# Gauge Symmetry

## Why is it interesting?

Gauge symmetries are at the heart of the best theory of fundamental interactions, the standard model of particle physics. Theories that make use of gauge symmetry are commonly called Yang-Mills theories.

In addition to this application, gauge symmetry can also be useful to understand finance. This is shown, for example, in

• Physics of Finance by Illinski; see also the book by the same author with the same title.

Gauge symmetry principles are regularly invoked in the context of justification, as deep physical principles, fundamental starting points in thinking about why physical theories are the way they are, so to speak.

“On continuous symmetries and the foundations of modern physics” by CHRISTOPHER A. MARTIN

Gauge theories are interesting because they are the most general class of renormalizable field theories, and we certainly need a field theory to be renormalizable if it is to function calculably all the way from $m_p ~ 10^{19}$ GeV down to atomic mass scales.Fermion masses and Higgs representations in SU(5) by John Ellis and Mary K. Gaillard

Symmetry dictates interaction

Cheng-Ning Yang

We do not yet have a full picture of how nature overcomes the dichotomy between simple fundamental laws and complex emergent phenomena. But particle physics has made huge progress in this direction and the key words are gauge theory. Gauge theory is the essential concept out of which the Standard Model is built: a concept that has all the features of a fundamental principle of nature. It is elegant (based on symmetry considerations), robust (no continuous deformations of the theory are generally allowed), and predictive (given the field content, all processes are described by a single coupling constant). In short, it has all the requirements for a physicist to see simplicity in it. The magic about gauge theory lies in the richness of its structure and its ability to produce, out of a simple conceptual principle, a great variety of different manifestations. Long-range forces, short-range forces, confinement, dynamical symmetry breaking are all phenomena described by the same principle. The vacuum structure of gauge theory is unbe- lievably rich, with θ-vacua, instantons, chiral and gluon condensates, all being expressions of the same theory. The phase diagram at finite temperature and density exhibits a variety of new phenomena and states of matter. In short, gauge theory is an exquisite tool to make complexity out of simplicity.

Related Concepts

## Layman

When we describe things in physics, we have always some freedom in our description. For example, it doesn't matter what coordinate system we choose. It makes no difference where we choose the origin of the coordinate system or how it is oriented.

The computations can be different in different coordinate systems and usually, one picks a coordinate system where the computation is especially simple. However, the physics that we are describing, of course, doesn't care about how we describe it. It stays the same, now matter how we choose our coordinate system.

In modern physics, we no longer describe what is happening merely through the position of objects at a given time, as we do it in classical mechanics. Instead, we use abstract objects called fields. The best theory of what is happening in nature at the most fundamental level is Quantum Field Theory. Like the electromagnetic field, these fields can get excited (think: we can produce a wave or ripple of the field). For example, when we excite the electron field we “produce” an electron.

The fields themselves are abstract mathematical entities that are introduced as convenient mathematical tools. With these new mathematical entities comes a new kind of freedom. Completely analogous to how we have the freedom to choose the orientation and the location of the origin of our coordinate system, we now have freedom in how we define our fields.

The freedom to “shift” or “rotate” our fields is called gauge symmetry. It is important to note that this symmetry is completely independent from the rotational and translational symmetry of our coordinate systems. When we “shift” or “rotate” a field we do not refer to anything in spacetime, but instead we “shift” and “rotate” merely our description of a given field.

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## Student

Here is the idea behind gauge theories in a nutshell:

Consider a field $\Psi$ representing electrically charged matter. The free field obeys the Dirac equation which is just the Euler–Lagrange equation(s) for the Lagrangian (density) $L_{Dirac} = \bar \Psi(i\gamma^\mu ∂μ - m)\Psi$.The corresponding action is clearly invariant under so-called ‘global’ U (1) phase transformations: $\Psi \to e^{iq\Lambda} \Psi$, $\bar \Psi \to e^{-iq\Lambda} \bar \Psi$ with $\Lambda$ a constant. It follows from Noether’s first theorem that when the equations of motion are satisfied there will be a corresponding conserved current.

