Quantum Mechanics from Two Physical Postulates

Abstract

For an arbitrary preparation, quantum mechanical descriptions refer to the complementary contexts set by incompatible measurements. We argue that an arbitrary preparation, therefore, should be described with respect to such a context by its degrees of disturbance (represented by real numbers) and their probability distribution (postulate 1). Measurement contexts thus provide reference frames for the preparation space of a physical system; a preparation being described by a point in this space with the aforementioned as its coordinates relative to a given measurement apparatus. However, all measurement contexts are equivalent with regard to the description of a given preparation; there is no preferred measurement (postulate 2). In the framework provided by the preparation space, we show that quantum mechanics emerges naturally from the above postulates in a new formulation which is manifestly canonical; provided the degrees of disturbance are identified with the quantum phases of the preparation with respect to (the basis furnished by) the measurement apparatus.

A geometric approach to the canonical reformulation of quantum mechanics

“…Furthermore, if the reference frame is such that phases are absent in the functional form of w, then, upon performing the phase integration, (17) reduces to ρ ij = ρ i δ ij , where ρ i is given by (14). This corresponds to the diagonal representation of ρ, with {ρ i } as its eigenvalues.…”

“…Therefore, the entropy too reduces to its familiar expression S = −tr(ρ lnρ). Finally, to complete the correspondence we should prove the equivalence of the von Neumann equation for ρ ij and the Liouville-like equation for w. One approach would be to substitute (17) into the von Neumann equation and obtain the Liouville-like equation (9) as the necessary and sufficient condition. However, a simpler proof is provided by noting that, since, (2π) n w = (n + 1)!…”

“…However, a simpler proof is provided by noting that, since, (2π) n w = (n + 1)! < ρ > −n!, which is most easily derived from (17) in the diagonal representation, we havė…”

“…Although the canonical formulation is well known [8,9,10,11,12,13,14,15,16], our emphasis on the fundamental significance of the line element bears valuable insight for understanding the foundations of quantum mechanics, provided the line element can be obtained from an independent premise. Such a premise, from which the line element follows as the sum of the above two terms, is given in reference [17]. Finally, in section III we discuss the canonical formulation of quantum statistical mechanics in the preparation space.…”

Геометрический Подход К Канонической Переформулировке Квантовой Механики

“…Хо-тя каноническая формулировка хорошо известна [3], поставленный нами акцент на фундаментальном значении, имеющем элемент длины, придает значительную глу-бину пониманию основ квантовой механики при условии, что элемент длины можно получить из независимых предпосылок. Такие предпосылки, из которых следует, что элемент длины представляет собой сумму двух указанных выше слагаемых, да-ны в работе [4]. Наконец, в разделе 3 мы обсуждаем каноническую формулировку квантовой статистической механики в пространстве приготовлений.…”