[MAP] Correction (Re: Hamiltonian-gauge potentials

[Robert D Ryne]

What I wrote below is not correct. As Kirk just pointed out to me, and  
Alex wrote earlier, it is true only in the linear approximation.
As Alex wrote earlier,
> But the eigen emittances are invariant under linear symplectic  
> transformations, and therefore also under gauge transformations in  
> the linear approximation.


Rob


On Mar 17, 2011, at 4:29 PM, Robert D Ryne wrote:

> 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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>> MAP-l at lists.bnl.gov
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>

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