## Why the name "Second Quantization"?

 Quote by George Jones I would like to participate, but I think and write *very* slowly,
No problem, me too.

 so I don't know how much this will happen. Regards, George

For example, you already have written a nice text in this thread that can serve our goals.

regards
marlon
 Mentor Physics Monkey and Marlon (and any others), you might interested in some ramblings of mine on sci.physics.research about the Dirac case. First this then this. I would like to find the time to clean these up, latex them, and post them on Physics Forums. Regards, George
 Recognitions: Homework Help Science Advisor Thanks for the reply, George, I can see it was just some confusion over terminology.

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 Quote by George Jones Actually, infinite-dimensional unitary irreducible linear representations of the universal covering group of the restricted Poincaré group. Regards, George

If it's unitary, it's no longer classical, by virtue of the Wigner theorem. The representation space is no longer a vector space, but a (rigged) Hilbert space which is typical for a quantum theory.

The group i was implying was $\mbox{SL(2,\mathbb{C})} \rtimes S$. It's interesting that "classical fields" are really nonunitary finite dimensional (ir)reducible representations of the $\mbox{SL(2,\mathbb{C})}$ group, but for the sake of completeness it's the universal covering group of the restricted Poincaré group (and not the restricted Lorentz group) which is important when passing from a classical to a quantum description of fields.

Daniel.
 I would like to acknowledge all of you for the answers and for the debate. DaTario

Mentor
 Quote by dextercioby If it's unitary, it's no longer classical, by virtue of the Wigner theorem. The representation space is no longer a vector space, but a (rigged) Hilbert space which is typical for a quantum theory.
Sorry for the late respopnse - I was (and still am to some degree) snowed under by work, and I have been trying to avoid Physics Forums.

I am having trouble unpacking the above sentences. Which Wigner theorem? Also, I read these sentences as saying that Hibert spaces aren't vector spaces, and (the triple of) space in a rigged Hibert spaces aren't vector spaces.

???

 The group i was implying was $\mbox{SL(2,\mathbb{C})} \rtimes S$. It's interesting that "classical fields" are really nonunitary finite dimensional (ir)reducible representations of the $\mbox{SL(2,\mathbb{C})}$ group, but for the sake of completeness it's the universal covering group of the restricted Poincaré group (and not the restricted Lorentz group) which is important when passing from a classical to a quantum description of fields. Daniel.
But all representations of the Poincare group lift to representations of its universal cover, so if you're saying that classical fields are representations of the Poincare group, then they're also representation of its cover. Of course, things don't work the other way round.

Unitarity is mathematical property that is very useful in quantum theory, but, as a mathematical property, it can also apply to classical representations.

In any case, what I'm interested in is the dimension of the representation. Classical fields are tensor fields, and tensor fields are modules over the ring of scalar fields. Addition conditions (e.g. wave equations) required by calssical fields restric themt to vector spaces, but I'm hard pressed to see how these spaces could be finite-dimensional.

As I hinted at in another post, sloppy physics books sometimes refer to finite-dimensionality, but this refers to the dimension of representations used to construct fields, not to the dimensionality of the actual representation spaces of which fields are members.

The easiest way for you to turn my into a believer in the finite-dimensionality of the spaces of classical fields would be to give an explicit example. In the interests of simplicity of presentation, the example doesn't have to a field that really exits, but it could correspond to reality.

Until I see an explicit example to the contrary, I will continue to believe and state that classical fields are elements of infinite-dimensional representation spaces.

Regards,
George