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Fermionic operator Created 2026-10-06 Updated 2026-10-14

A fermionic operator is odd under fermion parity: if

P = (- 1)^{N}

is the parity operator, then

P A P = - A

. A fermionic creation operator or fermionic annihilation operator is an example. Interchanging odd insertions in a time-ordered product contributes the corresponding fermionic sign. Odd parity alone does not imply that every pair of such operators has zero anticommutator.

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Free scalar four-point function Created 2026-10-06 Updated 2026-10-14

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For a centered free real scalar field, its vacuum time-ordered product has expectation

⟨ T ϕ_{1} ϕ_{2} ϕ_{3} ϕ_{4} ⟩ = G_{12} G_{34} + G_{13} G_{24} + G_{14} G_{23}

, where

G_{ij}

is the Feynman propagator between insertions

i, j

. These are the three complete pairings in the Wick theorem. All remaining normal-ordered products have zero vacuum expectation. No fermionic interchange signs occur for a bosonic field.

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Normal-ordered product 2026-10-06

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A normal-ordered product of free-field oscillator operators places all creation operators to the left of all annihilation operators, including the fermionic sign required to reorder fermions. Its vacuum expectation vanishes if any nontrivial oscillator factors remain. The Wick theorem expresses a free-field time-ordered product through these products and all Wick contractions.

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Operator formalism 2026-10-06

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The operator formalism describes a quantum system using states in a Hilbert space and observables represented by self-adjoint operators with suitable domains on that Hilbert space. Evolution and correlation functions are computed from Hamiltonian operators and their time-ordered products. For free quantized fields, the oscillator construction gives a Fock space. The operator-path-integral equivalence relates these matrix elements to integrals over field histories.

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 59 / 3 / Solution Created 2026-10-03 Updated 2026-10-14

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The real scalar field has canonical momentum

π = \dot{ϕ}

. Its Legendre transform gives

H = \int d^{3} x [\frac{1}{2} π^{2} + \frac{1}{2} (\nabla ϕ)^{2} + \frac{1}{2} m^{2} ϕ^{2} + \frac{λ}{4 !} ϕ^{4}] .

(1)

The first three terms constitute

H_{0}

, the free massive Hamiltonian operator. The remaining term is

H_{I} = λ \int ϕ^{4} d^{3} x /4! = - \int L_{I} d^{3} x

. This sign is a consequence of having no time derivatives in the interaction: the canonical momentum is unchanged, while the interaction enters the Hamiltonian density with the opposite sign to the Lagrangian density.

In the interaction picture, operators evolve with

H_{0}

and states evolve with

H_{I} (t)

constructed from the free interaction-picture field. Their evolution operator satisfies

i \partial_{t} U_{I} (t, t_{0}) = H_{I} (t) U_{I} (t, t_{0}), U_{I} (t_{0}, t_{0}) = I .

(2)

Integrating once gives

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

. Repeated substitution orders the integrations as

t \geq t_{1} \geq \dots \geq t_{n} \geq t_{0}

. Replacing the ordered simplex by the full integration cube introduces

1/ n!

and a time-ordered product, which proves the Dyson series

U_{I} (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})} .

(3)

Taking the asymptotic limits with the usual adiabatic or regulated scattering prescription yields

S = T exp [- i \int_{- \infty}^{\infty} H_{I} (t) d t] = T exp [i \int d^{4} x L_{I} (ϕ_{I} (x))] .

(4)

The field in this expression is the free field evolving in the interaction picture, not an independently prescribed classical field. For quantum fields the expansion is a regulated perturbation series; the time-ordering symbol cannot be dropped because the operators at different times need not commute.

Use relativistically normalized two-particle states

∣ p_{1}, p_{2} ⟩ = a^{†} (p_{1}) a^{†} (p_{2}) ∣0 ⟩

, with

⟨ p ∣ q ⟩ = (2 π)^{3} 2 E_{p} δ^{3} (p - q)

. The connected first-order scattering contribution is

⟨ p_{3}, p_{4} ∣ (S - I) ∣ p_{1}, p_{2} ⟩_{conn} = - \frac{iλ}{4 !} \int d^{4} x ⟨ p_{3}, p_{4} ∣ : ϕ_{I} (x)^{4} : ∣ p_{1}, p_{2} ⟩ .

(5)

Here the connected S-matrix uses a normalized vacuum and physical external one-particle states. At this order, normal ordering the vertex separates the four-external-leg contribution from vacuum and tadpole pieces; those disconnected or external-mass contributions are not the two-to-two scattering amplitude.

Two fields annihilate the incoming particles and two create the outgoing ones. There are

(2 4)

choices of the annihilating fields, two assignments of the incoming momenta and two assignments of the outgoing momenta. Hence the total factor is

(2 4) 2! 2! = 4!

, the external-leg factorial cancellation at a quartic scalar vertex. Each attachment contributes its plane-wave factor, so

⟨ p_{3}, p_{4} ∣ : ϕ_{I} (x)^{4} : ∣ p_{1}, p_{2} ⟩_{conn} = 4! e^{i (p_{3} + p_{4} - p_{1} - p_{2}) \cdot x} .

(6)

The spacetime integral is the four-momentum conservation delta function, giving

⟨ p_{3}, p_{4} ∣ (S - I) ∣ p_{1}, p_{2} ⟩_{conn} = i (2 π)^{4} δ^{4} (p_{3} + p_{4} - p_{1} - p_{2}) T, T = - λ + O (λ^{2}) .

(7)

In particular, the tree scattering amplitude is independent of angle.

For completeness, derive the relativistic two-body phase space in the centre-of-mass frame. Write

s = 2 E_{*}

p_{*}^{2} = E_{*}^{2} - m^{2}

. The spatial delta function sets the two final momenta opposite, and the energy delta function is

δ (s - 2 p^{2} + m^{2})

. Its radial Jacobian is

E_{*} / (2 p_{*})

. Thus

d Φ_{2} = \frac{1}{16 π ^{2}} \frac{p ^{2} d p}{E _{p}^{2}} δ (s - 2 E_{p}) d Ω = \frac{p _{*}}{16 π ^{2} s} d Ω.

(8)

The invariant flux factor is

F = 4 p_{*} s

for equal initial masses. Using

d σ = ∣ T ∣^{2} d Φ_{2} / F

, the labeled angular density is

λ^{2} / (64 π^{2} s)

. The outgoing quanta belong to one real scalar field, so they are identical. A full-sphere integration counts each event twice and requires the identical final-state symmetry factor

1/2!

. Consequently

σ = \frac{1}{2} \int_{S^{2}} \frac{λ ^{2}}{64 π ^{2} s} d Ω = \frac{λ ^{2}}{32 π s} + O (λ^{3}), s > 4 m^{2} .

(9)

Equivalently one may integrate the labeled density over a hemisphere representing each event once. The final-state factor is separate from the vertex factorial: applying it at the vertex instead would give the wrong elastic scattering from a quartic scalar contact interaction cross-section.

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 62 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

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Use Minkowski spacetime with signature

(+, -, \dots, -)

and

ℏ = 1

, with a vacuum time-ordering prescription. At a regulator, define the normalized generating functional

Z [J] = \frac{\int D ϕ e ^{i (S [ϕ] + \int J ϕ)}}{\int D ϕ e ^{i S [ϕ]}}, Z [0] = 1.

(1)

The normalized vacuum correlation functions of a time-ordered product are

G_{n} (x_{1}, \dots, x_{n}) = \frac{1}{i ^{n}} \frac{δ ^{n} Z}{δ J ( x _{1} ) \dots δ J ( x _{n} )}_{J = 0} .

(2)

The connected generating functional is

W [J] = - i lo g Z [J], C_{n} = i^{1 - n} \frac{δ ^{n} W}{δ J ( x _{1} ) \dots δ J ( x _{n} )}_{J = 0} .

(3)

The logarithm selects connected source diagrams, or equivalently cumulants. With the assumed vanishing one-point function, derivatives of

Z = e^{iW}

give the centered four-point cumulant decomposition:

G_{1234} = C_{1234} + C_{12} C_{34} + C_{13} C_{24} + C_{14} C_{23} .

(4)

In detail, four differentiations either act on one connected block or form one of three pair partitions. Contributions with blocks of size one vanish. If

W_{ij}

and

W_{1234}

denote source derivatives at zero, the same relation is

G_{1234} = i W_{1234} - W_{12} W_{34} - W_{13} W_{24} - W_{14} W_{23}

, since

C_{2} = - i W_{2}

and

C_{4} = i W_{4}

Let

φ (x) = δ W / δ J (x)

be the source-dependent mean field. On a locally invertible source-to-field branch, define the quantum effective action by the Minkowski Legendre transform

Γ [φ] = W [J] - \int d^{d} x J (x) φ (x) .

(5)

Its variation is

δ Γ = - \int J δ φ

, so

δ Γ/ δ φ = - J

. Differentiating the two inverse source-field maps yields the Minkowski inverse-Hessian relation for an effective action:

\int d^{d} z \frac{δ ^{2} Γ}{δ φ ( x ) δ φ ( z )} \frac{δ ^{2} W}{δ J ( z ) δ J ( y )} = - δ^{(d)} (x - y) .

(6)

Thus

Γ^{(2)} = - (W^{(2)})^{- 1}

. The inverse is understood at the regulator and on a nonsingular fluctuation sector; it is not an assertion that every source-field map is globally invertible.

For a free real scalar field, integrate by parts to write

S [ϕ] = \frac{1}{2} ϕ A ϕ, A = - □ - m^{2}, A_{ϵ} = A + i ϵ,

(7)

where repeated spacetime variables are integrated. Specify the propagator convention explicitly:

Δ_{F} (x - y) = \int \frac{d ^{d} p}{( 2 π ) ^{d}} \frac{i e ^{- i p (x - y)}}{p ^{2} - m ^{2} + i 0}, Δ_{F} = i A_{ϵ}^{- 1} .

(8)

Completing the Gaussian path integral around

ϕ = - A_{ϵ}^{- 1} J

cancels the source-independent determinant between numerator and denominator. Therefore

Z [J] = exp [- \frac{i}{2} J A_{ϵ}^{- 1} J], W [J] = - \frac{1}{2} J A_{ϵ}^{- 1} J = \frac{i}{2} J Δ_{F} J .

