Re: dirac equation in curved spacetime
- From: Rock Brentwood <markwh04@xxxxxxxxx>
- Date: Wed, 31 Oct 2007 05:51:38 +0000 (UTC)
On Sep 16, 12:11 pm, vivishek <vivishek.sud...@xxxxxxxxx> wrote:
Is there a single, well-accepted form of the dirac equation for a free
particle in curved spacetime?
The Dirac matrices gamma^0, gamma^1, gamma^2 and gamma^3 are not
actually tied to a spacetime frame, but to a local Minkowski frame. So
to transplant the Dirac equation into curved spacetime requires
transplanting the local Minkowski frame into a field of inertial
frames.
That requires identifying within each tangent space which quadruples
of axes constitute orthonormal frames, and to do so in such a way that
a consistent spinor "square root" can be taken. The ability to do this
identification consistently throughout the entire spacetime manifold
is non-trivial and requires an extra topological condition.
The dirac frames are orthonormal and are locally invariant under the
Lorentz group. The gravity field is identified as the local gauge
field for the Lorentz group. The Dirac equation acquires an extra set
of terms corresponding to the gauge potential. It is the same equation
as that coupling the spinor field to a gauge field.
The gauge generators are S^{ij} = h-bar/2 gamma^i gamma^j for i < j,
and i,j = 0,1,2,3. The Lie algebra of the generators is that of the
Lorentz group, taking the Lie bracket to be {S,S'} = (SS' - S'S)/(h-
bar). The "charge" is just h-bar/2, and is identified as the spin of
the Dirac field. I might be missing factors of i here, which you may
try to look for or correct. The end result is that the S^{ij} should
be a representation of the Lie algebra for the Lorentz group.
The gravitational potential is A = 1/2 omega_{ij} S^{ij}, where the
omega's are the "spin coefficient" 1-forms, which are directly related
to the connection coefficients. The field is F = dA + A^2 = 1/2 R_{ij}
S^{ij}, where the R_{ij} are the 2-forms related to the Riemannian
curvature.
The conversion from the local frame to the "world frame" is given by
gamma^i <-> h^i_m dx^m. The components h^i_m are the components of the
"vierbein" or orthonormal frame quadruple. They produce the metric
through the relation eta_{ij} h^i_m h^j_n = g_{mn}. This is what
"taking the square root" is in reference to -- the h's are the square
root of the metric, the g's. The eta terms are the components of the
Minkowski metric. They also double over as being related to the
structure coefficients for the Lie algebra corresponding to the
Lorentz group.
The Lagrangian 4-form for pure gravity is, up to proportion, L =
epsilon_{ijkl} h^i ^ h^j ^ R^{kl}, where h^i = h^i_m dx^m, produces
the field equations for pure gravity, where the variation of the
action takes the h fields and omega fields as fundamental. So, it's a
composite of a Lorentz gauge field, plus an extra "mediating" field,
h, that embodies directly the identification of which local frames are
inertial. The contribution from the Dirac field is as expected:
psi^bar (i h-bar D - A) psi, where this A includes the gravity-as-
Lorentz-gauge field and the sum of all the other gauge fields
interacting with the Dirac field.
Extra terms for gravity can be added: those proportional to
epsilon_{ijkl} h^i ^ h^j ^ h^k ^ h^l (for the cosmological constant)
and to h^i ^ h^j ^ R_{ij} (for a parity-violating contribution to
gravity).
In a quantized theory, the gauge part of gravity could be quantized
along with the rest of the gauge fields as a gauge field. But it's not
a Yang-Mills field or Yang-Mills-Higgs field, since its Lagrangian is
not quadratic in the field strengths. So, I'm not sure how or whether
it renormalizes. The pure SU(2) gauge field representation of gravity
is renormalizable, so this may entail something similar here.
The h part of the field cannot quantize in any normal way! That's
because it mediates between the full set of possible world frames and
the distinguished subset of inertial frames. In order to quantize a
field, one needs to first prescribe an underlying causal structure and
underlying field of inertial frames. If you allow the definition of
"inertiality" to fluctuate (that is, allow for h to have quantum
fluctuations) then what you're actually doing is fluctuating between
inertial and non-inertial frames.
The boom is lowered, however, as soon as you realise that the state
space in one frame does not coherently superpose with the state space
founded on another frame that is not inertial with respect to the
first. This is the upshot of the Unruh-Davies effect, and the more
general statement was essentially that given by Beckenridge (sp?) in
1973. So, there are no quantum superpositions between those states
corresponding to one h and those corresponding to another h, if the
two sets of values for the h field entail two definitions of
"inertiality" that are not inertial with respect to one another.
This discrepancy is a good part of what lies behind the perennial
difficulty of fully quantizing gravity -- i.e., there may not be a
fully quantum theory of gravity.
This issue is discussed in further depth (particularly in part 3.3)
under the review:
Time in Quantum Theory and General Relativity
http://federation.g3z.com/Physics/index.htm#QG2007_1
.
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