scholarly journals A Note on the Representation of Clifford Algebra

2020 ◽  
Author(s):  
Ying-Qiu Gu
Keyword(s):  
Clifford Algebra ◽  
Practical Calculation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Normal Representation ◽  
Riemann Geometry ◽  
Minkowski Space Time ◽  
Pauli Matrices ◽  
Dimensional Minkowski Space ◽  
Real Clifford Algebra

In this note we construct explicit complex and real matrix representations for the generators of real Clifford algebra $C\ell_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. We find two classes of representation, the normal representation and exceptional one. The normal representation is a large class of representation which can only be expanded into $4m+1$ dimension, but the exceptional representation can be expanded as generators of the next period. In the cases $p+q=4m$, the representation is unique in equivalent sense. These results are helpful for both theoretical analysis and practical calculation. The generators of Clifford algebra are the faithful basis of $p+q$ dimensional Minkowski space-time or Riemann space, and Clifford algebra converts the complicated relations in geometry into simple and concise algebraic operations, so the Riemann geometry expressed in Clifford algebra will be much simple and clear.

A Note on the Representation of Clifford Algebras

2021 ◽  
Vol 62 ◽  
pp. 29-52
Author(s):  
Ying-Qiu Gu ◽  
Keyword(s):  
Clifford Algebras ◽  
Dimensional Space ◽  
Number System ◽  
Space Time ◽  
Coordinate Frame ◽  
Practical Calculation ◽  
3 Dimensional ◽  
Covariant Derivatives ◽  
Dimensional Minkowski Space ◽  
Dimensional Space Time

In this note we construct explicit complex and real faithful matrix representations of the Clifford algebras $\Cl_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. In the cases $p+q=4m$, the representation is unique in equivalent sense, and the $1+3$ dimensional space-time corresponds to the simplest and best case. Besides, the relation between the curvilinear coordinate frame and the local orthonormal basis in the curved space-time is discussed in detail, the covariant derivatives of the spinor and tensors are derived, and the connection of the orthogonal basis in tangent space is calculated. These results are helpful for both theoretical analysis and practical calculation. The basis matrices are the faithful representation of Clifford algebras in any $p+q$ dimensional Minkowski space-time or Riemann space, and the Clifford calculus converts the complicated relations in geometry and physics into simple and concise algebraic operations. Clifford numbers over any number field $\mathbb{F}$ expressed by this matrix basis form a well-defined $2^n$ dimensional hypercomplex number system. Therefore, we can expect that Clifford algebras will complete a large synthesis in science.

Interacting Fields

2021 ◽  
pp. 237-252
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Expansion ◽  
Field Theories ◽  
Green Functions ◽  
Quantum Field ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space ◽  
Feynman Rules ◽  
Wightman Axioms

We present a simple form of the Wightman axioms in a four-dimensional Minkowski space-time which are supposed to define a physically interesting interacting quantum field theory. Two important consequences follow from these axioms. The first is the invariance under CPT which implies, in particular, the equality of masses and lifetimes for particles and anti-particles. The second is the connection between spin and statistics. We give examples of interacting field theories and develop the perturbation expansion for Green functions. We derive the Feynman rules, both in configuration and in momentum space, for some simple interacting theories. The rules are unambiguous and allow, in principle, to compute any Green function at any order in perturbation.

scholarly journals Elastic shocks in relativistic rigid rods and balls

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180858 ◽  
Author(s):  
João L. Costa ◽  
José Natário
Keyword(s):  
Free Boundary ◽  
Minkowski Space ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Space Time ◽  
Spherically Symmetric ◽  
Soft Phase ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space

We study the free boundary problem for the ‘hard phase’ material introduced by Christodoulou in (Christodoulou 1995 Arch. Ration. Mech. Anal. 130 , 343–400), both for rods in (1 + 1)-dimensional Minkowski space–time and for spherically symmetric balls in (3 + 1)-dimensional Minkowski space–time. Unlike Christodoulou, we do not consider a ‘soft phase’, and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

scholarly journals Thermal Casimir effect for rectangular cavities inside (D+1)-dimensional Minkowski space–time revisited

2014 ◽  
Vol 29 (08) ◽  
pp. 1450043 ◽  
Author(s):  
Rui-Hui Lin ◽  
Xiang-Hua Zhai
Keyword(s):  
Free Energy ◽  
High Temperature ◽  
Boundary Conditions ◽  
Minkowski Space ◽  
Casimir Effect ◽  
Side Length ◽  
Temperature Dependent ◽  
Dependent Part ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space

