Action and Phase

In a real-time path integral, a history $q(t)$ contributes an amplitude with phase

$$\exp\left[ \frac{i}{\hbar}S[q] \right].$$

The classical action therefore plays two roles:

• in classical mechanics, its stationary paths obey the equations of motion;
• in quantum mechanics, differences in action become relative phases and interfere.

This is the precise content behind the slogan that the classical path emerges from a sum over histories. Nonclassical histories are not assigned zero amplitude. In a semiclassical regime, their rapidly varying phases tend to cancel, while neighborhoods of stationary-action paths remain coherent to leading order.

This page is the canonical home for that physical phase logic. Action Principles owns the variational derivation, Hamilton–Jacobi Theory Preview connects principal functions to wavefunction phases, Classical Action and Quantum Phase owns the local gradient, wavefront, and gauge-covariance dictionary, Stationary Phase owns the asymptotic formulas, and Maslov Index owns the caustic phase correction.

Action in Classical Mechanics

For generalized coordinates $q^a(t)$ and a Lagrangian $L(q,\dot q,t)$, the action between fixed endpoint times is

$$S[q] = \int_{t_i}^{t_f}dt\, L(q,\dot q,t).$$

Action has dimensions of energy times time, the same dimensions as $\hbar$. A path is classical when the first variation vanishes under admissible variations with fixed endpoints:

$$q^a(t) \longrightarrow q^a(t)+\eta^a(t), \qquad \eta^a(t_i)=\eta^a(t_f)=0.$$

The first variation is

$$\begin{aligned} \delta S &= \left[ \frac{\partial L}{\partial\dot q^a} \eta^a \right]_{t_i}^{t_f}\\ &\quad+ \int_{t_i}^{t_f}dt\, \left( \frac{\partial L}{\partial q^a} - \frac{d}{dt} \frac{\partial L}{\partial\dot q^a} \right) \eta^a. \end{aligned}$$

The endpoint term vanishes for fixed endpoints. Requiring $\delta S=0$ for arbitrary interior variations gives the Euler–Lagrange equations

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

The action is stationary, not necessarily minimal. A classical path may be a minimum, maximum, or saddle of the action functional. Conjugate points and caustics can change the signature of the second variation without changing the classical equation of motion.

The detailed assumptions about endpoint data, boundary terms, and admissible variations belong to Action Principles.

Phase in Quantum Mechanics

The dimensionless quantity in the quantum weight is

$$\varphi[q] = \frac{S[q]}{\hbar}.$$

For two histories, the observable interference depends on their relative phase

$$\Delta\varphi = \frac{S[q_1]-S[q_2]}{\hbar}.$$

A common additive constant in every action changes the full transition amplitude by one overall phase and does not change probabilities. Action differences matter because they change relative phases between alternatives.

The real-time factor has unit magnitude:

$$\left\lvert e^{iS[q]/\hbar} \right\rvert =1$$

for a real action. It is not a probability density on path space. The measure, normalization, boundary conditions, and possible operator-ordering terms are additional parts of the path integral, as explained in Time Slicing.

Total derivatives and endpoint phases

Two Lagrangians related by

$$L'(q,\dot q,t) = L(q,\dot q,t) + \frac{d}{dt}F(q,t)$$

give the same Euler–Lagrange equations. Their actions differ by

$$S'[q] = S[q] + F(q_f,t_f) - F(q_i,t_i).$$

Thus a fixed-endpoint propagator changes by the endpoint phase

$$K' = e^{iF(q_f,t_f)/\hbar} K e^{-iF(q_i,t_i)/\hbar}.$$

Physical predictions remain unchanged when states and operators are transformed consistently. This is a useful warning: a classically irrelevant boundary term can still be visible in the phase convention of a quantum amplitude.

Real time and imaginary time

After an appropriate Wick rotation, the weight becomes schematically

$$e^{-S_E/\hbar},$$

which is exponentially damped rather than oscillatory. That is a different contour and boundary-value problem, not permission to reinterpret the original real-time phase as a probability. The canonical treatment is Euclidean and Imaginary-Time Path Integrals.

Stationary Action and Constructive Interference

Let $q_{mathrm{cl}}$ be a classical path with the required endpoints, and write a neighboring history as

$$q(t) = q_{mathrm{cl}}(t)+\eta(t), \qquad \eta(t_i)=\eta(t_f)=0.$$

Expanding the action gives

$$S[q] = S[q_{mathrm{cl}}] + \delta S[q_{mathrm{cl}};\eta] + \frac12\delta^2S[q_{mathrm{cl}};\eta] + \cdots.$$

Because $q_{mathrm{cl}}$ is stationary,

$$\delta S[q_{mathrm{cl}};\eta]=0.$$

The phase is therefore insensitive to first-order path variations. Nearby histories have similar phases over a neighborhood whose scale is controlled by the second variation and by $\hbar$. Their amplitudes add coherently to leading order.

