maverick280857
- 1,774
- 5
Hi everyone,
In chapter 5 of Lewis Ryder's book on QFT, the expression for the propagator as a path integral is derived. Equation 5.7, which is the expression for the propagator over a small path [itex](q_{j+1} t_{j+1};q_{j}t_{j})[/itex], reads
[tex]\langle q_{j+1} t_{j+1} |q_{j}t_{j}\rangle = \frac{1}{2\pi\hbar}\int dp \exp{\left[\frac{i}{\hbar}p(q_{j+1}-q_j)\right]} - \frac{i\tau}{\hbar}\langle q_{j+1}|H|q_{j}\rangle[/tex]
where [itex]\tau = t_{j+1}-t_{j}[/itex]. This expression holds quite generally, but equation 5.13, which reads
[tex]\langle q_{f} t_{f} |q_{i}t_{i}\rangle = \int \frac{\mathcal{D}q\mathcal{D}p}{h}\exp{\frac{i}{\hbar}\left[\int dt p\dot{q}-H(p,q)\right]}[/tex]
is derived under the assumption that H is of the form
[tex]H = \frac{p^2}{2m} + V(q)[/tex]
This allows us to express the propagator as a function of the action S[q(t)] in the above expression.
But what if H is not of this form? What does the propagator look like there? I suppose it depends on the specific case (the author points out one example of a Lagrangian [itex]L = f(q)\dot{q}^2/2[/itex] which requires the introduction of an effective action different from [itex]\int L dt[/itex]), but are there any general rules or classes of systems where one can write the above expression, but which do not have the canonical form of H given above?
The author also states that Feynman began with the above expression for the propagator, which is not a very rigorous thing to do, given the counterexample in the previous paragraph.
Thanks.
In chapter 5 of Lewis Ryder's book on QFT, the expression for the propagator as a path integral is derived. Equation 5.7, which is the expression for the propagator over a small path [itex](q_{j+1} t_{j+1};q_{j}t_{j})[/itex], reads
[tex]\langle q_{j+1} t_{j+1} |q_{j}t_{j}\rangle = \frac{1}{2\pi\hbar}\int dp \exp{\left[\frac{i}{\hbar}p(q_{j+1}-q_j)\right]} - \frac{i\tau}{\hbar}\langle q_{j+1}|H|q_{j}\rangle[/tex]
where [itex]\tau = t_{j+1}-t_{j}[/itex]. This expression holds quite generally, but equation 5.13, which reads
[tex]\langle q_{f} t_{f} |q_{i}t_{i}\rangle = \int \frac{\mathcal{D}q\mathcal{D}p}{h}\exp{\frac{i}{\hbar}\left[\int dt p\dot{q}-H(p,q)\right]}[/tex]
is derived under the assumption that H is of the form
[tex]H = \frac{p^2}{2m} + V(q)[/tex]
This allows us to express the propagator as a function of the action S[q(t)] in the above expression.
But what if H is not of this form? What does the propagator look like there? I suppose it depends on the specific case (the author points out one example of a Lagrangian [itex]L = f(q)\dot{q}^2/2[/itex] which requires the introduction of an effective action different from [itex]\int L dt[/itex]), but are there any general rules or classes of systems where one can write the above expression, but which do not have the canonical form of H given above?
The author also states that Feynman began with the above expression for the propagator, which is not a very rigorous thing to do, given the counterexample in the previous paragraph.
Thanks.