Latent Symmetry

The general Lagrangian gauge invariant under a group $G$ involving both bosonic $(\phi)$ and fermionic $(\psi)$ fields can be written as

$${\cal L} = - \frac{1}{4} \vec{F_{\mu\nu}} \vec{F^{\mu\nu}... ...,D_{\mu}\psi - \bar{\psi} (m_{\psi} + g \phi) \psi -V(\phi)$$ (228)

Here $V(\phi)$ is some selfinteraction (renormalizable, we hope) of the field $\phi$ and $D_{\mu}$ is the gauge covariant derivative

$$D_{\mu} \equiv \partial_{\mu} - i g T_{k} W_{\mu k}$$ (229)

with $T_{k}$ the generators of the gauge group $G$ and $W_{\mu k}$ the gauge fields which define

$$F_{k}^{\mu\nu} = \partial^{\mu}W_{k}^{\nu} - \partial^{\nu}W_{k}^{\mu} +g C_{k\ell m} W_{\ell}^{\mu} W_{m}^{\nu}$$ (230)

where the gauge group structure constant $C_{k\ell m}$ appear.

The Lagrangian (230) does not contain a mass term for the gauge field because a term of this type like

$${\cal L}_{m} = \frac{1}{2} m_{k\ell} W_{k}^{\mu} W_{\mu \ell}$$ (231)

with $m_{k\ell}$ being a mass matrix, is not invariant under the gauge transformation

$$\delta W_{k}^{\mu}(x) = -\frac{1}{g} \partial^{\mu}\alpha_{k}(x) - C_{k\ell m} \alpha_{\ell}(x) W_{m}^{\mu}$$ (232)

On the other hand, a mass term as (233) destroys the renormalizability of the theory as it was stressed in connection with the hypothesis of the intermediate vector boson.

Nevertheless, being the objective the formulation of a theory of the weak interactions, if the gauge bosons are going to play the part of mediators of this interaction, they should necessarily be massive to account for the short range of those forces.

The solution of this dilemma related to the mass of the gauge fields is in the ``latent" symmetry concept. It is found that the mass of the gauge fields can be induced by a breaking of the symmetry as long as it remains somehow latent (or hidden).

A latent, hidden or spontaneously broken symmetry is a symmetry present in the Lagrangian of the theory but which is not respected by the vacuum expectation values of the field. That is to say, it is a symmetry whose realization is in some sense $\grave{a}\; la$ Nambu-Goldstone. However, being related in this case to gauge invariance, starts the so called Higgs mechanism.

The notion of latent symmetry is already found in the level of quantum (or even classical) mechanics as there is no reason beforehand for an invariance of the Hamiltonian of a system to be also an invariance of its ground state.

The concept of latent symmetry has non trivial consequences in systems of infinite extension.

Let us take for example the case of a magnet. The typical Heisenberg Hamiltonian of exhange interactions to model that system is

$$H = - K \sum_{<i,j>} \vec{s}_{i}\cdot\vec{s}_{j}$$ (233)

which posses rotational invariance. Nevertheless, the system ground state, for low enough temperatures, corresponds to the spins aligned in a particular direction. That is to say, the ground state breaks the rotational symmetry. But, on the other hand, the Hamiltonian rotational symmetry is latent in the sense that it cannot be predicted beforehand in which of all the possible directions will the spins align. This is due to the fact that any direction corresponds to the same energy and therefore a rotation leads from an asymmetric solution to a degenerate one. The necessity of an infinite system lies in the fact that only in that case there will be a infinite moment of inertia and besides the the spacing between leveles of different angular impulse $J$ will be zero. In other words, it is necessary to suppose infinite $J$ to be able to have any particular direction and all of them corresponding to the same energy. In this way the latent symmetry will then be done.

It must be noticed that when a latent symmetry, as the one in the example, appears, there exists in general a critical point for the temperature $T=T_{c}$, below which the ground state becomes degenerate.

In order to generalize the concept to quantum field theory we speak of internal symmetries of the corresponding Hamiltonian and vacuum state instead of ground state.