[MAP] Liouville's theorem and electromagnetic fields

[Kirk T McDonald]

Rob,

1.  "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)."

This kind of thinking is what follows from emphasis on the canonical/symplectic point of view.

The astonishing result of Swann (now 78 years old) is that if you are willing to "think outside this box", you will find that phase volume has nothing to do with the vector potential.

2.  It sounds like you agree with Scott Berg that it's the right thing to use (E,t) rather than (p_z,z) as "canonical" longitudinal coordinates when sampling at fixed z rather than fixed t.

Nonetheless, it would be a service to mankind if this insight could be documented in a manner that ordinary mortals can understand.

I note that you evade the important question of how this works in the presence of electromagnetic fields.

--Kirk


From: Robert D Ryne 
Sent: Thursday, March 10, 2011 7:33 PM
To: Yuri Alexahin ; Kirk T McDonald 
Cc: MAP List ; alex dragt ; Alex Dragt ; Alex Dragt 
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields


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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