Archive for October, 2011

Discrete Quantum Gravity

October 19, 2011 1 comment

Stanley Gudder, University of Denver

Q. GR deals with smooth functions on spacetime. QM deals with self adjoint operators on a Hilbert space. How are they related?

A. GR and QM both have bad singularities. Maybe we shouldn’t be looking at the continuum but the discrete picture!

We discuss the causal set approach to unify gravity and quantum mechanics. Let us begin with light cones. We know about the future and past lightcones of an event, say a.

Source: Wikipedia

Introduction and definitions

We talk about the causal structure (M,<) of a Lorentzian spacetime (M,g) . M is a partially ordered set(Poset).

For a, b \in M, we say a< b if b is in the causal future of a.

In the discrete situation, the smallest length is the Planck length l_p \sim 1.6 \times 10^{-35} and the smallest time interval is l_t \sim 5.4 \times 10^{-43}

We call a finite poset a ‘causet

\mathbb{P}_m = All causets of cardinality m.

\mathbb{P} = \bigcup_m \mathbb{P}_m

If a<b , we call a the ancestor and b the successor.

If a<b, we call a the parent and b the child if
!\exists c \ni a<b<c

a is maximal if !\exists b \ni a<b

X \in \mathbb{P}_m, Y \in \mathbb{P}_{m+1}

X produces Y if Y is obtained from X by adjoining a single element to X that is maximal in Y. We call X the producer and Y the offspring.

A path in \mathbb{P} is a string
\omega= \omega_1 \omega_2 \ldots \ni \omega_i \in \mathbb{P}_i, \omega_{i+1} \in \mathbb{P}_{i+1}

Each such path represents a ‘universe’. Every l_t unit of time we have a new element in the path. Note that we have 2 notions of time here, the ‘chronological time’ and the ‘geometric time’.

An n-path is
\omega= \omega_1 \omega_2 \ldots \omega_n

\Omega = \{\omega|\omega= \omega_1 \omega_2 \ldots \}

\Omega_n = \{\omega|\omega= \omega_1 \omega_2 \ldots \omega_n \}

Cylinder(\omega_0) = \{\omega \in \Omega| \omega= \omega_0 \ldots\}

For A \subseteq \Omega_n,  cyl(A)= \bigcup_{\omega \in A} cyl(\omega)

a_n = \{cyl(A): A \subseteq \Omega_n\}

We have the heirarchy
a_1 \subseteq a_2 \subseteq a_3\subseteq \ldots

C(\Omega) = \bigcup a_n. This is an algebra of the subsets of \Omega.

If X \rightarrow Y in r isomorphic ways,
we write m(X \rightarrow Y) = r, where m is the multiplicity.

Source: DQG by Stan Gudder

Classical Sequential Growth Processes

C= (C_0, C_1, \ldots)
C_i \geq 0 are the coupling constants.

For r\leq s \in \mathbb{N} define
\lambda_c(s,r) = \displaystyle\sum\limits_{k=0}^s \begin{pmatrix} s-r\\k-r\end{pmatrix} C_k

X \in \mathbb{P}_m, Y \in \mathbb{P}_{m+1}, X \rightarrow Y

We define the transition probability as

p_c(X \rightarrow Y) = m(X \rightarrow Y) \frac{\lambda_c(\alpha, \pi)}{\lambda_c(m, 0}

\alpha is the number of ancestors
\pi is the number of elements adjoined to X to get Y.

It is not obvious that this is a probability but this can be shown.

For the part of the talk from here onward, I will just sketch the outline here. We can define a corresponding quantum sequential growth process which leads to a theory of quantum gravity. I would encourage interested reader to read the original papers listed below.

Further readings and references-

Photonic properties of non-crystalline solids

October 15, 2011 2 comments

Paul Steinhardt, Princeton

Photonic crystals are semi-conductor devices for light, that is, with them we can get allowed and forbidden frequency bands. They are not the topic of discussion of this talk. Instead we ask ourselves,

Q. What do we know about the photonic properties of non-crystalline solids?

Q. Why talk about non-crystalline solids?

Schroedinger – Maxwell analogy

Schroedinger equation Maxwell’s Equations
Quantum Classical
“Scalar” (spin neglected) Vector. TM + TE modes. Different band gaps for each mode.
Complicated by e-e interaction Linear. No \gamma - \gamma interaction.
Fundamental scale No fundamental scale limit. If the problem is solved for one scale, it is solved for all. We can choose whichever scale is convenient to work with experimentally.( Ignoring absorption, which is frequency dependent and can’t be scaled any way we want.)
Massive quanta. Parabolic dispersion Massless quanta. Linear dispersion
Atomic/molecular structures Can design continuous structures

