Path Integral Basics (Why dimension increases in the integrals?)

[Elwin.Martin]

Alright, I have a kind of dumb question:

Why do I distinguish between dq and dq_i when considering the propagation from q_i to q to q_f?

For example, if we want the wave function at some q_f and t_f given q_i and t_i, we may write:
ψ(q_f,t_f)=∫K(q_ft_f;q_it_i)ψ(q_i,t_i)dq_i

Why do we distinguish between dq_i and say, dq, of an intermediate point q?

I understand that a second integration arises, naturally, since another propagator is defined. I just don't get the need to distinguish? I feel like it's almost book-keeping, like we write out (or don't since we use product notation followed by a script D) our dq's so we don't have to write out more integrals...

I know that's not right though, because we treat q and q_i as different things...

Some direction would be great...I feel kind of dumb for whatever I'm missing.

edit:
I think I mis-titled this a little, but it's close enough, I hope.

 

[maverick_starstrider]

Alright, I have a kind of dumb question:

Why do I distinguish between dq and dq_i when considering the propagation from q_i to q to q_f?

For example, if we want the wave function at some q_f and t_f given q_i and t_i, we may write:
ψ(q_f,t_f)=∫K(q_ft_f;q_it_i)ψ(q_i,t_i)dq_i

Why do we distinguish between dq_i and say, dq, of an intermediate point q?

I understand that a second integration arises, naturally, since another propagator is defined. I just don't get the need to distinguish? I feel like it's almost book-keeping, like we write out (or don't since we use product notation followed by a script D) our dq's so we don't have to write out more integrals...

I know that's not right though, because we treat q and q_i as different things...

Some direction would be great...I feel kind of dumb for whatever I'm missing.

edit:
I think I mis-titled this a little, but it's close enough, I hope.

I'm not quite sure what you're asking but when deriving path integrals one takes the propagation (or the sum of all propagations) between states i and f and subdivides them, and then subdivides them again, and again until the path between two states has been turned from a finite path in phase space to an infinite sum of infinitesimal paths.