Feynman's two postulates tested with 1.4M photon paths, in high school physics

Yong-Li Wen and colleagues at South China Normal University published Direct experimental test of Feynman’s path integral postulates with single photons in Science Advances in August 2026.
From measurements that detect photons one at a time, they reconstructed the probability amplitude of every one of the 1,419,857 paths a photon could take and checked them against the two postulates Feynman set down in 1948. Levtech LAB covered it on September 2, and the full text is on PubMed Central.
A probability amplitude is an arrow-like quantity with a direction and a length. Add up all the arrows, square the length, and you get a probability.

I have never worked with path integrals and never took particle physics.
I tried to rebuild the whole thing within high school physics and math, using the paper itself, the Levtech article, and the same team’s 2023 paper.

How light travels in high school physics

High school physics teaches that light goes straight through air, bounces off a mirror with the angle of reflection equal to the angle of incidence, and bends when it enters water.
These three fold into one statement. Light takes the path that connects start and finish in the shortest time. That is Fermat’s principle.

Reflection gives equal angles because, among all paths that bounce off the mirror, that one is the shortest.
Refraction bends the path because light is slower in water, so going a bit farther through air before entering the water gets you there sooner than a straight line.
The usual analogy is a lifeguard on a beach. Running diagonally along the sand before entering the water is faster.

Strictly speaking it is not “shortest” but the path where, on a graph with path deformation on the horizontal axis and travel time on the vertical, the slope is zero.
Same as a graph going flat at a minimum. For a simple mirror or water surface, that flat point is the shortest-time path.

For particles, compare a quantity called action

Balls and electrons get the same kind of statement. Compute one number per path and pick the path where the slope is zero.
That number is built from kinetic and potential energy.

In high school energy conservation, a ball thrown upward trades kinetic energy $K$ for potential energy $U$ , and the sum $K + U$ stays constant.
Going up, $U$ grows and $K$ shrinks, and at the top $K$ is nearly zero.
Action is defined as the difference $K - U$ added up over the time from start to finish.

S = \int_{t_a}^{t_b} \left( \frac{1}{2} m v^2 - U \right) dt

$m$ is mass, $v$ and $U$ are the speed and potential energy at a given moment, $t_a$ and $t_b$ are the start and end times, and the integral means multiplying $K - U$ at each moment by a short time slice and adding them all.
Why the difference rather than the sum is a university mechanics derivation, so here it is just the score assigned to each path.

$K - U$ is large at moments when the ball is moving fast and low, and small when it is high and slow.
Fix the start and end points and times, then nudge the path in between. The speed changes so $K$ changes, and the height changes so $U$ changes.
The path actually observed in classical mechanics is the one where the graph of $S$ against path deformation has zero slope.
It goes by the name principle of least action, but as with light, it is not always a minimum.
Showing that projectile motion comes out of this needs calculus of variations, so I skip it, but it is known to give the same trajectory as the equations of motion.

The double slit and adding arrows

The waves unit in high school physics has double-slit interference.
Light through two narrow slits overlaps on a screen. Where the path difference is a whole number of wavelengths it is bright, and where it is off by half a wavelength it is dark.
Light intensity is proportional to the square of the wave amplitude.

Fire electrons one at a time and the same fringes appear.
Each electron lands at a single point on the screen as one particle, yet many of them build up fringes.
The electron does not split in half. Instead, the probability of landing at a spot is set by adding two probability arrows, one assigned to each slit.
Put a device in place to record which slit was used, and the electron interacts with it, so the interference conditions are no longer the same. Quantum mechanics treats this as an experimentally established rule.

An arrow is the offset between wave crests and troughs, recast as an angle.
If the path difference is a whole number of wavelengths the two arrows point the same way. Off by half a wavelength, they point opposite ways.
Same direction, they add and get longer. Opposite, they cancel and get shorter.
In the same form as intensity being amplitude squared, the squared length of the summed arrow is the probability of the electron arriving there.

Feynman extended this from two slits to infinitely many paths.
The procedure goes like this.

List every conceivable path from start to finish. Straight, roundabout, zigzag, all of them.
Assign one arrow of the same length to each path. The direction differs per path, and how it is set comes in postulate II in the next section.
Add all the arrows tip to tail.
Square the length of the result. That is the probability of the particle arriving at that endpoint. Rescale everything at the end so the probabilities over all endpoints sum to 100%.