(9)

The positive imaginary term in

A_{ϵ}

damps the oscillatory Gaussian. Together with vacuum projection at the time boundaries, it selects the Feynman poles and the time-ordered, in-out correlation function rather than a retarded inverse. It can also be obtained by continuation from the Euclidean vacuum integral. A denominator convention without the numerator

i

redistributes the displayed factors of

i

Finally

φ = - A_{ϵ}^{- 1} J

, hence

J = - A_{ϵ} φ

. Substitution into the Legendre transform gives the vacuum-normalized effective action of a free scalar field:

Γ [φ] = \frac{1}{2} φ A_{ϵ} φ ⟶ S [φ] (ϵ ↓ 0) .

(10)

All connected functions beyond order two vanish and all proper vertices beyond the classical quadratic kernel vanish. Vacuum normalization fixes the additive constant; without it, the quantum effective action equals the classical action up to a field-independent determinant contribution.

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 62 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

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Use units

ℏ = c = 1

and the Minkowski metric

g = diag (1, - 1, - 1, - 1)

. The canonical momentum of the real scalar field is

π = \partial L / \partial \dot{ϕ} = \dot{ϕ}

. Canonical quantization replaces the classical fields by operator-valued distributions and imposes the equal-time canonical commutation relations

[ϕ (t, x), π (t, y)] = i δ^{(3)} (x - y), [ϕ (t, x), ϕ (t, y)] = [π (t, x), π (t, y)] = 0.

(1)

These relations are imposed on an initial slice, with smeared fields understood. In the Heisenberg picture, simultaneous unitary time evolution of both operators preserves their commutators. Since the momentum is

\dot{ϕ}

, the requested equal-time commutator is

[ϕ (t, x), \dot{ϕ} (t, y)] = i δ^{(3)} (x - y)

The Legendre transform in mechanics, applied pointwise, gives the canonical Hamiltonian density of a real scalar field. Thus

H = \frac{1}{2} \int d^{3} x [π^{2} + ∣\nabla ϕ ∣^{2} + m^{2} ϕ^{2}] .

(2)

To obtain its dynamics rather than assume them, use the Heisenberg equation of motion

\dot{O} = i [H, O]

. The momentum-square term gives

[H, ϕ] = - iπ

. The field-square term gives

[H, π] = i m^{2} ϕ

, while differentiating the Dirac delta function in the gradient term and integrating by parts gives

[H, π] = - i \nabla^{2} ϕ + i m^{2} ϕ

. Consequently

\dot{ϕ} = π, \overset{π}{˙} = \nabla^{2} ϕ - m^{2} ϕ, (□ + m^{2}) ϕ = 0.

(3)

Boundary terms vanish for suitably decaying smeared fields. This is the Klein-Gordon equation as an operator identity.

A spatial Fourier transform reduces the Klein-Gordon equation to

\ddot{ϕ}_{p} + E_{p}^{2} ϕ_{p} = 0

, where

E_{p} = p^{2} + m^{2}

. Its two frequency branches and the Hermitian adjoint condition give

ϕ (x) = \int d Π_{p} [a (p) e^{- i p \cdot x} + a^{†} (p) e^{i p \cdot x}], d Π_{p} = \frac{d ^{3} p}{( 2 π ) ^{3} 2 E _{p}}, p^{0} = E_{p} .

(4)

The invariant-normalized scalar mode extraction can be written particularly simply as

a (p) = \int d^{3} x e^{i p \cdot x} [E_{p} ϕ (x) + iπ (x)], a^{†} (p) = \int d^{3} x e^{- i p \cdot x} [E_{p} ϕ (x) - iπ (x)] .

(5)

Substituting the mode expansion of a free field proves these inversion formulas: the opposite-frequency coefficient is proportional to

E_{p} - E_{q}

and vanishes after spatial Fourier inversion. Alternatively they are the corresponding Klein-Gordon inner products. Taking their commutators using the equal-time fields gives

(E_{p} + E_{q}) (2 π)^{3} δ^{(3)} (p - q)

in the mixed case. In the two equal-type cases the delta function enforces

q = - p

, multiplying

E_{p} - E_{q} = 0

. Therefore

[a (p), a^{†} (q)] = (2 π)^{3} 2 E_{p} δ^{(3)} (p - q), [a (p), a (q)] = [a^{†} (p), a^{†} (q)] = 0.

(6)

Conversely these mode commutators reproduce

[ϕ, \dot{ϕ}] = i δ^{(3)}

: the two frequency contributions each supply half the spatial Dirac delta function.

For the covariant oscillator Hamiltonian of a real scalar field, insert the mode expansion of a free field into

H

. Spatial integration sets the two momenta equal in the mixed terms and opposite in the two-creator or two-annihilator terms. The latter have coefficient

- E_{p}^{2} + p^{2} + m^{2} = 0

. The mixed terms give

H = \frac{1}{2} \int d Π_{p} E_{p} [a^{†} (p) a (p) + a (p) a^{†} (p)] = \int d Π_{p} E_{p} a^{†} (p) a (p) + E_{0}, E_{0} = \frac{1}{2} \int d^{3} p E_{p} δ^{(3)} (0) .

(7)

The constant

E_{0}

is the divergent zero-point energy; a finite box and a momentum cutoff make its meaning explicit. Normal ordering sets the free vacuum energy to zero and leaves all commutators unchanged. Directly applying the mode commutators to the mixed expression yields

[H, a (p)] = - E_{p} a (p), [H, a^{†} (p)] = E_{p} a^{†} (p) .

(8)

Choose the Fock vacuum by

a (p) ∣0 ⟩ = 0

. Each creation operator adds a boson of energy

E_{p}

and four-momentum

p

, while each annihilation operator removes one. Products of commuting creation operators give bosonic statistics from commuting creation operators and unrestricted occupation numbers. The real scalar field has one spin-zero species, identical to its antiparticle. Its one-particle states obey

⟨ q ∣ p ⟩ = (2 π)^{3} 2 E_{p} δ^{(3)} (p - q)

, precisely the relativistic normalization of a one-particle state.

For the Feynman propagator, put

z = x - y

and contract the two fields in the Fock vacuum. Only

a a^{†}

survives, giving the Wightman function

W (z) = ⟨ 0∣ ϕ (x) ϕ (y) ∣0 ⟩ = \int d Π_{p} e^{- i p \cdot z} .

(9)

The stipulated time-ordered product convention is

i Δ_{F} (z) = θ (z^{0}) W (z) + θ (- z^{0}) W (- z)

. In a contour integral over

p^{0}

, the Feynman i-epsilon prescription places the positive-energy pole just below the real axis and the negative-energy pole just above it. For

z^{0} > 0

close below, clockwise, obtaining

- i e^{- i E_{p} z^{0}} / (2 E_{p})

; for

z^{0} < 0

close above, obtaining

- i e^{i E_{p} z^{0}} / (2 E_{p})

. Changing

p \mapsto - p

in the second contribution therefore proves

Δ_{F} (z) = ϵ ↓ 0 lim \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{e ^{- i p \cdot z}}{p ^{2} - m ^{2} + i ϵ} .

(10)

Acting with

□ + m^{2}

multiplies the integrand by

- (p^{2} - m^{2})

. The remaining factor tends to

- 1

as a distribution, so Fourier inversion gives

(□ + m^{2}) Δ_{F} (z) = - δ^{(4)} (z) .

(11)

The sign also follows from the derivative jump of a free scalar time-ordered two-point function: the first time derivative of

i Δ_{F}

jumps by

⟨[\dot{ϕ}, ϕ]⟩ = - i δ^{(3)}

, so that of

Δ_{F}

jumps by

- δ^{(3)}

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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

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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}} .

(1)

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) .

(2)

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}) .

(3)

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})] .

(4)

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)) .

(5)

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}) .

(6)

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}) .

(7)

Thus the invariant connected amplitude is

T = - λ + O (λ^{2}) .

(8)

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}) .

(9)

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

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Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 47 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

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Use a Minkowski external source term

+ \int J χ

, where

χ

is the integration variable, and normalize the vacuum functional:

Z [J] = \frac{\int D χ e x p { i S [ χ ] + i \int d ^{4} x J ( x ) χ ( x )}}{\int D χ e ^{i S [χ]}}, W [J] = lo g Z [J] .

(1)

The usual vacuum boundary prescription is implicit. This is the logarithmic convention

Z = e^{W}

for the connected generating functional. It makes the given quadratic expression consistent with

Δ_{F} = ⟨ T χχ ⟩

. Another common convention is

Z = e^{i W_{c}}

, with

W_{c} = - iW

; mixing the two conventions would change the factors of

i

below.

Functional derivative generates the full and connected correlation functions:

G_{n} (x_{1}, \dots, x_{n}) = i^{- n} \frac{δ ^{n} Z}{δ J ( x _{1} ) \dots δ J ( x _{n} )}_{J = 0}, G_{n}^{c} (x_{1}, \dots, x_{n}) = i^{- n} \frac{δ ^{n} W}{δ J ( x _{1} ) \dots δ J ( x _{n} )}_{J = 0} .

(2)

These are time-ordered products in the vacuum, and

G_{1} = G_{1}^{c} = 0

by the stated assumption. Differentiating

e^{W}

four times partitions the four insertions into connected blocks. All partitions with a singleton vanish, leaving

G_{4} (1, 2, 3, 4) = G_{4}^{c} (1, 2, 3, 4) + G_{2}^{c} (1, 2) G_{2}^{c} (3, 4) + G_{2}^{c} (1, 3) G_{2}^{c} (2, 4) + G_{2}^{c} (1, 4) G_{2}^{c} (2, 3) .

(3)

The shorthand arguments stand for spacetime points; this is a functional cumulant identity, not an assumption of a free field.

To construct the quantum effective action, define the source-dependent classical field

φ (x) = ⟨ χ (x) ⟩_{J} = - i \frac{δ W}{δ J ( x )} .

(4)

Locally invert this relation to obtain

J [φ]

, assuming the response integral kernel is invertible with the chosen regulator and boundary prescription. Take its Legendre transform

i Γ [φ] = W [J [φ]] - i \int d^{4} x J [φ] (x) φ (x) .

(5)

The chain-rule terms proportional to

δ J

cancel, so

δ Γ/ δ φ = - J

. A field-independent constant is fixed separately and is immaterial to the vertices. At zero external source the classical field is zero, and there is no linear term.

