arXiv subjects

I. V. Kanatchikov

Publications and source records attributed to I. V. Kanatchikov.

At least 19 recordsLinked to original sources

arXiv2023

The De Donder-Weyl Hamiltonian formulation of TEGR and its quantization

The De Donder-Weyl (DW) Hamiltonian theory of fields treats space and time variables on equal footing. Its quantization, called precanonical quantization, leads to a hypercomplex generalization of quantum formalism to field theory as it follows from the quantization of Poisson-Gerstenhaber brackets defined on differential forms. Our recent work on precanonical quantization of general relativity is extended to the teleparallel equivalent of general relativity (TEGR) in tetrad Palatini formulation. The covariant precanonical Schrödinger equation for quantum TEGR and the relevant operators are constructed from the quantization of generalized Dirac brackets calculated using the constraints analysis generalized to the DW Hamiltonian theory. Our analysis of the ordering ambiguities in the precanonical Schrödinger equation allows us to estimate the contribution to the cosmological constant from the quantum TEGR and argue its consistency with the observed value, albeit with the current error of estimation of 13 orders of magnitude due to the theoretical uncertainties in the relation between the scale $\varkappa$ introduced by precanonical quantization and the mass gap in the pure gauge sector of QCD.

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arXiv2023

The Quantum Waves of Minkowski Spacetime And The Minimal Acceleration From Precanonical Quantum Gravity

We construct the simplest solutions of the previously obtained precanonical Schrödinger equation for quantum gravity, which correspond to the plane waves on the spin connection bundle and reproduce the Minkowski spacetime on average. Quantum fluctuations lead to the emergence of the minimal acceleration $a_0$ related to the range of the Yukawa modes in the fibers of the spin connection bundle. This minimal acceleration is proportional to the square root of the cosmological constant $Λ$ generated by the operator re-ordering in the precanonical Schrödinger equation. Thus the mysterious connection between the minimal acceleration $a_0$ in the dynamics of galaxies as described by Milgrom's MOND and the cosmological constant emerges as an elementary effect of precanonical quantum gravity. We also argue that the observable values of $a_0$ and $Λ$ can be obtained when the scale of the parameter $\varkappa$ introduced by precanonical quantization is subnuclear, in agreement with the previously established connection between the scale of $\varkappa$ and the mass gap in quantum SU(2) Yang-Mills theory.

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arXiv2023

Towards Precanonical Quantum Teleparallel Gravity

Quantization of the teleparallel equivalent of general relativity (TEGR) is discussed from the perspective of the space-time symmetric De Donder-Weyl (DW) Hamiltonian formulation with constraints and its quantization called precanonical quantization. The representations of operators and the covariant Schrödinger equation for TEGR are obtained from the quantization of generalized Dirac brackets calculated according to the analysis of constraints within the polysymplectic formulation of the DW Hamiltonian theory. We argue that the appropriate treatment of the operator ordering and the generalized Hermicity of operators results in an additional $c$-number term in the DW Hamiltonian operator which is identified with the cosmological constant and estimated to be consistent with its observed value.

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arXiv2019

On the precanonical structure of the Schrödinger wave functional in curved space-time

The functional Schrödinger equation in curved space-time is derived from the manifestly covariant precanonical Schrödinger equation. The Schrödinger wave functional is expressed as the trace of the multidimensional product integral of precanonical wave function restricted to a field configuration. The functional Schrödinger representation of QFT in curved space-time appears as a singular limiting case of a formulation based on precanonical quantization, which leads to a hypercomplex generalization of quantum formalism in field theory.

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arXiv2018

Precanonical structure of the Schrödinger wave functional in curved space-time

A relationship between the functional Schrödinger representation and the precanonical quantization of a scalar field theory is extended to an arbitrary curved space-time. The canonical functional derivative Schrödinger equation is derived from the manifestly covariant precanonical Schrödinger equation and the Schrödinger wave functional is expressed as the trace of the product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration when the ultraviolet parameter $\varkappa$ introduced in precanonical quantization is infinite. Thus the standard QFT in functional Schrödinger representation emerges from the precanonical formulation of quantum fields as a singular limiting case.

