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European Journal of Physics

Natural number dynamics: reconstructing physics from generalized atomicity

T. R. Robinson

Published Oct 23, 2020• 3 citations
Publisher

Abstract

Abstract A novel approach is presented in which some key basic equations of physics are reconstructed from a generalized atomicity viewpoint i.e., one in which a system is characterised simply by the number of items it contains and hence by the natural numbers alone. The natural numbers may then be associated with the eigenvalue spectrum of an operator. The non-negative nature of the natural numbers leads to the properties of creation and annihilation operators. By examining the parastatistical properties of the number operator, it is also shown that there is only one linear representation that allows a primitive system to be created. An interpretation of the nonlinear factorization of the number operator is also given in terms of the Hamiltonian for bound states of the Schrödinger equation for the case of general one-dimensional potentials. Introducing multiple categories of items yields the standard equations of many-body systems with interaction and scattering, and the existence of bosons and fermions.

DOI
10.1088/1361-6404/abc432

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Abstract A novel approach is presented in which some key basic equations of physics are reconstructed from a generalized atomicity viewpoint i.e., one in which a system is characterised simply by the number of items it contains and hence by the natural numbers alone. The natural numbers may then be associated with the eigenvalue spectrum of an operator. The non-negative nature of the natural numbers leads to the properties of creation and annihilation operators. By examining the parastatistical properties of the number operator, it is also shown that there is only one linear representation that allows a primitive system to be created. An interpretation of the nonlinear factorization of the number operator is also given in terms of the Hamiltonian for bound states of the Schrödinger equation for the case of general one-dimensional potentials. Introducing multiple categories of items yields the standard equations of many-body systems with interaction and scattering, and the existence of bosons and fermions.
In this chapter, the authors address the key goal of finding a representation of the natural numbers as a spectrum of eigenvalues of an operator. A possible representation of an operator that would be sure to have non-negative real eigenvalues would be one that has the form of a modulus. The representation based on H-form operators requires only a trivial modification to the notation. It is particularly relevant to a situation where the number n is specifically being used to count particles and is used in many-body quantum physics. The phasor operator representation leads to second way of representing the states of the number operator, that relies on functional analysis. There is a complementary representation to that of the u-dependent representation of the number operator and the number amplitude operators.
Matrix elements for fermion and boson operators describing the creation or annihilation of particles with multiple angular-momentum quantum numbers can be factored into products of matrix elements of generalized operators, one for each quantum number. The generalized operators have nonzero matrix elements between states of arbitrary permutational symmetry. This factorization greatly increases the efficiency of shell-model calculations. We provide a physical interpretation for these generalized operators and introduce a simple operator-reordering procedure.
Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose properties are given by the axioms for each type of number. The importance of the requirement of efficient implementability for physical models of the axioms is emphasized. Based on this, successor operations for each value of j corresponding to addition of k^{j-1} if j>0 and k^{j} if j<0 are defined. It follows from the efficient implementability of these successors, which is the case for all computers, that implementation of the addition and multiplication operators, which are defined in terms of polynomially many iterations of the successors, should be efficient. This is not the case for definitions based on the successor for j=1 only. This is the only successor defined in the usual axioms of arithmetic.
A new fundamental theory of composite particles in nonrelativistic many-body systems is developed. The theory is constructed making use solely of the physical Fock space for the basic elementary particles which comprise the many-body system. In this manner exchange symmetry in the elementary particles is exact. Physical composite particle creation and annihilation operators are introduced and these operators satisfy exact Bose (Fermi) commutation (anticommutati$\stackrel{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}{\mathrm{o}}$n) relations depending on whether the composites correspond to bosons or to fermions. Commuting physical occupation number operators for composite particles are then constructed in the usual manner from the composite creation and annihilation operators. These number operators are highly dressed in terms of the basic elementary particle operators. Finally, the foundations of statistical mechanics for bound composite particles are formulated in terms of the appropriate occupation number operators.
In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation (a†a^\daggera†) and annihilation (aaa) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator (a†x+ay)n(a^\dagger x+ay)^n(a†x+ay)n with nnn as the exponent and x, yx,\,yx,y are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.