Postulates and Theorems of Quantum Mechanics

Christian DE RONDE, César Massri

In this paper we derive a theorem which proves that the physical interpretation implied by the first postulate of quantum mechanics (QM) is inconsistent with the orthodox formalism. In order to expose this inconsistency we will analyze how the concept of ‘physical system’ is built within classical theories through the notion of invariance and explain in what sense a vector in Hilbert space is not capable of fulfilling these same mathematical conditions. Through an analysis of the mathematical formalism we derive a No Dynamical Invariance (NDI) theorem which proves that, contrary to what is claimed in the first postulate, a vector in Hilbert space cannot be interpreted coherently as the state of a physical (quantum) system. We conclude the paper by analyzing the consequences of the NDI theorem with respect to several ongoing debates in QM.

John P. Lowe

Alon E. Faraggi, Marco Matone

We consider the two main theorems in the derivation of the Quantum Hamilton--Jacobi Equation from the Equivalence Postulate (EP) of quantum mechanics. The first one concerns a basic cocycle condition, which holds in any dimension with Euclidean or Minkowski metrics and implies a global conformal symmetry underlying the Quantum Hamilton--Jacobi Equation. In one dimension such a condition fixes the Schwarzian equation. The second theorem concerns energy quantization which follows rigorously from consistency of the equivalence postulate.

John P. Lowe

Anthony Rizzi

Hegerfeldt's theorem shows that the localization of a particle (or field) is not possible under very general assumptions of quantum mechanics and special relativity. This theorem, and further developments and associated theorems force penetrating questions on the proper understanding of the theory and its use. In particular, the use of the collapse postulate is called into question. The existence of particles and even localized fields are called into question. And, this in turn challenges local measurements. It generally forces one to consider the profound problems and even wrong results that arise from the collapse postulate. We give a qualitatively new example of such a wrong result. Researchers and students alike are bothered by these fundamental issues. Students' ability to learn the material, and even their interest in physics, can be compromised by such disturbing conundra that seem to make physics, or at least QFT and its QM limit, seem incomprehensible and even inconsistent. This article resolves these issues in a way accessible to undergraduates, by applying the natural (minimal) interpretation of quantum mechanics (QM) (the ensemble interpretation) and explicating and clarifying the basic principles and premises used in QM and the Hegerfeldt theorem and its application to measurements.

John P. Lowe