Operators Of Physical Quantities Research Articles

Symmetries in Foundation of Quantum Theory and Mathematics

In standard quantum theory, symmetry is defined in the spirit of Klein’s Erlangen Program—the background space has a symmetry group, and the basic operators should commute according to the Lie algebra of that group. We argue that the definition should be the opposite—background space has a direct physical meaning only on classical level while on quantum level symmetry should be defined by a Lie algebra of basic operators. Then the fact that de Sitter symmetry is more general than Poincare symmetry can be proved mathematically. The problem of explaining cosmological acceleration is very difficult but, as follows from our results, there exists a scenario in which the phenomenon of cosmological acceleration can be explained by proceeding from basic principles of quantum theory. The explanation has nothing to do with existence or nonexistence of dark energy and therefore the cosmological constant problem and the dark energy problem do not arise. We consider finite quantum theory (FQT) where states are elements of a space over a finite ring or field with characteristic p and operators of physical quantities act in this space. We prove that, with the same approach to symmetry, FQT and finite mathematics are more general than standard quantum theory and classical mathematics, respectively: the latter theories are special degenerated cases of the former ones in the formal limit p → ∞ .

On the relation between the operators of physical quantities of a nonrigid molecule and the symmetry of its equilibrium configurations

The problem of changes in the expressions for effective operators of physical quantities (including an effective Hamiltonian) of a nonrigid molecule upon a change in its equilibrium configurations, in the case in which different possible geometries of such configurations correspond to different point groups, is considered for the example of the nonrigid dimethylacetylene molecule CH3C2CH3 in the ground electronic state.

A description of torsional motions of exchange and nonexchange types in the trimethylborane molecule B(CH3)3. Operators of physical quantities

Using the methods of a symmetry-group chain, the construction of a rigorous, combined description of the torsional motions of exchange and nonexchange types in the ground electronic state of the trimethylborane molecule B(CH3)3 is considered.

Description of torsional motion of identical tops in trimethylborane molecule B(CH3)3: II. Operators of physical quantities

The construction of a rigorous description of the torsional motion in a molecular system with three identical tops is considered. The construction is based on the methods of a symmetry group chain. The construction is outlined for the example of the trimethylborane molecule B(CH3)3.

Operators of physical quantities of nonrigid molecular systems with continuous axial symmetry groups

The construction of the effective operators of physical quantities is considered for nonrigid molecular systems whose geometrical symmetry of internal dynamics is determined by continuous axial groups. The construction is based on the methods of a symmetry group chain.

Geometry of the internal dynamics of C2H 3 + carbocation

Using the symmetry group chain methods, the internal dynamics of the simplest carbocation, C2H3+, is analyzed under the traditional assumptions that the equilibrium structures of the carbocation are planar and that the nonrigid motion between them is in-plane. This geometry of the internal dynamics is shown to agree with the data of the microwave spectroscopy on the splittings of rotational energy levels caused by the nonrigid motion. Previously, this statement was based on the model that violated the requirement of self-adjointness of operators of physical quantities.

Internal dynamics of (H2O)2 and (D2O)2 dimers: II. Effective operators of physical quantities with account of inversion and exchange motions

Using the symmetry group chain methods, a model of the internal dynamics of (H2O)2 and (D2O)2 dimers that takes into account their real geometries and two nonrigid motions with the lowest energy barriers is constructed. The motions are the inversion motion (or the acceptor switching) and the exchange motion of the acceptor and donor via an intermediate trans configuration. The model rigorously describes the interactions of various motions and leads to a simple algebraic scheme of calculation of both the positions of the levels in the energy spectrum and the intensities of transitions between them. It is important that the correctness of the model is limited only by the trueness of the choice of the symmetry of the internal dynamics.

Symmetry of torsional motion of identical tops in the CH2D-CH2D isotopic form of the ethane molecule

The symmetry of the torsional motion of identical tops in the CH2D-CH2D isotopical form of the ethane molecule is described by using the dynamic noninvariant group, i.e., the group that is larger than the symmetry group of the Hamiltonian of intramolecular motion in the Born-Oppenheimer approximation. As a result, the nuclear statistical weights of stationary states are determined with allowance for the motion of identical tops between energetically nonequivalent configurations and a procedure is developed for the qualitative construction of the operators of physical quantities describing the torsional-rotational spectrum of this system in an arbitrary vibronic state.

