[MAP] Hamiltonian-gauge potentials

[Robert D Ryne]

Kirk,

> On Mar 17, 2011, at 3:24 PM, Kirk T McDonald wrote:
> It remains that rms emittance (and eigen-emittance, I believe) is  
> not gauge invariant.   But we have to choose some gauge to proceed.


The eigen-emittances are gauge invariant.

More precisely, consider a numerical distribution of particles in the  
presence of electromagnetic fields. These can be defined, e.g., using  
Cartesian coordinates and velocities at some time, or using transverse  
coordinates and velocities along with energy and arrival time at some  
location z, etc. Next, express the numerical distribution using  
canonical variables. Some elements of the 6-vector describing each  
particle will depend on your choice of gauge (e.g. p_{x,canonical} and  
p_{y,canonical}). Next, compute the beam second moment matrix \Sigma.  
Some of the matrix elements will depend on your choice of gauge. Next,  
compute the moduli of the eigenvalues of J \Sigma (these are the eigen- 
emittances). Though the numerical distribution (the collection of 6- 
vectors) and the \Sigma matrix will depend on the gauge, the eigen- 
emittances will be independent of the choice of gauge.

Rob


On Mar 17, 2011, at 3:24 PM, Kirk T McDonald wrote:

> 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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