Extended gauge theories

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Extended gauge theories R. Aldrovandi Citation: Journal of Mathematical Physics 32, 2503 (1991); doi: 10.1063/1.529144 View online: http://dx.doi.org/10.1063/1.529144 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/32/9?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: On: Mon, 17 Mar 2014 13:04:04 Extended gauge theories R. AldrovandP) Laboratoire de Physique Thborique (Unitb Associke au CNRS no 769). Institut Henri Poincark, Universit6 Pierre et Marie Curie, II, rue Pierre et Marie Curie, 75231 Paris Cedex 05, France (Received 22 January 1991; accepted for publication 7 May 1991) A scheme inspired in Lie algebra extensions is introduced that enlarges gauge models to allow some coupling between space-time and gauge space. Everything may be written in terms of a generalized covariant derivative including usual differential plus purely algebraic terms. A noncovariant vacuum appears, introducing a natural symmetry breaking, but currents satisfy conservation laws alike those found in gauge theories. I. INTRODUCTION The phenomenological successesof the standard model encompassing the Weinberg-Salam model and QCD brought about an optimistic hope that gauge theories would provide the clue to the whole question of fundamental processes.It was anticipated that gravitation would also submit to ajoint picture and a more general, unified theory would be found including and surpassing both the standard model and General Relativity. However, as the years went by, it was realized that such high expectations were not being fulfilled. The large number of free parameters in electroweak theory, the difficulties with grand unification, the lasting frustation concerning the quark confinement problem and the dogged resistence of gravitation to submit to the standard gauge scheme have progressively changed those feelings. Experimental results are more then enough a guarantee that gauge theories are fundamental indeed, but theory would say that they must be somehow enlarged, the present dominating trends being those involving strings and supersymmetry. We wish here to examine the first steps into another, quite different kind of generalization of gauge theories, inspired in the theory of Lie algebra extensions. Experience in gravitational gauge modeling strongly suggests a coupling between spacetime and gauge space, while gauge models presuppose a strict local separation between them. The aim will be to find an acceptable compromise between these two conflicting positions. The general ideas are exposed in Sec. II, where attention is called to the fact that independence between space-time and gauge space is equivalent to the adjoint character of the gauge potential. It follows in Sec. III a purely descriptive resume on the subject of Lie algebra extensions,“2 in reality an adaptation of material on group extensions3v4 which is more abundant in the physical literature. In the next section some homological language’ is introduced and the previous results translated into it. The tone is rather pedagogical, introducing algebraic terminology via analogy with the supposedly known language of differential forms. Although modern treatments of the subject make use of general modules,6 we prefer to follow here the physically more intuitive J’ On sabatical leavefrom Instituto de Fisica Teorica, UNESP, SaoPaulo, Brazil. aproach using representations (or better, actions). This also allows us the consideration of the casein which neither algebra reduces to a module of the other and leads to a difference with respect to usual treatments, with the use of algebravalued cochains instead of the module-valued ones. “Noncommutative” modules are extensively used in noncommutative geometry,’ so that our approach is more akin in spirit to cyclic cohomology. Progressive introduction of bundle language hopefully paves the way to a later comparison with the geometrical approach to gauge theories. In order to keep notation and language at a reasonable level of simplicity, we adopt a rather free way of speaking, forgetting about sections, pull-backs, etc. whenever they are not essential. An algebraic derivative appears, in terms of which cohomology of representations has a treatment formally similar to the cohomology of differential forms. Group extensions have been used3 in the sixties to provide the formal proof of the socalled no-go theorems, or the theorems of McGlinn type.8 Such theorems forbade coupling between internal and spacetime symmetries at the algebraic level, an interdiction circumvented by gauge theories through the introduction of local vector fields. A brief outline on vector fields on manifolds and of the bundle structure of gauge theories is given in Sec. V. In this case, of course, usual differentials are also at work, and it turns out that everything can be written in terms of generalized derivatives, sums of usual and algebraic derivatives. Finally, the general field equations leading to what we claim to be the natural generalizations of Yang-Mills equations are given in Sec. VI. Generalized source currents