# Path integral in momentum space

1. Dec 20, 2007

### saki42

using the hamiltonian to derive the pathintegral is well known (see schulman), but i have only seen it for diagonal momenta and coupled coordinates:
G(x,t;y) = <x|exp(-itH/hbar)|y> using the trotter formula etc one arrives at:
G(x,t;y) = lim_N->infinity Int dx1...dx_N-1Prod_{j=0}^{N-1}<x_j+1|exp[-itT/(N hbar)|x_j>exp[-itV/(hbar N)]

with H=T+V and V diagonal in coordinate space

inserting a full set of momenta 1=int dp |p><p| one is able to solve this problem and express the argument of exp as iS/hbar.

BUT i am not dealing with a coordinate coupling (so <x_j+1|...|x_j> makes no sense. i have a T of the form T_L, T_R, T_D and T_(D,L), T_(D,R) where T_L has a full set of momentum states (normalized) called p_L and the same for T_R and T_D. the coupling (T_(D,L) and T_(D,R)) looks like p_D(p_L+p_R).
has anyone an idea how to reach S (argument of exp with factors) as done by the above method WITHOUT legendre transforming the hamiltonian from the start and deriving the path integral with the lagrangian????