After integration by parts, the quadratic action is

S_{0} [x] = - \frac{1}{2} \int d t x (t) (\partial_{t}^{2} + ω^{2}) x (t) .

With the pole prescription appropriate to the conventions in the question, its inverse kernel is

D (t) = \int \frac{d E}{2 π} \frac{i e ^{- i Et}}{ω ^{2} - E ^{2} - i ϵ} .

(2)

For

t > 0

close the contour in the lower half-plane and for

t < 0

close it in the upper half-plane. The enclosed pole in each case gives

D (t - t^{'}) = \frac{1}{2 ω} e^{iω ∣ t - t^{'} ∣},

(3)

which equivalently satisfies

(\partial_{t}^{2} + ω^{2}) D (t) = i δ (t)

The source-dependent Gaussian functional integral is evaluated by translating the integration variable by the classical sourced solution. Completing the square gives

Z_{0} [J] = Z_{0} [0] exp [- \frac{1}{2} \int d t d t^{'} J (t) D (t - t^{'}) J (t^{'})] .

(4)

Changes in the sign of the source term or of the path-integral phase move factors of

i

between

D

and the exponent but leave the contraction rules equivalent.

Solution

 0  0

Replace each occurrence of

x (t)

in the interaction by the functional derivative that inserts it. In standard Minkowski source conventions,

Z [J] = exp [- \frac{iλ}{3 !} \int d t (\frac{1}{i} \frac{δ}{δ J ( t )})^{3}] Z_{0} [J]

(1)

and hence

Z [J] = \sum_{n = 0}^{\infty} \frac{1}{n !} [- \frac{iλ}{3 !} \int d t (\frac{1}{i} \frac{δ}{δ J ( t )})^{3}]^{n} Z_{0} [J] .

(2)

The factors of

i

are adjusted together if one uses the source convention of part a directly.

This is a perturbation series, generally an asymptotic rather than convergent series because the number of Wick contractions grows factorially. For real cubic coupling the potential is also unbounded on one side, so the real-axis theory does not possess a stable nonperturbative ground state without a contour prescription or further stabilizing interactions.

c

 0  0

Solution

 0  0

The time-domain Feynman rules are:

each internal line joining times $t, t^{'}$ contributes $D (t - t^{'})$ ;
each cubic vertex contributes $- iλ$ and an integration $\int d t$ ;
divide by the graph's Feynman-diagram symmetry factor;
attach the external times to the corresponding lines and retain connected graphs.

There is no order-

λ

connected two-point correction. Through order

λ^{2}

, the two connected topologies are the two-vertex fish graph and the one-particle-reducible tadpole graph. With the displayed vertex convention,

⟨ T x (t_{1}) x (t_{2}) ⟩_{conn} = D (t_{1} - t_{2}) + \frac{( - iλ ) ^{2}}{2} \int d t d t^{'} D (t_{1} - t) D (t - t^{'})^{2} D (t^{'} - t_{2}) + \frac{( - iλ ) ^{2}}{2} \int d t d t^{'} D (t_{1} - t) D (t_{2} - t) D (t - t^{'}) D (0) + O (λ^{4}),

(1)

up to the common factors of

i

associated with the propagator convention. Vacuum normalization removes disconnected vacuum bubbles.

These integrals are ultraviolet finite in one time dimension: a harmonic-oscillator propagator behaves as

E^{- 2}

at large frequency, and the loop-frequency integrals have negative superficial degree of divergence. They are also infrared finite because

ω > 0

supplies a gap. The cubic instability affects nonperturbative convergence but does not create a divergence in these fixed-order integrals.

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 1

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