Abstract
The (free) graviton admits, in addition to the standard Pauli-Fierz description by means of a rank-two symmetric tensor, a description in which one dualizes the corresponding -curvature tensor on one column to get a -tensor, where D is the spacetime dimension. This tensor derives from a gauge field with mixed Yound symmetry called the ‘dual graviton’ field. The dual graviton field is related non-locally to the Pauli-Fierz field (even on-shell), in much the same way as a p -form potential and its dual -form potential are related in the theory of an abelian p -form. Since the Pauli-Fierz field has a Young tableau with two columns (of one box each), one can contemplate a double dual description in which one dualizes on both columns and not just on one. The double dual curvature is now a -tensor and derives from a gauge field with mixed Young symmetry, the ‘double dual graviton’ field. We show, however, that the double dual graviton field is algebraically and locally related to the original Pauli-Fierz field and, so, does not provide a truly new description of the graviton. From this point of view, it plays a very different role from the dual graviton field obtained through a single dualization. We also show that these equations can be obtained from a variational principle in which the variables to be varied in the action are (all) the components of the double-dual field as well as an auxiliary field with Young symmetry. By gauge fixing the shift symmetries of this action principle, one recovers the Pauli-Fierz action. Our approach differs from the interesting approach based on parent actions and covers only the free, sourceless theory. Similar results are argued to hold for higher spin gauge fields.
Highlights
This paper is dedicated to Peter Freund, who had precient intuitions about the importance of dual formulations of field theories and the role played by fields with mixed Young tableau symmetry in that context [1]
The graviton admits, in addition to the standard Pauli-Fierz description by means of a rank-two symmetric tensor, a description in which one dualizes the corresponding (2, 2)-curvature tensor on one column to get a (D − 2, 2)-tensor, where D is the spacetime dimension. This tensor derives from a gauge field with mixed Yound symmetry (D − 3, 1) called the ‘dual graviton’ field
We show that these equations can be obtained from a variational principle in which the variables to be varied in the action are the components of the double-dual field as well as an auxiliary field with (2, 1)Young symmetry
Summary
This paper is dedicated to Peter Freund, who had precient intuitions about the importance of dual formulations of field theories and the role played by fields with mixed Young tableau symmetry in that context [1]. Because of (1.2), one can introduce a dual (D − p − 2)-form potential B(D−p −2) such that H(D−1−p) = dB(D−p−2). Note that the relation between the p -form potential A(p ) and its dual B(D−p −2) is non-local. The dual curvature H(D−1−p ) is obtained by dualizing on the only column there is, and is described by a Young tableau with a single column with D − p − 1 boxes. The dual potential B(D−p −2) is described by a Young tableau with a single column with D − p − 2 boxes. Since the Pauli-Fierz field has a Young tableau with two columns and a curvature tensor that is described by a Young tableau with two columns, one can consider two different types of duality. Relevant to the covariant Lagrangian formulation, are in line with the light cone gauge considerations of [10]
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