GR [+ Spin] as Poincare' Gauge Theory (was: Gauge Bosons and Metrics)

whopkins@csd.uwm.edu

<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>lost.and.lonely.physicist@gmail.com wrote:\n> I understand that the gauge field for U[1] has only 1 index, whereas\n> the Christoffel symbol in GR has 3. The gauge field for SU[3] and\nSU[2]\n> has 2 indices, one for mu, one for summing over the generators, so\nthey\n> don\'t seem to have enough either.\n\nGoing further with my previous reply:\n\nthe indices on the general connection A_m^a are a spacetime index (m)\nand a Lie index (a). So, for U(1), since (a) only ranges over one\nindex value it\'s omitted, and only only talks about an A_m.\n\nThis has the unfortunate effect of suppressing the hidden metric and\nthe raising and lowering of indices:\nF^{mn}_{.} <--> {D/c, H/c)\nF_{mn}^{.} <--> (E, B)\nwith\nF^{mn}_{.} = k_{..} g^{mp} g^{nq} F^{.}_{pq}\nThe metric coefficient k_{..} is just (epsilon_0 c).\n\nFor the Riemannian connection, the indices are actually 2, not 3:\nGamma_{mn}^p <--> Gamma_m^{n/p}\nwith {n/p} the index of GL(4):\n[L^n_m,L^q_p] = delta^n_p e^q_m - delta^q_m e^n_p\n\nIn this light, the Cartan structure equations for\nOmega^n_p = Gamma_{mp}^n e^m\nTau^m = 1/2 Tau^m_{np} e^n ^ e^p = Torsion\nTheta^m_n = 1/2 R^m_{npq} e^p ^ e^q = Riemmannian Curvature\nwhich are, for a general frame (e^1,e^2,...):\nd(e^m) + Omega^m_n ^ e^n = Tau^m\nd(Omega^m_n) + Omega^m_p ^ Omega^p_n = Theta^m_n\nsuddenly appear in a new light. They\'re just the equations for a FIELD\nSTRENGTH\nF = dA + 1/2 [A,A]\nF^c_{mn} = d_m A^c_n - d_n A^c_m + f^c_{ab} A^a_m A^b_n\n(expressed in terms of the structure coefficients\n[Y_a,Y_b] = f^c_{ab} Y_c\nfor a Lie algebra basis (Y_1,Y_2,...)).\n\nBut the operator is not for the gauge group GL(4). It\'s for the\nINHOMOGENEOUS gauge group IGL(4), which extends GL(4) by adding\ngenerators for the translations (P_a, a=1,...,4):\n[P_a,P_b] = 0\n[P_a,L^c_d] = delta^c_a P_d\nThe gauge field is just the frame, itself, along with the connection\nforms:\nA = e^a P_a + Omega^a_b L^b_a.\nThe corresponding field strength, F, (may have to check for signs)\ncomes straight out of this:\nF = Tau^a P_a + Theta^a_b L^b_a.\nTorsion is the part of the field strength associated with translational\ndegrees of freedom; the Riemannian curvature that associated with\nrotational degrees.\n\nGeneral Relativity is, thus, a gauge theory for IGL(3,1).\n\nThe corresponding currents would be:\np_a -- momentum, coupling to Tau^a\ns^b_a -- spin, coupling to Theta^a_b.\n\nIf one wanted, instead, a gauge theory for the Lorentz group SL(3,1)\nand its inhomogeneous extension ISL(3,1), the Poincare\' group, then the\nframes would be restricted to [Minkowski] orthonormal forms:\ng^{mn} e^a_m e^b_n = eta^{ab} = diag(+,-,-,-).\n\nThen everything follows through as above. The corresponding connection\n-- called a spin connection -- consists simply of the covariant\nderivatives of the e\'s:\nOmega_m^a_b = d_b(e^a_m) - Gamma^p_{mn} e^b_p e^a_n\n\nand the total gauge field is:\nA = (e^a P_a + Omega^a_b L^b_a)\nwith field strength:\nF = (Tau^a P_a + Theta^a_b L^b_a)\nas before.\n\nThis additional structure is equivalent to providing a spin bundle on\nthe manifold.\n\nIf, further, one started out *only* with the gauge field:\nA = (e^a_m P_a + Omega_m^a_b L^b_a) dx^m\nis, and then DEFINED the metric as\ng_{mn} = eta_{ab} e^a_m e^b_n\nand the result would be a complete recovery of the Riemannian curvature\nand torsion through the correspondences:\nTau^m_{np} e^a_m = Tau^a_{np}\nR^m_{npq} e^a_m = e^b_n R^a_b_{pq}\nshowing that the key objects of GR come out for free from the gauge\nfield and field strength of a Poincare\' gauge theory.\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>lost.and.lonely.physicist@gmail.com wrote:
> I understand that the gauge field for U[1] has only 1 index, whereas
> the Christoffel symbol in GR has 3. The gauge field for SU[3] and
SU[2]
> has 2 indices, one for [itex]\mu,[/itex] one for summing over the generators, so
they
> don't seem to have enough either.