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arXiv2018

Schrödinger Functional of a Quantum Scalar Field in Static Space-Times from Precanonical Quantization

The functional Schrödinger representation of a scalar field on an $n$-dimensional static space-time background is argued to be a singular limiting case of the hypercomplex quantum theory of the same system obtained by the precanonical quantization based on the space-time symmetric De Donder-Weyl Hamiltonian theory. The functional Schrödinger representation emerges from the precanonical quantization when the ultraviolet parameter $\varkappa$ introduced by precanonical quantization is replaced by $\underlineγ{}_0δ^\mathrm{inv}(\mathbf{0})$, where $\underlineγ{}_0$ is the time-like tangent space Dirac matrix and $δ^\mathrm{inv}(\mathbf{0})$ is an invariant spatial $(n-1)$-dimensional Dirac's delta function whose regularized value at $\mathbf{x}=\mathbf{0}$ is identified with the cutoff of the volume of the momentum space. In this limiting case, the Schrödinger wave functional is expressed as the trace of the product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration and the canonical functional derivative Schrödinger equation is derived from the manifestly covariant Dirac-like precanonical Schrödinger equation which is independent of a choice of a codimension-one foliation.

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arXiv2017

On the spectrum of DW Hamiltonian of quantum SU(2) gauge field

The spectrum of masses of the colorless (color neutral) states of the DW (De Donder-Weyl) Hamiltonian operator of quantum SU(2) Yang-Mills field theory on $\mathbb{R}^D$ obtained via the precanonical quantization is shown to be purely discrete and bounded from below. The scale of the mass gap is estimated to be of the order of magnitude of the scale of the ultra-violet parameter $\varkappa$ introduced by precanonical quantization on dimensional grounds.

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arXiv2016

On the precanonical structure of the Schrödinger wave functional

We show that the Schrödinger wave functional may be obtained as the product integral of precanonical wave functions on the space of field and space-time variables. The functional derivative Schrödinger equation underlying the canonical field quantization is derived from the partial derivative covariant analogue of the Schrödinger equation, which appears in the precanonical field quantization based on the De Donder-Weyl generalization of the Hamiltonian formalism for field theory. The representations of precanonical quantum operators typically contain an ultraviolet parameter $\varkappa$ of the dimension of the inverse spatial volume. The transition from the precanonical description of quantum fields in terms of Clifford-valued wave functions and partial derivative operators to the standard functional Schrödinger representation obtained from canonical quantization is accomplished if $\frac{1}{\varkappa}\rightarrow 0$ and $\frac{1}{\varkappa}γ_0$ is mapped to the infinitesimal spatial volume element $\mathrm{d}\mathbf{x}$. Thus the standard QFT obtained via canonical quantization corresponds to the quantum theory of fields derived via precanonical quantization in the limiting case of an infinitesimal value of the parameter $\frac{1}{\varkappa}$.

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arXiv2016

Precanonical quantization and the Schrödinger wave functional revisited

We address the issue of the relation between the canonical functional Schrödinger representation in quantum field theory and the approach of precanonical field quantization proposed by the author, which requires neither a distinguished time variable nor infinite-dimensional spaces of field configurations. We argue that the standard functional derivative Schrödinger equation can be derived from the precanonical Dirac-like covariant generalization of the Schrödinger equation under the formal limiting transition $γ^0\varkappa\rightarrowδ(\mathbf{0})$, where the constant $\varkappa$ naturally appears within the precanonical quantization as the inverse of a small "elementary volume" of space. We obtain a formal explicit expression of the Schrödinger wave functional as a continuous product of the Dirac algebra valued precanonical wave functions, which are defined on the finite-dimensional covariant configuration space of the field variables and space-time variables.

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arXiv2015

Ehrenfest Theorem in Precanonical Quantization

We discuss the precanonical quantization of fields which is based on the De Donder--Weyl (DW) Hamiltonian formulation and treats the space and time variables on an equal footing. Classical field equations in DW Hamiltonian form are derived as the equations for the expectation values of precanonical quantum operators. This field-theoretic generalization of the Ehrenfest theorem demonstrates the consistency of three aspects of precanonical field quantization: (i) the precanonical representation of operators in terms of the Clifford (Dirac) algebra valued partial differential operators, (ii) the Dirac-like precanonical generalization of the Schrödinger equation without the distinguished time dimension, and (iii) the definition of the scalar product for calculation of expectation values of operators using the Clifford-valued precanonical wave functions.

