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Four complementary classical features of a large-amplitude coherent state are:

It is an eigenstate of the annihilation operator, $a ∣ z ⟩ = z ∣ z ⟩$ . Normally ordered field amplitudes therefore behave like a classical complex amplitude; normalized photon subtraction leaves the same state.
Free harmonic evolution preserves its form, $∣ z ⟩ \to ∣ z e^{- iω t} ⟩$ . The mean electric field follows a classical sinusoidal wave with an amplitude and phase set by $z$ .
Its photon number has Poisson distribution, with $Δ n / \overset{n}{ˉ} = 1/ \overset{n}{ˉ}$ and $g^{(2)} (0) = 1$ . Relative intensity fluctuations become small at large amplitude.
Its optical quadratures $X = (a + a^{†}) /2$ and $P = (a - a^{†}) / (2 i)$ have minimum, balanced uncertainties, $Δ X = Δ P = 1/2$ and $Δ X Δ P = 1/4$ . The fixed vacuum-scale fluctuation becomes small relative to the large phase-space displacement.

These properties identify a particular classical oscillation, rather than a state of definite photon number. Absolute quantum fluctuations remain even when their relative size is small.

Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 3 / a / Solution 2026-10-25

Let

∣ g ⟩, ∣ e ⟩

be the atomic levels,

σ_{+} = ∣ e ⟩ ⟨ g ∣

, and

a

the annihilation operator of the single field mode. Choose the joint ground energy to be zero. The resonant Jaynes-Cummings model is

H = ℏ ω a^{†} a + ℏ ω ∣ e ⟩ ⟨ e ∣ + ℏ g (a σ_{+} + a^{†} σ_{-}) .

The first term counts field photons, the second the atomic excitation, and the interaction exchanges one photon for one atomic excitation. A phase redefinition makes

g

real and positive. Resonance means

ω = ω_{e g}

; the total excitation

N = a^{†} a + ∣ e ⟩ ⟨ e ∣

is conserved. Adding an overall constant gives the equivalent convention using

σ_{z} /2

and a zero-point field term.

The model follows from the electric-dipole interaction and the rotating-wave approximation, retaining the energy-conserving terms while dropping

a σ_{-}

and

a^{†} σ_{+}

. We assume a single quantized monochromatic mode and negligible cavity leakage and atomic relaxation over the time considered. With this convention, the resonant population Rabi angular frequency in the one-excitation subspace is

2 g

Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 6 / b / Solution 2026-10-25

Use the resonant interaction picture and define

g > 0

by the one-atom Jaynes-Cummings model in part (a). Equal coupling to both atoms gives the Tavis-Cummings model

H_{I} = ℏ g [a (σ_{+}^{(1)} + σ_{+}^{(2)}) + a^{†} (σ_{-}^{(1)} + σ_{-}^{(2)})] .

The initial state has two excitations and is symmetric under atom exchange. Excitation conservation and this symmetry confine the evolution to

∣ A ⟩ = ∣ ee; 0 ⟩, ∣ B ⟩ = \frac{∣ e g ⟩ + ∣ g e ⟩}{2} \otimes ∣1 ⟩, ∣ C ⟩ = ∣ gg; 2 ⟩ .

The antisymmetric atomic spin singlet state is uncoupled and unpopulated. Applying the annihilation operator and creation operator gives

\frac{H _{I}}{ℏ} = g 020202020 .

The first off-diagonal entry combines two indistinguishable atomic emission paths; the second also includes the

2

enhancement for creating the second photon. Its eigenvalues are

0, \pm 6 g

. Put

Ω = 6 g

and

K = H_{I} /ℏ

, so

K^{3} = Ω^{2} K

. Its exponential is

e^{- i K t} = I - \frac{i s i n Ω t}{Ω} K + \frac{c o s Ω t - 1}{Ω ^{2}} K^{2} .

Applying this to

∣ A ⟩

yields

∣ ψ (t)⟩ = \frac{2 + cos Ω t}{3} ∣ ee; 0 ⟩ - \frac{i}{3} sin Ω t \frac{∣ e g ⟩ + ∣ g e ⟩}{2} \otimes ∣1 ⟩ + \frac{2}{3} (cos Ω t - 1) ∣ gg; 2 ⟩ .

All three states have the same bare energy on resonance, so restoring the free Schrödinger-picture evolution adds only a common global phase. The expression starts at

∣ ee; 0 ⟩

, has initial derivative

- i 2 g ∣ B ⟩

, and its three squared probability amplitudes sum to one.

Past exam of the laser science and quantum information processing course of the University of Oxford / 2022 / 3 / a / Solution 2026-10-25

For one electromagnetic normal mode,

a

and

a^{†}

are the annihilation operator and creation operator, with commutator

[a, a^{†}] = I

. Their action on normalized number states is

a ∣ n ⟩ = n ∣ n - 1 ⟩, a^{†} ∣ n ⟩ = n + 1 ∣ n + 1 ⟩ .

They remove or add one photon, whose energy is

ℏ ω

. The vacuum state

∣0 ⟩

is normalized and obeys

a ∣0 ⟩ = 0

. It has no photons, but retains zero-point energy

ℏ ω /2

for the quantum harmonic oscillator Hamiltonian

H = ℏ ω (a^{†} a + 1/2)

. The coherent state

∣ α ⟩ = D (α) ∣0 ⟩

is a displaced vacuum state.

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

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.

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}] .

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.

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} .

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)] .

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.

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) .

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) .

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} .

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} .

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) .

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)}

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.

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 51 / 4 / Solution Created 2026-10-03 Updated 2026-10-14

For

d \geq 2

, let

ω

be a Conformal Killing vector field, put

σ = \partial \cdot ω / d

, and assign the massless scalar field scaling dimension

Δ = (d - 2) /2

. In a finite conformal transformation with metric scale

Ω^{2}

, the transformation law is

ϕ^{'} (x^{'}) = Ω (x)^{- Δ} ϕ (x)

. Its infinitesimal fixed-coordinate form is

δ_{0} ϕ = - ω^{μ} \partial_{μ} ϕ - Δ σ ϕ .

For

L = (\partial ϕ)^{2} /2

, contract the Conformal Killing equation with

\partial_{μ} ϕ \partial_{ν} ϕ

. Direct differentiation then gives

δ_{0} L = - ω \cdot \partial L - (1 + Δ) σ (\partial ϕ)^{2} - \frac{Δ}{2} \partial_{μ} (ϕ^{2}) \partial^{μ} σ = - \partial_{μ} (ω^{μ} L) - \frac{Δ}{2} \partial_{μ} (ϕ^{2} \partial^{μ} σ) + \frac{Δ}{2} ϕ^{2} □ σ .

Here

2 (1 + Δ) = d

. Flat conformal Killing integrability gives

□ σ = 0

, with the Lorentzian version obtained by replacing the flat metric consistently. Thus the action variation is a boundary integral, vanishing for suitable boundary conditions. This proves conformal invariance with field dimension $(d - 2) /2$ , without using the field equation. The connected finite symmetry follows by integrating the infinitesimal generators. Equivalently its conformal currents follow from the improved stress-energy tensor of a free massless scalar.

