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[SOLVED] Re: Kaluza-Klein help needed -- non-Abelian generalization |
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[SOLVED] Re: Kaluza-Klein help needed -- non-Abelian generalization
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>markwh04@yahoo.com wrote:\n> John Baez wrote:\n> > h_{ij} = g_{ij}\n> > h_{i5} = A_i\n> > h_{55} = 1\n>\n> > HOWEVER, I don\'t find that the velocity of the particle around\n> > the circle is constant!\n>\n> You got stumped by this problem?\n>\n> That\'s easy. You\'ve got the wrong metric. It\'s\n>\n> h_{ij} = g_{ij} + K A_i A_j\n> h_{i5} = h_{5i} = K A_i\n> h_{55} = K\n>\n> The Lagrangian may be written (using the summation convention) as:\n> L = 1/2 (g_{ij} u^i u^j) + K/2 (A_i u^i + u^5)^2\n> The conjugate momenta are:\n[etc]\n\nThis can be worked out in a transparent fashion (at least when not\nreduced to ASCII form!) in the more general case, within the principal\nbundle formalism, using the *left-quotient* operator. The derivative of\nthe quotient completely captures in a much more general and clearer\ncontext the notion of a connection, making everything easier to work\nwith.\n\nSo, some background and notation first. Let P be acted on by a group\nG. This gives you a product operation with signature\nP.G -> P\nand properties\npe = p (e = group identity);\np(gh) = (pg)h,\nwith cosets\npG = { pg: g in G }\n\nOne can then define the set\np\\q = { g in G: pg = q }.\nThe stablizer of point p is just p\\p. The group acts transitively on P\nif all the p\\q are non-empty. It acts freely if p\\q has at most one\nmember -- in which case one may consider the quotient as a partial\noperation\np\\q: defined for all q in pG\nwith the properties:\np p\\q = q\np\\p = e\n(p\\q)^{-1} = q\\p\np\\q q\\r = p\\q\n(pg)\\(qh) = g^{-1} p\\q h.\n\nWhen these spaces are also manifolds, one can consider the actions of\nthese operators under derivatives. Recalling that the key property of\ntangent vectors on a manifold M is that for a curve r(t)\nr\'(t) is in T_{r(t)}(M),\nwe\'ll use the notation denoting by m\' (with subscripts) a general\nelement of T_m(M).\n\nWith the coset notation, a local SECTION on the space P is defined as a\nmap\ns: P/G = M -> P\nsuch that\ns(m)G = m.\nA principal bundle is thus a manifold P acted on by a symmetry group G\nin such a way that its cosets M = P/G can be arranged into a manifold,\nwhich can be coordinatized by differentiable local sections whose\ndomains cover M.\n\nIn the following, for a function f: M -> N, I\'ll use the notation Df:\nTM -> TN to denote its deriative. So, for f: R^m -> R^n, Df is just\nthe mxn Jacobian matrix of f. Its product with a vector v in TM is\ndenoted Df[v] in TN.\n\nThe product P.G -> P, naturally induces one with signatures:\np.T_g(G), T_p(P).g -> T_{pq}(G)\nwith the property\n(pg)\' = p\'g + pg\'.\n\nDifferentiating the relation,\ns(m)gG = s(m)G = m\none finds:\nds_m[m\']G = ds_m[m\']gG + s(m)g\'G = ds_m[m\']G = m\'.\nThus\nds_m[m\']G = m\'; s(m)g\' = 0.\n\nThe corresponding action for quotients\np\\T_q(P), T_p(P)\\q -> T_{p\\q}(G)\nis not given at the outset, but must be postulated; and is assumed to\nhave the property\n(p\\q)\' = p\'\\q + p\\q\'.