The Hilbert space of conditional clauses

Abstract

In the absence of a satisfactory interpretation of quantum theory, physical law lacks physical basis. This paper reviews the orthodox, or Dirac–von Neumann interpretation, and makes explicit that Hilbert space describes propositions about measurement results. Kets are defined as conditional clauses referring to measurements in a formal language. It is seen that these clauses are elements of a Hilbert space, such that addition is logical disjunction, the dual space consists of consequent clauses, and the inner product is a set of statements in the subjunctive mood. The probability interpretation gives truth values for corresponding future tense statements when the initial state is actually prepared and the final state is to be measured. The mathematical structure of quantum mechanics is formulated in terms of discrete measurement results at finite level of accuracy and does not depend on an assumption of a substantive, or background, space–time continuum. A continuum of kets, ∣x〉 for x ∈ ℝ3, is constructed from linear combinations of kets in a finite basis. The inner product can be expressed either as a finite sum or as an integral. Discrete position functions are uniquely embedded into smooth wave functions in such a way that differential operators are defined. It is shown that the choice of basis has no effect on underlying physics (quantum covariance). The Dirac delta has a representation as a smooth function. Operators do not in general have an integral form. The Schrödinger equation is shown from the requirements of the probability interpretation. It is remarked that a formal construction of quantum electrodynamics (qed) avoiding divergence problems has been completed using finite dimensional Hilbert space. I conclude that quantum mechanics makes statements about the world with clear physical meaning, such that space is emergent from particle interactions and has no fundamental role.