Gauge symmetry

In 1918, Herman Weyl tried to extend the invariance group of general relativity by assuming that physical laws should also be invariant under local scaling of the metric tensor $g_{\mu \nu}$, i.e. under

$$g_{\mu \nu}(x) \rightarrow \lambda(x) g_{\mu \nu}(x)$$ (34)

where $\lambda$ is some function of the coordinates.

He hoped that local scale invariance should be connected to the electromagnetic field just like invariance under local coordinate transformations is related to the gravitational field; in the sense that when $g_{\mu \nu}(x)$ is multiplied by $\lambda$, the electromagnetic potential $A_{\mu}$ was assumed to change by a $gauge$ $transformation$ to

$$A_{\mu} \rightarrow A_{\mu}+ \frac{\partial \lambda}{\partial x^{\mu}}$$ (35)

The attempt failed, but Weyl later showed the relation between charge conservation and the gauge invariance of the electromagnetic interactions.

The concept of gauge symmetry has however come to play a crucial role in particle physics. A $global$ symmetry transformation does not depend on positions in space and time, that is, under a global symmetry transformation, the change will be the same at each space-time point, and the field energy of a system remains unchanged under such a transformation.

If the global symmetry is altered, so as to become space-time dependent, we have a $local$ symmetry transformation, which acts differently at each space-time point. In this case, the field energy of the transformed system is changed and the original symmetry is lost.
If however there is yet another field present that "reads off" the local changes and compensates for them in such a way that the system behaves as under the global symmetry transformation, the symmetry is nevertheless maintained and the field energy is conserved.

In the case of an inner local symmetry, such a "compensating" field is called a gauge field. When this was discovered by Yang and Mills in 1954, it did not arouse great interest, since there was no way of applying these ideas to particle physics. One reason for this was that gauge field theories were believed to be non-renormalizable, another that experimental data did not support the existence of the predicted Yang-Mills quanta.

The first problem was solved in 1971, when gauge theories were shown to actually be renormalizable. The second problem was "solved" by the insight that gauge symmetries may appear with hidden or spontaneously broken symmetries.

The classical example of gauge invariance is found in electromagnetism, where the physically measurable fields are independent of whether the potential is defined as $A_{\mu}$ or as $A_{\mu}+\partial\lambda /\partial x^{\mu}$. This means that $A_{\mu}$ is a non measurable quantity. To choose which $A_{\mu}$ that is to be defined as the potential ``behind'' the measured electric and magnetic fields ${\bf {E}}$ and ${\bf {B}}$, is to choose a gauge.
In the quantum mechanical description of an electron as a wave in motion, a change of the electromagnetic potential implies a change in the phase of the electron field. The Lagrangian remains invariant if the translation in the electromagnetic field is supplemented with a well defined change in the phase of the electron field. In this case the gauge symmetry is local, the change of potential and the change of phase both being space-time dependent. In gauge theory the fields are introduced in a reverse order, the electromagnetic field entering as the gauge field compensating for the change of phase of the electron field under the $U(1)$ gauge transformations.

We thus start out with the free electron, which is described by the free-fermion Lagrangian density

$${\cal{L}}_0 = \bar{ \psi}(i\gamma_{\mu}\partial^{\mu}-m)\psi$$

which is invariant under the global phase transformation, where the phase of the field is changed by the same amount at each space-time point,

$$\psi {\hspace{4mm}}\rightarrow{\hspace{4mm}}\psi' = \psi e^{-i\alpha}$$ (36)

where $\alpha$ is a real number. This invariance corresponds to the conservation of the current $j^{\mu}= q\bar{ \psi}\gamma^{\mu} \psi$, which also implies the conservation of the charge $Q= q\int d^3 {\bf {x}}\psi^{\dagger}\psi$.

Then if we demand that the invariance should be local, with space-time dependent transformations

$$\psi {\hspace{4mm}}\rightarrow{\hspace{4mm}}\psi' = \psi e^{-iqf(x)}$$ (37)

the Lagrangian remains invariant if a gauge potential $A_{\mu}$ is introduced into theory, through the minimal substitution

$$\partial_{\mu} {\hspace{4mm}}\rightarrow{\hspace{4mm}}D_{\mu}= (\partial_{\mu}+iqA_{\mu})$$

where $q$ is the charge of the fermion annihilated by the field $\psi(x)$, and $A_{\mu}$ is the electromagnetic intermediator, transforming as $A$ $\rightarrow$ $A_{\mu}+\partial\lambda /\partial x^{\mu}$ under the gauge transformations.

The quantization of a gauge system is a rather involved procedure. Gauge theories are systems with constrained dynamical variables that are not in one-to-one correspondence with true dynamical degrees of freedom. The gauge symmetry implies an overcounting over all the field configurations, which must be removed by introducing suitable constraints.
This may be obtained by removing the redundant degrees of freedom by some gauge fixing conditions, whereby theory can be quantized.