order-theoretic structure in quantum mechanics


The following surveys how basic theorems about the standard foundation of quantum mechanics imply an accurate geometric incarnation of the “phase space in quantum mechanics” by an order-theoretic structure that combines with an algebraic structure to a ringed topos, the “Bohr topos”. While the notion of Bohr topos has been motivated by the Kochen-Specker theorem, the point here is to highlight that taking into account further theorems about the standard foundations of quantum mechanics, the notion effectively follows automatically and provides an accurate and useful description of the geometry of “quantum phase space” also in quantum field theory formulated in the style of AQFT.



physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics


algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT


The quantum phase space as a Jordan-geometry given by a ringed topos

Back when quantum mechanics was discovered in the first half of the 20th century, it was eventually formalized as mathematical physics and the traditional modern formulation emerged, where, in AQFT perspective, quantum observables are represented by suitable linear operators on a Hilbert space and more generally by elements of a C*-algebra of observables, and where quantum states are certain (namely positive and normalized) linear functionals on these C*-algebras of observables (states on star-algebras).

But one may still ask if the axioms in the definition of C*-algebra accurately capture the intended physics. This or similar questions were discussed back in the middle of the 20th century, when these notions were still in flux. Specifically in the 1930s Pascual Jordan argued (Jordan 32) that the associative algebra structure on the observables is more structure than supported by the physics of states and observation, that instead only the underlying structure of what is now called the Jordan algebra should matter.

Much later in 1978 this idea was formally validated by the Alfsen-Shultz theorem (Alfsen-Shultz 78). This states that the space of quantum states for given quantum observables depends indeed only on the underlying Jordan algebra structure. This is not too surprising: the definition of a state on an operator algebra does not even mention the associative algebra structure but mentions only the positivity structure, which is what the Jordan algebra captures.

Despite these insights, Jordan algebras found and find only marginal attention in mathematical physics. In a review from 2004 of the book of Alfsen and Shultz it says that back then Jordan algebras were hoped to shed light on conceptual problems of genuine quantum field theory, but that these hopes never materialized. However, more recent developments change this picture a bit, we come to that below.

Much more recently in 2010, the Harding-Döring-Hamhalter theorem sheds a new light on the role of Jordan algebra structure. This theorem states mild conditions under which a Jordan algebra structure on quantum observables is equivalently encoded in the poset of commutative subalgebras of the full C*-algebra.

These commutative subalgebras themselves are of course of old fame in quantum mechanics, they are the “classical contexts” given by tuples of quantum observables that all commute with each other and hence which can all be measured simultaneously without the uncertainty principle interfering. These classical contexts played a crucial role in the discussion of the foundation of quantum mechanics in the first half of the 20th century: back then people argued that for AA and BB two quantum observables which do not commute with each other it is unclear what it means physically to form their sum A+BA + B or their product ABA B, hence that a quantum state should be demanded to be a linear (and positive normed) functional on all commutative subalgebras, but not necessarily on the whole non-commutative algebra. Such a notion of “quantum state on all classical contexts” was called a quasi-state.

The issue of whether quasi-states are a more accurate description of quantum measurement was settled in 1957 by Gleason's theorem (Gleason 57), which says that given a Hilbert space of dimension greater than 2, then the quasi-states are automatically quantum states also on the full non-commutative algebra of observables. Typically this is viewed as making the notion of quasi-state obsolete, but since that is a formally weaker notion the opposite attitude makes sense: what is traditionally taken as the definition of quantum state is more accurately thought of as being a quasi-state, hence something that is intrinsically related not to a non-commutative algebra of observables, but to the “classical contexts” of its poset of commutative subalgebras.

Indeed, by combining the Alfsen-Shultz theorem with the Harding-Döring-Hamhalter theorem, we have (under the pertinent mild assumptions) that two algebras of observables have the same space of quantum states already when they have the same poset of commutative subalgebras. Notice that where Gleason's theorem only involves the commutative subalgebras themselves, this Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem crucially involves their order of inclusions, hence the actual poset structure of their inclusions.

Therefore there is an intrinsic order theoretic aspect in the standard foundations of quantum mechanics. But it is not just order theory, for it is not the poset structure of inclusions alone that matters in the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem, but that poset structure together with the actual commutative C*-algebra structure of each classical context.

There is an elegant way to combine these two aspects: a system of commutative algebras together with the order of their inclusions is equivalently a single algebra object internal to the sheaf topos over the poset. With some basics of topos theory in hand this is a trivial statement, but in view of the above it is worthwhile to make explicit: the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem (Harding-Döring 10, Hamhalter 11) says that the collection of quantum observables in quantum mechanics is accurately formalized by a single commutative C*-algebra internal to a sheaf topos over a poset.

It has been argued that this serves as a formalization of the views on quantum mechanics that in the middle of the 20th century Niels Bohr expressed in extensive but informal writing (see Scheibe 73): he said roughly that whatever quantum mechanics is, it must be expressible and must be expressed through classical contexts. In honor of this intuition, the above toposes deserve to be called Bohr toposes, following the term “Bohrification” in (Heunen-Landsman-Spitters 09).