Doing this with infinitely many arrows is the path integral.

Postulates I and II as equations

Feynman set down two postulates in his 1948 paper.
”Contribution” in the table means the one arrow assigned to one path.

Postulate	Statement	In arrows
I	Amplitude is the sum of every path’s contribution	Chain all the arrows and add
II	Equal magnitude for every path, phase is $S/\hbar$	Same length, pointing at $S/\hbar$ rad

$\hbar$ is Planck’s constant divided by $2\pi$ , about $10^{-34}$ J·s, where J·s is energy times time.
Action $S$ has the same unit, so dividing leaves a pure number.
Postulate II uses that number as the arrow’s angle in radians.
An action difference of $2\pi\hbar$ turns the arrow exactly one full circle.

To write arrows in formulas, use the complex plane.
A length-1 arrow at angle $\theta$ is $\cos\theta + i\sin\theta$ , abbreviated $e^{i\theta}$ . Here it is only a short name.
Combining postulates I and II, the final probability amplitude $K(b,a)$ , the sum of all path arrows from start $a$ to end $b$ , looks like this.

K(b,a) = N \sum_{\text{all paths}} e^{iS[\text{path}]/\hbar}

$S[\text{path}]$ is the action computed along that path, $e^{iS/\hbar}$ is the arrow at angle $S/\hbar$ , $\sum$ adds all the arrows, and $N$ is the coefficient that rescales the total at the end. They map onto steps 1 through 4.
There are infinitely many paths, so in practice the sum is taken as the limit of very finely spaced paths. That special sum is called the path integral.
$K(b,a)$ is called the propagator. It gives the probability amplitude for a particle that was at $a$ to show up at $b$ . This $K(b,a)$ is what the experiment measured.

Adding arrows and the classical path

If the arrow direction is $S/\hbar$ , the classical action principle follows from the postulates.

Call a neighboring path one that looks almost the same, with a single midpoint nudged sideways.
Near the zero-slope path, neighboring paths have almost equal $S$ , so the arrows around it line up and add to something long.

Away from the zero-slope path, a small nudge changes $S$ a lot, and within any fixed band of paths being summed the action changes by far more than $2\pi\hbar$ , so the arrows spin around many times. For everyday objects the action is many orders of magnitude larger than $\hbar$ , so this condition is easily met.
Neighboring arrows point every which way and cancel almost completely as a group.

Only the sum around the zero-slope path survives in relative terms, and that is the single path of classical mechanics.
In problems with several such paths, the neighborhoods of all of them survive.

The double slit follows the same procedure.
Pick a point on the screen. Add the arrows of all the fine-grained paths through the left slit into one arrow, and likewise combine the paths through the right slit into one.
Add the two, and the squared length is the brightness at that point.
The angle between the two changes with position, so where they line up and where they oppose show up as fringes.

What went untested for about 80 years

Path integrals are used across quantum mechanics and much of physics.

Even so, the paper’s abstract says neither of the 1948 postulates had ever been tested directly.
Getting the right answer from the calculation is one thing. Measuring the postulate in the middle, that every path’s arrow has equal length and a direction set by $S/\hbar$ , is another.
A correct sum does not confirm each arrow before the sum.

The Levtech article sums this up as a formulation used for decades without evidence for whether it is merely a calculational technique or something physically real.

The experiment that reconstructed 1,419,857 paths from photons

The paper measures a few quantities directly with the apparatus, reconstructs the path arrows from them by calculation, and then compares the reconstruction with the two postulates.

Making paths countable

The photons have a wavelength of 795 nm and travel straight along the optical axis.
Looking only at the transverse position $x$ , distance traveled stands in for time, and the motion obeys the same quantum equation as a free particle.
The paper divided this transverse position into 17 points from $-48.72$ μm to $+48.72$ μm.
Time was divided into 5 intervals, each the time for light to travel 15 mm.

Fix the start at the center position and time 0, and count each combination of one of the 17 points at the end of each of the 5 intervals as one path. That gives 17 to the fifth, 1,419,857 paths.
Fix the endpoint too and it is 17 to the fourth, 83,521.
One path is five moves chained together, so its arrow is the product of five propagators, the arrows for moving from one time step to the next. The product of two complex arrows has the lengths multiplied together and the angles added.
Start position and time combine 17×5 = 85 ways, and one measurement records all 17 arrival positions on the camera at once, so 85 settings supply every propagator, from which the 1.4 million arrows were reconstructed.