In one dimension the metric conformality condition is vacuous. The same action-variation calculation contains

- ϕ^{2} ω^{'''} /4

, so the free action is invariant under the projective subgroup

ω^{'''} = 0

, rather than arbitrary reparametrizations. The usual field-theory conformal-invariance claim above concerns

d \geq 2

d = 2

Δ = 0

, so the field transforms simply as a scalar. There is no improvement term and no derivative of the conformal scale in the variation. All local independent reparametrizations of the two null coordinates preserve the action; after Euclidean continuation they are holomorphic and antiholomorphic maps. The conformal algebra is therefore infinite-dimensional, in contrast to

d > 2

Now use Lorentzian coordinates with spatial period

ℓ

. The canonical momentum is

π = \dot{ϕ}

, and canonical quantization imposes

[ϕ (t, x), π (t, y)] = i δ_{ℓ} (x - y)

, with the periodic delta distribution

δ_{ℓ} (s) = ℓ^{- 1} \sum_{n \in Z} e^{2 πin s / ℓ}

. Solving the wave equation and imposing reality gives a convenient oscillator normalization:

ϕ (t, x) = q + \frac{pt}{ℓ} + \frac{i}{4 π} \sum_{n \neq = 0} \frac{1}{n} [α_{n} e^{- 2 πin (t - x) / ℓ} + α_{n} e^{- 2 πin (t + x) / ℓ}] .

Strict spatial periodicity excludes a winding term. The canonical commutation relation is equivalent to

[q, p] = i, [α_{m}, α_{n}] = m δ_{m + n, 0}, [α_{m}, α_{n}] = m δ_{m + n, 0}, [α_{m}, α_{n}] = 0,

with

α_{n}^{†} = α_{- n}

and the corresponding tilded relation. Both oscillator families commute with

q, p

. To verify the normalization, the zero-mode commutator supplies

i / ℓ

, while each oscillator family supplies

i (2 ℓ)^{- 1} \sum_{n \neq = 0} e^{2 πin (x - y) / ℓ}

. Their sum is exactly

i δ_{ℓ} (x - y)

. The positive-index modes are annihilation operators and the negative-index modes are creation operators, giving two independent bosonic Fock spaces together with the free-particle zero mode.

Define

α_{0} = α_{0} = p / 4 π

and the normal ordering quadratic generators

L_{m} = \frac{1}{2} \sum_{n \in Z} : α_{m - n} α_{n} :, L_{m} = \frac{1}{2} \sum_{n \in Z} : α_{m - n} α_{n} : .

These are the Fourier modes of the two chiral components of the stress-energy tensor. The oscillator operator commutators immediately give

[L_{m}, α_{n}] = - n α_{m + n}

. Applying this to both factors of

L_{n}

gives

(m - n) L_{m + n}

, except for a scalar term when

m + n = 0

, produced by reordering creators and annihilators. Its value is independent of the momentum sector. Evaluate it in the formal zero-momentum oscillator vacuum. For

m > 0

L_{- m} ∣0 ⟩ = \frac{1}{2} \sum_{r = 1}^{m - 1} α_{- r} α_{- (m - r)} ∣0 ⟩, ⟨ 0∣ L_{m} L_{- m} ∣0 ⟩ = \frac{1}{2} \sum_{r = 1}^{m - 1} r (m - r) = \frac{m ^{3} - m}{12} .

The two contractions of each pair explain the factor

1/2

. Hence the free-boson Virasoro central term yields

[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{m ^{3} - m}{12} δ_{m + n, 0}, c = 1 .

The tilded modes give a second commuting Virasoro algebra with

\overset{c}{ˉ} = 1

. The adjoint condition

L_{m}^{†} = L_{- m}

and the positive oscillator norms make this a unitary highest-weight Virasoro representation in each momentum sector. In a fixed momentum sector,

L_{0} = p^{2} / (8 π) + \sum_{n > 0} α_{- n} α_{n}

, so the oscillator ground state is a highest-weight state of weight

p^{2} / (8 π)

in either sector. For a noncompact field the zero mode is a free particle: definite momentum ground states are generalized states, rather than a normalizable zero-mode vacuum. The oscillator representation and central term remain well defined. With the standard cylinder vacuum-energy subtraction, the Hamiltonian is

H = \frac{2 π}{ℓ} (L_{0} + L_{0} - \frac{1}{12}) .

The constant is the Casimir energy of a two-dimensional conformal field theory; omitting it amounts to a different additive Hamiltonian convention, not a change of the plane Virasoro algebra.

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 49 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

Use a mostly-plus target metric tensor and the standard open-string mode expansion convention

α_{0}^{μ} = 2 α^{'} p^{μ}

. The integer string oscillator level is

N = \sum_{n \geq 1} α_{- n} \cdot α_{n}

, whose eigenvalues are nonnegative integers. Then

L_{0} = N + α^{'} p^{2}

, so the string mass-shell condition gives

M^{2} = - p^{2} = \frac{N - 1}{α ^{'}} .

The ground state is a tachyon, and the level-one state is massless. The physical string state conditions in old covariant string quantization are

(L_{0} - 1) ∣ ψ ⟩ = 0, L_{m} ∣ ψ ⟩ = 0 (m > 0) .

Only the positive-mode Virasoro constraints annihilate a ket; their negative-mode partners annihilate the corresponding bra. Imposing both signs on every ket would contradict the nonzero Virasoro central extension.

The central charge comes from normal ordering the quadratic oscillator expressions. Classically their Poisson brackets have no central term. Quantum mechanically, moving annihilation operators through creation operators produces double contractions. From the oscillator commutator one obtains

[L_{m}, α_{n}^{μ}] = - n α_{m + n}^{μ}

, which gives the noncentral part

(m - n) L_{m + n}

The coefficient is fixed by the free-boson Virasoro central term. On a formal zero-momentum vacuum, for

m > 0

L_{- m} ∣0 ⟩ = \frac{1}{2} \sum_{r = 1}^{m - 1} α_{- (m - r)} \cdot α_{- r} ∣0 ⟩ .

The two contraction pairings give

⟨ 0∣ L_{m} L_{- m} ∣0 ⟩ = \frac{D}{2} \sum_{r = 1}^{m - 1} r (m - r) = \frac{D}{12} m (m^{2} - 1) .

Here

η_{μν} η^{μν} = D

; the time coordinate contributes one, just as each spatial coordinate does. Comparing with the Virasoro algebra proves

c_{matter} = D .

This counts covariant target coordinates. It is distinct from the

D - 2

physical transverse polarizations. Including the central charge of reparameterization ghosts gives

c_{total} = D - 26

; cancellation of the worldsheet Weyl anomaly gives the usual critical dimension of the bosonic string

D = 26

A general level-one state is

∣ ϵ; k ⟩ = ϵ_{μ} α_{- 1}^{μ} ∣0; k ⟩ .

The zero-mode constraint requires

k^{2} = 0

. Since

[L_{1}, α_{- 1}^{μ}] = α_{0}^{μ}

, the remaining nontrivial positive-mode constraint is

k \cdot ϵ = 0

; all

L_{m}

with

m \geq 2

act trivially at this level. Thus the polarization vector is transverse to a nonzero null momentum.

There is also a null string state

L_{- 1} ∣0; k ⟩ = 2 α^{'} k \cdot α_{- 1} ∣0; k ⟩

. It is physical when

k^{2} = 0

and orthogonal to every physical polarization. The null-state quotient of a string therefore identifies

ϵ \sim ϵ + λk .

Choose

k = (E, E, 0, \dots, 0)

. Transversality sets

ϵ^{1} = ϵ^{0}

, and the null-state identification removes this common component. The remaining

D - 2

spatial components have positive norm. Hence the physical level-one states form a massless vector with $D - 2$ polarizations, or 24 at the critical dimension. The removed component is gauge redundancy. States violating the constraints, including independent timelike negative-norm polarizations, are unphysical; physical null states are removed by the quotient rather than counted as additional particles.

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 49 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

Use the Minkowski metric

η = diag (1, - 1, - 1, - 1)

and the free real scalar field Lagrangian density

L = \frac{1}{2} \partial_{μ} ϕ \partial^{μ} ϕ - \frac{1}{2} m^{2} ϕ^{2}

. Its conjugate canonical momentum is

π = \dot{ϕ}

. Spatial translation symmetry and Noether's theorem give the canonical stress-energy tensor

T^{μν} = \partial^{μ} ϕ \partial^{ν} ϕ - η^{μν} L, \partial_{μ} T^{μν} = (□ ϕ + m^{2} ϕ) \partial^{ν} ϕ = 0.