In fact, a Bohr topos is fairly trivial as far as toposes go, since, by the above, it is just a reflection – precisely: the “localic reflection” – of a purely order theoretic structure. But the key is that passing to the topos over the poset provides a home for the context-wise commutative C*-algebra structure which makes the Bohr topos have the additional structure of a ringed topos. This is the additional algebraic datum on top of the purely order theoretic datum in the Jordan algebra structure of quantum observables.

Ringed toposes have of course a long tradition in geometry (most famously in algebraic geometry). By Grothendieck‘s foundational work, laid out in the thesis of Monique Hakim, ringed toposes form a general foundation for structured geometry. More recently this was further strengthened and refined by Jacob Lurie by the notion of higher ringed toposes, which we will see appear in quantum field theory below. In as far as the quantum observables in quantum mechanics are supposed to be the dual of the phase space, it is therefore natural to have this “quantum phase space” be realized as a ringed topos.

Notice that this is a different geometric interpretation of “quantum phase space” than the traditional idea that quantum phase space is an object in non-commutative geometry! Here by the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem we see that it is not actually accurate to say that quantum phase space is dually given by a non-commutative C*-algebra, as in fact it is given dually just by a Jordan algebra. The ringed Bohr topos provides a natural geometric interpretation of this, one might call it “Jordan geometry”.

This geometric nature of the Bohr topos becomes more manifest as we consider its opposite category. By Gelfand duality this carries not an internal commutative C*-algebra but its Gelfand spectrum: an actual internal topological space. This internal topological space has been called the spectral presheaf (Butterfield-Hamilton-Isham 98).

While here we find this spectral presheaf as an accurate dual description of the space of quantum observables in quantum mechanics based on the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter theorem, interest in the spectral presheaf originally came from the observation that it provides a clear geometric formulation of yet another theorem about the foundations of quantum mechanics, namely of the Kochen-Specker theorem (Kochen-Specker 67).

This again amplifies the role of the “classical contexts” of commutative subalgebras: one may ask if there is a hidden variable description of quantum mechanics that allows to assign actual values to all quantum observables such that in all classical contexts this assignment behaves as an actual classical observable in that it provies a (star-)homomorphism of associative algebras from the commutative C*-algebra of the classical context to the real numbers. The Kochen-Specker theorem rules out such a hidden variable theory by stating that when the algebra of observables is that of bounded operators on a Hilbert space of dimension greater than 2, then such a “hidden variable” assignment cannot exist.

In (Butterfield-Isham 98 it was observed, that this statement has a natural geometric interpretation in the Bohr topos: it simply says that the spectral presheaf, hence the “Jordan-algebraic geometry” incarnation of quantum phase space, has no global element. This statement in turn is a characterization of how the quantum phase space is “exotic” as far as spaces go. It behaves like a non-trivial space, and yet there is no way to map a point into it as a whole, maps of points into it exist only locally.

In summary, the Bohr topos incarnation of the Jordan-Alfsen-Shultz-Harding-Döring-Hamhalter characterization of quantum observables not only accurately captures the nature of quantum observables, but also makes other subtle nature of quantum mechanics becomes more explicitly evident than in other formulations.

To see how one can get more out of the Bohr topos incarnation of the quantum phase space, it serves to pass from plain quantum mechanics to the more general context of quantum field theory. Here the original Haag-Kastler axioms of AQFT demand that to a region of spacetime is to be assigned the quantum observables as a C*-algebra/von Neumann algebra. But by the above discussion it is rather only the underlying Jordan algebra structure that matters, hence the Bohr topos. In light of this a local net of observables in AQFT is naturally regarded as a presheaf of ringed toposes on spacetime, assigning the respective Bohr topos of local observables to each local region of spacetime.

Such “Bohrification of local nets of observables” were analyzed in (Nuiten 12). There it was found that the natural structure of the transition functions of local nets of Bohr toposes of observables by geometric morphisms automatically capture the causal locality condition of local quantum field theory. This is now called “Nuiten’s lemma” in (Wolters-Halvorson 13). Moreover, (Nuiten 12) shows that a natural descent condition on spacetime nets of Bohr toposes is equivalent to strong locality of the quantum field theory, something slightly weaker than Einstein causality, which implies it. Since, by the above, the Bohr topos is the geometric incarnation of the Jordan algebra structure on quantum observables, one might see this as a reply to the alleged lack of implications (in the AS review, 04) fo Pascual Jordan‘s ideas from the 1930s to quantum field theory.

Be that as it may, notice that generally local systems of ringed toposes are to be expected to naturally encode quantum field theory on general grounds: the modern AQFT-style formulation of classical field theory/prequantum field theory via factorization algebras or similar assigns to subsets of spacetime the derived critical locus of local fields extremizing the given local action functional, hence the derived space of solutions to the Euler-Lagrange equations of motion: the covariant phase space (pre-quantum). In physics this derived critical locus is modeled explicitly as a BV-complex, but when realized in the full technical beauty of derived geometry it is in fact a higher ringed topos, as explained by Jacob Lurie. In view of this incarnation of classical field theory in AQFT-perspective as a net of higher ringed topos, it seems rather natural that under quantization it remains a net of ringed toposes, sending ringed toposes incarnating classical covariant phase spaces to ringed toposes incarnating their quantized version as quantum phase spaces.

