Dirac and Faddeev–Jackiw quantization of a five-dimensional Stüeckelberg theory with a compact dimension

• Dirac’s method for 5D Stueckelberg theory with a compact dimension is performed.
• By fixing the gauge in the effective theory we find present pseudo-Goldstone bosons.
• Dirac’s brackets are constructed for the zero-modes and the kk-excitations.
• The Faddeev–Jackiw quantization is performed.
• The equivalence between generalized Faddeev–Jackiw and Dirac’s brackets is shown.

A detailed Hamiltonian analysis for a five-dimensional Stüeckelberg theory with a compact dimension is performed. First, we develop a pure Dirac’s analysis of the theory; we show that after performing the compactification, the theory is reduced to four-dimensional Stüeckelberg theory plus a tower of Kaluza–Klein modes. We develop a complete analysis of the constraints, we fix the gauge and we show that there are present pseudo-Goldstone bosons. Then we quantize the theory by constructing the Dirac brackets. As complementary work, we perform the Faddeev–Jackiw quantization for the theory under study, and we calculate the generalized Faddeev–Jackiw brackets, we show that both the Faddeev–Jackiw and Dirac’s brackets are the same. Finally we discuss some remarks and prospects.