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Proceedings of the Edinburgh Mathematical Society

Certain Expansions in the Algebra of Quantum Mechanics

Neal H. McCoy

Published Jul 1, 1932• 22 citations
Publisher

Abstract

§ 1. Introduction. The algebra of quantum mechanics is characterized by the fact that the variables p, q obey all the laws of ordinary algebra except that multiplication is non-commutative and instead there exists a relation of the form where c is a real or complex scalar constant and is thus commutative with both p and q.
DOI
10.1017/s0013091500013870

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One of us has shown that the statistical properties of the measurements of a quantum mechanical system assume their simplest form when expressed in terms of a certain hypercomplex algebra which is commutative but not associative.1 This algebra differs from the non-commutative but associative matrix algebra usually considered in that one is concerned with the commutative expression ½(A × B + B × A) instead of the associative product A × B of two matrices. It was conjectured that the laws of this commutative algebra would form a suitable starting point for a generalization of the present quantum mechanical theory. The need of such a generalization arises from the (probably) fundamental difficulties resulting when one attempts to apply quantum mechanics to questions in relativistic and nuclear phenomena.
Abstract The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but satisfy instead certain quantum conditions. One can build up a theory without knowing anything about the dynamical variables except the algebraic laws that they are subject to, and can show that they may be represented by matrices whenever a set of uniformising variables for the dynamical system exists. It may be shown, however (see 3), that there is no set of uniformising variables for a system containing more than one electron, so that the theory cannot progress very far on these lines. A new development of the theory has recently been given by Schrödinger. Starting from the idea that an atomic system cannot be represented by a trajectory, i. e., by a point moving through the co-ordinate space, but must be represented by a wave in this space, Schrödinger obtains from a variation prin­ciple a differential equation which the wave function ψ must satisty. This differential equation turns out to be very closely connected with the Hamiltonian equation which specifies the system, namely, if H (qr, Pr - W = 0 is the Hamiltonian equation of the system, where the qr, Pr are canonical variables, then the wave equation for ψ is {H(qr, ih ∂/∂q) - W} ψ = 0.
Any particular form of mechanics (e.g. classical or quantum) makes use of a particular ‘formalism’ or set of rules governing the use of the symbols representing dynamical variables and the symbols (such as +, =) representing relations between dynamical variables. In the present paper an attempt is made to examine the physical content of this formalism, particularly that of quantum mechanics. This is done by building up a formalism as the direct expression of a number of physical postulates having direct operational meaning. It is shown that, in a simple case, with any two observables A and B can be associated a unique sum observable A + B and a unique (real) symmetric product observable A.B (= B.A). It is next shown that, if, like a classical Hamiltonian system, the system has the property that the time rate of change of an observable depends linearly on some observable H when the environment in which the system moves is varied, then a (real) skew product observable A × B (= − B × A) can be defined. Finally, if the equations of motion are ‘of second order', it is possible to express the second rate of change of any observable as an algebraic function of that observable and H. This leads to algebraic identities from which it follows that ‘complex multiplication’, defined by AB = A.B + iA × B, is associative (but not commutative). Observables are thus shown to possess the properties usually ascribed to them in quantum mechanics. These properties make possible a representation by Hermitian matrices.
Mathematically, the imaginary unit 𝑖 has long been regarded as an algebraic generator extending the real numbers to the complex field. Yet a century after the birth of quantum mechanics (1925–2025), the answer to its oldest question is now clear: quantum mechanics is not merely a mathematical structure—it is a physical reality, and all quantities appearing in its non-commutative relations are physical. Therefore, 𝑖 is not a mere algebraic extension but the universal operator demanded by the geometric completeness of non-commutative reality between position and momentum. The imaginary unit is a physical reality. The canonical relation [𝑥, 𝑝] = 𝑖ℏ is not merely algebraic—it is a geometric law, showing that the complex structure of quantum mechanics arises inevitably from physical reality itself. It stands without contradiction, without complexity. This is nature. In the (𝑥, 𝑝) plane, 𝑖 represents a 90° rotation and ℏ quantizes the curvature of that rotation. Complex numbers thus arise as the natural geometric language of non-commutative physics, providing a geometric resolution to the century-old question of physical completeness first raised by Einstein, Podolsky and Rosen (1935).
Abstract The general representation theory of quantum mechanics requires one to represent the states of a dynamical system by functions of a set of real variables q'r, each of which has a domain consisting of either discrete points or a continuous range of points, or possibly both together. A dynamical variable is represented by a function of two such sets of real variables q'r and q"r forming a generalized “matrix". In this paper we shall show that in certain cases it is advantageous to consider some of our variables q'r as complex variables and to suppose the representatives of states and dynamical variables to depend on them in accordance with the theory of functions of a complex variable. In the usual theory the domains of the s are the eigenvalues of certain observables q"r. This significance of the q"s of course gets lost when we consider them as complex variables, but we have, however, some beautiful mathematical features appearing instead, and we gain a considerable amount of mathematical power for the working out of particular examples.