Vanishing boundary flux then makes the four-momentum of a free real scalar field constant. In particular the contravariant spatial momentum is

P = - \int π \nabla ϕ d^{3} x .

The minus sign is from

\partial^{i} = - \partial_{i}

; it is essential for the following mode calculation.

In the quantized theory, insert the mode expansions into this integral, using a Hermitian symmetric ordering or normal ordering. Write

\int_{p} = \int d^{3} p / (2 π)^{3}

. The spatial integral is

(2 π)^{3} δ^{(3)} (p \pm k)

, leaving

P = \frac{1}{2} \int_{p} p (a_{p} a_{p}^{†} + a_{p}^{†} a_{p} - a_{- p} a_{p} - a_{- p}^{†} a_{p}^{†}) .

Each last term integrates to zero: replace

p

- p

and commute the two annihilation operators or the two creation operators. The mixed terms reorder by the canonical commutation relation. Their vacuum contribution is proportional to the integral of the odd vector

p

and vanishes with an inversion-symmetric regulator; normal ordering removes it directly. Thus the normal-ordered free scalar four-momentum has spatial part

P = \int_{p} p a_{p}^{†} a_{p} .

These manipulations can be made first with box modes and a symmetric momentum cutoff, then used on finite-particle wave packets.

The commutator derivation identity gives

[P_{i}, a_{q}^{†}] = \int_{p} p_{i} a_{p}^{†} [a_{p}, a_{q}^{†}] = q_{i} a_{q}^{†}, [P_{i}, a_{q}] = - q_{i} a_{q} .

For

U (y) = e^{- i P \cdot y}

, the commutator expansion for exponential conjugation therefore gives the translation of a scalar creation operator

U (y) a_{q}^{†} U (y)^{†} = e^{- i q \cdot y} a_{q}^{†} .

The vacuum has zero momentum, so

U ∣0 ⟩ = ∣0 ⟩

. Hence the one-particle momentum eigenstate obeys

U ∣ q ⟩ = e^{- i q \cdot y} ∣ q ⟩

: translation preserves its momentum and multiplies it by the associated phase.

The annihilation operator receives the opposite phase. Substituting both phases into the field expansion gives

U (y) ϕ (x) U (y)^{†} = ϕ (x + y) .

This is the specified conjugation convention for the spatial translation operator. On states, the active translation moves a wave packet by

y

, giving a position wavefunction

ψ (x - y)

. Conjugating the field by

U^{†}

instead would give the opposite argument shift. Infinitesimally the displayed result is

[P_{i}, ϕ] = i \partial_{i} ϕ

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 49 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

Direct multiplication of the gamma matrices in the chiral representation gives

γ^{5} = i γ^{0} γ^{1} γ^{2} γ^{3} = (- I_{2} 0 0 I_{2}) .

For example

σ^{1} σ^{2} σ^{3} = i I_{2}

by the Pauli matrix multiplication law, and the block multiplications give the displayed signs. Every

γ^{μ}

is off-diagonal in this representation, whereas the chirality matrix has opposite diagonal blocks. Therefore

γ^{5} γ^{μ} + γ^{μ} γ^{5} = 0

for all

μ

The left-handed chirality condition makes the plane-wave coefficient

λ (p) = (0 χ)

. With

p \cdot x = E x^{0} - p \cdot x

, the Dirac equation gives

\neq p λ (p) = 0 ⟺ (E I_{2} + σ \cdot p) χ = 0.

Since

(σ \cdot p)^{2} = ∣ p ∣^{2} I_{2}

, its determinant is

E^{2} - ∣ p ∣^{2}

. A nonzero spinor therefore exists exactly when

p^{2} = 0 .

For nonzero momentum, put

r = ∣ p ∣

and write its direction as

(sin ϑ cos φ, sin ϑ sin φ, cos ϑ)

. Normalized eigenvectors of

σ \cdot p

are

χ_{-} = (- e^{- i φ} sin (ϑ /2) cos (ϑ /2)), χ_{+} = (cos (ϑ /2) e^{i φ} sin (ϑ /2)) .

The general nonzero solution is

λ (p) = C (0 χ _{-})

for

E = r

and

λ (p) = C (0 χ _{+})

for

E = - r

, with arbitrary nonzero complex normalization

C

. At

p = 0

any constant two-component

χ

solves the equation, but the momentum direction and helicity are undefined. For the physical positive-energy branch this is the left-handed massless plane-wave spinor.

A rotation by angle

α

about

p

acts on the two-component spinor as

S (R) = e^{- i α σ \cdot p /2} .

The plane-wave argument is unchanged because the rotation fixes

p

, and

S (R) χ_{-} = e^{i α /2} χ_{-}

. Thus the positive-energy solution has helicity

h = - 1/2

, where a helicity-

h

state has rotation phase

e^{- i α h}

. The negative-energy coefficient has the opposite spinor phase.

In a quantized complex left-handed Weyl field, the positive-frequency term annihilates particles and the negative-frequency term creates antiparticles. Reinterpreting negative energy also reverses its spatial momentum; for the positive physical momentum the two spinor coefficients have the same left-chiral kernel. Because one multiplies an annihilation operator and the other a creation operator, the corresponding particle and antiparticle state phases are conjugate. The helicity content of a quantized Weyl field is consequently

h_{particle} = - \frac{1}{2}, h_{antiparticle} = + \frac{1}{2} .

There is one particle spin state at each nonzero momentum, rather than the two helicities supplied by a full massless Dirac field. The right-handed Weyl field interchanges these signs. This is the massless relation between chirality and helicity; it does not identify them for a massive field.

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 49 / 3 / Solution Created 2026-10-03 Updated 2026-10-14

Split the free interaction picture field as

ϕ = ϕ^{(+)} + ϕ^{(-)}

, where the first part contains annihilation operators and the second creation operators. Time ordering places the operator at the later time on the left, while normal ordering places all creation operators on the left. For

x^{0} > y^{0}

, commuting the annihilation part of the first field through the creation part of the second gives

ϕ (x) ϕ (y) =: ϕ (x) ϕ (y) : + [ϕ^{(+)} (x), ϕ^{(-)} (y)] .

The commutator is the c-number

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

For

y^{0} > x^{0}

, the same calculation interchanges

x, y

. Combining the two time orders proves the two-field Wick contraction identity

Tϕ (x) ϕ (y) =: ϕ (x) ϕ (y) : + D_{F} (x - y),

where

D_{F} (x - y) = ⟨ 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} .

The equality with the four-dimensional integral follows by closing the energy contour at the positive-energy pole for

x^{0} > y^{0}

and the negative-energy pole for the reverse order. This fixes the scalar Feynman propagator pole prescription; in this convention

D_{F}

itself includes the factor

i

Because these interactions have no derivatives,

H_{int} = - \int L_{int} d^{3} x

. Expand the Dyson series

S = T exp (i \int d^{4} z [\frac{μ}{3 !} ϕ (z)^{3} - \frac{λ}{4 !} ϕ (z)^{4}]) .

Apply the Wick theorem to each term together with the external insertions. Every paired field contributes a propagator, and the remaining normal-ordered vacuum expectation vanishes unless no fields remain. Representing the propagators by Fourier integrals and integrating each vertex position gives the four-momentum conservation delta function. The

3!

and

4!

contractions into labelled vertex slots cancel the factorials in the interaction density. The resulting scalar-field cubic and quartic Feynman vertices are

propagator: \frac{i}{p ^{2} - m ^{2} + i 0}, cubic vertex: i μ, quartic vertex: - iλ .

