The Dirac equation in a nonRiemannian manifold. I. An analysis using the complex algebra S. Marques Citation: Journal of Mathematical Physics 29, 2127 (1988); doi: 10.1063/1.527838 View online: http://dx.doi.org/10.1063/1.527838 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/29/9?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 12:48:38 http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1774531301/x01/AIP-PT/JMP_ArticleDL_022614/aipToCAlerts_Large.png/5532386d4f314a53757a6b4144615953?x http://scitation.aip.org/search?value1=S.+Marques&option1=author http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov http://dx.doi.org/10.1063/1.527838 http://scitation.aip.org/content/aip/journal/jmp/29/9?ver=pdfcov http://scitation.aip.org/content/aip?ver=pdfcov The Dirac equation in a non-Riemannian manifold. I. An analysis using the complex algebra s. Marques8 ) NASAIFermilabAstrophysics Center, Fermi National Accelerator Laboratory, MS. 209, P.O. Box 500, Batavia, Illinois 60510 (Received 9 December 1987; accepted for publication 27 April 1988) The Dirac wave equation is obtained in the non-Riemannian manifold ofthe Einstein­ Schrodinger nonsymmetric theory. A new internal connection is determined in terms of complex vierbeins, which shows the coupling of the electromagnetic potential with gravity in the presence of a spin-! field. I. INTRODUCTION The Einstein-Schrodinger (ES) nonsymmetric theory! was an attempt to geometrize, in an unitary way, the gravita­ tional and the electromagnetic fields. However, structure problems in the theory have not permitted a coherent inter­ pretation of the field equations. On the other hand, the tech­ nique of geometrizing fields introduced by Einstein contin­ ues to be a useful tool in classical field theory. Actually there has been some work2 where the electromagnetic field is made explicit; this is accomplished by first taking the skew-sym­ metric part of the metric as being proportional to the electro­ magnetic tensor and, then, adding a sourcelike term to the Lagrangian of the theory. This permits one to reobtain the Einstein-Maxwell equations through a correspondence principle. When we face the problem of developing a Dirac theory using the space-time manifold of the ES nonsymme­ tric theory, we use then complex vierbeins. This is equivalent to introducing an internal C space in the manifold of the general relativity theory. However, we have the problem of how to obtain a coherent mechanism that fits the correct rules for the transformation group. We can see, for example, that the way the problem was developed by Marques and Oliveira,4 in a study of the geometrical properties of C, Q, and 0 tangent spaces, does not permit the corresponding Dirac field equations to exist (we will consider the complex case presently). The reason is that the internal C connection was ignored when they introduced complex vierbeins. In­ stead, they generalize the tangent real connection (the one originating on the local real-tangent space) to a complex one. This induced them to generalize both the Lorentz group (to a pseudo-unitary group) and its representation U(L), for the generalized Dirac field theory. However, in spite of it being possible to show that, on the local tangent space, the trace of the symmetric part of the tangent connection should correspond to the electromagnetic field, there is no way to obtain the desirable correspondence to an (actual R) Dirac field theory. The problem can be solved by considering be­ sides the tangent connection, the connection corresponding to an internal C space. This forces us to maintain the Lorentz group as being that of the (local) space-time transforma­ tions on the (local) tangent space. On the other hand, we a) On leave from UNESP, Campus de IIba Solteira, Sao Pauio, Brazil. must also have an "internal" transformation, corresponding to the internal C space. In the space-time of general relativity it is possible to generalize the Dirac field equation by doing the transition yI'-yI'(x) , t/J.p-t/Jllp =t/J.P + Apt/J, where t/J(x) is now the electron