Path integral in quantum mechanics based on S-6 Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian: let s look at one piece first: P and Q obey: Probability amplitude for the particle to start at q at time t and end up at position q at time t is Campbell-Baker-Hausdorf formula complete set of momentum states where and are eigenstates of the position operator Q. In the Heisenberg picture: and we can define instantaneous eigenstates: Probability amplitude is then: 62 64 to evaluate the transition amplitude: let s divide the time interval T = t - t into N+1 equal pieces insert N complete sets of position eigenstates = to evaluate the transition amplitude: let s divide the time interval T = t - t into N+1 equal pieces insert N complete sets of position eigenstates = - we find: important for general form of hamiltonians with terms containing both P and Q in our case, it doesn t make any difference 63 65
What is it good for? Consider e.g.: taking the limit we get: the result can be simply written using path integral as: should be understood as integration over all paths in phase space that start at and end at (with arbitrary initial and final momenta) In simple cases when hamiltonian is at most quadratic in momenta, the integral over p is gaussian and can be easily calculated: lagrangian Similarly: time ordering is crucial! prefactor can be absorbed into the definition of measure Time-ordered products appeared in LSZ formula for scattering amplitudes! 66 68 Functional derivatives: Dirac delta function taking the limit we get: continuous generalization of they are defined to satisfy all the usual rules of derivatives (product rule,...) should be understood as integration over all paths in phase space that start at and end at (with arbitrary initial and final momenta) In simple cases when hamiltonian is at most quadratic in momenta: Consider modifying hamiltonian to: Then we have: And we find, e.g.: where is calculated by finding the stationary point of the p-integral by solving: for p and plugging the solution back to 67 69
thus we have: more examples: Similarly, for the replacement picks up the ground state as the final state in the limit. we can integrate over q and q which leads to a constant factor that can be absorbed into the normalization of the path integral. after we bring down as many qs and ps as we want we can set and return to the original hamiltonian: Thus, with the replacement the boundary conditions and we have: we don t have to care about 70 72 Finally, we want both initial and final states to be ground states and take the limits and : Adding perturbations: we can simply write (suppressing the ): looks complicated, we will use the following trick instead: is the ground-state wave function eigenstate of H corresponding eigenvalue Finally, if perturbing hamiltonian depends only on q, and we want to calculate only time-ordered products of Qs, and if H is no more than quadratic in P and if the term quadratic in P does not involve Q, then the equation above can be written as: is wave function of n-th state let s replace with and take the limit : every state except the ground state is multiplied by a vanishing factor! 71 73
Path integral for harmonic oscillator based on S-7 Consider a harmonic oscillator: ground state to ground state transition amplitude is: external force it is convenient to change integration variables: then we get: a shift by a constant equivalent to thus going to lagrangian formulation (integrating over p) we get: and the transition amplitude is: 74 76 using Furier-transformed variables: and setting for simplicity, we get But since, if there is no external force, a system in its ground state remain in its ground state. thus we have: and thus: E = E or, in terms of time-dependent variables: using inverse Fourier transformation where: 75 77
Comment: For even number of Qs we pair up Qs in all possible ways: is a Green s function for the equation of motion of the harmonic oscillator: you can evaluate it explicitly, treating the integral as a contour integral in the complex E-plane and using the residue theorem. Make sure you are careful about closing the contour in the correct half-plane for t > t and t < t and that you pick up the correct pole. in general: you should find: We can now easily generalized these results to a free field theory... 78 80 Let s calculate correlation functions of Q operators: for harmonic oscillator we find: For odd number of Qs there is always one f(t) left-over and the result is 0! Path integral for free field theory based on S-8 Hamiltonian density of a free field theory: similar to the hamiltonian of the harmonic oscillator dictionary between QM and QFT: classical field operator field classical source we repeat everything we did for the path integral in QM but now for fields; we divide space and time into small segments; take a field in each segment to be constant; the differences between fields in neighboring segments become derivatives; use the trick: multiplying by is equivalent to replacing with which we often don t write explicitly;... eventually we can integrate over momenta and 79 81
obtain path integral (functional integral) for our free field theory: path in the space of field configurations change of integration variables: Comments: lagrangian is manifestly Lorentz invariant and all the symmetries of a lagrangian are preserved by path integral a shift by a constant lagrangian seems to be more fundamental specification of a quantum field theory 82 84 to evaluate we can closely follow the procedure we did for the harmonic oscillator: Fourier transformation: But 83 85
Thus we have: For even number of we pair up in all possible ways: where we used inverse Fourier transformation to go back to position-functions is the Feynman propagator, a Green s function for the Klien-Gordon equation: in general: integral over zero s component can be calculated explicitly by completing the contour and using the residue theorem, the three momentum integral can be calculated in terms of Bessel functions Wick s theorem 86 88 Now we can calculate correlation functions: we find: For odd number of there is always one J left-over and the result is 0! 87