A note on fractal strings and spacetime

A short discussion is presented on the role of Mersenne primes and even perfect numbers in fractal strings, Cantorian-fractal spacetime, spin structures on Riemann surfaces and the classification of consistent string theories. Since in principle the number of Mersenne primes should be infinite this entails that the family of consistent quantum string theories in higher dimensions than 26 could be infinite as well and, consequently, this could be the physical reason why the number of quantum-spacetime dimensions is infinite. The Tsallis entropy of fractal strings is constructed followed by the exact expression for the fractal string mass spectrum.     

Hilbert cube model for fractal spacetime

A three-dimensional Hilbert cube has exactly three dimensions. It can mimic our spatial world on an ordinary observation scale. A four-dimensional Hilbert cube is equivalent to Elnaschie Cantorian spacetime. A very small distance in a very high observable resolution is equivalent to a very high energy spacetime which is inherently Cantorian, non-differentiable and discontinuous. This article concludes that spacetime is a fractal and hierarchical in nature. The spacetime could be modeled by a four-dimensional Hilbert cube. Gravity and electromagnetism are at different levels of the hierarchy. Starting from a simple picture of a four-dimensional cube, a series of higher dimensional polytops can be constructed in a self-similar manner. The resulting structure will resemble a Cantorian spacetime of which the expectation of the Hausdorff dimension equals to 4.23606799 provided that the number of hierarchical iterations is taken to infinity. In this connection, we note that Heisenberg Uncertainty Principle comes into play when we take measurement at different levels of the hierarchy.     

On the number of elementary particles in a resolution dependent fractal spacetime

We reconsider the fundamental question regarding the number of elementary particles in a minimally extended standard model. The main conclusion is that since the dimension of E-infinity spacetime is resolution dependent, then the number of elementary particles is also resolution dependent. For D = 10 of superstrings, D = 11 of M theory and D = 12 of F theory one finds N(SM) equal to (6)(10) = 60, (6)(11) = 66 and (6)(12) = 72 particles, respectively. This is in perfect agreement with prediction made previously by Mohamed Saladin El-Naschie and Marek-Crnjac.     

Real non-Archimedean structure of spacetime

Institute of Electronic Technology, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 86, No. 2, pp. 177–190, February, 1991.     

p-Adic model of quantum mechanics and quantum channels

A p-adic realization of the standard statistical model of quantum mechanics is constructed. Within this realization, a p-adic linear bosonic channel is defined, and its properties are analyzed. In particular, a criterion for the existence of a linear Gaussian bosonic channel is obtained, and its explicit construction is described. It is shown that the p-adic Gaussian bosonic channels possess an additivity property.     

Hyperbolic quantum mechanics

We start to develop the quantization formalism in a hyperbolic Hilbert space. Generalizing Born’s probability interpretation, we found that unitary transformations in such a Hilbert space represent a new class of transformations of probabilities which describe a kind of hyperbolic interference. The most interesting problem which prompted by our investigation is to find experimental evidence of hyperbolic interference. The hyperbolic quantum formalism can also be interesting as a new theory of probability waves that can be developed in parallel with the standard quantum theory. Comparative analysis of these two wave theories could be useful for understanding of the role of various structures of the standard quantum formalism. In particular, one of distinguishing feature of the hyperbolic quantum formalism is the restricted validity of the superposition principle.     

Quantum mechanics in Riemannian spacetime. II. Operators of observables

The formulation of the generally covariant analog of standard (nonrelativistic) quantum mechanics in a general Riemannian spacetime begun in earlier studies of the author is continued with the introduction of asymptotic (with respect toc ^–2) operators of the spatial position of a spinless particle and of the projection of its momentum onto an arbitrary spacetime direction. The connection between the position operator and the generalization of theV _1,3 Newton—Wigner operator is established. It is shown that the projection of the momentum onto the 4-velocity of the frame of reference (the energy operator) is unitarily equivalent to the Hamiltonian in the Schrödinger equation.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 412–423, March, 1992.     

Inertial fractal set and fractal structure of attractors for the ginzburg-landan equation

This paper proves that the Ginzburg-Landan partial differential equation admits an inertial fractal set whose fractal dimension is finite. Furthermore, We produce an exponentially approximating sequence of localizing compact fractal sets and a fractal structure of the attractor. This project is supported by the National Natural Science Foundation of China     

Stochastic model of quantum mechanics

Moscow Radio Engineering Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 82, No. 2, pp. 208–215, February, 1990.     

P-adic quantum mechanics and coherent states

V. A. Steklov Mathematics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 86, No. 2, pp. 210–220, February, 1991.     

Measurement and probability in quantum mechanics

Measurement-theoretical foundations of quantum probabilities are investigated in the form of measurement statistics and a statistical ensemble interpretation of quantum mechanics.     

Backbone fractal dimension and fractal hybrid orbital of protein structure

Fractal geometry analysis provides a useful and desirable tool to characterize the configuration and structure of proteins. In this paper we examined the fractal properties of 750 folded proteins from four different structural classes, namely (1) the α-class (dominated by α-helices), (2) the β-class (dominated by β-pleated sheets), (3) the (α/β)-class (α-helices and β-sheets alternately mixed) and (4) the (α + β)-class (α-helices and β-sheets largely segregated) by using two fractal dimension methods, i.e. “the local fractal dimension” and “the backbone fractal dimension” (a new and useful quantitative parameter). The results showed that the protein molecules exhibit a fractal behavior in the range of 1 ? N ? 15 (N is the number of the interval between two adjacent amino acid residues), and the value of backbone fractal dimension is distinctly greater than that of local fractal dimension for the same protein. The average value of two fractal dimensions decreased in order of α > α/β > α + β > β. Moreover, the mathematical formula for the hybrid orbital model of protein based on the concept of backbone fractal dimension is in good coincidence with that of the similarity dimension. So it is a very accurate and simple method to analyze the hybrid orbital model of protein by using the backbone fractal dimension.     

Necessary and sufficient postulates of quantum mechanics

We describe a system of axioms that, on one hand, is sufficient for constructing the standard mathematical formalism of quantum mechanics and, on the other hand, is necessary from the phenomenological standpoint. In the proposed scheme, the Hilbert space and linear operators are only secondary structures of the theory, while the primary structures are the elements of a noncommutative algebra (observables) and the functionals on this algebra, associated with the results of a single observation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 510–529, March, 2005