Stochastic Calculus and Processes in Non-Commutative Space-Time

Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there are infinitely many DSR constructions of the energy–momentum sector, each of whose can be promoted to the κ-Poincaré quantum (Hopf) algebra. Then we use the co-product of this algebra and the Heisenberg double construction of κ-deformed phase space in order to derive the non-commutative space–time structure and the description of the whole of DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space–time of the DSR theory is unique and related to the theory with non-commutative space–time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space–time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space–time, its intrinsic length parameter ℓ becomes observer-independent.

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On Characterization of Non-commutative Minkowski Space Time

Material Science Research India ◽

10.13005/msri/090117 ◽

2012 ◽

Vol 9 (1) ◽

pp. 123-127

Author(s):

Dharmendra Kumar ◽

Sunil KumarYadav

Keyword(s):

Geodesic Equation ◽

Space Time ◽

Gravity Models ◽

Commutative Case ◽

Force Equation ◽

Minkowski Space Time ◽

Tetrad Formulation ◽

Commutative Space ◽

Suitable Technique

The present study aims to derive modified geodesic equation in non-commutative space time. Snyder developed a model for non-commutative space time which provides a suitable technique of quantum structure of the space. We extend Tetrad formulation of general relativity to non-commutative case for complex gravity models. We derive geodesic equation on the k-space time in Non-commutative space, which is a generalization of Feynman’s approach. It has been shown that the homogeneous Maxwell’s equations may be derived by starting with the Newton’s force equation and generalized to relativistic. We show that the geodesic equation in the commutative space time is a suitable for generalization to κ -space time in κ -deformed space time. It shown that the κ-dependent correction to geodesic equation is cubic in velocities.

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Quantisation of Lorentz invariant scalar field theory in non-commutative space-time and its consequence

Nuclear Physics B ◽

10.1016/j.nuclphysb.2021.115633 ◽

2022 ◽

Vol 974 ◽

pp. 115633

Author(s):

E. Harikumar ◽

Vishnu Rajagopal

Keyword(s):

Field Theory ◽

Scalar Field ◽

Space Time ◽

Scalar Field Theory ◽

Invariant Scalar ◽

Commutative Space

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Ergodic Theory and the Structure of Non-commutative Space-Time

Journal of Physical Mathematics ◽

10.4172/2090-0902.1000229 ◽

2017 ◽

Vol 08 (02) ◽

Author(s):

Wang CHT ◽

James Moffat J ◽

Oniga O

Keyword(s):

Ergodic Theory ◽

Space Time ◽

Commutative Space

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Reformulation of elliptic equations for heat transfer and diffusion in solids with space-time algebra

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887822500153 ◽

2021 ◽

Author(s):

Victor L. Mironov

Keyword(s):

Heat Transfer ◽

Elliptic Equations ◽

Space Time ◽

Second Order ◽

Diffusion In Solids ◽

Heat Flows ◽

Vector Potentials ◽

Commutative Space ◽

The One ◽

And Diffusion

In this paper, we demonstrate the application of non-commutative space-time algebra of sedeons to generalize the system of equations describing heat transfer and impurity diffusion in solids at finite velocity. It is shown that by analogy with electrodynamics, these transfer processes can be described using a compact second-order sedeonic equation for generalized scalar and vector potentials. On the one hand, this equation is reduced to the system of first-order differential equations for vortex-less mass and heat flows, and on the other hand, it can be transformed to the second-order elliptical equations for the profiles of temperature and impurity concentration. The comparison of peculiarities in transfer within the frames of parabolic and elliptic equations is discussed.

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