I discussed Berry phase back in February, (Berry or Pancharatnam-Berry or Quantum phase) from a vector / density matrix point of view, but I thought it would be nice to describe Berry phase from the point of view of the U(1) gauge symmetry of quantum mechanics. From a density matrix point of view, the U(1) gauge symmetry is what arises in the state vector formalism, from the requirement that all physical observables be capable of being written in the density matrix formalism.
Let’s consider functions of three normalized spin-1/2 SU(2) spinors: |a), |b), and |c) in the bra-ket notation. (And I’m using round brackets to avoid a WordPress screwup.) Since the state vectors of quantum mechanics are unchanged by phase, we can also choose three arbitrary fixed real numbers, to respectively multiply these spinors. The bras and kets take the complex conjugate phases so the arbitrary phase transformation on these objects is as follows:
Quantum mechanic’s physical predictions will be unchanged by the substitutions. This symmetry is the U(1) symmetry of the SU(3)xSU(2)xU(1) symmetry of elementary particles and it was the first gauge symmetry. In short, letting the phase depend on position, and requiring that the theory of electrons be unchanged, defines a new field which turns out to be the photon.
This blog celebrated its 1st nominal birthday today with a new monthly record for view:
Views of the blog have increased month to month at a rate of about 20 page views per day per month. I don’t own a watch and hate to have to deal with time periods less than around two weeks. So in my natural units, page views are increasing at around 130 pages per square fortnight.
Filed under Aging, Blogroll
Lee Smolin recently put up an arXiv article, Matrix universality of gauge and gravitational dynamics, 0803.2926. The first sentence of the abstract is “A simple cubic matrix model is presented,which has truncations that, it is argued, lead at the classical level to a variety of theories of gauge fields and gravity.” A cubic matrix model gives nonlinear field equations. From the Smolin paper:
The dynamics cannot be linear because we want its solutions to reproduce those of non-linear field equations. The simplest non-linear dynamics are quadratic equations, which arise from a cubic action. The simplest possible non-linear action for matrices is
This raises the issue of what happens when one attempts to represent the quantum numbers of the elementary particles as the solutions to quadratic equations. In this post we will show that a very simple set of quadratic equations do give the quantum numbers for the fermions.
Kea recently wrote about Rowland’s being banned on arXiv. Rowland’s efforts are kind of similar to mine except that he works with the nilpotents (BB = 0) of a Clifford algebra while I work with the idempotents (BB = B). Some years ago, I had a conversation with Hestenes and discussed these things, and he mentioned that there is a close relationship between nilpotents and idempotents. Since then, I guess I’ve learned enough Clifford algebra to write about that relationship and so this post.
Let’s begin by picking up some intuition from the Pauli matrices, the well known matrix representation of the real Clifford algebra generated by three basis vectors with signature +++. As usual, we will call the three basis vectors x, y, z. The “presentation” of the algebra (i.e. the rules that distinguish it from the free algebra on x, y, and z) are that xx = yy = zz = 1, and that x, y, and z anticommute. The Pauli matrices are as follows:
A few idempotent and nilpotent 2×2 matrices are the following:
The above four are of interest because they exhaust the degrees of freedom of the 2×2 matrices. Of course there are an infinite number of idempotent 2×2 matrices and the same for the nilpotent. To get some intuition about the relationship between nilpotents and idempotents, write the above four matrices in terms of the Pauli spin matrices and unity. (The reader might be more familiar with this sort of thing in the Dirac algebra, where one often has cause for writing a 4×4 matrix in terms of the Dirac bilinears. This amounts to writing 4×4 matrices using a basis set of the 16 Dirac bilinears. The Pauli matrices are only 2×2 and so require a basis set of only four elements. We use the unit matrix and the three Pauli spin matrices.)
One finds a fairly diverse collection of characters hanging around the Crossroads Mall Chess Club, (which I sometimes inaccurately refer to as the “Overlake Mall Chess Club”). Mostly it’s men who love chess, or are retired or otherwise have too much time on their hands. In my case, it’s a love of watching others play chess. And one meets people there and one gets to know them. And they find out about one’s other hobbies, in my case physics, and they talk about their own.
In the case of Forrest LeDuc, his other hobby is divination. His regular employment is in the gold fields of north Idaho. Divination has undoubtedly been a central part of mining since before man knew how to smelt metals. I suppose that Neanderthals used divination to find flints, as well as game, other tribes, etc. Divination (or dowsing) is not taught in mining engineering, but the students, at least when I was a student 30 years ago, are exposed to divination by the miners, when they work summers in the mines. Despite centuries of suppression by the combined forces of the church and science, divining or dowsing is still in use. See the recent Mother Earth News article for a description.