wave function in the curved space-time, and Ap is the geometrical connection with rela­ tion to the internal space generated by the constant r matri­ ces {r). It is easy to show that Ap is given by Ap = ~q rV,rv.p] - {~v}[ rV,rp p , (1.1) where {~v} are the Christoffel symbols for the space-time connection. In terms of real vierbeins h ~ (and their inverses h ~), (1.1) is written as Ap = ( - 1140 (h vbh ~.P - {~v}h;h vb)lTab . (1.2) The function t/Ji (x), i = 1,00.,4, above, satisfies a Dirac equa­ tion defined on the curved space-time manifold of general relativity, which now has the form (1.3 ) where p is a mass coefficient. Thus the gravitational field is present in this equation through the connection Ap. The nonsymmetric manifold of the ES theory with the locally defined complex vierbeins referred to above will be used in this work. These vierbeins define new Fock-Ivan­ enko coefficients which permit the construction of the corre­ sponding Dirac equations related to the non-Riemannian manifold of the ES theory. In Sec. II we will present briefly the properties of the complex tangent space as well as the corresponding field equations obtained in the ES nonsym­ metric theory. In Sec. III the generalized Fock-Ivanenko coefficients will be determined, as well as the new Dirac equation. In Sec. IV we will proceed to their analysis. Throughout, we will use the p, v, ... indices as those on the non-Riemannian manifold; the a,b, ... indices will be those on the complex tangent space. II. THE COMPLEX TANGENT SPACE According to the correspondence principle there exists, at each point of the curved space-time of general relativity, a local tangent space5 with the structure of a flat space-time, with the metric given by the Minkowski tensor 'T/ab' Therefore, we must have the line element 2127 J. Math. Phys. 29 (9), September 1988 0022-2488/88/092127-05$02.50 © 1988 American Institute of Physics 2127 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 12:48:38 dfil = g,..v dx'"' dxv = 1'/ab dxa dxb, locally, where, g,..v = gv,..· In the ES nonsymmetric theory, the metric of curved space-time has the symmetry property ~v = gv,..' g,..v = g,..v + ik"v. Defining complex vierbeins e; (and their v inverse e';; ), we have that6 (2.1 ) (2.2) where 1'/ ab (and its inverse 1'/ab) is the metric of the tangent space which we take here as the Minkowski tensor. The met­ ric g,..v and its inverse g'"'v are such that g'"'Vguv = ~, where the order of indices are significant. From there, we obtain the orthogonality conditions for the complex vierbeins: e*;eIt = e;e*~ = 8: , e*;e~ = e;e*~ = 8;. (2.3) (2.4 ) As is well known, the transformation law for vectors in the complex tangent space, local to a curved space-time, is defined by (2.5) where LOb are the Lorentzian rotation matrices, which have the property L T1'/L = 1'/ (2.6) and, as e; (x) is a complex function, e; = e;R + ie;,. Then (2.7) is the conjugate of e;. This means that we have attached to the Minkowskian tangent space, an "internal space," the C space. The "internal" transformation law of an object ofthe C space, K, is K' = U(1)K, (2.8) where U( 1) stands for a unitary 1 X 1 (local) transforma­ tion matrix, U( 1) = eiq,(x>, and (2.9) where U( 1) = U -I ( 1) = e - iq,(x). A more general transfor­ mation law for the complex vierbeins now should be e';(x) = U(1)Lab(x)e!(x). (2.10) The covariant derivative of the vierbeins e; and e*; on this tangent space are now given by (2.11 ) (2.12) where Ava b is the tangent connection related to the Minkow­ skian space and Cv is the "internal connection." Their trans­ formation laws are, respectively, A~ = LAL -I - L.vL -I (space-time transformations) , (2.13 ) C~ = U(1)CvU- I (1) - U,v(l)U- 1(1) (internal transformations) . (2.14 ) Here Cv transforms as a vector under space-time transfor­ mations. Considering the particular case where we have only the internal transformations represented by the matrices 2128 J. Math. Phys., Vol. 29, No.9, September 1988 U( 1) = 1 + il/J, the internal connection Cv transforms in first order, as C~=Cv+il/J.v' (2.15) which