For the two-point vertex, differentiate the classical field once. If

G = G_{2}^{c}

, then

\frac{δ φ ( x )}{δ J ( y )} = i G (x, y), \frac{δ J ( x )}{δ φ ( y )} = - i G^{- 1} (x, y), \int d^{4} z G (x, z) G^{- 1} (z, y) = δ^{(4)} (x - y) .

(6)

The vertex coefficients in this problem are derivatives of

i Γ

, not of

Γ

alone. Therefore

Γ_{2} (x, y) = - G^{- 1} (x, y) .

(7)

For the three-point vertex, the derivative of the connected two-point integral kernel at a general external source is

\frac{δ G ( a , b )}{δ J ( c )} = i G_{3}^{c} (a, b, c) .

(8)

Combine this with

δ J / δ φ = - i G^{- 1}

and differentiate the inverse identity,

δ G^{- 1} = - G^{- 1} (δ G) G^{- 1}

. This gives the logarithmic-source inverse-kernel vertices relation

Γ_{3} (x, y, z) = \int d^{4} a d^{4} b d^{4} c G^{- 1} (x, a) G^{- 1} (y, b) G^{- 1} (z, c) G_{3}^{c} (a, b, c) .

(9)

All kernels here are finally evaluated at zero external source. The connected three-point function is amputated by its three inverse quantum field theory propagators.

For the specified quadratic

W

, the external source response is

φ = i Δ_{F} J

, hence

J = - i Δ_{F}^{- 1} φ

. Substitution in the transform gives

Γ [φ] = \frac{i}{2} \int d^{4} x d^{4} y φ (x) Δ_{F}^{- 1} (x - y) φ (y), Γ_{2} = - Δ_{F}^{- 1}, Γ_{n} = 0 (n \geq 3) .

(10)

For the standard free quantum field theory propagator

Δ_{F} (p) = i / (p^{2} - m^{2} + i 0)

, this is the usual quadratic free action, up to the vacuum prescription. The generating functional is Gaussian, so connected functions beyond second order vanish and there are no interaction vertices or nontrivial scattering amplitudes. Full higher even-point functions can still be nonzero: they are products of two-point functions, as the four-point formula shows.

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Use signature

(+ - - -)

and covariant one-particle state normalization

⟨ p^{'} ∣ p ⟩ = (2 π)^{3} 2 E_{p} δ^{3} (p - p^{'})

. The future scattering condition uses the outgoing asymptotic scalar field:

ϕ (x) ⟶ Z ϕ_{out} (x) (x^{0} \to + \infty), (\partial^{2} + m^{2}) ϕ_{out} = 0.

(1)

As with the incoming condition, this is understood in suitably smeared scattering matrix elements, not as unrestricted pointwise strong convergence of interacting and free local operators.

The free incoming field has expansion

ϕ_{in} (x) = \int \frac{d ^{3} k}{( 2 π ) ^{3} 2 E _{k}} [a_{in} (k) e^{- ik x} + a_{in}^{†} (k) e^{ik x}], [a_{in} (k), a_{in}^{†} (p)] = (2 π)^{3} 2 E_{k} δ^{3} (k - p) .

(2)

The creation operator defines

∣ p ⟩ = a_{in}^{†} (p) ∣0 ⟩

, with

a_{in} ∣0 ⟩ = 0

, so

⟨ p ∣ ϕ_{in} (x) ∣0 ⟩ = e^{i p x}

. On the other hand, translation invariance gives

ϕ (x) = e^{i P x} ϕ (0) e^{- i P x}, ⟨ p ∣ ϕ (x) ∣0 ⟩ = e^{i p x} ⟨ p ∣ ϕ (0) ∣0 ⟩ .

(3)

Matching the stable one-particle state component in the asymptotic condition fixes the constant overlap. Choose its state phase so that it is real and nonnegative. Then

⟨ p ∣ ϕ (0) ∣0 ⟩ = Z .

(4)

The same stable particle appears in the outgoing description, with compatible one-particle phases. Thus

Z

is the squared overlap measured by wave-function renormalization.

Insert a complete set of four-momentum eigenstates between the fields. For

x^{0} > 0

, translation invariance and Hermiticity give

⟨ 0∣ ϕ (x) ϕ (0) ∣0 ⟩ = \sum_{α} e^{- i p_{α} x} ∣ ⟨ α ∣ ϕ (0) ∣0 ⟩ ∣^{2} .

(5)

For

x^{0} < 0

, reversing the time-ordered product gives the same sum with

e^{i p_{α} x}

. Therefore, with continuous-state integrals implicit in the completeness sum,

i Δ (x) = \sum_{α} ∣ ⟨ α ∣ ϕ (0) ∣0 ⟩ ∣^{2} [θ (x^{0}) e^{- i p_{α} x} + θ (- x^{0}) e^{i p_{α} x}] .

(6)

Here

θ

is the Heaviside step function.

For the usual spectral representation, take a Lorentz-invariant vacuum, a positive-norm Hilbert space and the future-cone spectrum condition. Also take

⟨ 0∣ ϕ ∣0 ⟩ = 0

, as for the unbroken real scalar field, or subtract this constant and use the connected correlation function. Otherwise the vacuum intermediate state adds a disconnected constant not represented by the ordinary mass-shell measure below.

Let

W (x) = ⟨ 0∣ ϕ (x) ϕ (0) ∣0 ⟩

for that centered field, and use the Fourier transform convention

W (x) = \int d^{4} p (2 π)^{- 4} e^{- i p x} W (p)

. Define the Källén–Lehmann spectral density by the distributional equality

W (p) = (2 π)^{4} \sum_{α} ∣ ⟨ α ∣ ϕ (0) ∣0 ⟩ ∣^{2} δ^{4} (p - p_{α}) = 2 π θ (p^{0}) ρ (p^{2}) .

(7)

Lorentz invariance makes the weight on each future mass shell a function or measure of

p^{2}

; the spectrum condition gives

p^{2} \geq 0

. Its nonnegativity follows from the displayed squared matrix elements. Thus

W (x) = \int_{0}^{\infty} d σ ρ (σ) \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 p ^{2} + σ} e^{- i p x} .

(8)

Applying time ordering to this expression mass shell by mass shell proves the Källén–Lehmann spectral representation:

Δ (x) = \int_{0}^{\infty} d σ ρ (σ) Δ_{F} (x, σ), Δ_{F} (p, σ) = \frac{1}{p ^{2} - σ + i 0} .

(9)

The factor

i

is outside

Δ_{F}

under the convention in the question.

The stable one-particle state part of completeness is

\int d^{3} p [(2 π)^{3} 2 E_{p}]^{- 1} ∣ p ⟩ ⟨ p ∣

. Its overlap is

Z

, so its contribution to

W

Z

times the free mass-

m

two-point function. Hence

ρ (σ) = Z δ (σ - m^{2}) + ρ_{rest} (σ), ρ_{rest} \geq 0.

(10)

Other isolated particles as well as multiparticle states can belong to

ρ_{rest}

. The full scalar propagator accordingly has isolated pole

Z / (p^{2} - m^{2} + i 0)

, making

Z

its pole residue.

Finally use canonical field normalization: the canonical momentum is

\dot{ϕ}

and the equal-time canonical commutation relation is

[\dot{ϕ} (t, x), ϕ (t, y)] = - i δ^{3} (x - y)

. The vacuum commutator

C (x) = W (x) - W (- x)

has the same spectral decomposition. For each free mass shell,

\partial_{x^{0}} C_{σ} (x) ∣_{x^{0} = 0} = - \frac{i}{2} \int \frac{d ^{3} p}{( 2 π ) ^{3}} (e^{i p \cdot x} + e^{- i p \cdot x}) = - i δ^{3} (x),

(11)

independently of

σ

. Comparing with the interacting equal-time commutator gives the canonical scalar spectral sum rule:

\int_{0}^{\infty} ρ (σ) d σ = 1, 0 \leq Z \leq 1.

(12)

For the particle genuinely interpolated by the stated asymptotic field,

Z > 0

; the general nonnegative bound allows a field with zero overlap. The total weight and the upper bound require this canonical normalization and a well-defined distributional equal-time limit. They do not follow from the asymptotic condition alone: rescaling

ϕ

rescales both

Z

and the entire spectral measure.

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For one Grassmann variable, define the Berezin integral by

\int d θ 1 = 0

and

\int d θ θ = 1

, extending linearly. Since

θ^{2} = 0

, these rules determine it on every polynomial. With several Grassmann variables, integrate from the rightmost differential first. Equivalently,

\int d θ_{n} \dots d θ_{1}

extracts the coefficient of

θ_{1} \dots θ_{n}

in that order. Repeated indices give zero, while permuting distinct generators gives the fermionic sign of that permutation. Hence

\int d θ_{n} \dots d θ_{1} θ_{i_{1}} \dots θ_{i_{n}} = ε_{i_{1} \dots i_{n}}, ε_{1 \dots n} = 1,

(1)

where

ε

is the Levi-Civita symbol.

For the doubled variables, fix the paired measure orientation by

\int \prod_{i = 1}^{n} d \overset{ˉ}{θ}_{i} d θ_{i} θ_{1} \overset{ˉ}{θ}_{1} \dots θ_{n} \overset{ˉ}{θ}_{n} = 1.

(2)

The exponential in the Grassmann Gaussian integral has a finite Taylor expansion. Only its term of degree

2 n

survives integration. Each barred and unbarred index must occur exactly once; the

n!

orders of the even bilinears cancel the Taylor factorial. Reordering the unbarred indices gives the determinant permutation sign. Therefore

Z (B) = \int \prod_{i} d \overset{ˉ}{θ}_{i} d θ_{i} e^{- \overset{ˉ}{θ} Bθ} = \sum_{σ \in S_{n}} sgn (σ) \prod_{i} B_{iσ (i)} = det B .

(3)

This identity holds even for a singular matrix. The normalized expectations below require

B

invertible.

Put

C = B^{- 1}

. Differentiating with respect to

B_{ji}

inserts

- \overset{ˉ}{θ}_{j} θ_{i} = θ_{i} \overset{ˉ}{θ}_{j}

. The derivative of a determinant then gives the ordered Grassmann Gaussian contractions

⟨ θ_{i} \overset{ˉ}{θ}_{j} ⟩ = \frac{1}{det B} \frac{\partial det B}{\partial B _{ji}} = C_{ij} .

