Superselection sectors and $p+1$ form potentials in String Theory

Question

we often come across the phrase superselection sector. I have couple of questions from that.

First of all, I learned from some references about it (as a overview) and how is it related to superselection rules etc. There is of course this mathematical definition (from C* algebra, unfortunately I haven't gone through that), but I wanted to understand it physically. It seemed to me that basically if I construct two sectors (two hilbert spaces?) whose basis vectors don't cohere, then they belong to two different superselection sectors. My first question will be : Is that it or there are more into it than meets the eye! Please.

Secondly, I encountered it in String theory. I will probably mention the place (I have an unrelated question here too!) where I encountered it in string theory. Its the argument using which we can guess the presence of D branes from SUGRA action. We see that in the low energy ST action, there can be a term (some p+1 form potential, properly coupled etc.) Now I saw this argument that "this term has finite tension and hence its not finite energy excitation above the vacuum" (why? and how did we guess about finite tension?- Is it because of the very fact that this term exists in the ST action?) and also its suggested that these objects (multi form potentials) are in different superselection sector. So again, please let me know if there is any general comment on superselection sector that you want to make in this context and also about the arguments.

Answer 1

A superselection sector $H_S$ is one of the subspaces of the "total Hilbert space" $H$ such that one can prove that the transition amplitude from any state in $H_S$ into any state in another superselection sector $H_{S'}$ vanishes.

So in principle, one can divide the Hilbert space $H$ of an ordinary theory to superselection sectors with different eigenvalues of conserved quantities. We know that it's conserved so we can never get a state vector with a different values. However, that's not the dominant interpretation of the superselection sectors in particle physics because the sectors with a bigger energy or angular momentum can still contain regions with a smaller or different value of these quantities - plus another separated object that compensates it.

In particle physics, the superselection sectors are usually reserved for subspaces of the total Hilbert space whose states can't evolve into each other because it would require an infinite amount of energy than work. So typically, superselection sectors in quantum field theory or string theory refer to subspaces of the Hilbert space that correspond to different boundary conditions at infinity (in the space - inside the spacetime).

One would need to change an "infinite portion" of the space to switch from one sector to another.

So any "finite energy excitation" of one vacuum - with some values of the low-energy parameters - automatically belongs to a different superselection sector than a "finite energy excitation" of another vacuum - with another value of the low-energy parameters (such as the cosmological constant, particle masses, and couplings). Even in quantum field theory, one may find several stationary points of a field, e.g. a scalar field. It most of the spacetime find this field near one place, it's automatically a different sector than if the place is different.

Adding localized particles can't change the superselection sector in this sense.

However, if you add (stable) infinite branes whose total mass is infinite, you surely do change the superselection sector. The waves can propagate along the branes but the branes are attached to "infinity" at some loci and a finite being can't ever change it. Infinite branes or fluxes that are changed otherwise across the infinite space is the only way how I can imagine that your points about the superselection sectors and form fields are related.

In quantum cosmology, the division into superselection really disappears because any realistic Universe, regardless of the size, may be produced out of "almost nothing" - e.g. in inflation. But in many contexts, we study asymptotically flat, anti de Sitter, or other infinite Universes and the superselection sectors are sharply separated.

Answer 2

Glance through the first few pages of the book "PCT, Spin and Statistics, and all that". You can read the section on superselection in the amazon preview. And at $20 used, you should have a copy.

Quantum mechanics is about interference. Stuff that is in different superselection sectors cannot be mixed. An example is a quark and an electron. As the PCT book states, you can write down a mixture using math, but you cannot produce it in the real world. So superselection sectors are a restriction on the Hilbert space.