Srednicki — Quantum Field Theory · Problem 14.7

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · renormalization-anharmonic-oscillator

Problem

Renormalization of the anharmonic oscillator. Consider an anharmonic oscillator, specified by the lagrangian

L = \tfrac{1}{2}Z\dot{q}^2 - \tfrac{1}{2}Z_\omega \omega^2 q^2 - Z_\lambda \lambda \omega^3 q^4 . \tag{14.54}

We set

\hbar = 1

and

m = 1

;

\lambda

is then dimensionless. a) Find the hamiltonian

H

corresponding to

L

. Write it as

H = H_0 + H_1

, where

H_0 = \frac{1}{2}P^2 + \frac{1}{2}\omega^2 Q^2

, and

[Q,P] = i

. b) Let

|0\rangle

and

|1\rangle

be the ground and first excited states of

H_0

, and let

|\Omega\rangle

and

|I\rangle

be the ground and first excited states of

H

. (We take all these eigenstates to have unit norm.) We define

\omega

to be the excitation energy of

H

\omega \equiv E_I - E_\Omega

. We normalize the position operator

Q

by setting

\langle I|Q|\Omega\rangle = \langle 1|Q|0\rangle = (2\omega)^{-1/2}

. Finally, to make things mathematically simpler, we set

Z_\lambda

equal to one, rather than using a more physically motivated definition. Write

Z = 1 + A

and

Z_\omega = 1 + B

, where

A = \kappa_A \lambda + O(\lambda^2)

and

B = \kappa_B \lambda + O(\lambda^2)

. Use Rayleigh--Schr\"odinger perturbation theory to compute the

O(\lambda)

corrections to the unperturbed energy eigenvalues and eigenstates. c) Find the numerical values of

\kappa_A

and

\kappa_B

that yield

\omega = E_I - E_\Omega

and

\langle I|Q|\Omega\rangle = (2\omega)^{-1/2}

. d) Now think of the lagrangian of eq. (14.54) as specifying a quantum field theory in

d = 1

dimensions. Compute the

O(\lambda)

correction to the propagator. Fix

\kappa_A

and

\kappa_B

by requiring the propagator to have a pole at

k^2 = -\omega^2

with residue one. Do your results agree with those of part (c)? Should they?