For a free particle over a time $T=t_f-t_i$, the classical path is the straight line and

$$S_{mathrm{cl}} = \frac{m(q_f-q_i)^2}{2T}.$$

Writing $q=q_{mathrm{cl}}+\eta$ gives exactly

$$S[q] = S_{mathrm{cl}} + \frac{m}{2} \int_{t_i}^{t_f}dt\, \dot\eta^2.$$

The term linear in $\eta$ vanishes because $\ddot q_{mathrm{cl}}=0$ and the endpoints are fixed. Integrating the remaining Gaussian fluctuations supplies the prefactor of the Free-Particle Path Integral; it does not simply discard every nonclassical path.

Destructive Interference Away from Classical Paths

Away from a stationary path, a small deformation changes the action at first order. When a characteristic action scale $S_0$ satisfies

$$\frac{S_0}{\hbar}\gg1,$$

even modest changes of a history can rotate the phase through many cycles. Smooth families of such contributions then cancel strongly when integrated.

The finite-dimensional analogue is

$$I(\hbar) = \int dx\, a(x)e^{iS(x)/\hbar}.$$

If $S'(x)$ stays away from zero and endpoint contributions are controlled, repeated integration by parts suppresses that region by powers of $\hbar$. Near $S'(x)=0$, this argument fails and a stationary-phase contribution remains. The functional-integral statement is the infinite-dimensional analogue, interpreted through time slicing or another regulator.

Several cautions are essential:

• cancellation is asymptotic, not exact removal of every nonstationary history;
• endpoints, singularities, boundaries, and complex saddles can contribute;
• saying $S\gg\hbar$ is meaningless until the relevant action scale is specified;
• a stationary path can be degenerate, in which case the ordinary Gaussian approximation fails;
• decoherence and coarse graining are separate mechanisms that can suppress observable interference between branches.

The full one- and multidimensional estimates belong to Stationary Phase.

Multiple Classical Paths

The boundary-value problem can have more than one classical solution. In a semiclassical approximation, each branch contributes an amplitude of the form

$$K_{mathrm{sc}}(q_f,t_f;q_i,t_i) \sim \sum_{\gamma} A_{\gamma} \exp\left[ \frac{i}{\hbar}S_{\gamma} - i\frac{\pi}{2}\nu_{\gamma} \right],$$

where $\gamma$ labels classical paths, $A_\gamma$ contains fluctuation information, and $\nu_\gamma$ is a convention-dependent caustic index.

Stationary paths need not interfere constructively with one another. For two branches,

$$\begin{aligned} \left\lvert A_1e^{iS_1/\hbar} + A_2e^{iS_2/\hbar} \right\rvert^2 &= \lvert A_1\rvert^2 + \lvert A_2\rvert^2\\ &\quad+ 2\operatorname{Re} \left[ A_1A_2^* e^{i(S_1-S_2)/\hbar} \right]. \end{aligned}$$

The relative classical action controls the fringes, together with prefactor and Maslov phases. Multiple images in wave optics, trajectories around obstacles, periodic-orbit sums, and semiclassical tunneling continuations all exhibit versions of this structure.

The existence of several stationary branches is one reason the classical limit is not simply “keep one path.” Which interference survives depends on preparation, resolution, stability, environmental decoherence, and the observable being measured.

Maslov Phase Preview

The quadratic fluctuation operator around a classical path determines both the magnitude and phase of its semiclassical prefactor. When a fluctuation eigenvalue passes through zero at a caustic or conjugate point, the naive determinant vanishes and its square-root phase changes branch.

This produces a correction commonly written

$$\exp\left( -i\frac{\pi}{2}\nu_\gamma \right).$$

The integer $\nu_\gamma$ is called a Maslov or Morse-type index, with conventions varying across WKB, EBK quantization, semiclassical propagators, and trace formulas. It is not a decorative phase: omitting it gives incorrect interference and can spoil composition across caustics.

The full geometry, sign conventions, turning-point examples, and oscillator caustics belong to Maslov Index and Semiclassical Propagator.