Consider now ‘localizing’ these phase transformations, i.e. letting $\Lambda$ become an arbitrary function of the coordinates $\Lambda(x):$ $\Psi \to e^{iq\Lambda(x)} \Psi$, $\bar \Psi \to e^{-iq\Lambda(x)} \bar \Psi$. As it stands, the free field Lagrangian is clearly not invariant under such transformations, since the derivatives of the arbitrary functions, i.e. $∂μ \Lambda(x)$, will not vanish in general. The Lagrangian must be modified if the theory is to admit the local transformations as (variational) symmetries. In particular, we replace the free field Lagrangian with $$L_{interacting} = \bar \Psi(i\gamma^\mu ∂μ - m)\Psi - q A_μ \bar \Psi γ^\mu \Psi ≡ L_{Dirac} - J_μ A^μ ,$$ with $J_μ = q \bar \Psi \gamma_\mu \Psi$. This current is in fact the conserved current associated with the global U(1) invariance of the interacting theory. Towards securing local invariance we have introduced the field $A_μ$, the gauge potential. The particular form of coupling between the matter field and this gauge potential in $L_{interacting}$ is termed minimal coupling.

This modified Lagrangian is now invariant under the local phase transformations provided that the vector field $A_μ$ is simultaneously transformed according to $A_μ (x) → A_μ(x) - ∂μ \Lambda(x)$. electromagnetic potential.

“On continuous symmetries and the foundations of modern physics” by CHRISTOPHER A. MARTIN

The Lagrangian that we derive by demanding local invariance describes perfectly quantum electrodynamics. More specifically, this means the new term $- q A_μ \bar \Psi γ^\mu \Psi$ in the Lagrangian, that we wrote down to ensure local invariance, is exactly the correct term to describe, for example, how an electron interactions with a photon.

One often hears physicists say “to gauge a symmetry group.” It means to localize the group, or to make the transformation parameters vary spatio-temporally so that the system does not behave as a unit under the transformations. Local transformations allow a large degree of individuality for each point of a system by endowing it with an autonomous internal structure.The Poincare transformations become local in general relativity. The quantum phase transformation is localized and becomes exp[iO(x)] in quantum field theories. Local symmetries are ubiquitous in quantum field theories. The symmetry groups U(l), SU(2) x C/(7), and SU(3) for the electromagnetic, electro-weak, and strong interactions are all local transformations whose parameters are functions of spatio-temporal positions. These local symmetry groups are often called gauge groups and theories employing them gauge field theories.They determine the properties of the phase space at each spatio-temporal point individually. A local symmetry is more complicated than a global symmetry because itdemands the global invariance of the entire system under local transformations.Generally this requires the introduction of extra structures to reconcile thedifference of various local transformations. The extra structures are usually interpreted as interaction potentials, as discussed in the next chapter. From “How is Quantum Field Theory possible” by Auyang

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## Researcher

Gauge symmetry sometimes appears to be a curious shell game. One starts with some initial global symmetry algebra and makes it “local” via the introduction of new degrees of freedom, enlarging the symmetry algebra enormously; then, states that differ by gauge transformations are identified as the same physical state, effectively reducing the symmetry algebra. It is typically expected that the reduced symmetry algebra relating physical observables is the same as the initial algebra. In which case, the net effect of the gauge procedure, is to introduce new dynamical degrees of freedom (the gauge bosons). In the end, the advantage of the redundant description over a description involving only physical degrees is that the physical description is nonlocal. […] It has long been known that for gravity in asymptotically flat space [1, 2] or asymptotically AdS3 [3], the final physical symmetry algebra is an infinite-dimensional enhancement of the “global part” of the gauge group. Only recently, however, has it been realized that the enhancement also occurs for higher dimensional gravity, Maxwell theory, Yang–Mills theory, and string theory, and moreover, that the symmetry constrains the IR structure via nontrivial Ward identities [4–15].https://arxiv.org/abs/1510.07038

Gauge potentials take their values in the Lie algebra $\mathfrak{g}$ of the gauge group $\mathcal G$.

It is important to note that there is difference between a group $G$ and the corresponding gauge group $\mathcal G$.

$G$ is simply a set of symmetry transformations. In contrast, $\mathcal G$ is a group of smooth functions on spacetime that take values in $G$. More mathematically, the group of gauge transformations $\mathcal G$ means the bundles automorphisms which preserve the Lagrangian. (Source). The group $G$ is simply one fibre of the bundle, i.e. for example, $SU(2)$.

