Yu. G. Palii

The time surface term in quantum gravity

Abstract The role of the time surface term in the ADM Hamiltonian formulation of general relativity is investigated. We show that the variable contained in the time surface term (the scale factor) plays the role of a time-like variable. The conjugated variable represents the energy density in the reduced phase space, where the Schrodinger-like equation for a wave function is derived. The contribution from the surface term to the phase of the wave function allows us to define the phase time of the quantum Universe so that it coincides with the proper time as an invariant interval for the classical dust filled Universe. The quantum scenario of the evolution of the Universe filled in by the Weinberg-Salam fields is considered. The wave function of the early Universe as the functional from the Higgs fields and scale factor realizes the unitary irreducible representation of the SO (4, 1) group. The elementary particle masses are determined by the angles of the scale-scalar field mixing.

Generalized Hamiltonian dynamics of Friedmann cosmology with scalar and spinor matter source fields

The classical and quantum dynamics of the Friedmann-Robertson-Walker (FRW) universe with a massless scalar and massive fermion matter field as a source is discussed within the framework of the Dirac generalized Hamiltonian formalism. The Hamiltonian reduction of this constrained system is realized for two cases of minimal and conformal coupling between gravity and matter. It is shown that in both cases for all values of curvature, k = 0,±1, of maximally symmetric space there exists a time-independent reduced local Hamiltonian which describes the dynamics of the cosmic scale factor. The relevance of conformal timelike Killing vector fields in FRW spacetime to the existence of the time-independent Hamiltonian and the corresponding notion of conserved energy is discussed. The extended quantization with the Wheeler-deWitt equation is compared with the canonical quantization of unconstrained system. It is shown that quantum observables treated as expectation values of the Dirac observables properly describe the original classical theory.

Description of Friedmann observables in quantum universe

Abstract The solution of the problem of describing the Friedmann observables (the Hubble law, the redshift, etc.) in quantum cosmology is proposed on the basis of the method of gaugeless Hamiltonian reduction in which the gravitational part of the energy constraint is considered as a new momentum. We show that the conjugate variable corresponding to the new momentum plays the role of the invariant time parameter of evolution of dynamical variables in the sector of the Dirac observables of the general Hamiltonian approach. Relations between these Dirac observables and the Friedmann observables of the expanding Universe are established for the standard Friedmann cosmological model with dust and radiation. The presented reduction removes an infinite factor from the functional integral, provides the normalizability of the wave function of the Universe and distinguishes the conformal frame of reference where the Hubble law is caused by the alteration of the conformal dust mass.

Constraints on SU(2) ⊗ SU(2) invariant polynomials for a pair of entangled qubits

AbstractWe discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices

On the Hamiltonian reduction of geodesic motion on SU(3) to SU(3)/SU(2)

\mathfrak{P}_ +

Separability of two-qubit state in terms of local invariants

. Since elements of

On the homogeneous Gröbner basis for tensors

\mathfrak{P}_ +

Correction of Numerical Integration as an Optimal Control Problem

are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset,

A Friedmann universe in the quantization scheme of reduced phase space

\mathfrak{P}_ + \in \mathbb{R}^{15}

SU(6) Casimir invariants and SU(2) ⊗ SU(3) scalars for a mixed qubit-qutrit state

. We define