<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">Kirk,<div><br></div><div><blockquote type="cite">On Mar 17, 2011, at 3:24 PM, Kirk T McDonald wrote:</blockquote><div><blockquote type="cite"><div dir="ltr" bgcolor="#66ffff" text="#000000"><div dir="ltr"><div style="font-family: Arial; color: rgb(0, 0, 0); font-size: 10pt; "><div>It remains that rms emittance (and eigen-emittance, I believe) is not gauge invariant.   But we have to choose some gauge to proceed.</div></div></div></div></blockquote></div><div><br></div><div>The eigen-emittances <i>are</i> gauge invariant.</div><div><br></div><div>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 <i>canonical</i> variables. Some elements of the 6-vector describing each particle <i>will</i> 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 <i>will</i> 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 <i>independent</i> of the choice of gauge.</div><div><br></div><div>Rob</div><div><br></div><div><br><div><div>On Mar 17, 2011, at 3:24 PM, Kirk T McDonald wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div dir="ltr" bgcolor="#66ffff" text="#000000"> <div dir="ltr"> <div style="FONT-FAMILY: 'Arial'; COLOR: #000000; FONT-SIZE: 10pt"> <div>Folks</div> <div> </div> <div>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).</div> <div> </div> <div>For a general expression for the vector potential A in this gauge, see sec. 8 of</div> <div><a title="http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf" href="http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf">http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf</a></div> <div> </div> <div>For fields with time dependence e^{- i omega t), with nonzero wave number k = omega / c, then the vector potential is simply</div> <div>A = – i E / k.   (Gaussian units).</div> <div>Note that A = 0 wherever E = 0, in which regions B can only be static.</div> <div> </div> <div>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.]</div> <div> </div> <div>A static electric field has the vector potential </div> <div>A = – c(t – t_0) E.</div> <div>This seems a bit odd, but will not bother us.</div> <div> </div> <div>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.</div> <div> </div> <div>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.   </div> <div> </div> <div>Of course, the Hamiltonian gauge vector potential makes the canonical momentum p = p_mech + q A / c</div> <div>discontinuous at the surface of the rf cavity.</div> <div> </div> <div>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.</div> <div> </div> <div>However, if we switch to coordinates (x,y,t) the t-canonical momentum is</div> <div>p_t = – E_mech – q V = – E_mech</div> <div>in the Hamiltonian gauge, which is continuous at the cavity wall.</div> <div>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).</div> <div> </div> <div>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.</div> <div> </div> <div>It remains that rms emittance (and eigen-emittance, I believe) is not gauge invariant.   But we have to choose some gauge to proceed.</div> <div> </div> <div>It now looks like the Hamiltonian gauge is the one to use, along with coordinates (x,y,t) rather than (x,y,z).</div> <div> </div> <div>1. Time-dependent A_omega follows immediately from knowledge of E_omega, and is zero where E_omega and B_omega are zero.</div> <div> </div> <div>2.  Static A is same as in the Coulomb gauge (which is what we almost always use now).</div> <div> </div> <div>3.  No steps in canonical momenta so long as all rf cavities have flat faces where particles enter/exit.</div> <div> </div> <div>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.)</div> <div> </div> <div>---------------------------------------------------------</div> <div>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.</div> <div> </div> <div>--Kirk</div></div></div></div> _______________________________________________<br>MAP-l mailing list<br><a href="mailto:MAP-l@lists.bnl.gov">MAP-l@lists.bnl.gov</a><br>https://lists.bnl.gov/mailman/listinfo/map-l<br></blockquote></div><br></div></div></body></html>