# 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????

Considering the Problem (Interaction terms) for G in momentum space one gets <p_(M+1)|exp(-itp_M/(hbar N)[p_L+p_R]|p_M-1> so the p_M operator is no problem but the coupling of p_R (p_M+1) and p_L (p_M-1)

Last edited: Dec 20, 2007