(4)

Differentiate once more, using

\partial C_{ij} / \partial B_{l k} = - C_{i l} C_{kj}

. The result is

⟨ θ_{i} \overset{ˉ}{θ}_{j} θ_{k} \overset{ˉ}{θ}_{l} ⟩ = C_{ij} C_{k l} - C_{i l} C_{kj} .

(5)

The minus sign is the interchange sign in the fermionic Wick theorem. In particular

⟨ \overset{ˉ}{θ} Mθ ⟩ = - tr (MC)

. Reordering each barred-unbarred pair in the product and contracting the indices gives

⟨(\overset{ˉ}{θ} Mθ) (\overset{ˉ}{θ} Nθ)⟩ = tr (MC) tr (NC) - tr (MCNC) .

(6)

The covariance of Grassmann bilinears is the connected second term. Its Feynman diagram is one oriented fermion loop through both insertions; the first term has two separate fermion loops, one at each insertion. Each closed fermion loop contributes a minus sign, so two loops give a plus and one gives a minus.

Figure 1.
Connected and disconnected contractions of Grassmann bilinears
. **Details**.

For the Dirac field, write

D = γ \cdot \partial + m

and use signature

(-, +, \dots, +)

, consistent with the denominator in the question. A finite regulator turns the free fermionic path integral into the same Gaussian problem with

B = i D

, because

e^{i S} = e^{- i \overset{ˉ}{ψ} D ψ}

. The time-ordered product covariance is consequently

(i D_{F})^{- 1} = - i D_{F}^{- 1}

, where the subscript specifies the Feynman i-epsilon prescription.

With the Fourier transform convention

ψ (x) = \int d^{d} p (2 π)^{- d} e^{i p \cdot x} ψ (p)

, the derivative kernel is

iγ \cdot p + m

. The gamma matrices obey

(iγ \cdot p + m) (- iγ \cdot p + m) = (p^{2} + m^{2}) I,

(7)

since their anticommutator makes

(γ \cdot p)^{2} = p^{2} I

. Thus the mostly-plus Dirac propagator with a real derivative kernel is

⟨ T ψ (x) \overset{ˉ}{ψ} (0)⟩ = i S_{F} (x), S_{F} (p) = \frac{- iγ \cdot p + m}{- p ^{2} - m ^{2} + i 0} .

(8)

For

p^{2} = - p_{0}^{2} + p^{2}

, the denominator is

p_{0}^{2} - p^{2} - m^{2} + i 0

. Its positive-energy pole is below and negative-energy pole above the real axis, giving time ordering. Here

S_{F}

is defined without the leading factor

i

, as in the question; conventions that include that factor in the name Dirac propagator give the same covariance.

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Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 42 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

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Use

η_{μν} = diag (1, - 1, - 1, - 1)

and

ℏ = c = 1

. In the interaction picture, fields evolve with the free Hamiltonian, while interactions enter the state evolution through the Dyson series. Write the Klein-Gordon field as

ϕ = ϕ^{(+)} + ϕ^{(-)}

, with the positive-frequency part of a quantum field containing annihilation operators and

ϕ^{(-)} = (ϕ^{(+)})^{†}

containing creation operators. The mode normalization in the PDF corresponds to

[a_{p}, a_{q}^{†}] = (2 π)^{3} δ^{3} (p - q)

and

E_{p} = ∣ p ∣^{2} + m^{2}

The time-ordered product places fields with the latest time on the left. For two bosonic insertions,

T {ϕ (x) ϕ (y)} = θ (x^{0} - y^{0}) ϕ (x) ϕ (y) + θ (y^{0} - x^{0}) ϕ (y) ϕ (x) .

(1)

The normal-ordered product

: ϕ (x_{1}) \dots ϕ (x_{n}) :

instead moves every creation operator to the left of every annihilation operator, without adding the commutators generated by that rearrangement. Its vacuum expectation value is zero whenever a nonempty set of free-field factors remains.

The mode commutator gives the positive-frequency Wightman function

D^{+} (x - y) = [ϕ^{(+)} (x), ϕ^{(-)} (y)] = ⟨ 0∣ ϕ (x) ϕ (y) ∣0 ⟩ = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 E _{p}} e^{- i p \cdot (x - y)} .

(2)

Define the Feynman propagator here to be the two-point function itself, rather than that function divided by

i

. Then

Δ_{F} (x - y) = ⟨ 0∣ T {ϕ (x) ϕ (y)} ∣0 ⟩ = θ (x^{0} - y^{0}) D^{+} (x - y) + θ (y^{0} - x^{0}) D^{+} (y - x) .

(3)

Equivalently,

Δ_{F} (x - y) = \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{i e ^{- i p \cdot (x - y)}}{p ^{2} - m ^{2} + i 0} .

(4)

The positive-energy pole is below and the negative-energy pole above the real axis. Closing the energy contour integral below for positive time separation and above for negative time separation gives the two terms above. Thus this is the scalar Feynman propagator pole prescription, and

(□ + m^{2}) Δ_{F} = - i δ^{4}

. In particular,

T {ϕ (x) ϕ (y)} =: ϕ (x) ϕ (y) : + Δ_{F} (x - y)

: the difference between time ordering and normal ordering is the Wick contraction.

For

n

free bosonic fields, the Wick theorem states

T {ϕ_{1} \dots ϕ_{n}} = P \sum (i, j) \in P \prod Δ_{F} (x_{i} - x_{j}) : r \in / V (P) \prod ϕ_{r} :,

(5)

where

P

runs over all sets of disjoint pairs of indices, including the empty set, and

V (P)

is the set of paired indices. The empty normal-ordered product equals one. This is an operator identity for free fields, understood with smearing or a regulator when products of operator-valued distributions require it.

Here is an inductive proof of the scalar Wick theorem. Sort the arguments chronologically so that

x_{1}^{0}

is the latest. For any remaining set

I

, commuting the annihilation part of

ϕ_{1}

through the creation parts in a normal-ordered product gives

ϕ_{1} : \prod_{r \in I} ϕ_{r} :=: ϕ_{1} \prod_{r \in I} ϕ_{r} : + \sum_{j \in I} [ϕ_{1}^{(+)}, ϕ_{j}^{(-)}] : \prod_{r \in I ∖ {j}} ϕ_{r} : .

(6)

The creation part of

ϕ_{1}

is already in the correct position. Every commutator on the right is a scalar, so no further operator rearrangement is needed for it. Because

x_{1}

is latest, that commutator is

Δ_{F} (x_{1} - x_{j})

. Assume the Wick theorem for the other

n - 1

fields and multiply its expansion by

ϕ_{1}

. The first term of the displayed identity gives every pairing in which index 1 is unpaired; the sum gives every pairing in which 1 is paired with exactly one

j

. Each final partial pairing has exactly one such construction. The one-field identity supplies the base case, proving the theorem by mathematical induction. Applying time ordering restores arbitrary argument order.

Taking the vacuum expectation value retains only complete pairings. Odd free correlators vanish; for example the free scalar four-point function is

Δ_{12} Δ_{34} + Δ_{13} Δ_{24} + Δ_{14} Δ_{23}

. These facts explain why propagators are the edges of Feynman diagrams.

For the interacting quartic scalar field theory, the interaction Hamiltonian density is

λ ϕ^{4} /4!

. With a single time ordering acting on external insertions and interaction vertices, the unnormalized numerator expands as

N_{n} = \sum_{v = 0}^{\infty} \frac{( - iλ ) ^{v}}{v ! ( 4 ! ) ^{v}} \int d^{4} z_{1} \dots d^{4} z_{v} ⟨ 0∣ T {ϕ (x_{1}) \dots ϕ (x_{n}) ϕ (z_{1})^{4} \dots ϕ (z_{v})^{4}} ∣0 ⟩ .

(7)

Apply the proved Wick theorem term by term. Each complete pairing connects external insertions and four-legged vertices by Feynman propagators. There are

4!

assignments of four distinct incident legs at a vertex, canceling its factorial. Permuting identical integrated vertices cancels the corresponding part of

v!

. Permutations still leaving the labeled graph unchanged give its Feynman-diagram symmetry factor. For example,

N_{4, connected}^{(1)} = - iλ \int d^{4} z \prod_{r = 1}^{4} Δ_{F} (x_{r} - z),

(8)

whereas the two-point tadpole diagram has coefficient

- iλ /2

, because choosing the two vertex legs contracted with the labeled external fields gives

4 \cdot 3

possibilities and the other two contract with each other.

Thus the position-space rules attach

Δ_{F}

to every edge,

- iλ \int d^{4} z

to every four-legged interaction vertex, and divide by the graph's Feynman-diagram symmetry factor. Fourier transformation gives the momentum-space Feynman rules:

line: \frac{i}{p ^{2} - m ^{2} + i 0}, vertex: - iλ, loop: \int \frac{d ^{4} ℓ}{( 2 π ) ^{4}} .

(9)

Enforce momentum conservation at each vertex, retaining the overall four-momentum delta function. External lines of these correlation functions remain unamputated. The unnormalized expression includes Vacuum Feynman diagrams; dividing by

N_{0} = ⟨ 0∣ S ∣0 ⟩

cancels those disconnected vacuum factors. They should not be discarded without that normalization.

There is a time-ordering convention in the printed correlation-function notation worth making explicit. The usual interacting correlator uses the single global time ordering in

N_{n}

above. If the displayed

T {ϕ_{1} \dots ϕ_{n}} S

is read literally with

S

outside that ordering, the external block and the vertex block are separately time ordered: their cross-contractions are

D^{+} (x_{i} - z)

because each external field stands to the left of every vertex field. The same pairing construction then uses

Δ_{F}

for lines within either block and

D^{+}

for lines between blocks; in momentum space the latter is

2 π θ (p^{0}) δ (p^{2} - m^{2})

. This literal product is different from the conventional globally time-ordered interacting correlator, and does not have the ordinary external Feynman propagators derived above.

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There are two closely related objects to distinguish. Inverting the quadratic Proca action gives the usual covariant Proca propagator. After integration by parts, its kernel is

K^{ab} = η^{ab} (□ - m^{2}) - \partial^{a} \partial^{b}, K^{ab} (q) = - (q^{2} + m^{2}) η^{ab} + q^{a} q^{b} .

