<p> --------------<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.</p> <p><strong><font color="#ff0000"><font color="#ff0000">But, the terms grad V, d A / d t and curl A </font>do not appear in an rms emittance calculation</font></strong>, 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</p> <p>So, it appears to me that the differences in <strong><font color="#ff0000">2nd moments of these quantities,</font></strong> <strong><font color="#3333ff">which form the rms emittance</font></strong>, <strong><font color="#ff0000">do not result in the kind of cancellation associated with gauge invariance.</font></strong></p> <div>If so, it becomes rather questionable what is the physical significance of rms emittance when electromagnetic fields are present (as in any particle accelerator).</div> <div>--------------</div> <div> </div> <div> </div> <div>p_mech = p_can - q A is <strong><em><font color="#ff0000" face="arial,helvetica,sans-serif"><u>gauge invariant</u></font></em></strong></div> <div> </div> <div>A --> A + grad f, then p_can_new ---> p_can_old <strong><font color="#ff0000">+ q grad f</font></strong> </div> <div> </div> <div>Consider linear uncoupled case, betatron motion ~ "Courant-Snyder ellipse".</div> <div>rms emittance = <x^2> <p^2> - < xp >^2</div> <div> </div> <div>This is the area of a non-upright ellipse <font color="#ff0000"><strong><em><u>centered at the origin.</u></em></strong> </font></div> <div>This formula explicitly assumes <strong><font color="#ff0000"><x> = 0, <p> = 0.</font></strong></div> <div> </div> <div>The emittance is a <strong><em><font color="#ff0000">variance</font></em></strong>. If the beam centroid is not zero, then</div> <div> <div>rms emittance = <(x - <x>)^2> <(p - <p>)^2> - < (x - <x>)(p - <p>) >^2</div> <div> </div></div> <div>x_new = x_old</div> <div>p_new = p_old + q grad f</div> <div> <div><p_new> = <p_old> + q grad f</div></div> <div><strong><font color="#ff0000"> <div><strong><font color="#ff0000"> <div><strong><font color="#ff0000">p_new - <p_new> = p_old - <p_old></font></strong></div></font></strong></div></font></strong> </div> <div><font color="#3333ff"><font size="4" face="times new roman,serif"><strong><em>emittance_new = emittance_old</em></strong></font> </font></div> <div> <div> <div> </div> <div><em>et voila!</em></div> <div> </div> <div>Dragt has explained that in general the emittances <em>must be calculated with respect to the normal modes (for linear dynamics)</em></div> <div>or more generally with respect to the appropriate phase-space hyperplanes <strong><font color="#ff0000">(eigen-emittances)</font></strong>.</div> <div>This includes linear transverse x-y coupling and radial-longitudinal coupling.</div> <div>The quoted statement ~ "2nd moments" makes numerous simplifying assumptions.</div></div></div>