Abstract

In four dimensions, the Minkowski metric leads to the 16 dimensional Clifford algebra C(1,3), Dirac equation is using four of these 16 matrices that form a basis of this algebra, a new operator is defined using all of these matrices and also generalized for a curved space

Multilevel operator

We are using Pauli matrices σ, electromagnetic four-potential A_μ and charge e with [1].

                                                                                                            (1)

In four dimensions, Minkowski's metric leads to the Clifford algebra C Dirac matrices

                                                                   (2)

Multilevel operator Dⁿ acts on level n, n is the number of matrices in the product of the algebra members, for example, D³ acts on . Total multilevel operator D^mul = D⁰ +D¹ +D² +D³ +D⁴, the action of D^mul on the spinor function vanishes

                                                                                                                                                                                                   (4)

                                                                                                                                                                               (5)

                                                                                                                                           (6)

                                                                                                                                                                                  (7)

                                                                       (8)

                                                   (9)

Multilevel operator D^mul(η_μν ,A_μ ,e) can be generalized for a curved space with four-potential P, field charge q and covariant derivative[3] (;_μ) instead of derivative (,_μ) in the definition of

                                                                                                                                                                           (10)

                                                                                                                                                                               (11)

                                                                                                                                                                                                  (12)

                                                                                                                                                                               (13)

                                                                                                                                         (14)

                                                                                                                                       (15)

                                                                                                                                                                     (16)

                                                                                                                                  (17)

For gravity is the new gravitomagnetic tensor. is the torsion tensor [4].

with                                                                                                                         (18)

with the Riemann-Christoffel tensor [5].                                                                                                (19)

                                                                (20)

with                                                                                                            (21)

                                                                                                                                                         (22)

                                                                                                                     (23)

Conclusions

Multilevel operator has been generalized for a curved space with a general four-potential P. For gravity is the new gravitomagnetic tensor and torsion tensor appears in its definition.

In a flat space and operators vanish. In a curved space the curvature tensor appears in levels 3 and 4.

The appearance of torsion tensor and curvature tensor in multilevel operator means that this operator is a fundamental operator in Quantum Field Theory.

References

1. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 637.
2. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 650-652. 
3. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 380. 
4. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 384. 
5. Marian Fecko (2006 Differential Geometry and Lie Groups for Physicists. Cambridge University Press 404.