- #1
Opus_723
- 178
- 3
I am going through Shankar's treatment of Feynman Integrals right now, and I have one lingering doubt that I can't quite seem to work out.
I was pretty happy with the idea of discretizing time, then doing independent sums over xi at each time. But Shankar simply says that we can consider the sums over xi to be integrals. I don't quite follow this. Normally when you pass from a sum to an integral there has to be some infinitesimal factor in each term of your sum. Shankar just says that the sum over the phase factors becomes an integral in xi over the phase factors, but I don't see where the "dx" comes from in order to let us do that. I have a suspicion that something deeper is going on, but I can't quite grasp it.
I've actually noticed this sort of thing in a couple of other places. For example, the completeness relation for operators with discrete spectra seems to pick up an infinitesimal "d_" that doesn't seem to have a counterpart in the completeness relation for operators with discrete spectra. This may be completely unrelated to the path integrals. But I get the feeling that I'm missing something either very obvious or very subtle, because this sort of thing keeps coming up in QM. Could anyone help me clear this up?
I was pretty happy with the idea of discretizing time, then doing independent sums over xi at each time. But Shankar simply says that we can consider the sums over xi to be integrals. I don't quite follow this. Normally when you pass from a sum to an integral there has to be some infinitesimal factor in each term of your sum. Shankar just says that the sum over the phase factors becomes an integral in xi over the phase factors, but I don't see where the "dx" comes from in order to let us do that. I have a suspicion that something deeper is going on, but I can't quite grasp it.
I've actually noticed this sort of thing in a couple of other places. For example, the completeness relation for operators with discrete spectra seems to pick up an infinitesimal "d_" that doesn't seem to have a counterpart in the completeness relation for operators with discrete spectra. This may be completely unrelated to the path integrals. But I get the feeling that I'm missing something either very obvious or very subtle, because this sort of thing keeps coming up in QM. Could anyone help me clear this up?