The Math of Gauge Theories

With a bit of a delay I am resuming the posts on gauge theory and today I will talk about the math involved. 

In gauge theory you consider the base space-time as a manifold and you attach at attach point an object or what is called a fiber forming what it is called a fiber bundle. The picture which you should have in mind is that if a rug.

The nature of the fibers is unimportant at the moment, but they should obey at least the properties of a linear space. 

Physically think of the fibers as internal degrees of freedom at each spacetime point, and a physical configuration would correspond to a definite location at one point long the fiber for each fibers. 

The next key concept is that of a gauge group. A gauge group is the group of transformations which do not affect the observables of the theory. 

Mathematically, the gauge symmetry depends on how we relate points between nearby fibers and to make this precise we only need (only) one critical step: define a covariant derivative.

Why do we need this? Because an arbitrary gauge transformation does not change the physics and the usual ordinary derivative sees both infinitesimal changes to the fields, and the infinitesimal changes to an arbitrary gauge transformation. Basically we need to compensate for the derivative of an arbitrary gauge transformation.

If d is the ordinary derivative, let's call D the covariant derivative and their difference (which is a linear operator) is called either a differential connection, a gauge field, or a potential:

A(x) = D - d

D and d act differently: d "sees" the neighbourhood behaviour but ignores the value of the function on which it acts, and D acts on the value but is blind to the neighbourhood behaviour.   

The condition we will impose on D is that is must satisfy the Leibniz identity because it is derivative:

D(fg) = (Df)g+f(Dg)

which in turn demands:

A(fg) = (Af)g+f(Ag)

In general only one part of A may be used to compensate for gauge transformations, and the remaining part represent an external field that may be interpreted as potential. When no external potentials are involved, A usually respects integrability conditions. Those conditions depend on the concrete gauge theory and we will illustrate this in subsequent posts.

When external fields are present, the integrability conditions are not satisfied and this is captured by what is called a curvature. The name comes from general relativity where lack of integrability is precisely the space-time curvature.

The symmetry properties arising out of curvature construction gives rise to algebraic identities.

Next in gauge theories we have the homogeneous and inhomogeneous differential equations. As example of homogeneous differential equations are the Bianchi identities in general relativity and the two homogeneous Maxwell's equations. The inhomogeneous equations are related to the sources of the fields (current in electrodynamics, and stress-energy tensor in general relativity).

So to recap, the steps used to build a gauge theory are:

1. the gauge group
2. the covariant derivative giving rise to the gauge field
3. integrability condition
4. the curvature
5. the algebraic identities
6. the homogeneous equations
7. the inhomogeneous equations

In the following posts I will spell out this outline first for general relativity and then for electromagnetism. Technically general relativity is not a gauge theory because diffeomorphism invariance cannot be understood as a gauge group but the math similarities are striking and there is a deep connection between diffeomorphism invariange and gauge theory which I will spell out in subsequent posts. So for now please accept this sloppiness which will get corrected in due time.

The Bohm-Aharaonov effect

Today we come back to gauge theory and continue on Weyl's ideas. With the advent of quantum mechanics Weyl realized that he could reinterpret his change in scale as a change in the phase of the wavefunction. Suppose we make the following change to the wavefunction:

\(\psi \rightarrow \psi s^{ie\lambda/\hbar}\)

The overall phase does not affect the Born rule and we did not change the physics (here \(\lambda\) does not depend on space and time and it is called a global phase transformation). Let's make this phase change depend on space and time: \(\Lambda = \Lambda (x,t) \) and see where it leads. 

To justify this assume we are studying charged particle motion in an electromagnetic field and suppose that \(\Lambda\) corresponds to a gauge transformation for the electromagnetic field potentials \(A\) and \(\phi\):

\(A\rightarrow A + \nabla \Lambda\)

\(\phi \rightarrow \phi - \partial_t \Lambda\)

This should not change the physics and in particular it should not change Schrodinger's equation. To make Schrodinger's equation invariant under a local \(\Lambda\) change we need to add  \(-eA\) to the momentum quantum operator:

\(-i\hbar \nabla \rightarrow -i\hbar \nabla -eA\)

And the Schrodinger equation of a charged particle in an electromagnetic field reads:

\([\frac{1}{2m}{(-i\hbar\nabla -eA)}^2 + e\phi +V]\psi = -i\hbar\frac{\partial \psi}{\partial t}\)

But why do we have the additional \(eA\) term to begin with? It's origin is in Lorentz force. If \(B = \nabla \times A\) and \(E = -\nabla \phi - \dot{A}\), the Lagrangian takes the form:

\(L = \frac{1}{2} mv^2 - e\phi + ev\cdot A\)

which yields the canonical momenta to be:

\(p_i = \partial{\dot{x}_i} = mv_i + eA_i\)

and adding \(-eA\) to the momenta in the Hamiltonian yields Lorentz force from Hamlton's equations of motion. 

Coming back to Schrodinger's equation we notice that the electric and magnetic fields E and B do not enter the equation, but instead we have the electromagnetic potentials. Suppose we have a long solenoid which has inside a non zero magnetic field B, and outside zero magnetic field. Outside the solenoid, in classical physics we cannot detect any change if the current flows or not through the wire. However the vector potential is not zero outside the solenoid (\(\nabla\times A = 0\) does not imply \(A=0\)) and the Schrodinger equation solves differently when \(A = 0\) and \(A\ne 0\). 

From this insight Bohm and Aharonov came up with a clever experiment to put this to the test: in a double slit experiment, after the slits they proposed to add a long solenoid. Record the interference pattern with no current flowing through the solenoid and repeat the experiment with the current creating a magnetic field inside the solenoid. Since the electrons do not enter the solenoid, from classical physics we should expect no difference, but in quantum mechanics the vector potential is not zero and the interference pattern shifts. Unsurprisingly the experiment confirms precisely the theoretical computation.

There are several important points to be made. First, there is no classical explanation of the effect: E and B are not fundamental, but \(\phi\) and \(A\) are. It is mind boggling that even today there are physicists who do not accept this and continue to look for effects rooted in E and B. Second, the gauge symmetry is not just a accidental symmetry of Maxwell's equation but a basic physical principle which turns out to govern all fundamental forces in nature. Third, the right framework for gauge theory is geometrical and we will explore this in depth in subsequent posts. Please stay tuned.

Due to travel, the next post is delayed 2 days.