[Quick] States of interacting and noninteracting Hamiltonianby QuantumDevil Tags: hamiltonian, interacting, noninteracting, states 

#1
Dec2507, 02:50 PM

P: 29

Have multiparticle state of full Hamiltonian and oneparticle state of free Hamiltonian 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? P.S. these states are eigenvectors of appropriate Hamiltonians 



#2
Dec2507, 07:47 PM

HW Helper
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#3
Dec2607, 09:48 AM

P: 29





#4
Dec2607, 04:35 PM

Sci Advisor
P: 1,732

[Quick] States of interacting and noninteracting Hamiltonianinteracting 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
Dec2807, 06:53 AM

P: 29

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. 



#6
Dec2807, 04:18 PM

Sci Advisor
P: 1,732

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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