We reconsider the thermal scalar Casimir effect for p-dimensional rectangular cavity inside (D+1)-dimensional Minkowski space–time and clarify the ambiguity in the regularization of the temperature-dependent part of the free energy. We derive rigorously the regularization of the temperature-dependent part of the free energy by making use of the Abel–Plana formula repeatedly and get the explicit expression of the terms to be subtracted. In the cases of D = 3, p = 1 and D = 3, p = 3, we precisely recover the results of parallel plates and three-dimensional box in the literature. Furthermore, for D>p and D = p cases with periodic, Dirichlet and Neumann boundary conditions, we give the explicit expressions of the Casimir free energy in both low temperature (small separations) and high temperature (large separations) regimes, through which the asymptotic behavior of the free energy changing with temperature and the side length is easy to see. We find that for D>p, with the side length going to infinity, the Casimir free energy tends to positive or negative constants or zero, depending on the boundary conditions. But for D = p, the leading term of the Casimir free energy for all three boundary conditions is a logarithmic function of the side length. We also discuss the thermal Casimir force changing with temperature and the side length in different cases and find that when the side length goes to infinity, the force always tends to be zero for different boundary conditions regardless of D>p or D = p. The Casimir free energy and force at high temperature limit behave asymptotically alike that they are proportional to the temperature, be they positive (repulsive) or negative (attractive) in different cases. Our study may be helpful in providing a comprehensive and complete understanding of this old problem.

ZETA-FUNCTION REGULARIZATION APPROACH TO FINITE TEMPERATURE EFFECTS IN KALUZA-KLEIN SPACE-TIMES

1992 ◽  
Vol 07 (29) ◽  
pp. 2669-2683 ◽  
Author(s):  
ANDREI A. BYTSENKO ◽  
LUCIANO VANZO ◽  
SERGIO ZERBINI
Keyword(s):  
Zeta Function ◽  
Finite Temperature ◽  
Space Time ◽  
Discrete Torsion ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space ◽  
High Temperature Expansion ◽  
Kernel Approach ◽  
The One ◽  
Kaluza Klein

In the framework of heat-kernel approach to zeta-function regularization, the one-loop effective potential at finite temperature for scalar and spinor fields on Kaluza-Klein space-time of the form [Formula: see text], where MP is p-dimensional Minkowski space-time is evaluated. In particular, when the compact manifold is [Formula: see text], the Selberg trace formula associated with discrete torsion-free group Γ of the n-dimensional Lobachevsky space Hn is used. An explicit representation for the thermodynamic potential valid for arbitrary temperature is found. As a result a complete high temperature expansion is presented and the roles of zero modes and topological contributions is discussed.

A STRING MODEL FOR D-DIMENSIONAL DE SITTER SPACE–TIME AND EQUATIONS OF MOTION

2005 ◽  
Vol 20 (26) ◽  
pp. 6065-6081
Author(s):  
PAUL BRACKEN
Keyword(s):  
Evolution Equations ◽  
Gordon Equation ◽  
Equations Of Motion ◽  
String Model ◽  
De Sitter Space ◽  
Space Time ◽  
System Of Equations ◽  
De Sitter ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space

De Sitter space–time is considered to be represented by a D-dimensional hyperboloid embedded in (D+1)-dimensional Minkowski space–time. The string equation is derived from a string action which contains a Lagrange multiplier to restrict coordinates to de Sitter space–time. The string system of equations is equivalent to a type of generalized sinh–Gordon equation. The evolution equations for all the variables including the coordinates and their derivatives are obtained for D=2,3 and 4.

scholarly journals Expression of a Real Matrix as a Difference of a Matrix and its Transpose Inverse

2019 ◽  
Vol 11 (1) ◽  
pp. 1-6
Author(s):  
Mil Mascaras ◽  
Jeffrey Uhlmann
Keyword(s):  
Control Theory ◽  
Computational Simulation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Articulated Figures ◽  
The Difference

In this paper we derive a representation of an arbitrary real matrix M as the difference of a real matrix A and the transpose of its inverse. This expression may prove useful for progressing beyond known results for which the appearance of transpose-inverse terms prove to be obstacles, particularly in control theory and related applications such as computational simulation and analysis of matrix representations of articulated figures.

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