(1)

The matrix inverse with the Feynman i-epsilon prescription gives

Δ_{ab}^{cov} (q) = \frac{- i}{q ^{2} + m ^{2} - i 0} (η_{ab} + \frac{q _{a} q _{b}}{m ^{2}}) .

(2)

Multiplication by

K^{ab}

gives

i δ_{b}^{a}

in the distributional limit. The numerator agrees with the polarization sum for a massive vector boson at the poles, but is not a transverse linear projection at arbitrary four-momentum.

For the literal canonical time-ordered product of

A_{a}

and

A_{b}

, the nondynamical component

A_{0}

produces the Proca time-ordering contact term. In the chosen time coordinate the full answer is

Δ_{ab} (x, y) = \int \frac{d ^{4} q}{( 2 π ) ^{4}} e^{i q \cdot (x - y)} [\frac{- i ( η _{ab} + q _{a} q _{b} / m ^{2} )}{q ^{2} + m ^{2} - i 0} - \frac{i}{m ^{2}} δ_{a}^{0} δ_{b}^{0}] .

(3)

To see the local term directly, the three physical polarization vectors give

\sum_{λ} ϵ_{λ 0}^{2} = ∣ q ∣^{2} / m^{2}

. The canonical

00

two-point correlation function therefore has Fourier transform

Δ_{00} (q) = \frac{- i ∣ q ∣ ^{2}}{m ^{2} ( q ^{2} + m ^{2} - i 0 )} .

(4)

By contrast,

η_{00} + q_{0}^{2} / m^{2} = ∣ q ∣^{2} / m^{2} - (q^{2} + m^{2}) / m^{2}

, so the covariant expression contains an additional

i / m^{2}

. The mixed and spatial components have no additional contact term. Thus

Δ_{ab}^{cov} (x, y) = Δ_{ab} (x, y) + \frac{i}{m ^{2}} δ_{a}^{0} δ_{b}^{0} δ^{(4)} (x - y) .

(5)

T

is used to mean covariant time ordering, commonly denoted

T^{*}

, the conventional answer is instead just

Δ^{cov}

. Both conventions have the same propagating poles and agree away from coincidence; explicitly separating them respects the printed definition as an ordinary time-ordered product.

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For the real scalar field, the mostly-plus Minkowski metric gives a positive kinetic energy. Its canonical momentum and Hamiltonian operator are

π = \frac{\partial L}{\partial ϕ ˙} = \dot{ϕ}, H = \frac{1}{2} \int d^{3} x [π^{2} + (\nabla ϕ)^{2} + m^{2} ϕ^{2}] .

(1)

The Euler-Lagrange field equation is

(□ - m^{2}) ϕ = 0

, or

(\partial_{t}^{2} - \nabla^{2} + m^{2}) ϕ = 0

. Canonical quantization of a real scalar field promotes the fields to Hermitian operators and replaces the equal-time Poisson brackets by canonical commutation relations:

[ϕ (t, x), π (t, y)] = i δ^{(3)} (x - y), [ϕ (t, x), ϕ (t, y)] = [π (t, x), π (t, y)] = 0.

(2)

The Heisenberg equation of motion then gives

\dot{ϕ} = π

and

\overset{π}{˙} = \nabla^{2} ϕ - m^{2} ϕ

, so the classical field equation remains an operator identity. Products at coincident points need a regulator; equivalently, the field is an operator-valued distribution.

Resolve the free field into positive- and negative-frequency plane waves. With

E_{k} = ∣ k ∣^{2} + m^{2}

and

k \cdot x = - E_{k} t + k \cdot x

ϕ (x) = \int \frac{d ^{3} k}{( 2 π ) ^{3} 2 E _{k}} [a (k) e^{ik \cdot x} + a^{†} (k) e^{- ik \cdot x}] .

(3)

Reality pairs the two terms by Hermitian conjugation. The equal-time canonical commutation relations are equivalent to

[a (k), a^{†} (k^{'})] = (2 π)^{3} δ^{(3)} (k - k^{'}), [a, a] = [a^{†}, a^{†}] = 0.

(4)

For example, the two mixed terms in

[ϕ, π]

each supply half the Dirac delta function. Thus the normalization

1/ 2 E_{k}

is fixed, not optional. Each independent mode is a quantum harmonic oscillator, with

a

and

a^{†}

its annihilation operator and creation operator.

Choose the Fock vacuum by

a (k) ∣0 ⟩ = 0

. Repeated creation operators construct the bosonic Fock space. Substitution into the Hamiltonian operator yields

H = \int \frac{d ^{3} k}{( 2 π ) ^{3}} E_{k} a^{†} (k) a (k) + E_{0}, E_{0} = \frac{V}{2} \int \frac{d ^{3} k}{( 2 π ) ^{3}} E_{k},

(5)

where a finite volume

V

can be used as a regulator. Normal ordering sets the flat-space vacuum reference to zero. The spatial momentum operator is

P = \int \frac{d ^{3} k}{( 2 π ) ^{3}} k a^{†} (k) a (k) .

(6)

In particular,

[H, a^{†} (k)] = E_{k} a^{†} (k)

and

[P, a^{†} (k)] = k a^{†} (k)

. A one-particle state therefore has positive energy, momentum

k

, and invariant mass

m

. A scalar field transforms in the trivial spin representation, so its particles have spin zero. Because the field is real there is only one set of oscillators: the particle is its own antiparticle, with no distinct conserved particle-minus-antiparticle charge. The free theory does conserve its occupation-number sum, though generic scalar interactions need not.

A momentum eigenstate extends throughout space. A localized particle is described by a superposition such as

∣ f ⟩ = \int \frac{d ^{3} k}{( 2 π ) ^{3}} f (k) a^{†} (k) ∣0 ⟩, \int \frac{d ^{3} k}{( 2 π ) ^{3}} ∣ f (k) ∣^{2} = 1.

(7)

Its wave packet evolves through the phase

e^{- i E_{k} t}

; a narrow packet has group velocity

v = k / E_{k}

, whose magnitude is at most one. Localization and dispersion concern these superpositions, rather than classical trajectories attached to individual field modes.

Relativistic spacetime behavior is encoded by microcausality. Directly from the oscillator commutators,

[ϕ (x), ϕ (y)] = \int \frac{d ^{3} k}{( 2 π ) ^{3} 2 E _{k}} [e^{ik \cdot (x - y)} - e^{- ik \cdot (x - y)}] .

(8)

This is a Lorentz-invariant distribution and vanishes at equal time. Any spacelike separation can be transformed to an equal-time separation, so

[ϕ (x), ϕ (y)] = 0

for spacelike

x - y

. Thus local operations at spacelike-separated points commute. The vacuum Wightman function need not vanish outside the light cone, but that correlation does not transmit a controllable signal. The Feynman propagator is

⟨ 0∣ Tϕ (x) ϕ (y) ∣0 ⟩ = \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{- i e ^{i p \cdot (x - y)}}{p ^{2} + m ^{2} - i 0},

(9)

which propagates positive energy forward and negative energy backward in the time-ordered product. For causal response one uses the retarded Green function, whose support lies in or on the future light cone.

Finally, commuting creation operators imply

a^{†} (k_{1}) a^{†} (k_{2}) ∣0 ⟩ = a^{†} (k_{2}) a^{†} (k_{1}) ∣0 ⟩ .

(10)

The multiparticle state is symmetric under the particle exchange operator, which is Bose statistics. For a normalized single mode, the occupation states are

(a^{†})^{n} ∣0 ⟩ / n!

with

n = 0, 1, 2, \dots

: any number of identical bosons can occupy it. There is no Pauli exclusion principle for these particles. This construction agrees with the Spin-statistics theorem relating integer spin to bosonic exchange symmetry under relativistic locality and positive-energy assumptions. It demonstrates the required scalar case without assuming that theorem as the quantization prescription.

The quantized real scalar field describes positive-energy, mass- $m$ , spin-zero bosons that are their own antiparticles; commuting creation operators give Bose statistics, and local field commutators enforce microcausality.

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 43 / 1 / Solution Created 2026-10-03 Updated 2026-10-06

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Use natural units and the Minkowski metric

η = diag (1, - 1, - 1, - 1)

. A real scalar field assigns a real variable

ϕ (t, x)

to each spatial point. Its Lagrangian density can be taken to be

L = \frac{1}{2} \partial_{μ} ϕ \partial^{μ} ϕ - V (ϕ) = \frac{1}{2} \dot{ϕ}^{2} - \frac{1}{2} ∣\nabla ϕ ∣^{2} - V (ϕ), S [ϕ] = \int d^{4} x L .

(1)

The principle of stationary action gives the Euler-Lagrange equation

□ ϕ + V^{'} (ϕ) = 0

. With

V (ϕ) = m^{2} ϕ^{2} /2

, this is the Klein-Gordon equation. An additional nonlinear part of

V

describes interactions.

The canonical momentum is

π = \partial L / \partial \dot{ϕ} = \dot{ϕ}

. The Legendre transform in mechanics gives the canonical Hamiltonian density of a real scalar field

H = π \dot{ϕ} - L = \frac{1}{2} π^{2} + \frac{1}{2} ∣\nabla ϕ ∣^{2} + V (ϕ), H = \int d^{3} x H .

(2)

The Hamiltonian equations

\dot{ϕ} = π

and

\overset{π}{˙} = \nabla^{2} ϕ - V^{'} (ϕ)

recover the same field equation. In canonical quantization, the fields become operators satisfying the equal-time canonical commutation relations

[ϕ (t, x), π (t, y)] = i δ^{3} (x - y), [ϕ, ϕ] = [π, π] = 0.

(3)

A spatial lattice makes the analogy with many coupled quantum-mechanical coordinates precise. Each lattice field value is a coordinate, with its own conjugate momentum. The path integral is another representation of the same quantum evolution.

To see its origin, first consider one coordinate with

H = p^{2} / (2 M) + V (q)

. Split a time interval into

N

steps of length

ε

and insert position and momentum resolutions of the identity. The short-time kernel is

⟨ q_{j + 1} ∣ e^{- i ε H} ∣ q_{j} ⟩ = \int \frac{d p _{j}}{2 π} exp {i p_{j} (q_{j + 1} - q_{j}) - i ε [\frac{p _{j}^{2}}{2 M} + V (q_{j})]} + O (ε^{2}) .

