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Energy-Momentum and Gauge Conservation Laws

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Energy-Momentum and Gauge Conservation Laws View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Energy-momentum and gauge conservation laws 1 2 G.Giachetta, L.Mangiarotti, and G.Sardanashvily 1 Department of Mathematics and Physics, University of Camerino, Camerino 62032, Italy 2 Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia We treat energy-momentum conservation laws as particular gauge conservation laws when generators of gauge transformations are horizontal vector elds on bre bundles. In particular, the generators of general covariant transformations are the canonical horizontal prolongations of vector elds on a world manifold. This is the case of the energy-momentum conserva- tion in gravitation theories. We nd that, in main gravitational mo dels, the corresp onding energy-momentum ows reduce to the generalized Komar sup erp otential. We show that the sup erp otential form of a conserved ow is the common prop erty of gauge conservation laws if generators of gauge transformations dep end on derivatives of gauge parameters. At the same time, dep endence of a conserved ow on gauge parameters make gauge conservation laws form-invariant under gauge transformations. PACS numb ers: 03.50.-z, 04.20.Cv, 11.15.-q, 11.30.-g I. INTRODUCTION Energy-momentum in eld theory is a vast sub ject which can be studied from dif- ferent viewp oints. The problem lies in the fact that the canonical energy-momentum tensor fails to b e a true tensor, while the metric one is appropriate only to theories with a background geometry. In gravitation theories, an energy-momentum ow reduces to a sup erp otential dep ending onaworld vector eld as a gauge parameter. Analyzing the energy-momentum problem, we follow the general pro cedure of ob- taining di erential conservation laws in gauge theory [1-6]. Let Y ! X be a bre i bundle over a world manifold X co ordinated by(x ;y ), where x are co ordinates on X . By gauge transformations are generally meant automorphisms of Y ! X [7]. To ob- tain di erential conservation laws, it suces to consider one-parameter groups of gauge transformations whose generators are pro jectable vector elds i j u = u (x )@ + u (x ;y )@ (1) i on Y . If the Lie derivative L L of a Lagrangian L vanishes, wehave the weak conserva- u tion law d T 0 (2) 1 i of the corresp onding symmetry ow T along a vector eld u. If u = u @ is a vertical i vector eld, (2) is a familiar Nother conservation law, where T is a Nother current. This is the case of internal symmetries. If u is a horizontal prolongation on Y of a vector eld on X , called hereafter a world vector eld, we have an energy-momentum conservation law [5, 8, 9]. Of course, di erent horizontal prolongations of vector elds lead to di erent energy-momentum ows T . Let us emphasize that the expression (2) is linear in the vector eld u. Therefore, one can consider sup erp osition of conservation laws along di erent vector elds. In particular, every vector eld u (1) on Y pro jected onto a world vector eld can be seen as the sum u = u + v of a horizontal prolongation u of on Y and a vertical vector eld v on Y . It follows that every conservation law (2) can be represented as a sup erp osition of a Nother conservation law and an energy-momentum conservation 0 of the same world vector eld law. Conversely, two horizontal prolongations u and u 0 di er from each other in a vertical vector eld u u . Hence, the energy-momentum 0 ows along u and u di er from each other in a Nother current. One can not single out in a canonical way the Nother part of an energy-momentum ow. Therefore, if internal symmetries are broken, an energy-momentum ow is not conserved in general. In particular, a generic gravitational Lagrangian is invariant under general covariant transformations, but not under vertical gauge transformations of the general linear group GL(4; R). As a consequence, only the energy-momentum ow, corresp onding to the canonical horizontal prolongation of vector elds on X , i.e., the generators of general covariant transformations, is generally conserved in gravitation theory. This energy- momentum ow p ossesses the following two imp ortant p eculiarities. (i) It reduces to a sup erp otential, i.e., T d U ; U = U : (3) We will see that this