We constructed a symplectic realization of the dynamic structure of two interacting spin-two fields in three dimensions. A significant simplification refers to the treatment of constraints: instead of performing a Hamiltonian analysis $\grave{a}\, la$ Dirac, we worked out a method that only uses properties of the pre-symplectic two-form matrix and its corresponding zero-modes to investigate the nature of constraints and the gauge structure of the theory. For instance, we demonstrate that the contraction of the zero-modes with the potential gradient, yields explicit expressions for the whole set of constraints on the dynamics of the theory, including the symmetrization condition and an explicit relationship between the coupling and cosmological constants. This way, we further identify the necessary conditions for the existence of a unique non-linear candidate for a partially massless theory, using only the expression for the interaction parameters of the model. In the case of gauge structure, the transformation laws for the entire set of dynamical variables are more straightforwardly derived from the structure of the remaining zero-modes; in this sense, the zero-modes must be viewed as the generators of the corresponding gauge transformations. Thereafter, we use an appropriate gauge-fixing procedure, the time gauge, to compute both the quantization brackets and the functional measure on the path integral associated with our model. Finally, we confirm that three-dimensional bi-gravity has two physical degrees of freedom per space point. With the above, we provide a new perspective for a better understanding of the dynamical structure of theories of interacting spin-two fields, which does not require the constraints to be catalogued as first- and second-class ones as in the case of Dirac's standard method.
We derive an explicit expression for the star product reproducing the $κ$-Minkowski Lie algebra in any dimension $n$. The result is obtained by suitably reducing the Wick-Voros star product defined on $\mathbb{C}^{d}_θ$ with $n=d+1$. It is thus shown that the new star product can be obtained from a Jordanian twist.
It is known that any Poisson manifold can be embedded into a bigger space which admites a description in terms of the canonical Poisson structure, i.e., Darboux coordinates. Such a procedure is known as a symplectic realization and has a number of important applications like quantization of the original Poisson manifold. In the present paper we extend the above idea to the case of quasi-Poisson structures which should not necessarily satisfy the Jacobi identity. For any given quasi-Poisson structure $Θ$ we provide a recursive procedure of the construction of a symplectic manifold, as well as t…
The Faddeev-Jackiw symplectic formalism for constrained systems is applied to analyze the dynamical content of a model describing two massive relativistic particles with interaction, which can also be interpreted as a bigravity model in one dimension. We systematically investigate the nature of the physical constraints, for which we also determine the zero-modes structure of the corresponding symplectic matrix. After identifying the whole set of constraints, we find out the transformation laws for all the set of dynamical variables corresponding to gauge symmetries, encoded in the remaining ze…
A free quantum field theory with Lorentz symmetry is derived for spin-half symplectic fermions in 2+1 dimensions. In particular, we show that fermionic spin-half fields may be canonically quantized in a free theory with a Klein-Gordon Lagrangian. This theory is shown to have all the required properties of a consistent free quantum field theory, namely causality, unitarity, adherence to the spin-statistics theorem, CPT symmetry, and the Hermiticity and positive definiteness of the Hamiltonian. The global symmetry of the free theory is Sp(4) $\simeq$ SO(5). Possible interacting theories of both …
We give a geometrical description of gravitational theories from the viewpoint of symmetries and affine structure. We show how gravity, considered as a gauge theory, can be consistently achieved by the nonlinear realization of the conformal-affine group in an indirect manner: due the partial isomorphism between $CA\left( 3,1\right) $ and the centrally extended $Sp\left( 8\right) $, we perform a nonlinear realization of the centrally extended (CE)$Sp\left( 8\right) $ in its semi-simple version. In particular, starting from the bundle structure of gravity, we derive the conformal-affine Lie alge…
We give the principal realization of the twisted Yangians of orthogonal and symplectic types. The new bases are interpreted in terms of discrete Fourier transform over the cyclic group Z_N.
We consider a four dimensional space-time symmetry which is a non trivial extension of the Poincaré algebra, different from supersymmetry and not contradicting {\sl a priori} the well-known no-go theorems. We investigate some field theoretical aspects of this new symmetry and construct invariant actions for non-interacting fermion and non-interacting boson multiplets. In the case of the bosonic multiplet, where two-form fields appear naturally, we find that this symmetry is compatible with a local U(1) gauge symmetry, only when the latter is gauge fixed by a `t Hooft-Feynman term.
