Feynman's path integral formulation

See this video for intuition and also this other one. It explain why it coincides with the principle of least action in the limit.

For a particle

See @baez1994gauge, page 137.
Suppose that we have a quantum-mechanical particle that starts out in the state $ψ (x)$ at $t = 0$ , that is,

| Ψ (0) ⟩ = \int ψ (x) | x ⟩ d x .

And we wish to compute its state $ϕ (x)$ at some other time $t = T$ , i.e.,

| Ψ (T) ⟩ = \int ϕ (x) | x ⟩ d x .

Let

P (a \to b) = {γ : [0, T] \to R^{3} : γ (0) = a, γ (T) = b}

denote the space of all paths that start at the point $a$ at time 0 and end at the point $b$ at time $T$ .
The probability of finding the particle at a given point $b$ at time $T$ is the sum of the amplitudes of all the paths that end at that point.

A_{total} = \int_{P (a \to b)} A [γ] D γ,

where $D γ$ is some sort of mysterious "Lebesgue measure" on the space $P (a \to b)$ . The probability amplitude of an individual path is given by the exponential of the action of the path divided by $ℏ$ .

A [γ] = \exp (\frac{i}{ℏ} S [γ])

where $S$ is the action of the path. The action is defined as

S [γ] = \int_{0}^{T} L (γ, \dot{γ}) d t

where $L$ is the Lagrangian of the system. If we discretize the process in "finite jumps", all this should be concluded (or, at least, visualized) from quantum approach to particle motion- an example. I have to develop this...

The quantity $A_{total}$ is usually denoted by $K (a, b, T)$ and called the propagator.

Finally, the evolved state $ϕ (x)$ is given by

\begin{aligned} ϕ (x) & = \int_{R^{3}} K (x^{'}, x, T) ψ (x^{'}) d x^{'} \\ = \int_{R^{3}} (\int_{P (x \to x^{'})} e^{\frac{i}{ℏ} S (γ)} D γ) ψ (x^{'}) d x^{'} . \end{aligned}

What Happens in the Classical Limit?

In the classical limit, we let Planck’s constant $ℏ \to 0$ . Let's see what this does to the integrand:

A [γ] = e^{\frac{i}{ℏ} S [γ]}

As $ℏ \to 0$ , even small differences in action $S [γ]$ lead to wildly varying phases. That is, for most paths, tiny changes in $γ$ cause large changes in $S [γ]$ , so their contributions interfere destructively. Therefore, the dominant contributions to the integral come from where the phase is stationary, i.e., where $δ S = 0$ .

But guess what condition defines those stationary points?

δ S = 0 \Rightarrow Euler-Lagrange equations \Rightarrow Classical equations of motion

So only paths that extremize the action — i.e., the classical paths — have nearby paths with similar phase, and thus constructive interference.

All other (non-classical) paths vary rapidly in phase and cancel each other out.
Therefore: in the limit $ℏ \to 0$ , the path integral is dominated by the classical path $γ_{classical}$ , and:

K (a, b, T) \approx const \times e^{\frac{i}{ℏ} S [γ_{classical}]}

This is the classical limit of quantum mechanics, recovering Newtonian trajectories from quantum fuzziness.

Application: refraction of a laser beam

Let’s apply this path integral framework to the refraction of a laser beam, as seen in the picture where the beam passes from air into a glass block and back into air.
![400][https://media.sciencephoto.com/image/c0219912/800wm]

In the Feynman path integral formulation, light (or a photon) travels from one point to another by taking all possible paths. Each path contributes an amplitude, and the total amplitude at the destination determines the probability of detecting the light there.

Now, why do we see the beam? The path integral tells us that the amplitude—and thus the intensity—is highest along the classical path (the refracted path). However, light itself isn’t visible unless it reaches our eyes. In the air and glass, tiny particles (dust or imperfections) scatter some of the light toward us. They act like a global particle detector. Since the intensity is maximized along the classical path due to constructive interference, more light scatters from points along this path, making the beam visible as a sharp, bright line. Away from this path, the amplitude drops off rapidly due to destructive interference, so little scattering occurs, and the beam doesn’t appear diffuse.

In summary, the path integral formulation shows that the laser beam’s path—its refraction at each interface—is the classical path where the action is extremized, leading to constructive interference. This recovers the classical optics result of Snell’s law in the limit $ℏ \to 0$ . The visibility of the beam comes from scattering along this high-intensity path, revealing the trajectory determined by the path integral’s dominant contribution.

Application: The Classical Path of a Tennis Ball

Imagine throwing the tennis ball from point A to point B. In the path integral picture, the ball could take any path: a straight line, a zig-zag, a loop, or even a path that shoots up into space and back down. Each path has an associated action, a quantity from classical mechanics defined as the integral of the Lagrangian (kinetic energy minus potential energy) over time. The classical path—the parabolic trajectory—is the one that minimizes the action, according to the principle of least action.

In the path integral, paths close to this classical path have similar actions, so their phases (related to the action via the quantum phase factor $e^{i S / ℏ}$ , where $S$ is the action and $ℏ$ is the reduced Planck constant) don’t vary much. These paths interfere constructively. Paths far from the classical one, like a loop or zig-zag, have very different actions, so their phases vary rapidly and interfere destructively, canceling out.

When you watch a tennis ball in flight, sunlight (or ambient light) bounces off the ball and reaches your eyes. At each moment, the ball interact with photons coming from the sunlight. This cloud of photons is like a global particle detector. I we were throwing billions of tennis balls, and recording the reflected photons, we will see a sharp trajectory, but like in the case of the laser beam, we would have a "kind off" diffusion around the path, only that infinitesimally small.

For a field

How does this connect to Quantum Field Theory?

The path integral formulation extends naturally to fields (quantum fields). Instead of summing over all possible paths of a particle, we sum over all possible field configurations. The action $S$ now involves integrals over space and time, and the Lagrangian $L$ is replaced by a Lagrangian density $L$ which is a function of the fields and their derivatives.
The amplitude for a field configuration is given by

A = \exp (\frac{i}{ℏ} S [ϕ])

where $ϕ$ represents the fields, and

S [ϕ] = \int d^{4} x L (ϕ, \partial_{μ} ϕ)

where $d^{4} x$ represents integration over spacetime.