(4)

Multiplying the kernels and integrating over intermediate positions gives the phase-space path integral

K (q_{f}, t_{f}; q_{i}, t_{i}) = \int D q D p exp [i \int_{t_{i}}^{t_{f}} (p \overset{q}{˙} - H (p, q)) d t] .

(5)

The endpoints of

q

are fixed. The momentum integrals are Gaussian integrals; completing the square produces the configuration-space path integral

K (q_{f}, t_{f}; q_{i}, t_{i}) = \int_{q_{i}}^{q_{f}} D q e^{i S [q]}, S [q] = \int_{t_{i}}^{t_{f}} (\frac{M}{2} \overset{q}{˙}^{2} - V (q)) d t .

(6)

At finite slicing its normalization contains

(M / (2 πi ε))^{N /2} \prod_{j = 1}^{N - 1} d q_{j}

. This fixes the composition law and the initial delta-function kernel. One sums over all paths, not merely solutions of the classical equation. Restoring

ℏ

replaces the weight by

e^{i S /ℏ}

; stationary phase explains the emergence of classical trajectories.

For the field, use scalar field configuration eigenstates

∣ φ ⟩

, satisfying

ϕ (x) ∣ φ ⟩ = φ (x) ∣ φ ⟩

. Insert their completeness relations on every time slice. This gives

⟨ φ_{f} ∣ e^{- i H (t_{f} - t_{i})} ∣ φ_{i} ⟩ = \int D ϕ D π exp [i \int d^{4} x (π \dot{ϕ} - H)] .

(7)

The endpoint field configurations are fixed. Integrating the Gaussian momentum variables leaves the scalar field path integral

N \int D ϕ e^{i S [ϕ]}

. The functional measure means a regulated product over the field variables. A spacetime lattice or another ultraviolet cutoff makes this product finite before the continuum limit; interacting continuum calculations may require renormalization. The oscillatory Minkowski weight is an amplitude, not a positive probability density.

For vacuum expectation values, the boundaries must select the vacuum rather than arbitrary field configurations. Long imaginary-time evolution suppresses excited states:

e^{- T H} ∣ n ⟩ = e^{- T E_{n}} ∣ n ⟩

, so after normalization only the lowest-energy component remains as

T \to \infty

. This is vacuum projection by imaginary time. The corresponding Feynman i-epsilon prescription in the real-time integral specifies the vacuum boundary conditions and the poles of the propagator. With

t = - i τ

, the Euclidean path integral has the weight

e^{- S_{E}}

, where

S_{E} = \int d τ d^{3} x [\frac{1}{2} (\partial_{τ} ϕ)^{2} + \frac{1}{2} ∣\nabla ϕ ∣^{2} + V (ϕ)] .

(8)

It is often a useful regulated starting point; analytic continuation returns the vacuum time-ordered quantities.

Introduce a classical source

J (x)

and define the normalized vacuum generating functional

Z [J] = \frac{\int D ϕ e x p i ( S [ ϕ ] + \int d ^{4} x J ( x ) ϕ ( x ) )}{\int D ϕ e ^{i S [ϕ]}}, Z [0] = 1,

(9)

with the same vacuum prescription in numerator and denominator. A functional derivative brings down

i ϕ (x)

. The order of the time slices makes the operator insertion time-ordered. Thus source differentiation inserts time-ordered field operators:

⟨ Ω∣ T {ϕ (x_{1}) \dots ϕ (x_{r})} ∣Ω ⟩ = \frac{1}{i ^{r}} \frac{δ ^{r} Z [ J ]}{δ J ( x _{1} ) \dots δ J ( x _{r} )}_{J = 0} .

(10)

The denominator removes vacuum diagrams and gives normalized expectation values. It is essential that these are time-ordered products; differentiating this vacuum functional does not directly give every possible operator ordering.

The free theory illustrates the method. Its quadratic kernel is

K = - □ - m^{2}

with the vacuum pole prescription, and completing the square gives the Gaussian evaluation of a free scalar generating functional

Z_{0} [J] = exp [- \frac{1}{2} \int d^{4} x d^{4} y J (x) Δ_{F} (x - y) J (y)], Δ_{F} (x - y) = \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{i e ^{- i p \cdot (x - y)}}{p ^{2} - m ^{2} + i 0} .

(11)

Two source derivatives give

Δ_{F} = ⟨ Ω∣ T {ϕ (x) ϕ (y)} ∣Ω ⟩

. Higher derivatives give all pairings, the content of Wick theorem. For an interaction

S_{int}

, one may use path-integral perturbation by source derivatives:

Z [J] = \frac{e x p ( i S _{int} [ \frac{1}{i} \frac{δ}{δ J} ] ) Z _{0} [ J ]}{e x p ( i S _{int} [ \frac{1}{i} \frac{δ}{δ J} ] ) Z _{0} [ J ] ∣ _{J = 0}} .

(12)

Expanding this expression generates Feynman diagrams and their Wick contractions. The connected generating functional

W [J] = - i lo g Z [J]

retains connected contributions; in particular

δ W / δ J = ⟨ ϕ ⟩_{J}

. These functionals turn the computation of field-operator expectations into source differentiation of an ordinary regulated integral.

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Use signature

(+ - - -)

and take the covariant spatial components

A_{i}

as canonical coordinates. The canonical quantization of the electromagnetic field begins with the canonical momenta

Π^{μ} = \frac{\partial L}{\partial ( \partial _{0} A _{μ} )} = - F^{0 μ}, Π^{0} = 0, Π^{i} = F_{0 i} = \dot{A}_{i} - \partial_{i} A_{0} .

(1)

Thus the primary momentum constraint of the electromagnetic potential is

Π^{0} = 0

A_{0}

has no independent velocity. The Hamiltonian obtained by the Legendre transform in mechanics, up to a boundary term, is

H = \int d^{3} x [\frac{1}{2} Π^{i} Π^{i} + \frac{1}{4} F_{ij} F_{ij} - A_{0} \partial_{i} Π^{i}] .

(2)

Preserving the primary constraint requires the Gauss law constraint in gauge theory,

\partial_{i} Π^{i} = 0

. It also follows by varying

A_{0}

. These are two first-class constraints; they generate the gauge freedom and remove two canonical pairs from the four potential components. The reduced phase space has four dimensions per spatial mode, hence two propagating photon degrees of freedom.

Impose Coulomb gauge,

\partial_{i} A_{i} = 0

. With no charges, Gauss's law then gives

\nabla^{2} A_{0} = 0

; vanishing boundary conditions set

A_{0} = 0

. This is radiation gauge. The remaining components are transverse and obey the massless wave equation. Let

ϵ_{i}^{(r)} (k)

r = 1, 2

, be orthonormal transverse polarization vectors. Their photon polarization completeness relation is

k_{i} ϵ_{i}^{(r)} = 0, \sum_{r = 1}^{2} ϵ_{i}^{(r)} ϵ_{j}^{(r) *} = P_{ij}^{T} (k) := δ_{ij} - \frac{k _{i} k _{j}}{∣ k ∣ ^{2}} .

(3)

The canonical transverse photon field is the Hermitian operator

A_{i}^{T} (x) = \sum_{r = 1}^{2} \int \frac{d ^{3} k}{( 2 π ) ^{3} 2 ω _{k}} [ϵ_{i}^{(r)} a_{r} (k) e^{- ik \cdot x} + ϵ_{i}^{(r) *} a_{r}^{†} (k) e^{ik \cdot x}], ω_{k} = ∣ k ∣,

(4)

where

[a_{r} (k), a_{s}^{†} (k^{'})] = (2 π)^{3} δ_{rs} δ^{3} (k - k^{'}), [a_{r}, a_{s}] = [a_{r}^{†}, a_{s}^{†}] = 0.

(5)

The field and its conjugate momentum have the transverse equal-time commutator

[A_{i}^{T} (t, x), Π^{T j} (t, y)] = i δ_{ij}^{T} (x - y), δ_{ij}^{T} (x) = \int \frac{d ^{3} k}{( 2 π ) ^{3}} P_{ij}^{T} (k) e^{i k \cdot x} .

(6)

This is the quantized reduced bracket, or equivalently the Dirac bracket after imposing the constraints and gauge conditions. The normal-ordered Hamiltonian is

\sum_{r} \int d^{3} k ω_{k} a_{r}^{†} a_{r} / (2 π)^{3}

. Its excitations are photons; circular combinations of the two transverse polarizations have helicity

+ 1

and

- 1

. The scalar and longitudinal potential components do not create additional physical photons.

The Feynman propagator is the vacuum expectation of a time-ordered product. The mode expansion directly gives the radiation-gauge photon propagator, with

z = x - y

D_{ij}^{F} (z) = \int \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{P _{ij}^{T} ( k )}{2 ω _{k}} e^{i k \cdot z} [θ (z^{0}) e^{- i ω_{k} z^{0}} + θ (- z^{0}) e^{i ω_{k} z^{0}}] = \int \frac{d ^{4} k}{( 2 π ) ^{4}} \frac{i P _{ij}^{T} ( k )}{k ^{2} + i 0} e^{- ik \cdot z} .

(7)

The first expression comes from the creation-annihilation commutator; the second is its contour-integral representation. The positive-energy pole lies below the real axis and the negative-energy pole above it. In this reduced free-field description, the temporal operator is zero. A photon propagator must specify its gauge; the spatial transverse propagator is not the same tensor as the covariant four-potential propagator.

For the commonly used covariant form, add the gauge fixing term

- (\partial_{μ} A^{μ})^{2} / (2 ξ)

. The resulting Fourier-space kinetic operator is

K^{μν} (k) = - k^{2} η^{μν} + (1 - ξ^{- 1}) k^{μ} k^{ν} .

(8)

The inversion of the gauge-fixed Maxwell kinetic operator gives

K^{μν} D_{ν ρ}^{F} = i δ^{μ}_{ρ}

. In Feynman gauge,

ξ = 1

, the photon propagator is

D_{μν}^{F} (x - y) = \int \frac{d ^{4} k}{( 2 π ) ^{4}} \frac{- i η _{μν}}{k ^{2} + i 0} e^{- ik \cdot (x - y)} .

