# What is gauge invariance

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

Gauge invariance is a form of symmetry.

An experiment here today will work the same way over there tomorrow and with the apparatus pointing in a different direction.

This is called "global invariance" … the laws of physics are invariant under translations, both in space and time, and under rotations … and translations and rotations are global symmetries of space-time.

Similarly, the physics of a particle in a field is invariant under certain local transformations of the phase, provided that the ordinary derivative $\partial^{\mu}$ is replaced by the covariant derivative $D^{\mu}$.

Local phase transformations which depend on any form of charge (not necessarily electric) are for historical reasons called gauge transformations, and the invariance of field physics under them is called gauge symmetry or gauge invariance.

Equations

Extended explanation

Internal symmetry:

Particles generally have "internal" symmetries (a form of spin), the "rotations" of which form a "gauge group" of transformations.

This group is generated by infinitesimal transformations, which it is convenient to label by a parameter $\eta$, which is typically one-dimensional or three-dimensional.

Gauge fields:

Associated with this gauge group is a gauge field or fields. For example, if the infinitesimal parameter is one-dimensional, there will be one field, $A^{\mu}$, and if it is three-dimensional, there will be three fields, $\mathbf{W}^{\mu}\ =\ (W_1^{\mu},W_2^{\mu},W_3^{\mu})$.

Associated with the gauge field, or fields, is a covariant derivative, $D^{\mu}$, equal to the ordinary derivative, $\partial^{\mu}$, plus a multiple of the field, or of a "summary" of the fields, respectively.

This is basically a consequence of Noether's theorem, which states that every symmetry has an associated conserved current.

Locality:

The local (non-global) nature of gauge invariance is of fundamental importance. It was the essential step in the creation of the electroweak theory:

… The difference between a neutron and a proton is then a purely arbitrary process. As usually conceived, however, this arbitrariness is subject to the following limitation: once one chooses what to call a proton, what a neutron, at one space-time point, one is then not free to make any choices at other space-time points.

It seems that this is not consistent with the localised field concept that underlies the usual physical theories. In the present paper we wish to explore the possibility of requiring all interactions to be invariant under independent rotations of the isotopic spin at all space-time points …
(Yang and Mills, 1954, quoted by Aitchison and Hey)​

SU(2) isospin (weak force):

For example, an "isospin" is associated with every particle which feels the weak force. The group of isospin transformations is SU(2), whose infinitesimal elements are of the form $T_{\mathbf{\eta}}\ =\ \mathbf{I}\ +\mathbf{\tau}\cdot\mathbf{\eta}$, for infinitesimal ordinary three-dimensional vectors $\mathbf{\eta}$.

This infinitesimal local isospin transformation $T_{\mathbf{\eta}}$, where $\mathbf{\eta}$ depends on position, adds a phase $g\,\mathbf{\tau}\cdot\mathbf{\eta}/2$ to the wave function of any particle with "weak charge" $g$ in three "weak fields" $\mathbf{W}^{\mu}\ =\ (W_1^{\mu},W_2^{\mu},W_3^{\mu})$ with covariant derivative $$D^{\mu}\ =\ \partial^{\mu}\ +\ \ ig \mathbf{\tau}\cdot\mathbf{W}^{\mu}/2$$:

$$\psi '\ =\ (1\ +\ ig \mathbf{\tau}\cdot\mathbf{\eta}/2)\psi$$

which must be compensated by locally changing both the covariant derivative and the three fields:

$$D'^{\mu}\ =\ (1\ +\ ig \mathbf{\tau}\cdot\mathbf{\eta}/2)D^{\mu}$$

$$\mathbf{W}'^{\mu}\ =\ \mathbf{W}^{\mu}\ -\ \partial^{\mu}\mathbf{\eta}\ -\ g\,(\mathbf{\eta}\times\mathbf{W^{\mu}})$$

SU(1) phase rotation (electromagnetic force):

The group of ordinary phase rotations is SU(1), the group of the rotations on a circle, whose infinitesimal elements are of the form $T_{\eta}\ =\ 1\ +\ i\eta$, for infinitesimal ordinary scalars $\eta$.

This infinitesimal local transformation, where $\eta$ depends on position, adds a phase $q\,\eta$ to the wave function of any particle with electric charge $q$ in an electromagnetic field $A^{\mu}$ with covariant derivative $$D^{\mu}\ =\ \partial^{\mu}\ +\ iq\,A^{\mu}$$:

$$\psi '\ =\ (1\ +\ iq \eta)\psi$$

which must be compensated by locally changing both the covariant derivative and the field:

$$D'^{\mu}\ =\ (1\ +\ iq \eta)D^{\mu}$$

$$A'^{\mu}\ =\ A^{\mu}\ -\ \partial^{\mu}\eta$$

(For finite $\eta$, $T_{\eta}\ =\ e^{i\eta}$, and $\psi '\ =\ e^{iq \eta}\psi$ and $D'^{\mu}\ =\ e^{iq \eta}D^{\mu}$)

U(1) x SU(2) (electroweak force):

The electroweak interaction is a combination of the electromagnetic and weak interactions.

It has four fields, $\mathbf{W}^{\mu}\ =\ (W_1^{\mu},W_2^{\mu},W_3^{\mu})$ and $B^{\mu}$.

The electroweak theory is complicated by the fact that it treats left-hand and right-handed helicities differently, by recognising an additional charge, the hypercharge, $y$.

$$\hat{D}^{\mu}\ =\ \partial^{\mu}\ +\ \ ig \mathbf{\tau}\cdot\hat{\mathbf{W}}^{\mu} \ +\ \ ig' y\hat{B}^{\mu}/2$$

U(1) × SU(2) × SU(3) (the standard model):

See http://en.wikipedia.org/wiki/Standard_model

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