me Fermi National Accelerator Laboratory 

FERMILAB-Pub-89/147-A 

June 1989 

A NEW WAY TO INTERPRET THE DIRAC EQUATION /A//,/' 4 

,' ,J 

I^ 

IN A NON-RIEMANNIAN MANIFOLD 

f,;; 11- 1 

Sirley Marques-Bonham' /; -/ ,'< 

NA SA/Fe rmila b A strop h ysics Center 

Fermi National Accelerator Laboratory, M.S. 209 

J/?J 7-37 

/ ,'3 

,> P.O. Box 500 

Batavia, Illinois 6051 0, US. A. 

and 

Departamento de Campos e Partt/culas 

Centro Brasileiro de Pesquisas Fakcas 

Rua Xavier Sigaud, 150 

22.290 - Rio de Janeiro, RJ - Brazil 

Abstract 

The idea of internal mass terms introduced in ref. [l], is shown not to be an appropriate 

hypothesis when it is placed in connection with the components of the generalized (matrix) 

vierbeins being proportional to the Riemannian (gravitational) vierbeins. It would result 

in an undesirable canceling of the Electromagnetic and the Yang-Mills components in the 

generalized metric. Another hypothesis is introduced where the wave function 11 is Taylor 

expanded in a small parameter p. 

IZ1SA-CB-185464) A PEL: YAY OC Ili3EBYBET THE ~89-25tt3 ZIBAC EQDATICA Ib A BCI-BIECLLL~I~ BAIIVCLD (lermi Eational AcceiErator I&.) 5 p 

CSCL 12A Unclas 

(33164 02 19934 

'This work was rupported by FAPERJ, Brazil, at the Centro Brssileiro de Pesquisss Fisicss, Rio de 

Janeiro, Brazil. This work was also supported in part by the DOE and by the NASA, grant number 

NAGW-1340, at Fermilab. 

a Oporrtod by Unlversities Research Association Inc. under contract with the United States Department of Energy 
I. Introduction. 

In the second part of the work entitled: “The Dirac Equation in a non-Riemannian manifold”, 

ref. [l], a generalized Dirac equation was obtained as an application of the geometrical properties of a tangent space local to a non-Ftiemannian manifold of the Einstein-Schroedinger 

(ES) nonsymmetric theory [Z]. This generalized Dirac equation describes a spin 1/2 particle 

of mass m, with a wave function +(z), placed in the above non-Ftiemannian manifold, and 

under the influence of an (n-dimensional) Yang-Mills field [4]. This is written as: 

where, 

V,,=a,,+A,,+C~+r, , 

v~=~,,+A,-c,,-~, . (1.2) 

These are the (covariant) differential operators acting on the wave function +, where A,,, C,,, 

I?,, are the connections for the Dirac, complex and Yang-Mills - internal spaces, respectively. 

In (1.1) the generalized Dirac y-matrices are defined as: 

7p = Ef7” 9 

where the generalized (matrix) vierbeins can be expanded in terms of the internal axes as: 

where n is the dimension of the symmetry group, here SU(n) (as in the Yang-Mills theory). 

With the idea of making a possible interpretation of the theory, two suppositions were 

then made. One of them is that the mass term p in the Dirac equation also be a matrix-like 

term: 

P = POTO + Piri * (1.5) 

1 
and 

This has the effect splitting the Dirac equation into na parts, one corresponding to each of 

the Dirac-space axes. 

The second supposition determines the amplitude of each of the (split) Dirac equations. 

This is done by the following definitions': 

kti = pi = i(pX)lmi 

i = I, ..., na - 1 

In thie case p ie considered a parameter, and A is a constant with the (minimal) value of l/p, 

e 

2ii x - - = 2.58 x 103%~1 , 

where the maximal value for I p I is I p I=[ 

in ref. [3]. 

I= 3.8 x 10-3acm, in the normalization used 

Therefore, when the above quantities are placed in the Dirac equation (l.l), we obtain: 

In the limit of the parameter p -+ 0, we would get the standard Dirac equation in the 

presence of gravitation, electromagnetism, and Yang-Mills fields. 

Suppose now that: 

':OR = nfo = nfi - hf , 

and 

mo=mi~poR , 

for each i, and where hf and poR are taken as the vierbeins and mass term of General 

Relativity. This situation will produce projections of the Dirac equation on the internal space 

axes proportional to the one obtained in GR. The intensity, or amplitude, of this projection 

will be determined by the parameter p. However, it is easy to show that this special choice 

cancels the antisymmetrical parts of the generalized metric of the theory, which means the 

'With these definitions we split the generalised Dirac equation in n2 + 1 parts. 

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electromagnetic and Yang-Mills terms of the matrix metric would be eliminated. This is not 

desirable, as it would imply the canceling of the related terms in the Lagrangian density 

used for the Action principle. 

XI, An interpretation of the Dirac equation through 

the expansion of +. 

Let us propose now, that the mass term be maintained on the unitary axis (TO), and that 

the wave function be expanded in powera of the parameter p, which is supposed to be small. 

The expansion of + can be written as: 

Also, the Dirac equation (1.1) can be expanded in terms of its components on the internal 

axis as: 

TOk~o~7~vp$l+ ~~~k~o&Vp+ - TOP+ + T;kriTaVp+ = 0 (2.2) 

Now placing (2.1) in (2.2), and separating terms in orders of p, we obtain the following 

system of equations: 

p3 : ... and so on... 

Therefore, in the limit p + 0 we would obtain the Dirac equation as in General Relativity. 

The advantage of this proposition in comparison to the former splitting of the mass 

term, is that the zero-order wave function $0 corresponds exactly to the GR-Dirac equation 

in presence of the Electromagnetism and Yang-Mills fields, when we take kroR as being 

equivalent to the vierbeins of GR. The other equations should be interpreted as being the 

result of a more exact Dirac equation, that is corrections to the Dirac equation obtained in 

a Gauge theory local to a curved Riemannian space-time. 

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ACKNOWLEDGMENTS: 

The author is very grateful to Jean-Michel Maillet (Theory Group, CERN), for conversations on this subject during his visit to the Theory Group at Fermilab. 

References 

[l] S. Marques, Fermilab-Pub-88/50-A, submited to the J. of Math Phys.. 

[2] A. Einstein,. Rev. Mod. Phys. 20, 35 (1948); “The meaning of Relativity”, Princenton 

Univ., N. Jersey (1955), Appendix 2; E. Schroedinger, “Space-Time Structures”, 

Cambridge Univ. Press, London (1954); M.A. Tonellat, “Les Thdories Unitaires de 

I’hlectromagnCtisme e de la Gravitation”, Gauthier Villars. bteur, Paris (1965); 

W.B. Bonnor,Proc. R. SOC. 226 (A), 336 (1954); J.W. Moffat, D.H. Bod, Phys. 

Rev. D11, 1375 (1975). 

’ 

I 

[3] K. Borchsenius, Phys. Rev. D 13,3554 (1976); J.W. Maluf, S. Okubo, Phys. Rev. D31, 

1327 (1985). 

~ [4] C.N. Yang, R.L. Mills, Phys. Rev. 96, 191 (1954). 

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