satisfy, rather surprisingly, a conservation law analogous to the covariant divergenceless of currents in gauge theories. II. GAUGE SPACE AND SPACE-TIME Separation between space-time and internal space is inherent to the subjacent geometrical structure in the gauge scheme, a differentiable fiber bundle which is a smooth manifold combining “internal” gauge space and space-time in such a way that the total space is locally a direct product of both. In very simple words, disregarding sections and pullbacks, around any point of the bundle there exists a neighborhood on which a “separated” basis {X,} = {X0,X,> is defined for the local vector fields, the first m ( = space-time dimension) fields X0 representing a basis for space-time 2503 J. Math. Phys. 32 (9), September 1991 0022-2488/91/092503-10$03.00 @ 1991 American Institute of Physics 2503 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: On: Mon, 17 Mar 2014 13:04:04 fields and the remaining Xi representing a basis for the gauge group algebra. In such “direct product” basis’ the members will have commutation relations of the form [X0&] =fCo*xc, [X,26] = 0, (2.1) [X,,X, ] = cj,,q. The right-hand term in the first relation represents merely the possible anholonomicity of the space-time basis, which we shall keep for a while because it gives a similar aspect to the two algebras involved. The vanishing of the second commutator signals the separation between space-time and the gauge space, the direct-product character of the association of the algebras generated by the X,‘s and the X,‘s: In pedant language, the fields Xi’s respond to the action of the X0’s according to the null representation. Equations (2.1) summarize the basic, underlying background into which the gauge potentials are to be inserted. In the presence of a connection (or gauge potential) a = X,aj the X@‘sare replaced by the covariant derivatives XL = X0 - CCX’~,, in terms of which the commutation relations become [x:,x;] =fC&X; - Fj&Xj, [x;,xj] = 0, (2.2) [X,,X,] = C’,,x,. This new basis is sometimes called the “horizontal lift” basis. The fields {X,) keep forming an ideal of the total field algebra and a representation of the gauge group algebra. The coefficients F" uh=Xo(ahh)-XXb(ah,)--fc,,ah,-CC~a',a/h (2.3) represent the connection curvature (gauge field strength). The vanishing of the second commutator requires now that X, (aj, ) = Cihiako, (2.4) a condition also used to obtain (2.3). Of course, this only says that a belongs to the adjoint representation and is the expected behavior of a connection under infinitesimal gauge transformations, but it is important to notice this relation between a's adjointness and the independence between “internal” gauge space and space-time. Fields on manifolds generate one-dimensional local transformation groups. A field Y responds to the action generated by a field X according to the Lie derivative L, Y = [X, Y ] and the commutation relations describe the actions of transformations generated by the fields on each other. The second relation says that space-time fields do not “feel” gauge transformations and gauge fields do not “feel” space-time transformations. Anholonomicity being unessential, the X, in (2.1) may be seenas representing the generators of the group of translations on space-time. The presence of a connection modifies these translation generators to Xi, thereby altering space-time homogeneity. Through the Lie derivative, the presence of the fields X, will affect every tensor on the bundle. Ethos of such effect will arrive at the associated bundles, formed by space-time combined with group multiplets to which particles are attributed. Each multiplet will carry a representation of the gauge group, The covariant derivative on such associated bundles will have, instead of the fields Xi, their representative operators acting on the given representation. The gauge generators Xi, kept unmodified, do not respond to space-time transformations at all. We would of course expect something different were we to build a gauge model for the space-time symmetries themselves. In reality, the local underlying geometrical structure of gaugetheories iscompletely fixed by the above commutation relations, becauseof the Jacobi identities. The Jacobi identi- ty for three space-time fields Xi, X ;, X : gives the Bianchi identity [x~,[G?-q] -I- [x:,[xhG]] = -{X;W”,,) +XfW,,,) + [X&crx~]] +X;V’“,,) +fd,,F’,, +fda,Fj,, +fdbcFjud~X=O, or, in invariant language, (2.5) dF+ [a$] =o. (2.6) The Jacobi identity for three fields Xi, Xi, and Xi gives [x~~[xL?xj]] + [xi,[xL>x;l]] f [x~,[X,lX~]] = - fK,(Fj,,) + C’ihFh,,)Xj = 0, (2.7) which simply statesthe covariance of the curvature under gauge transformations: also F belongs to the adjoint repre- sentation Consequently, the commutation rules (2.2) do contain the basic geometrical background, to which of course dynamics is to be added. We can introduce dynamics through a Lagrangian or by the “duality rule” which states” that the field equations are, in the sourceless case,just (2.5), (2.6) written for the dual Fof F, X,Fiah+fcabFjac~- Cjkiak,FiUb = x : F'"b + f', q-j", = 0, which is the same as dFi- [a&=0. (2.8) Noether source currents are then added to the right-hand side. This rule has the advantage of giving the correct Geld equations even when no Lagrangian is present,” as is the casewhen the gauge group is non-semisimple.