Monday, September 4, 2017

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.