\nThe connection one-form is just\nomega_p[p\'] = p\\p\'.\nThe horizontal lift of a vector m\', m = pG, to a point p is\nL_p[m\'] = s(m) ds_m[m\']\\p + ds_m[m\'] s(m)\\p\nor more briefly\nL = s ds\\p + ds s\\p\nThe section-relative connection is\nB = s\\ds\nor more precisely:\nB_m[m\'] = s(m)\\ds_m[m\'].\n\nIts action under a global transformation s -> sg is\nB -> (sg)\\d(sg) = g^{-1} s\\ds g = g^{-1} B g\nand its action under a local transformation\ns(m) -> s(m) x(m)\nis\nB -> (sx)\\d(sx) = x^{-1} s\\(ds x + s x\')\n= x^{-1} s\\ds x + x^{-1} x\'\n= x^{-1} B x + x^{-1} x\'.\n\nIn a similar way, one may define natural actions of the group on the\ncotangent space through the properties:\ng.T*_p(P).h -> T*_{gph}(P)\n(gwh)[p\'] = w[g^{-1} p\' h^{-1}]\nas well as a quotient\np\\T*_q(P) -> T*_{p\\q}(P)\np\\w[q\'] = w[p q\'].\nThis time, it\'s the product that requires the additional structure of\nthe connection, with\np.T*_g(G) -> T*_{pg}(P)\np.a[q\'] = a[p\\q\'].\n\n(One can also define operations for the tangent and cotangent spaces:\nT_p(P)\\T*_q(P) -> R\np\'\\w = w[p\']\n).\n\nSo with the background out of the way...\n\nGiven a metric k on G that is invariant under left and right products\nk_{hgf}(h g\'_1 f, h g\'_2 f) = k_g(g\'_1, g\'_2)\nand a metric g on the base space M, define the bundle metric on P by\nh = g + k(omega, omega)\nor more precisely:\nh_p(p\'_1, p\'_2) = g_{pG}(p\'_1G, p\'_2G) + k_{pG}(p\\p\'_1, p\\p\'_2).\nIn the general case, the scaling of k may be positionally dependent on\nthe points in the base space M.\n\nGiven a section s, one may decompose P and TP as follows:\np = s(m) g; where m = pG; g = s(pG)\\p\nwith\np\' = ds[m\'] g + s g\'.\nOne finds that\np\'G = ds[m\']gG + s g\'G = m\' + 0 = m\',\nand\np\\p\' = g^{-1} s\\(ds g + s g\') = g^{-1} B_m[m\'] g + g^{-1} g\'\nor\np\\p\' = g^{-1} u g\nintroducing the horizontal velocity\nu = B_m[m\'] + v\nand gauge velocity\nv = g\' g^{-1}.\n\nIn terms of the decomposition, the metric can be written as:\nh_p(p\'_1, p\'_2) = g_m(m\'_1, m\'_2) + k_m(u\'_1, u\'_2).\n\nIn component form, this becomes:\nh_{AB}p\'_1^A p\'_2^B = g_{mn} m\'_1^m m\'_2^n + k_{ab} u\'_1^a u\'_2^b.\nusing the summation convention on the indices.\n\nThe components of the total metric are thus:\nh_{mn} = g_{mn} + k_{ab} B^a_m B^b_n\nh_{mb} = k_{ab} B^a_m\nh_{an} = k_{ab} B^b_n\nh_{ab} = k_{ab}.\n\nSo, from this you can readily find the geodesic equations of motion.\nThese arise from the Lagrangian:\nL(p,p\') = 1/2 h_p(p\',p\').\n\nVariation of L gives you the desired results.\n\nWorking within the decomposition, p = s(m) g as above, define the\nvariation of g by\nDelta(g) = D g.\nthen\nDelta(g^{-1}) = -g^{-1} D.\nand the variation of v will be:\nDelta(v) = Delta(g\' g^{-1})\n= (Delta g)\' g^{-1} - g\' g^{-1} D\n= (Dg)\' g^{-1} - v D\n= D\' g g^{-1} + D v - v D\n= D\' + [D,v].\n\nThe variations with respect to D give you the charge and precession\nequation (Wong\'s equation) for the charge. The variation with respect\nto m\' gives you the ordinary momentum and the Lorentz force law coupled\nwith the charge.\n\nFirst, employing the derivative notation alluded to before:\nDelta(u) = Delta(B_m[m\'] + v)\n= DB_m[Delta(m)][m\'] + B_m[Delta(m\')] + D\' + [D,v].