Abstract
From a theoretical point of view, classical electrodynamics admits a series of conceptually consistent generalizations. We have already analyzed an important one in Chap. 18 dedicated to massive vector fields, corresponding – at the quantum level – to the possibility of replacing the intermediate vector boson of the interaction, i.e. the photon, with a massive particle. Other generalizations of physical interest are:
-
the Bianchi identity can be modified to take into account the presence of magnetic charges;
-
the charged particles can be replaced with extended charges occupying a p-dimensional volume, so-called p-branes;
-
Maxwell’s equations can be formulated in a space-time of arbitrary dimensions;
-
the electromagnetic field can be subject to a self-interaction, a famous example being represented by the Born-Infeld Lagrangian, see Ref. [1].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For a textbook of differential geometry paying particular attention to the language of differential forms in connection to its applications to physics, see, for instance, Ref. [3].
- 2.
For simplicity we assume that the reparameterization \(\lambda \rightarrow \lambda '(\lambda )\) preserves the orientation of the brane, in which case by definition we have \(\det K>0\). In fact, in general the measure would transform as \(d^n\lambda '= d^n\lambda /|\det K|\).
- 3.
- 4.
We recall that polar coordinates are singular at the North and South Poles, i.e. for \(\vartheta =0\) and \(\vartheta =\pi \), and consequently at these points the induced metric is singular, too, and in particular its determinant g vanishes.
- 5.
The dimensions of the spaces where these theories live are determined by internal consistency conditions. Actually, in the case of superstring theories there is a plethora of possible consistent theories, but their maximum space-time dimension is \(D=10\). The structure of M-theory, which should unify all superstring theories in a space-time with one spatial dimension more, i.e. \(D=11\), has not yet been established in a definitive way.
References
M. Born, L. Infeld, Foundations of the new field theory. Proc. R. Soc. Lond. 144, 425 (1934)
R. Bott, L.W. Tu, Differential Forms in Algebraic Topology (Springer, New York, 1982)
Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, Analysis, Manifolds and Physics (North-Holland, Amsterdam, 1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Lechner, K. (2018). Electrodynamics of p-Branes. In: Classical Electrodynamics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-91809-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-91809-9_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91808-2
Online ISBN: 978-3-319-91809-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)