Comment on "Can disorder really enhance superconductivity?"

The paper by Mayoh and Garcia-Garcia [arXiv:1412.0029v1] is entitled "Can disorder really enhance superconductivity?". In our opinion, the answer given by the authors is not satisfactory, and we present the alternative picture. Our reply to the comment [arXiv:1502.06282] is added in the end, in order to reveal a series of untrue statements contained in it.Comment: Latex, 5 pages, reply to [arXiv:1502.06282] is added in the en

Computer Model of a "Sense of Humour". I. General Algorithm

A computer model of a "sense of humour" is proposed. The humorous effect is interpreted as a specific malfunction in the course of information processing due to the need for the rapid deletion of the false version transmitted into consciousness. The biological function of a sense of humour consists in speeding up the bringing of information into consciousness and in fuller use of the resources of the brain.Comment: 10 pages, 3 figures included; continuation of this series to appea

Triviality, Renormalizability and Confinement

According to recent results, the Gell-Mann - Low function \beta(g) of four-dimensional \phi^4 theory is non-alternating and has a linear asymptotics at infinity. According to the Bogoliubov and Shirkov classification, it means possibility to construct the continuous theory with finite interaction at large distances. This conclusion is in visible contradiction with the lattice results indicating triviality of \phi^4 theory. This contradiction is resolved by a special character of renormalizability in \phi^4 theory: to obtain the continuous renormalized theory, there is no need to eliminate a lattice from the bare theory. In fact, such kind of renormalizability is not accidental and can be understood in the framework of Wilson's many-parameter renormalization group. Application of these ideas to QCD shows that Wilson's theory of confinement is not purely illustrative, but has a direct relation to a real situation. As a result, the problem of analytical proof of confinement and a mass gap can be considered as solved, at least on the physical level of rigor.Comment: Latex, 15 page

On 't Hooft's representation of the \beta-function

It is demonstrated, that 't Hooft's renormalization scheme (in which \beta-function has exactly the two-loop form) is generally in conflict with the natural physical requirements and specifies the type of the field theory in an arbitrary manner. It violates analytic properties in the coupling constant plane and provokes misleading conclusion on accumulation of singularities near the origin. It artificially creates renormalon singularities, even if they are absent in the physical scheme. The 't Hooft scheme can be used in the framework of perturbation theory but no global conclusions should be drawn from it.Comment: LaTex, 9 pages, 2 figures include

Upper critical dimension in the scaling theory of localization

It is argued that the Thouless number g(L) is not the only parameter relevant in scale transformations, and that the second parameter connected with off-diagonal disorder should be introduced. A two-parameter scaling theory is suggested that explains a phenomenon of the upper critical dimension from the viewpoint of scaling ideas.Comment: Latex, 8 pages, 2 figure

Numerical results for the Anderson transition. Comment

Answer to cond-mat/0106005 and cond-mat/0106006 and additional notes are given concerning to my previous comment (cond-mat/0105324)

Conductance distribution in 1D systems: dependence on the Fermi level and the ideal leads

The correct definition of the conductance of finite systems implies a connection to the system of the massive ideal leads. Influence of the latter on the properties of the system appears to be rather essential and is studied below on the simplest example of the 1D case. In the log-normal regime this influence is reduced to the change of the absolute scale of conductance, but generally changes the whole distribution function. Under the change of the system length L, its resistance may undergo the periodic or aperiodic oscillations. Variation of the Fermi level induces qualitative changes in the conductance distribution, resembling the smoothed Anderson transition.Comment: Latex, 22 pages, 11 include

The Berry Phase for Simple Harmonic Oscillators

We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found by integration with the help of a computer algebra system.Comment: 7 pages, one figur

A thorny path of field theory: from triviality to interaction and confinement

Summation of the perturbation series for the Gell-Mann--Low function \beta(g) of \phi^4 theory leads to the asymptotics \beta(g)=\beta_\infty g^\alpha at g\to\infty, where \alpha\approx 1 for space dimensions d=2,3,4. The natural hypothesis arises, that asymptotic behavior is \beta(g) \sim g for all d. Consideration of the "toy" zero-dimensional model confirms the hypothesis and reveals the origin of this result: it is related with a zero of a certain functional integral. This mechanism remains valid for arbitrary space dimensionality d. The same result for the asymptotics is obtained for explicitly accepted lattice regularization, while the use of high-temperature expansions allows to calculate the whole \beta-function. As a result, the \beta-function of four-dimensional \phi^4 theory is appeared to be non-alternating and has a linear asymptotics at infinity. The analogous situation is valid for QED. According to the Bogoliubov and Shirkov classification, it means possibility to construct the continuous theory with finite interaction at large distances. This conclusion is in visible contradiction with the lattice results indicating triviality of \phi^4 theory. This contradiction is resolved by a special character of renormalizability in \phi^4 theory: to obtain the continuous renormalized theory, there is no need to eliminate a lattice from the bare theory. In fact, such kind of renormalizability is not accidental and can be understood in the framework of Wilson's many-parameter renormalization group. Application of these ideas to QCD shows that Wilson's theory of confinement is not purely illustrative, but has a direct relation to a real situation. As a result, the problem of analytical proof of confinement and a mass gap can be considered as solved, at least on the physical level of rigor.Comment: Review article, 30 pages, 15 figures. arXiv admin note: substantial text overlap with arXiv:1102.4534, arXiv:0911.114

Density of States near the Anderson Transition in a Space of Dimensionality d=4-epsilon

Asymptotically exact results are obtained for the average Green function and the density of states in a Gaussian random potential for the space dimensionality d=4-epsilon over the entire energy range, including the vicinity of the mobility edge. For N\sim 1 (N is an order of the perturbation theory) only the parquet terms corresponding to the highest powers of 1/epsilon are retained. For large N all powers of 1/epsilon are taken into account with their coefficients calculated in the leading asymptotics in N. This calculation is performed by combining the condition of renormalizability of the theory with the Lipatov asymptotics.Comment: 11 pages, PD