Moreover, one can argue that the “true” gauge symmetry is given by a subset of $\mathcal G$ called $\mathcal G_\star$:

\begin{align} \mathcal{G}_\star &= \big \{ \text{ set of all } g(x) \text{ such that } g \to 1 \text{ as } |x| \to \infty \big \} \notag \\ \mathcal{G} &= \big \{ \text{ set of all } g(x) \text{ such that } g \to \text{ constant element of $G$, not necessarily $1$ as } |x| \to \infty \big \} \end{align}

This comes about when one considers Gauss law to identify physical states. Such physical states are invariant under $\mathcal{G}_\star$ and thus this subgroup connects physically redundant variables in the theory.

Since the elements of $\mathcal G$ go only to a constant, which is not necessarily $1$ at spatial infinity, we have

\begin{align} \mathcal{G} / \mathcal{G}_\star \sim \text{ set of constant g's } \sim G \notag \\ \mathcal{G} / \mathcal{G}_\star \sim G \text{ is the Noether symmetry of the theory defined by the charges} \end{align}

All transformations $g(x)$ which go to a constant at spatial infinity that is not $1$ act as a Noether symmetry of the theory.

For more on this see, section 10.3 and chapter 16 in Quantum Field Theory - A Modern Perspective by V. P. Nair.

We denote the space of all connections by $\mathcal A$ (= the space of all gauge potentials $A_i$). This space is a an affine space, which simply means that any potential $A_i$ can be written as $A_i^{(0)} + h_i$, where $A_i^{(0)}$ is a given fixed potential and $h_i$ is an arbitrary vector field that takes values in the Lie algebra. Geometrically this means that, any two points in $\mathcal A$ can be connected by a straight line.

For two potentials $A_i^{(1)}$ and $A_i^{(2)}$, we can define the following sequence of gauge potential configurations

$$A_i(x,\tau) = A_i^{(1)}(1-\tau) + \tau A_i^{(2)} ,$$

where $0 \leq \tau \geq 1$ parametrizes the straight line between the two configurations. (Topologically this space is rather trivial).

The configuration space of the theory is $\mathcal C = \mathcal A / \mathcal G_\star$.

Now, to get physically sensible results we must be careful with these different notions:

Integration should therefore be carried out on the quotient space $G=\mathcal A/\mathcal{G}_\star$. Now $\mathcal A$ is a linear space but $\mathcal{G}_\star$ is only a manifold and has to be treated with more respect. Thus for integration purposes a Jacobian term arises which, in perturbation theory, gives rise to the well-known Faddeev-Popov “ghost” particles. Nonperturbatively it seems reasonable that global topological features of $\mathcal{G}_\star$ will be relevant.

Geometrical Aspects of Gauge Theories by M. F. Atiyah

Recommended Resources

• What is a gauge? by Terence Tao
• Quantum Field Theory: A Modern Perspective by V. Parameswaran Nair
• Geometry and topology of chiral anomalies in gauge theories by R. Rennie
• “Some elementary gauge theory concepts” by Hong-Mo Chan, Sheung Tsun Tsou
• Why gauge Symmetry by J. Gegelia

Example1
Example2:

## FAQ

Is gauge symmetry really a symmetry?
No!