\nThus\nDelta(L) = Delta(1/2 g_m(m\',m\') + 1/2 k_m(u,u))\n= 1/2 Dg_m[Delta(m)](m\',m\') + 1/2 Dk_m[Delta(m)](u,u)\n+ g_m(m\',Delta(m\')) + k_m(u,Delta(u)).\n\nPicking out the variation with respect to D\', one finds the\ncorresponding momentum\np_D = k_m(u, ())\nwhich is just u with its (Lie) index lowered.\n\nPicking out the variation with respect to D, one finds the force term\nF_D:\nF_D = k_m(u, [(),v]) = p_D[[(),v]].\n\nSince p_D is a cotangent vector in T*G, it\'s natural to extend the Lie\nbracket to the cotangent space. So, let\'s consider first how the Lie\nbracket is defined. For a Lie vector e\' in the tangent space T_e(G) =\nL, one has\ng e\' g^{-1} in T_{g e g^{-1})(G) in L.\nTherefore, L is closed under the adjoint operator\nad_g(e\') = g e\' g^{-1},\nand derivatives of this operator make sense. The Lie bracket is then\njust\n[g\',e\'] = (g e\' g^{-1})\'.\nExplicitly:\n[g\'(s), e\'(t)] = d^2(g(s) e(t) g(s)^{-1})/ds dt.\n\nSo, over the cotangent space T_e*(G) = L*, one has the coadjoint action\ndefined in the same way\nco_g(w) = g w g^{-1}\nand Lie bracket\n[g\', w] = (g w g^{-1})\'.\nApplying this to a Lie vector e\', one finds\n[e_1\',w][e_2\'] = (e_1 w e_1^{-1})\'[e_2\']\n= (e_1 w e_1^{-1}[e_2\'])\'\n= (w[e_1^{-1} e_2\' e_1])\'\n= w[(e_1^{-1} e_2\' e_1)\'].\nSince\ne_1^{-1}\'(t) = -e_1^{-1}(t) e_1\'(t) e_1^{-1}(t)\nwhich evaluated at the given t where e_1(t) = e is just\ne_1^{-1}\'(t) = -e_1\'(t),\nthen the Lie bracket for e_1^{-1} is the negative of that for e_1.\nThus\n[e_1\',w][e_2\'] = -w[[e_1\',e_2\']]\n= w[[e_2\',e_1\']].\nOr, in more prosaic notation:\n[a,w][b] = w[[b,a]], for lie vectors a, b in L.\n\n\nSince the metric k is assumed to be invariant under action of G to the\nleft and right, one has:\nd/ds (k_{m}(g(s) h_1\' g(s)^{-1}, g(s) h_2\' g(s)^{-1}))\n= d/ds (k_h(h_1\', h_2\') = 0.\nApplying this to the individual vectors using the Leibnitz property,\none gets\nd/ds (k(g h_1\' g^{-1}, g h_2\' g^{-1}))\n= k(d(g h_1\' g^{-1})/ds, g h_2\' g^{-1})\n+ k(g h_1\' g^{-1}, d(g h_2\' g^{-1})/ds)\n= k([g\', h_1\'], h_2\') + k(h_1\', [g\', h_2\']).\nThus for Lie vectors, a, b, c in L:\nk([a,b],c) + k(b,[a,c]) = 0.\nIn particular, since the metric is symmetric, then for b = c,\nk([a,b],b) = 0.\nTherefore, the force can be rewritten as:\nF_D = k_m(u, [(),v])\n= k_m(u, [(),u - B])\n= k_m(u, [(),u]) - k_m(u, [(),B])\n= -k_m(u, [(),B])\n= -p_D([(),B])\n= -[B,p_D] = [p_D, B].\n\nThe Wong equation for the precession of the charge is thus\nd(p_D)/dt = [p_D, B[dm/dt]].\n\nThe variation with respect to m\' yields the momentum\np = g_m(m\', ()) + k_m(u, B_m())\n= g_m(m\', ()) + p_D(B_m())\n= g_m(m\', ()) + p_D.B_m.\nIn component form\np_m = g_{mn} m\'^n + p_D_a B^a_m,\nwhich is the usual expression for the canonical momentum in the\npresence of a Lorentz force given by the potential Lie vector B coupled\nto the charge Lie vector p_D.\n\nThe corresponding force is the variation with respect to m:\nF = 1/2 Dg_m[](m\',m\') + 1/2 Dk_m[](u,u) + k_m(u,DB_m()[m\']).\n= 1/2 Dg_m[](m\',m\') + 1/2 Dk_m[](u,u) + p_D(DB_m()[m\']).\n\nThe corresponding equation of motion is dp/ds = F. Differentiating p,\nwe find\ndp/ds = Dg_m[m\'](m\',()) + g_m(m\'\',()) + p_D\'[B_m()] + p_D[dB_m[m\'][]]\n= Dg_m[m\'](m\',()) + g_m(m\'\',()) + p_D[[B[m\'],B[]]] +\np_D[dB_m[m\'][]].