# What algebraic structure does the collection of all physical quantities form?

What algebraic structure -- by which I'm referring to abstract algebra theoretic ones such as ring, field, module, etc. -- does the collection of all physical quantities form?

An related and/or variant question is -- what algebraic structure does the collection of all (or some reasonable subset of) units form?

• This might be the first abstract algebra question I've seen on Physics SE. – Jimmy360 Mar 31 '15 at 4:47
• @Jimmy360: If you type "C* algebra" in the search bar, you get 2675 results. Or try "Lie group", "Lie algbera", etc. – Nikolaj-K Mar 31 '15 at 9:34

The physical observables both in classical and quantum mechanics are thought to have, in modern mathematical terms, an involutive algebra structure.

This point of view has been developed first and foremost for quantum mechanics. The observables of quantum mechanics are a suitable $C^*$-algebra. By Gel'fand-Naimark isomorphism any $C^*$ algebra is isometrically isomorphic to an algebra of bounded operators on a Hilbert space, therefore the interpretation of QM observables as operators in a Hilbert space is natural. The unbounded operators, unluckily, are not directly contained in the algebra; they are introduced by two different notions:

• they can be affiliated to the $C^*$ algebra, i.e. an unbounded normal operator is affiliated to an algebra if its spectral projections are all contained in the algebra (and possibly also its associated unitary group if self-adjoint);

• the completeness of the algebra w.r.t. the operator norm is relaxed, and we content with a $*$-algebra (usually with locally convex topology); many results such as the Gel'fand-Naimark-Segal construction can be generalized to $*$-algebras, however many subtleties involving domains, irreducible representations, commutation and so on has to be taken into account.

Given a $C^*$ algebra $V^*$, two related spaces are also important: the dual space $V^*$, and the bicommutant $V^{''}$. The dual space $V^*$ contains the quantum states $E_V$: they are the positive functionals with norm one, and form a convex subset. The bicommutant $V^{''}$ is a von Neumann algebra, and has a unique predual $V_*$, useful to define the normal states $E_V\cap V_*$ (density matrices).

The quantum evolution is usually given by a proper subset of the algebra of linear endomorphisms of $V_*$ that are isometric and preserve positivity (and therefore by duality you get an endomorphism of observables, and again by duality an endomorphism of general states).

The reasoning above can be extended to classical systems, restricting to commuting involutive algebras; the classical evolution however, is usually non-linear.

(Philosophical) Addendum: The original question is about the collection of all physical quantities; despite this concept being quite vague, I would say that any reasonable "collection of all physical quantities" is not a set but a proper class, therefore it cannot have a "common" algebraic structure (that are defined on sets); at most it may be a category or a conglomerate, but since our knowledge of the physical world is so far incomplete, you cannot know what the collection of all physical quantities really should be; surely you may imagine it is sufficiently large not to be a set.