Ryan Westafer

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.

When I was at #pcampatl, a speaker from a certain private space flight company discussed very new products and the art of “selling space rides to 7 year-olds.”

It’s an intriguing idea, that you might spend money/effort to help kids believe they will one day get to ride into space. Perhaps the wealthy will buy space rides instead of Porches. This is the art of selling a product to future customers — 20 years from now!

Google's Marketing to Children

It appears Google may be at this by having many many children draw its logo, in hopes of being selected. One drawing is selected, maybe several are featured, but *every* drawing constitutes a labor of pride commissioned by a young child, spending maybe an hour or two drawing Google’s logo. It’s something to think about. Very clever, indeed.

Engineers work with the real world and physicists with the imaginary world; together we sustain the world.

The analogy to real and imaginary can be taken farther if you like; consider the solutions of a 2nd order ODE (general oscillator with damping).  Regardless, take a look at the descriptions below.  Are they appropriate?  What else could we add?

Engineering

• Local
• First-order
• Conditionally or locally stable
• Results at optimal cost
• …

Physics

• Universal
• Nth-order
• Unconditionally stable
• Results at any cost
• …

There are some partial or complete overlaps of the two fields, and I think that’s both important and exciting.

Sustainability

When two or more seemingly complementary fields work together, we obtain sustainable solutions.

Consider the money spent on science.  How and when does it benefit mankind?  Consider the total expenditures for the Large Hadron Collider and associated experiments and staff.  If this money was spent on programs in Africa or southeast Asia, would it help the world more?  These questions are hard to assess or answer, but I think the issues are important to our future.

There is an interesting analogy I think describes the general process of the Large Hadron Collider quite well: a cosmological super computer.

Simulated Higgs event

The facility is designed to create an extreme differential of energy density such that the least common particles and interactions (dissipation processes) can be observed.  It is important to note that “least common” refers to frequency of observation, not necessarily predominance in our universe.  Scientists will be essentially creating and annihilating mini-universes.  The LHC is like a heat engine, just like typical natural processes: a nonuniform distribution of energy dissipates and becomes uniform.  It’s essentially a  “big bang” seeded with baryonic matter (colliding protons).  Whats more, scientists have setup super-cooled solenoids to reveal great detail of the scattering events produced.

This is rather like a super computer.  A great amount of heat is dissipated to accomplish a programmed/predicted result.  The analogy is that scientists are inducing the universe to compute in a way we have not yet observed.  It may be the most powerful computer ever made, though we don’t yet know the complete language of the machine.  In engineer-speak, we might say it’s measurement of a very complicated (high dimensional) impulse response.  (And the events are likely the best approximations of impulses ever created by man; unrivaled energy density)

Dark Matter?

While they’re not really dark matter as typically defined, and certainly not at zero absolute temperature, some of the detectors utilize cryo-cooled baryonic matter.  The generalized blackbody luminosity and thermal noise are significantly reduced, increasing the signal-to-noise ratio (SNR).  This boosts the system’s information output, for, in information theoretic terms, the capacity of a communication channel is given by .  So the information gleaned from this computing operation depends upon the bandwidth (various types of detectors or essentially configurations of matter) and their temperatures, since the SNR may be formulated in terms of detector temperature.  So, the supercooled masses seem like our best approximation of so-called “dark matter,” a component of the “receiver” for the emissions of highly luminous matter.  So, in a way, the energy cascades from the colliding protons are like information cast out from a computer monitor to our eyes, and the heat pumped out of the cryogenic refrigerators and dissipated in the surrounding environment is something like what a desktop computer does in one’s office.

The emissions will be scrutinized to create/validate more complete models of particle physics so we can make more complete predictions.  Only time will tell, but is this useful for humanity or is it an elaborate exercise?  Will the validations be useful?  Won’t they be obvious?  I think the negative results and unexpected results will be the most fruitful.  This verification of the highly successful Standard Model seems unnecessary until we make further practical use of the esoteric high-energy interactions.

More…

1.  Technically, I think an estimate of effective bandwidth would require computation of the transactional “overlap” integrals for the interactions of various detector states with the collision states.

2.  Additional complications arise due to adiabatic vs. diabatic system evolution.

This might seem trivial and obvious, but there are some interesting examples to think about.

Loss is physical; it’s a fundamental fact scientists accept axiomatically through the 2nd Law. We observe it many forms of dissipation: friction, condensation, crystallization, etc.  It’s fun to think of how these things also relate to the emergence of “order” and “disorder.”  The fractal pattern of a snowflake can occur just by the crystallization of water molecules: we see how a new form is produced in the course of dissipation.  Chaotic molecules bouncing around begin to stick together and localize in space – due to some random speck of dust or fluctuation in energy.  The molecule energies are converted from kinetic to potential (inertial) and their positional variance is reduced.

In another (opposite) example, consider a spinning toy.  It starts as a collection of orbital angular momentum vectors all approximately parallel: one-dimensional in macroscopic angular momentum space.  As it slows, the angular momentum becomes multidimensional: first by “wobble” or ever-larger sweeping orbits on a tabletop, then by a chaotic skidding dance of loops and bounces, and eventually, rest.  Such transition from one dimension to many is the nature of mode conversion, dissipation, and dimensional expansion.

So, as energy dissipates, it is manifest as loss for a given system because it leaves the system’s boundaries.  So, to keep the original system energy intact (as if to conserve energy), we must infinitesimally alter the system boundary to include the energy which exited.  This implies expansion because the boundary must adjust “outward.”

Sometimes I find ambiguity in the terms “order” vs. “disorder” and “entropy” vs. “negentropy.”  Because the terms are opposites, they behave in the same way, but with opposite character.  Is the fractal pattern of a snowflake “order-like” or disorder-like?”  It’s a pattern and can be defined by simple rules, but if order is thought-of as uniformity, the chaotic water molecules which made the snowflake appeared to be uniform when viewed at the same scale as the snowflake.  So, this is a relative issue of scale.  It seems we ought to ask order “at what scale?”  Or disorder “at what scale?”  – and “relative to what?”

In conjunction with the relative issue of scale, we have the relative issue of perspective.  If we define the boundary of one system, then we have defined another (its exterior).  The phenomena of expansion and contraction are analogous, but they are views of opposite perspective.

Assumptions

• We do not add matter or energy to the system by perturbation of the system boundary.  We only minimally perturb the boundary to retain the photon or whatever energy was emitted.  This becomes difficult as a photon may go on to interact with other systems, but we could (in theory) track its energy as mode conversion occurs through various interactions.
• 2nd Law of Thermodynamics: entropy increases, and its increase corresponds to the incapacity to do work.  Sounds like cooling to me.  If cooling is a universal law, then we must question how we are able to heat things.  In every heating process, heat is transferred.  The source cools while the sink warms.  Still, this is not purely reciprocal.
• It seems appropriate to consider expansion in existing dimensions (over which the boundary is defined), or also internalized expansion – achieving higher dimensionality rather than expanse of existing dimensions.