\nThus\nDg_m[m\'](m\',()) + g_m(m\'\',()) + p_D[[B[m\'],B[]]] + p_D[dB_m[m\'][]]\n= 1/2 Dg_m[](m\',m\') + 1/2 Dk_m[](u,u) + p_D(DB_m[][m\']).\n\nDefining the Yang-Mills force\nG(v, w) = DB_m[v][w] - DB_m[w][v] + [B_m[v], B_m[w]]\nand Christoffel coefficients\nGamma(u,v,w) = 1/2 (Dg[u](v,w) + Dg[v](u,w) - Dg[w](u,v))\nthe equations of motion become\ng_m(m\'\', ()) + Gamma(m\', m\', ()) = G((), m\') + 1/2 Dk_m[](u,u).\n\nApplying this to the vector m\', and noting that\n(g(m\', m\'))\' = 2 g(m\'\', m\') + 2 Gamma(m\', m\', m\')\nwe get\n(g(m\', m\'))\' = Dk_m[m\'](u, u).\n\nIf the gauge metric has an inverse, k^{-1}, one may define the charge\nmagnitude\n|p_D|^2 = k^{-1}(p_D, p_D).\n= p_D[u]\n= k(u, u)\nand\nDk_m[m\'](u, u) = -D{k^{-1}_m)(p_D, p_D).\nAdjoint invariance applies to this metric as well\nk^{-1}([a,b], c) + k^{-1}(b, [a,c]) = 0.\nTherefore, under the derivative, its action becomes\n(|p_D|^2)\' = D(k^{-1})[m\'](p_D, p_D) + 2 k^{-1}(p_D, [p_D, B])\n= D(k^{-1})[m\'](p_D, p_D)\n= -Dk_m[m\'](u, u).\n\nThus, one arrives at the constant of motion:\ng(m\', m\') + |p_D|^2.\nIf the gauge metric is constant, then each part is a constant of motion\nseparately.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>markwh04@yahoo.com wrote:
> John Baez wrote: > > [itex]h_{ij} = g_{ij}[/itex] > > [itex]h_{i5} = A_i[/itex] > > [itex]h_{55} = 1[/itex] > > > HOWEVER, I don't find that the velocity of the particle around > > the circle is constant! > > You got stumped by this problem? > > That's easy. You've got the wrong metric. It's > > [itex]h_{ij} = g_{ij} + K A_i A_j[/itex] > [itex]h_{i5} = h_{5i} = K A_i[/itex] > [itex]h_{55} = K[/itex] > > The Lagrangian may be written (using the summation convention) as: > [itex]L = 1/2 (g_{ij} u^i u^j) + K/2 (A_i u^i + u^5)^2[/itex] > The conjugate momenta are: [etc] This can be worked out in a transparent fashion (at least when not reduced to ASCII form!) in the more general case, within the principal bundle formalism, using the *left-quotient* operator. The derivative of the quotient completely captures in a much more general and clearer context the notion of a connection, making everything easier to work with. So, some background and notation first. Let P be acted on by a group G. This gives you a product operation with signature P.[itex]G -> P[/itex] and properties [itex]pe = p (e =[/itex] group identity); p(gh) [itex]= (pg)h,[/itex] with cosets [itex]pG = {[/itex] pg: g in G } One can then define the set [itex]p\q = { g[/itex] in G: [itex]pg = q }[/itex]. The stablizer of point p is just [itex]p\p.[/itex] The group acts transitively on P if all the p\q are non-empty. It acts freely if p\q has at most one member -- in which case one may consider the quotient as a partial operation [itex]p\q:[/itex] defined for all q in pG with the properties: [itex]p p\q = qp\p = e(p\q)^{-1} = q\pp\q q\r = p\q(pg)\(qh) = g^{-1} p\q h[/itex]. When these spaces are also manifolds, one can consider the actions of these operators under derivatives. Recalling that the key property of tangent vectors on a manifold M is that for a curve r(t) r'(t) is in [itex]T_{r(t)}(M),[/itex] we'll use the notation denoting by m' (with subscripts) a general element of [itex]T_m(M)[/itex]. With the coset notation, a local SECTION on the space P is defined as a map [itex]s: P/G = M -> P[/itex] such that s(m)G = m. A principal bundle is thus a manifold P acted on by a symmetry group G in such a way that its cosets [itex]M = P/G[/itex] can be arranged into a manifold, which can be coordinatized by differentiable local sections whose domains cover M. In the following, for a function f: [itex]M -> N,[/itex] I'll use the notation Df: [itex]TM -> TN[/itex] to denote its deriative. So, for [itex]f: R^m -> R^n, Df[/itex] is just the mxn Jacobian matrix of f. Its product with a vector v in TM is denoted Df[v] in TN. The product P.[itex]G -> P,[/itex] naturally induces one with signatures: p.[itex]T_g(G), T_p(P)[/itex].[itex]g -> T_{pq}(G)[/itex] with the property (pg)' = p'g + pg'. Differentiating the relation, s(m)gG [itex]= s(m)G = m[/itex] one finds: [itex]ds_m[m']G = ds_m[m']gG + s(m)g'G = ds_m[m']G = m'[/itex]. Thus [itex]ds_m[m']G =[/itex] m'; s(m)g' = . The corresponding action for quotients [itex]p\T_q(P), T_p(P)\q -> T_{p\q}(G)[/itex] is not given at the outset, but must be postulated; and is assumed to have the property [itex](p\q)' = p'\q + p\q'[/itex]. The connection one-form is just [itex]\omega_p[p'] = p\p'.[/itex] The horizontal lift of a vector m', m = pG, to a point p is [itex]L_p[m'] = s(m) ds_m[m']\p + ds_m[m'] s(m)\p[/itex] or more briefly [itex]L = s ds\p + ds s\p[/itex] The section-relative connection is [itex]B = s\ds[/itex] or more precisely: [itex]B_m[m'] = s(m)\ds_m[m'].[/itex] Its action under a global transformation [itex]s -> sg[/itex] is [itex]B -> (sg)\d(sg) = g^{-1} s\ds g = g^{-1} B g[/itex] and its action under a local transformation s(m) [itex]-> s(m) x(m)[/itex] is [itex]B -> (sx)\d(sx) = x^{-1} s\(ds x + s x')= x^{-1} s\ds x + x^{-1} x'= x^{-1} B x + x^{-1} x'[/itex]. In a similar way, one may define natural actions of the group on the cotangent space through the properties: g.[itex]T*_p(P)[/itex].[itex]h -> T*_{gph}(P)[/itex] (gwh)[p'] [itex]= w[g^{-1} p' h^{-1}][/itex] as well as a quotient [itex]p\T*_q(P) -> T*_{p\q}(P)p\w[q'] =[/itex] w[p q']. This time, it's the product that requires the additional structure of the connection, with p.[itex]T*_g(G) -> T*_{pg}(P)[/itex] p.a[q'] [itex]= a[p\q'][/itex]. (One can also define operations for the tangent and cotangent spaces: [itex]T_p(P)\T*_q(P) -> Rp'\w =[/itex] w[p'] ). So with the background out of the way... Given a metric k on G that is invariant under left and right products [itex]k_{hgf}(h g'_1 f, h g'_2 f) = k_g(g'_1, g'_2)[/itex] and a metric g on the base space M, define the bundle metric on P by [itex]h = g + k(\omega, \omega)[/itex] or more precisely: [itex]h_p(p'_1, p'_2) = g_{pG}(p'_1G, p'_2G) + k_{pG}(p\p'_1, p\p'_2)[/itex]. In the general case, the scaling of k may be positionally dependent on the points in the base space M. Given a section s, one may decompose P and TP as follows: [itex]p = s(m) g;[/itex] where m = pG; [itex]g = s(pG)\p[/itex] with [itex]p' =[/itex] ds[m'] [itex]g + s g'[/itex]. One finds that p'G = ds[m']gG + s g'G [itex]= m' + =[/itex] m', and [itex]p\p' = g^{-1} s\(ds g + s g') = g^{-1} B_m[m'] g + g^{-1} g'[/itex] or [itex]p\p' = g^{-1} u g[/itex] introducing the horizontal velocity [itex]u = B_m[m'] + v[/itex] and gauge velocity [itex]v = g' g^{-1}.