scholarly journals Geometrical Applications of Split Octonions

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Merab Gogberashvili ◽  
Otari Sakhelashvili
Keyword(s):  
Quantum System ◽  
Lie Group ◽  
Free Particle ◽  
Explicit Representation ◽  
Time Intervals ◽  
Extra Time ◽  
Heisenberg Uncertainty ◽  
Left Right Asymmetry ◽  
Zero Norm ◽  
Split Octonions

It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3 + 1)-theory (e.g., number of dimensions, existence of maximal velocities, Heisenberg uncertainty, and particle generations). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie groupG2. This group generates specific rotations of (3 + 4)-vector parts of split octonions with three extra time-like coordinates and in infinitesimal limit imitates standard Poincare transformations. In this picture translations are represented by noncompact Lorentz-type rotations towards the extra time-like coordinates. It is shown how theG2algebra’s chirality yields an intrinsic left-right asymmetry of a certain 3-vector (spin), as well as a parity violating effect on light emitted by a moving quantum system. Elementary particles are connected with the special elements of the algebra which nullify octonionic intervals. Then the zero-norm conditions lead to free particle Lagrangians, which allow virtual trajectories also and exhibit the appearance of spatial horizons governing by mass parameters.

scholarly journals On deformations of classical mechanics due to Planck-scale physics

2020 ◽  
Vol 29 (10) ◽  
pp. 2050070
Author(s):  
Olga I. Chashchina ◽  
Abhijit Sen ◽  
Zurab K. Silagadze
Keyword(s):  
Quantum System ◽  
Evolution Equations ◽  
Hidden Variables ◽  
Classical Mechanics ◽  
Planck Scale ◽  
Leibniz Rule ◽  
Uncertainty Relations ◽  
Interesting Possibility ◽  
Heisenberg Uncertainty ◽  
Space Formulation

Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

The Duflo non-commutative Fourier transform for the Lorentz group

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2040011
Author(s):  
Giacomo Rosati
Keyword(s):  
Fourier Transform ◽  
Phase Space ◽  
Quantum System ◽  
Lie Group ◽  
Lorentz Group ◽  
Commutative Algebra ◽  
Cotangent Bundle ◽  
Irreducible Representations ◽  
Algebra Representation

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

Symmetries of the Quantum State Space and Group Representations

1998 ◽  
Vol 10 (07) ◽  
pp. 893-924 ◽  
Author(s):  
Gianni Cassinelli ◽  
Ernesto de Vito ◽  
Pekka Lahti ◽  
Alberto Levrero
Keyword(s):  
State Space ◽  
Quantum System ◽  
Symmetry Group ◽  
Quantum State ◽  
Lie Group ◽  
Universal Covering ◽  
Unitary Representations ◽  
Group Representations ◽  
Simply Connected ◽  
Connected Lie Group

The homomorphisms of a connected Lie group G into the symmetry group of a quantum system are classified in terms of unitary representations of a simply connected Lie group associated with G. Moreover, an explicit description of the T-multipliers of G is obtained in terms of the ℝ-multipliers of the universal covering G* of G and the characters of G*. As an application, the Poincaré group and the Galilei group, both in 3+1 and 2+1 dimensions, are considered.

scholarly journals THE q-EUCLIDEAN ALGEBRA Uq(eN) AND THE CORRESPONDING q-EUCLIDEAN LATTICE

1996 ◽  
Vol 11 (05) ◽  
pp. 863-885 ◽  
Author(s):  
GAETANO FIORE
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Hopf Algebra ◽  
Quantum System ◽  
Classical Limit ◽  
Free Particle ◽  
Simple Lattice ◽  
Euclidean Lattice ◽  
Euclidean Algebra ◽  
Suitable Notion

We present the Euclidean Hopf algebra Uq(eN) dual of [Formula: see text] and describe its fundamental Hilbert space representations,6 which turn out to be rather simple “lattice-regularized” versions of the classical ones, in the sense that the spectra of squared momentum components are discrete and the corresponding eigenfunctions normalizable. These representations can be regarded as describing a quantum system consisting of one free particle on the quantum Euclidean space. A suitable notion of the classical limit is introduced, so that we recover the classical continuous spectra and generalized (nonnormalizable) eigenfunctions in that limit.

Quantum equations from Brownian motion

2011 ◽  
Vol 89 (2) ◽  
pp. 185-191 ◽  
Author(s):  
B. S. Rajput
Keyword(s):  
Uncertainty Principle ◽  
Free Particle ◽  
Continuous Limit ◽  
Order Effects ◽  
Heisenberg Uncertainty Principle ◽  
Classical Counterpart ◽  
Quantum Equation ◽  
Heisenberg Uncertainty ◽  
Lattice Random Walks ◽  
Quantum Equations

The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.

scholarly journals Octonionic geometry and conformal transformations

2016 ◽  
Vol 13 (07) ◽  
pp. 1650092 ◽  
Author(s):  
Merab Gogberashvili
Keyword(s):  
Cosmological Constant ◽  
Minkowski Space ◽  
Lie Group ◽  
Space Time ◽  
Conformal Transformations ◽  
Group Of Automorphisms ◽  
Minkowski Space Time ◽  
Split Octonions

We describe space-time using split octonions over the reals and use their group of automorphisms, the noncompact form of Cartan’s exceptional Lie group G2, as the main geometrical group of the model. Connections of the G2-rotations of octonionic 8D space with the conformal transformations in 4D Minkowski space-time are studied. It is shown that the dimensional constant needed in these analysis naturally gives the observed value of the cosmological constant.

STOCHASTIC DIFFERENTIAL EQUATIONS FOR THE CONTINUOUS SPONTANEOUS LOCALIZATION MODEL

2000 ◽  
Vol 15 (30) ◽  
pp. 1833-1842
Author(s):  
L. F. SANTOS ◽  
C. O. ESCOBAR
Keyword(s):  
Wave Function ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Quantum System ◽  
Open Quantum System ◽  
Random Potential ◽  
Free Particle ◽  
Localization Model ◽  
Spontaneous Localization ◽  
Continuous Spontaneous Localization

We extend Vink's method [J. C. Vink, Phys. Rev.A48, 1808 (1993)], developed for an isolated quantum system, to an open quantum system consisting of a free particle interacting with its surrounding through a random potential, which causes the spontaneous localization of its wave function. We then obtain the stochastic differential equations (SDE) underlying its evolution. These SDE help us to observe the effects of the environment upon the movement of the particle.

scholarly journals Linear entropy and squeezing of the interaction between two quantum system described by su (1, 1) and su(2) Lie group in presence of two external terms

AIP Advances ◽  
2017 ◽  
Vol 7 (1) ◽  
pp. 015013 ◽  
Author(s):  
M. Sebawe Abdalla ◽  
E. M. Khalil ◽  
A. S.-F. Obada ◽  
J. Peřina ◽  
J. Křepelka
Keyword(s):  
Quantum System ◽  
Lie Group ◽  
Linear Entropy
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