[MAP] Is rms emittance gauge invariant?

[Sateesh Mane]

 --------------
A gauge transformation
A -> A + grad f
V -> V + d f / d t
leaves the fields
E = - grad V - d A / dt
B = curl A
unchanged.

*But, the terms grad V, d A / d t and curl A do not appear in an rms
emittance calculation*, which involves A and V (in case we use coordinates
x
y
t
p_x = p_mech_x + q A_x
p_y = p_mech_y + q A_y
p_t = E_mech + q V

So, it appears to me that the differences in *2nd moments of these
quantities,* *which form the rms emittance*, *do not result in the kind of
cancellation associated with gauge invariance.*
If so, it becomes rather questionable what is the physical significance of
rms emittance when electromagnetic fields are present (as in any particle
accelerator).
--------------


p_mech = p_can - q A   is *gauge invariant*

A --> A + grad f, then p_can_new ---> p_can_old *+ q grad f*

Consider linear uncoupled case, betatron motion ~ "Courant-Snyder ellipse".
rms emittance = <x^2> <p^2> - < xp >^2

This is the area of a non-upright ellipse *centered at the origin.*
This formula explicitly assumes  *<x> = 0,  <p> = 0.*

The emittance is a *variance*.  If the beam centroid is not zero, then
 rms emittance = <(x - <x>)^2> <(p - <p>)^2> - < (x - <x>)(p - <p>) >^2

x_new = x_old
p_new = p_old + q grad f
 <p_new> = <p_old> + q grad f
*
p_new - <p_new> = p_old - <p_old>
*
*emittance_new = emittance_old*

*et voila!*

Dragt has explained that in general the emittances *must be calculated with
respect to the normal modes (for linear dynamics)*
or more generally with respect to the appropriate phase-space hyperplanes *
(eigen-emittances)*.
This includes linear transverse x-y coupling and radial-longitudinal
coupling.
The quoted statement ~ "2nd moments" makes numerous simplifying assumptions.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: https://lists.bnl.gov/mailman/private/map-l/attachments/20110315/6623b046/attachment.html