[/itex] In terms of the decomposition, the metric can be written as: [itex]h_p(p'_1, p'_2) = g_m(m'_1, m'_2) + k_m(u'_1, u'_2)[/itex]. In component form, this becomes: [itex]h_{AB}p'_1^A p'_2^B = g_{mn} m'_1^m m'_2^n + k_{ab} u'_1^a u'_2^b[/itex]. using the summation convention on the indices. The components of the total metric are thus: [itex]h_{mn} = g_{mn} + k_{ab} B^{a_m} B^{b_n}h_{mb} = k_{ab} B^{a_m}h_{an} = k_{ab} B^{b_n}h_{ab} = k_{ab}[/itex]. So, from this you can readily find the geodesic equations of motion. These arise from the Lagrangian: L(p,p') [itex]= 1/2 h_p(p',p')[/itex]. Variation of L gives you the desired results. Working within the decomposition, [itex]p = s(m) g[/itex] as above, define the variation of g by [itex]\Delta(g) = D g[/itex]. then [itex]\Delta(g^{-1}) = -g^{-1} D.[/itex] and the variation of v will be: [itex]\Delta(v) = \Delta(g' g^{-1})= (\Delta g)' g^{-1} - g' g^{-1} D= (Dg)' g^{-1} - v D= D' g g^{-1} + D v - v D= D' +[/itex] [D,v]. The variations with respect to D give you the charge and precession equation (Wong's equation) for the charge. The variation with respect to m' gives you the ordinary momentum and the Lorentz force law coupled with the charge. First, employing the derivative notation alluded to before: [itex]\Delta(u) = \Delta(B_m[m'] + v)= DB_m[\Delta(m)][m'] + B_m[\Delta(m')] + D' +[/itex] [D,v]. Thus [itex]\Delta(L) = \Delta(1/2 g_m(m',m') + 1/2 k_m(u,u))= 1/2 Dg_m[\Delta(m)](m',m') + 1/2 Dk_m[\Delta(m)](u,u)+ g_m(m',\Delta(m')) + k_m(u,\Delta(u))[/itex]. Picking out the variation with respect to D', one finds the corresponding momentum [itex]p_D = k_m(u, ())[/itex] which is just u with its (Lie) index lowered. Picking out the variation with respect to D, one finds the force term [itex]F_D:F_D = k_m(u, [(),v]) = p_D[[(),v]][/itex]. Since [itex]p_D[/itex] is a cotangent vector in [itex]T*G,[/itex] it's natural to extend the Lie bracket to the cotangent space. So, let's consider first how the Lie bracket is defined. For a Lie vector e' in the tangent space [itex]T_e(G) =[/itex] L, one has g [itex]e' g^{-1}[/itex] in [itex]T_{g e g^{-1})(G)[/itex] in L. Therefore, L is closed under the adjoint operator [itex]ad_g(e') = g e' g^{-1},[/itex] and derivatives of this operator make sense. The Lie bracket is then just [g',e'] [itex]= (g e' g^{-1})'[/itex]. Explicitly: [g'(s), e'(t)] [itex]= d^2(g(s) e(t) g(s)^{-1})/ds dt.[/itex] So, over the cotangent space [itex]T_e*(G) = L*,[/itex] one has the coadjoint action defined in the same way [itex]co_g(w) = g w g^{-1}[/itex] and Lie bracket [g', [itex]w] = (g w g^{-1})'[/itex]. Applying this to a Lie vector e', one finds [itex][e_1',w][e_2'] = (e_1 w e_1^{-1})'[e_2']= (e_1 w e_1^{-1}[e_2'])'= (w[e_1^{-1} e_2' e_1])'= w[(e_1^{-1} e_2' e_1)'].[/itex] Since [itex]e_1^{-1}'(t) = -e_1^{-1}(t) e_1'(t) e_1^{-1}(t)[/itex] which evaluated at the given t where [itex]e_1(t) = e[/itex] is just [itex]e_1^{-1}'(t) = -e_1'(t),[/itex] then the Lie bracket for [itex]e_1^{-1}[/itex] is the negative of that for [itex]e_1.