Now comes the tricky part. The following is *very important* to
understand, because it is a common feature to nearly all differential
formulations of any sort of potential-based field theory, quantum or
classical.

We *know* from our extensive study of elementary physics that there
must be some freedom in the choice of
and
. The fields are
physical and can be ``directly'' measured, we know that they are *unique* and cannot change. However, they are both defined in terms of
*derivatives* of the potentials, so there is an *infinite
family* of possible potentials that will *all* lead to the *same
fields*. The trivial example of this, familiar from kiddie physics, is
that the electrostatic potential is only defined with an arbitrary
additive constant. No *physics* can depend on the choice of this
constant, but some choices make problems more easily solvable than
others. If you like, experimental physics depends on potential *differences*, not the absolute magnitude of the potential.

So it is now in grown-up electrodynamics, but we have to learn a new
term. This freedom to add a constant potential is called *gauge
freedom* and the different potentials one can obtain that lead to the
same physical field are generated by means of a *gauge
transformation*. A gauge transformation can be broadly defined as any
formal, systematic transformation of the potentials that leaves the
fields invariant (although in quantum theory it can be perhaps a bit
more subtle than that because of the additional degree of freedom
represented by the quantum phase).

As was often the case in elementary physics were we freely moved around
the origin of our coordinate system (a gauge transformation, we now
recognize) or decided to evaluate our potential (differences) from the
inner shell of a spherical capacitor (another choice of gauge) we will
*choose a gauge* in electrodynamics to make the solution to a
problem as easy as possible or to build a solution with some desired
characteristics that can be enforced by a ``gauge condition'' - a
constraint on the final potentials obtained that one can show is within
the range of possibilities permitted by gauge transformations.

However, there's a price to pay. Gauge freedom in non-elementary physics
is a *wee bit broader* than ``just'' adding a constant, because
gradients, divergences and curls in multivariate calculus are not simple
derivatives.

Consider
.
must be unique, but many
's exist
that correspond to any given
. Suppose we have one such
. We
can obviously make a new
that *has the same curl* by adding
the gradient of *any scalar function
*. That is:
= = (+ &Lambda#Lambda;) = '

We see that:
' = + &Lambda#Lambda;
is a *gauge transformation* of the vector potential that leaves the
field invariant.

Note that it probably isn't true that
can be *any* scalar
function - if this were a math class I'd add caveats about it being
nonsingular, smoothly differentiable at least one time, and so on. Even
if a physics class I might say a word or two about it, so I just did.
The point being that before you propose a
that isn't, you at
least need to think about this sort of thing. However, great physicists
(like Dirac) have subtracted out irrelevant infinities from potentials
in the past and gotten away with it (he invented ``mass
renormalization'' - basically a gauge transformation - when trying to
derive a radiation reaction theory), so don't be too closed minded about
this either.

It is also worth noting that this only shows that this is *a*
possible gauge transformation of
, not that it is sufficiently
general to encompass *all* possible gauge transformations of
.
There may well be tensor differential forms of higher rank that cannot
be reduced to being a ``gradient of a scalar function'' that still
preserve
. However, we won't have the algebraic tools to think
about this at least until we reformulate MEs in relativity theory and
learn that
and
are *not, in fact, vectors!* They are
components of a second rank tensor, where both
and
combine
to form a first rank tensor (vector) in four dimensions.

This is quite startling for students to learn, as it means that there
are *many* quantities that they might have thought are vectors that
are not, in fact, vectors. And it matters - the tensor character of a
physical quantity is closely related to the way it transforms when we
e.g. change the underlying coordinate system. Don't worry about this
quite yet, but it is something for us to think deeply about later.

Of course, if we change
in arbitrary ways,
will change as
well! Suppose we have an
and
that leads to some particular
combination:
= -&phis#phi;- t
If we transform
to
by means of a gauge transformation (so
is preserved), we (in general) will still get a different
:
' & = & -&phis#phi;- 't

& = & -&phis#phi;- t(+ &Lambda#Lambda;)

& = & - &Lambda#Lambda;t &ne#ne;
as there is no reason to expect the gauge term to vanish. This is
baaaaad. We want to get the *same*
.

To accomplish this, *as* we shift
to
we must *also*
shift
to
. If we substitute an unknown
into the
expression for
we get:
' & = & -&phis#phi;' - t(+ &Lambda#Lambda;)

' & = & -&phis#phi;' - t -
&Lambda#Lambda;t

We see that in order to make
(so it doesn't vary with the
gauge transformation) we have to *subtract* a compensating piece to
to form
:
&phis#phi;' = &phis#phi;- &Lambda#Lambda;t
so that:
' & = & -&phis#phi;' - 't = -&phis#phi;+
&Lambda#Lambda;t - t -
&Lambda#Lambda;t

& = & -&phis#phi;- t =

In summary, we see that a fairly general gauge transformation that
preserves *both*
and
is the following pair of *simultaneous* transformations of
and
. Given an arbitrary
(but well-behaved) scalar function
:
&phis#phi;' & = & &phis#phi;- &Lambda#Lambda;t

' & = & + &Lambda#Lambda;
will leave the derived fields invariant.

As noted at the beginning, we'd like to be able to *use* this gauge
freedom in the potentials to choose potentials that are easy to evaluate
or that have some desired formal property. There are two choices for
gauge that are very common in electrodynamics, and you should be
familiar with both of them.