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arXiv2015

On precanonical quantization of gravity in spin connection variables

The basics of precanonical quantization and its relation to the functional Schrödinger picture in QFT are briefly outlined. The approach is applied to quantization of Einstein's gravity in vielbein and spin connection variables and leads to a quantum dynamics described by the covariant Schrödinger equation for the transition amplitudes on the bundle of spin connection coefficients over the space-time, that yields a novel quantum description of space-time geometry. A toy model of precanonical quantum cosmology based on the example of flat FLRW universe is considered.

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arXiv2015

De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity

The De Donder-Weyl (DW) covariant Hamiltonian formulation of Palatini first-order Lagrangian of vielbein (tetrad) gravity and its precanonical quantization are presented. No splitting into the space and time is required in this formulation. Our recent generalization of Dirac brackets is used to treat the second class primary constraints appearing in the DW Hamiltonian formulation and to find the fundamental brackets. Quantization of the latter yields the representation of vielbeins as differential operators with respect to the spin connection coefficients, and the Dirac-like precanonical Schrödinger equation on the space of spin connection coefficients and space-time variables. The transition amplitudes on this space describe the quantum geometry of space-time. We also discuss the Hilbert space of the theory, the invariant measure on the spin connection coefficients, and point to the possible quantum singularity avoidance built in in the natural choice of the boundary conditions of the wave functions on the space of spin connection coefficients.

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arXiv2014

On precanonical quantization of gravity

Precanonical quantization is based on the mathematical structures of the De Donder-Weyl Hamiltonization of field theories. The resulting formulation of quantum gravity describes the quantum geometry of space-time in terms of operator-valued distances and the transition amplitudes between the values of spin connection at different points of space-time, which obey the covariant precanonical analogue of the Schrödinger equation. In the context of quantum cosmology the theory predicts a probability distribution of a cosmological spin-connection field, which may have an observable impact on the large scale structures in the universe.

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arXiv2003

Precanonical quantization of Yang-Mills fields and the functional Schroedinger representation

Precanonical quantization of pure Yang-Mills fields, which is based on the covariant De Donder-Weyl (DW) Hamiltonian formalism, and its connection with the functional Schrodinger representation in the temporal gauge are discussed. The YM mass gap problem is related to a finite dimensional spectral problem for a generalized Clifford-valued magnetic Schrödinger operator in the space of gauge potentials which represents the DW Hamiltonian operator.

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arXiv2002

Geometric (pre)quantization in the polysymplectic approach to field theory

The prequantization map for a Poisson-Gerstenhaber algebra of dynamical variables represented by differential forms within the polysymplectic formulation of the De Donder--Weyl covariant Hamiltonian field theory is presented and the corresponding prequantum Schroedinger equation for a non-homogeneous form valued wave function is derived. This is the first step toward understanding the procedures of covariant precanonical field quantization from the point of view of geometric quantization.

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arXiv2001

Precanonical Quantization and the Schroedinger Wave Functional

A relation between the Schroedinger wave functional and the Clifford-valued wave function which appears in what we call precanonical quantization of fields and fulfills a Dirac-like generalized covariant Schroedinger equation on the space of field and space-time variables is discussed. The Schroedinger wave functional is argued to be the trace of the positive frequency part of the continual product over all spatial points of the values of the aforementioned wave function restricted to a Cauchy surface. The standard functional differential Schroedinger equation is derived as a consequence of the Dirac-like covariant Schroedinger equation.

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arXiv2001

Covariant Geometric Prequantization of Fields

A geometric prequantization formula for the Poisson-Gerstenhaber bracket of forms found within the DeDonder-Weyl Hamiltonian formalism earlier is presented. The related aspects of covariant geometric quantization of field theories are sketched. In particular, the importance of the framework of Clifford and spinor bundles and superconnections in this context is underlined.

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arXiv2001

Precanonical Quantum Gravity: quantization without the space-time decomposition

A nonpertubative approach to quantum gravity using precanonical field quantization originating from the covariant De Donder-Weyl Hamiltonian formulation which treats space and time variables on equal footing is presented. A generally covariant ``multi-temporal'' generalized Schroedinger equation on the finite dimensional space of metric and space-time variables is obtained. An important ingredient of the formulation is the ``bootstrap condition'' which introduces a classical space-time geometry as an approximate concept emerging as the quantum average in a self-consistent with the underlying quantum dynamics manner. An independence of the theory from an arbitrarily fixed background is ensured in this way. The prospects and unsolved problems of precanonical quantization of gravity are outlined.

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