IJPAM: Volume 78, No. 6 (2012)

ON LEVI-CIVITA'S ALTERNATING SYMBOL,
SCHOUTEN'S ALTERNATING UNIT TENSORS,
CPT, AND QUANTIZATION

Evert Jan Post$^1$, Stan Sholar$^2$, Hooman Rahimizadeh$^3$, Michael Berg$^4$
$^1$The University of Houston
4800, Calhoun Road, Houston, Texas, 77004, USA
$^2$The Boeing Company
$^3$Department of Mathematics
Loyola Marymount University
1 LMU Drive, Los Angeles, CA 90045, USA


Abstract. The purpose of the present article is to demonstrate that by adopting a unifying differential geometric perspective on certain themes in physics one reaps remarkable new dividends in both microscopic and macroscopic domains. By replacing algebraic objects by tensor-transforming objects and introducing methods from the theory of differentiable manifolds at a very fundamental level we obtain a Kottler-Cartan metric-independent general invariance of the Maxwell field, which in turn makes for a global quantum superstructure for Gauss-Ampère and Aharonov-Bohm ``quantum integrals.'' Beyond this, our approach shows that postulating a Riemannian metric at the quantum level is an unnecessary concept and our differential geometric, or more accurately topological yoga can substitute successfully for statistical mechanics.

Received: January 18, 2012

AMS Subject Classification: 58-XX, 53-XX, 82-XX,

Key Words and Phrases: differential geometric, Maxwell field, Kottler-Cartan metric-independent general invariance, differentiable manifolds, Gauss-Ampère and Aharonov-Bohm ``quantum integrals''

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 78
Issue: 6