is a common prop erty of gauge conservation laws if a vector eld u e dep ends on the derivatives of gauge parameters. Indeed, the canonical prolongation of a world vector eld = @ dep ends on the derivatives @ of the comp onents of which play the role of gauge parameters of in nitesimal general covariant transformations. (ii) A gravitational sup erp otential (3) dep ends on the comp onents of a world vector eld . This is also a common prop erty of gauge conservation laws which make them gauge-covariant, i.e., form-invariant under gauge transformations. Only in the Ab elian case of electromagnetic theory, an electric current is free from gauge parameters. A direct computation shows that General Relativity,Palatini formalism, metric-ane gravitation theory, and gauge gravitation theory in the presence of fermion elds lead 2 to the same gravitational sup erp otential. This is the generalized Komar sup erp otential ! @L @L (D + S ); (4) U ( )= @R @R where R and S are the curvature and the torsion of a world connection K , and D is the corresp onding covariant derivative. In the particular case of the Hilb ert{ Einstein Lagrangian L and a Levi{Civita connection K , the sup erp otential (4) reduces to the familiar Komar sup erp otential. Throughout the pap er, we follow the convention where the covariant derivative reads D = @ K : II. GAUGE CONSERVATION LAWS To obtain gauge conservation laws, we follow the rst variational formula of Lagran- gian formalism. Given a gauge eld system describ ed by sections of a bre bundle Y ! X , its space of elds and their rst order partial derivatives is the nite-dimensional rst order jet 1 manifold J Y of Y . Its elements are elds identi ed by their values and the values of their rst order derivatives at p oints of X (see, e.g., [9] for a detailed exp osition). This i i i space is provided with the adapted co ordinates (x ;y ;y ), where y are the derivative i i co ordinates such that y = @ . The transformation law of the derivative co ordinates is ! 0i 0i @y @y @x 0i j y = y + : j 0 @y @x @x A Lagrangian L of a eld system on a bre bundle Y ! X is de ned as a density i i 4 L = L(x ;y ;y )d x 1 Y . We will use the notation on the space J = @ L; i i 4 ! = d x; ! = @ c!; ! = @ c! : Let u be a pro jectable vector eld (1) on a bre bundle Y , treated as the generator 1 of a one-parameter group of gauge transformations. Its prolongation on the space J Y reads 1 i i i J u = u @ + u @ +(d u y @ u )@ i i 3 i where d = @ + y @ denote the total derivatives. The rst variational formula provides i the canonical decomp osition of the Lie derivative i i i L L! =[@ u L+(u @ +u @ +(d u y @ u )@ )L]! (5) u i i of a Lagrangian L in accordance with the variational problem. This decomp osition reads i i i @ u L +[u @ +u @ +(d u y @ u )@ ]L = (6) i i i i (u y u ) L d T ; i where i i L =(@ d @ )L; d = @ + y @ + y @ ; (7) i i i i i are the variational derivatives and i i T = T ! ; T = (u y u ) u L; (8) i is said to b e a symmetry ow along the vector eld u. It should b e emphasized that the ow (8) is de ned mo dulo the terms i i d (c (y u u )); i where c are arbitrary skew-symmetric functions on Y [9]. Here, we leave aside these i b oundary terms which are indep endent of a Lagrangian, but they may be essential if one examines integral conservation laws. On the shell L =(@ d @ )L=0 i i i where solutions of the Euler{Lagrange equations live, the rst variational formula (6) leads to the weak identity i i i @ u L +[u @ +u @ +(d u y @ u )@ ]L (9) i i i i d [ (u y u ) u L]: i If the Lie derivative L L (5) vanishes, i.e., the Lagrangian L is invariant under the u corresp onding one-parameter group of gauge transformations, the weak identity (9) is brought into the weak conservation law i i 0 d [ (u y u)u L]; (10) i of the ow T (8) along the vector eld u. 4 The weak identity (10) leads to the di erential conservation law d T ()= 0 on solutions of the Euler{Lagrange equations L()= 0: i This di erential conservation law implies the integral conservation law Z T ()! =0; (11) @M where M is a compact 4-dimensional sub-manifold of X with a b oundary @M . We will see that, in gauge and gravitation mo dels, a conserved ow takes the sp ecial form T = W + d U (12) where the term W is expressed in the variational derivatives (7), i.e., W 0, and T on-shell reduces to a sup erp otential U (3).
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