- #1

paweld

- 255

- 0

[tex]

S = \int d^4 x \frac{1}{2} \left[ \partial_\mu \phi \partial^\mu \phi - m^2 \phi^2 - \frac{\lambda}{4!}\phi^4\right]

[/tex]

Using relation between Poisson bracket in classical mechanics and

comutators of quantum operators we have:

[tex]

[\partial_0\hat{\phi}(t,x_1),\hat{\phi}(t,x_2)] = - i \delta(x_1-x_2)

[/tex]

Unfortunatelly in case of interacting field we cannot decompose field in terms

of basic solution (modes) and we don't have operators which create and anihilate

modes. So in case of interaction field construction of Hilbert space on which

[tex]\hat{\phi} [/tex] acts in not straightforward. What is the Hilbert space

on which field operator acts and how the action of it on some basic states

looks like.

Some people claim that in case of interaction we have to assume that the spectrum

of full hamiltonian is the same as spectrum of hamiltonian without interaction

(so that the theory poses particle interpretation). However even if this assumption

is correct we still don't know how [tex]\hat{\phi} [/tex] on these states, and whole

theory is underdetermined.