- #1
paweld
- 255
- 0
Let's consider self-interacting scalar field described by action:
[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.
[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.