[Quick] States of interacting and non-interacting Hamiltonian

  • #1
Eigenstates of interacting and non-interacting Hamiltonian

Have multi-particle state of full Hamiltonian and one-particle state of free Hamiltonian 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?

P.S. these states are eigenvectors of appropriate Hamiltonians
 
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Answers and Replies

  • #2
olgranpappy
Homework Helper
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Have multi-particle state of full Hamiltonian and one-particle state of free Hamiltonian non-zero scalar product?
The multi-particle state of the full hamiltonian can be expanded as a sum of free single-particle and many-particle states with various coefficients. Are any of the coefficient in front of the free single-particle states non-zero?

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.
if that were the case then the scalar product would obviously be zero. but are you sure that is the case? If they were eigenvectors of *the same* operator with different eigenvalues, then they would certainly be orthogonal, but they are eigenvectors of different operators.

Do you know any reference discussing this problem?

P.S. these states are eigenvectors of appropriate Hamiltonians
 
  • #3
Are any of the coefficient in front of the free single-particle states non-zero?
I don't know. Therefore I asked question in the first message.
 
  • #4
strangerep
Science Advisor
3,168
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Have multi-particle state of full Hamiltonian and one-particle state of free Hamiltonian 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.
[These states are eigenvectors of appropriate Hamiltonians]

Do you know any reference discussing this problem?
[This is definitely not a "quick" discussion topic. :-)] If you're talking about full multiparticle
interacting QFT in 4D, the Fock space constructed by polynomials of (free) creation operators
acting on a vacuum state is disjoint from the Hilbert space corresponding to eigenstates
of the full interacting Hamiltonian. The latter Hilbert space is not merely a subspace of
the free Fock space, but a completely disjoint space (hence scalar products between states
from the two spaces are conventionally defined to be zero).

If you search for "Haag's theorem" and "unitarily inequivalent representations" that
should turn up more info on this, at least in the context of orthodox formulations of QFT.
 
  • #5
I dont' actually understand the notion "full multiparticle
interacting QFT in 4D" introduced by You and the difference between it and "full interacting Hamiltonian". (Maybe the reason is that I'm not native English speaker).
Could you explain it in more detailed way or give me some references?

By "full interacting Hamiltonian" I don't mean interaction term, of course.
 
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  • #6
strangerep
Science Advisor
3,168
1,013
I dont' actually understand the notion "full multiparticle
interacting QFT in 4D" introduced by You and the difference between it and "full interacting Hamiltonian". (Maybe the reason is that I'm not native English speaker).
Could you explain it in more detailed way or give me some references?

By "full interacting Hamiltonian" I don't mean interaction term, of course.
I'll write [itex]H_0[/itex] to mean the free Hamiltonian. That is, the Hamiltonian in the
absence of interactions.

I'll write [itex]H_{int}[/itex] to mean the interaction. (Some people write [itex]V[/itex] for
this instead.)

Then the full interacting Hamiltonian is [itex]H = H_0 + H_{int}[/itex]

By "multiparticle Hamiltonian", I meant a Hamiltonian that describes the dynamical
evolution of many particles (rather than just a single particle). Actually, this is a poor
term, and I should have said "infinite degrees of freedom" instead of "multiparticle".

"4D" just means that it's for 3+1 spacetime.

Peskin & Schroeder is the obvious reference that immediately comes to mind
for basic QFT stuff. That should clarify the distinction between the "free" and
"interaction" parts of the Hamiltonian.

But P&S don't talk much about the disjointness between "free" and "interacting"
Hilbert spaces for infinite degrees of freedom. Try Umezawa for that.
 

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