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Statistical homogeneity forces a scalar two-point function to have momenta

(k, - k)

. Statistical isotropy then makes it a function only of

∣ k ∣

, so it is parity even without using perturbation theory.

For a scalar three-point function, momentum conservation gives

k_{1} + k_{2} + k_{3} = 0

, so all three vectors lie in one plane. A rotation by

π

around the normal to that plane sends every

k_{a}

- k_{a}

. Rotational invariance therefore identifies a triangle with its parity reverse, proving nonperturbatively that the scalar primordial bispectrum is parity even.

A rotationally invariant local parity-odd three-scalar vertex must contain a Levi-Civita symbol contracted with three spatial momenta. Its momentum-space factor is proportional to

ϵ_{ijk} k_{1}^{i} k_{2}^{j} k_{3}^{k} = k_{1} \cdot (k_{2} \times k_{3}) = 0,

because momentum conservation makes the momenta linearly dependent. The equivalent position-space expression is a total divergence; its spatial integral vanishes under the stated vanishing boundary condition. Thus a parity-odd interaction of three scalar fields contributes nothing to the action.

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 11B / i / Solution Created 2026-09-24 Updated 2026-09-25

In components,

[\nabla \times \nabla ϕ]_{i} = ϵ_{ijk} \partial_{j} \partial_{k} ϕ = 0,

because the Levi-Civita symbol is antisymmetric in

j, k

while equality of mixed partial derivatives makes

\partial_{j} \partial_{k} ϕ

symmetric. Hence

\nabla \times \nabla ϕ = 0 .

(2)

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 3B / ii / Solution Created 2026-09-24 Updated 2026-09-25

Use the Levi-Civita symbol and the product rule. First,

\nabla \cdot (F \times G) + + + = \partial_{i} (ϵ_{ijk} F_{j} G_{k}) = ϵ_{ijk} (\partial_{i} F_{j}) G_{k} + + ϵ_{ijk} F_{j} \partial_{i} G_{k} = G \cdot (\nabla \times F) - F \cdot (\nabla \times G) .

For the second identity, the contraction

ϵ_{ijk} ϵ_{k l m} + = δ_{i l} δ_{jm} - δ_{im} δ_{j l}

(2)

gives

+ [\nabla \times (F \times G)]_{i} + + + = ϵ_{ijk} \partial_{j} (ϵ_{k l m} F_{l} G_{m}) = \partial_{j} (F_{i} G_{j} - F_{j} G_{i}) = F_{i} \partial_{j} G_{j} - G_{i} \partial_{j} F_{j} + + G_{j} \partial_{j} F_{i} - F_{j} \partial_{j} G_{i} .

(3)

In vector notation this is

+ \nabla \times (F \times G) + = F (\nabla \cdot G) - G (\nabla \cdot F) + + (G \cdot \nabla) F - (F \cdot \nabla) G .

(4)

These are the divergence and curl of a cross product identities.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 307 / 3 / i / Solution Created 2026-09-24 Updated 2026-09-25

Both terms in

B_{μ} = W_{μ} - \frac{1}{4} \overset{ˉ}{Q} \overset{σ}{ˉ}_{μ} Q

commute with momentum. For the first,

[W_{μ}, P_{ρ}] = \frac{1}{2} ϵ_{μν λσ} [M^{ν λ}, P_{ρ}] P^{σ} = 0,

because the two resulting products of momenta are symmetric in indices contracted with the Levi-Civita symbol. For the second,

[P_{ρ}, Q] = [P_{ρ}, \overset{ˉ}{Q}] = 0

. Therefore

[B_{μ}, P_{ρ}] = 0 .

(2)

Pseudotensor 2026-10-06

Under an orthogonal coordinate transformation

R

, a Cartesian pseudotensor has the usual tensor transformation law multiplied by

det R

. Thus proper rotation matrices act as on ordinary tensors, while an orientation-reversing change contributes an extra minus sign. Interpreted as a rank-three pseudotensor, the Levi-Civita symbol has the same numerical components in every orthonormal frame, because its ordinary three-index transformation already contributes a factor

det R

Rotation invariance of the Levi-Civita symbol 2026-10-06

The determinant transformation formula for the Levi-Civita symbol shows that it is invariant under every proper rotation matrix, for which

det R = 1

. Under an orthogonal reflection its sign reverses. This distinguishes proper-rotation isotropy from the pseudotensor transformation convention when both orientations are included.