- #1
pellman
- 683
- 6
From a QM (not QFT) context, one particle, we start with a hamiltonian H(q,p) and develop something like
\langle q'',T|q',0\rangle \approx \int e^{-i\sum_{l=0}^{N}[H(q_l,p_l)-p_l\dot{q}_l]\delta t}\prod_{j=1}^N{dq_j}\prod_{j=0}^N{\frac{dp_k}{2\pi}}
where \delta t = T/(N+1) and \dot{q}_j \equiv (q_{j+1}-q_j)/\delta t and where the approximation becomes equality for small delta-t or, equivalently, large N. In that case we write
e^{-i\sum_{l=0}^{N}[H(q_l,p_l)-p_l\dot{q}_l]\delta t}\rightarrow e^{-i\int_0^T{[H(q(t),p(t))-p(t)\dot{q}(t)}]dt}
Now generally when we look at a Riemann sum, we are dealing with a given function f(x) and looking at a sum over f(x_i)\delta x for finer and finer slices \delta x.
However, here the individual q_l's are not part of a given function q(t) nor do I see any reason to expect them to approach anything like an integrable function q(t). Indeed, since each q_l is a variable of integration over the entire real line, I can't see them settling down to anything like a piece-wise continuous function for large N.
For any discrete set of time slices for a free particle, given that a particle at q_j at time t_j, then there is a non-zero probability that it be found at any other value of q at time t_j + \delta t. The smaller we make \delta t, the more jagged most of the potential paths become. I don't see how they approach something we can integrate over.
All of my QFT books make no mention of this. Can someone recommend a link or text which covers this limit with more rigor?
\langle q'',T|q',0\rangle \approx \int e^{-i\sum_{l=0}^{N}[H(q_l,p_l)-p_l\dot{q}_l]\delta t}\prod_{j=1}^N{dq_j}\prod_{j=0}^N{\frac{dp_k}{2\pi}}
where \delta t = T/(N+1) and \dot{q}_j \equiv (q_{j+1}-q_j)/\delta t and where the approximation becomes equality for small delta-t or, equivalently, large N. In that case we write
e^{-i\sum_{l=0}^{N}[H(q_l,p_l)-p_l\dot{q}_l]\delta t}\rightarrow e^{-i\int_0^T{[H(q(t),p(t))-p(t)\dot{q}(t)}]dt}
Now generally when we look at a Riemann sum, we are dealing with a given function f(x) and looking at a sum over f(x_i)\delta x for finer and finer slices \delta x.
However, here the individual q_l's are not part of a given function q(t) nor do I see any reason to expect them to approach anything like an integrable function q(t). Indeed, since each q_l is a variable of integration over the entire real line, I can't see them settling down to anything like a piece-wise continuous function for large N.
For any discrete set of time slices for a free particle, given that a particle at q_j at time t_j, then there is a non-zero probability that it be found at any other value of q at time t_j + \delta t. The smaller we make \delta t, the more jagged most of the potential paths become. I don't see how they approach something we can integrate over.
All of my QFT books make no mention of this. Can someone recommend a link or text which covers this limit with more rigor?
