Feynman phases around stationary action

The physical question

What does stationary action really say about the interference of nearby paths?

A path integral weights alternative histories by a complex phase determined by their action. A full field of paths is infinite-dimensional; this lab chooses one normalized sine deformation of the classical free-particle path and evaluates the resulting finite-dimensional Fresnel integral exactly. The restriction is explicit and makes stationary-phase cancellation reproducible.

Before changing a parameter: For S(a)=Sclassical+c a², predict which deformation a has zero first derivative and how many oscillations appear as c increases.

Prerequisites: the corresponding lecture-note treatment of eigenstates, phases, normalization and elementary differential equations. Work through the calculation on paper before exploring parameter sweeps.

Quantitative predictions

Live computed observables

Quadratic action and path phase

Every point is one deformation amplitude. The oscillations accelerate away from stationary action at a=0.

Cosine of the phase accumulated by sine-deformed paths

For a free particle, the action difference scales as a². The phase derivative vanishes at a=0.

Complex amplitude accumulation

Integrate the complex path weights from the left edge of the family toward the right; destructive contributions bend the trajectory.

Real and imaginary parts of cumulative complex Gaussian-weighted Fresnel integral versus deformation bound

This is a one-mode path-family integral, not a discretized full propagator. Its final complex value is benchmarked by an exact Gaussian integral.

Stationary-phase suppression

The normalized coherent resultant shrinks when phase curvature or path-family width increases.

Coherent amplitude magnitude compared with the fully in-phase Gaussian reference

|A|/A₀=(1+4c²σ⁴)⁻¹/⁴ for a quadratic phase and Gaussian weighting.

Recover the mathematics

Derivation and physical meaning

For a free particle over time T with endpoints xᵢ,x𝒇, define x(t)=xcl(t)+a sin(πt/T), where xcl(t) is the straight classical trajectory joining the endpoints. The deformation vanishes at both boundaries, so all paths share the same endpoints. Inserting into S[x]=∫mẋ²/2 dt gives a vanishing cross term and

S(a)=Scl + mπ²a²/(4T), Scl=m(x𝒇−xᵢ)²/(2T).

With ℏ set to one for the dimensionless phase display, let c=mπ²/(4T). The relative quantum phase becomes exp(i c a²). A smooth Gaussian envelope exp[−a²/(2σ²)] regularizes and specifies how we sample this one-dimensional family. The resulting integral is analytically known:

A(c,σ)=∫ℝ da exp[−a²/(2σ²)+i c a²] = √(2π)σ / √(1−2icσ²).

Thus |A|/[√(2π)σ]=(1+4c²σ⁴)^−1/4. The numerical quadrature independently sums real and imaginary contributions and can be compared against the exact complex Gaussian result. As the phase winds more quickly at large |a|, contributions away from a=0 cancel. This is the stationary-phase mechanism, rather than a claim that only one trajectory exists.

The overall factor exp(iScl/ℏ) can be restored afterward. For a true Feynman propagator every independent fluctuation mode and its measure must be incorporated, including normalization and boundary conditions. Here the selected one-mode slice is a controlled experiment illustrating phase cancellation and the Gaussian stationary-phase calculation.

Check an exact limiting case

If c→0 every deformation contributes almost the same phase and |A| tends to √(2π)σ. If cσ²≫1 the coherent magnitude is suppressed approximately as (2cσ²)⁻¹/². If σ→0 the family concentrates at its stationary path and its relative phase becomes constant.

Scope and validity of the model

Free particle, fixed endpoints, one sinusoidal deformation coordinate, Gaussian envelope as an explicit pedagogical regulator and ℏ=1 for display. The full propagator's functional measure and determinant are not evaluated; the exact one-dimensional integral is the independent benchmark.

Independent investigations

Explore and explain

Locate stationary action. Inspect dS/da=2ca at several deformations. Explain why all paths have the same endpoints while only a=0 is stationary.
Follow destructive interference. Increase cσ² and track the cumulative real and imaginary amplitudes. Compare the final integral against the exact complex Gaussian formula.
Distinguish a path slice from the propagator. Explain which additional ingredients enter a full multi-mode path integral and why the present restricted integral is nevertheless a precise test of the stationary-phase mechanism.

Self-check: What is the action derivative at the undeformed path a=0?

Zero for any positive c Equal to c Infinite when the path is classical

For a deeper challenge, reproduce the displayed model independently with a symbolic or numerical notebook and ask an AI tutor for a critical review of each approximation, boundary condition and sign.

Synthesis and references

What the experiment establishes

A path integral is organized by phase interference across histories. This laboratory displays the mechanism without claiming that one fluctuation mode is the entire functional integral, and gives students an exact complex benchmark for testing quadrature and stationary-phase intuition.

Lecture notes: Corresponding lecture-note chapter ↗.