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

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.)

Quantum Operators as Observables in Quantum Theories

Remark 2.1   representations 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 Schrödinger 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 Frechét nuclear spaces or `rigged' Hilbert spaces (Hilbert bundles).

Basic mathematical definitions in QOA:

• Von Neumann algebra
• Hopf algebra
• groupoids
• Haar systems associated to measured groupoids or locally compact groupoids.
• C*-algebras and quantum groupoids

[more to come]

Bibliography

1

E. M. Alfsen and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkh'́auser, Boston–Basel–Berlin (2003).

2

I. Baianu : Categories, Functors and Automata Theory: A Novel Approach to Quantum Automata through Algebraic–Topological Quantum Computations., Proceed. 4th Intl. Congress LMPS, (August-Sept. 1971).

3

I. C. Baianu, J. F. Glazebrook and R. Brown.: A Non–Abelian, Categorical Ontology of Spacetimes and Quantum Gravity., Axiomathes 17,(3-4): 353-408(2007).

4

F.A. Bais, B. J. Schroers and J. K. Slingerland: Broken quantum symmetry and confinement phases in planar physics, Phys. Rev. Lett. 89 No. 18 (1–4): 181-201 (2002).

5

M. Chaician and A. Demichev: Introduction to Quantum Groups, World Scientific (1996).

6

A. Connes: Noncommutative Geometry, Academic Press 1994.

7

L. Crane and I.B. Frenkel. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. Topology and physics. J. Math. Phys. 35 (no. 10): 5136-5154 (1994).

8

W. Drechsler and P. A. Tuckey: On quantum and parallel transport in a Hilbert bundle over spacetime., Classical and Quantum Gravity, 13:611–632 (1996).

9

V. G. Drinfel'd: Quantum groups, In Proc. Int. Cong. of Mathematicians, Berkeley 1986, (ed. A. Gleason), Berkeley, 798–820 (1987).

10

G. J. Ellis: Higher dimensional crossed modules of algebras, J. of Pure Appl. Algebra 52 (1988), 277–282.

11

P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups, Comm.Math.Phys., 196: 591-640 (1998)

12

P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19–52 (1999)

13

P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang–Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999), pp. 89–129, Cambridge University Press, Cambridge, 2001.

14

B. Fauser: A treatise on quantum Clifford Algebras. Konstanz, Habilitationsschrift. (2002).

15

J. M. G. Fell. 1960. “The Dual Spaces of C*–Algebras.”, Transactions of the American Mathematical Society, 94: 365–403 (1960).

16

F.M. Fernandez and E. A. Castro.: (Lie) Algebraic Methods in Quantum Chemistry and Physics., Boca Raton: CRC Press, Inc (1996).

17

R. P. Feynman: Space–Time Approach to Non–Relativistic Quantum Mechanics, Reviews of Modern Physics, 20: 367-387 (1948). [It is also reprinted in (Schwinger 1958).]

18

Gel'fand, I. and Naimark, M., 1943, On the Imbedding of Normed Rings into the Ring of Operators in Hilbert Space, Recueil Mathématique [Matematicheskii Sbornik] Nouvelle Série, 12 [54]: 197-213. [Reprinted in C*-algebras: 1943–1993, in the series Contemporary Mathematics, 167, Providence, R.I. : American Mathematical Society, 1994.]

19

R. Gilmore: “Lie Groups, Lie Algebras and Some of Their Applications.”, Dover Publs., Inc.: Mineola and New York, 2005.

20

P. Hahn: Haar measure for measure groupoids., Trans. Amer. Math. Soc. 242: 1-33(1978).

21

P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc. 242:34-72(1978).

22

R. Heynman and S. Lifschitz. 1958. “Lie Groups and Lie Algebras”., New York and London: Nelson Press.

23

C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008)