[MAP] Hamiltonian-gauge potentials

Kirk T McDonald kirkmcd at Princeton.EDU
Thu Mar 17 18:24:49 EDT 2011


Folks

I believe that what Alex Dragt has been trying to tell us is that for applications involving Hamiltonian dynamics it is most favored to use the so-called Hamiltonian gauge, in which the scalar potential V is zero everywhere.   Further, this ties in nicely with the use of coordinates (x,y,t) rather than (x,y,z).

For a general expression for the vector potential A in this gauge, see sec. 8 of
http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf

For fields with time dependence e^{- i omega t), with nonzero wave number k = omega / c, then the vector potential is simply
A = – i E / k.   (Gaussian units).
Note that A = 0 wherever E = 0, in which regions B can only be static.

For a static magnetic field, the vector potential is the same as that in the Coulomb gauge (and in the Lorenz gauge).  [In some static cases, such a toroidal magnets, the static vector potential will be nonzero in regions where B is zero; but in regions where a Fourier component B_omega is zero for nonzero omega, A_omega will be zero also.]

A static electric field has the vector potential 
A = – c(t – t_0) E.
This seems a bit odd, but will not bother us.

An unusual feature of the Hamiltonian gauge vector potential is that is is not continuous at a perfectly conducting surface – which did bother me for quite a while.

However, in using A to compute B, the (ill-defined) normal derivative at a perfectly conducting surface is not needed to deduce the tangential B.   

Of course, the Hamiltonian gauge vector potential makes the canonical momentum p = p_mech + q A / c
discontinuous at the surface of the rf cavity.

This is troublesome if we use coordinates (x,y,z) and have an accelerating cavity with E_z, such that p_z takes a step on entering or leaving the cavity.

However, if we switch to coordinates (x,y,t) the t-canonical momentum is
p_t = – E_mech – q V = – E_mech
in the Hamiltonian gauge, which is continuous at the cavity wall.
Also, canonical momenta p_x and p_y are continuous if the particles enter and leave the cavity through faces at constant z (of a good conducting material).

All this is serendipitous for the cooling sections of a muon collider, where we are almost certain to use cavities with flat (Be) faces where the particles enter and exit.

It remains that rms emittance (and eigen-emittance, I believe) is not gauge invariant.   But we have to choose some gauge to proceed.

It now looks like the Hamiltonian gauge is the one to use, along with coordinates (x,y,t) rather than (x,y,z).

1. Time-dependent A_omega follows immediately from knowledge of E_omega, and is zero where E_omega and B_omega are zero.

2.  Static A is same as in the Coulomb gauge (which is what we almost always use now).

3.  No steps in canonical momenta so long as all rf cavities have flat faces where particles enter/exit.

4.  No scalar potential to worry about, so p_t = – E_mech, as in our present software. (It doesn’t hurt anything to define p_t = + E_mech, as we actually do.)

---------------------------------------------------------
5.  Although phase volume is the same where or not we include the potentials in the momenta, this is not true for rms emittance (or eigen-emittance).   So we should stop using ECAL9, and switch to emitcalc (with a better approximation to the vector potential) and/or equivalent programs developed by Tom Roberts, Chris Rogers, et al.

--Kirk
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