PlanetPhysics/Observables and States

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Introduction

The notions of observables and states are fundamental to mechanics. In this entry, we shall begin with the conceptual background to these ideas, then proceed to examine how these notions work in classical, statistical, and quantum mechanics.

The basis for these notions lies in making numerical measurements on physical systems and comparing the observed values with predicted theoretical values. The value measured will depend on the quantity being measured and upon the initial and boundary conditions imposed on the system. To account for this dependence, we introduce observables and states --- an observable is a mathematical entity in a theory which represents a measurement which can be made on the physical system described by that theory and a state is a mathematical entity which encodes conditions placed on that system. A theory of a system will provide the set of observables and the set of states for that system, describe how they evolve with time, and specify how to obtain numerical values by combining states and observables.

To make this discussion concrete, we may consider an elementary example --- the freely falling body. Here, examples of observables would include the height and velocity of the object. The state of the system may be specified by stating the initial height and velocity or by specifying the height at an initial time and at a final time. Given such a specification, we can then compute the values of velocity and position at any time using these formulae

${\begin{matrix}h-h_{0}&={1 \over 2}g(t-t_{0})^{2}\\v-v_{0}&=g(t-t_{0}).\end{matrix}}$

The values so obtained may then be compared with experiment.

In addition to the height and velocity, there are other observables such as energy. However, it is possible to express these observables algebraically in terms of the height and velocity. (Note that this requires use of the equations of motion.)

${\begin{matrix}E&={1 \over 2}mv^{2}+mgh\end{matrix}}$

Quantum Operators as Observables in Quantum Theories

quantum operator algebras (QOA) in quantum field theories are defined as the algebras of observable operators, and as such, they are also related to the von Neumann algebra;

quantum operators are usually defined on Hilbert spaces, or in some QFTs on Hilbert space bundles or other similar families of spaces.

\htmladdnormallink{representations {http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of Banach $*$-algebras} (that are defined on Hilbert spaces) are closely related to C* -algebra representations which provide a useful approach to defining quantum space-times.

Quantum operator algebras in quantum field theories: QOA Role in QFTs

Important examples of quantum operators are: the Hamiltonian operator (or Schr\"odinger operator), the position and momentum operators, Casimir operators, unitary operators and spin operators. The observable operators are also self-adjoint . More general operators were recently defined, such as Prigogine's superoperators. The observable corresponding to the Hamiltonian operator of a closed, conservative system is its energy.

Another development in quantum theories was the introduction of Frech\'et nuclear spaces or `rigged' Hilbert spaces (Hilbert bundles).

Basic mathematical definitions in QOA:

• Von Neumann algebra
• Hopf algebra
• groupoids
• \htmladdnormallink{Haar systems {http://planetphysics.us/encyclopedia/Groupoid.html} associated to measured groupoids or locally compact groupoids.}
• C*-algebras and quantum groupoids

[more to come]