The problem with gauge symmetry is that it is not a symmetry in the sense of quantum mechanics. A symmetry is the invariance of the Hamiltonian under transformations of quantum states, which are elements of a Hilbert space. Gauge symmetry is not a symmetry because the corresponding transformation does not change the quantum states. Gauge symmetry acts trivially on the Hilbert space and does not relate physically distinct states. A gauge transformation is like a book by James Joyce: it seems that something is going on, but nothing really happens. Gauge symmetry is the statement that certain degrees of freedom do not exist in the theory. This is why gauge symmetry corresponds only to as a redundancy of the theory description. The non-symmetry nature of gauge symmetry explains why gauge symmetry, unlike global symmetry, cannot be broken by adding local operators to the action: gauge symmetry is exact at all scales. The only way to “break” gauge symmetry is adding to the theory the missing degrees of freedom, but this operation is not simply a deformation of the theory (as the case of adding local operators to an action with global symmetry) but corresponds to considering an altogether different theory. The non-symmetry nature of gauge symmetry also explains trivially the physical content of the Higgs theorem. For a spontaneously-broken global symmetry, an infinite number of vacuum states are related by the symmetry transformation. This leads to the massless modes dictated by the Goldstone theorem. In a spontaneously-broken gauge symmetry, there is a single physical vacuum and thus there are no massless Goldstones. Gauge symmetry does not provide an exception to the Goldstone theorem, simply because there is no symmetry to start with. For gauge symmetry, the word ‘symmetry’ is a misnomer, much as ‘broken’ is a misnomer for spontaneously broken symmetry. But as long as the physical meaning is clear, any terminology is acceptable in human language. The important aspect is that the mathematical language of gauge symmetry (both in the linear and non-linear versions) is extremely pow- erful in physics and permeates the Standard Model, general relativity, and many systems in condensed matter. As the redundancy of degrees of freedom is mathematically described by the same group theory used for quantum symmetries, the use of the word ‘symmetry’ seems particularly forgivable. Does this necessarily make gauge symmetry a fundamental element in the UV? The property of gauge symmetry of being – by construction – valid at all energy scales may naively suggest that gauge symmetry must be an ingredient of any UV theory from which the Standard Model and general relativity are derived. On the contrary, many examples have been constructed – from duality to condensed-matter systems – where gauge symmetry is not fundamental, but only an emergent property of the effective theory [41]. Gauge symmetry could emerge in the IR, without being present in the UV theory. If this is the case, gauge symmetry is not the key that will unlock the mysteries of nature at the most fundamental level. The concept of symmetry has given much to particle physics, but it could be that it is running out of fuel and that, in the post-naturalness era, new concepts will replace symmetry as guiding principles.

But there are several reasons not to accept this view. First of all terminology. When we say gauge symmetry, this is really a misnomer. It's a misnomer because in physics gauge symmetry is not a symmetry. It is not a symmetry of anything. Symmetry is a set of transformations that act on physical observables. They act on the Hilbert space. The Hilbert space is always gauge invariant. So the gauge symmetry doesn't even act on the Hilbert space. So it's not a symmetry of anything. […] Second, gauge symmetry can be made to look trivial. So, I'll give one trivial example and then I'll make it more elaborate… [explains the Stückelberg mechanism, where one introduces a Stückelberg field to make a non U(1) gauge invariant Lagrangian, gauge invariant] This is almost like a fake… This gauge symmetry is what we would call emergent, except that in this case it is completely trivial. The second thing which is wrong about gauge symmetry, which suggests that it's not fundamental is that, it started in condensed matter physics, people talked about spontaneous symmetry breaking. That was crucial in the context of superconductivity and superfluidity and so forth. And the recent Nobel price in physics was also associated with spontaneous gauge symmetry breaking. That of Higgs, and Englert. This is all very nice and physicists love to talk about spontaneous symmetry breaking, but this is a bit too naive. First of all I've already emphasized that a gauge symmetry is not a symmetry. And since it is not a symmetry, how could it possibly be broken. You can break a symmetry that exists, but you cannot break a symmetry that does not exist. Second, the phenomenon of spontaneous symmetry breaking is often associated with the fact that the system goes to infinity. Concretely in quantum mechanics, you never have symmetry breaking. It is only in quantum field theory or statistical mechanics, where we have volume going to infinity we have an infinite number of degrees of freedom and there we have this phenomenon of spontaneous symmetry breaking. That's not true for gauge theories. For gauge theories, we have a lot of symmetry. At every point of space we have a separate symmetry. But the number of degrees of freedom that transform under a given symmetry transformation is always finite. Nothing goes off to infinity. So the gauge symmetry cannot be spontaneously broken. The ground state is always unique. Or if you wish, all these would-be separate ground states are all related to each other by a gauge transformation. […] I said that gauge symmetry cannot be ultimate symmetry because it's so big, there is a separate transformation at every point in space. So the breaking of a gauge theory cannot happen, I can use a phrase from the financial crisis in 2008 that a gauge symmetry is so big, it's too big to fail.

Duality and emergent gauge symmetry - Nathan Seiberg

Gauge symmetry is deep

•Largest symmetry (a group for each point in spacetime)

•Useful in making the theory manifestly Lorentz invariant, unitary and local (and hence causal)

But

•Because of Gauss law the Hilbert space is gauge invariant.( More precisely, it is invariant under small gauge transformation; large gauge transformations are central.)

