[MAP] Liouville's theorem and electromagnetic fields

Kirk T McDonald kirkmcd at Princeton.EDU
Thu Mar 10 17:09:30 EST 2011


Folks,

There is a technical question as to how we should be calculating emittance for beams in electromagnetic fields.

The formal theory of Liouville’s theorem is clear that the invariant volume in phase space is to be calculated with the canonical momentum
gamma m v + e A / c
and not the mechanical momentum m v.

This is awkward in two ways:
1.   We don’t always know the vector potential of our fields
2.   The vector potential is subject to gauge transformations, so canonical momentum is not gauge invariant.

The second issue is disconcerting in that it suggests that phase-space volume, and emittance, are not actually invariant  -- with respect to gauge transformations.

Hence, it is useful to note a very old paper,
W.F.G. Swann, Phys. Rev. 44, 233 (1933)
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”.

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.

---------------
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.

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.

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.....

--Kirk

PS  I have placed Swann’s paper as DocDB 560
http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase
user = ionization pass = mucollider1

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.

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).

Is there any written reference that explains this “well known” fact?

How is this prescription affected by electromagnetic fields?

The vector potential of even a simple rf accelerating cavity has an A_z component (which is zero on axis, but nonzero off it).
http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf
Note that the vector potential is nonzero outside the cavity, even though the E and B fields are zero there!

Do we know how to include A_z in our longitudinal emittance calculations?
-------------- next part --------------
An HTML attachment was scrubbed...
URL: https://lists.bnl.gov/mailman/private/map-l/attachments/20110310/a6bf31b9/attachment.html 


More information about the MAP-l mailing list