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

Abstract

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.

Symplectic Approach of Three-Dimensional Palatini Theory Plus a Chern-Simons Term

“…Furthermore, the matrix (22) is still singular; however, we have shown that there are no more constraints and that the theory has a gauge symmetry. In order to construct a symplectic tensor, we need to fix the gauge [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], and thus we will fix the temporal gauge, say, 0 = 0 = 0, which means thaṫ= 0 anḋ= 0. In this manner, the fixing gauge will be added to the symplectic Lagrangian via Lagrange multipliers, Θ and Ξ .…”

“…In this manner, with the antecedents mentioned above, in this paper, we will study the P-CS theory from a symplectic point of view. For this aim, we will use the symplectic formalism of Faddeev-Jackiw [FJ] [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], due basically to the fact that the FJ approach is more economical than Dirac's method. In fact, the FJ is a symplectic description where all relevant information of the theory can be obtained through a symplectic tensor, which is constructed from the symplectic variables that are identified from the Lagrangian.…”

Symplectic analysis of three-dimensional Abelian topological gravity

“…In fact, the FJ method is a symplectic approach, namely, all relevant information of the theory can be obtained through an invertible symplectic tensor, which is constructed by means of the symplectic variables that are identified as the degrees of freedom. Because of the theory is singular there will be constraints, and FJ has the advantage that all constraints are at the same footing, since it is not necessary to perform the classification of the constraints in primary, secondary, first class or second class as in Dirac's method is done [31][32][33][34][35][36][37][38][39][40][41][42]. When the symplectic tensor is obtained, then its components are identified with the FJ generalized brackets; Dirac's brackets and FJ brackets coincide to each other.…”

Dirac's and Generalized Faddeev-Jackiw brackets for Einstein's theory in G $\rightarrow 0$ limit

“…In this manner, it is possible to obtain all the Dirac results, say, in Dirac's approach we can construct the Dirac brackets by means two ways; eliminating only the second class constraints remaining the first class ones or we can fix the gauge and convert the first class constraints into second class ones, in any case, we can reproduce these Dirac's results by means of the [FJ] framework. In fact, for the former we use the configuration space as symplectic variables, for the later we use the phase space [12,13].…”