[MAP] Liouville's theorem and electromagnetic fields

[Robert D Ryne]

I have not yet read the papers mentioned. But here are some brief  
comments. Alex Dragt and I (cc to Alex) have been thinking about this  
a lot in the past months.

The natural quantities to be computed are called "eigen-emittances."
To compute them properly they need to be derived from a beam 2nd  
moment matrix, Sigma, formed using canonical variables.
The eigen-emittances are invariant under linear symplectic  
transformations.

The eigen-emittances can be computed in various ways, but the simplest  
is to compute the eigen-values of J Sigma, where J is the fundamental  
symplectic 2-form; the eigen-emittances are the modulii of the eigen- 
values of J Sigma (which are pure imaginary and in +/- pairs). If one  
is interested in calculating the symplectic matrix that transforms  
Sigma to Williamson normal form, Alex Dragt has an algorithm to do  
this and has implemented it in the MaryLie code.

Though the entries of Sigma will depend on the choice of gauge, the  
eigen-emittances themselves are gauge invariant. We can't just set the  
vector potential to zero inside elements where it is nonzero, and  
expect to calculate the correct eigen-emittances (as was suggested  
below).

>>>> 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?
>>>>


The above follows directly from whether we use the time t as the  
independent variable or the Cartesian coordinate z as the independent  
variable. When using the time, the longitudinal variables are  
(z,p_{z,canonical}). When using z, the longitudinal variables are (t, - 
E) where t is arrival time at location z, and where E is the total  
energy of a particle when it reaches location z, i.e. E=\gamma m c^2 +  
q \Phi.

Rob

On Mar 10, 2011, at 4:29 PM, Yuri Alexahin wrote:

> Hi Kirk,
>
> Thank you for digging out these interesting papers.
> Of course the Poincare invariants remain the same no matter what  
> momenta are used.
> But this is not what we calculate from tracking or measurement data  
> using standard definition.
> So a clarification is still needed of what and how we should  
> calculate.
>
> Yuri
>
> ----- Original Message -----
> From: Kirk T McDonald <kirkmcd at Princeton.EDU>
> Date: Thursday, March 10, 2011 4:09 pm
> Subject: [MAP] Liouville's theorem and electromagnetic fields
> To: MAP List <map-l at lists.bnl.gov>
> Cc: Kirk McDonald <kirkmcd at Princeton.EDU>
>
>
>> 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?
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