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 Incoming links: Complex scalar quartic contact vertex

Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 62 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

For the complex scalar field, the canonical momenta are

π = \dot{ϕ}^{†}

and

π^{†} = \dot{ϕ}

. Split

H = H_{0} + H_{I}

. The interaction picture is obtained from the Schrödinger picture by

∣ Ψ_{I} (t)⟩ = e^{i H_{0} t} ∣ Ψ_{S} (t)⟩

and

O_{I} (t) = e^{i H_{0} t} O_{S} e^{- i H_{0} t}

, choosing a common time origin. The fields therefore obey the free Klein-Gordon equation, while the states evolve through

H_{I} (t)

. The canonical quantization of a complex scalar field gives

ϕ_{I} (x) = \int d Π_{p} [a (p) e^{- i p \cdot x} + b^{†} (p) e^{i p \cdot x}], d Π_{p} = \frac{d ^{3} p}{( 2 π ) ^{3} 2 E _{p}} .

The adjoint expansion contains

a^{†}

and

b

. Particle and antiparticle modes are independent; their nonzero commutators are

[a (p), a^{†} (q)] = [b (p), b^{†} (q)] = (2 π)^{3} 2 E_{p} δ^{(3)} (p - q) .

All cross-species commutators, and all pairs of creation operators or of annihilation operators, vanish. These formulas reproduce

[ϕ (x), π (y)] = [ϕ^{†} (x), π^{†} (y)] = i δ^{(3)} (x - y)

and the remaining vanishing equal-time commutators.

There are no derivative interactions, so the interaction Hamiltonian is

H_{I} (t) = - \int d^{3} x L_{I} (x)

. Differentiating the interaction picture state gives

i \partial_{t} ∣ Ψ_{I} ⟩ = H_{I} (t) ∣ Ψ_{I} ⟩

. Its time-evolution operator satisfies

U (t, t_{0}) = I - i \int_{t_{0}}^{t} d t_{1} H_{I} (t_{1}) U (t_{1}, t_{0}) .

Iteration gives ordered integration regions

t_{1} > \dots > t_{n}

. Each is

1/ n!

of the full region with a time-ordered product, yielding the Dyson series

U (t, t_{0}) = \sum_{n = 0}^{\infty} \frac{( - i ) ^{n}}{n !} \int_{t_{0}}^{t} d t_{1} \dots d t_{n} T [H_{I} (t_{1}) \dots H_{I} (t_{n})] .

With the usual asymptotic scattering prescription, taking

t_{0} \to - \infty

and

t \to + \infty

proves

S = T exp (i \int d^{4} x L_{I} (x)) .

Time ordering is essential because different interaction Hamiltonians need not commute.

At first order, use the connected tree scattering amplitude, equivalently the normal-ordered quartic vertex with the vacuum and one-particle mass contributions removed. This qualification matters if the local product is interpreted literally without normal ordering: single-vertex tadpole diagrams and vacuum diagrams are not the requested connected two-particle amplitude. The complex scalar quartic contact vertex is obtained from

S - I = - \frac{iλ}{4} \int d^{4} x : ϕ^{† 2} (x) ϕ^{2} (x) : + O (λ^{2}) .

In a fully connected matrix element, an incoming particle and outgoing antiparticle attach to the two

ϕ

factors; an outgoing particle and incoming antiparticle attach to the two

ϕ^{†}

factors. Each pair has two assignments, giving

2! 2! = 4

. The external relativistic normalization of a one-particle state cancels each mode measure: for example

[ϕ (x), a^{†} (p_{1})] = e^{- i p_{1} x}

and

[b (q_{2}), ϕ (x)] = e^{i q_{2} x}

. Consequently the connected integrand is

4 e^{- i (p_{1} + q_{1} - p_{2} - q_{2}) \cdot x}

, and

⟨ p_{2} q_{2} ∣ (S - I) ∣ p_{1} q_{1} ⟩_{conn} = - iλ (2 π)^{4} δ^{(4)} (p_{1} + q_{1} - p_{2} - q_{2}) + O (λ^{2}) .

Thus the invariant connected amplitude is

T = - λ + O (λ^{2}) .

The Dirac delta function expresses four-momentum conservation.

For the identity contribution, commute

a (p_{2})

through

a^{†} (p_{1})

and independently

b (q_{2})

through

b^{†} (q_{1})

; all terms leaving a creation operator against the left Fock vacuum or an annihilation operator against the right Fock vacuum vanish. Hence

⟨ p_{2} q_{2} ∣ I ∣ p_{1} q_{1} ⟩ = (2 π)^{6} (2 E_{p_{1}}) (2 E_{q_{1}}) δ^{(3)} (p_{2} - p_{1}) δ^{(3)} (q_{2} - q_{1}) .

There is no exchange term between the two distinct particle and antiparticle species.

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