Each vertex also has

(2 π)^{4} δ^{(4)} (\sum p)

, each independent internal momentum is integrated with

d^{4} p / (2 π)^{4}

, and repeated equivalent contractions give the usual Feynman-diagram symmetry factor. For the unamputated correlation functions, retain the propagators joining external insertions to vertices. The displayed unnormalized numerator includes disconnected vacuum bubbles; dividing by

⟨ 0∣ S ∣0 ⟩

cancels them. Amputated scattering amplitudes additionally use the LSZ reduction formula.

For stability, examine the classical scalar potential

V (ϕ) = \frac{1}{2} m^{2} ϕ^{2} - \frac{μ}{6} ϕ^{3} + \frac{λ}{24} ϕ^{4} .

The tree-level boundedness criterion is $λ > 0$ , with arbitrary real cubic coupling. If

λ = 0

, boundedness requires

μ = 0

and

m^{2} \geq 0

; if

λ < 0

, or

λ = 0

with

μ \neq = 0

, the potential is unbounded below. Thus a positive quartic coupling permits a stable ground-state vacuum, although it need not be the perturbative vacuum at

ϕ = 0

For

λ > 0

and

m^{2} \geq 0

, the stability of a cubic-quartic scalar potential determines when zero is a global minimum:

V (ϕ) = ϕ^{2} [\frac{λ}{24} (ϕ - \frac{2 μ}{λ})^{2} + \frac{m ^{2}}{2} - \frac{μ ^{2}}{6 λ}] .

The square bracket is nonnegative for every field value exactly when

μ^{2} \leq 3 λ m^{2} .

For a nontrivial equality case there is a second degenerate minimum at

ϕ = 2 μ / λ

. If

m^{2} > 0

and

μ^{2} > 3 λ m^{2}

, zero remains a local minimum but is a false vacuum, with a lower minimum elsewhere. If

m^{2} = μ = 0

and

λ > 0

, the origin is the quartic minimum despite its vanishing quadratic curvature. These are classical or tree-level stability statements; renormalized couplings and the quantum effective potential refine the quantum analysis.

Figure 1.
Stable, degenerate and metastable scalar vacua
. **Details**.

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 53 / 4 / Solution Created 2026-10-03 Updated 2026-10-14

An antilinear involution need not preserve norms, so the required physical assumption is that momentum reversal maps a complete normalized basis to a normalized basis. For a spinless sector take

\hat{T} ∣ p ⟩ = η (p) ∣ - p ⟩

with

∣ η ∣ = 1

and relativistic normalization

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

. Any extra degeneracy labels must likewise be mapped by a norm-preserving permutation or unitary matrix. The condition

\hat{T}^{2} = 1

requires

η (- p) η (p)^{*} = 1

in the scalar basis.

For arbitrary normalizable wave packets

∣ ϕ ⟩ = \int d Π_{p} f (p) ∣ p ⟩

and

∣ ψ ⟩ = \int d Π_{p} h (p) ∣ p ⟩

, where

d Π_{p} = d^{3} p / [(2 π)^{3} 2 E_{p}]

, antilinearity conjugates their coefficients. Using the normalized reversed basis gives

⟨ \hat{T} ϕ ∣ \hat{T} ψ ⟩ = \int d Π_{p} f (p) h (p)^{*} ∣ η (p) ∣^{2} = [\int d Π_{p} f (p)^{*} h (p)]^{*} .

Completeness proves $⟨ \hat{T} ϕ ∣ \hat{T} ψ ⟩ = ⟨ ϕ ∣ ψ ⟩^{*}$ for all states, so this quantum time-reversal operator is antiunitary. The same proof with sums over extra labels uses the unitarity of their reversal matrix.

For a Hermitian interaction-picture Hamiltonian, the condition of time-reversal symmetry is

\hat{T} V (t) \hat{T}^{- 1} = V (- t), V (t)^{†} = V (t) .

Expand the scattering matrix as a Dyson series, using ordered integration rather than commuting operators:

S = 1 + \sum_{n \geq 1} (- i)^{n} \int_{t_{1} > \dots > t_{n}} d t_{1} \dots d t_{n} V (t_{1}) \dots V (t_{n}) .

Conjugation by

\hat{T}

conjugates

(- i)^{n}

i^{n}

but preserves the product order. Substitute

s_{j} = - t_{j}

; the integration domain becomes

s_{1} < \dots < s_{n}

and

\hat{T} S \hat{T}^{- 1} = 1 + \sum_{n \geq 1} i^{n} \int_{s_{1} < \dots < s_{n}} d s_{1} \dots d s_{n} V (s_{1}) \dots V (s_{n}) .

On the other hand, taking the adjoint of the original series conjugates its coefficient and reverses the operator order. Relabel

s_{1} = t_{n}, \dots, s_{n} = t_{1}

in each adjoint integral; it is exactly the same series. Thus

\hat{T} S \hat{T}^{- 1} = S^{†} .

Equivalently, time ordering turns into anti-time ordering after reversing the time arguments. One can first use a finite symmetric time interval and the same time-symmetric switching prescription before taking the scattering limit; the argument never assumes

[V (t), V (s)] = 0

Antiunitarity relates amplitudes by

⟨ \hat{T} f ∣ \hat{T} S \hat{T}^{- 1} ∣ \hat{T} i ⟩ = ⟨ f ∣ S ∣ i ⟩^{*} .

Using

\hat{T} S \hat{T}^{- 1} = S^{†}

and taking a complex conjugate gives the time-reversal reciprocity of the scattering matrix

⟨ f ∣ S ∣ i ⟩ = ⟨ \hat{T} i ∣ S ∣ \hat{T} f ⟩ .

The reverse process starts with the reversed final state and ends with the reversed initial state. Intrinsic state phases can appear when these reversed states are rewritten in a separately chosen basis, but the corresponding squared amplitudes agree. Equal amplitudes do not mean identical numerical cross sections when the reversed process has different flux or phase-space factors.

For a spin-zero complex scalar field, choose the transformation convention

\hat{T} ϕ (t, x) \hat{T}^{- 1} = η_{T} ϕ (- t, x)

. Its mode expansion of a free field is

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

Apply

\hat{T}

to the expansion. Its antiunitarity conjugates both numerical plane waves, giving coefficients

\hat{T} a \hat{T}^{- 1}

multiplying

e^{i E_{p} t - i p \cdot x}

and

\hat{T} b^{†} \hat{T}^{- 1}

multiplying

e^{- i E_{p} t + i p \cdot x}

. In

η_{T} ϕ (- t, x)

, replace

p

- p

to obtain precisely these same plane waves. Comparing independent modes proves time reversal of a complex scalar field:

\hat{T} a (p) \hat{T}^{- 1} = η_{T} a (p_{P}), \hat{T} b^{†} (p) \hat{T}^{- 1} = η_{T} b^{†} (p_{P}), p_{P} = (E_{p}, - p) .

Taking adjoints gives

\hat{T} b (p) \hat{T}^{- 1} = η_{T}^{*} b (p_{P})

. Positive energy is preserved, and an annihilation operator remains an annihilation operator. Time reversal does not exchange particles with antiparticles.

The coefficient

η_{T}

is an intrinsic time-reversal phase. Applying

\hat{T}

twice gives

\hat{T}^{2} a \hat{T}^{- 2} = η_{T}^{*} η_{T} a

, so

∣ η_{T} ∣ = 1

; it need not be restricted to

\pm 1

. Write

η_{T} = e^{i ϑ}

and redefine

a^{'} = e^{i ϑ /2} a

. Because the phase is conjugated by

\hat{T}

\hat{T} a^{'} \hat{T}^{- 1} = e^{- i ϑ /2} η_{T} a (p_{P}) = a^{'} (p_{P}) .