is the same as a gauge transformation law for an elec­ tromagnetic potential. We also have the relation R p a sab_o (216) ,..vyep - vy be,.. - , . where R ~vy is the curvature in the non-Riemannian space­ time written in terms of the nonsymmetic affinity, and Svy is the curvature over the complex tangent space, Svy = Av.y - Ay.v - [Av,Ay] , (2.17) which is skew symmetric with respect to the curved-space indices and anti-Hermitian in the tangent space indices. The internal curvature can be also obtained: Pvy = Cv•y - Cy.v ' (2.18) Then, in the particular case of (2.15), the internal curvature can be considered to correspond to the Maxwell electromag­ netic tensor. One of the field equations of the ES nonsymmetric theo­ ry, obtained through a variational principle, is g" ';a = 0, + - where the symbol (;) means that the connection used in this equation is the Schrodinger connection, 1 OP,..v' o = OP"P = O. (The notation used in this work is the same ,.. v used by Marques and Oliveira in Ref. 4. It has been kept the tranditional use of the "+ -" covariant derivative used in the ES nonsymmetric theory, Ref. 1. See also, M. A. Tonel­ lat, Ref. 9.) (In general, 0 P,.. a is a nonsymmetric connection such that 0 *P,..a = 0 P a,..') This equation corresponds to the following vierbein equations: e~)a = (e*~a)* = e;.a - OP,..ae; + Aa abe! + Cae; = 0, (2.19) where (2.1) was used. Taking the inverse equation: " v g+ -;a = 0, and (2.2), we have the corresponding equations for the inverse vierbeins " (")* e*+. = e'"' = e*'" + 0'" e*P ala ala a.a pa a - Aa bae*~ - Cae*~ = O. We can rewrite Eqs. (2.13) and (2.14) as a. _ a 0'" e':. A b -1.t - 0 e"Ja - e,...a - pa a - a ae'b - , " e* +. = e*'" + 0'" e*P + A b e*'" = 0 ala a.a pa a a a b , where (2.20) (2.21 ) (2.22) Aa ab = Aa ab + 8:Ca , A* a ab = Aa °b - 8:Ca . (2.23) From (2.19) and (2.20) we obtain the relation (2.24) and from (2.23), Aa = Re[e;e*;'a] =Re[ -e~+ae*~], (2.25) Ca = ;(Im [e;e*;'a ] ) = i(Im [ - e~-i-ae*~]> . (2.26) Taking (2.25), we can expand it in terms of real and imaginary parts. We then obtain for Aa S. Marques 2128 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 12:48:38 = (e~ eIb + e~ eIb + e~ (Jpa'" eli: + e~ (Jpa'" eli: R R,a r-l I,a r-R _ R ,-1 _ I - e~ (J ':aeli: + e~ (J ':aeli: ) rl V R r-R V I (2.27) or Aa = Re [ - e;ae*t ] = ( - e~ eIb - e~ eIb + (J~aepa eIb + (J~aepa eIb r-R,a R r-l,a J !::....- R R !:..- I I + (J ~aepa eIb - (J ~aepa eIb ) . vR1 viR (2.28) Analogously, from (2.26), we obtain for Ca Ca = i(lm [e;e*;;a]) = i(e~ eIb - e~ eIb + e~ (J':aeli: + e~ (J':aeli: r-I R,a r-R l,a r-R V R r-I V J (2.29) or = i(e;R.aelb/ - e;/.aelbR - (J~ae;ReIb/ + (J~ae;/elbR - - (2.30) Suppose we have a theory where the antisymmetrical part ofthe space-time connection is zero, (JP"v = 0, but still v with complex vierbeins, i.e., a theory where we have a com- plex, antiysmmetrical part for the metric. We now obtain for the tangent and internal connections, and Av = (e;RelbR.V + e;/eIb/.v + e;Rr~veli:R + e;/r~ve'b/) = ( - e;R.velbR - e;/.velb/ + e;R r~velbR + e;/ r~velb/) (2.31) Cv = i(e;/elbR.V - e;ReIb/.V + e;/r~ve'bR - e;Rr~veli:/) = i(e;R.Velb/ - e;/.VelbR - e;Rr~velb/ + e;/r~velbR) , (2.32) where we used the notation r~v for the symmetrical connec­ tion. We can see that the relation of the (complex) metric, with the new complex vierbeins, adds new extra terms to the tangent connection. Also, the internal connection has a rela­ tion with the vierbeins, which would not exist if the vierbeins are real. It is noticeable, from (2.31) and (2.32), that the same happens in a "complex theory" without a complex tor­ sion term. It easy to conclude, as in Ref. 2, that the Einstein­ Maxwell theory is reached in a convenient limit such as to eliminate the complex part of the metric and therefore the corresponding complex ones for the vierbeins. However, some years ago, this fact was criticized by theoretical analy­ sis,3 which does not change the power of a geometrical analy­ sis. Thinking from a geometrical point of view, we will go forward, obtaining the (Dirac) field equations and see what we can get in this "complex theory." 