QFT, meet Dr. Nyquist
Dispersion in 1st Brillouin Zone
On the “momentum masquerade” …
Tonight I was planning a phononic crystal (PnC) photomask and lamenting photolithography feature size limitations. Couldn’t we use the aliasing effect of the lattice to equivalently work with smaller wavelengths in a larger structure? After all, wavenumbers above the Brillouin zone boundary are “folded” into the first Brillouin zone. Our physical structure samples the solutions of greater momentum.
I’ve long wondered where this momentum aliasing effect fits into physics. Even a fundamental minimum length, e.g. the Planck length, would imply a great universal masquerade of momenta. Such a view seems to agree with a few principles of a “holographic” universe.
Have a look at this 2004 paper pointing out Quantum Field Theory’s inherent momentum masquerade. A key component is nonlinearity and convolution due to the appearance of products under the Fourier transform.
The Aliasing Problem in Lattice Field Theory
There could be many unusual perspectives on this point, but I can’t give any time to them now. One involves CPT symmetry and the effect of time reversal on a dispersion relation.
[Side Notes]
At first look there is a curious contradiction between the dispersion and the assumption of time harmonic modes. Each point on dispersion plots such as the one provided above is a solution to a “quasi time-invariant” eigenproblem, if that phrase makes sense. More concretely, the time dependence has been factored out, and is singly parameterized by the eigenvalue (frequency). How can a time harmonic signal exhibit the envelope attenuation necessitated by the Kramers Kronig relations for this dispersive medium? We might plot this graph in 3D, with complex eigenvalues to help solve this seeming contradiction. That’s right. For physical signals, we can solve the complex eigenproblem where damping is allowed. We then allow various losses. But couldn’t this also be allowed by linear superposition of time harmonic modes? We can form arbitrary signals (solutions) by superposing many time harmonic ones. So maybe there is no contradiction and this method has arbitrarily reduced the problem to a basis of perfectly periodic solutions, with the caveat that all solutions are required. This notion is further validated by the “nonlocality” of the Kramers-Kronig relations which may be implemented by the Hilbert transform over all frequencies.