•Hence:gauge symmetry is not asymmetry.

• It does not act on anything.

• A better phrase is gauge redundancy.

Gauge symmetries cannot break

•Not a symmetry and hence cannot break

•For spontaneous symmetry breaking we need an infinite number of degrees of freedom transforming under the symmetry. Not here.

•This is the deep reason there is no massless Nambu-Goldstone boson when gauge symmetries are “broken.”

Gauge symmetries cannot break For weakly coupled systems (e.g. Landau-Ginsburg theory of superconductivity, or the weak interactions) the language of spontaneous gauge symmetry breaking is appropriate and extremely useful[Stueckelberg,Anderson,Brout, Englert,Higgs].

Global symmetries can emerge as accidental symmetries at long distance. Then they are approximate. Exact gauge symmetries can be emergent.

Examples of emergent gauge symmetry…

Gauge symmetries are properly to be thought of as not being symmetries at all, but rather redundancies in our description of the system 1. The true configuration space of a (3 + 1)- dimensional gauge theory is the quotient $\mathcal{A}^3/\mathcal{G}^3$ of gauge potentials in $A_0=0$ gauge modulo three-dimensional gauge transformations. When gauge degrees of freedom become anomalous, we find that they are not redundant after all.

Hamiltonian Interpretation of Anomalies by Philip Nelson and Luis Alvarez-Gaume

From the modern point of view, then, gauge symmetry is merely a useful redundancy for describing the physics of interacting massless particle of spin 1 or 2, tied to the specific formalism of Feynman diagrams, that makes locality and unitarity as manifest as possible.

Gauge invariance is not physical. It is not observable and is not a symmetry of nature. Global symmetries are physical, since they have physical consequences, namely conservation of charge. That is, we measure the total charge in a region, and if nothing leaves that region, whenever we measure it again the total charge will be exactly the same. There is no such thing that you can actually measure associated with gauge invariance. We introduce gauge invariance to have a local description of massless spin-1 particles. The existence of these particles, with only two polarizations, is physical, but the gauge invariance is merely a redundancy of description we introduce to be able to describe the theory with a local Lagrangian. A few examples may help drive this point home. First of all, an easy way to see that gauge invariance is not physical is that we can choose any gauge, and the physics is going to be exactly the same. In fact, we have to choose a gauge to do any computations. Therefore, there cannot be any physics associated with this artificial symmetry.

Quantum Field Theory and the Standard Model by Matthew Schwartz

What's the interpretation of gauge symmetry?
The best summary can be found starting at page 7 in http://philsci-archive.pitt.edu/223/1/Objectivity.pdf

Especially: “local gauge invariance in quantum theory does not imply the existence of an external electromagnetic field”!

Do we really care about the gauge group or only about the corresponding Lie algebra?
Oftentimes, the notion “gauge group” is used a bit sloppy. Perturbative effects only depend on the local structure of the group and hence only on the Lie algebra. There are many groups with the same Lie algebra and in many situations it is not clear which group is the “correct” group that describes the physics. From experiments, we only learn something about the Lie algebra. (This is, for example, mentioned in Physics and Geometry by EDWARD WITTEN).

For example, there are thirteen groups with the same Lie algebra as the famous $SU(3) \times SU(2) \times U(1)$ gauge symmetry of the standard model. In addition, there are good reasons to believe that the correct gauge group of the standard model is not $SU(3) \times SU(2) \times U(1)$ , but rather $S(U(3) \times U(2) )$.

Is it possible to derive gauge symmetry from something else?
Yes:

Gauge Symmetry from Lorentz Symmetry

This is done, for example, in Vol. 1 of Weinberg's Quantum Field Theory book in section 5.9.

Weinberg shows that a massless spin 1 vector field $A_\mu$ cannot be a four-vector under Lorentz transformations. Instead, he derives that under a Lorentz transformation $\Lambda$ a massless spin 1 vector field $A_\mu$ transforms as follows:

$$U(\Lambda) A_\mu(x) U^{-1}(\Lambda) = \Lambda^\nu_\mu A_\nu(\Lambda x) + \partial_\mu \Omega(x,\Lambda),$$

where $\Omega(x,\Lambda)$ is some function of the creation and annihilation operators. Therefore, he concludes that in order to get a Lorentz invariant theory it is not enough to write down terms in the Lagrangian that are invariant under the “naive” transformation $A_\mu \to \Lambda^\nu_\mu A_\nu$, but additionally the terms must be invariant under $A_\mu \to A_\mu + \partial_\mu \Omega$. This second part of the transformation is the well known gauge transformation of $A_\mu$. In this sense, the gauge symmetry follows from the Lorentz symmetry.

A summary of Weinberg's argument with an easier notation can be found in this article.

This emergence of gauge symmetry was also discussed nicely from a bit different perspective in this recent paper by Nima Arkani-Hamed, Laurentiu Rodina, Jaroslav Trnka.

“Maxwell’s theory and Einstein’s theory are essentially the unique Lorentz invariant theories of massless particles with spin $j =1$ and $j =2$”.

Photons and Gravitons in Perturbation Theory: Derivation of Maxwell's and Einstein's Equations by Steven Weinberg

Take note that there is a close connection between this kind of argument and the famous Weinberg-Witten Theorem.

Gauge Symmetry from Asymptotic Freedom

This point of view is formulated by Wilczek in his article “What QCD tells us about nature - and why we should listen”:

Summarizing the argument, only those relativistic field theories which are asymptotically free can be argued in a straightforward way to exist. And the only asymptotically free theories in four space-time dimensions involve nonabelian gauge symmetry, with highly restricted matter content. So the axioms of gauge symmetry and renormalizability are, in a sense, gratuitous. They are implicit in the mere existence of non-trivial interacting quantum field theories.

Does gauge symmetry really "dictate interaction"?
Definitely not alone.

This modification is not uniquely dictated by the demand of local gauge invariance. There are infinitely many other gauge-invariant terms that might be added to the Lagrangian if gauge invariance were the only input to the argument. In order to pick out the minimal modification uniquely, we must bring in, besides gauge invariance and knowledge of field theories generally, the requirements of Lorentz invariance, simplicity, and, importantly, renormalizability. (For example, a Pauli term is Lorentz invariant and gauge invariant but not renormalizable.) The minimal modification is then the simplest, renormalizable, Lorentz and gauge-invariant Lagrangian yielding second-order equations of motion for the coupled system (O’Raifeartaigh, 1979). The point is simply that, in the context of the gauge argument, the requirement of local gauge invariance gets a lot of its formal muscle in combination with other important considerations and requirements.

“On continuous symmetries and the foundations of modern physics” by CHRISTOPHER A. MARTIN

The Pauli Term is $\frac{m_0}{\Lambda_0^2}\bar\Psi \gamma^{\mu \nu} F_{\mu \nu} \Psi$, where $\gamma^{\mu \nu}$ is $[\gamma^\mu,\gamma^\nu]$. It is non-renormalizable, because the factors coming from $\frac{m_0}{\Lambda_0^2}$ in higher order of perturbation theory, have to be compensated by more and more divergent integrals.

Moreover, any theory can be made gauge invariant by the “Stueckelberg trick”:

While many older textbooks rhapsodize about the beauty of gauge symmetry, and wax eloquent on how “it fully determines interactions from symmetry principles”, from a modern point of view gauge invariance can also be thought of as by itself an empty statement. Indeed any theory can be made gauge invariant by the “Stuckelberg trick”–elevating gauge transformation parameters to fields–with the “special” gauge invariant theories distinguished only by realizing the gauge symmetry with the fewest number of degrees of freedom.

This is similar to the discussion about the role of “general covariance” in general relativity. According to Einstein, “general covariance” is the symmetry principle at the heart of general relativity. However, it was quickly noted by Kretschmann that any theory can be formulated in a general covariant way.

What are alternatives to the gauge paradigm?
See, for example, spin-helicity formalism and the references there.

Loop Formulation:

See Anandan (1983) who argues that from both the physical and mathematical points of view, the holonomy contains all the relevant (gauge-invariant) information. Specifically, the connection can be constructed (up to gauge transformation) from a knowledge of the holonomies. Formalizing gauge theories in terms of holonomies associated with (non-local) loops in space appears, though, to require a revamped conception of the notion of a physical field (see Belot, 1998).

