arXiv subjects
Publications and source records attributed to O. Yu. Shvedov.
Different constructions for Hilbert state space for constrained systems are investigated. Properties of Gaussian states analogous to quantum mechanical Gaussian wave functions are studied. Their evolution for quadratic Hamiltonian case are discussed. A notion of Maslov complex germ is introduced for systems with linear constraints.
Different approaches to axionatic field theory are investigated. The main notions of semiclassical theory are the following: semiclassical states, Poincare transformations, semiclassical action form, semiclassical gauge equivalence and semiclassical field. If the manifestly covariant approach is used, the notion of semiclassical state is related to Schwinger sourse, while the semicalssical action is presented via the R-function of Lehmann, Symanzik and Zimmermann. Semiclassical perturbation theory is constructed. Its relation with the S-matrix theory is investigated. Semiclassical electrodynamics and non-Abelian gauge theories are studied, making us of the Gupta-Bleuler and BRST approaches.
Quantum systems with constraints are often considered in modern theoretical physcics. All realistic field models based on the idea of gauge symmetry are of this type. A partial case of constraints being linear in coordinate and momenta operators is very important. Namely, when one applies semiclassical methods to an arbitrary constrained system, the constraints in "general position case" become linear. In this paper, different mathematicals constructions for the Hilbert space space for the constraint system are discussed. Properties of Gaussian and quasi-Gaussian wave functions for these systems are investigated. An analog of the notion of Maslov complex germ is suggested. Properties of Hamiltonians being quadratic with respect to the coordinate and momenta operators are discussed. The Maslov theorem (it says that there exists a Gaussian eigenfunction of the quantum Hamiltonian iff the classical Hamiltonian system is stable) is generalized to the constrained systems. The case of infinite number degrees of freedom (constrained Fock space) is also discussed.
A manifestly covariant formulation of quantum field Maslov complex-WKB theory (semiclassical field theory) is investigated for the case of scalar field. The main object of the theory is "semiclassical bundle". Its base is the set of all classical states, fibers are Hilbert spaces of quantum states in the external field. Semiclassical Maslov states may be viewed as points or surfaces on the semiclassical bundle. Semiclassical analogs of QFT axioms are formulated. A relationship between covariant semiclassical field theory and Hamiltonian formulation is discussed. The constructions of axiomatic field theory (Schwinger sources, Bogoliubov $S$-matrix, Lehmann-Symanzik-Zimmermann $R$-functions) are used in constructing the covariant semiclassical theory. A new covariant formulation of classical field theory and semiclassical quantization proposal are discussed.
Semiclassical perturbation theory is investigated within the framework of axiomatic field theory. Axioms of perturbation semiclassical theory are formulated. Their correspondence with LSZ approach and Schwinger source theory is studied. Semiclassical S-matrix, as well as examples of decay processes, are considered in this framework.
The main notions of semiclassical scalar electrodynamics in different gauges (Hamiltonian, Couloumb, Lorentz) are discussed. These are semiclassical states, Poincare transformations, fields, observables, gauge equivalence. General properties of these objects are formulated as axioms of semiclassical theory; they are heuristically justified. In particular, a semiclassical state may be viewed as a set of classical background field and quantum state in the external background. Superpositions of these "elementary" states can be also considered. Set of all "elementary" semiclassical states forms a semiclassical bundle, with base being classical space and fibres being quantum states in the external background. Quantum symetry transformations (Poincare and gauge transformations) are viewed semiclassically as automorphisms of the semiclassical bundle. Specific features of electrodynamics are investigated for different gauges.
The state space and observables for the leading order of the large-N theory are constructed. The obtained model ("theory of infinite number of fields") is shown to obey Wightman-type axioms (including invariance under boost transformations) and to be nontrivial (there are scattering processes, bound states, unstable particles etc). The considered class of exactly solvable relativistic quantum models involves good examples of theories containing such difficulties as volume divergences associated with the Haag theorem, Stueckelberg divergences and infinite renormalization of the wave function.
The main difficulty of quantum field theory is the problem of divergences and renormalization. However, realistic models of quantum field theory are renormalized within the perturbative framework only. It is important to investigate renormalization beyond perturbation theory. However, known models of constructive field theory do not contain such difficulties as infinite renormalization of the wave function. In this paper an exactly solvable quantum mechanical model with such a difficulty is constructed. This model is a simplified analog of the large-N approximation to the $Φϕ^aϕ^a$-model in 6-dimensional space-time. It is necessary to introduce an indefinite inner product to renormalize the theory. The mathematical results of the theory of Pontriagin spaces are essentially used. It is remarkable that not only the field but also the canonically conjugated momentum become well-defined operators after adding counterterms.
One of the axioms of quantum field theory is the property of unitarity of the evolution operator. However, if one considers the quantum electrodynamics in the external field in the leading order of perturbation theory, one will find that the evolution transformation is a non-unitary canonical transformation of creation and annihilation operators. This observation was one of the arguments for the hypothesis that one should choose different representations of the canonical commutation relations at different moments of time in the exact quantum field theory. In this paper the contradiction is analyzed for the case of a simple quantum mechanical model being an analog of the leading order of the large-N field theory. On the one hand, this model is renormalized with the help of the constructive field theory methods; the Hilbert space and unitary evolution operator are constructed. On the other hand, the leading order of the evolution transformation in the strong external field is shown to be non-unitary. Thus, unitarity of evolution in the exact theory is not in contradiction with non-unitarity of the approximate theory.
