<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div></div><div><br></div><div><br></div><div>I hope these comments answer some of your questions.</div><div><br></div><div>Best regards,</div><div><br></div><div>Alex Dragt</div><div><br></div><div>***************************************************</div><div><br><div><div>On Mar 14, 2011, at 12:33 PM, Kirk T McDonald wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div>Alex,<br><br>Phase volume is independent of gauge, because a gauge transformation is a canonical transformation.<br><br>This does NOT immediately imply that rms emittance is gauge invariant --  since rms emittance is (unfortunately) not an exact measure of phase volume.<br><br>Did Alex Dragt really claim that rms emittance is gauge invariant?<br><br>--------------<br>A gauge transformation<br>A -> A + grad f<br>V -> V + d f / d t<br>leaves the fields<br>E = - grad V - d A / dt<br>B = curl A<br>unchanged.<br><br>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<br>x<br>y<br>t<br>p_x = p_mech_x + q A_x<br>p_y = p_mech_y + q A_y<br>p_t = E_mech + q V<br><br>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.<br><br>If so, it becomes rather questionable what is the physical significance of rms emittance when electromagnetic fields are present (as in any particle accelerator).<br><br>--Kirk<br><br><br><br><br>-----Original Message----- From: Alexey Burov<br>Sent: Monday, March 14, 2011 8:00 AM<br>To: <a href="mailto:map-l@lists.bnl.gov">map-l@lists.bnl.gov</a><br>Subject: Re: [MAP] Liouville's theorem, kinematic invariants,and dynamic invariants<br><br>It was already mentioned by Alex Dragt that emiitances are independent<br>on the gauge transformations, since they are canonical.<br><br>On 3/13/11 10:57 PM, Valeri Lebedev wrote:<br><blockquote type="cite">Dear All,<br></blockquote><blockquote type="cite">I was impressed with intensity of the discussion and a large number of e-mails and would like to add a few more words.<br></blockquote><blockquote type="cite">1. First, there is no uncertainty with choice of the vector potential in the real applications. One has to keep in mind that the reason we would like to know the emittances is that we want to use this beam in a collider and we need to know the emittances and Twiss parameters of the beam out-coming the cooling section. That means that the computed emittances have to coincide with usual emittances in the regions where magnetic field is zero. For obvious reason the vector potential has to be equal to zero in these regions and uncertainty disappears.<br></blockquote><blockquote type="cite">2. For some reason a necessity to know the Twiss parameters of out-coming beam was not discussed, but, I would like to note, that the knowledge of Twiss parameters is the same important as knowledge of emittances if one wants to prevent the emittance growth in the course of beam transfer to the collider and to minimize required apertures and, consequently, non-linearities in the course of beam transport and acceleration from cooling section to the collider.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">These problems are addressed in my and Alex Bogacz paper and I cannot agree that it is too complicated to be understood by a general folks. As far as I understand all problems are addressed there. Otherwise we do not have a correct language to discuss cooling.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Valeri<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">_______________________________________________<br></blockquote><blockquote type="cite">MAP-l mailing list<br></blockquote><blockquote type="cite"><a href="mailto:MAP-l@lists.bnl.gov">MAP-l@lists.bnl.gov</a><br></blockquote><blockquote type="cite"><a href="https://lists.bnl.gov/mailman/listinfo/map-l">https://lists.bnl.gov/mailman/listinfo/map-l</a><br></blockquote>_______________________________________________<br>MAP-l mailing list<br><a href="mailto:MAP-l@lists.bnl.gov">MAP-l@lists.bnl.gov</a><br>https://lists.bnl.gov/mailman/listinfo/map-l <br></div></blockquote></div><br></div></body></html>