Going further with my previous reply:

the indices on the general connection [itex]A_m^a[/itex] are a spacetime index (m)
and a Lie index (a). So, for U(1), since (a) only ranges over one
index value it's omitted, and only only talks about an [itex]A_m[/itex].

This has the unfortunate effect of suppressing the hidden metric and
the raising and lowering of indices:
[itex]F^{mn}_{.} <--> {D/c, H/c)F_{mn}^{.}[/itex] <--> (E, B)
with
[itex]F^{mn}_{.} = k_{..} g^{mp} g^{nq} F^{.}_{pq}[/itex]
The metric coefficient [itex]k_{..}[/itex] is just [itex](\epsilon_0 c).[/itex]

For the Riemannian connection, the indices are actually 2, not 3:
[itex]\Gamma_{mn}^p[/itex] <--> [itex]\Gamma_m^{n/p}[/itex]
with [itex]{n/p}[/itex] the index of GL(4):
[itex][L^{n_m},L^{q_p}] = \delta^n_p e^{q_m} - \delta^q_m e^{n_p}[/itex]

In this light, the Cartan structure equations for
[itex]\Omega^n_p = \Gamma_{mp}^n e^m\Tau^m = 1/2 \Tau^m_{np} e^n ^ e^p =[/itex] Torsion
[itex]\Theta^m_n = 1/2 R^{m_}{npq} e^p ^ e^q =[/itex] Riemmannian Curvature
which are, for a general frame [itex](e^1,e^2,...):d(e^m) + \Omega^m_n ^ e^n = \Tau^md(\Omega^m_n) + \Omega^m_p ^ \Omega^p_n = \Theta^m_n[/itex]
suddenly appear in a new light. They're just the equations for a FIELD
STRENGTH
[itex]F = dA + 1/2[/itex] [A,A]
[itex]F^{c_}{mn} = d_m A^{c_n} - d_n A^{c_m} + f^{c_}{ab} A^{a_m} A^{b_n}[/itex]
(expressed in terms of the structure coefficients
[itex][Y_a,Y_b] = f^{c_}{ab} Y_c[/itex]
for a Lie algebra basis [itex](Y_1,Y_2,...)).[/itex]

But the operator is not for the gauge group GL(4). It's for the
INHOMOGENEOUS gauge group IGL(4), which extends GL(4) by adding
generators for the translations [itex](P_a, a=1,...,4):[P_a,P_b] = [P_a,L^{c_d}] = \delta^c_a P_d[/itex]
The gauge field is just the frame, itself, along with the connection
forms:
[itex]A = e^a P_a + \Omega^a_b L^{b_a}.[/itex]
The corresponding field strength, F, (may have to check for signs)
comes straight out of this:
[itex]F = \Tau^a P_a + \Theta^a_b L^{b_a}.[/itex]
Torsion is the part of the field strength associated with translational
degrees of freedom; the Riemannian curvature that associated with
rotational degrees.

General Relativity is, thus, a gauge theory for IGL(3,1).

The corresponding currents would be:
[itex]p_a --[/itex] momentum, coupling to [itex]\Tau^as^{b_a} --[/itex] spin, coupling to [itex]\Theta^a_b[/itex].

If one wanted, instead, a gauge theory for the Lorentz group SL(3,1)
and its inhomogeneous extension ISL(3,1), the Poincare' group, then the
frames would be restricted to [Minkowski] orthonormal forms:
[itex]g^{mn} e^{a_m} e^{b_n} = \eta^{ab} = diag(+,-,-,-)[/itex].

Then everything follows through as above. The corresponding connection
-- called a spin connection -- consists simply of the covariant
derivatives of the e's:
[itex]\Omega_m^a_b = d_b(e^{a_m}) - \Gamma^p_{mn} e^{b_p} e^{a_n}[/itex]

and the total gauge field is:
[itex]A = (e^a P_a + \Omega^a_b L^{b_a})[/itex]
with field strength:
[itex]F = (\Tau^a P_a + \Theta^a_b L^{b_a})[/itex]
as before.

This additional structure is equivalent to providing a spin bundle on
the manifold.

If, further, one started out *only* with the gauge field:
[itex]A = (e^{a_m} P_a + \Omega_m^a_b L^{b_a}) dx^m[/itex]
is, and then DEFINED the metric as
[itex]g_{mn} = \eta_{ab} e^{a_m} e^{b_n}[/itex]
and the result would be a complete recovery of the Riemannian curvature
and torsion through the correspondences:
[itex]\Tau^m_{np} e^{a_m} = \Tau^a_{np}R^{m_}{npq} e^{a_m} = e^{b_n} R^{a_}{b_}{pq}[/itex]
showing that the key objects of GR come out for free from the gauge
field and field strength of a Poincare' gauge theory.

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