Physicists in Guangzhou Confirm Feynman’s Path-Integral Rules With Single Photons, but Not That They Take Every Path

On August 26, 2026, physicists at South China Normal University in Guangzhou published what they described as the first direct experimental test of the two postulates behind the Feynman path integral. The team reported in Science Advances that it had reconstructed probability amplitudes for 1,419,857 possible paths of single photons across a discrete grid.

Richard Feynman set out the postulates in a 1948 paper in Reviews of Modern Physics as the basis of his path-integral formulation of quantum mechanics. The first, in the form the Guangzhou team tested, states that the probability of a particle arriving at a point is the squared magnitude of a sum of complex amplitudes, one amplitude for each path the particle could take.

Because the amplitudes add as a coherent superposition, the result generally differs from the sum of the paths’ separate probabilities. The second postulate states that every path contributes an amplitude of the same magnitude, with a phase equal to the path’s classical action divided by the reduced Planck constant.

Yong-Li Wen and Li-Man Tian are the paper’s joint first authors, and Enke Wang, Hui Yan and Shi-Liang Zhu are its corresponding authors. Yan and Zhu are also affiliated with Hefei National Laboratory.

The team used heralded single photons with a wavelength of 795 nanometers and treated their sideways spread in free space as the motion of a quantum particle in one dimension. It divided a strip about 97 micrometers wide into 17 positions and the propagation into five steps of 15 millimeters each.

For each step, the team measured the propagator, the complex amplitude for a photon to move from one position to another. The measurement used photon polarization as a pointer inside a stabilized interferometer, with an intensified camera registering individual photons.

Multiplying five single-step propagators gave the amplitude of one path. With the starting position fixed and 17 possible positions after each of the five steps, the grid contains 1,419,857 paths.

The team reported that squaring the sum of the amplitudes reproduced the distribution of photons on the camera, while a sum of the paths’ individual probabilities, which leaves out interference, did not. The summed-amplitude result differed from the theoretical prediction by a mean absolute percentage error of 4.45%. The paper gives the fidelity for this first postulate as 94.9% in its introduction and 94.4% in its results section.

For the second postulate, the reconstructed amplitudes showed equal magnitudes and phases that followed the classical action, with a fidelity of 94.7%. The squared amplitudes of individual paths, which the paper calls path probabilities, scattered by a mean absolute percentage error of 17.4%, against 18.0% in a simulation the authors based on their measured propagator errors.

The experiment extends the group’s 2023 Nature Photonics paper, in which it measured single-photon propagators and used them to demonstrate the quantum principle of least action. The authors wrote that changes to the setup raised propagator fidelity from 87.6% to 98.5%.

In their discussion, the authors wrote that each photon’s behavior was consistent with a superposition of all the paths, which they said supports treating the paths as physically real rather than as mathematical devices. They added that more refined methods are needed to settle the question. Zhu, a professor at South China Normal University, told Physics World that the team had provided evidence for the view that “quantum paths reflect physical reality rather than being mere mathematical artefacts.”

Phys.org reported the study on August 31 and Physics World on September 25. The authors wrote that the method could be extended to interacting systems, to tests of entangled histories and indefinite causal order, and to studies of quantum-to-classical transitions and decoherence. The data are available on Dryad.

My Analysis

Several people have asked me since August whether this paper proved that particles take all paths at once. It didn’t, and no experiment of this design could. Wen and colleagues measured the free-space propagator of single photons with 98.5% fidelity, up from 87.6% in their own 2023 setup, and their values match the predictions of quantum mechanics.

Interpretations of quantum mechanics in which a photon follows exactly one path, or no definite path at all, assign the same probabilities to these measurements, and classical wave optics predicts the same propagators. The Guangzhou team made a careful measurement of a quantity that all of those accounts predict identically, and no one can learn from such a measurement whether the paths are real.

What the team measured and what it computed

Wen and colleagues measured the single-step propagator for 85 combinations of starting position and step, 17 positions at each of five steps, and read each one out across 17 end positions on the camera. That gives 1,445 complex numbers. Each value was estimated from many repeated photon detections in four polarization settings and normalized by a separate intensity measurement. Each step used its own optical configuration, set by moving the slit and the camera.

