Abstract
The supersymmetric version of the Stueckelberg Lagrangian of the massive vector superfield leads to an example of higher-order derivative model. The canonical quantization yields massive states which compose two irreducible representations, one physical supermultiplet and another spurious. The origin of the different spurious states are investigated, especially those originating from the higher-order derivative terms. The spurious superfield is found to be decoupled when the supercurrent satisfies some appropriate conditions.
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Equation (3.3) is an extension of the Legendre transformation to the canonical coordinates of the particles system described by a higher-order derivative Lagrangian (see B. Doubrovine, S. Novikov, A. Fomenko Geometrie contemporaine Methodes et applications, part 2, p. 300, Moscou: Edition Mir 1985. For a Lagrangian of the type:\(\begin{gathered} L = \int {d^3 x\mathcal{L}(\phi ,\partial _\mu ,\phi ,\partial _\mu \partial _\nu \phi )} \hfill \\ = \int {d^3 x\mathcal{L}(\phi ,\partial i\phi ,\partial _i \partial _j \phi ,\partial _0 \phi ,\partial _i \partial _0 \phi ,\partial _0 \partial _i \phi ,\partial _0 \partial _0 \phi )} \hfill \\ \end{gathered} \) where the fields\(\phi ,\partial _0 \phi = \dot \phi \) and\(\partial _0 \partial _0 \phi = \ddot \phi \) are independent and∂ i φ,∂ i ∂ j φ and∂ i \(\dot \phi \) are considered as functions ofφ and\(\dot \phi \). Then, we obtain:\(\begin{gathered} \Pi _\phi = \delta L/\delta \dot \phi - d(\delta L/\delta \ddot \phi )dt = \partial \mathcal{L}/\delta \dot \phi - 2\partial _i (\partial \mathcal{L}{\text{/}}\partial \partial _i \dot \phi ) \hfill \\ - d(\partial \mathcal{L}/\partial \ddot \phi )dt, \hfill \\ \end{gathered} \) which agrees with the second-order canonical momentum definition given in [8]
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Guerdane, M., Lagraa, M. Canonical quantization of the massive vector supermultiplet: an example of higher-order derivative model. Z. Phys. C - Particles and Fields 51, 675–681 (1991). https://doi.org/10.1007/BF01565595
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DOI: https://doi.org/10.1007/BF01565595