“Problem Set Four”

(1) Consider a simple harmonic oscillator in one dimension. The Lagrangian is

L = ∫ dt (\frac{m}{2} (\frac{dx}{dt})^{2} − \frac{1}{2}mω^{2}x^{2}) .  (1)

The Hamiltonian is accordingly

H = \frac{p^{2}}{2m} + \frac{1}{2}mω^{2}x^{2}  (2)

where

p = −i\frac{d}{dx}  (3)

acting on L^{2}(R).

Calculate the kernel of the operator e^{−itH}, that is the matrix element

⟨b|e^{−iHt}|a⟩,  (4)

using path integrals.

That is, find the classical path x_{0}(t) that starts at a at time 0 and ends at b at time t. Write x(t) = x_{0}(t) + x̃(t). By substituting x(t) in L, write the action as a quadratic function of x̃(t) plus a known function of a, b and t. Do the path integral over x̃(t) by interpreting it as a determinant which you must study somehow.

In fact after dividing out formally by an infinite but t-independent constant - or if you prefer, upon using something like a zeta function definition of the determinant - you should be able to reduce to an infinite sum that will look something like

∏_{n=1}^{∞} (1 + \frac{p^{2}}{n^{2}} )  (5)

for some p. This product in fact defines a certain elementary function of p.

(2) Consider the scalar field theory with Lagrangian

L = ∫ d^{n}x (\frac{1}{2}|dϕ|^{2} + \frac{1}{2}m^{2}ϕ^{2} + \frac{g}{3!}ϕ^{3})  (6)

in n-dimensional Euclidean space.

Calculate the contribution of order g^{3} to the three-point function ⟨ϕ(x)ϕ(y)ϕ(z)⟩. (It is best to work in momentum space, that is to set z = 0 and take the Fourier transform with respect to x and y.)

(a) Demonstate that the result is finite in n ≤ 6, and for n = 6, describe the precise form of the divergence – that is, introduce a cutoff Λ, and describe just what sort of Λ-dependence you get for large Λ.

Renormalize the result for n = 6. That is, carry out the necessary renormalization and write down the renormalized, finite correction to the three point function.

(b) For n < 6 make the result as explicit as you can. Use Feynman’s identity

\frac{1}{ABC} = ∫_{0}^{1} dα dβ dγ δ(α + β + γ − 1)\frac{1}{(αA + βB + γC)^{3}} .  (7)

After completing the square, you should be able to reduce the integral over the loop momentum q to

∫ dα dβ dγ δ(α + β + γ − 1) ∫ d^{n}q \frac{1}{(q^{2} + b^{2})^{3}}  (8)

for some b (which is a polynomial in the three external momenta and in α and β). Set n = 4 and carry out the q integral explicitly.

For extra credit, get as far as you reasonably can with the α and β integrals.

(3) Now consider a theory in n-dimensional Euclidean space of a scalar ϕ and a spinor ψ with Lagrangian

L = ∫ d^{n}x (\frac{1}{2}|dϕ|^{2} + \frac{1}{2}M^{2}ϕ^{2} + \frac{i}{2} (ψ, i(D + m)ψ) + \frac{g}{2}ϕ(ψ, ψ)) .  (9)

(ψ takes values in a representation of Spin(n) that is associated with a real representation of Spin(1, n − 1). The precise details depend a bit on n, but this should not affect the calculation you will have to do. ( , ) is a quadratic form on the spin representation. D is the Dirac operator.)

The free propagator of the ψ field is in momentum space 1/(Γ · k + im), where Γ · k is physicist’s notation for Clifford multiplication by k.

The free propagator of ϕ is 1/(k^{2} + M^{2}).

L is written so that there is only one interaction vertex, a cubic vertex with two ψ’s and one ϕ. The factor to be included at such a vertex in a Feynman diagram is just −g (times the identity matrix or natural bilinear form ( , ) on the fermion ψ, depending on how you look at it).

Compute the correction of order g^{2} to the fermion two point function ⟨ψ(x)ψ(0)⟩. (Again, it is best to take the Fourier transform. Also, it is best to write the two point function as 1/(Γ · k + im + iΣ) and think of computing a correction to Σ, which will be given by an “amputated one loop diagram.” Σ will be of the form w(k^{2})Γ · k + iv(k^{2}) and the problem is to study both functions w and v.)

(a) Show that the result is finite for n < 4.

(b) Show that (for n < 4) the result is analytic in k^{2} for k^{2} > −(m + M)^{2}. (We are writing the formulas in Euclidean signature here; in Lorentz signature with metric + − − − −− this would read k^{2} < (m + M)^{2}.) How does this compare to what you would have expected? In particular, your result is holomorphic near k^{2} = −m^{2}, so that the correction changes the value of the fermion mass (by an amount that is small if g is small) but not the existence of a discrete state.

(c) For n = 4, what renormalizations are necessary to remove the infinity that you have found? Write a formula for the renormalized, finite, correction to the two point function, that is, to Σ. For n = 4, does the renormalized two point function have the same analytic behavior found in (b) for n < 4?