2 distinct scattering mechanisms:

  • Bragg scattering: ‘large scale’ resonance of the array
  • Mie scattering: ‘small scale’ resonance of scatterers
To get the largest band gap, we find conditions such that Bragg and Mie scattering reinforce each other. The experimental challenge is that the TE and TM modes prefer different dielectric patterns to achieve an optimum bandgap. The search is for “complete band gaps“, which are the overlap of TE and TM, so that they are band gaps for both polarization modes. The optimum configuration was found to be cylinders connected by a trivalent network of walls.
In general the band gap for TE < TM. The complete gap is, obviously, narrower than both. The figure of merit of the band gap is delta(f)/f, where delta(f) is the width of the band gap.
The major disadvantage found was that the band gap is anisotropic, which meant less symmetry. ( Symmetry beyond 6-fold is forbidden). There has been no theoretical framework to optimize band gaps. Most of the results were achieved by brute force simulations on computer or by running experiments.
Non-crystalline solids:
Until 1984 solids were thought to be either crystalline or disordered. Today we know of quasicrystals and many new classes of disordered solids.
Q. Why even consider non-crystalline solids?
  • Higher symmetry may give wider band gaps, at least for some dielectric constant ratios.
  • New types of structures, different modes and defects can find new applications.

Octogonal quasicrystal

In their pape rin 1984, Paul Steinhardt and D. Levine talk about quasicrystals,  the natural extension of the notion of a crystal to structures with quasiperiodic, rather than periodic, translational order. In their paper in Nature in 2005,  W. Man et al show that quasicrystals can have optical bandgaps rendering them useful to technology and also talk about the experimental measurement of the photonic properties of icosahedral quasicrystals.
It was found that quasicrystals beat diamond, the previous record holder, in terms of symmetry and wider band gaps. To find a more evenly distributed network, Delaunay tesselation was used.
Q. What if we applied it to isotropic disordered point pattern such as Poisson or hyperuniform?
Type Figure of merit
Crystal 19.3%
Quasicrystal 16.5%
Disordered 10.2% (isotropic rings – Bragg pattern)

We see that have a tradeoff between band gap and isotropy.

Further readings:

Optics, Self-Assembly, and Evolution of Aperiodic Color Producing Structures from Birds and Insects

October 15, 2011 1 comment

Richard Prum, Yale.

“I know what you are! You are a bird-watcher in q-space!”

When we think of colors in nature, the things that comes to mind are pigments and dyes. However, nature makes extensive use of nano structures to produce the vibrant hues that captivate us. These structural colors make use of interference patterns of light and hence depends on the angle of observation, unlike those produced by pigments. This is known as Iridescence. In most birds these are made by melanin-keratin nano-structures in feather barbules.

The structures are periodic in 1D, 2D or less commonly, 3D.

Velvet Asity

The male Velvet Asity, or Philepitta castanea, produces the brilliant green and blue colors from by the hexagonal array of parallel collagen fibers in its caruncle tissue. Since this is periodic, we can use Bragg’s law to describe the interference.


But in some birds quasi-ordered arrays were found. A simple description using Bragg’s law would no longer work. This was found in mammals too, like the vervet monkey.


The speaker presented a 2D fourier analysis of spatial variation in the refractive index of the feather barb of Cotinga Maynana. The colors are non-iridescent under diffusive light illumination like in nature, but iridescent under artificial directional lighting often used in the laboratory.

There also exist  channel-type nanostructures consisting of beta-keratin bars and air channels in elongate and tortuous forms. Sphere-type nanostructures consist of spherical air cavities in a beta-keratin matrix.

Q. How does spongy medullary keratin self-assemble?

A. Phase separation by Spinodal decomposition (SD)


SD is an unstable phase separation. Phase separation in spongy cells is “arrested” at a specific size to produce a specific color.

Phase separation in 3D is seen in butterflies, which get their colors from mathematical objects called gyroids.

A gyroid is a minimal surface, meaning its mean curvature vanishes. Minimal surfaces include, but are not limited to, surfaces of minimum area subject to various constraints.  The planes of a gyroid’s surface never intersect. Gyroids also contain no straight lines, and can never be divided into symmetrical parts

In butterflies this is formed by hydrophilic-hydrophobic interactions in copolymers.


Further readings and references :