Z
3-Graded exterior differential calculus and gauge theories of higher order |
| |
Authors: | Richard Kerner |
| |
Institution: | (1) Laboratoire de Gravitation et Cosmologie Relativistes, Université Pierre-et-Marie-Curie, CNRS-D0 769, Tour 22, 4-ème étage, Boîte 142, 4, Place Jussieu, 75005 Paris, France |
| |
Abstract: | We present a possible generalization of the exterior differential calculus, based on the operator d such that d3=0, but d20. The entities dx
i
and d2
x
k
generate an associative algebra; we shall suppose that the products dx
i
dx
k
are independent of dx
k
dx
i
, while theternary products will satisfy the relation: dx
i
dx
k
dx
m
=jdx
k
dx
m
dx
i
=j
2dx
m
dx
m
dx
i
dx
k
, complemented by the relation dx
i
d2
x
k
=jd2
x
k
dx
i
, withj:=e2i/3.We shall attribute grade 1 to the differentials dx
i
and grade 2 to the second differentials d2
x
k
; under the associative multiplication law the grades add up modulo 3.We show how the notion ofcovariant derivation can be generalized with a 1-formA so thatD:=d+A, and we give the expression in local coordinates of thecurvature 3-form defined as :=d2
A+d(A
2)+AdA+A
3.Finally, the introduction of notions of a scalar product and integration of theZ
3-graded exterior forms enables us to define the variational principle and to derive the differential equations satisfied by the 3-form . The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensorF
ik
and its covariant derivativesD
i
F
km
. |
| |
Keywords: | 53-XX 15-XX 81-XX |
本文献已被 SpringerLink 等数据库收录! |
|