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Free field equations, with various spins, for space–time algebras with second-rank tensor (instead of the usual vector) momentum are constructed. Similar algebras are appearing in superstring/M theories. Special attention is paid to gauge invariance properties, in particular the spin-two equations with gauge invariance are constructed for dimensions 2+2 and 2+4, and the connection with Einstein equation and diffeomorphism invariance is established.

It has been proposed that the Poincaré and some other symmetries of noncommutative field theories should be twisted. Here we extend this idea to gauge transformations and find that twisted gauge symmetries close for arbitrary gauge group. We also analyse twisted-invariant actions in noncommutative theories.

The explicit form of linearized gauge invariant interactions of scalar and general higher even spin fields in the AdS_D space is obtained. In the case of general spin-ℓ a generalized "Weyl" transformation is proposed and the corresponding "Weyl" invariant action is constructed. In both cases the invariant actions of the interacting higher even spin gauge field and the scalar field include the whole tower of invariant actions for couplings of the same scalar with all gauge fields of smaller even spin. For the particular value of ℓ = 4, all results are in exact agreement with Ref. 1.

We analyze the constraints of Christ–Lee model by means of modified Faddeev–Jackiw formalism in Cartesian as well as polar coordinates. Further, we accomplish quantization à la Faddeev–Jackiw by choosing appropriate gauge conditions in both the coordinate systems. Finally, we establish gauge symmetries of Christ–Lee model with the help of zero-modes of the symplectic matrix.

The Bell inequality is thought to be a common constraint shared by all models of local hidden variables that aim to describe the entangled states of two qubits. Since the inequality is violated by the quantum mechanical description of these states, it purportedly allows distinguishing in an experimentally testable way the predictions of quantum mechanics from those of models of local hidden variables and, ultimately, ruling the latter out. In this paper, we show, however, that the models of local hidden variables constrained by the Bell inequality all share a subtle, though crucial, feature that is not required by fundamental physical principles and, hence, it might not be fulfilled in the actual experimental setup that tests the inequality. Indeed, the disputed feature neither can be properly implemented within the standard framework of quantum mechanics and it is even at odds with the fundamental principle of relativity. Namely, the proof of the inequality requires the existence of a preferred absolute frame of reference (supposedly provided by the lab) with respect to which the hidden properties of the entangled particles and the orientations of each one of the measurement devices that test them can be independently defined through a long sequence of realizations of the experiment. We notice, however, that while the relative orientation between the two measurement devices is a properly defined physical magnitude in every single realization of the experiment, their global rigid orientation with respect to a lab frame is a spurious gauge degree of freedom. Following this observation, we were able to explicitly build a model of local hidden variables that does not share the disputed feature and, hence, it is able to reproduce the predictions of quantum mechanics for the entangled states of two qubits.

In this paper, we quantize the Friedberg–Lee–Pang–Ren (FLPR) model, using an admissible gauge condition, within the framework of modified Faddeev–Jackiw formalism. Further, we deduce the gauge symmetries and establish off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetries. We also show that the physical states of the theory are annihilated by the first class constraints which is consistent à la Dirac formalism.

In this paper, we show that the mass-shell constraints in the gauged twistor formulation of a massive particle given in [Deguchi and Okano, Phys. Rev. D 93, 045016 (2016) [Erratum 93, 089906(E) (2016)]] are incorporated in an action automatically by extending the local $U(2)$ transformation to its inhomogeneous extension denoted by $IU(2)$. Therefore, it turns out that all the necessary constraints are incorporated into an action by virtue of the local $IU(2)$ symmetry of the system.

Gauge-invariant systems in unconstrained configuration and phase spaces, equivalent to second-class constraints systems upon a gauge-fixing, are discussed. A mathematical pendulum on an (n-1)-dimensional sphere S^n-1 as an example of a mechanical second-class constraints system and the O(n) nonlinear sigma model as an example of a field theory under second-class constraints are discussed in details and quantized using the existence of underlying dilatation gauge symmetry and by solving the constraint equations explicitly. The underlying gauge symmetries involve, in general, velocity dependent gauge transformations and new auxiliary variables in extended configuration space. Systems under second-class holonomic constraints have gauge-invariant counterparts within original configuration and phase spaces. The Dirac's supplementary conditions for wave functions of first-class constraints systems are formulated in terms of the Wigner functions which admit, as we show, a broad set of physically equivalent supplementary conditions. Their concrete form depends on the manner the Wigner functions are extrapolated from the constraint submanifolds into the whole phase space.