‘2 From (2.8) follows a severe constraint on the source current J, dj- [a,& =O. (2.9) This property ensures the gauge invariance of the total (gauge field plus sources) system. The commutation relations above suppose a very special association, on the bundle, of the two algebras {X,} and (Xi) of the representatives of space-time fields and gauge group left-invariant fields. Algebra associations are the object of the theory of Lie algebra extensions, We shall see below how this theory, besides providing some new insight on gauge theories in general, suggests a modification of the above relations to [XlZi] =fcabXf -F”abXj -P/,X,> [x:,x,] = CjujXj, (2.10) 2504 J. Math. Phys., Vol. 32, No. 9, September 1991 R. Aldrovandi 2504 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: On: Mon, 17 Mar 2014 13:04:04 [X,,X,] = c&q. The second relation admits now a coupling between gauge and space-time transformations, of which the additional “curvature*‘,8 is a measure. The (modified) translation generators act on the gauge generators no more through the null representation, but according to a representation determined by the coefficients Cj,,. Some coupling of the sort must exist if gravitation is to be given by a gauge theory related to space-time symmetries, obviously not indifferent to translations. III. LIE ALGEBRA EXTENSIONS Let L be a Lie algebra with generators J, and commutation relations [Jct,J,] =fCobJc. (3.1) Consider a vector space Von which a representationp of L is defined. p is a mapping of L into the set of automorphisms of K p:L+Aut( V), p:J, -p(J, 1. It is helpful to take a basis {X,) on Vand seeeachp (J, ) as a matrix with elements [p(Jo )I’, = C/i, with the representation acting according to (3.2) p(Jo 1Cx,1 = C’aixJ- (3.3) More precisely, p is an action of L on V, that is, a mapping p:L 8 v- v, leading each pair (J,, Xi ) into C’,X,EV. It is preferable to use the word actioninstead of representation(which we shall nevertheless be using rather freely for its intuitive value), since J, -+p(J, ) is not aprioria homomorphism. Whenp is a homomorphism, Vis the carrierspaceof the representation. As long as V is simply a vector space, V is an L module, but we shall below drop this condition. Let US recall that a Lie algebra L consists of an underly- ing vector space on which an antisymmetric internal oper- ation [, ] L is defined which satisfies the Jacobi identity. In particular, any vector space like Vabove may be considered as a commutative Lie algebra, generated by matrices X, such that [X0X’] F’= 0. (3.4) It is more economical to consider once and for all the general case coming out when Vis not simply a vector space, or a commutative algebra, but a nontrivial algebra by itself. This means that we have instead [x,,xJ] ,’ = ‘ /‘ ,J& (3.5) with C ly some structure constants. Of course, V is then no more a simple L module and we actually have the action of an algebra on the other (which modern authors prefer to call an operation’” ). A particular example occurs when L= V and the matrices (3.2) with elements [p(J, )I’, =fCoh gen- erate the adjoint representation of L. Consider now the direct srlm L $ V of the underlying vector space of L and the underlying vector space of V. How can we combine the algebras L and V to get a larger Lie algebra E with underlying vector space L $ V? The direct product case is well known, which appears when, once absorbed into E, both L and Vconstitute subalgebras ignoring each other: every element of L commutes with every element of V. The general answer is, however, that L and V may be combined in various ways to give different algebras E (the extendedalgebras), depending mainly on the action of L on E. We want to obtain E as an “extension of L by V” and now proceed to discuss under which conditions E deserves such a name. To begin with, once L and Vare “immersed” in the larger vector space L EI V, the operation [ ,lE in E does not necessarily coincide with those ofL and V.Furthermore, once [ ,] E is defined, there will be conditions on the structure coefficients, coming from the Jacobi identities. And then, when such conditions are satisfied, the particular algebra E obtained depends on the action of L on V, on how the origi- nal representation behaves when considered in the enlarged space. A first condition, justifying the expression “extension by V,” is that V be simply included in L EI V. More formally, the mapping i: V-E is an inclusion and preserves the algebra V.This means that i is an algebra isomorphism and the E bracket [ ,] E will, when restricted to the {X,}, coincide with that of (3.5): [Xi,XjlE = [Xi,Xj] y = C”,X,Ei( v). (3.6) This is the action of i( V) on i( V) itself. We shall identify i( V) = Vas long as it does not lead to confusion. Concerning the insertion of L into E, things are more interesting when the mapping a:L + E, o:J, --+X0, taking L into E is not necessarily an