Path Integral Formulation

Feynman's path integral is an alternative — but equivalent — formulation of quantum mechanics. Instead of evolving a state vector with the Schrödinger equation (see the postulates), it expresses transition amplitudes as a sum over all possible classical trajectories, weighted by a phase determined by the classical action.

To make the logical structure transparent, every subsection below is tagged as one of:

• (Definition) — a mathematical object or notation introduced for use later.
• (Postulate) — a primitive assumption of the path-integral formulation. These are the axioms; everything else follows.
• (Derived) — a result that can be proved from the postulates plus the standard postulates of QM.

The path integral is built on two postulates (P1 and P2 below). Everything else in this document is either a definition or a derived consequence.

1. Preliminaries

The path integral uses several concepts from classical mechanics and functional analysis that go beyond the Hilbert-space framework of the Mathematical Preliminaries. Everything in this section is a definition or a recap of classical mechanics — no quantum-mechanical content yet.

1.1 Lagrangian Mechanics (Definition / classical recap)

In the Lagrangian formulation of classical mechanics, the dynamics of a system with generalized coordinates $q(t)$ are encoded in a Lagrangian

$$L(q,\dot{q},t)=T-V,$$

where $T$ is the kinetic energy and $V$ the potential energy. The action is the time integral of the Lagrangian along a trajectory:

$$S[q(t)]={\int }_{t_i}^{t_f}dtL(q(t),\dot{q}(t),t).$$

Hamilton's principle of stationary action (a postulate of classical mechanics, not of QM) states that the classical trajectory between fixed endpoints is the one for which $S$ is stationary, yielding the Euler–Lagrange equations

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}-\frac{\partial L}{\partial q}=0.$$

1.2 Functionals and Functional Derivatives (Definition)

A functional is a map from a space of functions to numbers. The functional derivative $\delta F/\delta q(t)$ is defined by the variation

$$\delta F[q]=\int dt\frac{\delta F}{\delta q(t)}\delta q(t)+O(\delta q^2).$$

For the action,

$$\frac{\delta S}{\delta q(t)}=\frac{\partial L}{\partial q}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}.$$

1.3 Functional Integration (Definition)

A functional integral generalizes ordinary integration to integration over a space of functions:

$$\int \mathcal{D}q(t)F[q(t)].$$

The symbol $\mathcal{D}q(t)$ is defined by a limiting procedure (time-slicing):

1. Discretize time into $N$ slices $t_n=t_i+nϵ$ with $ϵ=(t_f-t_i)/N$.
2. Replace the path $q(t)$ by its values $q_n=q(t_n)$ at the slice points.
3. Replace $\int \mathcal{D}q$ by ${\prod }_n\int d{q}_n$ together with an appropriate normalization factor (fixed by Postulate P1 below so that the resulting evolution is unitary).
4. Take $N\to \infty$, $ϵ\to 0$.

For oscillatory integrands $e^{iS/\hslash }$ this limit is formal; it acquires rigorous mathematical meaning only after Wick rotation (§1.4), where it becomes the Wiener measure of Brownian motion.

1.4 Wick Rotation (Definition)

A Wick rotation is the analytic continuation $t\to -i\tau$ (with $\tau$ real). It maps the Lorentzian time axis to the Euclidean one and converts oscillatory factors into exponentially damped ones:

$$e^{iS[q]/\hslash }⟶e^{-S_E[q]/\hslash },$$

where $S_E$ is the Euclidean action, obtained from $S$ by replacing $t\to -i\tau$ and changing the sign of the kinetic term so that $S_E$ is positive-definite.

1.5 Stationary-Phase / Saddle-Point Approximation (Definition / mathematical lemma)

For an integral $I(\hslash )=\int dx{e}^{if(x)/\hslash }$ in the limit $\hslash \to 0$, the integrand oscillates rapidly except near stationary points $f^{\prime }(x_0)=0$. Expanding to quadratic order,

$$I(\hslash )\approx \sum _{x_0}\sqrt{\frac{2\pi i\hslash }{f^{\prime \prime }(x_0)}}{e}^{if(x_0)/\hslash }.$$

The infinite-dimensional generalization applies to path integrals: in the limit $\hslash \to 0$ they are dominated by paths satisfying $\delta S/\delta q(t)=0$. This is a fact about oscillatory integrals, not about quantum mechanics.