2129 J. Math. Phys., Vol. 29, No.9, September 1988 III. THE GENERALIZATION OF THE FOCK-IVANENKO COEFFICIENTS The Dirac constant ymatrices satisfy the anticommuta­ tion relations {Ya'Yb} = 217ab 14' (3.1) {ya,i'} = 217ab14' (3.2) where 17ab (and its inverse 17ab ) is the Minkowski tensor with signature + 2, and to the relation ya,b = O. The set formed with combinations of Y matrices, {rJ = {14 , Ya' Uab = (i/2) [Ya' Yb]' Ys = YOYIY2Y3' YsYa}' composes a linearly independent set in the internal space of the Dirac wavefunctions ¢. Now, multiplying (3.1) bye*~ and e!, and using (2.1), we obtain (3.3 ) where g,..v is now the ES nonsymmetric metric. In (3.4) we have defined Y,.. and 1',.. by a _ .0 _. e,.. Ya - Y,.., e,.. Ya - Y,.. . (3.4) Analogously, multiplying (3.2) by e*~ and e~, we obtain {jI',yv} = 2g"v14 , where (3.5) ~ya = 1', e*~ya = jI', (3.6) and the relation (2.2) was used. The covariant derivative of l' (x) over the non-Riemannian manifold of non symmetric theory, is given by Y"lv = Y,...v - {lP,..vYp + [!:i.v'Y,..] , (3.7) + where !:i.,.. is the internal connection, corresponding to the space of generalized ymatrices (or, also, of Dirac wave func­ tions space), and {lP,..v is a more general space-time affinity (that, at least in principle, includes the internal connection C,.. ). Taking the identity (3.5) and the Eq. (2.19), we have that Y"lv = (e: Ya)1 = (e:lv)Ya = 0, (3.8) + -+-' v -+- since Ya is a constant matrix. In the same way, we obtain r"lv = (e*:lv)Ya = 0 . - - Expanding (3.8) and (3.9) we have Y"lv = Y,...v - (JP,..vYP + [!:i.v,Y,..] + CvY,.. = 0, + r,.jv = r,..,v - (JPv,.. rp + [!:i.v,r,..] - Cvr,.. = O. (3.9) (3.10) (3.11 ) We can observe then, from (3.10) and (3.11), that we obtain a relation similar of that of general relativity, i.e., !:i.v = (l/4i) A? Uab = (l/4i)Re[e;e*'":;v ]uab , or (3.12) !:i.v = (l/4i)Re[ - e;ve*,..b JUab , (3.13) where it was used (2.21) for Av. Ifwe now consider ¢(x) as the wave function of a spino! particle of mass m, placed in a non-Riemmanian manifold of ES nonsymmetric theory, ~(x) = ¢t Yo will be the wave S. Marques 2129 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 12:48:38 function of its antiparticle, and the corresponding Dirac wave equations are, respectively, yP(ap + ~p + Cp )'" - /1:'" = 0 , - ~(ap + ~p - Cp)Y' - Il~ = 0, where Il = mc/ll. (3.14 ) (3.15 ) The new operator (a,p + ~p + Cp ) comes from the co­ variant derivation of the function ",(x), which, besides being an object that transforms, locally, under the representation of Lorentz group [U(L)], also transforms under the (inter­ nal) UO) group. Equations (3.14) and (3.15) describe par­ ticles placed in a curved non-Riemannian space-time of the ES nonsymmetric theory, since the connections ~p and Cp are now related to complex vierbeins, as well as the complex space-time connection. Another way to obtain Eqs. (3.14) and (3.15) is through a minimal action principle. In this case the action is A = f X' d 4x, where the Lagrangian is given by X' =1=i[W("',p + ~p"'+ Cp"') + (~,p + ~~p - ~Cp )Y'''' - Il~"'] . (3.16 ) From (3.15), the wave equation for the charge conjugate function, ",C , is ( 3.17) where "'C = C~T, and C is the charge conjugate matrix. Therefore, if the wave equation of a particle is constructed with the set yP and (~p + Cp ), the wave equation for its "charge conjugate" will be constructed with the set Y' and (~p-Cp)' Let us write now the internal connection Cp as Cp = ileAl' (x) . (3.18) Then, after (2.15), we can interpret e as the electric charge for the electron, AI' (x) as the electromagnetic potential, and 1 will be a constant such that it balances units. Equations (3.14) and (3.17) can be written now as yP(ap + ~p + ie/Ap)'" - Il'" = 0, IV. CONCLUSION ( 3.19) (3.20) We have learned that complexifying the space-time manifold of general relativity is equivalent to attaching it an internal C space. The new metric is no longer symmetric and its antisymmetric part should be proportional to the electro­ magnetic tensor [see