IJPAM: Volume 39, No. 3 (2007)

A REPRESENTATION OF REAL AND COMPLEX
NUMBERS IN QUANTUM THEORY

Paul Benioff$^1$
Physics Division
Argonne National Laboratory
9700, South Cass Avenue, Argonne, IL 60439, USA
e-mail: pbenioff@anl.gov


Abstract.A quantum theoretic representation of real and complex numbers is described as equivalence classes of Cauchy sequences of quantum states of finite strings of qubits. There are $4$ qubit types, each with associated single qubit annihilation creation operators that give the state and location of each qubit type on a $2$ dimensional integer lattice. The string states, defined as finite products of creation operators acting on the vacuum state, correspond to complex rational numbers with real and imaginary components. These states span a Fock space $\mathcal F.$ Arithmetic relations and operations are defined for the string states. Cauchy sequences of these states are defined, and the arithmetic relations and operations lifted to apply to these sequences. Based on these, equivalence classes of these sequences are seen to have the requisite properties of real and complex numbers. The representations have some interesting aspects. Quantum equivalence classes are larger than their corresponding classical classes, but no new classes are created. There exist Cauchy sequences where each state in the sequence is an entangled superposition of the real and imaginary components, yet the sequence is a real number. Except for superposition state coefficients, the construction is done with no reference to the real and complex number base of $\mathcal F.$

Received: May 23, 2007

AMS Subject Classification: 81Q99, 40A05, 11B99, 81P68

Key Words and Phrases: quantum representations of real and complex numbers, Cauchy sequences, states of finite strings of qubits

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 39
Issue: 3