1.6 Time Ordering (Definition)

The time-ordering operator $T$ rearranges a product of operators in order of decreasing time argument:

$$T\hat{A}(t_1)\hat{B}(t_2)=\{\begin{array}{ll}\hat{A}(t_1)\hat{B}(t_2)& \text{if }{t}_1>t_2,\\ \hat{B}(t_2)\hat{A}(t_1)& \text{if }{t}_2>t_1,\end{array}$$

with an extra sign $(-1{)}^P$ for each pair-exchange of fermionic operators.

1.7 Propagator (Definition)

The propagator (or transition amplitude) is the position-basis matrix element of the time-evolution operator:

$$K(x_f,t_f;x_i,t_i)\equiv ⟨x_f|\hat{U}(t_f,t_i)|x_i⟩=⟨x_f|e^{-i\hat{H}(t_f-t_i)/\hslash }|x_i⟩.$$

This is purely a definition in terms of objects already present in the canonical formulation (see postulates).

2. The Postulates of the Path Integral

These two statements are the entire content of the path-integral formulation. Everything in §3 is derived from them (combined with the standard postulates of QM).

Postulate P1 — Amplitude as a Sum Over Paths

The propagator $K(x_f,t_f;x_i,t_i)$ is given by a functional integral over all continuous paths $x(t)$ connecting $(x_i,t_i)$ to $(x_f,t_f)$:

$$\boxed{K(x_f,t_f;x_i,t_i)={\int }_{x(t_i)=x_i}^{x(t_f)=x_f}\mathcal{D}x(t)\exp \left(\frac{i}{\hslash }S[x(t)]\right)}$$

where $S[x(t)]={\int }_{t_i}^{t_f}dtL(x,\dot{x},t)$ is the classical action along the path. Each path contributes a complex number of unit modulus with phase $S/\hslash$.

Concretely, using the time-slicing definition (§1.3), for a non-relativistic particle with $L=\frac{1}{2}m{\dot{x}}^2-V(x)$,

$$K=\underset{N\to \infty }{\lim }{\left(\frac{m}{2\pi i\hslash ϵ}\right)}^{N/2}\int \prod _{n=1}^{N-1}d{x}_n\exp \left(\frac{i}{\hslash }\sum _{n=0}^{N-1}ϵL\left(\frac{x_{n+1}+x_n}{2},\frac{x_{n+1}-x_n}{ϵ}\right)\right).$$

The normalization factor $(m/2\pi i\hslash ϵ{)}^{N/2}$ is part of the postulate — it is fixed by requiring that the resulting evolution be unitary.

Postulate P2 — Composition (Superposition of Alternatives)

The amplitude for a process composed of two successive sub-processes is the integral over intermediate alternatives of the product of sub-amplitudes:

$$\boxed{K(x_f,t_f;x_i,t_i)=\int dxK(x_f,t_f;x,t)K(x,t;x_i,t_i),t_i<t<t_f.}$$

Note. P2 is not logically independent of P1 — given the time-slicing definition of $\mathcal{D}x$, P2 follows from P1. In Feynman's original formulation P1 and P2 were stated as independent axioms, with P2 (the "sum over alternatives" rule) playing the conceptual role of replacing the classical-probability composition law. We list both because the conceptual content is split between them.

3. Derived Results

Everything below is a consequence of the postulates above (together with the canonical postulates of QM where applicable).