Einstein, Ref. 1, Eqs. (11 )-( 17) in Sec. III]. Through complex vierbeins, it is possible to obtain a (complex) tangent space local to the nonsymmetric curved space of the ES type. Using these concepts we obtained here Dirac field equations for a spin-! particle. The internal com­ plex connection corresponds to the electromagnetic poten­ tial. We observe that it is possible to define the complex vier­ beins as (4.1 ) 2130 J. Math. Phys., Vol. 29, No.9, September 1988 where K is considered now as a parameter, and A is a con­ stant. We use here, as in Ref. 2, Aa:~_c3e = 1.82Xl(p6 statvolt. L2 flG cm In a limit where the parameter K-.O, we obtain from (2.27) and (2.28) the real connection Av from general relativity. Using the above expression for the vierbeins, we can display the interesting behavior of the new Dirac equations that ap­ pears when we split up yP (x) in terms of its real and imagi­ nary parts, and also suppose a complex mass term: Il = IlR + illl , where we again can take III = KAm. Then, from (3.19). [ e';;R y"(ap + ~p + ielAp) '" + VIlR "'] +ikA. [n~y"(ap +~p +ielAp)"'+m"'] =0. (4.2) In the limit of the parameter K-.O, we should get the normal Dirac equations in the presence of gravitational and electro­ magnetic fields. Therefore, it means that we can get another identical set of Dirac equations if we take n~ == e';; - h ~, and R m == Il R' where h ~ and Il R will be vierbeins and the mass term of general relativity theory. ACKNOWLEDGMENTS The author thanks Mark Rubin, Richard Holman, Mar­ celo Gleiser, and Tom R. Taylor for helpful discussions. Also, the author is very grateful to Patricio S. Letelier and Robert Geroch, who read and critized the manuscript. This work was supported by National Research Council (CNPq ) at Universidade Federal do Mato Grosso, Brazil. This work was also supported in part by the DOE and by the NASA at Fermilab, and by CAPES-MEC Brazil. APPENDIX: COMMENTS ON A MORE GENERAL TRANSFORMATION LAW IN A TANGENT SPACE ASSOCIATED TO A COMPLEX INTERNAL SPACE Let us consider, instead of (2.10), a more general trans­ formation law for objects in the complex tangent space. Con­ sidering, for instance, the vectors e;, it can be defined as e/;(x) =LOb(x)e!(x) , (AI) e/*; (x) = L *abe*! (x) . (A2) The complex matrix Lab now is a kind of pseudo-Lorentz matrix that follows the relation V7JL = 7J. (A3) The covariant derivative of e; on this complex tangent space is then defined as (A4) (AS) where the affinity is complex. Its transformation law is A'=LAL-1-LL- 1 I' I' .1" A~* = L *A*L*-I - L!.L*-I . (A6) It is directly shown (see Ref. 4 that, through the Einstein field equations for the nonsymmetric theory (a complex the­ ory), g/. "" = 0, we obtain the same corresponding field + - equations for the vierbeins described in Eqs. (2.21) and S. Marques 2130 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 12:48:38 (2.22). However, we must also have 7Ju bl,. = 0, where the + - "minus" sign corresponds to the complex conjugate of the affinity AIL: 7Ju+ b}.,. = 7Jab.1L - AIL ca 7Jcb - A * IL \ 7Jac = ° . (A 7) As 7Jab lowers indices, we have that AIL is anti-Hermitian with respect to the index of the tangent space. Then, we have AlLab = A,,"b + iA"ab . v ~- (A8) The expansion of L in first order is, from (A.I )-(A.3), L~I+E+ijl, L-l~I-E-ijl, (A9) where E = E(X) are infinitesimal rotation matrices as before andjl = jl(x) are symmetric infinitesimal matrices. We can write the latter as jlab = (a + Ii Tr jl)ab , (A1O) where a is a symmetric trace-free matrix. Considering then, a particular transformation such that L~I+!K, K=Trjl, (All) the affinity Aa of this complex theory transforms as A'a = Aa - (i/4)K.a , Tr A'a = Tr Aa - iKa , (AI2) which is similar to the gauge transformation of an electro­ magnetic potential. (In the same way, we can show that the complex part of a nonsymmetric tangent curvature obtained with the above Aa will be related to the Maxwell electromag­ netic tensor.) Now, from (A 7), (3.8), and (3.9), we can easily obtain a relation between Aa and the connection Aa : A ab ac [,,(I) ] a eJ.