Thank Nanoscale for bringing to my attention a long standing puzzle in mesocopic scale condensed matter physics, the “0.7 anomaly”. The problem is the behavior of the conductance (inverse of resistance) for a quantum point contact (QPC). Two experimental papers showing the effect are 0706.0792 and cond-mat/0005082, which see. Shot noise decreases at the 0.7 anomaly just as it does at the integer conductance points, see cond-mat/0311435. A recent perturbation analysis of the situation is given by 0707.1989. This blog post gives a non perturbational calculation.
Such a point contact is a region which is so small that it can hold at most a single electron. One controls the size of the conductance region by applying a voltage to a gate. One expects that the conductance will be a multiple of G0 = 1/13K ohms; this works for high conductance values (increasing to saturation) but an anomaly appears near cutoff.
This really is basic physics and should be well understood. The computer that I (and you, assuming you’re reading this in 2008) are using are built from CMOS logic gates. The QPC effect occurs when such a gate is operated near its cutoff point. There are billions of billions of these gates currently in operation on this planet (operating in cutoff and saturation); by all expectations they should have been well understood many decades ago.
However, it turns out that the situation is difficult to analyze with the usual tools of quantum mechanics. One ends up with a highly nonlinear situation involving “quasibound” states. Since the density matrix formalism I work with is specifically designed to solve highly nonlinear bound state QFT problems without recourse to the usual perturbation theory, this is a natural place to explore its application. In addition, this could be a good first application of MUB (mutually unbiased bases) to quantum theory.
The Fort Pillow Massacre dates to April 12, 1864, the fourth year of the American Civil War, on the banks of the Mississippi river in west Tennessee. I bought this 530 page hardback by Andrew Ward at Half Price Books at a bit of a steal for $3 or so:
The book is peculiarly interesting in that it reminds one that our current arguments over the rules of war, that is, the definition of enemy combatants, and the mistreatment of prisoners, were also the subject of great controversy 140 years ago.
Perhaps a sign of my maturity, or perhaps some other thing, it no longer bothers me when youngsters (those aged, say 20 to 30) make appalling statements regarding history. Instead, I simply assume that they’re just trying to piss me off, and I ignore it. This sentiment was brought up when I showed the book to a chess player at the Overlake Mall just after buying it. “The Civil War? Oh that’s one of my favorites. That was when the slaves were freed. Or was that World War I?” Under the assumption that the comment was rooted in ignorance rather than in wishing to see me blow a gasket, I write this book review.
I’ve finally just now figured out how to combine the quark / lepton weak hypercharge and weak isospin quantum numbers with the generation numbers. This should allow the Koide mass formula to be extended to the quarks!
The weak hypercharge and weak isospin quantum numbers for the elementary fermions are:
, with the quantum numbers for the antiparticles with opposite handedness given by the negatives of the above.
The Koide mass formulas for the charged and neutral leptons can be derived from making the assumption that these particles are color neutral composite particles built from three preons that I’ve usually called “snuarks” and they are taken from a set of three mutually unbiased bases for the Pauli algebra.
In the density matrix language, a particle is not represented by a state vector, but instead by a state matrix. A state matrix gives the transition amplitudes for the states that are bound together. The diagonal entries correspond to the amplitude for the propagation of one of the snuarks without change. The off diagonal entries give amplitudes for the various ways a snuark can switch states. Consequently, in the language of quarks and gluons, the off diagonal entries represent the gauge bosons of the theory while the diagonal entries give the valence quarks. Using “r,g,b” as the indices for the matrix, the (r,g) entry gives the amplitude for transitions from G to R; in the quark / gluon language, this would be the action of a R/G gluon.
Suppose we have an object X composed of three spin-1/2 fermions, R, G, and B. I should mention that “spin-1/2” means SU(2) here, in particular the usual 2-dimensional Pauli spin. The three fermions can each have spin +- 1/2, what spin states can object X have? This is a problem learned in undergraduate quantum mechanics; the answer is that 2x2x2 = 4+2+2, that is, one obtains a spin-3/2 quadruplet, and two spin-1/2 doublets. If that didn’t make any sense, then the wikipedia article on 2×2 = 3+1 spin might help, but this explanation is longer and better.
One easy way to derive the decomposition for n spin-1/2 particles uses Pascal’s triangle, and the fact that the simple representations of SU(2) have quantum numbers that run from -n/2 to +n/2, and have multiplicity 1 at each location. This fact is proved on Wikipedia in the usual method of the place (short and difficult to follow), but is probably well known to most of the readers. If we are to graph the quantum numbers of the various representations, we need one dimension, i.e. a number line, which we will draw horizontally. Then the various simple representations of SU(2) have quantum numbers as follows:
In the above, symbols “1” give the multiplicity of the eigenvalue. The spin-3/2 states, which are the “4” of 2x2x2 = 4+2+2, are circled in red. For the simple representations of SU(2) these are all the same so we could have used a dot, but farther down we’ll be dealing with higher multiplicities. Each representation is labeled by its highest spin state. The various representations have been assembled this way so that they are at least a little reminiscent of Pascal’s triangle.