We present free field realizations for the associated vertex operator algebras of a number of four-dimensional $\mathcal{N}=2$ superconformal field theories. Our constructions utilize an exceptionally small set of chiral bosons whose number matches the complex dimensionality of the Higgs branch of the superconformal field theory. In the case of theories whose Higgs branches support additional degrees of freedom (free vector multiplets or decoupled interacting SCFTs), the corresponding "free field realizations" include additional ingredients: symplectic fermions in the case of vector multiplets…
In this work we study representations of the Poincare group defined over symplectic manifolds, deriving the Klein-Gordon and the Dirac equation in phase space. The formalism is associated with relativistic Wigner functions; the Noether theorem is derived in phase space and an interacting field, including a gauge field, approach is discussed.
The electromagnetic field in Poisson gauge theory: the groupoidal approach
Universidad de Alcalá（阿尔卡拉大学）；Universidade Federal do ABC（ABC联邦大学）；Università di Napoli Federico II（那不勒斯费德里科二世大学）；INFN Sez. di Napoli（意大利国家核物理研究所那不勒斯分部）
We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the $(p,q)$ 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we iden…
We describe three perspectives on higher quantization, using the example of magnetic Poisson structures which embody recent discussions of nonassociativity in quantum mechanics with magnetic monopoles and string theory with non-geometric fluxes. We survey approaches based on deformation quantization of twisted Poisson structures, symplectic realization of almost symplectic structures, and geometric quantization using 2-Hilbert spaces of sections of suitable bundle gerbes. We compare and contrast these perspectives, describing their advantages and shortcomings in each case, and mention many ope…
We examine an interacting dark matter--variable vacuum energy model for a spatially flat Friedmann-Roberston-Walker spacetime, focusing on the appearance of cosmological singularities such as \emph{big rip, big brake, big freeze}, and \emph{ big separation} along with abrupt events (\emph{infinite $γ$- singularity} and \emph{new w-singularity}) at late times. We introduce a phenomenological interaction which has a nonlinear dependence on the total energy density of the dark sector and its derivative, solve exactly the source equation for the model and find the energy density as function of the…
We quantize the interaction of gravity with Yang-Mills and spinor fields, hence offering a quantum theory incorporating all four fundamental forces of nature. Using canonical quantization we obtain solutions of the Wheeler-DeWitt equation in a vector bundle and the method of second quantization leads to a symplectic vector space $(V,\om)$ and a corresponding CCR representation for the bosonic components and a CAR representation for the fermionic part. The solution space of the Wheeler-DeWitt equation is invariant under gauge transformations and under isometries in the spacelike base space $\so…
Constrained Symplectic Quantization I: the Quantum Harmonic Oscillator
Gran Sasso Science Institute（格兰萨索科学研究所）；INFN-Laboratori Nazionali del Gran Sasso（意大利国家核物理研究所格兰萨索国家实验室）；INFN Roma1（意大利国家核物理研究所罗马一分部）；Sapienza University（罗马大学）；Physics and Astronomy Department ”Galileo Galilei”, Università di Padova（帕多瓦大学伽利略伽利莱物理学与天文学系）
Decoupling the chiral dynamics in the canonical approach to the WZNW model requires an extended phase space that includes left and right monodromy variables. Earlier work on the subject, which traced back the quantum qroup symmetry of the model to the Lie-Poisson symmetry of the chiral symplectic form, left some open questions: - How to reconcile the monodromy invariance of the local 2D group valued field (i.e., equality of the left and right monodromies) with the fact that the latter obey different exchange relations? - What is the status of the quantum group symmetry in the 2D theory in whic…
We describe a new realization of both global and local supersymmetry acting in spaces of commuting and anticommuting differential forms. Unlike the standard supersymmetry, it has Lorentz scalar transformation parameters. It is related but is not reducible to standard supersymmetry. Reformulation of the Standard Model with the new supersymmetry, called scalar supersymmetry, can be achieved with the particle content of the SM. BRST symmetry is extended to include scalar supersymmetry multiplets. Linear realization of scalar supersymmetry with free fields or with fields interacting with backgroun…
In this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group ${\rm GL}_r(\mathcal{K}_{\mathbb{P}^1_x})$ with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at $\infty \in \mathbb{P}_{1}$. The restriction of the family is that the matrix elements in the defining representation are linear functions of $x$. We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs…
We study "field space entanglement" in certain quantum field theories consisting of N number of free scalar fields interacting with each other via kinetic mixing terms. We present exact analytic expressions for entanglement end Renyi entropies between arbitrary number of scalar fields by which we could explore certain entanglement inequalities. Other entanglement measures such as mutual information and entanglement negativity have also been studied. We also give some comments about possible holographic realizations of such models.