Thus the scalar intrinsic phase can be absorbed into the annihilation-operator convention. Redefining

b^{'} = e^{- i ϑ /2} b

simultaneously implements the corresponding whole-field rephasing

ϕ^{'} = e^{i ϑ /2} ϕ

and removes the phase from both mode transformations.

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 48 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

Use Minkowski spacetime with

g_{μν} = diag (1, - 1, - 1, - 1)

and units

ℏ = c = 1

. The transformation rotates the two components in their internal plane. In particular,

δ (ϕ_{1}^{2} + ϕ_{2}^{2}) = 2 α (ϕ_{1} ϕ_{2} - ϕ_{2} ϕ_{1}) = 0, δ (\partial_{μ} ϕ_{1} \partial^{μ} ϕ_{1} + \partial_{μ} ϕ_{2} \partial^{μ} ϕ_{2}) = 0.

Thus both terms in the Lagrangian density are invariant: this is an internal rotation symmetry of two real scalar fields, rather than a transformation of spacetime. The equal masses are essential to this symmetry.

The Noether theorem gives the Noether current

j^{μ} = \sum_{k = 1}^{2} \frac{\partial L}{\partial ( \partial _{μ} ϕ _{k} )} \frac{δ ϕ _{k}}{α} = ϕ_{2} \partial^{μ} ϕ_{1} - ϕ_{1} \partial^{μ} ϕ_{2} .

Indeed, the cross terms in its divergence cancel, and the two Klein-Gordon equations imply

\partial_{μ} j^{μ} = ϕ_{2} □ ϕ_{1} - ϕ_{1} □ ϕ_{2} = - m^{2} ϕ_{2} ϕ_{1} + m^{2} ϕ_{1} ϕ_{2} = 0.

For fields decaying sufficiently at spatial infinity, there is no outward current flux. The conserved Noether charge is therefore

Q = \int d^{3} x (ϕ_{2} \dot{ϕ}_{1} - ϕ_{1} \dot{ϕ}_{2}) .

In canonical quantization, the canonical momenta are

π_{k} = \dot{ϕ}_{k}

, with equal-time canonical commutation relations

[ϕ_{k} (x), π_{l} (y)] = i δ_{k l} δ^{3} (x - y), [ϕ_{k}, ϕ_{l}] = [π_{k}, π_{l}] = 0.

Consequently

Q = \int d^{3} x (π_{1} ϕ_{2} - π_{2} ϕ_{1})

. Operators of different components commute, so this expression is Hermitian without an ordering correction. Write

\int_{p} = \int d^{3} p / (2 π)^{3}

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

and

a_{k p} = a_{p}^{k}

. Substituting the oscillator expansions and using the Dirac delta function from the spatial integral gives

Q = - \frac{i}{2} \int_{p} [(a_{1 p} - a_{1, - p}^{†}) (a_{2, - p} + a_{2 p}^{†}) - (a_{2 p} - a_{2, - p}^{†}) (a_{1, - p} + a_{1 p}^{†})] .

The terms with two annihilation operators cancel after

p \mapsto - p

; the terms with two creation operators cancel in the same way. In the remaining terms use

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

There is no commutator constant between the distinct components. The result is already in normal ordering:

Q = - i \int_{p} (a_{2 p}^{†} a_{1 p} - a_{1 p}^{†} a_{2 p}) .

For the charged oscillator basis of a scalar doublet, compute

[Q, a_{1 p}^{†}] = - i a_{2 p}^{†}, [Q, a_{2 p}^{†}] = i a_{1 p}^{†}, b_{\pm p}^{†} = \frac{a _{1 p}^{†} \mp i a _{2 p}^{†}}{2} .

It follows that

[Q, b_{\pm p}^{†}] = \pm b_{\pm p}^{†}

. The Fock vacuum satisfies

Q ∣0 ⟩ = 0

, so $b_{+ p}^{†} ∣0 ⟩$ has charge $+ 1$ , and $b_{- p}^{†} ∣0 ⟩$ has charge $- 1$ . To obtain a normalized one-particle state, replace the sharp momentum by

\int_{p} f (p) b_{\pm p}^{†} ∣0 ⟩

, where

\int_{p} ∣ f (p) ∣^{2} = 1

. The charge is unchanged. Equivalently,

Q = \int_{p} (b_{+ p}^{†} b_{+ p} - b_{- p}^{†} b_{- p}) .

The generator convention is

δ ϕ_{k} = i α [Q, ϕ_{k}]

, which reproduces the original rotation signs.

For stability of a two-scalar quartic potential, the relevant potential energy density is

V = \frac{m ^{2}}{2} (ϕ_{1}^{2} + ϕ_{2}^{2}) + V_{4}, V_{4} = λ (ϕ_{1}^{2} - ϕ_{2}^{2})^{2} + 2 (λ + μ) ϕ_{1}^{2} ϕ_{2}^{2} .

Along either axis boundedness requires

λ \geq 0

; along

ϕ_{1} = ϕ_{2}

it requires

λ + μ \geq 0

. Conversely the displayed sum is nonnegative whenever both conditions hold. With

m^{2} > 0

the origin is then the unique global minimum, including the equality cases. Hence the stable-vacuum conditions are

λ \geq 0, μ \geq - λ .

(12)

Strict positivity of the quartic term away from the origin would instead require

λ > 0

and

μ > - λ

. That stronger condition is unnecessary for a positive mass term. If

m = 0

, the same non-strict conditions give boundedness, but the equality cases have flat directions and do not give an isolated classical vacuum.

Finally, the interaction changes the variation of the potential energy by

δ V_{4} = 4 α (λ - μ) ϕ_{1} ϕ_{2} (ϕ_{1}^{2} - ϕ_{2}^{2}) .

(13)

The interacting Euler-Lagrange equations consequently give

\partial_{μ} j^{μ} = - 4 (λ - μ) ϕ_{1} ϕ_{2} (ϕ_{1}^{2} - ϕ_{2}^{2}) .

(14)

Thus the same Noether charge is conserved for all field configurations precisely when $μ = λ$ . Then

V_{4} = λ (ϕ_{1}^{2} + ϕ_{2}^{2})^{2}

preserves the internal rotation symmetry of two real scalar fields; imposing stability additionally requires

λ \geq 0

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 10 / 7 / Solution Created 2026-10-03 Updated 2026-10-14

Use the unitary normalization of the Fourier transform,

F f (ξ) = (2 π)^{- n /2} \int_{R^{n}} e^{- i x \cdot ξ} f (x) d x, G g (x) = (2 π)^{- n /2} \int_{R^{n}} e^{i x \cdot ξ} g (ξ) d ξ .

The Schwartz space

S (R^{n})

consists of smooth function s for which every Schwartz seminorm

sup_{x} ∣ x^{α} \partial^{β} f (x) ∣

is finite. Such functions and all their derivatives are integrable. Differentiating under the integral and integrating by parts give

\partial_{ξ}^{α} F f = (- i)^{∣ α ∣} F (x^{α} f), ξ^{β} F f = (- i)^{∣ β ∣} F (\partial^{β} f) .

Consequently

sup_{ξ} ∣ ξ^{β} \partial_{ξ}^{α} F f (ξ) ∣ \leq (2 π)^{- n /2} ∥ \partial^{β} (x^{α} f) ∥_{1} .