The 1,419,857 path amplitudes were computed afterwards, as every product of five measured numbers with one number from each step. No photon was observed traveling a five-step path, and none could have been, because the five factors in each path amplitude come from five separate optical configurations.

The path count follows from the grid the team chose. Sampling 34 positions per step would have taken more measurements and produced 45 million paths, and sampling 170 positions would have produced 142 billion. The photons in the apparatus would behave the same way in every case.

In free space, the single-step propagator depends only on how far the photon moves sideways in one step. In theory, then, all 1,445 measured values sample one function at 17 jump sizes, from zero to 16 grid spacings.

Postulate I as a matrix product, postulate II as Fresnel diffraction

For the first postulate, the team summed the amplitudes of the 83,521 paths that connect the starting point to each end point and compared the squared magnitude with the camera image. On the discrete grid, that sum is one entry in the product of five 17-by-17 matrices, one matrix per step: $\sum_j \phi_j = \left(K_5 K_4 K_3 K_2 K_1\right)_{x_f,\,x_0}$

Agreement with the camera image shows that the measured single-step propagators compose correctly and that amplitudes add where classical probabilities would not. That second property is interference, which Thomas Young demonstrated for light in a paper read to the Royal Society in November 1803. Classical electromagnetism predicts the same interference, since it adds field amplitudes before squaring them. The sum of path probabilities the authors compared against is a model with the interference removed, and no wave theory predicts it for coherent light.

In this setup, the second postulate amounts to a statement about a single step. Every single-step amplitude should have the same magnitude and, apart from a phase shared by every jump, a phase of $$\pi(\Delta x)^2/(\lambda d)$$ for a sideways jump $$\Delta x$$ over a step of length $$d$$. That expression is the kernel of Fresnel diffraction, the formula optical engineers use to predict how light spreads beyond a slit. The paper’s own discrete Lagrangian, which assigns the photon’s sideways motion an effective mass of $$2\pi\hbar/(\lambda c)$$, produces exactly that phase.

Once each step has an equal magnitude and an action phase, every product of five steps has both properties by arithmetic, since magnitudes multiply to a constant and phases add up to the path’s total action. Checking the 1.4 million paths is therefore not an independent test of the second postulate beyond the 1,445 single-step values. The path-level check shows how the step errors accumulate. Wen and colleagues simulated the scatter they should expect from the step errors alone and got 18.0%, against the 17.4% they found in the squared path amplitudes.

The authors argue that equal magnitudes and action phases do not follow naturally from the Schrödinger picture. Feynman showed in 1948 that the wave function built from his postulates satisfies the Schrödinger equation. For a free particle, the case in this experiment, the propagator derived from the Schrödinger equation has a constant magnitude and, apart from a constant, a phase equal to the classical action divided by $$\hbar$$. Sakurai and Napolitano work through that calculation in section 2.6 of Modern Quantum Mechanics.

Interpretations of quantum mechanics that predict the same propagators

The authors write that Feynman’s propagator formulation is equivalent to the Schrödinger equation, and Zhu said the same to Physics World. Mathematically equivalent formulations give the same prediction for every measurement both describe. The Schrödinger equation predicts these propagators without any paths. So does Bohmian mechanics applied to the effective Schrödinger equation the authors use for the photon’s sideways motion, and under that theory each photon follows exactly one trajectory, guided by its wave function. Pilot-wave treatments of the full electromagnetic field need more structure, but the paper’s own model is the one at issue here.

In 2011, Aephraim Steinberg’s group at the University of Toronto sent photons one at a time from a NIST quantum-dot source through a two-slit interferometer and reconstructed their average trajectories. In this weak measurement, a calcite crystal shifted each photon’s polarization slightly, by an amount that depended on its direction of travel, before a detector recorded its position. Averaging over many photons gave the trajectories. Steinberg said the trajectories were consistent with the de Broglie–Bohm interpretation, in which each photon takes one path, and Physics World named the result its Breakthrough of the Year.

