Path integrals in quantum mechanics Phys V3500/G8099 handout #1 References: there s a nice discussion of this material in the first chapter of L.S. Schulman, Techniques and applications of path integration. Path integrals in quantum mechanics Consider the standard 1-d quantum mechanics problem of a particle moving in a potential (but let s set = 1). H = p2 2m + V T + V The basic quantity of interest is the time evolution operator U(t 1, ) = e ih(t 1 ) or rather its matrix elements in a position basis G(x 1, t 1 x 0, ) = x 1 e ih(t 1 ) x 0. Knowing G is tantamount to solving the Schrodinger equation since ψ(x 1, t 1 ) = x 1 ψ(t 1 ) = x 1 e ih(t 1 ) ψ( ) = dx 0 x 1 e ih(t 1 ) x 0 x 0 ψ( ) = dx 0 G(x 1, t 1 x 0, )ψ(x 0, ) Let s begin by studying time evolution over very short intervals, setting t 1 = +. Then we have x 1 e ih x 0 x 1 1 i(t + V ) x 0 x 1 e it e iv x 0
where the error in the last line is O( 2 ). I ll assume the error is negligible; for a more careful discussion see the Trotter product formula in Schulman. Inserting a complete set of momentum eigenstates x 1 e ih x 0 dp x 1 e it p p e iv x 0 = dp e ip2 /2m e iv (x 0) 1 2π eip(x 1 x 0 ) Completing the square in the exponent dp x 1 e ih i x 0 2m (p m (x 1 x 0 )) 2 e im 2 (x 1 x 0 ) 2 e iv (x 0) 2π e Shifting variables of integration and performing the Gaussian integral dp x 1 e ih i x 0 2m p2 e im 2 (x 1 x 0 ) 2 e iv (x 0) 2π e = 1 ( ) 1/2 2πm e im 2 (x 1 x 0 ) 2 e iv (x 0) 2π i ( m ) h 1/2 i 1 = m( e x 1 x 0 2 ) 2 i V (x 0 ) 2πi The quantity in square brackets is very intriguing. Suppose we tried to make sense of the action S = t1 dt 1 2 mẋ2 V (x) for paths connecting (x 0, ) to (x 1, t 1 = + ). Adopting a finite-difference approximation for the particle velocity, it seems reasonable to say that t1 dt 1 2 mẋ2 1 ( ) 2 2 m x1 x 0. Also adopting a Riemann-sum definition of dt V, with a single term in the Riemann sum, it seems reasonable to say that t1 dt V V (x 0 ). This means the quantity in square brackets can be thought of as the classical Lagrangian, and the exponent can be thought of as i times the classical action!
So far this has all been for evolution over infinitesimal time intervals. But extending these results to finite time intervals is easy. All we have to do is insert lots of complete sets of position eigenstates. x f e ih(t f t i ) x i = dx 1 dx N 1 x N e ih(t f t i )/N x N 1 x N 1 e ih(t f t i )/N x N 2 x N 2 x 1 x 1 e ih(t f t i )/N x 0 This is exact for any N, where we ve set x 0 = x i and x N = x f. Sending N we have repeated evolution over infinitesimal time intervals, and making use of our previous result { [ ( x f e ih(t f t i m ) N/2 N ( ) 2 ) 1 x i = lim dx 1 dx N 1 exp i N 2πi 2 m xi x i 1 V (x i 1)]} where = (t f t i )/N. Now we define a measure for integrating over paths ( m ) N/2 Dx( ) = lim dx 1 dx N 1 N 2πi and identify the action for a path as N [ S[x( )] = lim ( ) 2 1 N 2 m xi x i 1 V (x i 1)]. i=1 Then we can write the matrix elements of the time evolution operator as x f e ih(t f t i ) x i = Dx( ) e is[x( )] x(t i )=x i x(t f )=x f We have expressed the amplitude for a particle to go from position x i at time t i, to position x f at time t f, in terms of a sum over all paths connecting (x i, t i ) to (x f, t f ). This is known as the Feynman path integral or sum-over-histories. Note that the classical path, i.e. the path satisfying the classical equations of motion, does not play a special role here! On the contrary all paths are equally likely, in the sense that the probability of any given path amplitude 2 e is[path] 2 1. (Question what is the significance of the classical trajectory?) This leads to a general attitude towards quantum theory, which we ll take over wholeheartedly into string theory: integrate over everything that isn t fixed by your initial or final conditions. i=1
Your friend, the Gaussian path integral There s basically only one path integral anyone knows how to evaluate. It s a generalization of the ordinary one-dimensional integral π dx e ax2 = a. This formula is valid if a is a positive real number; by analytic continuation it also holds in the complex a plane. As an intermediate step to path integrals let A be an N N real symmetric positivedefinite matrix, and let s try to make sense of R N d N x e xt Ax. We can diagonalize A with an orthogonal transformation, A = R T ΛR where R is orthogonal and Λ = diag.(λ 1,..., λ N ) is diagonal with real positive entries. Then setting y = Rx we have d N x e xt Ax = d N y e yt Λy = π N/2 Defining our integration measure Dx = d N x/π N/2 we have Dx e xt Ax = det 1/2 A. πn/2 = 1/2 (λ 1 λ N ) det 1/2 A. By analytic continuation we can extend this to complex symmetric A. Nothing here seems to depend on N, so we might as well send N and consider integration over an infinite-dimensional space. In particular let s consider a function space { } x( ) : [0, 1] R with inner product ( ) x( ), y( ) = 1 0 dt x(t)y(t).
(I ll be sloppy about exactly what space of functions we re considering). Consider a real symmetric positive operator O with eigenvalues Ox n (t) = λ n x n (t). Then with a suitable integration measure we have Dx( ) e R 1 0 dt x(t)ox(t) = det 1/2 O n=1 λ 1/2 n. We ll often extend this to complex symmetric operators. Two comments, A multitude of sins can be swept under the rug by saying with a suitable integration measure. For most operators of interest the determinant (the infinite product) diverges. In QFT you ll learn how to handle this with renormalization.