The Higgs vacuum

Question

Srednicki's "Quantum Field Theory", an electronic copy of which is freely available here, seems to state on p 205 that the states eq. (32.3) which differ by a phase factor that can range through [0,2$\pi$) are mutually orthogonal. But if the underlying Hilbert space is separable this does not seem possible. Who can enlighten me?

Answer 1

For the benefit of others who read this, note that if $\mathcal H$ is a separable Hilbert space, then there exists a countable, orthonormal basis for $\mathcal H$. Notice that this does not immediately imply that there cannot exist an uncountable basis for $\mathcal H$, but this nonetheless turns out to be true as a consequence of the dimension theorem;

Let $V$ be a vector space, then any two bases for $V$ have the same cardinality.

See also the following math.SE posts:

1. https://math.stackexchange.com/questions/232166/showing-the-basis-of-a-hilbert-space-have-the-same-cardinality

2. https://math.stackexchange.com/questions/450106/uncountable-basis-and-separability

Now, for your question. Let's suppose that there are an uncountable number of orthogonal vacuua $|\theta\rangle$ in the Hilbert space where $\theta\in [0,2\pi)$, then we have the following possibilities

1. The Hilbert space of the theory is not separable. In this case, there is no contradiction.

2. The Hilbert space of the theory is separable. In this case, there is a contradiction, and we need a resolution.

As far as I am aware, most axiomatizations of QFT assume that the Hilbert space of the theory is separable, but there is discussion in the literature about relaxing this assumption. I'll attempt to dig up some references.

Let's therefore assume separability and look for a resolution. The standard resolution is that when constructing the Hilbert space of the theory, one chooses only one of these (physically equivalent) vacuua to be the vacuum of the Hilbert space, then one constructs the rest of the physical Hilbert space about this vacuum. The rest of the vacua are not elements of the Hilbert space of the theory.

There is another perspective on this which is interesting. Let's suppose that there is some larger, non-separable Hilbert space $\mathcal H_\mathrm{big}$ containing all of the vacua $|\theta\rangle$ and which is an orthogonal direct sum all of the Hilbert spaces $\mathcal H_\theta$ that could have been generated from each of the possible vacua and used as the physical Hilbert space of the theory.
\begin{align} \mathcal H_\mathrm{big} = \bigoplus_{\theta\in[0,2\pi)} \mathcal H_\theta \end{align} Then we view each of the Hilbert spaces $\mathcal H_\theta$ as a superselection sector of the larger Hilbert space $\mathcal H_\mathrm{big}$. In this case, if the physical system occupies a state $|\psi\rangle$ in a given superselection sector $\mathcal H_\theta$, then the state of the system will remain in the sector for all times under the Hamiltonian evolution, so we may as well view "the" Hilbert space of the system as simply the superselection sector it started in. In a sense, this is essentially the same as originally having picked a vacuum on which to build the Hilbert space because the different superselection sectors don't "talk" to each other.

The following physics.SE post is useful for understanding superselection sectors:

What really are superselection sectors and what are they used for?

I also found the following nLab page on superselection theory to be illuminating:

http://ncatlab.org/nlab/show/superselection+theory

Answer 2

Not 100% sure but here is an attempt to a solution:

1)First of all, it's not clear to me why the underlying Hilbert space needs to be separable. Is this mentioned somewhere in the book and is there a physical reason for this requirement?

2)Nevertheless, I will assume here that the Hilbert space of physical states needs indeed to be separable. A possible way out of the contradiction is the following: There is an uncountable number of solutions that minimize the potential but all of them are physically equivalent. You can imagine that these solutions live in some kind of space if you want to, but this is not the Hilbert space of physical states. Out of these solutions we arbitrarily pick one and define it as the true (physical) vacuum. But any choice is physically equivalent and all of them lead to the same (single) physical state.