algebra homomorphism. Because we wish to consider the extensions coming through the given representation p, the automorphisms representing the action of L on i( V) are written [X0,X,]. =p(J,)(X,) = CiuiTG( V). (3.7) This expression and (3.6) say that i( V) is a normal subalge- bra (an ideal) of the extended algebra. The commutation rules of E will be (3.6), (3.7) and the expression for [X, ,X, ] E. The X,‘s would provide a linear (or uector) rep- resentation of L if they just mimic the behavior of the JO’s, [X,&]. =fcotlL (3.8) but in fact this is not necessarily the case for a general action of L on a(L). The general relation emerging is of the form [xo,xb] E =fCohXc -Bjubxj9 (3.9) the last term measuring the homomorphism breaking. This expression may be viewed as depicting automorphisms of E acting on the XU’s. In this case, when acting on its own representatives X,, the automorphisms induced by the action ofL do not necessarily yield results restricted to u(L). This sub- space is not necessarily a closed subalgebra: the /?‘U,,‘s are 2505 J. Math. Phys., Vol. 32, No. 9, September 1991 R. Aldrovandi 2505 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: On: Mon, 17 Mar 2014 13:04:04 precisely some constant (in the present purely algebraic context) components measuring the departure of the action from a(L). For linear representations, p’,, = 0 and CTwill be a Lie algebra homomorphism. When V is commutative, representations satisfying ( 3.9) are usually called projective representations in Physics literature.‘4*15 We shall eventually usethe same terminology in the present more general context. deficit, the real short-term interest rate, the expected inflation rate, or the world interest rate, a negative relationship with the nominal effective exchange rate, and an unclear relationship with the percent change in real GDP. In comparison, the equilibrium long-term interest rate in the standard closedeconomy loanable funds model (Hoelscher, 1986) can be written as (10) The equilibrium long-term interest rate in the standard open-economy loanable funds model (Cebula, 1988, 1994, 1997a, 1997b, 1998, 1999, 2000, 2003) is given by (11) where NCF is the net capital inflow. The sign of NCF should be negative as an increase in the net capital inflow to Slovakia would shift the supply of loanable funds to the right and reduce the equilibrium long-term interest rate. 3. EMPIRICAL RESULTS The data were collected from the October 2009 edition of the International Financial Statistics, which is published by the International Monetary Fund. The dependent variable is Slovakia’s government bond yield. Because the data for the government deficit are only available during 2006.Q1 – 2007.Q4 with 8 observations, B is represented by the ratio of government borrowing to GDP. Due to incomplete data for the Treasury bill rate, the real short-term interest rate is represented by the real money market rate to test a potential substitution effect. Y is represented by the percent change in real GDP at the 2005 price. The expected inflation rate is represented by the lagged inflation rate based on the consumer price index. To reduce multicollinearity, the lagged EU government bond yield is chosen to represent the world interest rate. ε is represented by the nominal effective exchange rate. An increase in the nominal effective exchange rate means appreciation of the Slovak currency. NCF is represented by the ratio of the net capital inflow to GDP where the net capital inflow is the sum of the portfolio, direct and other investments in the financial account. The data for the government bond yield before 2000.Q3 and the data for the money market rate 62 Government Borrowing and the Long-Term Interest Rate after 2008.Q4 are not available. Hence, the sample ranges from 2000.Q3 to 2008. Q4. As shown in Table 1, based on the unrestricted cointegration rank test, there are 2 cointegrating relations. Therefore, there is a long-term stable relationship among the variables. Table 1. Unrestricted Cointegration Rank Test (Maximum Eigenvalue) Hypothesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Statistic Critical Value None * 0.920727 81.11551 46.23142 At most 1 * 0.715977 40.27837 40.07757 At most 2 0.605054 29.72820 33.87687 At most 3 0.504136 22.44650 27.58434 At most 4 0.295154 11.19285 21.13162 At most 5 0.228419 8.298043 14.26460 At most 6 Notes: 0.028020 0.909452 3.841466 Max-eigenvalue test indicates 2 cointegrating relations at the 5% level. * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Prob.