# Math. Axioms of Quantum Mechanics: $C^*$- vs. $W^*$-Algebras

In the general formulation of Quantum Mechanics, a physical system is described by a $C^*$-algebra $\mathcal{A}$, where the observables correspond to the self-adjoint elements of the algebra, and the states correspond to the normalized positive linear functionals acting on $\mathcal{A}$. All at all possible outcomes of a measurement of an observable $A \in \mathcal{A}$ are given by its spectrum $\sigma(A)$, and the probability distribution of $A$ in the state $\rho$ is given by the probability distribution $d\rho_A$ induced on $\sigma(A)$ by the Gel'fand isomorphism. Furthermore, the expectation value of $A$ in $\rho$ is then simply given by the dual pair $\rho(A)$ (cp. for example Walter Thirring: "Quantum Mathematical Physics: Atoms, Molecules and Large Systems").

However, in almost every case, one considers a physical system as not only a $C^*$-, but a $W^*$-algebra (i.e. an abstract von Neumann algebra). There are various ways to differentiate the $W^*$- from the more general $C^*$-algebras; for this application, the following property is probably the most interesting one:

A $C^*$-algebra $\mathcal{A}$ is a $W^*$-algebra iff every norm-bounded ascending filter (/net) has a supremum in $\mathcal{A}$, where the (semi-) order is induced by the notion of positivity in $\mathcal{A}$.

From a pragmatic (and purely mathematical) standpoint, there are many much more powerful statements one can make in $W^*$-algebras that do not hold in general $C^*$-algebras. However, how does one motivate this additional axiom from a physical perspective?

In a somewhat related note, when one deals with $W^*$-algebras, the states of interest are almost always just the normal (i.e. supremum-preserving) states (see e.g. the time-evolution in the Schrödinger picture, which is defined only for the normal states). What is the reason for this? Are the other states not "physically relevant"? If yes, how so?

PS: If one could give some references dealing with these kind of questions, I would be very thankful - sources motivating the axioms of QM are surprisingly hard to come by (even the otherwise excellent Thirring skips over the justifications for the assumptions that I mentioned.)

These are delicate issues.

Both $C^*$-algebras and $W^*$-algebras (also known as von Neumann algebras) are used to describe the observables of a quantum system. Properly speaking, observables are self-adjoint elements of the algebra.

$W^*$ algebras are relevant in physics since they are isomorphic to concrete $C^*$-algebras of bounded operators in complex Hilbert spaces with the further property that they are closed with respect to weak and strong operator topology and satisfy the so-called double commutant theorem. Finally they are generated (in the complex case) by their set of orthogonal projectors interpretable as the set of elementary observables of the quantum system, providing this way a natural interpretation of the spectral theorem and a justification of the correspondence

The closure of von Neumann algebras with respect to the strong operator topology is the feature that makes them more interesting than simple $C^*$-algebra of operators. Simultaneously it makes them very rigid structures not suitable to describe some quantum systems in general situations (e.g., locally covariant QFT in curved spacetime).

I mention two consequences.

(a) If a (bounded) self-adjoint operator belong to a $C^*$-algebra of operators representing the observables of a quantum system, in general, its projection-valued measure does not belong to the algebra -- it would happen if the algebra were von Neumann -- because it is obtained from the algebra with operations which are not uniformly continuous but just strongly continuous.

(b) Similarly if a symmetry is described by a strongly continuous unitary group of operators of a $C^*$-algebra of concrete operators, generally speaking, the projection-valued measure of the generator of the group does not belong to the $C^*$-algebra. It happens if the algebra is von Neumann instead.

Also superselection rules enter the theory differently depending on the choice between $C^*$-algebras and von Neumann algebras to describe the observables of the system.

The use of non-normal states (of either $C^*$ or $W^*$ algebra) is sometimes compulsory, since there are mathematical results forcing two irreducible representations of the same algebra to be non-unitarily equivalent and these representations are constructed upon different algebraic states through the GNS construction. As a consequence the said (pure) states cannot belong to the same folium. Haag's theorem is the most known result of this sort. However there are other cases in QFT and in statistical mechanics discussing extended systems and spontaneously broken symmetry.

References. There are many papers on these subjects, but usually they are too mathematically minded or mostly directed towards QFT.

Haag's book: Local Quantum Physics (Second Revised and Enlarged Edition). Springer, Berlin (1996) mentions several general results though the book is evidently interested in QFT.

Emch's book: Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience, New York (1972) is another (maybe a bit hold) source. There also Jordan algebras are exploited as crucial tools.

Strocchi's introductive textbook: An Introduction To The Mathematical Structure Of Quantum Mechanics: A Short Course For Mathematicians. World Scientific, Singapore (2005). This book is directly related to quantum mechanics but it is also its limit.

A classic reference written as an encyclopedia is Beltrametti-Cassinelli's book Beltrametti, E.G., Cassinelli, G.: The logic of quantum mechanics. Encyclopedia of Mathematics and its Applications, vol. 15, Addison-Wesley, Reading, Mass.,(1981).

A modern treatise where several problems related to those you raised (very complete but perhaps sometimes written into a too concise fashion) Blank, J., Exner, P., Havlicek, M.: Hilbert Space Operators in Quantum Physics, second edition, Springer, Berlin (2007).

A bit discussion on all these issues appears in my lectures of some years ago https://arxiv.org/abs/1508.06951

A much more extended discussion -- including a list of axioms for QM -- can be found in my 2013 book http://www.springer.com/it/book/9788847028340 (I am sure that the file of the book stays in library genesis!) and especially in the 2nd corrected and enlarged edition http://www.springer.com/it/book/9783319707051 in print, where I have added several sections on these subjects.

A recent book containing chapters of several authors (I wrote Ch.5 together with I.Khavkine) about applications of algebraic formalism to QFT is Brunetti R., Dappiaggi, C., Fredenhagen, K., and Yngvason, J. (Eds): Advances in Algebraic Quantum Field Theory, Springer-Verlag, Berlin, (2015).

• Thank you for your extensive answer! I will definitely look into the references you mentioned! Oct 16, 2017 at 16:35