[/itex] Thus [itex][e_1',w][e_2'] = -w[[e_1',e_2']]= w[[e_2',e_1']].[/itex] Or, in more prosaic notation: [a,w][b] = w[[b,a]], for lie vectors a, b in L. Since the metric k is assumed to be invariant under action of G to the left and right, one has: [itex]d/ds (k_{m}(g(s) h_1' g(s)^{-1}, g(s) h_2' g(s)^{-1}))= d/ds (k_h(h_1', h_2') =[/itex] . Applying this to the individual vectors using the Leibnitz property, one gets [itex]d/ds (k(g h_1' g^{-1}, g h_2' g^{-1}))= k(d(g h_1' g^{-1})/ds, g h_2' g^{-1})+ k(g h_1' g^{-1}, d(g h_2' g^{-1})/ds)= k([g', h_1'], h_2') + k(h_1', [g', h_2']).[/itex] Thus for Lie vectors, a, b, c in L: k([a,b],c) [itex]+ k(b,[a,c]) = .[/itex] In particular, since the metric is symmetric, then for [itex]b = c,k([a,b],b) = .[/itex] Therefore, the force can be rewritten as: [itex]F_D = k_m(u, [(),v])= k_m(u, [(),u - B])= k_m(u, [(),u]) - k_m(u, [(),B])= -k_m(u, [(),B])= -p_D([(),B])= -[B,p_D] = [p_D, B].[/itex] The Wong equation for the precession of the charge is thus [itex]d(p_D)/dt = [p_D, B[dm/dt]][/itex]. The variation with respect to m' yields the momentum [itex]p = g_m(m', ()) + k_m(u, B_m())= g_m(m', ()) + p_D(B_m())= g_m(m', ()) + p_D[/itex].[itex]B_m[/itex]. In component form [itex]p_m = g_{mn} m'^n + p_{D_a} B^{a_m},[/itex] which is the usual expression for the canonical momentum in the presence of a Lorentz force given by the potential Lie vector B coupled to the charge Lie vector [itex]p_D.[/itex] The corresponding force is the variation with respect to m: [itex]F = 1/2 Dg_m[](m',m') + 1/2 Dk_m[](u,u) + k_m(u,DB_m()[m']).= 1/2 Dg_m[](m',m') + 1/2 Dk_m[](u,u) + p_D(DB_m()[m']).[/itex] The corresponding equation of motion is [itex]dp/ds = F[/itex]. Differentiating p, we find [itex]dp/ds = Dg_m[m'](m',()) + g_m(m'',()) + p_D'[B_m()] + p_D[dB_m[m'][]]= Dg_m[m'](m',()) + g_m(m'',()) + p_D[[B[m'],B[]]] +p_D[dB_m[m'][]][/itex]. Thus [itex]Dg_m[m'](m',()) + g_m(m'',()) + p_D[[B[m'],B[]]] + p_D[dB_m[m'][]]= 1/2 Dg_m[](m',m') + 1/2 Dk_m[](u,u) + p_D(DB_m[][m']).[/itex] Defining the Yang-Mills force G(v, [itex]w) = DB_m[v][w] - DB_m[w][v] + [B_m[v], B_m[w]][/itex] and Christoffel coefficients [itex]\Gamma(u,v,w) = 1/2 (Dg[u](v,w) + Dg[v](u,w) - Dg[w](u,v))[/itex] the equations of motion become [itex]g_m(m'', ()) + \Gamma(m', m', ()) = G((), m') + 1/2 Dk_m[](u,u)[/itex]. Applying this to the vector m', and noting that (g(m', m'))' [itex]= 2 g(m'', m') + 2 \Gamma(m', m', m')[/itex] we get (g(m', m'))' [itex]= Dk_m[m'](u, u)[/itex]. If the gauge metric has an inverse, [itex]k^{-1},[/itex] one may define the charge magnitude [itex]|p_D|^2 = k^{-1}(p_D, p_D).= p_D[u]= k(u, u)[/itex] and [itex]Dk_m[m'](u, u) = -D{k^{-1}_m)(p_D, p_D).[/itex] Adjoint invariance applies to this metric as well [itex]k^{-1}([a,b], c) + k^{-1}(b, [a,c]) = .[/itex] Therefore, under the derivative, its action becomes [itex](|p_D|^2)' = D(k^{-1})[m'](p_D, p_D) + 2 k^{-1}(p_D, [p_D, B])= D(k^{-1})[m'](p_D, p_D)= -Dk_m[m'](u, u).[/itex] Thus, one arrives at the constant of motion: g(m', m') [itex]+ |p_D|^2[/itex]. If the gauge metric is constant, then each part is a constant of motion separately. |
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