Why we do quantum mechanics on Hilbert spaces
نویسنده
چکیده
You will go through great pains to learn the profound mathematics of Hilbert spaces and operators on them. What in experiment suggests the specific form of quantum mechanics with its “postulates”? Why should measurable quantities be represented by operators on a Hilbert space? Why should the complete information about a system be represented by a vector from a Hilbert space? It looks like we make a lot of assumptions for setting up quantum mechanics. The arguments below will show, that we make one less than we make for classical mechanics, and that this intails all the strangeness. It is a bit like in general relativity: by omitting one postulate from geometry, we enter a whole new space of possibilities. The short overview presented below is a severely diluted representation of the discussion in: F. Strocchi, Mathematical Structure of Quantum Mechanics W. Thirring, Mathematical Physics III: Quantum Mechanics
منابع مشابه
Hermitian metric on quantum spheres
The paper deal with non-commutative geometry. The notion of quantumspheres was introduced by podles. Here we define the quantum hermitianmetric on the quantum spaces and find it for the quantum spheres.
متن کاملQuantum Mechanics beyond Hilbert Space
When describing a quantum mechanical system, it is convenient to consider state vectors that do not belong to the Hilbert space. In the rst part of this paper, we survey the various formalisms have been introduced for giving a rigorous mathematical justiication to this procedure: rigged Hilbert spaces (RHS), scales or lattices of Hilbert spaces (LHS), nested Hilbert spaces, partial inner produc...
متن کاملConvexity and the Euclidean Metric of Space-Time
We address the reasons why the “Wick-rotated”, positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point o...
متن کاملThe Mathematical Formalism of Quantum Mechanics
The prerequisites for Physics 221A include a full year of undergraduate quantum mechanics. In this semester we will survey that material, organize it in a more logical and coherent way than the first time you saw it, and pay special attention to fundamental principles. We will also present some new material. Physics 221B will largely consist of new material. We begin by presenting some of the m...
متن کاملQuantum Mechanics for Totally Constrained Dynamical Systems and Evolving Hilbert Spaces
We analyze the quantization of dynamical systems that do not involve any background notion of space and time. We give a set of conditions for the introduction of an intrinsic time in quantum mechanics. We show that these conditions are a generalization of the usual procedure of deparametrization of relational theories with hamiltonian constraint that allow to include systems with an evolving Hi...
متن کامل