(9)

Equivalently,

□ D_{μν}^{F} = i η_{μν} δ^{4} (x - y)

with the vacuum pole prescription. For general

ξ

, the momentum-space numerator is

η_{μν} - (1 - ξ) k_{μ} k_{ν} / (k^{2} + i 0)

A covariant canonical realization uses four polarization oscillators with

[a_{r}, a_{s}^{†}] = - (2 π)^{3} η_{rs} δ^{3} (k - k^{'})

. The resulting indefinite inner product is auxiliary. In Gupta-Bleuler quantization, impose

(\partial_{μ} A^{μ})^{(+)} ∣ phys ⟩ = 0

and take the Gupta-Bleuler null-state quotient. The scalar-longitudinal combination is thereby removed from the physical state space, leaving the same two transverse photon states. Thus the four-component Feynman-gauge numerator does not imply four physical polarization states.

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 301 / 1 / Solution Created 2026-10-03 Updated 2026-10-20

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Use units

ℏ = c = 1

, the Minkowski metric

η_{ab} = diag (1, - 1, - 1, - 1)

, and the Fourier transform convention

f (x) = \int d^{4} p e^{- i p \cdot x} f (p) / (2 π)^{4}

. Here gamma matrices satisfy

{γ^{a}, γ^{b}} = 2 η^{ab} I_{4}

, and

\overset{ˉ}{ψ} = ψ^{†} γ^{0}

is the Dirac adjoint. A single Dirac field describes both electrons and Positrons; these are the particle and antiparticle sectors of that field.

The free Maxwell Lagrangian and Dirac action, with a linear covariant gauge condition for the photon, give

S_{0} = \int d^{4} x [- \frac{1}{4} F_{ab} F^{ab} - \frac{1}{2 α} (\partial_{a} A^{a})^{2} + \overset{ˉ}{ψ} (i γ^{a} \partial_{a} - m) ψ], F_{ab} = \partial_{a} A_{b} - \partial_{b} A_{a} .

(1)

Before gauge fixing, the Maxwell Lagrangian has zero modes in field theory along

A_{a} \mapsto A_{a} + \partial_{a} λ

, so its quadratic operator cannot be inverted on all potentials. In the linear Lorenz gauge, the Faddeev-Popov determinant is

det □

. It is independent of

A

and may be absorbed into the normalization; the corresponding Faddeev-Popov ghost fields have no interacting vertices in this Abelian linear gauge. A residual gauge symmetry is removed by the specified boundary conditions.

The free generating functional is

Z_{0} [J, \overset{η}{ˉ}, η] = N \int D A D \overset{ˉ}{ψ} D ψ exp (i S_{0} + i \int d^{4} x [J_{a} A^{a} + \overset{η}{ˉ} ψ + \overset{ˉ}{ψ} η]), Z_{0} [0] = 1.

(2)

The Dirac field variables and their sources are independent Grassmann fields in this path integral. They anticommute; replacing them by ordinary commuting fields would give the wrong statistics and the wrong functional determinant. The bosonic Gaussian functional integral contributes an inverse square root of a determinant, and the Grassmann Gaussian integral contributes a determinant. If

K_{A}

is the photon quadratic operator and

K_{D} = i γ^{a} \partial_{a} - m

, completing the square gives

Z_{0} = exp [- \frac{i}{2} \int J K_{A}^{- 1} J - i \int \overset{η}{ˉ} K_{D}^{- 1} η] = exp [- \frac{1}{2} \int J D_{F} J - \int \overset{η}{ˉ} S_{F} η] .

(3)

In the last expression

D_{F} = i K_{A}^{- 1}

and

S_{F} = i K_{D}^{- 1}

are the photon propagator and Dirac propagator, with Feynman i-epsilon prescription. The products include the appropriate spacetime integrals and index contractions. Functional derivatives with respect to

J

, and consistently ordered left or right Grassmann derivatives with respect to the fermionic sources, generate the time ordering of the corresponding fields. In Feynman gauge,

α = 1

, the momentum-space two-point functions are

D_{F}^{ab} (p) = \frac{- i η ^{ab}}{p ^{2} + i 0}, S_{F} (p) = \frac{i ( \neq p + m )}{p ^{2} - m ^{2} + i 0}, \neq p = γ^{a} p_{a} .

(4)

The Feynman i-epsilon prescription specifies vacuum boundary conditions rather than an arbitrary inverse of the differential operator.

The covariant photon propagator uses four potential components. Its operator formalism counterpart is Gupta-Bleuler quantization: impose

(\partial_{a} A^{a})^{(+)} ∣ phys ⟩ = 0

and take the Gupta-Bleuler null-state quotient. This leaves a positive physical state space with two transverse photon polarization vectors. The temporal and longitudinal oscillator components occur in intermediate covariant expressions; the Ward identity removes their dependence from physical amplitudes. The free Dirac field uses the canonical anticommutation relations, producing the same fermionic signs as its Grassmann fields in the path integral.

The operator-path-integral equivalence can be seen directly with a regulator. Divide time into small intervals and insert complete sets of field-coordinate states for bosons, and resolutions in fermionic coherent states for fermions. The bosonic matrix elements produce the phase-space factor

exp i \sum [π Δ ϕ - H Δ t]

; integrating out the quadratic canonical momentum produces the bosonic action. The fermionic coherent state overlaps produce the first-order term

i ψ^{†} \dot{ψ}

and the Berezin integral measure. Multiplying the short-time kernels recovers the path integral. Projecting the remote endpoints onto the Fock vacuum with an infinitesimal damping selects the same Feynman propagators as the operator formalism. Field insertions become time-ordered products under this construction. Conversely, their quadratic generating functional obeys the canonical free-field equations and has precisely the oscillator two-point functions, so its higher free correlators agree by the Wick theorem. This establishes the equivalence for the regulated free theory and order by order in the perturbation series.

To couple the matter field electromagnetically, promote its global phase symmetry to the local transformation

ψ \mapsto e^{- i e λ (x)} ψ, \overset{ˉ}{ψ} \mapsto \overset{ˉ}{ψ} e^{i e λ (x)}, A_{a} \mapsto A_{a} + \partial_{a} λ .

(5)

The gauge covariant derivative

D_{a} = \partial_{a} + i e A_{a}

obeys

D_{a}^{'} ψ^{'} = e^{- i e λ} D_{a} ψ

. Replacing

\partial_{a}

D_{a}

in the Dirac action therefore gives the invariant matter density

\overset{ˉ}{ψ} (i γ^{a} D_{a} - m) ψ = \overset{ˉ}{ψ} (i γ^{a} \partial_{a} - m) ψ - e \overset{ˉ}{ψ} γ^{a} ψ A_{a} .

(6)

The electromagnetic field tensor is unchanged by the local transformation. Thus the unfixed quantum electrodynamics action is gauge-invariant. The Dirac current

j^{a} = \overset{ˉ}{ψ} γ^{a} ψ

is conserved by the matter equations, and the electron and Positron excitations carry opposite charges. The added gauge fixing density selects a representative and is not itself invariant under arbitrary local transformations; it does not change gauge-invariant observables. At a free fermion vertex,

q_{a} γ^{a} = (\neq p + \neq q - m) - (\neq p - m) .

(7)

For external on-shell Dirac spinors the corresponding current contraction vanishes. The quantum extension is the Ward identity, which makes physical amplitudes insensitive to adding a multiple of the photon momentum to its polarization vector. A gauge-compatible regularization preserves this vector-current identity.

With

L_{int} = - e \overset{ˉ}{ψ} γ^{a} ψ A_{a}

, expand

exp (i \int L_{int})

in powers of

e

. A term of order

n

contains

n

spacetime integrations and

1/ n!

. Applying the Wick theorem pairs the free fields: an

A

A

Wick contraction supplies a photon propagator, and a

ψ

\overset{ˉ}{ψ}

Wick contraction supplies an oriented Dirac propagator. Each insertion supplies an interaction vertex. The permutations of contractions cancel the expansion factorials except for the Feynman-diagram symmetry factor. Interchanging Grassmann fields produces the fermionic sign, including a minus sign for every closed fermion loop. This is how Feynman diagrams arise from expectation values, rather than an extra dynamical assumption.

The normalized vacuum generating functional removes components with no external insertions. In the operator formalism, for an interacting-vacuum expectation value of an inserted product

O

, the same cancellation appears as

⟨ Ω∣ T O ∣Ω ⟩ = \frac{⟨ 0∣ T { O _{I} e x p ( i \int d ^{4} x L _{int, I} )} ∣0 ⟩}{⟨ 0∣ T e x p ( i \int d ^{4} x L _{int, I} ) ∣0 ⟩} .

(8)

Vacuum projection and the Feynman i-epsilon prescription are implicit. The linked-cluster theorem exponentiates all connected vacuum bubbles into the same factor in numerator and denominator, so it cancels. Normalize by $Z [0]$ to remove every vacuum component. This cancellation of vacuum bubbles still leaves products of disconnected diagrams that each contain external insertions. If only connected correlators are wanted, differentiate the connected generating functional

W = - i lo g [Z [J, \overset{η}{ˉ}, η] / Z [0]]

The resulting momentum-space QED Feynman rules for the bare theory can be stated in Feynman gauge as follows:

An internal oriented fermion line of four-momentum $p$ contributes $i (\neq p + m) / (p^{2} - m^{2} + i 0)$ .
An internal photon line contributes $- i η_{ab} / (p^{2} + i 0)$ .
A one-photon two-fermion interaction vertex contributes $- i e γ^{a}$ , with its spinor and photon indices attached to the incident lines.
Each interaction vertex conserves four-momentum. With all incident momenta treated as incoming, include $(2 π)^{4} δ^{4} (\sum p)$ ; after using these constraints, integrate every independent loop four-momentum as $\int d^{4} ℓ / (2 π)^{4}$ .
Contract the gamma matrix and Dirac propagator factors in their order along the fermion line. A closed fermion loop has a trace and a factor $- 1$ . A relative odd permutation of external fermions also contributes $- 1$ .
Divide each labelled diagram by its Feynman-diagram symmetry factor and sum the allowed diagrams. Omit vacuum components as explained above.
For an amputated scattering amplitude, an incoming electron has $u_{s} (p)$ and an outgoing electron has $\overset{u}{ˉ}_{s} (p)$ ; an incoming Positron has $\overset{v}{ˉ}_{s} (p)$ and an outgoing Positron has $v_{s} (p)$ . An incoming photon has $ϵ_{a} (p)$ and an outgoing photon has $ϵ_{a}^{*} (p)$ . The external polarization vectors are physical and transverse, and external four-momenta are on shell. This last step follows from the LSZ reduction formula and amputation of external propagators; an unamputated correlator keeps its external quantum field theory propagators.

