Interaction-picture and gell-mann low

[geoduck]

Suppose you want to evaluate:

$$\langle \Omega | T\phi(x) \phi(y) | \Omega \rangle$$

where [itex]\Omega [/itex] is the ground state of the full Hamiltonian, and the fields are in the Heisenberg representation. Assume [itex]x_0 > y_0 [/itex] for now. Then it's straightforward to show that

$$\langle \Omega | T\phi(x) \phi(y) | \Omega \rangle=\langle \Omega | S^\dagger(t,0)\phi_I(x) S(x_0, y_0) \phi_I(y) S(y_0,0) | \Omega \rangle$$

where the fields are in the interaction representation [itex] \phi_I(x)=\phi_F(x)[/itex], [itex]S(t,t')=U(t)U^\dagger(t') [/itex], and [itex]U(t)=e^{iH_0t}e^{-iHt}[/itex], where subscript [itex]F [/itex] stands for the free-field.

According to Wikipedia, the Gell-Mann Low formula is:

$$ |\Psi^{(\pm)}_\epsilon \rangle = \frac{ U_{\epsilon I} (0,\pm\infty) |\Psi_0 \rangle}{\langle \Psi_0 | U_{\epsilon I}(0,\pm\infty)|\Psi_0\rangle} $$

I want to use this in my equation to have an expression in terms of the noninteracting vacuum [itex]|\Psi_0\rangle [/itex] instead of the interacting one [itex]|\Omega \rangle[/itex].

Is it correct to say that [itex]|\Psi^{(-)}_\epsilon \rangle [/itex] is equal to my [itex]|\Omega \rangle [/itex], and [itex]\langle \Psi^{(+)}_\epsilon | [/itex] is equal to my [itex]\langle \Omega| [/itex]?

If I do this, then I get:

$$\langle \Omega | T\phi(x) \phi(y) | \Omega \rangle=\frac{\langle \Psi_0 |U^\dagger_{\epsilon I} (0, \infty) S^\dagger(x_0,0)\phi_I(x) S(x_0, y_0) \phi_I(y) S(y_0,0) U_{\epsilon I} (0, -\infty) |\Psi_0 \rangle}{\langle \Psi_0 | U_{\epsilon I}(0,\infty)|\Psi_0\rangle^* \langle \Psi_0 | U_{\epsilon I}(0,-\infty)|\Psi_0\rangle}$$

Now assume that their [itex]U_{\epsilon I}[/itex] corresponds to my [itex]S [/itex]. Then using the property that [itex]S^\dagger(t,t')=S(t',t) [/itex] and [itex]S(t,t')S(t',t'')=S(t,t'') [/itex] I get:

$$\langle \Omega | T\phi(x) \phi(y) | \Omega \rangle=\frac{\langle \Psi_0 |S(\infty,x_0)\phi_I(x) S(x_0, y_0) \phi_I(y) S(y_0, -\infty) |\Psi_0 \rangle}{\langle \Psi_0 | S(\infty,0)|\Psi_0\rangle \langle \Psi_0 | S(0,-\infty)|\Psi_0\rangle}$$

This is almost the Green's function written in the interaction picture, except the denominator is wrong. The correct expression is:

$$\langle \Omega | T\phi(x) \phi(y) | \Omega \rangle=\frac{\langle \Psi_0 |TS(\infty,-\infty)\phi_I(x) \phi_I(y) |\Psi_0 \rangle}{\langle \Psi_0 | S(\infty,-\infty)||\Psi_0\rangle}$$

I don't think the two denominators are equal. In general you need a complete set of unperturbed energy eigenstates, and not just one, to make the two denominators equal.

Also, what is the meaning of choosing two different vacuums for the bra and ket, [itex]|\Psi^{(\pm)}_\epsilon \rangle [/itex], for the interacting ground state? I thought we usually assume the interacting ground state is nondegenerate? But it seems that it's doubly degenerate.

 

[eljose79]

it is important to carefully evaluate and analyze equations and theories to ensure their accuracy and validity. In this case, it is correct to say that $|\Psi^{(-)}_\epsilon \rangle$ is equivalent to $|\Omega \rangle$ and $\langle \Psi^{(+)}_\epsilon |$ is equivalent to $\langle \Omega |$. This is because the Gell-Mann Low formula is used to transform the state $|\Omega \rangle$ into the non-interacting vacuum $|\Psi_0 \rangle$.

However, the expression you have derived for $\langle \Omega | T\phi(x) \phi(y) | \Omega \rangle$ is not entirely correct. As you mentioned, the denominator should involve a complete set of unperturbed energy eigenstates, and not just one. Additionally, the choice of two different vacuums for the bra and ket, $|\Psi^{(\pm)}_\epsilon \rangle$, for the interacting ground state is not necessary. In fact, the interacting ground state is typically assumed to be non-degenerate.

It is important to carefully consider all aspects of a theory and its derivations to ensure that the final result is accurate and consistent. In this case, it may be helpful to revisit the derivation and make any necessary corrections to arrive at the correct expression for $\langle \Omega | T\phi(x) \phi(y) | \Omega \rangle$.