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 Incoming links: Time-sliced equivalence of phase-space and configuration-space path integrals

Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 62 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

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Use canonical quantization with

[\overset{q}{^}, \overset{p}{^}] = i ℏ

\overset{p}{^} = - i ℏ \partial_{q}

, and

\hat{H} = \overset{p}{^}^{2} / (2 m) + V (\overset{q}{^})

. Assume a real potential energy with the regularity and lower-bound conditions needed for a self-adjoint Hamiltonian and its Trotter product formula; singular potentials require their domain and limiting prescription to be specified separately. The transition amplitude is

K (β, α; T) = ⟨ β ∣ e^{- i T \hat{H} /ℏ} ∣ α ⟩

Take

δ = T / N

q_{0} = α

q_{N} = β

, and choose the left-endpoint potential energy convention. The precise time-sliced phase-space path integral is

K_{N} = \int \prod_{j = 1}^{N - 1} d q_{j} \prod_{j = 1}^{N} \frac{d p _{j}}{2 π ℏ} exp {\frac{i}{ℏ} \sum_{j = 1}^{N} [p_{j} (q_{j} - q_{j - 1}) - δ (\frac{p _{j}^{2}}{2 m} + V (q_{j - 1}))]} .

(1)

The endpoint positions are fixed and the momenta are unconstrained. Its continuum notation is

K = \int_{q (0) = α}^{q (T) = β} D q D p exp [\frac{i}{ℏ} \int_{0}^{T} (p \overset{q}{˙} - H (p, q)) d t],

(2)

where the functional measure means the displayed limit, not an unspecified product of unnormalized measures.

At each slice, complete the square in momentum. With the square-root branch selected by continuation from positive Euclidean time,

\int \frac{d p}{2 π ℏ} exp [\frac{i}{ℏ} (p Δ q - \frac{δ p ^{2}}{2 m})] = (\frac{m}{2 πi ℏ δ})^{1/2} exp [\frac{im ( Δ q ) ^{2}}{2ℏ δ}] .

(3)

Thus the time-sliced equivalence of phase-space and configuration-space path integrals holds already at finite

N

K_{N} = (\frac{m}{2 πi ℏ δ})^{N /2} \int \prod_{j = 1}^{N - 1} d q_{j} exp {\frac{i}{ℏ} \sum_{j = 1}^{N} [\frac{m ( q _{j} - q _{j - 1} ) ^{2}}{2 δ} - δ V (q_{j - 1})]} .

(4)

This defines the configuration-space path integral

K = \int_{q (0) = α}^{q (T) = β} D q exp [\frac{i}{ℏ} \int_{0}^{T} (\frac{m}{2} \overset{q}{˙}^{2} - V (q)) d t] .

(5)

There are

N

Gaussian prefactors but only

N - 1

coordinate integrations. Omitting that distinction would change the kernel normalization. The paths integrated in the limit need not be differentiable; the kinetic action notation stands for the lattice difference expression.

To identify this with the operator amplitude, normalize

⟨ q ∣ p ⟩ = (2 π ℏ)^{- 1/2} e^{i pq /ℏ}

. The kernel of one product factor is exactly

⟨ q_{j} ∣ e^{- i δ \overset{p}{^}^{2} / (2 m ℏ)} e^{- i δ V (\overset{q}{^}) /ℏ} ∣ q_{j - 1} ⟩ = \int \frac{d p _{j}}{2 π ℏ} e^{i p_{j} (q_{j} - q_{j - 1}) /ℏ - i δ p_{j}^{2} / (2 m ℏ) - i δ V (q_{j - 1}) /ℏ} .

(6)

Insert

N - 1

position resolutions of the identity between factors. Their product is precisely

K_{N}

. The Trotter product formula then gives

(e^{- i T \overset{p}{^}^{2} / (2 m N ℏ)} e^{- i T V (\overset{q}{^}) / (N ℏ)})^{N} ⟶ e^{- i T \hat{H} /ℏ},

(7)

so both functional integrals represent the same canonical kernel. One can define the finite integrals first at

T = - i τ

τ > 0

, where the Gaussian momentum and coordinate kernels converge, and then continue to the real-time boundary value. Alternatively use an equivalent oscillatory damping prescription. Operator convergence fixes the distributional kernel limit; it need not imply pointwise convergence at every endpoint for every admissible potential energy. There is no operator-ordering ambiguity for this separated kinetic-plus-potential Hamiltonian once the common slicing convention is fixed.

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