# Superselection sectors as finite lattice limit

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if the continuum theory is in any sense a limit of a lattice theory, how can  a non-zero VEV ever emerge during this "taking continuum limit" process?

I had answered: Probably by inventing suitable projectors to subspaces of the lattice Hilbert spaces where the limiting states are well-defined and give the expectation in an appropriate superselection sectors. He then asked:

but at what stage of the limit taking does one introduce such projection?

edited Jan 14, 2016

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I haven't seen it done and cannot tell precisely how to do it. But here is an overview of what I know about the context (and at the end a guess of an answer to your question):

Nonperturbative quantum field theory is often done approximately on finite lattices with periodic boundary conditions . This simplifies many things and leads to computational tractability of certain QFT problems that are not accessible to renormalized perturbation theory - at the cost of hiding other features of QFT.

In particular, since asymptotic states have no meaning, there are neither mathematically well-defined scattering processes nor superselection sectors. Unlike in real QFT, the mass spectrum is discrete (excluding scattering states).

Asymptotic states need unbounded space to exist. Thus to obtain them in a lattice setting one needs to perform an infinite volume limit. Mathematically, this is very difficult, and has been successfully performed only in space-time dimensions $<4$. For example, the rigorous lattice approach of Balaban to nonabelian gauge theory is complete except for this last step, which nobody knows how to perform. This is one of the reasons why Yang-Mills quantization was selected as one of the Millennium problems.

In the infinite volume limit, all problems resurface that were swept under the carpet by reducing quantum field theory to a system with finitely many degrees of freedom. For scattering theory one needs to know the asymptotic particle content (aka bound states). Getting these wrong and substituting them by the Lagrangian particle content gives rise to severe infrared problems. Finite lattices allow one to obtain reasonable approximations to the bound state spectrum through calculations in imaginary time, but scattering needs more, and I do not know how to treat it in a lattice context.

Other infrared phenomena develop because of non-unique asymptotic conditions in the infinite volume limit. One doesn't have this when all physical particles are massive - then the asymptotic fields decay sufficiently fast at infinity sufficiently to make all relevant integrals well-behaved. But when there are massless physical degrees of freedom, there are physically inequivalent asymptotic conditions for the massive fields, referred to a superselection sectors. Moreover, the massive particles in the theory are then not particles of the kind discussed in QFT textbooks but so-called infraparticles. These two phenomena are related in some poorly understood way.

I am concentrating in the following on the superselection structure since this determines the possible representations of the observable algebra. Each superselection sector carries a different (i.e., inequivalent) such representation. In particular, representations in which a symmetry of the (often bigger) field algebra is broken are labelled by order parameters, and different order parameters correspond to different superselection structures.

So the question is how to retrieve one of these by imposing conditions on a sequence of periodic lattices whose volume tends to infinity. Clearly one must break the symmetry involved in the periodicity, and take the limit in a piece of the volume whose size is much smaller than the periods, but still becomes arbitrarily large. On the boundary of this piece one can specify boundary conditions that have a sensible infinite volume limit characterizing the superselection sector. The projection I was talking of is the projection that restricts the Hilbert space to states referring to this small volume only and satisfy the additional boundary conditions.

I cannot tell whether one can carry this out. But the setting seems to me plausible enough to merit consideration.

answered Jan 12, 2016 by (15,468 points)

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