** 0.0000 0.0475 0.1446 0.1984 0.6280 0.3492 0.3403 Table 2 plots the residual histogram and presents the normality test for the error terms. As shown, the Jarque-Bera statistic of 4.01 is smaller than the critical value of 9.21 at the 1% level or 5.99 at the 5% level. Hence, the null hypothesis of a normal distribution of the error terms cannot be rejected. Table 3 reports the estimated regression and related statistics. The Newey-West generalized least squares (GLS) method is employed in order to yield consistent estimates for the covariance and standard errors. As shown, 91.5% of the variation in the government bond yield can be explained by the right-hand side variables with significant coefficients. Except for the coefficient of the growth rate of real GDP, all other coefficients are significant at the 1% or 5% level. The government bond yield is positively associated with the ratio of government borrowing to GDP, the real money market rate, the expected inflation rate, the EU government bond yield, and it is negatively affected by the nominal effective exchange rate. 63 Economic Annals, Volume LV, No. 184 / January − March 2010 Table 2. The Jargue-Bera Normality Test of the Regression Residuals Table 3. Estimated Regression of the Government Bond Yield for Slovakia based on the Extended Loanable Funds Model Variable C B RS Y πe R* ε Coefficient -0.822985 1.602431 0.232864 0.009744 0.979523 1.479321 -0.017698 Std. Error 0.776459 0.673581 0.072405 0.012127 0.337383 0.276401 0.006003 t-Statistic -1.059920 2.378974 3.216114 0.803528 2.903296 5.352087 -2.948262 Prob. 0.2986 0.0247 0.0034 0.4287 0.0073 0.0000 0.0065 R-squared 0.930380 Adjusted R-squared 0.914909 Akaike inform. criterion 1.477811 Schwarz criterion 1.792062 F-statistic 60.13645 Prob (F-statistic) 0.000000 Sample period 2000.Q3 – 2008.Q4 N Notes: 34 C is the constant. B is the ratio of government borrowing to GDP. RS is the real money market rate. Y is the percent change in real GDP. πe is the expected inflation rate. R* is the EU government bond yield. ε is the nominal effective exchange rate. 64 Government Borrowing and the Long-Term Interest Rate Several different versions are considered to determine whether the outcomes may vary. If the 10-year U.S. government bond yield replaces the EU government bond yield, its positive coefficient will be significant at the 5% level, the positive coefficient of the nominal effective exchange rate will be insignificant, and other results will be similar. If the lagged nominal effective exchange rate replaces the nominal effective exchange rate, its negative coefficient is significant at the 10% level, and other results are similar. If the SKK/USD exchange rate replaces the nominal effective exchange rate, its positive coefficient will be significant at the 1% level, but the coefficients of the ratio of government borrowing to GDP, the real money market rate, and the expected inflation rate will be insignificant at the 10% level. To save space, details are not printed here and will be available upon request. When the standard closed-economy loanable funds model in equation (10) is considered in empirical work, the value of the adjusted R2 is 0.845, and the sign and significance of all the coefficients are similar to those reported in Table 3. When the standard open-economy loanable funds model in equation (11) is considered, the value of the adjusted R-squared is 0.830, the positive coefficient of the ratio of the net capital inflow to GDP is insignificant at the 10% level, and other results are similar to the closed-economy loanable funds model. Hence, the inclusion of the EU government bond yield and the nominal effective exchange rate increases the value of adjusted R-squared of the regression and improves the explanatory power of the behaviour of the Slovak government bond yield. 4. SUMMARY AND CONCLUSIONS This paper has applied an extended open-economy loanable funds model to examine whether the Slovak long-term interest rate would be affected by government borrowing and other selected macroeconomic variables. The results show that more government borrowing would raise the government bond yield and that a higher real money market rate, a higher expected inflation rate, a higher EU government bond yield, and a lower nominal effective exchange rate would raise the Slovak government bond yield. In the standard closed-economy loanable funds model without including the EU government bond and the nominal effective exchange rate, similar results for other variables are found. In the standard open-economy loanable funds model, except that the positive coefficient of the ratio of the net capital inflow to GDP is insignificant at the 10% level, other results are similar to those found in the standard closed-economy loanable funds model. Hence, the EU government bond yield and the nominal 65 Economic Annals, Volume LV, No. 184 / January − March 2010 effective exchange rate incorporated in this study increase the explanatory power of the behaviour of the government bond yield. There are several policy implications. The significant coefficient of the ratio of government borrowing to GDP implies that pursing expansionary fiscal policy to stimulate the economy would raise the long-term government bond yield and crowd out part of private investment expenditures. It suggests that the multiplier effect of increased government deficit spending would not change much due to crowding-out. In the open-economy loanable funds model, the world interest rate and the exchange rate need to be considered as international investors search for better returns in determining the supply of loanable funds to Slovakia or other countries. 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