What the apparatus measures directly

A camera only counts photons, a real number. The direction of a complex arrow does not come out of it directly.
Polarization, the light’s vibration direction, is used as a meter needle instead.
The photon’s polarization is split into two components. One component passes through a 15 μm slit to fix the starting position, and the other passes untouched.
After free-space propagation, polarization is measured in four orientations at each arrival position (two diagonal, two circular) and the photon counts compared. Their differences give the arrow’s horizontal and vertical components, the real and imaginary parts.
The arrow’s length comes from the photon count at that spot.

The main parts are an interferometer that splits and recombines the polarizations, a cylindrical lens that squeezes light in one direction plus the 15 μm slit, two lens systems that relay the light distribution to the next stage, and a high-sensitivity camera that counts single photons.
The same team measured propagators with the same polarization method in their 2023 paper, but the propagator fidelity (agreement between measured and theoretical values) was 87.6%, which the paper says was not enough to test the postulates.
This time they strengthened the coupling to polarization to improve signal-to-noise, raised the precision of the lens systems and the path-length stability of the interferometer, and reworked vibration control, reaching 98.5%.
The discrepancy between measured and theoretical propagators, computed over the 17 arrival positions and averaged over the 85 settings, was 8.17±3.50%, down from 24.3% in the earlier paper.

Comparing with postulate I

At each arrival position, the paper compares the arrival distribution counted directly by the camera, the distribution from adding all reconstructed arrows and squaring, and the classical sum that squares each length without adding the arrows.
If postulate I holds, the directly counted distribution matches the second and departs from the third.

The directly counted distribution matched the summed-arrow distribution, and the classical sum was far off.
The summed-arrow distribution differed from theory by 4.45% on average.
Fidelity here is, at each arrival position, the overlap between the sequence of 83,521 reconstructed arrows leading there and the sequence computed from theory, averaged over the 17 positions. The introduction gives 94.9% and the relevant passage in the body gives 94.4%, and I cannot tell which is right.

Comparing with postulate II

The 1,419,857 arrows are sorted by path length and by action.
Action is measured with the straight classical path as zero. Since angles repeat every $2\pi$ , action is folded every $2\pi\hbar$ and one full turn is split into 100 bins. In each bin the mean squared length and mean angle of the arrows were taken.
Mean squared length was flat regardless of path length or action, and the angle grew in proportion to $S/\hbar$ . That is just what postulate II states, equal lengths with the angle set by action.

The spread of per-path probabilities was 17.4%. That is the absolute difference between each path’s probability and the all-path mean, divided by the average of the two, averaged over all paths.
Recomputing with noise of the same size as the propagator measurement error gives 18.0%, about the same as observed, so the team attributes the spread to measurement error.
The fidelity between the sequence of all path arrows and the sequence computed from theory was 94.7%.

Item	Value
Paths reconstructed	1,419,857 (17⁵)
Propagator fidelity (2023 paper)	87.6%
Propagator fidelity (this paper)	98.5%
Measured vs theoretical propagator discrepancy (mean over 85 settings)	8.17±3.50%
Postulate I, summed-arrow distribution vs theory	4.45%
Postulate I fidelity	94.9% (94.4% in the body)
Postulate II, spread of per-path probabilities	17.4% (18.0% in error simulation)
Postulate II fidelity	94.7%

How to read the fidelities and errors

Fidelity is a score from 0 to 100% for the overlap between two sequences of arrows, approaching 100% as the two sequences match in overall shape and direction.
98.5% scores each measured propagator against its theoretical value. 94.9% (94.4% in the body) scores the 83,521 reconstructed arrows at each arrival position against theory, averaged over 17 positions. 94.7% scores all 1.4 million reconstructed arrows against theory. The three numbers grade different things.

The differences and spreads use what the paper calls MAPE (mean absolute percentage error), the absolute difference between a measured value and its comparison target, divided by the average of the two, averaged over everything. 0% means agreement. Standard MAPE divides by the measured value alone, so this is a symmetric variant with a different denominator.
4.45% is the difference between the summed-arrow arrival distribution and theory. 17.4% is the difference between per-path probabilities and the all-path mean.
17.4% is the deviation from postulate II’s equal-length claim, but it is the same size as the measurement-error simulation.

The paper concludes that both postulates are confirmed by these results.