Concept of Gauge Invariance

I agonized over this post. I thought I could explain the concept of “gauge invariance” in simple terms but I couldn’t. This subject requires a lot of background. So, I will not attempt to present a tutorial here. Instead, I will point you to the educational resources on the web and provide short descriptions from these sources. At the end, I have few comments just to add color.

Resources

[1] “Gauge theories” by Gerard ‘t Hooft (Scholarpedia article)

[2] “Gauge invariance” by Jean Zinn-Justin and Riccardo Guida (Scholarpedia article)

[3] “Gauging What’s Real: The Conceptual Foundations of Contemporary Gauge Theories” by Richard Healey (this is an important book, I hope it does not go out of print)

[4] “Historical roots of gauge invariance” by J.D. Jackson

[5] “Gauge Symmetry in Quantum Mechanics” (UCSD)

[6] “What is gauge?” by Terrance Tao (excellent article)

[7] “Gauge theory” – Wikipedia

[8] “Introduction to gauge theory” – Wikipedia

[9] “The Yang-Mills Model” by Sheldon Lee Glashow

[10] “The Arbitrariness of Local Gauge Symmetry” by Alexandre Guay

[11] “Classical Symmetries and Conservation Laws”

[12] “What, in simplest terms, is gauge invariance?” – Physics StackExchange

[13] “Symmetry and Symmetry Breaking” by Katherine Brading, Elen Castellani, and Nicholas Teh (SEP article, Section 2.5 discusses gauge symmetry)

[14] “Yoichiro Nambu: breaking the symmetry” by Sumit R. Das and Spenta R. Wadia  (CERN Courier article)

[15] “Gauge Theory” by David Tong (431 pages of pure excellence)

[16] https://web2.ph.utexas.edu/~coker2/index.files/gaugef.htm

Short descriptions

“A gauge theory in the most general sense is then any theory whose physical content is preserved by such a (possibly) location-dependent transformation of variables.” [3]

“Gauge symmetry is, in many ways, an odd foundation on which to build our best theories of physics. It is not a property of Nature, but rather a property of how we choose to describe Nature. Gauge symmetry is, at heart, a redundancy in our description of the world. Yet it is a redundancy that has enormous utility, and brings a subtlety and richness to those theories that enjoy it. This course is about the quantum dynamics of gauge theories. It is here that the utility of gauge invariance is clearest. At the perturbative level, the redundancy allows us to make manifest the properties of quantum field theories, such as unitarity, locality, and Lorentz invariance, that we feel are vital for any fundamental theory of physics but which teeter on the verge of incompatibility. If we try to remove the redundancy by fixing some specific gauge, some of these properties will be brought into focus, while others will retreat into murk. By retaining the redundancy, we can flit between descriptions as is our want, keeping whichever property we most cherish in clear sight.” [15]

“Gauge theories refers to a quite general class of quantum field theories used for the description of elementary particles and their interactions. The theories are characterized by the presence of vector fields, and as such are a generalization of the older theory of Quantum Electrodynamics (QED) that is used to describe the electromagnetic interactions of charged elementary particles with spin 1/2. Local gauge invariance is a very central issue. An important feature is that these theories are often renormalizable when used in 3 space- and 1 time dimension.” [1]

“A rule of thumb is that local gauge invariance requires all derivatives in our equations to be replaced by covariant derivatives.” [1]

“In summary, gauge symmetry attains its full importance in the context of quantum mechanics. In the application of quantum mechanics to electromagnetism, i.e., quantum electrodynamics, gauge symmetry applies to both electromagnetic waves and electron waves. These two gauge symmetries are in fact intimately related. If a gauge transformation θ is applied to the electron waves, for example, then one must also apply a corresponding transformation to the potentials that describe the electromagnetic waves. Gauge symmetry is required in order to make quantum electrodynamics a renormalizable theory, i.e., one in which the calculated predictions of all physically measurable quantities are finite. ” [8]

“if we try to construct a gauge-symmetric theory of identical, non-interacting particles, the result is not self-consistent, and can only be repaired by adding electric and magnetic fields that cause the particles to interact.” [8]

““Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple: a gauge is nothing more than a “coordinate system” that varies depending on one’s “location” with respect to some “base space” or “parameter space”, a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations). By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed. Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis.” [6]

Adding color

• If your starting point is Quantum Field Theory which is a theoretical framework where matter fields and force-carrying fields are separate mathematical entities then there is a fundamental problem. In such a framework it is difficult to represent the constraints dictated by Nature (conservations laws and other constraints such as the Lorentz invariance).  Unification of all fields is a distant dream. A half-way solution is to couple the matter fields to the force-carrying fields using the mathematical concept of “covariant derivative”. This mathematical procedure allows us to encode the knowledge of constraints.
• As an aside, don’t forget that conservation laws are empirical observations that stood the test of time. Conservation laws are not derived from theory. On the contrary, the theories are supposed to be consistent with the conservation laws and other constraints such as the Lorentz invariance. Noether’s theorem and gauge theories are such mathematical constructs. We don’t derive the conservation of charge from Noether’s theorem; Noether’s theorem tells us how to construct a theory to be consistent with the conservation of charge. Many people miss this subtlety.