Understanding Superselection: Rules & Interactions with Environment

[Imparcticle]

What is superselection?

I have only learned that "superselection rules are induced by interactions with the environment." Can someone eleborate?

 

[humanino]

Superselection as I have been using it refers to rules stating the impossibility to superpose specific states. For instance, it is impossible to have a superposition of boson and fermion states, or to superpose different electric charge states. This is in contrast with the fundamental linearity of QM. Also, the impossibility to have macroscopic objects superposed into spatially separated states, or localisation of macroscopic systems, is also related to superselection, via the decoherence principle.

It is often argued that there are two types of superselection, referred to as hard and soft. Hard superselection applies for instance for the charge, the mass, or the spin, and can rigorousy be demonstrated from symmetry considerations. Soft superselection refer to a dynamical process, indeed induced by environment, and has applications for instance in solid state physics. The interaction with the environment is introduced by a random noise term in the Hamiltonian, yielding a disturbance or phase randomization, which in turn provokes decoherence. This is basically a transition from quantum to classical probability, due to non-diagonal elements of the density matrix averaging to zero whereas the diagonal elements do not. Non-diagonal elements are scalar product between states having different random phases, whereas the (identical) states in the scalar product for the diagonal elements have a coherent (the same) phase and do not average to zero.

"Lectures on decoherence", A. Armour

 

[seratend]

Have you got a good demonstration on a hard superselection case (charge, mass or spin)? It will be a pleasure to know such one :shy: . The demos (I know) have always introduced at the final step an argument that is external to the theory and seems not to be required (thus unvalidating the demo).

The soft superselections are almost always used in texts as a short cut to explain the result of a measurement in QM (like the selection of a preferred basis in the decoherence model of measurement - subject that is currently under investigation). I don't konw any soft superselection rule (at least that seems serious), so rigorous demonstration for such ones are also wecome as they can help to understand the measurement process in the quantum theroy.

 

[humanino]

Nobody else answered... I am not home, I don't have access to my books, but I thought it was basically the same as in Schur's lemma. We have a Hilbert space that is the sum of several representations of a C*-algebra, and each sector has different charges. Googling gave me that

John Baez[/URL]]You can think of mathematicians as physicists who only do trivial
observations. So the C*-algebra of observables for a mathematician
consists only of scalar multiples of the identity. These are the
observables that don't depend on anything about the state of the
universe! This little C*-algebra forms a copy of the complex numbers
sitting in the center of the larger C*-algebra of observables used by
the physicist. The only laws of physics the mathematician can express
are trivial ones like 1 + 1 = 2, which involve observables living in the
mathematician's C*-algebra.

When the physicist's C*-algebra has a nontrivial center, things get a
little more interesting: we have the C*-algebra of the mathematician,
the C*-algebra of the physicist, and the center of the latter algebra,
which we could call "the C*-algebra of the classical physicist". The
classical physicist can ignore noncommutativity, but is restricted to
talking about very special things - observables that commute with all
others. Such quantities must be conserved and Lorentz-invariant, for
starters! Famous examples include the total electric charge of the
universe, or the total lepton number, or the total baryon number - in
models where these quantities are conserved.

I did not manage to find anything else relevant for you.