IC/8O/1O5
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
A PROTON MASS DOUBLET FROM GENERAL RELATIVITY
Mendel Sachs INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL. SCIENTIFIC AND CULTURAL ORGANIZATION 1980 MIRAMARETRIESTE
I. INTRODUCTION IC/8O/105 1) In support of the thesis of my research programme , which attempts to explain the behaviour of elementary matter from the axiomatic basis of general International Atonic Energy Agency relativity, I will demonstrate in this paper, in a semi-quantitative fashion, and that the earlier determined value (from a special relativistic form of the United Nations Educational Scientific and Cultural Organization theory) for a new fundamental constant (length), ,•* , that is necessarily INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS incorporated in the general form of the non-linear spinor formalism, is compatible with the magnitude of the proton mass, according to its explicit derivation from the globally covariant formalism.
It was found that when fully exploiting the algebraic (group) basis, as well as the geometrical basis of general relativity, one is led to a spinor field relation between the mass of elementary matter and its. explicit coupling to the matter if its environment - in full accord with the Mach principle - in A PROTON MASS DOUBLET FROM GENERAL RELATIVITY * terms of the curvature of space time that represents the material of the observed matter environment.
Mendel Sachs •• The irreducible group representations of general relativity theory International Centre for Theoretical Physics, Trieste, Italy. Imply that the basic matter fields are the (two component) spinor solutions of coupled non-linear field equations. In the linear limit, where the coupling vanishes, the latter matter field equations take on the formal structure of
ABSTRACT quantum mechanics. Conventionally, the inertia! mass is inserted into the latter in the form of the parameter A = Mc/fi. But in general relativity, This paper discusses features of the proton mass according to the incorporating the Mach principle, A is derivable from the field solutions author's non-linear, spinor field theory in general relativity. Within its of the metrical equations, which in turn are coupled to the matter field 2) context, where mass doublets are generally predicted for all spinor matter equations. In this regard, it was found that \ is a positive-definite field fields, it is shovn, in a semi-quantitative fashion, that 1) in addition to The manifestation of inertia of elementary matter follows from the the normal (stable) proton, there is a heavier proton that has a mass of "non-flat" features of apace time that represent its physical coupling to order 193 GeV, and 2) a fundamental constant, a ~ 2.087 * 10"11* cm, that other matter. Thus, In principle, when all other matter vanishes, the inertial vas determined earlier from a more general version of electrodynamics in mass of the observed matter correspondingly vanishes, specifically following which a short-range part plays a role in the nucleon domain, leads to a the approach of the curvature of space time towards flatness. prediction of the normal proton mass that is within 1.0% of its empirical value. It was found, in particular, that the mass of elementary matter relates to the spin-affine connection field !) , in accordance with the eigenfunction equation
MIRAMAKE - TRIESTE July 1980 (1)
The ftenerally variable eigenvalues \ are the squares of the masses of •- To be submitted.for publication. elementary particles in the limit as the tvo-component spinor solutions | I|I^ •• Permanent address: Department.
-2- the non-linear field theory, the distribution of values of X , correspondingly The electron-muon mass doublet was the first example investigated . 2 approach the discrete set of values {^ } - thus predicting a mass spectrum of It was found that the primary source of coupling that gives rise to the masses elementary particles. of the electron (and muan) is its electromagnetic interaction vith the components In the general equation (l), the spin-affine connection field SI of of the particle-antiparticle pairs of the physical vacuum. [The latter set of a (generalized) Riemannian space time, is defined in terras of the covaxiant countable pairs, in a particular state, derived from this non-linear field derivative of the two-component apinor variable in the curved space time theory, is .quite different from the "physical vacuum" concept of quantum field theory - though it plays similar roles in special cases .]
