1 The Quantum Anharmonic Oscillator

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1 1 The Quantum Anharmonic Oscillator Perturbation theory based on Feynman diagrams can be used to calculate observables in Quantum Electrodynamics, like the anomalous magnetic moment of the electron, and the predictions agree with experiment with impressive accuracy. The perturbation series in QED is mathematically not well defined. (1) In each order there are ultraviolet divergences that must be removed, making it impossible to calculate corrections to the mass and charge of the electron and the normalization of the particle states. () The perturbation series does not converge for any finite value of the renormalized expansion parameter and must be interpreted as an asymptotic series. This means that predictions can be improved by computing additional terms, but at some point the series breaks down. The asymptotic nature of the QED perturbation series was first discussed clearly by Dyson[1]. To understand perturbation theory, Feynman diagrams, and the breakdown of perturbation theory, it is helpful to start with the simplest quantum field theory that cannot be solved exactly. The quantum anharmonic oscillator can be viewed as a scalar field theory in 1 spacetime dimension x µ = x 0 = ct. Space consists of a single point. The classical Lagrangian is L(φ, φ) = 1 φ m φ λφ 4. The Lagrangian has dimension [M] = [L] 1 in natural units c = = 1, so φ has dimension [L] 1/. and the quartic coupling constant has dimension [L] 3. The theory has a well defined ground state if the quartic coupling constant λ 0. There is no stable ground state if λ < 0. 1

2 Canonical Quantization To quantize a classical field theory, the canonical momenta conjugate to the field variables are found, and a Legendre transformation used to derive the classical Hamiltonian. π(t) = L φ = φ, H = φπ L = 1 π + m φ + λφ 4. The field and canonical momentum are promoted to operators acting on the Hilbert space. The operator properties are determined by replacing the classical Poisson brackets with Heisenberg commutators Ĥ = 1 ˆπ + m ˆφ + λ ˆφ 4, {φ, π} PB = φ π φ π π φ [ ] φ π = 1 ˆφ, ˆπ = i. The quantum harmonic oscillator ground state is determined by the eigenvalue equation Ĥ 0 = E 0 (λ) 0. The ground state energy can be calculated to very high order in perturbation theory. Bender and Wu studied this problem in a series of articles, one of which[] presents results using Feynman diagrams and compares them with results obtained using a recursion relation for the Rayleigh-Schrödinger perturbation series. A very convenient way to calculate the series is to introduce ladder (creation and annihilation) operators ˆφ = a + a m, ˆπ = a a i /m, [a, a ] = 1.

3 In this Fock space representation, also called second quantization, parameter m is viewed as the mass of a particle, the ground state 0 is a vacuum state with no particles, and the n-th excited n state has n non-interacting particles (with energy mc = ω in conventional untis). The ˆφ 4 term in the Hamiltonian represents interactions, and the shift in energy of the harmonic oscillator states is evaluated using Feynman rules and Feynman diagrams. The first order perturbation theory shift in the ground state energy is given by E 0 = λ 4 0 ˆφ 4 0 = Expanding ( a + a ) 4 gives a sum of 4 = 16 terms. λ 16m 0 ( a + a ) 4 0. Wick s Theorem The expectation value can be evaluated using Wick s Theorem. Consider any term with any number of creation and annihilation operators. A normal ordered product is defined by moving all creation operators to the left and annihilation operators to the right. A contraction is the commutator involved any exchange. Wick s theorem states that the original product equals the normal ordered product plus a sum of normal ordered terms obtained by making contractions in all possible ways. Products of operators: aa = N(aa ) + [a, a ] = a a + 1, a = N(a ) + [a, a] = N(a ), a = N(a ) + [a, a] = N(a ). 3

4 Example with 3 operators: a a = N(a a ) + [a, a]n(a ) + N(a)[a, a ] + [a, a ]N(a) = a a + a. General case of 4 operators from Peskin-Schroeder Eq. (4.39) It is now easy to evaluate the first order correction. Note that the vaccum expectation value of any normal ordered product of one or more operators is zero. Therefore only the 3 terms on the last line with two contractions each will contribute. Each of these terms must have two creation and two annihilation operators so that both contractions are non-zero. There are exactly 6 such terms and ways of contracting each of them. aaa a aa aa a aaa aa a a a aa a a a aa 0 ˆφ 4 0 = 1, E 0 = λ 3λ 1 = 16m 4m. 4

