scholarly journals A special irreducible matrix representation of the real Clifford algebra C(3,1)

1999 ◽  
Vol 40 (7) ◽  
pp. 3616-3631 ◽  
Author(s):  
K. Scharnhorst
Keyword(s):  
Clifford Algebra ◽  
Matrix Representation ◽  
Irreducible Matrix ◽  
The Real ◽  
Real Clifford Algebra

scholarly journals Particle-antiparticle duality from an extra timelike dimension

2019 ◽  
Vol 79 (9) ◽  
Author(s):  
Marcos R. A. Arcodía ◽  
Mauricio Bellini
Keyword(s):  
Clifford Algebra ◽  
Extra Dimension ◽  
Complex Structure ◽  
Minkowski Spacetime ◽  
Gauge Condition ◽  
Extra Time ◽  
The Real ◽  
Dirac Particles ◽  
Real Clifford Algebra

Abstract It is a well known fact that the usual complex structure on the real Clifford Algebra (CA) of Minkowski spacetime can be obtained by adding an extra time-like dimension, instead of the usual complexification of the algebra. In this article we explore the consequences of this approach and reinterpret known results in this new context. We observe that Dirac particles and antiparticles at rest can be interpreted as eigenstates of the generator of rotations in the plane formed by the two time-like coordinates and find an effective finite scale for the extra dimension when no EM fields are present (without postulating compactness). In the case of non-vanishing EM fields, we find a gauge condition to preserve such a scale.

O(d, d)-Symmetry and Ernst Formulation of Einstein–Kalb–Ramond Theory

1997 ◽  
Vol 12 (21) ◽  
pp. 1573-1582 ◽  
Author(s):  
Alfredo Herrera-Aguilar ◽  
Oleg Kechkin
Keyword(s):  
Matrix Representation ◽  
Two Dimensions ◽  
Target Space ◽  
Subsequent Reduction ◽  
R Matrix ◽  
The Real ◽  
Ernst Potential

A Ernst-like matrix representation of (3+d)-dimensional Einstein–Kalb–Ramond theory is developed. The analogy with the Einstein and Einstein–Maxwell–Dilaton–Axion theories is discussed. The subsequent reduction to two dimensions is considered. It is shown that, in this case, the theory allows two different Ernst-like d×d-matrix formulations: the real nondualized target space, and the Hermitian dualized nontarget space one. The O(d, d)-symmetry is written in an SL (2,R) matrix-valued form in both cases. The Kramer–Neugebauer transformation, which algebraically maps the nondualized Ernst potential onto the dualized one, is presented.

scholarly journals ON THE REAL MATRIX REPRESENTATION OF PT-SYMMETRIC OPERATORS

2014 ◽  
Vol 54 (2) ◽  
pp. 113-115 ◽  
Author(s):  
Francisco M. Fernández
Keyword(s):  
Matrix Representation ◽  
Real Matrix ◽  
Matrix Representations ◽  
The Real ◽  
Symmetric Operators

We discuss the construction of real matrix representations of PT-symmetric operators. We show the limitation of a general recipe presented some time ago for non-Hermitian Hamiltonians with antiunitary symmetry and propose a way to overcome it. Our results agree with earlier ones for a particular case.

XVII.—Invariant Matrices and the Geometry of Numbers

1957 ◽  
Vol 64 (3) ◽  
pp. 223-238
Author(s):  
K. Mahler
Keyword(s):  
Recent Work ◽  
Linear Group ◽  
Affine Space ◽  
Matrix Representation ◽  
Convex Bodies ◽  
Linear Mapping ◽  
Geometry Of Numbers ◽  
Arithmetical Property ◽  
The Real ◽  
Invariant Matrices

SynopsisWith every matrix representation of the (real) full linear group can be associated a multi-linear mapping of one affine space, Rn, into another, RN. This mapping is studied from the viewpoint of the geometry of numbers of convex bodies, and a general arithmetical property of such mappings is proved. The result generalizes my recent work on compound convex bodies.

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