en>TakuyaMurata at 17:37, 30 January 2014

2014-01-30T17:37:18Z

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In [[theoretical physics]], the '''Curtright field''' (named after [[Thomas Curtright]])<ref>{{cite doi|10.1016/0370-2693(85)91235-3 |noedit}}</ref> is a [[tensor]] [[quantum field]] of mixed symmetry, whose gauge-invariant dynamics are [[Hodge dual|dual]] to those of the general relativistic [[graviton]] in higher (''D''>4) spacetime dimensions. Or at least this holds for the linearized theory.<ref>{{cite doi|10.1088/1126-6708/2003/06/060|noedit}}</ref><ref>{{cite doi|10.1103/PhysRevD.88.064032|noedit}}</ref>
For the full nonlinear theory, less is known. Several difficulties arise when interactions of mixed symmetry fields are considered, but at least in situations involving an infinite number of such fields (notably string theory) these difficulties are not insurmountable.

In four [[spacetime]] [[dimension]]s, the field is not dual to the graviton, if massless,
but it can be used to describe ''massive'', pure [[spin (physics)|spin]] 2 [[quantum|quanta]].<ref>{{cite doi|10.1016/0550-3213(80)90174-1|noedit}}</ref> Similar descriptions exist for other massive higher spins, in ''D''≥4.<ref>{{cite doi|10.1088/1126-6708/2008/09/058|noedit}}</ref>

The simplest example of the linearized theory is given by a rank three Lorentz tensor <math>T_{\alpha\beta\mu}</math> whose indices carry the permutation symmetry of the [[Young diagram]] corresponding to the [[integer partition]] 3=2+1. That is to say, <math>T_{\alpha\beta\mu}=T_{[\alpha\beta]\mu}</math> and <math>T_{[\alpha\beta\mu]}=0</math> where indices in square brackets are totally antisymmetrized. The corresponding field strength for <math>T_{\alpha\beta\mu}</math> is
<math> F_{\alpha\beta\gamma\mu}=3\partial_{[\gamma}T_{\alpha\beta]\mu}.</math> This has a nontrivial trace <math> F_{\alpha\beta}=\eta^{\gamma\mu}F_{\alpha\beta\gamma\mu}</math> where <math>\eta^{\gamma\mu}</math> is the [[Minkowski metric]] with signature <tt>(+,−,−,...)</tt>.

The action for <math>T_{\alpha\beta\mu}</math> in ''D'' spacetime dimensions is bilinear in the field strength and its trace.
:<math> S=\frac{-1}{6} \int d^{D}x (F_{\alpha\beta\gamma\mu}F^{\alpha\beta\gamma\mu}-3F_{\alpha\beta}F^{\alpha\beta}).</math>
This action is gauge invariant, assuming there is zero net contribution from any boundaries, while the field strength itself is not. The gauge transformation in question is given by
:<math> \delta T_{\alpha\beta\mu}=\partial_{[\alpha}S_{\beta]\mu}+\partial_{[\alpha}A_{\beta]\mu}-\partial_{\mu}A_{\alpha\beta}</math>
where ''S'' and ''A'' are arbitrary symmetric and antisymmetric tensors, respectively.

An infinite family of mixed symmetry [[gauge field]]s arises, formally, in the zero tension limit of [[string theory]],<ref>{{cite doi| 10.1016/0550-3213(86)90525-0|noedit}}</ref> especially if ''D''>4. Such mixed symmetry fields can also be used to provide alternate local descriptions for [[massive particle]]s, either in the context of strings with nonzero tension, or else for individual particle quanta without reference to string theory.

==See also==
* [[Kalb–Ramond field]]
* [[P-form electrodynamics]]

==References==
{{reflist}}

{{DEFAULTSORT:Kalb-Ramond Field}}
[[Category:String theory]]
[[Category:Bosons]]
[[Category:Hypothetical particles]]
[[Category:Gauge bosons]]

{{particle-stub}}

Draft:Moduli stack of principal bundles - Revision history

en>TakuyaMurata at 17:37, 30 January 2014