Abstract

In four dimensions, the Minkowski metric η_μν = diag (+1,—1,—1,—1) leads to the 16-dimensional Clifford algebra C(1,3), Dirac's equation [1] 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. This new multilevel operator generalizes the Dirac's equation, the value of the generalize Dirac's operator is calculated in the Schwarzschild's metric

Key Words

Gravitomagnetic Tensor; Gravitational Magnetic Field; Energy Momentum 1-Form; Clifford Algebra; Dirac Equation; Dirac Operator; Gravity and Quantum Mechanics Unification; Multilevel Operator; Schwarzschild's Metric

Introduction

Dirac's equation is the relativistic wave equation derived by physicist Paul Dirac in 1928. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrodinger equation which described wave functions of only one complex value.

Dirac's operator is just the tip of the iceberg, the tip of a generalized operator that is obtained by operating on all members of the Clifford algebra basis and not just on four of them.

The Schwarzschild's metric is named in honour of Karl Schwarzschild, who found the exact solution in 1915 and published it in January 1916, a little more than a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial at space solution. Schwarzschild died shortly after his paper was published, as a result of a disease he developed while serving in the German army during World War I. Johannes Droste in 1916 independently produced the same solution as Schwarzschild.

Schwarzschild's metric is an exact solution to the Einstein's field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant is all zero.

The new generalized Dirac's operator, the multilevel operator, is calculated in the Schwarzschild's metric, torsion tensor and new gravitomagnetic tensor appear in level 2, curvature tensor appears in levels 3 and 4.

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 η_μν = diag (+1,-1,-1,-1) leads to the Clifford algebra C(1,3)[2],  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 γ¹⁰, γ¹¹, γ¹², and γ¹³. Total multilevel operator = D⁰ +D¹ +D² +D³ +D⁴, the action of on the spinor function vanishes ψ=0                                      3

                                                                                                                                                                                                                                            4

                                                                                                                                                                                                                        5

                                                                                                                                                                               6

                                                                                                                                                                                                                           7

                                                                                                                                                         8

                                                                                                                                           9

Multilevel operator can be generalized for a curved space with four-potential P, field charge q and covariant derivative [3] instead of derivative  in the definition of

                                                                                                                                                                                                                      10

                                                                                                                                                                                                                           11

                                                                                                                                                                                                                                             12

                                                                                                                                                                                                                         13

 with                                                                                                                                                                                     14

                                                                                                                                                                                  15

                                                                                                                                                                                                                16

                                                                                                                                                                             17

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

                                                                                                                                                               18

 with  the Riemann-Christoffel tensor [5].                                                                                                                                       19

                                                                                                         20

                                                                                                                                               21

                                                                                                                                                                                         22

                                                                                                                                                               23

Gravitomagnetic Tensor Defined in Schwarzschild's Metric

We are using  with , this metric is defined by [6]

                                                                                                                                        24

                                                                                                                                 25

                                                                                                                          26

                                                                                                                                                                                                                               27

                                                                                                                                                                                                                       28

                                                                                                                                                                                                                           29

                                                                                                                                                                                                                              30

                                                                                                                                                                                                                                    31

                                                                                                                                                                                                                                       32

                                                                                                                                                                                                                          33

                                                                                                                                                                                                                          34

                                                                                                                                                                                                                      35

                                                                                                                                                                                                                     36

                                                                                                                                                                                                                 37

                                                                                                                                                                                                                 38

                                                                                                                                                                                                  39

, from equations (16) and (17)                                                                                                                   40

                                                                                                                                                                    41

                                                                                                                                                         42

                                                                                                                                                        43

                                                                                                                                                       44

                                                                                                                                                                    45

                                                                                                                                                                 46

                                                                                                                                                                   47

Energy-momentum form is a 1-form [7]                                                                                             

                                                                                                                                                                                           48

dp is a 2-form

                                                                                     49

                                                                                                                                                                                                                      50

                                                                                                                                                                                                                      51

                                                                                                                                                                                                                      52

Comparing equations (41-43) and (50-52) we can infer

                                                                                                                                                                                                                                       53

 is related to the scalar 0 -form  is related to the Energy-momentum 1-form, is related to the Electromagnetic 2 -form, is related to
3-form [8].

                                                                                                                                                                                                                  54

is related to L 4-form [9].                                                                                                                

                                                                                                                                                                                          55

 is the proyector matrix, historically

                                                                                                                         56

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.

, have been calculated for Schwarzschild's metric, then , the gravitomagnetic tensor has been obtained

Each, where n is the number of γ matrices in the product of the algebra members, is related to an n-form.

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.
6. P.A.M. Dirac (1996) General Theory of Relativity. Princeton University 30-32.
7. J.A. Wheeler., C. Misner., K.S. Thorne (2017) Gravitation. Princeton University 91.
8. Roger Penrose (2006) El camino a la realidad. Random House Mondadori, 603.
9. J.A. Wheeler., C. Misner., K.S. Thorne (2017) Gravitation. Princeton University 119.