“On continuous symmetries and the foundations of modern physics” by CHRISTOPHER A. MARTIN

For more on the loop space formulation of quantum field theory, have a look at the small book “Some Elementary Gauge Theory Concepts” by Sheung Tsun Tsou, Hong-Mo Chan

Constrained Hamiltonian Formalism

See, for example, “Tracking down gauge: an ode to the constrained Hamiltonian formalism” by JOHN EARMAN

Where does gauge symmetry come from?

What Nielsen imagines is that the whole cosmos is just at the point of a phase transition between two phases. He and his colleagues, such as Don Bennett, try to demonstrate that many of the observed properties of the elementary particles arise simply from this fact, independently of whatever the fundamental laws of physics are. They want to say that, just as bubbles are universally found in liquids that are boiling, the fundamental particles we observe may be simply universal consequences of the universe being balanced at the point of a transition between phases. If so, their properties may to a large extent be independent of whatever fundamental law governs the world.

[…]

In fact, Nielsen and his colleagues do claim some successes for the hypothesis of random fundamental dynamics. Among them is the fact that all the fundamental interactions must be gauge interactions, of the type described by Yang-Mills theory and general relativity. This means that the world would appear at large scales to be governed by these interactions, whether or not they are part of the fundamental description of the world at the Planck scale. This last claim is, in fact, rather well accepted among particle theorists. It has been independently confirmed by Steven Shenker and others.

The Life of the Cosmos by Lee Smolin

Such a point of view is supported, for example, by observations in condensed matter physics:

Well, all the asymptotic behavior and renormalization group fixed points that we look at in condensed matter theory seem to grow symmetries not necessarily reflecting those of the basic, underlying theory. In particular, I will show some experiments tomorrow, where, in fact, one knows for certain that the observed symmetry grows from a totally unsymmetric underlying physics. Although as a research strategy I think what you say about postulating symmetry is totally unarguable, one can remark, in opposition, that it is only the desperate man who seeks after symmetry! If we truly understand a theory, we should see symmetry coming out or, on the other hand, failing to appear. So I am certainly not criticizing you on strategy. But you recognize - you put it very nicely, and I was relieved to hear it - that the renormalization group principle works in a large space, there are many fixed points, and there are many model field theories. So I am still unclear as to the origin of your faith that string theory should give us the standard model rather than some other type of local universe.

Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao

Why do we need gauge symmetry?

Is gauge symmetry an autonomous concept, logically independent of other leading principles of physics? On the contrary, it appears to be mandatory, in the theory of vector particles, to insure consistency with special relativity and quantum mechanics. For if the transverse parts of the vector field produce excitations that have a normal probabilistic interpretation (i.e., the square of their amplitude is the probability for their presence), then Lorentz invariance implies that the longitudinal parts produce excitations that are, in the jargon of quantum theory, ghosts. That is to say, the square of their amplitudes is minus the probability for their presence, so that when we contemplate their production we are confronted with the specter of negative probabilities, which on the face of it are senseless. Gauge invariance saves the day by insuring that the longitudinal modes decouple, i.e. that transition amplitudes to excite such modes actually vanish. Thus gauge invariance is required, in order to insure that no physical process is assigned a negative probability. Yang-Mills Theory In, Beyond, and Behind Observed Reality by Frank Wilczek

So what does this mean? What’s the point of having a local symmetry if we can just choose a gauge (in fact, we have to choose a gauge to do any computations) and the physics is the same? There are two answers to this question. First, it is fair to say that gauge symmetries are a total fake. They are just redundancies of description and really do have no physically observable consequences. In contrast, global symmetries are real features of nature with observable consequences. For example, global symmetries imply the existence of conserved charges, which we can test. So the first answer is that we technically don’t need gauge symmetries at all. The second answer is that local symmetries make it much easier to do computations. You might wonder why we even bother introducing this field Aµ which has this huge redundancy to it. Instead, why not just quantize the electric and magnetic fields, that is Fµν, itself? Well you could do that, but it turns out to be more of a pain than using Aµ. To see that, first note that Fµν as a field does not propagate with the Lagrangian L = − 1 4 Fµν 2 . All the dynamics will be moved to the interactions. Moreover, if we include interactions, either with a simple current AµJµ or with a scalar field φ ⋆Aµ∂µφ or with a fermion ψ¯γµAµψ, we see that they naturally involve Aµ. If we want to write these in terms of Fµν we have to solve for Aµ in terms of Fµν and we will get some crazy non-local thing like Aν = ∂ν Fµν. Then we’d have to spend all our time showing that the theory is actually local and causal. It turns out to be much easier to deal with a little redundancy so that we don’t have to check locality all the time. Another reason is that all of the physics of the electromagnetic field is not entirely contained in Fµν. In fact there are global properties of Aµ that are not contained in Fµν but that can be measured. This is the Aharanov-Bohm effect, that you might remember from quantum mechanics. Thus we are going to accept that using the field Aµ instead of Fµν is a necessary complication. So there’s no physics in gauge invariance but it makes it a lot easier to do field theory. The physical content is what we saw in the previous section with the Lorentz transformation properties of spin 1 fields.

