Eigenstates of interacting and non-interacting Hamiltonian

Quantum_Devil

Hi!

Have multi-particle state of full Hamiltonian and one-particle state
of free Hamiltonian a non-zero scalar product? Intuitively one can say
that scalar product of such states should be zero because each of
these states mentioned above belongs to different (orthogonal)
subspaces of the Fock space.
Do you know any reference discussing this problem?


Quantum_Devil.

Igor Khavkine

On Dec 26, 11:02 am, Quantum_Devil <quantumde...@wp.pl> wrote:

> Have multi-particle state of full Hamiltonian and one-particle state
> of free Hamiltonian a non-zero scalar product? Intuitively one can say
> that scalar product of such states should be zero because each of
> these states mentioned above belongs to different (orthogonal)
> subspaces of the Fock space.

This question is a bit tricky. See, in single particle quantum
mechanics (aka QM with a single degree of freedom), one often deals
with a given Hilbert space of states with position (x) and momentum
(p) operators defined on it, with the canonical commutation relations.
On the same Hilbert space, one is also given a Hamiltonian operator
(H) and sometimes a perturbation (V), both defined in terms of x and
p. Because most of the operators in question will be unbounded, one
has to be careful with questions of domain and self-adjointness. For
example, if H and H+V (or rather, their self-adjoint extensions) share
the same domain, then one could take two corresponding eigenvectors
and look at their scalar product, since they both would be in the same
common domain and in the same ambient Hilbert space.

In quantum field theory (aka QM with infinitely many degrees of
freedom), on the other hand, the situation is more complicated. Often,
the "perturbation" V is so singular, that its impossible to have both
H and H+V extended to self-adjoint operators on the same Hilbert
space. In that case, eigenstates of H and H+V would live in separate
Hilbert spaces and it wouldn't make much sense in taking their inner
product. Alternatively, one could just adopt the convention that any
two vectors belonging to separate Hilbert spaces should be orthogonal.

It's worth remarking what the phrase "separate Hilbert spaces" often
encountered in physics literature actually means. Mathematically, all
separable Hilbert spaces are isometric to each other. However, in QM,
one implicitly considers a Hilbert space together with x and p
operators satisfying canonical commutation relations (in field theory,
these are actually operator valued distributions, assigning a "field
operator" to each point in space-time; in field theory, the field
values are the dynamical variables, instead of position and momentum).
So, it is not always possible to transform some set of canonically
commuting x and p operators into another set of canonically commuting
operators x' and p', using a unitary transformation or an isometry.
These are called unitarily inequivalent representations of the
canonical commutation relations, often referred to as "separate/
distinct Hilbert spaces" instead.

The operators H and V are always defined in terms of x and p. So, for
a particular representation of the canonical commutation relations, H
may be extended to a self-adjoint operator, while H+V may not, or vice
versa. Unfortunately, that's precisely the situation when H is the
free field Hamiltonian (which is well defined in the Fock
representation) and V is a local interaction; H+V is not well defined
unless one uses a representation different from the Fock one. That is
the content of Haag's theorem.

Another way to look at this question is from the point of view of
regulated (or truncated) field theory. For example, when field theory
is discretized on a finite lattice, it becomes a system with a finite
number of degrees of freedom. Then both H and H+V will be defined on
the same Hilbert space (lapsing back into physics terminology :-). The
inner product between respective eigenstates of H and H+V can then be
examined in the limit of the lattice size and lattice density going to
infinity. If Haag's theorem applies in this limit, then the inner
product should go to zero in the same limit.

> Do you know any reference discussing this problem?

You may want to look at Haag's book _Local Quantum Physics_. I'm sure
there are more references if you have more specific questions.

Hope this helps.

Igor