Hidden gauge symmetry in spontaneously broken phase?

Question

Throughout this post I will be using Weinberg's notation. When a global symmetry group $G$ is spontaneously broken down to a subgroup $H$, it is often useful to reparametrize whatever fields that make up your theory in terms of fields which better reflect the low energy degrees of freedom. This reparameterization is often referred to as the coset construction of the low energy effective action.

In the coset construction, there a few objects that can be constructed, which ensure that whatever action you write down with them will remain invariant under the full group $G$, even though it is only manifestly invariant under $H$. In the interest of making this question compact, I won't go through all of the justifications of the construction, but simply state the results that are relevant.

In the end, you will arrive at three objects: $H$-matter fields $\tilde{\psi}$, a notion of an $H$-covariant derivative $\mathcal{D}_{\mu}=\partial_{\mu}+it_iE_{i\mu}$, where $t_i$ are the generators of $H$, and $E_{i\mu}$ is built out of the goldstone fields, and $D_{a\mu}$ fields which encode the derivative of the goldstone fields in an $H$-covariant way. The theorem is that any $H$ invariant lagrangian built out of $\tilde{\psi}$, $\mathcal{D}_{\mu}\tilde{\psi}$, and $D_{a\mu}$ is invariant with respect to $G$.

In this construction, we arrive at $E_{i\mu}$ which looks very much like an $H$-gauge field, in that it transforms inhomogeneously with respect to $g\in G$ $$t_iE_{i\mu}\to h(\xi,g)\big(t_iE_{i\mu}\big)h^{-1}(\xi,g)+i(\partial_{\mu}h(\xi,g))h^{-1}(\xi,g)$$ Where $h(\xi,g)\in H$ depends on the Goldstone fields $\xi_a(x)$. For completeness, here are the other transformations under $g\in G$

$$\tilde{\psi}\to h(\xi,g)\tilde{\psi}$$ $$x_aD_{a\mu}\to h(\xi,g)\big(x_aD_{a\mu}\big)h^{-1}(\xi,g)$$

With $x_a$ the broken generators. Essentially, the global $G$ invariance "looks" like an $H$-gauge symmetry (but it is not).

Although the original symmetry (global $G$) is not a gauge symmetry, everything is constructed in a way so that if we instead transform all of the fields with an $H$-gauge transformation

$$t_iE_{i\mu}\to h(x)\big(t_iE_{i\mu}\big)h^{-1}(x)+i(\partial_{\mu}h(x))h^{-1}(x)$$ $$\tilde{\psi}\to h(x)\tilde{\psi}$$ $$x_aD_{a\mu}\to h(x)\big(x_aD_{a\mu}\big)h^{-1}(x)$$

The action is invariant! My question is, is this a real symmetry? Are there real consequences of this? Or is this just some artifact that doesn't mean anything?

Answer 1

This is a verbose placeholder for an answer, as Weinberg himself in his QTFvII, Ch 19.5, p 195 et seq, beats the SU(2)×SU(2) σ-model of Gell-Mann and Levy to a pulp. For simplicity, you may eliminate the σ, and thus move on an O(4)/O(3) hypersphere parameterized by three projective coordinates, his Goldstone πs, or ζs, a manifold manifestly isospin invariant (your H~O(3)).

(19.5.9), (.11, .12, .14, .15, .16, .18, .39-.49) take you exactly where you want to go.

All quantities, including the artfully defined covariant derivatives, transform like isovectors and isospinors globally, and (Weinberg's discovery, ref 25) also locally by particular , not arbitrary, field-dependent-gauge local transformations. That is, the three broken axial transformations are hidden into specific field-dependent isorotations and are thereby automatically accounted for! The point is, if you leave the gauge field jazz and rhetoric aside for a moment, gauge fields are a completion/machine ensuring that covariant derivatives of fields transform linearly, despite their spacetime dependence.

The gauge field itself is a red herring, once a covariant derivative transforming homogeneously is at hand. (But if you were hyperfocussed on it, it of course follows the streamlining of Weinberg, CCWZ II where it was introduced at the end.)

Without introducing new superfluous degrees of freedom, Weinberg crafts covariant derivatives which transform linearly under these three SSBroken axial transformations (cosets), just like the unbroken isorotations! He never needs to introduce "real" gauge fields with superfluous degrees of freedom, and you never need to fix a gauge, or wonder about unphysical states. This is a theory of three pions and isospinor fermions.

In summary, this is just a compact, elegant rewriting of the conventional σ-model, emphasizing geometrical features, and automatically accommodating the SSB, which leads to dozens of physical consequences and constraints, of course. Forming your isoscalar effective Lagrangian using nothing but these covariant derivatives, you automatically, almost magically, achieve axial invariance as well.

I'll give you the bare essentials using π as Weinberg's ζ, so it is dimensionless here, having absorbed the pion decay constant F, and leaving the matter fields to you and Weinberg.

The hypersphere O(4)/O(3) is $$ \vec \phi^2+ \sigma^2=1, \qquad\leadsto \qquad \sigma=\pm\sqrt{1-\vec\phi^2}. $$ You change to projective Goldstone coordinates, $$ \vec \phi =\frac{2\vec \pi}{1+\pi^2}, ~~~~ \sigma= \frac{1-\pi^2}{1+\pi^2}, $$ which, of course, satisfy the above hypersphere constraint (check this).
The three vector generators of isospin yield $$ \delta \vec \phi= \vec \theta\times \vec \phi , ~~~\leadsto ~~~ \delta \vec \pi= \vec \theta\times \vec \pi, $$ while the three coset space SSBroken axials yield $$ \delta \vec \phi = \vec \epsilon \sigma, ~~ \delta \sigma =- \vec \epsilon \cdot \vec \phi , \leadsto \\ \delta \vec \pi = \vec \epsilon (1-\pi^2)/2 +(\epsilon \cdot \pi )\vec \pi , $$ all of them preserving the constraint, and the bare kinetic term $$\partial_μ\vec \phi\cdot \partial_μ \vec \phi + \partial_μ \sigma \partial_μ \sigma=4\frac{\partial_μ \vec \pi \cdot \partial_μ \vec \pi }{(1+\pi^2)^2}\equiv \vec{ D}_μ \cdot \vec{ D}_μ .$$ So, Weinberg, "amazingly", discovered that the "pion covariant derivative" $\vec{ D}_μ\equiv 2 \partial_μ \vec \pi /(1+\pi^2) $ is a global isovector, of course, but it also transforms "locally" as a "local" isovector under the particular field-dependent axial transformation, $$ \delta \vec{ D}_μ= (\vec \epsilon \times \vec \pi ) \times \vec{ D}_μ, $$ so this kinetic term is also axially invariant (of course: it came out of a mere change of variables!) by dint of the special gauge invariance it was designed to possess... The upshot is that pion functions in the effective Lagrangian are automatically invariant if only all derivatives are covariant, and isospin invariance is imposed.

Since the 60s, Bando, Kugo, et al. have extended and generalized these ideas into a heuristic phenomenology of the KSFR relation, but it is suited to index-happy formalists.