Is there quantum chaos in the prime numbers?
نویسندگان
چکیده
A statistical analysis of the prime numbers indicates possible traces of quantum chaos. We have computed the nearest neighbor spacing distribution, number variance, skewness, and excess for sequences of the first N primes for various values of N . All four statistical measures clearly show a transition from random matrix statistics at small N toward Poisson statistics at large N . In addition, the number variance saturates at large lengths as is common for eigenvalue sequences. This data can be given a physical interpretation if the primes are thought of as eigenvalues of a quantum system whose classical dynamics is chaotic at low energy but regular at high energy. We discuss some difficulties with this interpretation in an attempt to clarify what kind of physical system might have the primes as its quantum eigenvalues.
منابع مشابه
Quantum-like Chaos in Prime Number Distribution and in Turbulent Fluid Flows
Recent studies by mathematicians and physicists have identified a close association between the distribution of prime numbers and quantum mechanical laws governing the subatomic dynamics of quantum systems such as the electron or the photon. It is now recognised that Cantorian fractal space-time fluctuations characterise dynamical systems of all space-time scales ranging from the microscopic su...
متن کاملRandom numbers and random matrices: Quantum chaos meets number theory
The statistical analysis of the eigenvalues of quantum systems has become an important tool in understanding the connections between classical and quantum physics. The statistical properties of the eigenvalues of a quantum system whose classical counterpart is integrable match those of random numbers. The eigenvalues of a chaotic classical system have statistical properties like those of the ei...
متن کاملQuantum Knots and Riemann Hypothesis
In this paper we propose a quantum gauge system from which we construct generalized Wilson loops which will be as quantum knots. From quantum knots we give a classification table of knots where knots are one-to-one assigned with an integer such that prime knots are bijectively assigned with prime numbers and the prime number 2 corresponds to the trefoil knot. Then by considering the quantum kno...
متن کاملInvestigation of strong force influence on behavior of nuclear energy levels in Calcium and Titanium isotopes: Based on quantum chaos theory
The atomic nucleus is a complex many-body system that consists of two types of fermion (neutron and proton). They are in the strong interaction. The statistical properties of energy levels and influence of strong force between these fermions are well described by random matrix theory. Resonance of energy levels depends on the Hamiltonian symmetry placed in one of the GOE, GUE and GSE ensembles ...
متن کاملA Novel Approach for Optimization of Transportation Problem in Chaos Environment
Nature is characterized by its chaotic behavior. Mathematics is considered one of the appropriate tools to achieve the best definition of possible its chaos variables and process. Classical mathematics deals with the numbers as static and meaningless, but chaos mathematics deals with it as dynamic evolutionary, and value- added. This paper attempts to introduce the transportation problem re...
متن کامل