# Fell's Thm and Haag's Thm: How do we know there are interactions?

1. ### David Baker

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I've been thinking recently about a weird puzzle in algebraic and
constructive QFT. Namely, when one combines the conclusions of two
central theorems, one arrives at a seemingly absurd result: that free
and interacting quantum fields are experimentally indistinguishable.

Fell's theorem implies that any state can be approximated arbitrarily
closely (in the weak*-topology) by Fock states. Measuring the
expectation values of any finite set of observables in the C*-algebra
of observables, up to some finite degree of accuracy, will only
determine the state to within a weak*-neighborhood.

This would seem to imply that no finite number of experiments can
determine whether a given state is a Fock state or not.

Haag's theorem implies that no interacting state is in the same folium
as any free state, and therefore that no interacting state is a Fock
state.

So, if (by Fell's Thm) we can't rule out the possibility that any
given state is a Fock state, then (by Haag's Thm) we can't rule out
the possibility that any given state is a free/noninteracting state.

Something has to be wrong with this reasoning, but what?

2. ### djschubes@yahoo.com

0
[SOLVED] Fell's Thm and Haag's Thm: How do we know there are interactions?

On Jul 30, 8:59 am, David Baker <djba...@princeton.edu> wrote:
> I've been thinking recently about a weird puzzle in algebraic and
> constructive QFT. Namely, when one combines the conclusions of two
> central theorems, one arrives at a seemingly absurd result: that free
> and interacting quantum fields are experimentally indistinguishable.
>
> Fell's theorem implies that any state can be approximated arbitrarily
> closely (in the weak*-topology) by Fock states. Measuring the
> expectation values of any finite set of observables in the C*-algebra
> of observables, up to some finite degree of accuracy, will only
> determine the state to within a weak*-neighborhood.
>
> This would seem to imply that no finite number of experiments can
> determine whether a given state is a Fock state or not.
>
> Haag's theorem implies that no interacting state is in the same folium
> as any free state, and therefore that no interacting state is a Fock
> state.
>
> So, if (by Fell's Thm) we can't rule out the possibility that any
> given state is a Fock state, then (by Haag's Thm) we can't rule out
> the possibility that any given state is a free/noninteracting state.
>
> Something has to be wrong with this reasoning, but what?

My own understanding of this is shaky, so I'd be happy if someone else
would correct me or clarify. But I believe you are confusing the
approximation by Fock states with approximating the dynamics with free
dynamics.

I take Haag's theorem to mean that the free Hamiltonian isn't defined
on the interacting Hilbert space....i.e. the interacting states can't
be expressed as a superposition of free particle states. But these
free particle states can still be considered as abstract states that
give an expectation value to observables in the C* algebra, even
though they aren't in the folium. An interacting state can be
'approximated' by these abstract states in the sense of Fell's
theorem, but that doesn't mean that the interacting states obey the
dynamics of the free Hamiltonian.