John Preskill teaches a course on Quantum Computation at Caltech, and has made his lecture notes available online.
Category Archives: Physics
Applying Dynamical Systems to Statistical Mechanics
The best part about this site is that I can link to interesting articles that I don’t have time to read. One of the early inspirations for the study of dynamical systems was statistical mechanics, particularly the ergodic hypothesis. Things have now come full circle with applications of dynamical systems to statistical mechanics. Here’s a survey article on the subject by David Ruelle: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics.
Philosophy of Symmetry Breaking
As a follow-up to yesterday’s post, I came across an entry on symmetry breaking at the Stanford Encyclopedia of Philosophy. It’s interesting to see philosophy directly engaging with the implications of modern physics.
They also have two entries on the philosophical underpinnings of statistical mechanics: a general entry, and a more specific one on Boltzmann. The actual justification for statistical mechanics and the reduction of thermodynamics to statistical mechanics is apparently fairly controversial.
Symmetry Breaking
Sean at Cosmic Variance has written a nice introduction to symmetry breaking, one of the oddest (and most successful) ideas in particle physics.
Baez Week 222
Week 222 of This Week’s Finds in Mathematical Physics is up.
Semiring analogies
Sigfpe has been posting on his blog about one of my minor obsessions, semirings. A semiring is a ring without subtraction. There are lots of semirings that arise in both pure and applied mathematics. Here are some examples that show just how common they are:
- Any collection of sets closed under union and intersection form a semiring, with union as addition and intersection as multiplication (or vice versa).
- Regular languages form a semiring. The sum of two regular languages is the union, while the product is juxtaposition.
- The reals with positive infinity added form a semiring where the “sum“ of two numbers is the minimum, and the “product“ is addition. (Including positive infinity is not strictly necessary, but it serves as a “zero“ for the min operation). This known as the min-plus semiring.
The last example has an interesting extra property: you can take infinite “sums“ by taking the infimum of an infinite set of elements. You can set up an extended analogy between the reals with usual arithmetic operations and the min-plus semiring. In this analogy, integration becomes optimization. You can even extend the analogy as far as the Fourier transform. The min-plus analogue of the Fourier transform is the Legendre transform that arises in classical mechanics. Sigfpe explains the analogy here.
He has also posted about an application of semirings to understanding the game Tetris.
Atomic Orbitals
There’s an incredibly informative discussion thread about atomic orbitals at Brad de Long’s weblog.
Introductory Lectures on Quantum Field Theory
Spotted some tempting reading via It’s equal but it’s different: an introduction to quantum field theory, called (incredibly) Introductory Lectures on Quantum Field Theory.
More Lévy
According to Wikipedia, the analogue of a random walk for a Lévy distribution is called a Lévy flight. I presume it’s called a “flight” because paths are no longer guaranteed to be continuous, but can have sudden jumps.
I also spotted a survey paper, More “Normal” than Normal: Scaling Distributions and Complex Systems, which argues that in physical applications, heavy-tailed distributions such as the Lévy distribution are more natural than the normal distribution. This seems to be emerging conventional wisdom in some circles, but I don’t know how true it is.
Dipoles
As a mathematician, something that I always envied physicists is the uninhibited way they use mathematics.
The classic example is the Dirac delta function, which is a function that’s zero everywhere except the origin, but has area one. The fact that no such function exists is only a minor inconvenience. Delta functions can be made rigorous as a distribution, but the concept well predates its formal definition. For example, Green’s functions, which are defined in terms of the delta function, date from 1828.
A more dramatic example is the concept of a dipole. A dipole is the limit of two electric charges of opposite charge as the distance between them goes to zero. It can also be thought of as the limit of the difference of two Dirac delta functions, or even the derivative of a delta function. Dipoles are used to approximate the effect of magnets from a long distance. In terms of distributions, a dipole is the derivative operator on the space of smooth functions, but this is far from the physical intuition.