(8212) Correlator for free particle

Consider a particle of mass \( m \), that is free to move in 1D. To be specific assume propagation during the time interval \( 0 < t < t_f \) from \( x=0 \) to \( x=x_f \). Define:
\( Z = \langle x_f | U(t_f) |0 \rangle \)
\( \langle\langle x(t_1) \rangle\rangle = \langle x_f | U(t_f-t_1) \hat{x} U(t_1) |0 \rangle \)
\( \langle\langle x(t_2) x(t_1) \rangle\rangle = \langle x_f | U(t_f-t_2) \hat{x} U(t_2-t_1) \hat{x} U(t_1) |0 \rangle \)
\( \overline{x(t_1)} \equiv \langle\langle x(t_1) \rangle\rangle / Z \)
\( \overline{x(t_2) x(t_1)} \equiv \langle\langle x(t_2) x(t_1) \rangle\rangle / Z \)
\( \hat{x}(t) \equiv U^{-1} \hat{x} U \)
\( \langle \hat{x}(t_2) \hat{x}(t_1) \rangle \equiv \langle 0 | \hat{x}(t_2) \hat{x}(t_1) | 0 \rangle \)

(1) Given that the particle experiences a field of force \( F(t) \) find its propagator \( Z[F] \). In the subsequent items assume that \( F(t)=0 \).

(2) Using generating function formalism calculate \( \overline{x(t_1)} \).

(3) Using generating function formalism calculate \( \overline{x(t_2) x(t_1)} \).

(4) Repeat the calculation of \( \overline{x(t_1)} \) using an elementary approach. Explain how formally you can deduce from this calculation a diverging result for \( \langle x_f | \hat{x}(t_1) |0 \rangle \).

(5) Relate the symmetrized version of the correlator \( \langle \hat{x}(t_2) \hat{x}(t_1) \rangle \) to the spreading \( \langle (\hat{x}(t_2) - \hat{x}(t_1))^2 \rangle \). Explain how a finite result can be obtained by appropriate normalization and regularization of the "0" state.

Tip: The last item allows a shortcut via the Heisenberg formalism.

Note: The Gell-Mann-Low theorem regarding the calculation of correlators is discussed in Section [27] of the lecture notes.