Null boundary phase space: slicings, news & memory

We construct the boundary phase space in D-dimensional Einstein gravity with a generic given co-dimension one null surface $$ \mathcal{N} $$ N as the boundary. The associated boundary symmetry algebra is a semi-direct sum of diffeomorphisms of $$ \mathcal{N} $$ N and Weyl rescalings. It is generated by D towers of surface charges that are generic functions over $$ \mathcal{N} $$ N . These surface charges can be rendered integrable for appropriate slicings of the phase space, provided there is no graviton flux through $$ \mathcal{N} $$ N . In one particular slicing of this type, the charge algebra is the direct sum of the Heisenberg algebra and diffeomorphisms of the transverse space, $$ \mathcal{N} $$ N v for any fixed value of the advanced time v. Finally, we introduce null surface expansion- and spin-memories, and discuss associated memory effects that encode the passage of gravitational waves through $$ \mathcal{N} $$ N , imprinted in a change of the surface charges.

Generalized uncertainty principle: from the harmonic oscillator to a QFT toy model

Several models of quantum gravity predict the emergence of a minimal length at Planck scale. This is commonly taken into consideration by modifying the Heisenberg uncertainty principle into the generalized uncertainty principle. In this work, we study the implications of a polynomial generalized uncertainty principle on the harmonic oscillator. We revisit both the analytic and algebraic methods, deriving the exact form of the generalized Heisenberg algebra in terms of the new position and momentum operators. We show that the energy spectrum and eigenfunctions are affected in a non-trivial way. Furthermore, a new set of ladder operators is derived which factorize the Hamiltonian exactly. The above formalism is finally exploited to construct a quantum field theoretic toy model based on the generalized uncertainty principle.

Exponential Formulas, Normal Ordering and the Weyl-Heisenberg Algebra

We consider a class of exponentials in the Weyl-Heisenberg algebra with exponents of type at most linear in coordinates and arbitrary functions of momenta. They are expressed in terms of normal ordering where coordinates stand to the left from momenta. Exponents appearing in normal ordered form satisfy differential equations with boundary conditions that could be solved perturbatively order by order. Two propositions are presented for the Weyl-Heisenberg algebra in 2 dimensions and their generalizations in higher dimensions. These results can be applied to arbitrary noncommutative spaces for construction of star products, coproducts of momenta and twist operators. They can also be related to the BCH formula.

Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed D=4 quantum inhomogeneous conformal Hopf algebras Uθ(su(2,2)⋉T4) and Uθ¯(su(2,2)⋉T¯4), where T4 describes complex twistor coordinates and T¯4 the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of D=4 Heisenberg-conformal algebra (HCA) su(2,2)⋉Hℏ4,4 (Hℏ4,4=T¯4⋉ℏT4 is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length λp, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in Hℏ4,4. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra Uθ(su(2,2)⋉T4).

Translational symmetries of quadratic Lagrangians

In this paper, we show that a quadratic Lagrangian, with no constraints, containing ordinary time derivatives up to the order [Formula: see text] of [Formula: see text] dynamical variables, has [Formula: see text] symmetries consisting in the translation of the variables with solutions of the equations of motion. We construct explicitly the generators of these transformations and prove that they satisfy the Heisenberg algebra. We also analyze other specific cases which are not included in our previous statement: the Klein–Gordon Lagrangian, [Formula: see text] Fermi oscillators and the Dirac Lagrangian. In the first case, the system is described by an equation involving partial derivatives, the second case is described by Grassmann variables and the third shows both features. Furthermore, the Fermi oscillator and the Dirac field Lagrangians lead to second class constraints. We prove that also in these last two cases there are translational symmetries and we construct the algebra of the generators. For the Klein–Gordon case we find a continuum version of the Heisenberg algebra, whereas in the other cases, the Grassmann generators satisfy, after quantization, the algebra of the Fermi creation and annihilation operators.

Testing short distance anisotropy in space

The isotropy of space is not a logical requirement but rather is an empirical question; indeed there is suggestive evidence that universe might be anisotropic. A plausible source of these anisotropies could be quantum gravity corrections. If these corrections happen to be between the electroweak scale and the Planck scale, then these anisotropies can have measurable consequences at short distances and their effects can be measured using ultra sensitive condensed matter systems. We investigate how such anisotropic quantum gravity corrections modify low energy physics through an anisotropic deformation of the Heisenberg algebra. We discuss how such anisotropies might be observed using a scanning tunnelling microscope.

Detecting scaling in phase transitions on the truncated Heisenberg algebra

We construct and analyze a phase diagram of a self-interacting matrix field coupled to curvature of the non-commutative truncated Heisenberg space. The model reduces to the renormalizable Grosse-Wulkenhaar model in an infinite matrix size limit and exhibits a purely non-commutative non-uniformly ordered phase. Particular attention is given to scaling of model’s parameters. We additionally provide the infinite matrix size limit for the disordered to ordered phase transition line.