Spinor Info

In recent weeks, there has been a lot of discussion in the physics community about a curious observation: an abundance of energetic positrons observed by a satellite named PAMELA. According to some people, this unexpected abundance of positrons is likely caused by annihilation of massive dark matter particles, constituting “smoking gun” evidence that dark matter really exists.

Of course it is possible that this abundance is due to some conventional astrophysics, such as pulsars doing this or that. This is a subject of on-going dispute.

One thing I do not see discussed is that the anticipated behavior is apparently based on a model developed several years ago that uses as many as eleven adjustable parameters yet nevertheless, does not produce a spectacularly good fit of even the low energy data. I wonder if I am missing something.

PAMELA positron fraction and theoretical models.

Looks like Stephen Hawking is coming to Waterloo. I may not be an adoring fan, but I am certainly an admirer: being able to overcome such a debilitating disease and live a creative life is no small accomplishment even when you don’t become a world class theoretical physicist in the process.

A new paper by Sean Carroll asks this question in its title “What if time really exists?” I feel reassurred that Carroll thinks it does (hmmm, let me check my watch… yup, I think it exists, too) but the fact that a paper with this title appears in an archive of theoretical physics papers perhaps illustrates what is so wrong with physics today. To quote Carroll, “when something is so obvious and important, declaring that it isn’t real is sure to win points for boldness”, but have physicists really become this shallow?

I’ve been reading a lot about gauge theories lately. I once wrote what I thought was a fine and concise description of the principle of gauge invariance, but I needed Schrödinger’s equation for it, which made my explanation both non-classical and non-relativistic.

Finally, with the help of a Wikipedia article no less, I think I managed to understand how a gauge theory can come into being without involving any quantum physics. It’s simple, really, surprisingly so.

What you need is a (classical) field theory. A field theory is specified by its Lagrangian, which, crudely speaking, is just the difference between the kinetic and potential energy. For a scalar field φ, the kinetic energy of the field is the square of its gradient, ∂_μφ∂^μφ. The potential term can be nothing (massless particle in empty space) or it can contain a mass term in the form m²φ².

Where things begin to get interesting is when we allow φ to be a complex field. In this case, rather than writing φ², we must now write φφ^*, where φ^* is the complex conjugate of φ. The same thing happens in the kinetic term. So now the Lagrangian reads,

L = ∂_μφ∂^μφ^* – m²φφ^*.

The reason why this is so interesting is that if we change the phase of φ by a set amount (i.e., multiply φ by e^iψ) the conjugate’s phase changes by the opposite amount (i.e., φ^* it gets multiplied by e^–iψ). Their product, therefore, multiplied by e^iψe^–iψ = 1, remains unchanged. In other words, our Lagrangian is invariant under a global rotation in the complex plane. Right there, this has an important implication: as per Noether’s theorem, a global symmetry implies the existence of a conserved current.

But what if the symmetry is not global but local? Meaning that we rotate φ in the complex plane as before, but the angle of rotation is not the same everywhere? Clearly, φφ^* still remains unchaged just as before, but the same is not true for ∂_μφ∂^μφ^*; the derivative operator brings new terms into the Lagrangian.

These new terms are best dealt with by changing the derivative operator into a covariant derivative: ∂_μ → D_μ = ∂_μ + A_μ, where A_μ is an arbitrary vector field.

Or maybe not so arbitrary. We can make A_μ anything we want, of course, but that also means that we can demand that A_μ satisfy a field equation. Perhaps the field equations of electromagnetism… why not? (After all, every vector field satisfies the field equations of electromagnetism.)

The difference between the original Lagrangian (written using the ordinary derivative ∂_μ) and the new Lagrangian (written using the covariant derivative operator D_μ) is the interaction Lagrangian that describes how the φ field interacts with itself through a vector field A_μ. By making the complex φ field locally gauge invariant, we have, in effect, invented the electromagnetic vector potential A_μ.

This is, after all, what gauge theories do: they turn a local symmetry into a force. The local symmetry can be geometric in nature (e.g., a rotation) or it can be an internal symmetry of a field that is not described by simple real numbers. In the present example, the field was made up of complex numbers, and the symmetry was that of the complex plane. This symmetry group is U(1), which is an Abelian group: two rotations in the complex plane, executed one after the other, produce the same result regardless of the order in which they are executed.

In many physically important cases, the symmetry is non-Abelian. The most profound consequence of this is that in place of the gauge field A_μ, which is “inert”, we get gauge field(s) that interact with themselves. In practical terms, when the theory is Abelian, like electromagnetism, the gauge field A_μ represents photons, which are uncharged; but when the theory is non-Abelian, like electroweak theory, the gauge fields are non-Abelian, carry charge, and interact with each other.