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## Renormalization of the vacuum angle in quantum mechanics, Berry phase and continuous measurements

The vacuum angle θ renormalization is studied for a toy model of a quantum

particle moving around a ring, threaded by a magnetic flux θ. A novel type

of renormalization group (RG) transformation is introduced by coupling the

particle to an additional slow variable, which may also be viewed as a part of a

device, which ‘measures’ the particle’s position with finite accuracy. Then the

renormalized θ appears as a magnetic flux in the effective action for the slow

‘pointer’ variable. This ‘measurement-induced’ renormalization is shown to

have the same properties as the θ renormalization due to instantons in quantum

field theories and leads to the RG flow diagram, similar to that of the quantum

Hall effect, with observable effective θ vanishing in the limit of small coupling

between the particle and the measuring device.

We present a possible approach to the study of the renormalization group (RG) flow based entirely on the information theory. The average information loss under a single step of Wilsonian RG transformation is evaluated as a conditional entropy of the fast variables, which are integrated out, when the slow ones are held fixed. Its positivity results in the monotonic decrease of the informational entropy under renormalization. This, however, does not necessarily imply the irreversibility of the RG flow, because entropy is an extensive quantity and explicitly depends on the total number of degrees of freedom, which is reduced. Only some size-independent additive part of the entropy could possibly provide the required Lyapunov function. We also introduce a mutual information of fast and slow variables as probably a more adequate quantity to represent the changes in the system under renormalization and evaluate it for some simple systems. It is shown that for certain real space decimation transformations the positivity of the mutual information directly leads to the monotonic growth of the entropy per lattice site along the RG flow and hence to its irreversibility.

*The article looks at the possible role of measurement in a quantum-mechanical description of physical reality. The widely spread interpretations of quantum phenomena are considered as indicating an apparent connection between conscious processes (such as observation) and the properties of the microcosm.*

In this paper an asymptotic expansion of ergodic integrals for suspension flows over Vershik automorphisms is obtained and a limit theorem for these flows is given.

Bibliography: 49 titles.

We show that the complex solution for the classical equations has direct physical meaning, if one uses quasi-classical approach. This combination of the quantum and classical logic, when applied to the Newton system, provides a real, “visible to a naked eye” quantization of the vortex momenta.

The importance of circular systems within radio-schemes increases in the process of transition from marco-to mirco- and then to nano-levels. It is worth mentioning, that on the macro-level any electric chain (since the time of Kirchhoff) as well as the simple radio circuit is usually replaced for necessary calculations by the model system of the interfaced closed chains, cycles (circles) and is being afterwards calculated, investigated or being really assembled, each cycle separately. Theoretically each fixed “coloured” point of a circle can define the function of any circle's element (the resistor, the condenser, the source, the rectifier, the transistor, etc.) or it can correspond to the material (conductor, insulator and so forth). The analysis and the computer modelling of the circles and their nano-structural elements are the purpose of this work.

We explain the Dirac–Segal approach to quantum field theory. We study local observables in this approach and the theory of deformations. We found out that this theory of deformation in the second-order coincides with the renormalization of the same theory, would it be considered in Polyakov approach. We conjecture that it is still true to all orders.

A method based on the spectral analysis of thermowave oscillations formed under the effect of radiation of lasers operated in a periodic pulsed mode is developed for investigating the state of the interface of multilayered systems. The method is based on high sensitivity of the shape of the oscillating component of the pyrometric signal to adhesion characteristics of the phase interface. The shape of the signal is quantitatively estimated using the correlation coefficient (for a film–interface system) and the transfer function (for multilayered specimens).

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.