3.1 Wavefunction Evolution → Schrödinger Equation (Derived)

Given an initial wavefunction $\psi (x_i,t_i)=⟨x_i|\psi (t_i)⟩$, inserting a position completeness relation and applying P1 gives

$$\psi (x_f,t_f)=\int d{x}_{i}K(x_f,t_f;x_i,t_i)\psi (x_i,t_i).$$

Expanding the time-sliced form of $K$ for an infinitesimal $ϵ$ and keeping terms to $O(ϵ)$ reproduces the Schrödinger equation $i\hslash {\partial }_t\psi =\hat{H}\psi$. Hence Postulate 5 of the canonical formulation is derivable from P1 — the path integral is a complete alternative to canonical quantization, not an addition to it.

3.2 Classical Limit (Derived)

Applying the stationary-phase approximation (§1.5) to P1 in the limit $\hslash \to 0$, the integral is dominated by paths satisfying

$$\frac{\delta S}{\delta x(t)}=0⟺\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}-\frac{\partial L}{\partial x}=0.$$

These are precisely the classical Euler–Lagrange equations. Classical mechanics therefore emerges from the path integral in the $\hslash \to 0$ limit; this is a theorem, not an additional postulate.

3.3 Euclidean (Imaginary-Time) Path Integral (Derived)

Wick-rotating P1 with $t\to -i\tau$ gives the Euclidean propagator

$$K_E(x_f,{\tau }_f;x_i,{\tau }_i)=\int \mathcal{D}x(\tau )\exp \left(-\frac{1}{\hslash }{S}_E[x(\tau )]\right),$$

with Euclidean action $S_E[x(\tau )]=\int d\tau [\frac{1}{2}m{\dot{x}}^2+V(x)]$. This is mathematically a genuine measure (the Wiener measure in the free case) and is the foundation for non-perturbative methods (instantons, lattice quantization, the QFT–statistical-mechanics correspondence).

3.4 Correlation Functions (Derived)

Time-ordered vacuum expectation values of Heisenberg-picture operators have the path-integral representation

$$⟨0|T\hat{x}(t_1)\cdots \hat{x}(t_n)|0⟩=\frac{\int \mathcal{D}xx(t_1)\cdots x(t_n)e^{iS[x]/\hslash }}{\int \mathcal{D}x{e}^{iS[x]/\hslash }}.$$

The time-ordering on the operator side appears automatically because the $x(t_i)$ under the integral are ordinary commuting numbers.

3.5 Generating Functional (Derived)

A compact way to encode all correlation functions is the generating functional

$$Z[J]=\int \mathcal{D}q\exp \left(\frac{i}{\hslash }S[q]+\frac{i}{\hslash }\int dtJ(t)q(t)\right).$$

Time-ordered correlators are obtained as functional derivatives:

$$⟨0|T\hat{q}(t_1)\cdots \hat{q}(t_n)|0⟩=\frac{1}{Z[0]}{\frac{(-i\hslash {)}^n{\delta }^{n}Z[J]}{\delta J(t_1)\cdots \delta J(t_n)}|}_{J=0}.$$

This formalism generalizes directly to quantum field theory.

4. Equivalence with the Canonical Formulation

The path-integral and canonical (operator) formulations of quantum mechanics are mathematically equivalent: each can be derived from the other.

• Canonical → Path integral: P1 can be proved by repeated insertion of position-basis completeness relations into $⟨x_f|e^{-i\hat{H}(t_f-t_i)/\hslash }|x_i⟩$ and using the Trotter product formula. From this perspective P1 is a theorem, not a postulate.
• Path integral → Canonical: Conversely, taking P1 (and P2) as primitive, one derives the Schrödinger equation (§3.1), the Hilbert-space structure, and the Born rule. From this perspective the canonical postulates are theorems.

Whether to call P1 a "postulate" or a "theorem" therefore depends on which formulation one takes as foundational. The two are interchangeable axiomatic starting points for the same physical theory; their advantages are complementary:

• Canonical formulation: transparent treatment of states, measurement, and the Hilbert-space structure; well-suited to non-relativistic problems and perturbation theory in the interaction picture.
• Path integral: manifest classical limit, natural treatment of symmetries (especially gauge symmetries), straightforward generalization to field theory and curved backgrounds, and a direct bridge to statistical mechanics via Wick rotation.