Two years later, Konstantin Bliokh and colleagues at RIKEN showed that the Toronto measurements also have a classical-optics description. The averaged momentum values in the Toronto data trace the Poynting vector, the local flow of electromagnetic energy in a classical light field. Their result concerns those linear-optics observables and does not extend to every weak measurement. The Toronto group reconstructed flow lines that match Bohmian trajectories, and the Guangzhou group reconstructed amplitudes for many paths. Both datasets agree with standard quantum mechanics, and no one can use either dataset to choose between those pictures of the photon.

Wen and colleagues sliced the propagation by position. The same transition amplitude can be calculated in other representations, including momentum space, where the intermediate steps look entirely different. A physicist who splits an amplitude into intermediate contributions has not thereby learned which of those mathematical pieces are physically real.

The signals used to reconstruct the propagators are first-order interference measurements, which give the same normalized result for one photon at a time as for a laser beam in the same optical mode. In optical terms, each propagator is the field that the slit’s image produces 15 millimeters downstream, measured against the unmodified beam in the interferometer’s other arm. An attenuated laser in the same mode, with the camera triggering adapted, would in principle return the same propagators, and anyone can predict them with a textbook Fresnel calculation.

The photon source itself is nonclassical. The team measured a heralded second-order correlation of 0.234, and classical light cannot go below 1 on that measure. Classical optics also has no account of antibunching, entanglement or Bell-inequality violations, and none of those effects enters the propagator data.

The physical-reality claim in the paper and the coverage

Wen and colleagues made the claim in the paper itself. In their discussion, they wrote that weak measurement lets them construct the trajectories of millions of single photons. The team estimated each single-step propagator from many repeated detections and multiplied the estimates, and it followed no individual photon at any point.

Their summary says the experiments “provide evidence for the physical reality of all paths,” then adds that more refined methods are needed to answer the question. Physics World quoted Zhu without the caveat. Physics World also restated the first postulate as a statement that a particle does not travel along a single trajectory.

Feynman’s 1948 postulate gives the probability that an ideal measurement finds a particle’s path within a region of space-time, as the squared sum of contributions from every path in that region. It does not say that the particle travels every one of those paths.

Physics World’s headline said the single-photon measurements confirm Feynman’s vision of quantum mechanics. Phys.org’s said physicists had finally put the path integral to the test.

The advance in propagator measurement

The authors report that propagator fidelity rose from 87.6% to 98.5% and that the mean propagator error fell from 24.3% to 8.17%. Wen told Physics World that the team reached that accuracy through four changes: stronger signals, a custom high-precision imaging system, real-time normalization against a reference beam, and nanometer-scale mechanical stability. According to the authors, small errors multiply across the five steps, and without those gains the reconstructed path distribution would have been close to random.

The authors propose extending the method to interacting systems, systems in curved spacetime, entangled histories and indefinite causal order. In systems whose dynamics are hard to model, measured propagators could test approximations and benchmark numerical simulations.

The authors’ “first direct test” label describes that operational achievement, reconstructing path amplitudes from measured single-step propagators, and I know of no earlier experiment that did so at this scale. As a test of quantum theory, the Guangzhou team checked the free-particle propagator, which the Schrödinger equation predicts and which Fresnel diffraction measurements have confirmed for light since the nineteenth century.

I would have published the measurement and dropped the third point of the discussion, where the reality claim appears.

Testable alternatives to the Schrödinger equation

Physicists can tell accounts of quantum mechanics apart experimentally only when the accounts predict different results, and theories that modify the dynamics do. The Diósi–Penrose model adds a gravity-related collapse of superpositions, and it predicts that charged particles emit faint radiation as their motion diffuses. In a study published online in September 2020, Sandro Donadi, Angelo Bassi and colleagues reported a search for that radiation at the Gran Sasso underground laboratory in Italy and ruled out the model’s natural, parameter-free version.

Interpretations that keep the standard dynamics and assign the same probabilities to these measurements, including the one in which every path is physically real, predict the same free-space propagator. Measuring that propagator more precisely cannot separate them. A test of path reality would need an account that predicts a different result.