= 3 1(1 + n 1(1 (2) It was found that the mass eigenvalue for the heavy electron (muon) ,V U V is proportional to the fine structure constant a, and the normal electron mass
The invariant metric of this space time is the differential is proportional to a . Thus, the ratio m /m was found to be of the order of 3/2a ~206. The factor 3/2 arose from the positronium-like states of e~ - e
ds (3) of the background matter that couples to the observed electron/union . This theory similarly predicts that the proton spinor field must lie vhere q^ is the fundamental 16-component metrical field. It is, geometrically, the lower mass value of a mass doublet. Assuming again that the primary a four-vector, but, algebraically, each of its four components is a quarternion. source that gives rise to the space-time curvature (in terms of !i ) that is responsible for the proton mass, is its electromagnetic coupling of the scalar rather than the real number scalar (g dx ) % y of the conventional particle-antiparticle pairs, p - p~" of the physical vacuum, It is similarly Einstein formalism. The correspondence predicted, as in the case of the e-u doublet, that there must exist a heavy proton with inertial mass that is the order of 206 times that of the g normal proton mass. Since M ~ 0-938 GeV, its heavy sister should have a mass M" — 193 GeV. The lifetime of p' could then b.e estimated in a way as well as correspondences between the quarternion variables and all of the similar to that of the muon, dependent on the lifetime of excited proton- w tensors of a Riemannian space,were found and the field equations in q were antiproton pairs of the environment of an observed proton . shovo to incorporate Einstein's tensor field equations (in addition to six extra metrical field equations 3'' ')• y
It follows from the vanishing of the covariant derivatives of the q III. THE SHORT-RAHGE ELECTROMAGNETIC POTENTIAL AND K fields that the spin-affine correction has the explicit form It was found earlier X' that in the jth (coupled) matter field p T r q- (k) equation the electromagnetic Interaction operator is made up of two parts,
4 =
II. PROTOH MASS DOUBLET (5a) Because the mass operator on the left-hand side of Eq.(l) is a two- dimensional matrix field, with two-component spinor basis functions | + ^ , it generally predicts two distinct (squared) mass eigenvalues. Thus, general relativity theory predicts that spinor matter fields In curved space time i t J a = 1 must Incorporate mass doublets. That is to say, for each spinor particle, (5b) there must exist another one that is physically identical to it, except for mass (and stability). -3- J) is the usual term in the Lagrangian density that gives the Lorentz force potential to the form(e /r) where b = 32irag_ is the same order of -l>t ' density (predicting the Coulomb potential), except for 1) the automatic magnitude aa g^-^10 cm. This result followed by taking rejection of all self-energy terms (i = j) and 2) G{x-x') is uniquely (in this theory) the symmetric Green,'s function for D'Alembert's equation. K * (8) J/_ is an electromagnetic interaction that arises because of the where factorization of the Maxwell field formalism into a pair of tvo uncoupled two-component spinor equations <$; - j.-v K - (9)
v a 3 f> = e * T i (6) + and rr y* + Ygy Y3z)/r, where Yk are the Dirac matrices. The states •' of were found to have the exact form that is necessarily incorporated in this field theory. FQ are linear combinations of Dirac matrices and a « 1,2 denote the two separate equations ...... , ."t>/2r in the two-component spinor solutions f^. The latter interaction comes from a Lagrangian density that where * are the exact, states of Thus it was found that. I with thin automatically incorporates a new fundamental constant, g , that has the alteration) dimension of length. In the course of this research programme, it was found that the added electromagnetic interaction JL only plays an important <-«|l/r|*> <#|e-b/r'H*> role at small distances - that are the order of magnitude of ^ - This constant vas derived from the application of the total interaction (Eq.