5 Recursion Relations for Perturbative Coefficients Bender and Wu scale the field φ for convenience and define φ x, ˆπ i d dx and write the eigenvalue equation in the Schrödinger representation as a differential equation [ x Ĥ 0 = d d x x + 1 ] 4 λx4 Φ(x) = E(λ)Φ(x), to be solved with boundary condition Φ(± ) = 0. An exact analytic solution of this equation has never been found. Using perturbation theory, the ground state energy can be expanded in a power series E 0 (λ) m = 1 + ( ) n λ A m 3 n, n=1 where the first term is the ground state energy of the simple harmonic oscillator, and the coefficients A n are dimensionless real numbers. A recursion relation for the coefficients A n can be derived by expressing the ground state wave function as a power series [ ( ) n λ Φ(x) = 1 + B m 3 n (x)] e x /4, n=1 5

6 where the ground state wave function of the simple harmonic oscillator has been factored out. To satisfy the Schrödinger equation order by order in λ, it is sufficient to take the coefficient functions B n (x) to be polynomials of degree n n ( ) x B 0 = 1, B n (x) = ( 1) n j B n,j. Plugging these series expansions into the Schrödinger equation and collecting terms of order λ n gives the recursion relation j=1 jb n,j = (j + 1)(j + 1)B n,j+1 + B n 1,j n 1 Bender and Wu show that the coefficients in the series for E 0 are given by A n = ( 1) n+1 B n,1 = 1 4 ni n k=1 B n k,1 B k,j. where I n is a positive integer divisible by 3. To solve the recursion relation start with first order n = 1 with wave function polynomial ( ) ( ) x x B 1 (x) = B 1,1 B 1,, and the recursion relations for j = 1, : B 1,1 = 6B 1, 4B 1, = B 0,0 6

7 which can be solved for At second order n = B 0,0 = B 0 = 1, B 1, = 1 4 B 0,0 = 1 4, B 1,1 = 3B 1, = 3 4, A 1 = B 1,1 = 3 4. B (x) = we can use the 4 recursion relations for j = 1,, 3, 4 which can be solved for ( ) ( ) x x ( ) x 3 ( ) x 4 B,1 + B, + B,3 + B,4, j = 1 : B,1 = 6B, B 1,1 j = : 4B, = 15B,3 B 1,1 B 1, j = 3 : 6B,3 = 8B,4 + B 1, j = 4 : 8B,4 = B 1, B,4 = 1 3, B,3 = 13 48, B, = 31 3, B,1 = 1 8, A = B,1 = 1 8. Bender and Wu solved these recursion relations to obtain the first 75 terms in the perturbation series and used the results to show that the series diverges. The first 9 coefficients are 7

8 We will use Mathematica to obtain these results. 8

9 Feynman Diagram Calculation Bender and Wu summarize a Feynman diagram calculation of the first 3 coefficients A 1, A, A 3 in Appendix B. The diagrams to be computed and the Feynman rules for the theory are: Feynman Rules A factor of 1 E m +iɛ for each propagator A factor of 4λ for each vertex For each independent loop integrate i π de Multiply by the symmetry number S.N. 9

10 Order 1 The symmetry factor is the inverse of the number of transformations that map the diagram into itself without breaking any lines. For this diagram, the two loops can be interchanged, giving a factor of. The start and end a loop line can be interchanged, giving a factor of. The total number of symmetry transformations is = 8 and the symmetry factor is 1/8. Apply the Feynman rules: [ 1 i ] 8 4λ 1 de π E m + iɛ The denominator is the Green function of the free field Euler-Lagrange equation [ ] d dt + m φ(t) = 0 with Feynman boundary conditions ɛ 0 +. The integral can be evaluated using contour integration 1 de E m + iɛ = 1 [ ] 1 de m E m + iɛ 1 = πi E + m iɛ m. The value of the order 1 Feynman diagram is the first order correction to the ground state energy 3λ 4m = E 0. 10

11 References [1] F.J. Dyson, Divergence of Perturbation Theory in Quantum Electrodynamics Phys. Rev. 85, (195). [] C.M. Bender and T.T. Wu, Anharmonic Oscillator Phys. Rev. 184, (1969). 11