Does the phenomenon of super oscillation challenge energy conservation?

The QuantaMagazine popular article "Puzzling Quantum Scenario Appears Not to Conserve Energy" (see this) told about puzzling observations the quantum physicists Sandu Popescu, Yakir Aharonov and Daniel Rohrlich made 1990 (see this). These findings challenge energy conservation at the level of quantum theory.

The experiment of authors starts from the purely mathematical observation that a function can behave faster than any of the Fourier components in its Fourier transform when restricted to a volume smaller than the domain of Fourier transform. This is rather obvious since representing the restricted function as a Fourier transform in the smaller domain one obtains faster Fourier components. This phenomenon is called super oscillation.

Does this phenomenon have a quantum counterpart? The naive replacement of Fourier coefficients with oscillation operators for photons need not make sense. If one makes the standard assumption that classical states correspond to coherent states, also super-oscillations should correspond to a coherent state.

Coherent states are eigenstates of the annihilation operator and proportional to exponential exp(α a^†)|0>, where "0" refers to the ground state an a^† to creation operator. These states contain N-photon states with an arbitrarily large photon number. For some number of photons the probability is maximum.

This raises several questions.

1. Coherent states are not eigen-states of energy: can one really accept this? This kind of situation is encountered also in the model of superconductivity assuming coherent state of Cooper pairs having an ill-defined fermion number.
2. Could the super oscillation correspond to the presence of N-photon states with a large number of photons? Could the state of n parallel photons behave like a Bose-Einstein condensate having N-fold total energy in standard physics or its modification, such as TGD?

Authors tested experimentally whether the super-oscillation has a quantum counterpart. In an ideal situation one would have a single photon inside an effectively 1-D box. One opens the box for time T and inserts a mirror inside the box to the region where super oscillation takes place and the photon looks like a short wavelength photon. The mirror reflects the photon with some probability out of the box. If T is long one expects that the procedure does not affect the photon appreciably. What was observed were photons with the energy of a super photon rather than energy of any of its low energy components.

In the experiment described in the popular article, red light would correspond to photons with energy around 2 eV and gamma rays to photons with energies around MeV, a million times higher energy. The first guess of standard quantum theorists would be that the energies of mirrored photons are the same as for the photons in the box. Second guess would be that, if the coherent state corresponds to the super oscillation as a classical state, then the measured high energy photons could correspond to or result from collinear n-photon states present in the coherent state.

In the TGD framework zero energy ontology (ZEO) provides a solution to the problem related to the conservation of energy. In ZEO, quantum states are replaced by zero energy states as pairs of states assignable to the boundaries of causal diamond (intersection of light-cones with opposite time directions) with opposite total quantum numbers. By Uncertainty Principle this is true for Poincare charges only at an infinite volume limit for the causal diamond but this has no practical consequences. The members of the pair are analogs of initial and final states of a particle reaction. In ZEO, it is possible to have a superposition of pairs for which the energy of the state at either boundary varies. In particular, coherent states have a representation which does not lead to problems with conservation laws.

What about the measurement outcome? The only explanation for the finding that I can invent in TGD is based on the hierarchy phases of ordinary matter labelled by effective Planck constants and behaving like a hierarchy of dark matter predicted by the number theoretical vision of TGD.

1. Dark photons with h_eff= nh₀ > h can be formed from ordinary photons with h_eff= h. The energy would be by a factor h_eff/h larger than for an ordinary photon with the same wavelength. Note that dark photons play a key role in the TGD based view of living matter.

   TGD also predicts dark N-photons as analogs of Bose-Einstein condensates. They are predicted by number theoretic TGD and there is empirical evidence for them (see this). This would require a new kind of interaction and number theoretical view about TGD predicts this kind of interaction based on the notion Galois confinement giving rise to N-photons as Galois confined bound states of virtual photons with energies give by algebraic integers for an extension of rationals defined by a polynomial defining the space-time region considered.

   I have proposed an analogous energy conserving transformation of dark photon or dark N-photon to ordinary photon as an explanation for the mysterious production of bio-photons in biomatter. The original model for dark photons is discussed here. Now the value of h_eff could be much larger: as large as h_eff ≈ 10¹⁴: in this case the wavelength would be of order Earth size scale.

2. What comes to mind is that an N-photon state present in the coherent state can transform to a single photon state with N-fold energy. In the standard model this is not possible. On the other hand, in the experiments, discussed from the TGD point of view in here, it is found that N-photon states behaving like a single particle are produced. Could the N-photon states present in a coherent state be Galois confined bound states or could they transform to such states with some probability?

   In the recent case, the dark photons would have the same wavelength as red photons in the box but energy would be a million times higher. Could a dark photon or N-photon with Nh_eff/h ≈ 10⁶ be reflected from the mirror and transform to an ordinary photon with gamma ray energy.

   One must notice that the real experiment must use many-photon states N-photons might be also formed from N separate photons.

To sum up, new physics would be involved. ZEO is needed to clarify the issues related to energy conservation and the number theoretic physics predicting dark matter hierarchy is needed to explain the observation of high energy photons.

See the article TGD and condensed matter of the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD.