Barcelos-Neto ◽

Ashok Das

Keyword(s):

Path Integrals ◽

Space Time ◽

Curved Space ◽

Schwinger Model

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Asymptotic behavior of the effective potential of composite fields in a curved space-time

Soviet Physics Journal ◽

10.1007/bf00896543 ◽

1988 ◽

Vol 31 (2) ◽

pp. 163-167

Author(s):

I. L. Bukhbinder ◽

S. D. Odintsov

Keyword(s):

Asymptotic Behavior ◽

Effective Potential ◽

Space Time ◽

Curved Space

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Space-Time Coupling: Current Concept and Two Examples from Ultrafast Optics Studied Using Exact Solution of EM Equations

Symmetry ◽

10.3390/sym13040529 ◽

2021 ◽

Vol 13 (4) ◽

pp. 529

Author(s):

Nikolay L. Popov ◽

Alexander V. Vinogradov

Keyword(s):

Wave Equations ◽

Coherent Control ◽

Ultrafast Optics ◽

Spectral Width ◽

Space Time ◽

Current Approach ◽

Unipolar Pulse ◽

Number Of Cycles ◽

Control Of Quantum Systems ◽

Coupling Current

Current approach to space-time coupling (STC) phenomena is given together with a complementary version of the STC concept that emphasizes the finiteness of the energy of the considered pulses. Manifestations of STC are discussed in the framework of the simplest exact localized solution of Maxwell’s equations, exhibiting a “collapsing shell”. It falls onto the center, continuously deforming, and then, having reached maximum compression, expands back without losing energy. Analytical solutions describing this process enable to fully characterize the field in space-time. It allowed to express energy density in the center of collapse in the terms of total pulse energy, frequency and spectral width in the far zone. The change of the pulse shape while travelling from one point to another is important for coherent control of quantum systems. We considered the excitation of a two-level system located in the center of the collapsing EM (electromagnetic) pulse. The result is again expressed through the parameters of the incident pulse. This study showed that as it propagates, a unipolar pulse can turn into a bipolar one, and in the case of measuring the excitation efficiency, we can judge which of these two pulses we are dealing with. The obtained results have no limitation on the number of cycles in a pulse. Our work confirms the productivity of using exact solutions of EM wave equations for describing the phenomena associated with STC effects. This is facilitated by rapid progress in the search for new types of such solutions.

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