# IJPAM: Volume 39, No. 3 (2007)

**A REPRESENTATION OF REAL AND COMPLEX**

NUMBERS IN QUANTUM THEORY

NUMBERS IN QUANTUM THEORY

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
qubit types, each with associated single qubit annihilation
creation operators that give the state and location of each
qubit type on a 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 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

**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