Where Photons Have Been: Nowhere Without All Components of Their Wavefunctions

A nested interferometer experiment by Danan et al (2013) is discussed and some claims evaluated concerning the whereabouts of the photon, primarily in the context of time-symmetric interpretations of quantum theory including the Two-State Vector Formalism (TSVF) and the Transactional Interpretation (TI). It is pointed out that the TSVF account fails to predict the observed data based only on the first-order wavefunction component. It is shown that the Transactional Interpretation readily accounts for all the observed phenomena.

💡 Research Summary

The paper revisits the nested‑interferometer experiment performed by Danan et al. (2013), which has been widely cited in discussions about the “where was the photon?” question. In that experiment a pair of Mach‑Zehnder interferometers are equipped with vibrating mirrors; each mirror’s oscillation frequency is imprinted on the photon’s transverse mode. By analysing the power spectrum of the detector signal one finds that certain mirror frequencies disappear, leading some authors to claim that the photon was not present in those arms despite the mirrors being physically in the optical path.

The author first examines the Two‑State Vector Formalism (TSVF), a time‑symmetric interpretation that treats the forward‑evolving wavefunction and the backward‑evolving (post‑selected) wavefunction as a pair. According to the usual TSVF reading, a non‑zero product of the two vectors in a region signals the photon’s presence there. The paper shows, through explicit numerical simulations, that if one restricts the analysis to the first‑order component of the forward‑evolving wavefunction, the TSVF fails to reproduce the observed spectral features. In particular, the missing mirror frequencies and the higher‑order mixing terms that appear in the data cannot be generated by a linear‑order model; the TSVF prediction is essentially flat where the experiment shows clear peaks. This demonstrates that the TSVF’s reliance on only the leading‑order term is insufficient for this setup.

The author then turns to the Transactional Interpretation (TI). TI posits that quantum processes involve an “offer wave” (the usual forward‑propagating solution of the wave equation) and a “confirmation wave” (the time‑reversed solution) that travel back and forth until a “handshake” (transaction) is completed. In the context of the Danan experiment, the vibrating mirrors modulate the phase of the offer wave; the confirmation wave, returning from the detector, carries this modulation back through the interferometer. Because both waves are present, the interaction region naturally generates not only the first‑order modulation but also second‑ and higher‑order terms arising from the product of the two waves. By constructing a full‑wave model that includes these nonlinear mixing terms, the author reproduces the entire observed spectrum, including the frequencies that seemed to vanish under a naïve TSVF analysis. The TI therefore accounts for the experimental data without invoking any exotic “photon‑presence” criteria; the photon’s “whereabouts” are simply the spacetime regions where the offer and confirmation waves overlap and exchange energy.

The paper argues that the experimental design deliberately amplifies the role of the backward‑propagating component: the vibrating mirrors imprint a tiny phase shift on the forward wave, and the returning confirmation wave reinforces this shift, making the mixed frequencies observable. Consequently, any interpretation that neglects the backward component cannot capture the full physics. The author concludes that while TSVF offers an elegant time‑symmetric picture, its practical implementation must include higher‑order contributions to match reality, and that the Transactional Interpretation provides a more complete and experimentally consistent account of the nested‑interferometer results.

Finally, the paper suggests future work: applying the same analysis to other interferometric configurations, testing the robustness of the transaction picture under different post‑selection schemes, and exploring whether alternative time‑symmetric frameworks can be extended to incorporate the necessary nonlinear terms. Such studies would further clarify the role of forward and backward waves in quantum phenomena and help settle the ongoing debate about the photon’s “path” in complex interferometric experiments.