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<DIV>Folks,</DIV>
<DIV> </DIV>
<DIV>There is a technical question as to how we should be calculating emittance
for beams in electromagnetic fields.</DIV>
<DIV> </DIV>
<DIV>The formal theory of Liouville’s theorem is clear that the invariant volume
in phase space is to be calculated with the canonical momentum</DIV>
<DIV>gamma m v + e A / c</DIV>
<DIV>and not the mechanical momentum m v.</DIV>
<DIV> </DIV>
<DIV>This is awkward in two ways:</DIV>
<DIV>1. We don’t always know the vector potential of our
fields</DIV>
<DIV>2. The vector potential is subject to gauge transformations, so
canonical momentum is not gauge invariant.</DIV>
<DIV> </DIV>
<DIV>The second issue is disconcerting in that it suggests that phase-space
volume, and emittance, are not actually invariant -- with respect to gauge
transformations.</DIV>
<DIV> </DIV>
<DIV>Hence, it is useful to note a very old paper,</DIV>
<DIV>W.F.G. Swann, Phys. Rev. 44, 233 (1933)</DIV>
<DIV>which shows that the phase-space volume for a set of noninteracting
particles is the same whether or not the term e A / c is included in the
“momentum”.</DIV>
<DIV> </DIV>
<DIV>This result has the consequence that phase-space volume (and emittance) is
actually gauge invariant – although the location of a volume element in space
space is gauge dependent.</DIV>
<DIV> </DIV>
<DIV>---------------</DIV>
<DIV>This suggests that we could simply calculate emittances based only on the
mechanical momentum, and avoid having to worry about the accuracy of our model
for the vector potential.</DIV>
<DIV> </DIV>
<DIV>Of course, our calculations are actually of rms emittance, which is a
better representation of the “ideal” emittance if the phase-space volume is more
“spherical”, and not elongated/twisted.</DIV>
<DIV> </DIV>
<DIV>It could be that the shape of the phase-space volume is better for rms
emittance calculation if the vector potential, in some favored gauge, is
included in the calculation.....</DIV>
<DIV> </DIV>
<DIV>--Kirk</DIV>
<DIV> </DIV>
<DIV>PS I have placed Swann’s paper as DocDB 560</DIV>
<DIV><A title=http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase
href="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase">http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase</A></DIV>
<DIV>user = ionization pass = mucollider1</DIV>
<DIV> </DIV>
<DIV>See also the paper by Lemaitre that used Liouville’s theorem for cosmic
rays in the Earth’s atmosphere (using mechanical momentum). This may
well be the earliest paper about particle beams and Liouville’s theorem.</DIV>
<DIV> </DIV>
<DIV>PPS Scott Berg notes that when one evaluates emittance at a fixed
plane in space, rather than at a fixed time, it is better to use the
“longitudinal” coordinates (E,t) rather than (P_z,z).</DIV>
<DIV> </DIV>
<DIV>Is there any written reference that explains this “well known” fact?</DIV>
<DIV> </DIV>
<DIV>How is this prescription affected by electromagnetic fields?</DIV>
<DIV> </DIV>
<DIV>The vector potential of even a simple rf accelerating cavity has an A_z
component (which is zero on axis, but nonzero off it).</DIV>
<DIV><A title=http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf
href="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf">http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf</A></DIV>
<DIV>Note that the vector potential is nonzero outside the cavity, even though
the E and B fields are zero there!</DIV>
<DIV> </DIV>
<DIV>Do we know how to include A_z in our longitudinal emittance
calculations?</DIV></DIV></DIV></BODY></HTML>