The semiclassical theory for the large-N field models is developed from an unusual point of view. Analogously to the procedure of the second quantization in quantum mechanics, the functional Schrodinger large-N equation is presented in a third-quantized form. The third-quantized creation and annihilation operators depend on the field $ϕ({\bf x})$. If the coefficient of the $ϕ^4$-term is of order 1/N (this is a usual condition of applicability of the 1/N-expansion), one can rescale the third-quantized operators in such a way that their commutator will be small, while the Heisenberg equations will not contain large or small parameters. This means that classical equation of motion is an equation on the functional $Φ[ϕ(\cdot)]$. This equation being a nonlinear analog of the functional Schrodinger equation for the one-field theory is investigated. The exact solutions are constructed and the renormalization problem is analysed. We also perform a quantization procedure about found classical solutions. The corresponding semiclassical theory is a theory of a variable number of fields. The developed third-quantized semiclassical approach is applied to the problem of finding the large-N spectrum. The results are compared with obtained by known methods. We show that not only known but also new energy levels can be found.
Large-N field systems are considered from an unusual point of view. The Hamiltonian is presented in a third-quantized form analogously to the second-quantized formulation of the quantum theory of many particles. The semiclassical approximation is applied to the third-quantized Hamiltonian. The advantages of this approach in comparison with 1/N-expansion are discussed.
The Hamiltonian approach to the quantum field theory is considered. Since there are additional difficulties such as the Haag theorem and Stueckelberg divergences, renormalization of the time-dependent dynamical quantum field theory is much more complicated than renormalization of the S-matrix. It is necessary to consider the regularized theory with ultraviolet and infrared cutoffs and impose the conditions not only on the dependence of the Hamiltonian on the cutoffs (as usual) but also on the dependence of the initial states. It happens that one should consider the initial states to be singulary dependent on the cutoffs in order to avoid the Stueckelberg divergences. Different types of semiclassical approximations to quantum theory are discussed. It happens that the method of quantizing classical solutions to field equations corresponds not to the WKB-approach but to the complex-WKB theory. The problem of imposing conditions on the semiclassical initial states is discussed. Different prescriptions for choice of initial conditions are analysed.
The problem of renormalization of the semiclassical one-loop equations used in the non-equilibrium field theory is considered. Recently, the renormalizability of such equations has been justified for some special cases of classical field configurations. In this paper the general case of arbitrary spatially inhomogeneous field configuration is investigated. It is shown that for certain quantum states the divergences arising in one-loop equations can be eliminated by usual perturbation-theory counterterms.
Semiclassical approximation based on extracting a c-number classical component from quantum field is widely used in the quantum field theory. Semiclassical states are considered then as Gaussian wave packets in the functional Schrodinger representation and as Gaussian vectors in the Fock representation. We consider the problem of divergences and renormalization in the semiclassical field theory in the Hamiltonian formulation. Although divergences in quantum field theory are usually associated with loop Feynman graphs, divergences in the Hamiltonian approach may arise even at the tree level. For example, formally calculated probability of pair creation in the leading order of the semiclassical expansion may be divergent. This observation was interpretted as an argumentation for considering non-unitary evolution transformations, as well as non-equivalent representations of canonical commutation relations at different time moments. However, we show that this difficulty can be overcomed without the assumption about non-unitary evolution. We consider first the Schrodinger equation for the regularized field theory with ultraviolet and infrared cutoffs. We study the problem of making a limit to the local theory. To consider such a limit, one should impose not only the requirement on the counterterms entering to the quantum Hamiltonian but also the requirement on the initial state in the theory with cutoffs. We find such a requirement in the leading order of the semiclassical expansion and show that it is invariant under time evolution. This requirement is also presented as a condition on the quadratic form entering to the Gaussian state.
The problem of topology change transitions in quantum gravity is discussed. We argue that the contribution of the Giddings-Strominger wormhole to the Euclidean path integral is pure imaginary. This is checked by two techniques: by the functional integral approach and by the analysis of the Wheeler-De Witt equation. We present also a simple quantum mechanical model which shares many features of the system consisting of parent and baby universes. In this simple model, we show that quantum coherence is completely lost and obtain the equation for the effective density matrix of the ''parent universe''.
Wormholes -- solutions to the euclidean Einstein equations with non-trivial topology -- are usually assumed to make real contributions to amplitudes in quantum gravity. However, we find a negative mode among fluctuations about the Giddings-Strominger wormhole solution. Hence, the wormhole contribution to the euclidean functional integral is argued to be purely imaginary rather than real, which suggests the interpretation of the wormhole as describing the instability of a large universe against the emission of baby universes.
The problem of topology change transitions in quantum gravity is investigated from the Wheeler-de Witt wave function point of view. It is argued that for all theories allowing wormhole effects the wave function of the universe is exponentially large. If the wormhole action is positive, one can try to overcome this difficulty by redefinition of the inner product, while for the case of negative wormhole action the more serious problems arise.
The model consisting of gravitational, scalar and axionic fields is considered. It is shown that the action of the Lavrelashvili-Rubakov-Tinyakov wormhole can be made arbitrarily negative by varying the parameters of the model. This means that semiclassically calculated probability of transition through this wormhole is not exponentially small (as usual) but exponentially large.