Spectrum of the ethylene molecule with realization of its two isomeric forms: II. Operators of physical quantities

It is shown that the use of dynamic noninvariant groups within the concept of the symmetry-group chain allows one to describe the spectrum of a molecular system when it is necessary to take into account changes in its isomeric form on changing the electronic state. A description is constructed for the example of the ethylene molecule, whose equilibrium configuration for a particular electronic state realizes one of two isomeric forms depending on the electronic state. The second part of this work is devoted to the construction of the effective operators of physical quantities.

Exactly solvable quantum mechanical models with Stückelberg divergences

We consider an exactly solvable quantum mechanical model with an infinite number of degrees of freedom that is an analogue of the model of N scalar fields (λ/N)(ϕaa)2 in the leading order in 1/N. The model involves vacuum and S-matrix divergences and also the Stuckelberg divergences, which are absent in other known renormalizable quantum mechanical models with, divergences (such as the particle in a δ-shape potential or the Lee model). To eliminate divergences, we renormalize the vacuum energy and charge and transform the Hamiltonian by a unitary transformation with a singular dependence on the regularization parameter. We construct the Hilbert space with a positive-definite metric, a self-adjoint Hamiltonian operator, and a representation for the operators of physical quantities. Neglecting the terms that lead to the vacuum divergences fails to improve and, on the contrary, worsens the renormalizability properties of the model.

Configurational degeneracy of exchange type in the vibration-rotation spectrum of a nonrigid molecule

In the vibration-rotation spectrum of a nonrigid molecule, the degeneracy that is due to several independent energetically equivalent equilibrium configurations related by exchange transitions is described using the concept of configurational spin. For the CH3OH molecule with three independent configurations related by exchange transitions occurring by internal rotation, the algebraic and symmetric properties of this spin are analyzed, and effective operators of physical quantities are qualitatively constructed based on exchange degeneracy.

Finiteness of physics and its possible consequences

The modular analog of representations of the so(1,4) algebra for a system of two spinless particles is considered in the framework of approach (proposed by the author earlier), in which physical systems are described by the elements of a linear space over a finite field, and operators of physical quantities by linear operators in this space. The eigenvalues of the free mass operator and corresponding eigenvectors are found. It is shown that if the finite field under consideration satisfies some properties then the particles necessarily interact with each other. The corresponding interaction is universal, since it is fully defined by the masses of particles and characteristics of the finite field. The formula for the kernel of the interaction operator is derived, but since the kernel is defined by the sum over the Galois field and this sum cannot be calculated explicitly, then it cannot be determined how far the above interaction manifests itself in the usual conditions.

Modular representations as a possible basis of finite physics

The approach in which physical systems are described by the elements of a linear space over a finite field, and operators of physical quantities by linear operators in this space, is discussed. The mathematical formulation of the correspondence between such a description and the conventional one is given for a case when the characteristic of the finite field is sufficiently large. The above correspondence is considered in the examples of finite analogs of representations of so(2,3), so(1,4), and osp(1,4) algebras. Finite analogs of representations of infinite-dimensional algebras are briefly discussed.

Probability operator of a harmonic oscillator

A unique linear rule of constructing quantum operators defined by the probability operator\(\mathop F\limits^ \wedge \) for coordinates and momenta, is considered.\(\mathop F\limits^ \wedge \) is assumed to be a normalized, positive definite operator, establishing a dynamical correspondence between the classical and quantum Poisson brackets. It is shown that such an operator exists in the case of a harmonic oscillator. The principal implications of the suggested rule of constructing the operators of physical quantities are determined, in comparison with the corresponding results of conventional quantum mechanics.

Study of excited states of atoms by means of nonorthogonal radial orbitals

Rules are derived for graphical determination of the overlap integral of wave functions and matrix elements of operators of physical quantities. A calculation is made for four-electron atoms in the configuation 1s22s4s using nonorthogonal radial orbitals.

Qualitative construction of rotational operators of physical quantities in the theory of molecular spectra

Qualitative construction of rotational operators of physical quantities in the theory of molecular spectra