Nice. I will have a look into your work. But how does this relate to the question?

Philadelphia, PA

Dear Passon, 

You wrote:

Quantum Field Theory has the known problem of (unitary) in-equivalent representations.

--end quotation

I am trying to get a better grasp on your question, and I found that there is a lengthy discussion of the matter in the Stanford Encyclopedia of Philosophy, under the heading of "Quantum Field Theory." 

http://plato.stanford.edu/entries/quantum-field-theory/

Much of this depends on the details of the differences between QM and QFT, and it seems clear that QFT aims to include the quantum mechanics of both particles and fields. It might help to clarify what is meant by "representations." I take it that this is parallel to or a development/application of the Schrodinger equation.

The following paragraph from the Stanford paper struck me as interesting, and perhaps a key to understanding what is at stake:

The coexistence of UIRs can be readily understood looking at ferromagnetism (see Ruetsche 2006). At high temperatures the atomic dipoles in ferromagnetic substances fluctuate randomly. Below a certain temperature the atomic dipoles tend to align to each other in some direction. Since the basic laws governing this phenomenon are rotationally symmetrical, no direction is preferred. Thus once the dipoles have “chosen” one particular direction, the symmetry is broken. Since there is a different ground state for each direction of magnetization, one needs different Hilbert spaces—each containing a unique ground state—in order to describe symmetry breaking systems. Correspondingly, one has to employ inequivalent representations.

---end quotation

What is your take on this? Why, in the first place would there be "different ground states" for each direction of magnetization? Explaining something of the assumed relationships here, between directions of magnetization, ground states and Hilbert spaces might help the present question along. 

I'd be much interested in your comments.

H.G. Callaway

Dear Callaway, thanks for this remark and the quote! Very nice. I will have to look into that, best regards, op

Philadelphia, PA

Dear Passon, 

Thanks for your reply. 

Thinking about the quotation a bit, I am beginning to suspect that the "different ground states" differ only in "sign"--corresponding to the differing possible orientations, as the poles of a magnet differ by "north" and "south." If that is anywhere near right, then the different Hilbert spaces might conceivably be combined into one more comprehensive Hilbert space. If so, that might be interesting, because it suggest that the "inequivalent representations" may simply be regarded as aspects or developments of a more comprehensive representation --which is resolved once the rotational symmetry is broken. It seems, in any case, that some representation of the unbroken symmetry is needed to start, in making sense of this example. 

However, I'm not the mathematician here. I am thinking purely by analogy.

H.G. Callaway