Eigenstates of interacting and non-interacting Hamiltonian

  1. 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.
     
  2. jcsd
  3. 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
     
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