Expand the derivative by the product rule and insert the integrable weight

(1 + ∣ x ∣)^{- n - 1}

; the right side is bounded by finitely many Schwartz seminorms of

f

. This proves continuity

F : S \to S

. The identical argument with the opposite sign proves continuity of

G

For surjectivity and injectivity an inversion proof is needed. If

f \in S

, multiply

F f (ξ)

e^{- ε ∣ ξ ∣^{2} /2}

. Fubini's theorem and the elementary Gaussian integral give

G (e^{- ε ∣ ξ ∣^{2} /2} F f) (x) = \int_{R^{n}} f (y) ρ_{ε} (x - y) d y, ρ_{ε} (x) = (2 π ε)^{- n /2} e^{- ∣ x ∣^{2} / (2 ε)} .

The Gaussian has total integral one and its mass outside any fixed ball tends to zero. The right side therefore tends to

f (x)

, by uniform continuity of

f

. On the left,

F f \in S \subset L^{1}

, so dominated convergence gives

GF f (x)

. This proves

GF = I

. Reversing the signs in the same calculation proves

FG = I

. Thus

F : S (R^{n}) ⟶ S (R^{n}) is a continuous bijection, with inverse G .

Its inverse is continuous by the seminorm estimate already proved. This establishes the requested bijection without assuming Fourier inversion as an unexplained theorem.

Inversion gives

F^{2} f (x) = f (- x)

and the Parseval identity. Indeed, inserting the integral for

F f

into

\int F f \overline{F g}

and using

f, F g \in L^{1}

gives

\int f \overline{g}

. Density of Schwartz functions in

L^{2}

extends

F

to a unitary operator on

L^{2}

, the Plancherel theorem. Translation of

f

becomes multiplication by a phase, modulation becomes frequency translation, and

F (f * g) = (2 π)^{n /2} (F f) (F g), F (\partial_{j} f) = i ξ_{j} F f .

These formulas explain why the Fourier transform diagonalizes constant-coefficient differential operators. By transpose it also defines a bijection of the tempered distributions: with the complex-linear distributional pairing,

⟨ F T, ϕ ⟩ = ⟨ T, F ϕ ⟩

, since the Fourier kernel is symmetric in its variables.

The connection with creation and annihilation operators is especially transparent for the harmonic oscillator. On the common invariant dense domain

S

, set

a_{j} = \frac{x _{j} + \partial _{j}}{2}, a_{j}^{†} = \frac{x _{j} - \partial _{j}}{2} .

Integration by parts shows that these are formal Hilbert-space adjoints. They are closable unbounded operators; commutators below are computed on

S

, avoiding undefined products on arbitrary

L^{2}

vectors. The identity

[\partial_{j}, x_{k}] = δ_{jk} I

gives the canonical commutation relations

[a_{j}, a_{k}^{†}] = δ_{jk} I, [a_{j}, a_{k}] = [a_{j}^{†}, a_{k}^{†}] = 0.

It also gives

H = - Δ + ∣ x ∣^{2} = 2 \sum_{j} a_{j}^{†} a_{j} + n I .

The normalized vacuum

h_{0} (x) = π^{- n /4} e^{- ∣ x ∣^{2} /2}

satisfies

a_{j} h_{0} = 0

. Define the Hermite functions

h_{α} = \frac{( a _{1}^{†} ) ^{α_{1}} \dots ( a _{n}^{†} ) ^{α_{n}}}{α _{1} ! \dots α _{n} !} h_{0} .

Moving annihilation operators past creation operators with the commutation relations proves

a_{j} h_{α} = α_{j} h_{α - e_{j}}, a_{j}^{†} h_{α} = α_{j} + 1 h_{α + e_{j}}, H h_{α} = (2∣ α ∣ + n) h_{α} .

These same relations show that the Hermite functions are orthonormal: unequal occupations are orthogonal eigenvectors of one of the commuting symmetric number operators

a_{j}^{†} a_{j}

, and the ladder relations recursively give norm one.

For completeness, let

g \in L^{2}

be orthogonal to every

h_{α}

. The Hermite polynomials have nonzero leading monomials and triangular lower-degree terms, so these orthogonality relations say that all polynomial moments of

h (x) = g (x) e^{- ∣ x ∣^{2} /2} \in L^{1}

vanish. Its Fourier integral extends to an entire function of

ζ \in C^{n}

F (ζ) = \int g (x) e^{- ∣ x ∣^{2} /2} e^{- i x \cdot ζ} d x .

(12)

The Cauchy-Schwarz inequality, with the Gaussian absorbing every exponential linear weight and every polynomial, justifies differentiation uniformly on compact sets of

ζ

. Every Taylor coefficient at zero is a vanishing moment, so the entire function is zero. Fourier uniqueness on

L^{1}

follows here without circularity: if an

L^{1}

function has zero Fourier transform, convolving it with

ρ_{ε}

gives zero by the Gaussian integral and Fubini, and the approximate-identity limit in

L^{1}

gives the original function zero. Thus

h = 0

and

g = 0

. The Hermite functions form an orthonormal basis of

L^{2}

Finally, the elementary Gaussian integral gives

F h_{0} = h_{0}

. The derivative and multiplication identities above imply

F a_{j} = i a_{j} F, F a_{j}^{†} = - i a_{j}^{†} F,

(13)

where the operators on the right act on the frequency variable. Hence

F h_{α} = (- i)^{∣ α ∣} h_{α} .

(14)

Thus creation and annihilation organize both the oscillator's spectral decomposition and the four possible Fourier eigenvalues. On Schwartz functions the oscillator powers control weighted derivatives, and Hermite coefficients decay faster than every power of

1 + ∣ α ∣

; conversely this rapid coefficient decay implies all Schwartz seminorms are finite. The Fourier phases preserve this decay, providing another explanation of the Fourier automorphism of Schwartz space.

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 60 / 5 / a / Solution Created 2026-10-03 Updated 2026-10-14

Use units

ℏ = 1

. A Hermitian quadrature coupling of a cavity mode to a continuum of external modes can be written

H_{couple} = i \int d ω κ (ω) (a + a^{†}) \otimes (b_{ω}^{†} - b_{ω}),

with real coupling amplitude

κ

. The free Quantum Hamiltonian is

H_{0} = ω_{c} a^{†} a + \int ω b_{ω}^{†} b_{ω} d ω

. In the interaction picture, the two photon-exchange products

a b_{ω}^{†}

and

a^{†} b_{ω}

acquire phases with frequency

ω - ω_{c}

, whereas

a^{†} b_{ω}^{†}

and

a b_{ω}

oscillate at

ω + ω_{c}

For weak coupling near resonance, the rotating-wave approximation drops those rapidly oscillating pair-creation and pair-annihilation terms and retains

H_{I} (t) = i \int d ω κ (ω) {a \otimes b_{ω}^{†} e^{i (ω - ω_{c}) t} - a^{†} \otimes b_{ω} e^{- i (ω - ω_{c}) t}} .

For a broadband flat coupling

κ = γ / (2 π)

, define the envelope annihilation operator

b (t) = (2 π)^{- 1/2} \int b_{ω} e^{- i (ω - ω_{c}) t} d ω

. The Markov approximation extends the detuning integral over the full real line and gives

[b (t), b^{†} (s)] = δ (t - s)

. Then the cavity-reservoir interaction is

H_{I} (t) = i γ {a (t) \otimes b^{†} (t) - a^{†} (t) \otimes b (t)} .

The phases and normalization of the external field fix the sign convention. The creation operators and annihilation operators exchange one excitation between cavity and field, rather than creating or destroying a pair.

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 44 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

Use natural units

ℏ = c = 1

and the Minkowski metric

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

. The canonical momentum is

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

, and the Legendre transform in mechanics gives

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

For canonical quantization of a real scalar field, impose 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.

