Re: please help my confusion about particles and irreps.

Eugene Stefanovich wrote:
> You confusion is quite understandable, because most QFT textbooks
> don't do a good job in explaining how single particles are treated in
> QFT. The correct logic is as follows:
>
> 1. Single particle states live in Hilbert spaces corresponding to
> infinite dimensional irreducible unitary representations of the
> two-fold cover of the Poincare group.

To resolve both the issues raised in article preceding this: the
fundamental systems ("elementary particles or whatever you want to call
them") are the irreducible representations of the *combined* group: G x
Poincare'. I raised the issue of U(1) x Poincare', by the way, a short
ways back. Infraparticles are non-trivial representations which can't
be gotten at through the usual Wigner classification of Poincare'.

The reason the representations of the Poincare' part are
infinite-dimensional is because the group is not compact or even
semi-simple. Essentially, it has copies of the Heisenberg algebra
([p,q]=z;[p,z]=0=[q,z]) buried within it, which cannot have a finite
dimensional representation.

In particular, it's the [K,P] commutators which are the culprit, giving
you:
[K_i,P_j] = M delta_{ij}
where M = m + H/c^2 is the relativistic mass, m the rest mass (H the
kinetic energy). This generalizes the commutator for the (non-trivial
extension) of the Galilei group where you have
[K_i,P_j] = m delta_{ij}
which the Heisenberg relations ultimately come out of.

(In the relativistic case, m can be reabsorbed into the energy generate
by defining the total energy E = mc^2 + H, but that's an aside here).

So the particle space is really the product of the space for Poincare'
and a space for the group G, the latter being finite dimensional.

> 3. Systems with variable number of particles (the kind of systems we
> are dealing with in QFT) are described in the
> Fock space which is a direct sum of n-particle Hilbert spaces where
> n varies from 0 to infinity. In the Fock space one can define
> particle creation and annihilation operators, and each operator
> in the Fock space can be conveniently written as a function of
> these creation and annihilation operators.

For quantum fields, in the order of logical precedence: the field
operators come first, and the state spaces are defined therefrom. For
fields, the operators give you a system of infinite number of degrees
of freedom, and the Fock space is NOT the only representation space.

In general, Fock space is uniquely defined as the space where the
particle number is finite. It's the state space spanned from the
0-particle state by a finite number of creation operators applied to
it.

Non-Fock spaces also exists, each essentially having an infinite number
of particles (which is ok, since it's over an infinite space).

The reason this is pointed out is...

> 4. In order to describe dynamics (reactions, scattering, decays, etc.)
> one must define an "interacting" unitary representation of the
> Poincare group in the Fock space[...]

Haag's Theorem specifically precludes the existence of an non-trivial
interaction over Fock space! The interaction picture must necessarily
reside outside of Fock space in one of the non-trivial representations.

One of the reasons for the Algebraic Approach (of von Neumann, Haag,
et. al.) is to get away from the issue of representations working with
the field quantities, themselves -- exclusively as much as possible.

.

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