gauge invariance

Gauge freedom and four-potentials in the STA form of Maxwell’s equation.

March 27, 2022 math and physics play , , , , , , , , , , , , , , , , , , , , ,

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In a recent video on the tensor structure of Maxwell’s equation, I made a little side trip down the road of potential solutions and gauge transformations. I thought that was worth writing up in text form.

The initial point of that side trip was just to point out that the Faraday tensor can be expressed in terms of four potential coordinates
F_{\mu\nu} = \partial_\mu A_\nu – \partial_\nu A_\mu,
but before I got there I tried to motivate this. In this post, I’ll outline the same ideas.

STA representation of Maxwell’s equation.

We’d gone through the work to show that Maxwell’s equation has the STA form
\grad F = J.
This is a deceptively compact representation, as it requires all of the following definitions
\grad = \gamma^\mu \partial_\mu = \gamma_\mu \partial^\mu,
\partial_\mu = \PD{x^\mu}{},
\gamma^\mu \cdot \gamma_\nu = {\delta^\mu}_\nu,
\gamma_\mu \cdot \gamma_\nu = g_{\mu\nu},
&= \BE + I c \BB \\
&= -E^k \gamma^k \gamma^0 – \inv{2} c B^r \gamma^s \gamma^t \epsilon^{r s t} \\
&= \inv{2} \gamma^{\mu} \wedge \gamma^{\nu} F_{\mu\nu},
J &= \gamma_\mu J^\mu \\
J^\mu &= \frac{\rho}{\epsilon} \gamma_0 + \eta (\BJ \cdot \Be_k).

Four-potentials in the STA representation.

In order to find the tensor form of Maxwell’s equation (starting from the STA representation), we first split the equation into two, since
\grad F = \grad \cdot F + \grad \wedge F = J.
The dot product is a four-vector, the wedge term is a trivector, and the current is a four-vector, so we have one grade-1 equation and one grade-3 equation
\grad \cdot F &= J \\
\grad \wedge F &= 0.
The potential comes into the mix, since the curl equation above means that \( F \) necessarily can be written as the curl of some four-vector
F = \grad \wedge A.
One justification of this is that \( a \wedge (a \wedge b) = 0 \), for any vectors \( a, b \). Expanding such a double-curl out in coordinates is also worthwhile
\grad \wedge \lr{ \grad \wedge A }
\lr{ \gamma_\mu \partial^\mu }
\lr{ \gamma_\nu \partial^\nu }
A \\
\gamma^\mu \wedge \gamma^\nu \wedge \lr{ \partial_\mu \partial_\nu A }.
Provided we have equality of mixed partials, this is a product of an antisymmetric factor and a symmetric factor, so the full sum is zero.

Things get interesting if one imposes a \( \grad \cdot A = \partial_\mu A^\mu = 0 \) constraint on the potential. If we do so, then
\grad F = \grad^2 A = J.
Observe that \( \grad^2 \) is the wave equation operator (often written as a square-box symbol.) That is
&= \partial^\mu \partial_\mu \\
&= \partial_0 \partial_0
– \partial_1 \partial_1
– \partial_2 \partial_2
– \partial_3 \partial_3 \\
&= \inv{c^2} \PDSq{t}{} – \spacegrad^2.
This is also an operator for which the Green’s function is well known ([1]), which means that we can immediately write the solutions
A(x) = \int G(x,x’) J(x’) d^4 x’.
However, we have no a-priori guarantee that such a solution has zero divergence. We can fix that by making a gauge transformation of the form
A \rightarrow A – \grad \chi.
Observe that such a transformation does not change the electromagnetic field
F = \grad \wedge A \rightarrow \grad \wedge \lr{ A – \grad \chi },
\grad \wedge \grad \chi = 0,
(also by equality of mixed partials.) Suppose that \( \tilde{A} \) is a solution of \( \grad^2 \tilde{A} = J \), and \( \tilde{A} = A + \grad \chi \), where \( A \) is a zero divergence field to be determined, then
\grad \cdot \tilde{A}
\grad \cdot A + \grad^2 \chi,
\grad^2 \chi = \grad \cdot \tilde{A}.
So if \( \tilde{A} \) does not have zero divergence, we can find a \( \chi \)
\chi(x) = \int G(x,x’) \grad’ \cdot \tilde{A}(x’) d^4 x’,
so that \( A = \tilde{A} – \grad \chi \) does have zero divergence.


[1] JD Jackson. Classical Electrodynamics. John Wiley and Sons, 2nd edition, 1975.

Update to old phy356 (Quantum Mechanics I) notes.

February 12, 2015 math and physics play , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

It’s been a long time since I took QM I. My notes from that class were pretty rough, but I’ve cleaned them up a bit.

The main value to these notes is that I worked a number of introductory Quantum Mechanics problems.

These were my personal lecture notes for the Fall 2010, University of Toronto Quantum mechanics I course (PHY356H1F), taught by Prof. Vatche Deyirmenjian.

The official description of this course was:

The general structure of wave mechanics; eigenfunctions and eigenvalues; operators; orbital angular momentum; spherical harmonics; central potential; separation of variables, hydrogen atom; Dirac notation; operator methods; harmonic oscillator and spin.

This document contains a few things

• My lecture notes.
Typos, if any, are probably mine(Peeter), and no claim nor attempt of spelling or grammar correctness will be made. The first four lectures had chosen not to take notes for since they followed the text very closely.
• Notes from reading of the text. This includes observations, notes on what seem like errors, and some solved problems. None of these problems have been graded. Note that my informal errata sheet for the text has been separated out from this document.
• Some assigned problems. I have corrected some the errors after receiving grading feedback, and where I have not done so I at least recorded some of the grading comments as a reference.
• Some worked problems associated with exam preparation.