The idea of internal mass terms introduced in ref. (1), 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 psi is Taylor expanded in a small parameter p

Marques-Bonham, Sirley

English

The idea of internal mass terms is shown not 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 /psi/ is Taylor expanded in a small parameter p

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A new way to interpret the Dirac equation in a non-Riemannian manifold

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Fermi National Accelerator Laboratory me 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 , ; ;  1 1 -  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: Y A Y  OC I l i 3 E B Y B E T  THE ~ 8 9 - 2 5 t t 3  ZIBAC E Q D A T I C A  I b  A B C I - B I E C L L L ~ I ~  B A I I V C L D  
(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 
https://ntrs.nasa.gov/search.jsp?R=19890016292 2020-03-20T01:24:31+00:00Z
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 prop- 
erties 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: 
7 p  = 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 1 0 3 % ~ 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 
m o = m i ~ p o R  , 
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. 
2 
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: 
T O k ~ o ~ 7 ~ v p $ l +  ~ ~ ~ k ~ o & V p +  - TOP+ + T ; k r i T a V p +  = 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. 
3 
ACKNOWLEDGMENTS: 
The author is very grateful to Jean-Michel Maillet (Theory Group, CERN), for conver- 
sations 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. b t e u r ,  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). 
4 


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