QFT Ground State Analysis: Understanding e^(-iHT) |0>

[silverwhale]

Hello Everybody,

In page 86, in Peskin & Schroeders Introduction to QFT, the following expression is introduced to analyze $$\left | \Omega \right >;$$ the ground state of the interacting theory:

$$e^{-iHT} \left | 0 \right >.$$

Where |0> is the ground state of the free theory and H is the Hamiltonian of the interacting theory. What is this expression? It is not the dirac picture free theory ground state, and it can't be just a time translation of the free theory ground state. Well maybe it is but I never saw this in my QM days.

Furthermore, the expression is equated to

$$e^{-iHT} \left | 0 \right > = \Sigma_n e^{-iE_nT} \left | n \right > \left < n \right | 0 >.$$

Don't |n> and |0> belong to different Hilbert spaces? Am I missing something here?

Thanks for any clarification!

 

[Bill_K]

What is this expression? It is not the dirac picture free theory ground state, and it can't be just a time translation of the free theory ground state.

Sure, I think that's what it is, it's the evolution of the free ground state in the Schrödinger picture.

Don't |n> and |0> belong to different Hilbert spaces?

Not in this example. In fact they specifically assume that <Ω|0> is nonzero.

[By the way, some of Peskin & Schroder is available on Google Books, including p86!]

 

[strangerep]

Don't |n> and |0> belong to different Hilbert spaces?

In general, yes.

In this type of treatment, one makes such assumptions -- and is punished much later as various divergences emerge. That's why regularization+renormalization are needed later -- to try and correct for the fact that one is not using quite the right Hilbert space.

There's a similar sleight of hand in P&S later on p109 where they say "if the formula (4.88) could somehow be justified, we could use it to retrieve...". They attempt such justification later in ch7 with discussion about field strength renormalization, and several other things.

 

[silverwhale]

Got it! Many thanks.

So basically we have the ground state of the free theory in the shrödinger picture being at t_0 = 0, and we apply the time translation operator containing the (full) Hamiltonian as it should be.

And the second fact would be just that, well, they should be in different Hilbert spaces but we ignore this fact for now and try to reach expression 4.31!

Thanks!