Abstract
Despite being a principle of classical field theory, gauge invariance of electrodynamics revealed its deep significance and found its far-reaching interpretation only in relation to quantum mechanics of electrons and the Schrödinger equation. In this chapter, we study the generalization of the concept of a locally invariant gauge theory to non-Abelian gauge groups constructed by following the model of Maxwell theory. This generalization may seem a little academic at first glance because, besides the Maxwell field, it contains further massless gauge fields which are unknown to macroscopic physics. However, it becomes physically realistic if it is combined with the phenomenon of spontaneous symmetry breaking. Both concepts, non-Abelian gauge theory and spontaneous symmetry breaking, initially are purely classical concepts. At the same time, one lays the (classical) foundations for the gauge theories of the fundamental interactions which nowadays are generally accepted and whose validity has been confirmed by numerous experiments. This chapter describes the foundations for the construction of such a theory, within a classical (i.e. nonquantum) framework. Only when introducing fermionic particles (such as quarks and leptons) does the quantization of gauge theories become mandatory.
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.
The codimension of a subspace \(W\) of the finite-dimensional vector space \(V\) is \(\text{codim }W:=\dim V-\dim W\).
- 2.
- 3.
The factors \(\hbar\) and \(c\) which appear there are not essential. In a quantized theory, they can be reinserted at any stage. Alternatively, they may be replaced by \(1\) by the choice of natural units.
- 4.
C.N. Yang and R.L. Mills, Phys. Rev. 96 (1954) 191.
- 5.
This is in agreement with the notion of compactness in set theory: Every infinite-dimensional subset of a compact set \(M\) contains a series whose limit is an element of the set.
- 6.
There is no Lorentz invariance in this example because \(\psi\) obeys a nonrelativistic equation of motion. Lorentz transformations are replaced by rotations in \(\mathbb{R}^{3}\) with respect to which both Lagrange densities are invariant.
- 7.
A precise definition of a fibre bundle in differential geometry may be found, e. g., in [ME], Sect. 4.7.
- 8.
This geometric approach makes use, in an essential way, of the fact that \(*F\wedge F\) is a 4-form and, hence, can be integrated over a manifold with dimension \(4\). The dimension of the base manifold is essential in this context. Integration on arbitrary smooth manifolds is not developed in this book.
- 9.
P.W. Higgs, Phys. Lett. 12 (1964) 132 and Phys. Rev. 145 (1966) 1156; F. Engler and R. Brout, Phys. Rev. Lett. 13 (1964) 321; G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Phys. Rev. Lett. 13 (1964) 585; T.W.B. Kibble, Phys. Rev. 155 (1967) 1554.
- 10.
The action from the right is the conventional choice in differential geometry. Unfortunately, it is not in agreement with the practice in physics where one prefers to have symmetries act from the left. Right action, then, occurs only with contragredient transformation behaviour.
- 11.
In quantum mechanics, these operators are called “raising and lowering operators”. Note that using these linear combinations does not mean that one has complexified the Lie algebra of the structure group or the structure group itself.
- 12.
As one sees, the notion of “potential” is used in two different meanings: on the one hand as a gauge potential in the sense of electrodynamics, on the other hand as a potential or potential energy of the scalar fields in the sense of classical mechanics. This should not be a serious source of misunderstandings.
- 13.
The quotation marks are meant to emphasize that this term contains more than the kinetic energy of the scalar field.
- 14.
D.H. Constantinescu, L. Michel, L.A. Radicati, Journal de Physique 40 (1979) 147.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Scheck, F. (2012). Local GaugeTheories. In: Classical Field Theory. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27985-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-27985-0_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27984-3
Online ISBN: 978-3-642-27985-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)