Figure 1.
An oriented electron line meets a photon at the QED interaction vertex
. **Details**.

The illustrated interaction vertex has incoming fermion momentum

p

, incoming photon momentum

k

, and outgoing fermion momentum

p^{'} = p + k

. Its solid-line arrows indicate fermion flow. The photon wavy line has no fermion-flow arrow. The propagators, the vertex $- i e γ^{a}$ , momentum conservation, and the fermionic signs determine the perturbative amplitudes.

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 301 / 3 / Solution Created 2026-10-03 Updated 2026-10-06

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Use

η_{ab} = diag (1, - 1, - 1, - 1)

and

ℏ = 1

. The free real scalar field has action

S = \frac{1}{2} \int d^{4} x [\partial_{a} ϕ \partial^{a} ϕ - m^{2} ϕ^{2}] .

(1)

The canonical momentum is

π = \partial L / \partial \dot{ϕ} = \dot{ϕ}

, and the Hamiltonian operator is obtained by the Legendre transform of the density:

H = \frac{1}{2} \int d^{3} x [π^{2} + (\nabla ϕ)^{2} + m^{2} ϕ^{2}] .

(2)

Canonical quantization promotes

ϕ

and

π

to Hermitian operator-valued fields and imposes the equal-time canonical commutation relations

[ϕ (t, x), π (t, y)] = i δ^{3} (x - y), [ϕ (t, x), ϕ (t, y)] = [π (t, x), π (t, y)] = 0.

(3)

Their Heisenberg equation of motion gives

\dot{ϕ} = π

\overset{π}{˙} = \nabla^{2} ϕ - m^{2} ϕ

, hence the Klein-Gordon equation

(□ + m^{2}) ϕ = 0

Let

E_{p} = p^{2} + m^{2}

and

p \cdot x = E_{p} t - p \cdot x

. The Hermitian field mode expansion is

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

(4)

The scalar field oscillator inversion extracts

a (p) = e^{i E_{p} t} \int d^{3} x e^{- i p \cdot x} [\frac{E _{p}}{2} ϕ (t, x) + \frac{i}{2 E _{p}} π (t, x)],

(5)

and its adjoint obtained by Hermitian conjugation extracts

a^{†}

. Substitute these expressions into the equal-time canonical commutation relation. The two mixed field-momentum terms give

[a (p), a^{†} (q)] = e^{i (E_{p} - E_{q}) t} \frac{1}{2} (\frac{E _{p}}{E _{q}} + \frac{E _{q}}{E _{p}}) (2 π)^{3} δ^{3} (p - q) = (2 π)^{3} δ^{3} (p - q) .

(6)

For

[a (p), a (q)]

, the corresponding coefficient is a difference of those square roots and multiplies

δ^{3} (p + q)

; it vanishes because

E_{p} = E_{- p}

. Taking the adjoint gives

[a^{†}, a^{†}] = 0

. Thus these are bosonic annihilation operators and creation operators. The vacuum satisfies

a (p) ∣0 ⟩ = 0

, and normal ordering gives

: H := \int d^{3} p E_{p} a^{†} (p) a (p) / (2 π)^{3}

, after removing the constant zero-point energy. A one-particle excitation has energy

E_{p}

and spin zero.

The Feynman propagator

G_{F} (x - y) = ⟨ 0∣ Tϕ (x) ϕ (y) ∣0 ⟩

is the vacuum expectation value of the time-ordered product of two field insertions. For

x^{0} > y^{0}

, the field at

y

creates a one-particle excitation from the vacuum and the field at

x

annihilates it; the opposite time ordering reverses the roles. It is a propagation amplitude and correlation function of vacuum fluctuations, rather than a transition probability. From the oscillator expansion, only the

a

a^{†}

contraction survives, so with

t = x^{0} - y^{0}

and

r = x - y

G_{F} (t, r) = \int \frac{d ^{3} p}{( 2 π ) ^{3} , 2 E _{p}} [θ (t) e^{- i E_{p} t + i p \cdot r} + θ (- t) e^{i E_{p} t - i p \cdot r}] = \int \frac{d ^{3} p}{( 2 π ) ^{3} , 2 E _{p}} e^{i p \cdot r} e^{- i E_{p} ∣ t ∣} .

(7)

Here

θ

is the Heaviside step function, and changing

p

- p

in the second term gives the last line.

For the Fourier transform convention

G_{F} (ω, p) = \int d t d^{3} r e^{iω t - i p \cdot r} G_{F} (t, r)

, insert a positive damping factor

e^{- ϵ ∣ t ∣}

and integrate the positive and negative half-lines separately:

G_{F} (ω, p) = lim_{ϵ ↓ 0} \frac{i}{2 E _{p}} [\frac{1}{ω - E _{p} + i ϵ} - \frac{1}{ω + E _{p} - i ϵ}] = \frac{i}{ω ^{2} - p ^{2} - m ^{2} + i 0} .

(8)

The last equality is a distribution identity; the infinitesimals in the partial fractions need not have the same finite magnitude as the infinitesimal in the combined denominator. The Feynman i-epsilon prescription means that the positive-frequency pole is just below the real energy axis and the negative-frequency pole is just above it. Closing the contour integral below for

t > 0

, and above for

t < 0

, reproduces the oscillator result by the residue theorem. This prescription fixes which homogeneous solutions are added to the Green function. As an independent normalization check,

e^{- i E ∣ t ∣} / (2 E)

has derivative jump

- i

t = 0

, so

(□ + m^{2}) G_{F} (x) = - i δ^{4} (x) .

(9)

It is the expectation-value convention for

G_{F}

, including the numerator

i

, that determines this source normalization.

The vacuum is a centered free Gaussian state. The Wick theorem expresses the four-field time-ordered product as the sum of all pair Wick contractions plus terms containing a normal-ordered product. The latter terms have zero vacuum expectation value. There are exactly three complete pairings, with no fermionic signs for this bosonic field. Thus the free scalar four-point function is

⟨ 0∣ Tϕ (x_{1}) ϕ (x_{2}) ϕ (x_{3}) ϕ (x_{4}) ∣0 ⟩ = G_{F} (x_{1} - x_{2}) G_{F} (x_{3} - x_{4}) + G_{F} (x_{1} - x_{3}) G_{F} (x_{2} - x_{4}) + G_{F} (x_{1} - x_{4}) G_{F} (x_{2} - x_{3}) .

(10)

These formulas describe canonical quantization of a real scalar field; in particular, a real scalar field uses a single oscillator family rather than independent charged-particle and antiparticle families.

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The Heisenberg picture evolution of the annihilation operator follows from

[H, a_{p}] = - E_{p} a_{p}

. With

p^{0} = E_{p}

, this gives the mode expansion of a free field

ϕ (x) = \int_{p} \frac{1}{2 E _{p}} (a_{p} e^{- i p \cdot x} + a_{p}^{†} e^{i p \cdot x}) .

(1)

Define

Δ^{+} (z) = \int_{p} e^{- i p \cdot z} / (2 E_{p})

. The time-ordered product then gives the Feynman propagator

Δ_{F} (z) = θ (z^{0}) Δ^{+} (z) + θ (- z^{0}) Δ^{+} (- z) = \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{i e ^{- i p \cdot z}}{p ^{2} - m ^{2} + i 0} .

(2)

The last expression follows by closing the

p^{0}

contour integral below for

z^{0} > 0

and above for

z^{0} < 0

; the scalar Feynman propagator pole prescription supplies the required positive- and negative-frequency terms.

For

z^{0} \neq = 0

every term obeys the homogeneous Klein-Gordon equation. At

z^{0} = 0

, the equal-time commutator of the fields vanishes, so the propagator is continuous and there is no derivative of a Dirac delta function. Its first time derivative has the jump

[\partial_{z^{0}} Δ_{F}]_{0^{-}}^{0^{+}} = \int_{p} (- i) e^{i p \cdot z} = - i δ^{3} (z) .

(3)

The second time derivative therefore supplies precisely the source:

(□_{x} + m^{2}) Δ_{F} (x - y) = - i δ^{4} (x - y) .

(4)

Thus the defined Feynman propagator is a Green's function with source normalization

- i

;

i Δ_{F}

has source normalization

+ 1

. Keeping this factor avoids confusing the vacuum expectation value convention with conventions which define the propagator after dividing by

i

.^[ref]

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For the interacting vacuum state defined perturbatively by the same Feynman i-epsilon prescription, the normalized time-ordered product is

⟨ T r = 1 \prod m ϕ (x_{r}) ⟩ = \frac{1}{Z [ λ ; 0 ]} \frac{1}{i ^{m}} \frac{δ ^{m} Z [ λ ; j ]}{δ j ( x _{1} ) \dots δ j ( x _{m} )}_{j = 0} .

(1)

Each functional derivative inserts

i ϕ

, accounting for

i^{- m}

. The denominator implements cancellation of vacuum bubbles, while disconnected pieces carrying external insertions still contribute. The lower limit

i = i

in the printed product is a typographical error; the product here runs from 1 to

m

At order

V

in the cubic scalar field interaction term, the expansion of the generating functional supplies

1/ [V! (3!)^{V}]

. The

V!

counts permutations of indistinguishable integrated interaction vertices, and

3!

counts permutations of the three field slots at each interaction vertex. Applying the Wick theorem pairs fields into Feynman propagators. Several pairings produce the same Feynman diagram; their number

N_{G}

multiplies the expansion coefficient. For fixed labelled external points, the net combinatorial weight is

w_{G} = \frac{N _{G}}{V ! ( 3 ! ) ^{V}} = \frac{1}{S _{G}} .

(2)

Here

S_{G}

is the order of the graph's automorphism group when the external labels are fixed, with permutations of identical parallel lines and reversal of a self-loop included as applicable. This is the Feynman-diagram symmetry factor convention in which a Feynman diagram is divided by

S_{G}

. Each cubic interaction vertex then contributes

- iλ

, each line contributes

Δ_{F}

, and each internal vertex is integrated over its position. External labelled points are not divided by

m!

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