The Heisenberg picture equation

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

then gives

\dot{ϕ} = π

. For the other equation, the momentum term commutes with

π

; commuting the gradient term past

π

and integrating the derivative of the Dirac delta function by parts gives

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

. Consequently

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

The Fourier transform reduces this Klein-Gordon equation to oscillators of frequency

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

. Hermiticity of the real scalar field pairs their positive- and negative-frequency coefficients, so its general mode expansion can be normalized as

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}], p^{0} = E_{p} .

To verify its normalization rather than guess the oscillator algebra, use scalar field oscillator inversion at

t = 0

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

Its adjoint has the conjugate phase and the opposite sign in front of

iπ

. The two mixed field-momentum commutators give

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

The Fourier representation of the Dirac delta function sets the energies equal in the last step. Similarly,

[a (p), a (q)]

has a factor

E_{q} - E_{p}

multiplying

δ^{3} (p + q)

and vanishes; the two-creator commutator vanishes as well.

Substituting the expansion in

H

and integrating the spatial phases, the two-annihilator and two-creator coefficients cancel by

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

. The free real scalar Hamiltonian in oscillator variables is

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

It differs from

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

by its constant vacuum energy; normal ordering removes that constant. Using the oscillator commutators, for either Hamiltonian,

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

Choose the Fock vacuum annihilated by every

a (p)

. Each creation operator adds a particle of energy

E_{p}

and momentum

p

; its annihilation operator removes one. Their mutual commutativity gives bosonic Fock space and symmetric many-particle states. Thus the scalar mass-shell frequency has a particle interpretation, rather than merely describing a classical oscillation.

For

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

and

r = x - y

, vacuum contraction and time ordering give the Feynman propagator

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

In the scalar Feynman propagator pole prescription, the integral of

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

has poles at

E_{p} - i 0

and

- E_{p} + i 0

. Close clockwise below for

t > 0

and counterclockwise above for

t < 0

. The residue theorem gives exactly the corresponding term above, proving

D_{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} .

As a direct check of the distributional source,

g_{E} (t) = e^{- i E ∣ t ∣} / (2 E)

is continuous, but

g_{E}^{'} (0^{+}) - g_{E}^{'} (0^{-}) = - i

. The derivative jump of a free scalar time-ordered two-point function therefore gives

(\partial_{t}^{2} + E^{2}) g_{E} = - i δ (t)

. Spatial Fourier inversion yields

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

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 56 / 4 / a / Solution Created 2026-10-03 Updated 2026-10-14

Use a real free Klein-Gordon field on a prescribed four-dimensional globally hyperbolic spacetime, with signature

(- + + +)

and

ℏ = c = 1

. A mass

μ

and curvature coupling

ξ

give the quadratic action

S [ϕ] = - \frac{1}{2} \int d^{4} x - g (g^{ab} \nabla_{a} ϕ \nabla_{b} ϕ + (μ^{2} + ξ R) ϕ^{2}),

and the field equation

(□ - μ^{2} - ξ R) ϕ = 0

. Minimal coupling is

ξ = 0

; for a massless scalar in four dimensions, conformal coupling is

ξ = 1/6

. Global hyperbolicity supplies a Cauchy hypersurface

Σ

and a well-posed initial-value problem: suitable data

(ϕ, n^{a} \nabla_{a} ϕ)

Σ

determine a unique solution. One assumes appropriate support or decay conditions so no unaccounted flux crosses spatial infinity. The geometry is classical here; quantum backreaction is a further problem.

The real solution space has the conserved symplectic form on scalar-field solutions

Ω (ϕ_{1}, ϕ_{2}) = \int_{Σ} (ϕ_{1} n^{a} \nabla_{a} ϕ_{2} - ϕ_{2} n^{a} \nabla_{a} ϕ_{1}) d Σ.

Its complexification defines the Klein-Gordon inner product

(u, v)_{K G} = i \int_{Σ} n^{a} (u^{*} \nabla_{a} v - v \nabla_{a} u^{*}) d Σ.

The divergence of its current is

i (u^{*} □ v - v □ u^{*}) = 0

by the field equation, proving independence of the chosen Cauchy hypersurface. This Hermitian form is not positive definite on the whole complex solution space: conjugating a solution reverses its norm.

Choose a complete positive-norm mode space and basis

u_{i}

satisfying

(u_{i}, u_{j})_{K G} = δ_{ij}, (u_{i}^{*}, u_{j}^{*})_{K G} = - δ_{ij}, (u_{i}, u_{j}^{*})_{K G} = 0.

Here completeness means that every real solution expands in these modes and their conjugates. Equivalently, one chooses a compatible complex structure on the Klein-Gordon solution space; this is extra data, not fixed by the field equation. Quantization replaces the mode coefficients by annihilation operators and creation operators:

ϕ (x) = \sum_{i} (a_{i} u_{i} (x) + a_{i}^{†} u_{i} (x)^{*}), [a_{i}, a_{j}^{†}] = δ_{ij}, [a_{i}, a_{j}] = 0.

In a foliation with induced metric determinant

h

, the canonical momentum is

π = h n^{a} \nabla_{a} ϕ

. The normalized complete mode expansion implements the equal-time canonical commutation relations

[ϕ (x), π (y)] = i δ^{3} (x - y)

and vanishing field-field and momentum-momentum commutators on

Σ

. Propagation by the field equation gives microcausality: field observables commute at spacelike separation. Thus there is a local quantum field theory before one has selected a particle interpretation.

The chosen positive-mode space completes to a one-particle Hilbert space

H_{1}

. Its bosonic Fock space is

⨁_{n = 0}^{\infty} Sym^{n} H_{1}

. The Fock vacuum obeys

a_{i} ∣0 ⟩ = 0

; acting with

a_{i}^{†}

produces multiparticle states, and

N_{i} = a_{i}^{†} a_{i}

counts particles in mode

i

. The vacuum two-point function is

W (x, y) = \sum_{i} u_{i} (x) u_{i} (y)^{*}

. The Hadamard condition restricts physically admissible states by prescribing their local short-distance singularity; it permits renormalization of the stress-energy tensor but does not select one unique vacuum.

The particle ambiguity is that there is generally no distinguished choice of positive-mode space. A second complete normalized basis can be

v_{i} = \sum_{j} (α_{ij} u_{j} + β_{ij} u_{j}^{*}) .

The Bogoliubov transformation preserves the mode normalization precisely when

α α^{†} - β β^{†} = I, α β^{T} - β α^{T} = 0.

Its operators obey

b_{i} = (v_{i}, ϕ)_{K G} = \sum_{j} (α_{ij}^{*} a_{j} - β_{ij}^{*} a_{j}^{†})

. Consequently a state with no

a

-particles generally has

b

-particles if

β \neq = 0

. A unitary change of basis within the same positive-mode space has

β = 0

and merely relabels modes; a change mixing the positive and negative subspaces changes the vacuum and particle notion. Arbitrary coordinates called “time” do not resolve this choice. This is the vacuum ambiguity in a nonstationary spacetime, while the field equation and local commutator remain unchanged.

In the standard stationary setting, a complete future-directed timelike Killing vector field

K

generates a time-translation symmetry. With a stable positive energy operator and suitable boundary conditions, choose positive-frequency solutions by

i L_{K} u_{ω} = ω u_{ω}, ω > 0.

The positive stationary Hamiltonian selects a particle splitting: positive spectral modes define annihilation operators, and the normal-ordered Hamiltonian is

H_{K} = \sum_{i} ω_{i} a_{i}^{†} a_{i}

. Its ground state is the vacuum state in a stationary spacetime. The Killing symmetry transports this splitting unchanged, so no positive-negative mixing is induced by stationary evolution. Degeneracies allow unitary basis changes without changing the vacuum. Stationarity fixes the ground-state particle interpretation relative to this time flow; it does not forbid excited or thermal states.