Relation to the traditional non-commutative geometry

In the above we highlighted that by the fundamental theorems on the foundations of quantum mechanics – going back to insights of Pascual Jordan way back in 1930 and formally affirmed by more recent results by Alfsen-Shultz and Harding-Döring – it follows that, accurately speaking, quantum phase space is not really an object in noncommutative geometry, but rather in a kind of non-associative “Jordan geometry” which is naturally captured by the ringed Bohr topos over the poset of commutative subalgebras.

This observation indeed puts doubt on the long and widely held believe that the quantum phase space is an object in noncommutative geometry, a belief that in fact motivated much of the development of noncommutative geometry in the first place.

But that this is not really true was “well known” all along: it is pointed out for instance in the foundational text (Bates-Weinstein 97) on geometric quantization. On page 80 there is highlighted how given a classical phase space represented by a Poisson manifold (X,{,})(X, \{-,-\}), hence of a manifold that carries

  1. a commutative algebra of classical observables

  2. equipped with a compatible non-commutative Lie bracket {,}\{-,-\} (the Poisson bracket)

the quantization of this data is to be thought of as applying to these two items separately:

  1. a non-associative but commutative Jordan algebra structure of quantum observables is the deformation quantization of the commutative algebra structure of classical observables;

  2. a non-commutative Lie bracket structure is the deformation quantization of the Poisson bracket

and that if the quantization of both items is given by a single C*-algebra structure, then the non-associative commutative Jordan algebra structure is the one induced by the anticommutator

AB=12(AB+BA) A \circ B = \tfrac{1}{2}(A B + B A)

while the non-commutative algebra structure is that given by the commutator

[A,B]=12(ABBA) [A,B] = \tfrac{1}{2}(A B - B A)

See also at Jordan algebra – Origin in quantum physics. This splitting of the notion of quantization into a Lie-algebraic and a Jordan algebraic aspect is formalized in the notion of Jordan-Lie-Banach algebra.

To understand that this makes good sense notice that this decomposition is that into kinematics and dynamics of quantum mechanics:

  1. kinematics— the construction of the quantum observables and of the space of quantum states alone indeed does not involve the associative product of the C*-algebra;

  2. dynamics— the commutator Lie bracket structure is used to impose quantum Hamiltonian flows exp([A,])\exp([A,-]), hence (time) propagation along the trajectories generated by the observables AA.

But then observe in addition that when we pass from quantum mechanics to quantum field theory axiomatized as AQFT, then in fact time propagation is no longer implemented by inner automorphisms of the form exp([A,])\exp([A,-]). Indeed it is impossible for any realistic physical system with infinitely many degrees of freedom (such as a field) to have time evolution given by an inner automorphism. That this does work for quantum mechanics is really an artifact of the degenerate case of finitely many degrees of freedom considered there.

Instead, in AQFT the spacetime-evolution of the quantum fields is all encoded in the transition functions of the local net of observables. By the above discussion, this is however, accurately speaking, not actually to be thought of as a net of C*-algebras, but rather as a net of Jordan algebras, hence as a net of Bohr toposes. This way the need for the non-commutative algebraic structure disappears and only the non-associative commutative Jordan algebra structure remains.

In this context it may be noteworthy to recall what is a well-kept secret: despite much work on AQFT with the traditional Haag-Kastler axioms that demand to assign C*-algebras to regions of spacetime, there is to date not a single non-free field example in spacetime dimension greater than 3 of these axioms. (There are plenty of interesting example in dimension 2, though, see at conformal net and pointers given there.)

On the other hand, more recently the variant of the AQFT axioms known as factorization algebras has been shown to admit plenty of interesting examples of quantum field theory. Comparison of the axioms is not straightforward and should be taken with a grain of salt, but it is maybe noteworthy that a factorization algebra is indeed a net that assigns to a region of spacetime not an algebra structure. The algebra structure there is instead all encoded into the way that spacetime regions are included into each other.

List of theorems invoked

For reference, the following lists that theorems about the standard foundations of quantum mechanics that are being referred to above:


  • Pascual Jordan, Über eine Klasse nichtassociativer

    hyperkomplexer Algebren, Nachr. Ges. Wiss. Göttingen (1932), 569–575.

  • A.M. Gleason, Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics, Indiana Univ. Math. J. 6 No. 4 (1957), 885–893 (web)

  • Simon Kochen, Ernst Specker, The problem of hidden variables in quantum mechanics , Journal of Mathematics and Mechanics 17, 59–87 (1967), (pdf)

  • Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).

  • Erhard Scheibe, The logical analysis of quantum mechanics . Oxford: Pergamon Press, 1973.

  • Jan Hamhalter, Isomorphisms of ordered structures of abelian C *C^\ast-subalgebras of C *C^\ast-algebras, J. Math. Anal. Appl. 383 (2011) 391–399 (journal)

Last revised on December 23, 2017 at 18:41:43. See the history of this page for a list of all contributions to it.