International Journal of Modern Physics A Vol. 27, No. 32 (2012) 1250189 (28 pages) c  World Scientific Publishing Company DOI: 10.1142/S0217751X12501898 THE MANY FACES OF THE MASSLESS TENSOR GAUGE FIELD OF DEGREE TWO EUGEN-MIHAITA CIOROIANU Department of Physics, University of Craiova, 13 Al. I. Cuza Str., Craiova 200585, Romania manache@central.ucv.ro Received 4 October 2012 Revised 26 November 2012 Accepted 5 December 2012 Published 27 December 2012 Starting from a generic second-order Lagrangian density (that describes the dynamics of an Abelian tensor gauge field of degree two and depends on two arbitrary real constants) we first perform the Dirac analysis. This procedure reveals seven distinct situations (dictated by the values of the real parameters that label the second-order Lagrangian density) in each of them one determines the number of degrees of freedom and also a generating set of gauge transformations. Second, with the help of some auxiliary gauge/ matter tensor gauge fields of degree three, in each of the seven situations aforementioned, we construct the first-order Lagrangian density corresponding to the second-order one. Keywords : Massless tensor gauge fields; Dirac analysis; gauge symmetry. PACS number: 11.10.Ef 1. Motivation The motivation of this paper comes from an old attempt to unify gravity with electromagnetism proposed by Einstein and developed by himself 1 and others. 2,3 In these geometrical field theories of gravity the potentials constitute the components of a tensor gauge field of degree two with no symmetry. A better understanding of these models can be accomplished by analyzing the dynamics of their free versions. In view of this, our starting point is represented by a tensor gauge field of degree two A µ‖ν with no symmetry pertaining to the reducible representation ⊗ of the Lorentz group and subject to the gauge transformations δ ǫ A µ‖ν = ∂ µ ǫ ν . (1) 1250189-1