Re: funny capacitor (gauge invariance)

["John S. Denker" <jsd@MONMOUTH.COM>]

I wrote:
> >
> > we need a way to calculate the
> > physically-meaningful quantities.

Then at 11:12 PM 3/10/01 -0500, Ludwik Kowalski wrote:

> the gauge is not a physically meaningful quantity TO ME.

Good!  The gauge, by itself, is the quintessential _unobservable_
quantity.  Any proper physics formula will give an answer that is
independent of the choice of gauge.

*** Example 1:  The charge on the ordinary two-terminal capacitor depends
on the _difference_ between the two terminal voltages:
        Q = C (V1 - V2)                         (equation 1)
where
 -- V1 may be gauge dependent, and
 -- if so V2 must be gauge dependent in the same way, while
 -- the only thing that matters (V1-V2) which is manifestly gauge-independent.

> Couple of good examples based on familiar problems would be useful.

*** Example 2:  Consider the familiar voltmeter.  It has two leads, not
one!  This is important!  This is a fine example of gauge invariance.  As I
have said again and again:
        You have to measure voltage relative to something.
This "relativity" of voltage is one consequence of gauge invariance.

Demonstration:  Put together a circuit with several batteries and several
resistors.  Make the resistors big enough that they won't discharge the
batteries too rapidly;  things in the 50kOhm to 500kOhm range should be
fine.  For example (details don't matter):

           _____R1____________R2_________R3_____
          |             |           |           |
          b1            R4          R5          b3
          |_____R6______|_____b2____|____R7_____|
          |                                     |
          |                                     |
          |___________________R8________________|


This circuit has eight nodes.  Label them A through H.  Hook the black lead
of the voltmeter to node A and use the red lead to measure the voltages on
each of the eight nodes.  Write the results in a column labelled
ref_A.  Then move the black lead to node B and repeat the
measurements.  Write the results in a column labelled ref_B.

The numbers in the two columns will differ.  They will differ by a gauge
transformation.  But the resistors don't care.  The batteries don't
care.  You cannot build a circuit that cares about the gauge transformation.

So the physics of moving the black lead of the voltmeter (the voltage
reference) is one example of the physics of gauge transformation.

*** Example 3:  Gauge invariance is not limited to choosing a node of the
circuit.  Any voltage will do.  You can build a mutant voltmeter consisting
of a regular voltmeter plus an adjustable offset.  The offset can be
implemented as a potentiometer:

        *********************************************
        *                                           *
        *                                           *
        *      """""""""                            *
        *      "       "___________________________________
        *      "  Reg  "                            *
        *      "   VM  "____       _______          *
        *      "       "    |     R       |         *
        *      """""""""    |     R     -----       *
        *                   |     R      ---        *
        *                   |____>R     -----       *
        *                         R      ---        *
        *                         R     -----       *
        *                         R      ---        *
        *                         R_______|________________
        *                                           *
        *********************************************


Using this mutant voltmeter, you can change the gauge over some range by
moving the slider on the potentiometer.  This will affect the numbers
measured with this instrument -- but if you record an entire column of
numbers using some consistent slider-setting, that column of numbers will
be as good as any.  The elements of the circuit being measured don't care
about the gauge.  They only care about voltage differences.

You can repeat the measurement over and over, using a different gauge k(t)
in each column (t).   You will have no problems, provided gauge is kept
consistent within each column.

*** In all generality:
        del(phi) = del(phi + k(t))              (equation 2)
so the gauge k(t) cannot affect the observable electric fields, and
        del^2(phi) = del^2(phi + k(t))          (equation 3)
so the gauge cannot affect the observable charges.

Here k(t) can be any function depending on time but not depending on location.

> I have no trouble in accepting the "charge invariance"
> (the total net charge of the universe remains zero) because charge
> is a familiar physical quantity.

Good.  Charge is observable.  Gauge, by itself, is not.

BTW it should be called charge _conservation_.  Invariance is not
synonymous with conservation.  There is a deep principle of physics that
says conservation of _one_ thing is associated with invariance of some
_other_  thing.  This is called Nöther's theorem.
  http://www.emmynoether.com/noeth.htm
  http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Emmy.html
  http://www.scidiv.bcc.ctc.edu/Math/Noether.html

For instance, given two reference frames that differ only in the position
of the origin of coordinates, the laws of physics are the same in the two
frames.  This invariance w.r.t position is associated with conservation of
momentum.  But this is a tangent we need not pursue at the moment.

But before we leave this topic, I'll give you three guesses:  Fill in the
blank:
        Conservation of charge is associated with __________ invariance.

> But how can I accept the "gauge invariance" without knowing
> how to expressed the gauge numerically. We are dealing with
> a classical physics problem and I would like to know how "the
> amount of gauge ?" is defined in classical physics. Is this a
> dimensionless quantity? If not then what is its unit?

It has units of volts, just like electrostatic potential;  see e.g.
equation (2) above.