 Solution

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