Since the Higgs mechanism is a generic feature of gauge theories, we might as well introduce it in as simple of a context as possible. The theory we will study is known as ``Scalar QED-3" which is short for Scalar Quantum Electrodynamics in 2+1 dimensions. One thing you will find as you develop as a scientist is that the more complicated sounding the name, the simpler the theory. Scalar QED-3 is a ``toy model" for electromagnetism. It contains a field $\phi$ whose excitations represent the Higgs boson -- just as when we played with phonons we had a field whose excitations produced phonons. This field will live in 2 space and 1 time dimension, and be coupled to a vector and scalar potential -- $(A_t,A_x,A_y)$. We are working in 2+1 dimensions because there are fewer variables, which makes the math easier. [Though I bet by the end of this lecture you will still wish I found an easier model yet.] We will spread the arguments over 2 days.

The analog of the magnetic field here is a scalar -- all spatial loops are in the $x-y$ plane. The electric field is still a vector. We will write these fields as
\begin{eqnarray}
S_t&=&F_{xy}=\partial_x A_y-\partial_y A_x\\
S_y&=&F_{tx}=\partial_t A_x-\partial_x A_t\\
S_x&=& F_{yt}=\partial_y A_t-\partial_t A_y.
\end{eqnarray}
The magnetic field is $S_t$, while the electric field is $S_x$ and $S_y$. The minus signs here are easy -- just cyclic permutations

The field $\phi$ obeys a relativistic field equation:
\begin{equation}\label{klein}
(\partial_t-i A_t)^2\phi -(\partial_x-i A_x)^2\phi-(\partial_y-i A_y)^2\phi+m^2\phi=0.
\end{equation}
This is the simplest relativistic generalization of the Schrodinger equation.
The electric and magnetic fields obey analogs
of Maxwell's equations: $\nabla\cdot E=\rho/\epsilon_0$ and $\nabla\times B-\epsilon_0\partial_t E=\mu_0 J$. These read
\begin{eqnarray}\label{max}
\partial_x S_y-\partial_y S_x &=& \frac{\phi^* (\partial_t-i A_t)\phi-\phi(\partial_t+i A_t)\phi^*}{2i}\\\nonumber
-\partial_y S_t-\partial_t S_y &=& -\frac{\phi^* (\partial_x-i A_x)\phi-\phi(\partial_x+i A_x)\phi^*}{2i}\\\nonumber
\partial_t S_x+\partial_x S_t &=& -\frac{\phi^* (\partial_y-i A_y)\phi-\phi(\partial_y+i A_y)\phi^*}{2i}
\end{eqnarray}
The other Maxwell equations just come from the fact that the electric and magnetic fields are the appropriate derivatives of the vector and scalar potentials. Here (in 2+1 dimensions) there is just one such constituent equation
\begin{equation}
\partial_t S_t+\partial_x S_x+\partial_y S_y=0,
\end{equation}
which is essentially $\nabla\times E=-\partial_t B$.

It will be convenient to Eqs.~(\ref{max}) in terms of the potentials:
\begin{eqnarray}\label{poteq}
\partial_t [\partial_x A_x+\partial_y A_y]-(\partial_x^2+\partial_y^2)A_t &=& \frac{\phi^*\partial_t \phi-\phi \partial_t \phi^*}{2i}-|\phi|^2 A_t\\
\partial_x [-\partial_y A_y+\partial_t A_t]-(-\partial_y^2+\partial_t^2)A_x &=& -\frac{\phi^*\partial_x \phi-\phi \partial_x \phi^*}{2i}+|\phi|^2 A_x\\
\partial_y [\partial_t A_t-\partial_x A_x]-(\partial_t^2-\partial_x^2)A_y &=& -\frac{\phi^*\partial_y \phi-\phi \partial_y \phi^*}{2i}+|\phi|^2 A_y
\end{eqnarray}

By construction these equations have a gauge symmetry
\begin{eqnarray}
A_\nu&\to& A_\nu+\partial_\nu\Lambda\\
\phi&\to& e^{i\Lambda}\phi
\end{eqnarray}

\section{Small Oscillations}
We will first understand the small oscillations of these fields. We take $\phi$, and $A$ to be small, and linearize Eq.~(\ref{klein}) to find the Klein-Gordon equation
\begin{equation}
\partial_t^2\phi -\partial_x^2\phi-\partial_y^2\phi+m^2\phi=0.
\end{equation}
Taking $\phi(x,y,t)=\phi e^{-i\omega t+ i k_x x+i k_y y}$, we get
\begin{equation}
(-\omega^2+k^2+m^2)\phi=0.
\end{equation}
In other words, the excitations have the relativistic dispersion
\begin{equation}
\omega=\sqrt{k^2+m^2}.
\end{equation}
There are two modes here for each $k$ -- the real and imaginary part of $\phi$ can fluctuate independently.

If we linearizing Maxwell's equations we get rid of the right hand sides. Substituting the Ansatz
$S_\nu(x,y,t)=S_\nu e^{-i\omega t+i k_x x+i k_y y}$, we find
\begin{eqnarray}
k_x S_y - k_y S_x&=&0\\
-k_y S_t+\omega S_y &=&0\\
-\omega S_x+k_x S_t &=&0\\
-\omega S_t + k_x S_x + k_y S_y &=&0.
\end{eqnarray}
This is four equations, but only 3 unknowns. The first 3 equations, however are not linearly independent. If I multiply the first by $\omega$ ,the second by $k_x$ and the third by $k_y$, and add them I get zero. Therefore there are really 3 equations in 3 unknowns. Without any loss of generality I take $S_t=\omega$, which gives $S_x=k_x$ and $S_y=k_y$. Substituting this into the last equation I get
\begin{equation}
\omega^2=k^2.
\end{equation}
This is the analog of light. It travels with a relativistic dispersion. Note there is only one mode. Light is transverse -- and in 2D there is only one transverse mode.

It will be useful to repeat that last analysis using the vector potentials. We do not need the constituent equations, and Fourier transforming Eq.~(\ref{poteq}) yield
\begin{equation}
\left(
\begin{array}{ccc}
\omega^2-k_y^2&k_xk_y&\omega k_x\\
k_x k_y &\omega^2-k_x^2&\omega k_y\\
\omega k_x&\omega k_y& k_x^2+k_y^2
\end{array}
\right)
\left(
\begin{array}{c}
A_x\\
A_y\\
A_t
\end{array}
\right)=0
\end{equation}
The normal modes are found by setting the determinant equal to zero.

An even better approach is to look at the eigenvectors of this matrix. The normal modes are found by setting the eigenvalues equal to zero.
This matrix has three eigenvectors:

$(k_x \omega,k_y\omega, k^2)$

has eigenvalue $\omega^2+k^2$.
Since this is greater than zero, it can never be the normal mode.

$(k_x,k_y,-\omega)$

has eigenvalue $0$ for any $\omega$ or $k$. This corresponds to a gauge transformation.

$(-k_y,k_x,0)$

has eigenvalue $\omega^2-k^2$, and corresponds to the normal mode we just found.