Igor Volovich

Quantum theory and its stochastic limit

1. Notations and Statement of the Problem.- 2. Quantum Fields.- 3. Those Kinds of Fields We Call Noises.- 4. Open Systems.- 5. Spin-Boson Systems.- 6. Measurements and Filtering Theory.- 7. Idea of the Proof and Causal Normal Order.- 8. Chronological Product Approach to the Stochastic Limit.- 9. Functional Integral Approach to the Stochastic Limit.- 10. Low-Density Limit: The Basic Idea.- 11. Six Basic Principles of the Stochastic Limit.- 12. Particles Interacting with a Boson Field.- 13. The Anderson Model.- 14. Field-Field Interactions.- 15. Analytical Theory of Feynman Diagrams.- 16. Term-by-Term Convergence.- References.

On p-Adic Mathematical Physics

A brief review of some selected topics in p-adic mathematical physics is presented.

Number theory as the ultimate physical theory

At the Planck scale doubt is cast on the usual notion of space-time and one cannot think about elementary particles. Thus, the fundamental entities of which we consider our Universe to be composed cannot be particles, fields or strings. In this paper the numbers are considered as the fundamental entities. We discuss the construction of the corresponding physical theory. A hypothesis on the quantum fluctuations of the number field is advanced for discussion. If these fluctuations actually take place then instead of the usual quantum mechanics over the complex number field a new quantum mechanics over an arbitrary field must be developed. Moreover, it is tempting to speculate that a principle of invariance of the fundamental physical laws under a change of the number field does hold. The fluctuations of the number field could appear on the Planck length, in particular in the gravitational collapse or near the cosmological singularity. These fluctuations can lead to the appearance of domains with non-Archimedean p-adic or finite geometry. We present a short review of the p-adic mathematics necessary, in this context.

On the p-adic summability of the anharmonic oscillator

Abstract Some remarkable arithmetic properties of the anharmonic oscillator are considered. The concept of the p-adic summability is discussed. For the ground-state energy of the anharmonic oscillator the p-adic summability is shown.

On the Adelic String Amplitudes

Abstract Some remarkable properties of the adelic string amplitudes for the physical domain of the Mandelstam variables are considered. It is shown that the p-adic four-point functions are always negative. Also, a formula is obtained which expresses the product of moduli of the p-adic amplitudes and the Veneziano amplitude in terms of the zeta functions. This product is absolutely convergent unlike the divergent product of these amplitudes without moduli, recently considered by Freund and Witten. Using the zeta function representation, p-adic interpolation of the Veneziano amplitude is also considered.

Quantum group particles and non-archimedean geometry

Abstract The classical and quantum mechanics on the quantum plane are discussed. They correspond to q-deformation and h -q- deformations of the usual mechanics, respectively. A particle on the quantum line as the simplest example of such a kind of system is investigated in more detail. A new phenomenon occurs even for the free particle, namely, the mass for a free particle on the quantum line is not a number but a non-commutative element. Some relations between q-deformed quantum mechanics and p-adic quantum mechanics are also presented.

Non-extremal intersecting p-branes in various dimensions☆

Abstract Non-extremal intersecting p-brane solutions of gravity coupled with several antisymmetric fields and dilatons in various space-time dimensions are constructed. The construction uses the same algebraic method of finding solutions as in the extremal case and modified “no-force” conditions. We justify the “deformation” prescription. It is shown that the non-extremal intersecting p-brane solutions satisfy the harmonic superposition rule and intersections of non-extremal p-branes are specified by the same characteristic equations for the incidence matrices as for the extremal p-branes. We show that S -duality holds for non-extremal p-brane solutions. Generalized T -duality takes place under additional restrictions to the parameters of the theory, which are the same as in the extremal case.

QUANTIZATION OF THE RIEMANN ZETA-FUNCTION AND COSMOLOGY

Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein–Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat–Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zeta-function field theory are discussed.

New quantum algorithm for studying NP-complete problems

Ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using our new quantum algorithm.

Open and closed p-adic strings and quadratic extensions of number fields

Abstract In order to elaborate the hypothesis of three types of gravitons being related to three types of closed p -adic strings, the corresponding scattering amplitudes are constructed. The graviton scattering amplitude in the form of the convolution of two characters on the quadratic extension of a locally compact field can be constructed only for the closed type II superstring. The relation between closed and open p -adic string amplitudes change substantially in comparison to the ordinary case. Moreover, the p -adic amplitude for open strings is defined in many ways, depending on the particular ordering on the field of the p -adic numbers.