Phase Invariance

It is the simplest case: one parameter group symmetry. Given the Lagrangian of a complex scalar field

$${\cal L}(\phi,\partial_{\mu}\phi) = \partial_{\mu}\phi^{\ast} \partial^{\mu}\phi -m^{2} \phi^{\ast} \phi$$ (149)

invariant in front of the phase transformation, corresponding to the group $U(1)$

$$\phi \rightarrow \phi' = e^{i \alpha} \phi$$ (150)

which for the infinitesimal case it is reduced to

$$\phi \rightarrow \phi' \simeq(1 + i \alpha) \phi$$ (151)

so that

$$\delta \phi = i \alpha \phi$$ (152)

$$\delta \phi^{\ast} = - i \alpha \phi^{\ast}$$ (153)

The Noether theorem implies the conservation of the current

$$J_{\mu} = i [\partial_{\mu} \phi^{\ast} \phi - \phi^{\ast} \partial_{\mu}\phi]$$ (154)

which allows to define a conserved charge

$$Q = \int d^{3}x J_{0} = constant\; \Rightarrow \; \dot{Q} = 0$$ (155)

which is the ``charge" of the field.

Real fields can be introduced through the transformations

$$\phi_{1} = \frac{\phi +\phi^{\ast}}{\sqrt{2}}$$ (156)

$$\phi_{2} = \frac{\phi -\phi^{\ast}}{i \sqrt{2}}$$ (157)

in such a way that the Lagrangian is separated as

$${\cal L} = {\cal L}_{1} + {\cal L}_{2}$$ (158)

with

$${\cal L}_{k} = \frac{1}{2} \partial_{\mu}\phi_{k} \partial^{\mu}\phi_{k} - \frac{1}{2} m^{2} \phi_{k}^{2}\;;\;(k=1,2)$$ (159)

which is the Lagrangian of a real scalar field.

The phase transformation implies in this notation

$$\phi_{1} \rightarrow \phi'_{1} = \cos \alpha \phi_{1} - \sin \alpha \phi_{2}$$ (160)

$$\phi_{2} \rightarrow \phi'_{2} = \cos \alpha \phi_{2} + \sin \alpha \phi_{1}$$ (161)

that is to say, a rotation in the plane $(\phi_{1},\phi_{2})$.

The phase invariance under (152) means that the fields are not physically distinguishable. One can call $\phi_{1}$ or $\phi_{2}$ to any pair of orthogonal combinations of the original fields. This fact indicates the existence of a global symmetry. It is called global because the change of phase of the fields is the same for all the points $x$ of the space-time.

This idea of global symmetry can be extended to more complicated phase transformations, with group structure depending on more parameters. For example the case of the isospin, that is to say a symmetry group $SU(2)$.

In this case the particle fields can be grouped in multiplets and the phase transformation is, instead of (152), of the form

$$\psi(x) \rightarrow \psi'(x) = e^{i \theta_{a} t_{a}} \psi(x)\;;\;a=1,2,3$$ (162)

$t_{a}$: matrix of appropriated dimension characteristic of the symmetry group $SU(2)$.

$\theta_{a}$: constant parameters.