Natural Science Forum / Physics / Research / November 2007

One possibility is the real gravitational Dirac equation. It's the
same as the matrix version with a bit more geometry. It's called real
because because the uninterpreted scalar i is replaced with an
associative product of ortho-normal vectors See Spacetime Geometry
with Geometric Calculus at http://modelingnts.la.asu.edu/html/GCgravity.html
and go to equation (120)

Thus spake vivishek <vivishek.sudhir@gmail.com>

Its form is determined by manifest covariance, but of course it may be
treated either as a wave equation or an operator equation as usual. Here
is a selection of papers.

arXiv:0706.4413
Title: Quantum wave equations in curved space-time from wave mechanics
Authors: Mayeul Arminjon

arXiv:hep-th/0610207
Title: On bound states of Dirac particles in gravitational fields
Authors: Nicolas Boulanger, Fabien Buisseret, Philippe Spindel

arXiv:gr-qc/0603099
Title: Electromagnetic and gravitational self-force on a relativistic
particle from quantum fields in curved space
Authors: Chad R. Galley, B. L. Hu, Shih-Yuin Lin

arXiv:hep-th/0411016
Title: Symmetries and supersymmetries of the Dirac operators in curved
spacetimes
Authors: I. I. Cotuaescu, M. Visinescu

arXiv:gr-qc/0409080 [ps, pdf, other]
Title: $\bar{SL}(4,R)$ Embedding for a 3D World Spinor Equation
Authors: Djordje Sijacki

arXiv:gr-qc/0209096
Title: Gravity, torsion, Dirac field and computer algebra using MAPLE
and REDUCE

arXiv:gr-qc/0103056 [ps, pdf, other]
Title: Application of linear hyperbolic PDE to linear quantum fields in
curved spacetimes: especially black holes, time machines and a new semi-
local vacuum concept
Authors: Bernard S. Kay (York)

arXiv:gr-qc/0010065 [ps, pdf, other]
Title: The phase of a quantum mechanical particle in curved spacetime
Authors: P.M. Alsing, J.C. Evans, K.K. Nandi

arXiv:gr-qc/0008047 [ps, pdf, other]
Title: A new approach to electromagnetic wave tails on a curved
spacetime

arXiv:gr-qc/9708041 [ps, pdf, other]
Title: Gravity from Dirac Eigenvalues
Authors: Giovanni Landi, Carlo Rovelli

arXiv:gr-qc/9612034 [ps, pdf, other]
Title: General Relativity in terms of Dirac Eigenvalues
Authors: Giovanni Landi, Carlo Rovelli

Regards

I haven't the faintest idea. However, by chance I noticed that this

http://arxiv.org/abs/0709.0936

has just appeared, you may find it (or it's references) of interest.

No

So many like geometric structures (U, V, non-commutative) for the
spacetime and interactions with fermion and EM fields (e.g. Q-A
interaction) you choose.

For instance, it is usually thought that curved spacetime cannot deal
with spin. Therefore, a first step is generalization of Riemannian
structure (V space) to a Riemann-Cartan one (U).

Now spacetime is not just curved like in GR. The connection Gamma_ab^c
is not just the Christoffel but include the contorsion tensor K_ab^c.
(See equation 2.2 in [gr-qc/9309027])

A popular option is substitution R --> R - Q^2 in the Lagrangian of
GR. Here Q is the torsion (associated to the contorsion K).

Another posibility is to introduce different coupling constants for
curvature and torsion. Then add next term to GR Lagrangian: epsilon
{root -g} Q^2. Where epsilon is the coupling constant.

the Dirac equation in a Riemann-Cartan spacetime

gamma^b D_b phi = m phi

(see 3.29 in [gr-qc/9309027] for alternative). However, the covariant
derivative D may be computed using the connection Gamma_ab^c
containing tensor corrections to GR connection.

Above equation and the related Dirac Fock Ivanenko equation are under
study in unification and cosmology.

[Adittional references]

New method of integration for the Dirac equation on a curved space-
time. J. Math. Phys. 1992, 33, 2279. Bagrov, V. G; Obukhov, V. V.

http://arxiv.org/pdf/math-ph/0502001

On solutions of the Einstein-Cartan-Dirac theory. Class. Quantum Grav.
1985, 2, 919. Seitz, M.

http://arxiv.org/pdf/gr-qc/9309027

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

I have posted this:

http://groups.google.ca/group/sci.physics.research/msg/f798c23beba36a8f
news:fl7eg.2733$ho5.218228@news20.bellglobal.com
------------------------------------------------------------------------
This article has what they write to be a version of the Dirac equation
for curved spacetimes:
http://xxx.lanl.gov/abs/hep-th/0411016

Writing the parallelization as t:T(M) -> R^4 and v_k being dual
vector fields t^j(v_k) = d_jk, their equation is the same as,

(i gamma^j v_j + i/4 gamma^j gamma^k gamma^l w_kl(v_j) - m) Psi = 0

where w is antisymmetric in its two indices (w_kl = - w_lk) and is the
solution of,

dt^a + w^a_b /\ t^b = 0

i.e.,

w_kl(v_j) = ( dt_j(v_l, v_k) + dt_l(v_j, v_k) - dt_k(v_j, v_l) ) / 2
------------------------------------------------------------------------
w is the rotational part of the Cartan connection (t, w): R^4 ->
euc(3, 1).

I have not been able to prove that the physics are independent of the
local orientation of the frame (t in my notation) to more than first
order in changes to the local orientation of the frame.

On Nov 18, 9:47 am, mihai cartoaje <mcarto...@gmail.com> wrote:

I meant w is the rotational part of the torsion-free Cartan connection
(t, w): R^4 -> euc(3, 1).