The qualification is that asymptotic stationarity alone, in the sense used for a Kerr exterior, is insufficient for a global positive-energy construction:

K

becomes spacelike in an ergoregion of a stationary spacetime. The usual unambiguous construction applies to a strictly stationary spacetime with the stated stability and spectral assumptions, or to appropriate stationary asymptotic regions for scattering. Zero modes or instability need separate treatment. Likewise different observer time flows on different regions need not define the same particles.

For particle creation by a nonstationary spacetime, choose normalized positive-frequency modes

u_{j}^{in}

in the stationary early region and

u_{i}^{out}

in the stationary late region. Propagate both sets by the Klein-Gordon equation through the intermediate time-dependent geometry. Compute their overlaps on any common Cauchy hypersurface:

u_{i}^{out} = \sum_{j} (α_{ij} u_{j}^{in} + β_{ij} u_{j}^{in *}), α_{ij} = (u_{j}^{in}, u_{i}^{out})_{K G}, β_{ij} = - (u_{j}^{in *}, u_{i}^{out})_{K G} .

The initial in-vacuum has

a_{j}^{in} ∣ 0_{in} ⟩ = 0

. Since

a_{i}^{out} = \sum_{j} (α_{ij}^{*} a_{j}^{in} - β_{ij}^{*} a_{j}^{in †})

, contracting the in operators yields the particle number from Bogoliubov coefficients:

⟨ 0_{in} ∣ N_{i}^{out} ∣ 0_{in} ⟩ = j \sum ∣ β_{ij} ∣^{2}, ⟨ N_{out} ⟩ = i, j \sum ∣ β_{ij} ∣^{2} .

Thus the procedure is to fix asymptotic mode choices, solve the classical wave equation, project using the conserved inner product, and take squared negative-frequency overlaps. Particle energy comes from the time-dependent background, so particle creation does not contradict the linear deterministic field equation. For an initial state diagonal in in-mode occupation numbers

n_{j}

, with no anomalous correlations, the more general expectation is

\sum_{j} [∣ α_{ij} ∣^{2} n_{j} + ∣ β_{ij} ∣^{2} (n_{j} + 1)]

For continuous spectra, sums become integrals and normalized wave packets avoid meaningless squares of delta functions. A finite total count requires the appropriate convergence of the double sum or integral. The bosonic mode mixing implementability condition is that

β

be Hilbert-Schmidt; otherwise the two Fock representations need not be unitarily equivalent, even though the local field algebra and finite-mode particle calculations still make sense. This distinction between deterministic field evolution and a chosen representation is essential to quantum field theory in curved spacetime.

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

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) .

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)} .

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) .

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} .

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} :,

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} : .

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 ⟩ .

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),

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}} .

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.

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 54 / 4 / Solution Created 2026-10-03 Updated 2026-10-20

Work initially in units

G = c = ℏ = k_{B} = 1

and consider gravitational collapse settling to a nonextremal Schwarzschild black hole. Classically, outgoing rays sufficiently close to the forming event horizon suffer an arbitrarily large redshift. Quantum mechanically, the meaning of positive frequency at past null infinity differs from that at future null infinity; a Bogoliubov transformation between these two mode descriptions can therefore mix creation operators and annihilation operators. Hawking's argument computes that mixing and obtains a thermal occupation, even when the incoming state has no particles.

To see why the frequency mixing is universal, use final Schwarzschild time

t

, the Schwarzschild tortoise coordinate

r_{*}

, and retarded time

u = t - r_{*}

. Near the final event horizon,

r_{*} \sim (2 κ)^{- 1} lo g (r - 2 M)

, with Schwarzschild surface gravity

κ = 1/ (4 M)

. On a fixed advanced-time section,

u = v - 2 r_{*}

, so

r - 2 M

is proportional to

e^{- κ u}

. Tracing a ray back through the regular collapsing region maps this small separation smoothly to its separation from the last escaping incoming ray. If

U

is an affine incoming null coordinate and

U_{H}

labels that limiting ray, the Hawking exponential ray map is

U_{H} - U = A e^{- κ u} (1 + o (1)), A > 0.

The logarithm at a simple horizon root fixes the exponent. Smooth propagation through the earlier collapse changes

A

and phases but not the leading exponential redshift.

For a massless bosonic test scalar field, an outgoing mode of frequency

ω > 0

at future null infinity behaves as

e^{- iω u}

. Trace it backward in the high-frequency geometric optics approximation. Its relevant incoming dependence is, up to normalization and a phase,

p_{ω} (U) \sim Θ (U_{H} - U) (\frac{U _{H} - U}{A})^{ia}, a = \frac{ω}{κ} .

Here

Θ

is the Heaviside step function. Incoming rays with

U > U_{H}

do not escape. This truncated logarithmic phase is not purely positive frequency with respect to

U

. Decomposing it into incoming positive-frequency solutions

e^{- i ω^{'} U}

and negative modes

e^{+ i ω^{'} U}

gives the Bogoliubov transformation. With

x = U_{H} - U

, the two relevant Fourier transform integrals are

I_{-} (ω^{'}) = \int_{0}^{\infty} x^{ia} e^{- (ϵ + i ω^{'}) x} d x, I_{+} (ω^{'}) = \int_{0}^{\infty} x^{ia} e^{- (ϵ - i ω^{'}) x} d x, ϵ > 0.

The former is the positive-frequency coefficient and the latter is the negative-frequency coefficient; the Klein-Gordon inner product introduces the same frequency normalization into their modulus ratio. By the defining integral of the gamma function and analytic continuation in its Laplace parameter,

I_{\mp} = Γ (1 + ia) (ϵ \pm i ω^{'})^{- 1 - ia} .

For a complex number

z

in the right half-plane,

∣ z^{- 1 - ia} ∣ = ∣ z ∣^{- 1} e^{a a r g z}

. As

ϵ ↓ 0

, the two arguments tend to

+ π /2

and

- π /2

. This gives the thermal ratio of Hawking Bogoliubov coefficients:

\frac{∣ β _{ω ω^{'}} ∣ ^{2}}{∣ α _{ω ω^{'}} ∣ ^{2}} = e^{- 2 πω / κ} .

The regulator fixes the branches and is essential to the sign of the thermal exponent.

Expand the outgoing annihilator as

b = \int d ω^{'} (α_{ω ω^{'}}^{*} a_{ω^{'}} - β_{ω ω^{'}}^{*} a_{ω^{'}}^{†})

, including a complete set of incoming channels. The canonical identities for a bosonic Bogoliubov transformation express

[b, b^{†}] = 1

. For normalized outgoing wave packets, they give

\int (∣ α ∣^{2} - ∣ β ∣^{2}) d ω^{'} = 1

. In the incoming Fock vacuum, the outgoing mean of the particle number operator is

N_{ω} = \int ∣ β ∣^{2} d ω^{'}

. Combining the two identities with the thermal ratio, in the late-time narrow-frequency packet limit, yields

N_{ω} = \frac{1}{e ^{2 πω / κ} - 1}, T_{H} = \frac{κ}{2 π} = \frac{1}{8 π M} .

Wave packets avoid interpreting the divergent normalization of infinitely long continuum modes as a finite particle count. The temperature is the Hawking temperature, and the occupation is the zero-chemical-potential Bose-Einstein distribution. Restoring constants, the Schwarzschild black hole temperature is

T_{H} = ℏ c^{3} / (8 π GM k_{B})

for physical mass

M

The preceding thermal occupation describes the horizon-originating channel before exterior scattering. The exterior angular and curvature potential partly reflects it. For a scalar field, the number flux at future null infinity is consequently

\frac{d N}{d t d ω} = \frac{1}{2 π} \sum_{ℓ = 0}^{\infty} (2 ℓ + 1) \frac{Γ _{ℓ} ( ω )}{e ^{ω / T_{H}} - 1},