Eigenstates of interacting and noninteracting Hamiltonianby Quantum_Devil Tags: eigenstates, hamiltonian, interacting, noninteracting 
#1
Dec2707, 05:00 AM

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Hi!
Have multiparticle state of full Hamiltonian and oneparticle state of free Hamiltonian a nonzero 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
Dec2907, 05:49 AM

P: n/a

On Dec 26, 11:02 am, Quantum_Devil <quantumde...@wp.pl> wrote:
> Have multiparticle state of full Hamiltonian and oneparticle state > of free Hamiltonian a nonzero 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 selfadjointness. For example, if H and H+V (or rather, their selfadjoint 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 selfadjoint 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 spacetime; 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 selfadjoint 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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