Common Mistakes

• Saying the action is always minimized rather than stationary.
• Treating $e^{iS/\hbar}$ as a probability weight on real-time paths.
• Claiming that a particle literally follows every path as a set of simultaneous classical trajectories.
• Saying only the classical path contributes, rather than a coherent neighborhood of histories.
• Assuming all classical paths interfere constructively with one another.
• Quoting the limit $\hbar\to0$ without fixing physical action scales.
• Dropping measure and fluctuation prefactors because only the exponential looks important.
• Ignoring endpoint phases produced by total derivatives in the Lagrangian.
• Using ordinary stationary phase at a degenerate saddle or caustic.
• Omitting Maslov phases when comparing semiclassical branches.
• Confusing the oscillatory real-time weight with the damped Euclidean weight.

Cross-Links

• Why Path Integrals?
• From Propagators to Path Integrals
• Time Slicing
• Action Principles
• Hamilton–Jacobi Theory Preview
• Lagrangian Mechanics Review
• Stationary Phase
• Stationary Phase and the Classical Limit
• Free-Particle Path Integral
• Harmonic-Oscillator Path Integral
• Euclidean and Imaginary-Time Path Integrals
• Semiclassical Propagator
• Maslov Index

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.
• M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, 1990.
• L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed., Butterworth-Heinemann, 1976.
• V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989.

Exercises

1. Show that the action phase is dimensionless and find the condition for two equal-prefactor paths to interfere constructively.

Solution

The Lagrangian has dimensions of energy, so

$$[S] = [L][t] = \text{energy}\times\text{time} = [\hbar].$$

Therefore $S/\hbar$ is dimensionless. For equal real prefactors, constructive interference occurs when

$$\frac{S_1-S_2}{\hbar} = 2\pi n, \qquad n\in\mathbb Z.$$

Additional prefactor or Maslov phases must be included when they are present.

2. Derive the Euler–Lagrange equation from the first variation for fixed endpoints.

Solution

For $q\to q+\eta$,

$$\delta S = \int_{t_i}^{t_f}dt\, \left( \frac{\partial L}{\partial q}\eta + \frac{\partial L}{\partial\dot q}\dot\eta \right).$$

Integrate the second term by parts:

$$\begin{aligned} \delta S &= \left[ \frac{\partial L}{\partial\dot q}\eta \right]_{t_i}^{t_f}\\ &\quad+ \int_{t_i}^{t_f}dt\, \left( \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial\dot q} \right)\eta. \end{aligned}$$

The boundary term vanishes because $\eta(t_i)=\eta(t_f)=0$. Since the interior variation is arbitrary, stationarity requires

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

3. Verify the free-particle expansion $S[q]=S_{\mathrm{cl}}+(m/2)\int\dot\eta^2dt$.

Solution

Set $q=q_{\mathrm{cl}}+\eta$ in

$$S[q] = \frac{m}{2} \int dt\,\dot q^2.$$

Then

$$S[q] = S[q_{\mathrm{cl}}] + m\int dt\, \dot q_{\mathrm{cl}}\dot\eta + \frac{m}{2} \int dt\,\dot\eta^2.$$

Integrating the cross term by parts gives

$$m\left[ \dot q_{\mathrm{cl}}\eta \right]_{t_i}^{t_f} - m\int dt\, \ddot q_{\mathrm{cl}}\eta.$$

Both terms vanish: $\eta$ vanishes at the endpoints and the free classical path obeys $\ddot q_{\mathrm{cl}}=0$. The stated expansion follows.

4. Two semiclassical branches have equal real amplitudes $A$ and action difference $\Delta S$. Compute their intensity and identify constructive and destructive cases.

Solution

The summed amplitude is

$$\mathcal A = A e^{iS_1/\hbar} + A e^{iS_2/\hbar}.$$

Its squared magnitude is

$$\begin{aligned} \lvert\mathcal A\rvert^2 &= 2A^2 \left[ 1+\cos(\Delta S/\hbar) \right]\\ &= 4A^2 \cos^2\left( \frac{\Delta S}{2\hbar} \right). \end{aligned}$$

It is maximal when $\Delta S/\hbar=2\pi n$ and vanishes when $\Delta S/\hbar=(2n+1)\pi$. Unequal or complex prefactors modify the contrast and shift the phase condition.

5. Show how adding a total derivative to the Lagrangian changes a fixed-endpoint propagator and explain why transition probabilities remain unchanged.

Solution

If

$$L'=L+\frac{dF}{dt},$$

then

$$S'=S+F(q_f,t_f)-F(q_i,t_i).$$

Every fixed-endpoint path receives the same extra endpoint factor, so

$$K' = e^{iF_f/\hbar} K e^{-iF_i/\hbar}.$$

This is equivalent to rephasing the endpoint position states. When initial states, final states, and operators are transformed consistently, the overall endpoint phases cancel from physical probabilities and expectation values.