My COSMION 2004 Slides

General Relativity and the Dirac-Kahler Equation∗

Alexander Yu. Vlasov†

Federal Radiological Center, IRH197101, Mira Street 8St.–Petersburg, Russia

A. Friedmann Lab for Theoretical Physics191023, Griboedov Canal 30/32St.–Petersburg, Russia

The Dirac Equation

Dψ = mψ, ψ ∈ C4, D =

3∑µ=0

γµ i ∂

∂xµ, γµ ∈ M(4,C), γµγν + γνγµ = 2gµν.

The Dirac-Kahler Equation

dω − d?ω = mω, ω ∈ Λ(M) =

4⊕

k=0

Λk(M), ? : Λk(M) → Λ4−k(M).

D = (d− d?), d : Λk(M) → Λk+1(M), d? : Λk(M) → Λk−1(M).

Original papers:• Dirac EquationP. A. M. Dirac, Proc. Roy. Soc. Lond. A117 610 (1928).• Dirac-Kahler EquationD. D. Ivanenko and L. D. Landau, Z. Phys. 48 340 (1928),E. Kahler, Randiconti di Mat. Appl. 21 425 (1962).

Factorization of the Klein-Gordon Operator(¤−m2

)φ = 0

¤ = −3∑

µ,ν=0

gµν∂2

∂xµ∂xν, D2 = ¤, D2 = −(dd? + d?d) = ∆1,3 = ¤

∗See also http://arXiv.org/abs/math-ph/0403046†E-mail: [email protected]

GR and the Dirac-Kahler Equation A. Yu. Vlasov

Matrix, Clifford, Tensor Form of the Dirac Equation

DΨ = mΨ

ψ =

ψ1

ψ2

ψ3

ψ4

↔ Ψ =

ψ1 0 0 0ψ2 0 0 0ψ3 0 0 0ψ4 0 0 0

!

ψξT ≡ ψ ⊗ ξ︷︸︸︷

ξ1ψ1 ξ2ψ1 ξ3ψ1 ξ4ψ1




Isomorphisms of 16D linear spaces (without algebraic structures, e.g.,

multiplication): Λ(C4) ∼= Cl(4,C) ∼= M(4,C) ∼= C4 ⊗ C4 (¦)Cl(4,C) – Clifford algebra (Dirac-Hestenes equation, spinors as ideals and

equivalence classes in the Clifford algebra).Λ(C4) – Exterior forms (antisymmetric tensors), Grassmann algebra

(Dirac-Kahler equation, Dirac type tensor equation).Cl(4,C) ∼= M(4,C) (also as algebras) for any given matrix representation

of gamma matrices γk.Cl(4,C) ∼= Λ(C4) (as linear spaces) γi1 · · · γik ↔ dxi1 ∧ · · · ∧ dxik. (∗)M(4,C) ∼= C4 ⊗ C4 – may be formally identified with space of states of

two spinor systems (ψ ⊗ ξ).

Transformation PropertiesExterior forms ω ∈ Λ(M), differentials d and d? are covariant with respectto all linear transformations, group GL(4,R). So the Dirac-Kahler equationis general covariant. Usual Dirac equation is only Lorentz covariant.

Lorentz covariance for different forms of Dirac equation

L ∈ SO(3, 1), SL ∈ Spin(3, 1), ψ 7→ SLψ, Ψ 7→ SLΨS−1L (∗∗)

• For the Lorentz group SO(3, 1) ⊂ GL(4,R) natural transformationsof Λ(M) (as tensors) coincide with (∗∗) with respect to map (∗).

For model with two systems (∗∗) is: ψ ⊗ ξ 7→ (SLψ)⊗ (S−1L

Tξ), so

transformation of ψ is not “mixed” with auxiliary spinor ξ. The isomor-phisms (¦) of four linear spaces let us consider full group GL(4,R) and for

G /∈ SO(3, 1) state ψ becomes entangled with ξ:∑

α(Lα

Gψ)⊗ (RαGξ).


Singularities, Black, White, and Worm- Holes

One of “prima facie” (C. J. Isham) questions in quantum gravity:

It was suggested ‡, that the black hole evaporation may be related with

a transition from a pure state to a mixed state of a quantum system,

but how in principle to consider such a transition?

Gauge theories and gravity

For a space with a singularity there is a problem with definition of metric

on some subspaces. The (pseudo-)Riemannian manifold is the particular

example of the affinely (or linear) connected space. The linear connection

and the metric — are two different geometrical objects.

The linear connection is the particular example in more general theory of

connections on a principle bundles with an arbitrary structure group, but

this theory has some counterpart in the physics — the gauge theory.

In the general theory, it is considered a connection on a principal bundle

with some Lie group. For a linear connection the structure group is GL(4,R).

The important question — is the reduction of the structure group to some

subgroup.

For General Relativity, it is the reduction to the Lorentz group SO(3, 1)

— it is just the question, if it is possible to choose the atlas, there all

transitions between different maps may be described by transformations from

the considered subgroup. In General Relativity it is related with the question

about possibility to use only the transformation from the Lorentz group for

transition between different coordinate systems and due to a general theorem

of differential geometry, it is possible iff globally exist the Minkowski metric

and the tetrad field — it is possible geometrical treatment of the equivalence

principle in General Relativity.

‡(Before recent Hawking’s lecture in Dublin :)


r

t

rg

ds2 = g(r)dt2 − dr2

g(r)− r2dS2, g(r) = 1− rg

r.

Schwarzschild metric

t = t−∫

f (r)dr

g(r), r = t +

∫dr

f (r)g(r)

rrg

t f (r) =√

1− g2(r)

sin(ϑ) =

√rg

2r

ds2 = g(r)(dt 2 − dr2) + 2f (r)drdt− r2dS2.


ds2 =1

g(r)

(dt2 − (

1− g2(r))dr2

)− r2dS2

ds2 = g(r)(dt 2 − dr2)± 2√

1− g2(r)drdt− r2dS2

l = ±√

2r

rg− 1

ds2 =l2 − 1

l2 + 1(dt2 − l2dl2) +

4l2

l2 + 1dldt.− 1

4(l2 + 1)2dS2

My COSMION 2004 Slides

Education

Transcript of My COSMION 2004 Slides