We construct a sequence of nilpotent BRST charges in RNS superstring theory, based on new gauge symmetries on the worldsheet, found in this paper. These new local gauge symmetries originate from the global nonlinear space–time α-symmetries, shown to form a noncommutative ground ring in this work. The important subalgebra of these symmetries is U(3) × X₆, where X₆ is solvable Lie algebra consisting of six elements with commutators reminiscent of the Virasoro type. We argue that the new BRST charges found in this work describe the kinetic terms in string field theories around curved backgrounds of the AdS × CP_n-type, determined by the geometries of hidden extra dimensions induced by the global α-generators. The identification of these backgrounds is however left for the work in progress.

The problem of the gauge hierarchy is brought up in a hypercomplex scheme for a $U(1)$ field theory; in such a scheme, a compact gauge group is deformed through a $\gamma$-parameter that varies along a noncompact internal direction, transverse to the $U(1)$ compact one, and thus an additional $SO(1,1)$ gauge symmetry is incorporated. This transverse direction can be understood as an extra internal dimension, which will control the spontaneous symmetry breakdown, and will allow us to establish a mass hierarchy. In this mechanism, there is no brane separation to be estabilized as in the braneworld paradigm, however, a different kind of fine-tuning is needed in order to generate the wished electroweak/Planck hierarchy. By analyzing the effective self-interactions and mass terms of the theory, an interesting duality is revealed between the real and hybrid parts of the effective potential. This duality relates the weak and strong self-interaction regimes of the theory, due to the fact that both mass terms and self-coupling constants appear as one-parameter flows in $\gamma$. Additionally, the $\gamma$-deformation will establish a flow for the electromagnetic coupling that mimics the renormalization group flow for the charge in QED.

Given the $\beta$ functions of the closed string sigma model up to one loop in ${\alpha }^{\prime }$, the effective action implements the condition $\beta =0$ to preserve conformal symmetry at quantum level. One of the more powerful and striking results of string theory is that this effective action contains Einstein gravity as an emergent dynamics in space–time. We show from the $\beta$ functions and its relation with the equations of motion of the effective action that the differential identities are the Noether identities associated with the effective action and its gauge symmetries. From here, we reconstruct the gauge and space–time symmetries of the effective action. In turn, we can show that the differential identities are the contracted Bianchi identities of the field strength $H$ and Riemann tensor $R$. Next, we apply the same ideas to DFT. Taking as starting point that the generalized $\beta$ functions in DFT are proportional to the equations of motion, we construct the generalized differential identities in DFT. Relating the Noether identities with the contracted Bianchi identities of DFT, we were able to reconstruct the generalized gauge and space–time symmetries. Finally, we recover the original $\beta$ functions, effective action, differential identities, and symmetries when we turn off the $\tilde{x}$ space–time coordinates from DFT.

In general relativity, the allowed set of diffeomorphisms or gauge transformations at asymptotic infinity forms the BMS group, an infinite-dimensional extension of the Poincaré group. We focus on the structure of the BMS group in two distinct forms of Hamiltonian dynamics — the instant and front forms. Both similarities and differences in these two forms are examined while emphasizing the role of noncovariant approaches to symmetries in gravity.

We review the geometric formulation of the second Noether's theorem in time-dependent mechanics. The commutation relations between the dynamics on the final constraint manifold and the infinitesimal generator of a symmetry are studied. We show an algorithm for determining a gauge symmetry which is closely related to the process of stabilization of constraints, both in Lagrangian and Hamiltonian formalisms. The connections between both formalisms are established by means of the time-evolution operator.

A definition of space-time metric deformations on an n-dimensional manifold is given. We show that such deformations can be regarded as extended conformal transformations. In particular, their features can be related to the perturbation theory giving a natural picture by which gravitational waves are described by small deformations of the metric. As further result, deformations can be related to approximate Killing vectors (approximate symmetries) by which it is possible to parameterize the deformed region of a given manifold. The perspectives and some possible physical applications of such an approach are discussed.

Graded Lagrangian formalism in terms of a Grassmann-graded variational bicomplex on graded manifolds is developed in a very general setting. This formalism provides the comprehensive description of reducible degenerate Lagrangian systems, characterized by hierarchies of non-trivial higher-order Noether identities and gauge symmetries. This is a general case of classical field theory and Lagrangian non-relativistic mechanics.