[MAP] Liouville's theorem and electromagnetic fields

[Alexey Burov]

Kirk,

  they are not invariant. To avoid long discussion here, please have a 
look at Lebedev-Bogacz paper:
http://iopscience.iop.org/1748-0221/5/10/P10010/pdf/1748-0221_5_10_P10010.pdf, 

the very end of it, pp. 21-23. You see that the 2 emittances of e-beam 
born at the magnetized cathode, \epsilon_1 and \epsilon_2 may differ by 
orders of magnitude. This is actual case for e-beam of our e-cooler.

Alexey.

On 3/10/2011 4:42 PM, Kirk T McDonald wrote:
> Alexey,
> For the subspace (q,p) we have
> dq’ dp’ = J dq dp
> J = | dq’/dq  dq’/dp |
>       | dp’/dq  dp’/dp |
> Suppose p = m v + A   (in units where e/c = 1)
> and we transform
> q’ = q
> p’ = mv = p – A(q)
> Then the Jacobian is
> J = |       1     0 |
>       | –dA/dq  1 | = 1
> It looks to me like the partial phase volumes are also invariant under 
> the “transformation” of neglecting the vector potential.
> --Kirk
> *From:* Alexey Burov <mailto:burov at fnal.gov>
> *Sent:* Thursday, March 10, 2011 5:33 PM
> *To:* map-l at lists.bnl.gov <mailto:map-l at lists.bnl.gov>
> *Subject:* Re: [MAP] Liouville's theorem and electromagnetic fields
> One remark to Swann's paper:
> His theorem relates to the total emittance, not to the partial ones. 
> Partial emittances are sensitive to eA/c term.
>
> A possible way to get rid of eA/c inside solenoidal structures is to 
> make a fake 0-length edge of the solenoid at a place where emittances 
> are calculated; kicks from the edge solenoidal fields have to be taken 
> into account, of course.
>
> Alexey.
>
> On 3/10/2011 4:09 PM, Kirk T McDonald wrote:
>> 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 
>> <http://puhep1.princeton.edu/%7Emcdonald/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?
>>
>>
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