(5)) The next step is then to calculate the "short-rnnge" term <^I/P ' ' JJ.JiK "> to the hydrogen problem. It was found that uL has the effect of lifting determining where it approaches and exceeds the •••ffective interaction in Kq.iiOi the accidental degeneracy in the Dirac hydrogenic states, thus predicting It is seen that when r ^ 10~ cm, the ''ouloub force weakens aa e~b , the Lamb shift. The procedure vas to calculate the ratio (3S1y2 - SP^)/ while the short-range part < J^ ^ becomes relatively greater. Thus, this (SSi/o - 2P.,_) - which is independent of g^ to the accuracy required to generalized field theory of electrodynamics incorporates with the usual long- compare with the data. When it was discovered that this ratio agreed with range part of the electromagnetic force a short-range part that becomes -lit the experimental ratio, g_ was then determined from a fit to the Lamb significant at r ^10 cm - the domain of nucleon dimeneions. As momentum in splitting (2Si/o ~ ^i/p)' ** vas found to have the value transfer betveen interacting matter becomes increasingly greater (as r becomes smaller) the short-range force in this theory of electromagnetic forces (£.067 ± 0.001) * 10 cm . (T) totally dominates the long-range part- as is tht- case where the nuclear and weak interactions dominate the electromagnetic interaction. With this constant fixed, the theory with -, present, was applied 7) U 8) At short range, where experimental physic:: reveals that the proton to electrodynamical problems of e-p scattering , e-He scattering , has finite extent - expressed conventionally in If-rms of its form factors - p-nueleus scattering and to the low-lying energy levels of atomic helium while the electron may still be considered to be a "point particle", the One of the main points of this paper is to take note of the fact that magnitude of the spin-affine connection field is the order of 1/R . This the magnitude of * g_ is very close to the Compton wavelength of the proton, is because fl^ is a measure of the curvature of Lipace time that expresses the IB and to associate this with the expression (l) for the mass eigenvalues of coupling of other matter to the observed matter that gives the latter mass. When the short-range interaction . becomes significant, this is the primary elementary matter. c ? contribution to (1 for the proton, because of the nature of the nucleon. Tt An interesting development from the generalization (5) of the electro- happens, as discussed above, when r is of the onier of a . magnetic interaction is the (effective) alteration of the (1/r) Coulomb
-5- Mote that in the earlier analysis of the e-ii mass doublet , the REFEKENCES critical domain of curvature that determined the order of magnitude of the electron mass was ~10 cm - Indicating a smaller "form factor" radius for 1) A review of this work, up to 1971, was published in Int. J. Theor. Phya. the electron and muon, though not zero. a) i, U33 (1971); b) kj S+53 (1971); c) 5_, 35 (1972); d) -5_, l6l (1972).
With SI -*1/K for the nucleon {proton) mass field, Eq.(l) reveals 2) M. Sachs, Nuovo Cimento =|3E, 398 (1968). that 3) M. Sachs, Nuovo Cimento U7_, 759 (1967).
(11) U) M. Sachs, Huovo Cimento 55B, 199 (1968).
5) M- Sachs, Huovo Cumento JJL, 2l»7 (1972). Using the value in Eq. (7) for (» (determined empirically only from the Lamb splitting in hydrogen), Eq.(ll) then gives 6) M. Sachs, Nuovo Cimento IPS, 339 (1972). 7) M. Sachs and S.L. Schwebel, Nucl. Phys. U3_, 3QU (1963).
GeV 8) M. Sachs, Huovo Cimento 5JA, 56l (1968).
9) M. Sachs, Lettere al Nuovo Cimento £T_, 586 (1980). The experimental value is 10) H. Yu and M. Sachs, Int. J. Theor. Phys. 13, 73 (1975). 0.938 GeV
Thus, the theoretical estimate of the present analysis is within l.Qjt of the empirical value for the proton rasas.
While this analysis gives an order of magnitude result, a more detailed 3ttidy of the field equations vhose solutions yield the explicit variables in Eq.(l.) must yet be carried out to make a prediction about precise magnitudes - such as the linear approximation method, applied to the e-u doublet in earlier work , Still, the result obtained above should, on physical grounds, give the correct order of magnitude of the interaction domain that determines the proton mass. This is verified here in the close agreement obtained with the empirical proton mass.
ACKNOWLEDGMENTS
The author would like to, thank Professor Abdus Sal am, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
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