The electromagnetic field
- I. Introduction
- II. Elements of quantum mechanics
- III. The electromagnetic field
- IV. The qubit
- V. Quantum gates
- VI. States and operators
- VII. Shor’s algorithm
- VIII. Quantum teleportation
- IX. The Deutsch algorithm
- X. The no-cloning theorem
- XI. About anyons
- XII. About SU(2)
- XIII. Teleportation topology
- XIV. The Simon algorithm
- XV. Superdense coding
- XVI. Chern-Simons theory
- XVII. The toric code
- REF. Some pointers
This time we’ll describe some crucial elements we need later on:
- how electric and magnetic fields are unified in one vector field
- how electrons ‘feel’ hole in spacetime
- how the Wilson loop is a natural thing when looking at loops (and later on, knots).
Electromagnetism as described by Maxwell’s equations is actually a decoupling of a more fundamental field called the vector potential, denoted by
with
and in the example below we show how you can go from Maxwell’s equation to all of this. The essential idea you need to understand is that the whole of electromagnetism sits in this potential and (quantum) field theory deals with
This is even more striking when looking at the famous Aharonov-Bohm effect where one can have a vanishing magnetic field and still a noticeable influence on quantum behavior.
If you take a quantum particle travelling around a point-like magnetic field the path integral becomes
where
using Stoke’s trick. The surface integral is basically the flux from the magnetic field and because it’s concentrated in the center you can see that the factor is independent of the path taken. Hence,
with
you can see that the vector potential induces a phase-shift
Note the surprise here: you move something around in a place where there is no magentic field and still you can have a noticeable effect from something you circled around. The situation is a bit similar to the complex line integrals where poles lead to non-zero residues. Though in that case the function is singular and not the underlying space.
The phase-shift is crucial to understanding anyons and topological computing in general and there are a few key-concept to take away from here. First, contrary to the situation you have in complex analysis the function we integrate is not singular rather the underlying space has a singularity. The magnetic line punches a hole in the topology leading to, what is called, a non-zero winding number (or fundamental homology group
One also says that the fundamental group reflects the knottiness (naughtyness?) of the space. Similarly, if you look at the taurus a bit you will discover that the tarus has
From Maxwell to the vector potential
Let’s take the Maxwell equations in vacuum:
The fact that $ =0$ implies that there is a vectorial function with
means that there is a scalar
which automatically satisfies the third equation. The first equation can be transformed using
giving
Finally, the second equation gives
Now, lean back a bit and look at the last two equations. You can see that if you set (that is, require)
all of Maxwell’s equation reduce to
Even more, if you define the four-vector as
everything simplifies to
So, as stated earlier, one can reduce all of electromagnetism to dealing with the vector potential instead of using the electric and magnetic fields.
The simplification has far reaching consequences and deep roots in differential geometry. The concept of gauge invariance can be seen in the equation above. Replacing
does indeed not alter the equations and it means that the vector potential can always be shifted with the derivative of an arbitrary function. This idea, when extended to a local invariance, leads to gauge field theory and ultimately to the standard model of elementary particles.
Vector potential of a solenoid
How can one have a non-vanishing A-field with a zero magnetic field? Take a magnetic field centered in a cylindrical tube of Radius R along the z-axis. By defining
and
So, you have indeed a non-zero vector potential outside the magnetic tube. Note that the vector potential is continuous at
Minimal coupling
Minimal coupling is a prescription to couple the dynamics of one field to another. Usually it refers to how a matter-field (say, electrons) is coupled to electromagnetism.
The minimal coupling of the vector field with a free particle is essentially the substitution
leading to a factor
This simple rule has deep roots in differential geometry (covariant derivatives and vector bundles). It might appear a bit ad-hoc but there is a large body of refined maths underneath it. It’s also where you can venture into non-commutative geometry and exotic variations of Riemannian geometry.
Wilson loops and curvature
From the discussion above it should be clear that the integral
is rather important. It plays in fact a crucial role in the bridge between knots and QFT as we’ll see later. For now, let’s look at the parallel transport of the fibers (of a vector bundle). If
and if you continue transporting it you get
where the product has to be performed in such a way that the ordering of the
with
which is known as a Wilson loop. Note the similarity with the Aharonov-Bohm factor.
One of the very cool things you can do with this is looking at infinitesimal loops and one can show that the result is directly related to the curvature of