Realization Of SO Research Articles

Symmetries of the Dirac quantum walk and emergence of the de Sitter group

A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac walks, provide a description of the free evolution of relativistic quantum fields in the small wave-vector regime. The clash between the intrinsic discreteness of quantum walks and the continuous symmetries of special relativity is resolved by giving a definition of change of inertial frame in terms of a change of values of the constants of motion, which leaves the walk operator unchanged. Starting from the family of 1 + 1 dimensional Dirac walks with all possible values of the mass parameter, we introduce a unique walk encompassing the latter as an extra degree of freedom, and we derive its group of changes of inertial frames. This symmetry group contains a non-linear realization of SO+(2,1)⋉R3; since one of the two space-like dimensions does not correspond to an actual spatial degree of freedom but rather the mass, we interpret it as a 2 + 1 dimensional de Sitter group. This group also contains a non-linear realization of the proper orthochronous Poincaré group SO+(1,1)⋉R2 in 1 + 1 dimension, as the ones considered within the framework of doubly special relativity, which recovers the usual relativistic symmetry in the limit of small wave-vectors and masses. Surprisingly, for the Dirac walk with a fixed value of the mass parameter, the group of allowed changes of reference frame does not have a consistent interpretation in the limit of small wave-vectors.

A neutrino mass-mixing sum rule from SO(10) and neutrinoless double beta decay

Minimal SO(10) grand unified models provide phenomenological predictions for neutrino mass patterns and mixing. These are the outcome of the interplay of several features, namely: i) the seesaw mechanism; ii) the presence of an intermediate scale where B-L gauge symmetry is broken and the right-handed neutrinos acquire a Majorana mass; iii) a symmetric Dirac neutrino mass matrix whose pattern is close to the up-type quark one. In this framework two natural characteristics emerge. Normal neutrino mass hierarchy is the only allowed, and there is an approximate relation involving both light-neutrino masses and mixing parameters. This differs from what occurring when horizontal flavour symmetries are invoked. In this case, in fact, neutrino mixing or mass relations have been separately obtained in literature. In this paper we discuss an example of such comprehensive mixing-mass relation in a specific realization of SO(10) and, in particular, analyse its impact on the expected neutrinoless double beta decay effective mass parameter 〈mee〉, and on the neutrino mass scale. Remarkably a lower limit for the lightest neutrino mass is obtained (mlightest ≳ 7.5 × 10−4 eV, at 3 σ level).

Deformed twistors and higher spin conformal (super-)algebras in four dimensions

Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of deformations labelled by the spin t of an SU(2)_T group, which is the 6d analog of helicity in four dimensions. These deformations of the minrep of SO(6,2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6,2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5,1) and apply them to define higher spin algebras and superalgebras in AdS_7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS_7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS_7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS_7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8*|2N) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS_7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.

Deformed twistors and higher spin conformal (super-)algebras in six dimensions

Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of labelled by the spin t of an SU(2)T group, which is the 6d analog of helicity in four dimensions. These deformations of the minrep of SO(6,2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6,2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5,1) and apply them to define higher spin algebras and superalgebras in AdS7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8 ∗ |2N) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.

ISO(4,1) symmetry in the EFT of inflation

In DBI inflation the cubic action is a particular linear combination of the two, otherwise independent, cubic operators π̇3 and π̇(∂iπ)2. We show that in the Effective Field Theory (EFT) of inflation this is a consequence of an approximate 5D Poincar'e symmetry, ISO(4,1), non-linearly realized by the Goldstone π.This symmetry uniquely fixes, at lowest order in derivatives, all correlation functions in terms of the speed of sound cs. In the limit cs → 1, the ISO(4,1) symmetry reduces to the Galilean symmetry acting on π. On the other hand, we point out that the non-linear realization of SO(4,2), the isometry group of 5D AdS space, does not fix the cubic action in terms of cs.

Q-Deformed Bosons at Quasi-classical Limit of SO(3)

When a boson interacts with another to form a composite system with SO(3) dynamic symmetry, it is shown that there exists the q-deformed bosonic excitation satisfying the q-deformed Heisenberg commutation relation in the quasi-classical limit that the angular momentum j for SO(3) is large, but not infinite. In second quantization this quasi-excitation is associated with the boson realization of SO(3) Lie algebra. Physically, the phenomena of q-deformed excitation can happen in many models of quantum dynamics, such as super emission from a system of many identical two-level atoms, the spin wave in Heisenberg chain, the high spin precession and the coherent output of Bose–Einstein atoms in a trap. Especially, in these models, the deformation parameter q is well defined intrinsically by a conservative quantity, such as the total atomic number and the angular momentum.

DYNAMICAL HAMILTONIAN FOR PREASSIGNED TIME-EVOLUTION OF FERMIONIC SQUEEZING

We derive the general form of dynamical Hamiltonian generating the preassigned time-evolution of multimode fermionic squeezing. The fermionic squeezing is a quantum-mechanical image in fermionic Hilbert space corresponding to an orthogonal transformation in pseudo-classical Grassmann number space. The derivation is mainly based on the orthogonal properties of the transformation matrices. It is shown that some new fermionic operator realization of SO (2n) group and some new Hamiltonians can be obtained via this approach.

The Multicomponent Coherent States of the Lie Algebra SO(4)11The project supported by Natural Science Foundation of Hunan Province

The multicomponent coherent states associated with the Lie algebra SO(4) are presented. An inhomogeneous differential realization of SO(4) in this multicomponent coherent state space is obtained.

Three-body forces in the SO(8) model.

The SO(8) fermion pair model in its original version is unable to describe rigid axial rotation. Using the Hartree-Bogolyubov intrinsic state within an exact Schwinger realization of SO(8) we show that this can be remedied by introducing a cubic quadrupole interaction. Prolate and oblate minima are found and the degeneracy of the \ensuremath{\gamma} band with the ground-state band is removed. The new interaction can be understood from a renormalization of the quadrupole operator. An analogy with the liquid drop model, where the quadrupole moment is expanded in the shape coordinates ${\ensuremath{\alpha}}_{\ensuremath{\mu}}$, shows that the higher-order correction is of order 1/\ensuremath{\Omega}.