Abstract
The local free action principle for bosonic unconstrained higher spin gauge fields in $d$-dimensional (A)dS$_d$ spacetime has been already established by Segal. However, later on, Schuster and Toro, in 4-dimensional flat spacetime, extracted such an action from their bosonic continuous spin gauge theory, when the continuous spin parameter vanishes. On the other hand, similar to the Schuster-Toro's action, we found the fermionic continuous spin gauge theory in 4-dimensional flat spacetime. Thus it is noteworthy to take out its fermionic higher spin formulation in a limit, and generalize it to $d$-dimensional (A)dS$_d$ spacetime. Therefore, in this paper, we will present a similar action principle \`a la Segal for fermions in $d$-dimensional (A)dS$_d$ spacetime. Moreover, at the level of equations of motion, we will demonstrate how the Fronsdal and the Fang-Fronsdal equations in $d$-dimensional (A)dS$_d$ spacetime are related, respectively, to the Euler-Lagrange equations of the bosonic and fermionic higher spin actions mentioned above.
Highlights
From the massless limit of the Singh-Hagen formulation [1,2], the local constrained Lagrangian formulation describing free bosonic massless higher spin fields was established by Fronsdal (Fang and Fronsdal) in flat and anti–de-Sitter (AdS) spacetimes [3,4] ([5,6]) in a metriclike approach
The constrained Lagrangian formulations include some conditions on the gauge fields and parameters; along with them, there are unconstrained formulations comprised of no conditions on the gauge fields and parameters, in both the BRST and geometric approaches, in Minkowski and (A)dS spacetimes
An unconstrained Lagrangian formulation should be useful to make links between the higher spin theories and the BRST form of the string theory [23,47], as well as it is thought that the unconstrained Lagrangian formulation might be helpful for studying a possible Lagrangian formulation for the Vasiliev equations, describing interacting higher spin fields [48,49,50]
Summary
From the massless limit of the Singh-Hagen formulation [1,2], the local constrained Lagrangian formulation describing free bosonic (fermionic) massless higher spin fields was established by Fronsdal (Fang and Fronsdal) in flat and anti–de-Sitter (AdS) spacetimes [3,4] ([5,6]) in a metriclike approach (see [7] for a recent review). The constrained Lagrangian formulations include some conditions on the gauge fields and parameters; along with them, there are unconstrained formulations comprised of no conditions on the gauge fields and parameters, in both the BRST and geometric approaches, in Minkowski and (A)dS spacetimes (see, e.g., [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]).. In 2014, in their research on constructing a free action principle for bosonic continuous spin particles (CSPs), Schuster and Toro presented an unconstrained formulation for bosonic higher spin gauge fields in flat spacetime [57]. We were motivated to construct a Segal-like formulation for fermionic higher spin gauge fields in d-dimensional ðAÞdSd spacetime (explained below), which had not yet been discussed in the literature. These above-mentioned actions [56,57,58], as well as the present work, can be placed in the following Table by understanding the fact that, ala Segal, the bosonic and fermionic CSP actions in (A)dS spacetime (μ; Λ ≠ 0) have not yet been discovered. In order to be self-contained, a short discussion on the Segal action [56] will be presented in Appendix C
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