We might instead give a gauge-invariant interpretation, taking the physical state as specified completely by the gauge-invariant electric and magnetic field strengths. In this case, electromagnetism is deterministic since the gauge invariance that threatened determinism is in effect washed away from the beginning. However, in the case of non-trivial spatial topologies, the gauge-invariant interpretation runs into potential complications. The issue is that in this case there are other gauge invariants. So-called holonomies (or their traces, Wilson loops) – the line integral of the gauge potential around closed loops in space – encode physically significant information about the global features of the gauge field. The problem is that these gauge invariants, being ascribed to loops in space, are apparently non-local. But, coming full circle, providing a local description requires appeal to non-gauge-invariant entities such as the electromagnetic potential, whose very reality is in question according to the received understanding. The context for this discussion is the interpretation of the well-known Aharonov–Bohm (A–B) effect

“On continuous symmetries and the foundations of modern physics” by CHRISTOPHER A. MARTIN

Why do we localize the global symmetry?

Since $\Lambda$ is a constant, however, this gauge transformation must be the same at all points in space-time; it is a global gauge transformation. So when we perform a rotation in the internal space of $\phi$ at one point, through an angle $\Lambda$, we must perform the same rotation at all other points at the same time. If we take this physical interpretation seriously, we see that it is impossible to fulfil, since it contradicts the letter and spirit of relativity, according to which there must be a minimum time delay equal to the time of light travel. To get round this problem we simply abandon the requirement that $\Lambda$ is a constant, and write it as an arbitrary function of space-time, $\Lambda(x^\mu)$. This is called a local gauge transformation, since it clearly differs from point to point.

page 93 in Quantum Field Theory by Ryder

First, the initial and all-important demand of local as opposed to global gauge invariance is anything but self-evident, and presumably it must be argued for on some basis. Historically, the arguments surrounding the ‘demand’ as such are quite thin. The most prevalent form goes back to Yang and Mills’ remarks to the effect that ‘local’ symmetries are more in line with the idea of ‘local’ field theories. Arguments from a sort of locality, and especially those predicated specifically on the demands of STR (i.e. no communication-at-a-distance) (See for example Ryder (1996, p. 93)). , are somewhat suspect, however, and careful treading is needed. Most immediately, the requirement of locality in the STR sense – say, as given by the lightcone structure – does not map cleanly onto to the global/local distinction figuring into the gauge argument – i.e. $G_r$ vs. $G_{\infty r}$ . Overall, the question of how ‘natural’, physically, this demand is, is not uncontentious. This is especially so in light of the received view of gauge transformations which maintains that they have no physical significance or counterpart (more below). I will return briefly in the next section to considering possible

“On continuous symmetries and the foundations of modern physics” by CHRISTOPHER A. MARTIN

## History

Gauge theory was invented by Hermann Weyl, who tried to derive electrodynamics from scale invariance.

The original paper is: Hermann Weyl, Raum, Zeit, Materie: Vorlesungen über die Allgemeine Relativitätstheorie, Springer Berlin Heidelberg 1923

Good books on the history of gauge theories are:

• Lochlainn O'Raifeartaigh, The Dawning of Gauge Theory, Princeton University Press (1997)
• Lochlainn O'Raifeartaigh, Norbert Straumann, Gauge Theory: Historical Origins and Some Modern Developments Rev. Mod. Phys. 72, 1-23 (2000).
• Norbert Straumann, Early History of Gauge Theories and Weak Interactions (arXiv:hep-ph/9609230)
• Norbert Straumann, Gauge principle and QED, talk at PHOTON2005, Warsaw (2005) (arXiv:hep-ph/0509116)