The Electromagnetic Connection
Disclaimer: Nothing in this note is original.
Is this note we discuss the profound connection between the electromagnetic field and the parallel transportation of wave functions. This will endow the electromagnetic field a geometric interpretation.
Classical case
In the presence of electromagnetic fields, the charged particle will receive forces not only from a potential
Recall that the energy-momentum covector defined on the cotangent bundle of the configuration space is
We add to it the electromagnetic covector
then we have a new total energy-momentum 1-form, from which we can construct a new Lagrangian by Legendre transform, which will tell us how a charged particle moves in EM field.
Quantum case
Ignore the spin of a electron, it is described by a wave function
In Cartesian coordinates,
With the presence of electromagnetism, the following replacement is needed,
Furthermore, we can write the Schrodinger function in terms of the covariant derivatives if we introduce connections by
The gauge potential has a compact
The vector potential is not uniquely defined. We may adopt different gauge potentials in different patches, and at the overlapping region the gauge fields (connections) can be glued via gauge transformation. This is precisely the picture of complex line bundle we introduced before, whose covariant derivative is defined via the connection
We find that
We can again confirm that gauge transformation for
In summary, a wave function is not to be considered as a single-valued complex function of
This brings us back to the starting point of gauge theories in quantum mechanics, namely
Weyl’s principal of gauge invariance. If
satisfies the Schrodinger equation with
A summary.
- Fiber bundle: complex line bundle.
- Transition function:
. - Structure group:
. - Connection:
. - Curvature:
.
In curvilinear coordinates, what form should the Schrodinger equation take? Recall that when there is no electromagnetic field and the spacetime is flat, the Schrodinger equations is simply
In a curvilinear coordinates on an Riemannian manifold
where
refer to my blog here. Inserting this into the Schrodinger equation we get the curvilinear Schrodinger equation, which is left as a homework (to myself).
Global Potentials
If the space is
Theorem. Consider a region
where
Dirac monopole
When there is a monopole in spacetime, then we are forced to introduce the bundle and connection.
Assume there is a monopole sitting at the origin. The magnetic field generated is
where
which is obviously not defined on the negative
on
the potential is given by
and Maxwell’s equation holds everywhere on
where
Now the potential is not single valued unless
is satisfied.
In the language of curvature, we have
where
In summary, if there exists a monopole, then we must regard the wavefunction as a section of a complex line bundle.
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