GR [+ Spin] as Poincare' Gauge Theory (was: Gauge Bosons and Metrics)
whopkins_at_csd.uwm.edu
Date: 03/06/05
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Date: Sun, 6 Mar 2005 07:52:03 +0000 (UTC)
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 mu, 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 A_m^a 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 A_m.
This has the unfortunate effect of suppressing the hidden metric and
the raising and lowering of indices:
F^{mn}_{.} <--> {D/c, H/c)
F_{mn}^{.} <--> (E, B)
with
F^{mn}_{.} = k_{..} g^{mp} g^{nq} F^{.}_{pq}
The metric coefficient k_{..} is just (epsilon_0 c).
For the Riemannian connection, the indices are actually 2, not 3:
Gamma_{mn}^p <--> Gamma_m^{n/p}
with {n/p} the index of GL(4):
[L^n_m,L^q_p] = delta^n_p e^q_m - delta^q_m e^n_p
In this light, the Cartan structure equations for
Omega^n_p = Gamma_{mp}^n e^m
Tau^m = 1/2 Tau^m_{np} e^n ^ e^p = Torsion
Theta^m_n = 1/2 R^m_{npq} e^p ^ e^q = Riemmannian Curvature
which are, for a general frame (e^1,e^2,...):
d(e^m) + Omega^m_n ^ e^n = Tau^m
d(Omega^m_n) + Omega^m_p ^ Omega^p_n = Theta^m_n
suddenly appear in a new light. They're just the equations for a FIELD
STRENGTH
F = dA + 1/2 [A,A]
F^c_{mn} = d_m A^c_n - d_n A^c_m + f^c_{ab} A^a_m A^b_n
(expressed in terms of the structure coefficients
[Y_a,Y_b] = f^c_{ab} Y_c
for a Lie algebra basis (Y_1,Y_2,...)).
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 (P_a, a=1,...,4):
[P_a,P_b] = 0
[P_a,L^c_d] = delta^c_a P_d
The gauge field is just the frame, itself, along with the connection
forms:
A = e^a P_a + Omega^a_b L^b_a.
The corresponding field strength, F, (may have to check for signs)
comes straight out of this:
F = Tau^a P_a + Theta^a_b L^b_a.
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:
p_a -- momentum, coupling to Tau^a
s^b_a -- spin, coupling to Theta^a_b.
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:
g^{mn} e^a_m e^b_n = eta^{ab} = diag(+,-,-,-).
Then everything follows through as above. The corresponding connection
-- called a spin connection -- consists simply of the covariant
derivatives of the e's:
Omega_m^a_b = d_b(e^a_m) - Gamma^p_{mn} e^b_p e^a_n
and the total gauge field is:
A = (e^a P_a + Omega^a_b L^b_a)
with field strength:
F = (Tau^a P_a + Theta^a_b L^b_a)
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:
A = (e^a_m P_a + Omega_m^a_b L^b_a) dx^m
is, and then DEFINED the metric as
g_{mn} = eta_{ab} e^a_m e^b_n
and the result would be a complete recovery of the Riemannian curvature
and torsion through the correspondences:
Tau^m_{np} e^a_m = Tau^a_{np}
R^m_{npq} e^a_m = e^b_n R^a_b_{pq}
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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