## Abstract

We consider Hilbert’s problem of the axioms of physics at a qualitative or conceptual level. This is more pressing than ever as we seek to understand how both general relativity and quantum theory could emerge from some deeper theory of quantum gravity, and in this regard I have previously proposed a *principle of self-duality* or *quantum Born reciprocity* as a key structure. Here, I outline some of my recent work around the idea of quantum space–time as motivated by this non-standard philosophy, including a new toy model of gravity on a space–time consisting of four points forming a square.

This article is part of the theme issue ‘Hilbert’s sixth problem’.

## 1. Introduction

Relative realism is a general philosophy that reality is like pure mathematics, created by decisions to work within certain axioms or assumptions [1–3]. To the extent that we are not conscious of this, we experience the reality created by those axioms. To the extent that we *are* aware, we transcend that level of ‘reality’ but the fact that those axioms were possible, together with all the substructure they contain, is an element of a larger reality in which that was just one path we could have taken. This makes reality relative to your point of view, which is not necessarily a bad thing given the Copenhagen interpretation of quantum mechanics.

In this point of view, physical reality as we know it should be characterized or in some sense created by the decision to adopt certain axioms or assumptions. The difference with most mathematical subjects is that we do not *a priori* know what the axioms are but are working backwards to find them. My thesis in [1–3] was that if we eventually succeed, then we will in fact uncover a characterization of what it is to be a physicist. And knowing this, one can anticipate that one of the central axioms of physics should turn out to be rooted in the scientific method, which one can formulate as a dual relationship between theory and experiment.

It is not clear to what extent Hilbert himself would have agreed with the above. Clearly, in asking for ‘axioms of physics’ in his sixth problem, Hilbert took the view, as we do, that there could indeed be axioms. I would see this as more than just a reductionist view that many modern-day theoretical physicists take in any case that physics can be subject to fundamental principles and ultimately boiled down to a single unified theory guided by those principles. As explained in an excellent modern account [4], it is likely that Hilbert’s ambition in the sixth problem was to elevate such axioms to a casting role comparable to the complete and independent set of axioms that he eventually found in [5] for geometry. What we do with the assumption that axioms exist, however, is very different. In relative realism we see such axioms as casting physics as a branch of pure mathematics with physical reality ‘created’ by our adopting them, while Hilbert’s position was the exact opposite that geometry was an empirical subject more like physics with its reality already out there as discovered by experience, i.e. not that physics was more like geometry as pure mathematics but that geometry was more like physics as empirical. Hilbert saw the role of axioms for physics as a way to put it on a firmer footing, or, in his words as quoted in [4], to support and fortify ‘when signs appear that the loose foundations are not able to sustain the expansion of the rooms’. The metaphor here was that science is a growing edifice of different rooms in which the different branches operate. The physics here is already out there, we approach it patchily and then mathematics has a retrospective clarifying role, although also an explanatory role if, ideally, one finds axioms that lead to exactly the physics that is observed.

In fact, Hilbert took the view that mathematics in general is ‘not like a game whose tasks are determined by arbitrarily stipulated rules’ [4,6] but rather that good axioms are part of a pre-existing structure of ‘mathematical reality’ as I would put it. In my experience, many mathematicians would also agree with this, although many would not. For example, the French mathematician Dieudonné [7] famously argued the other side—that mathematics was more like a game of chess in which there is no absolute truth because the rules are arbitrary. In relative realism we do take the ‘mathematics is out there’ view but we *also* take the Dieudonné view except that we do not see it as perjorative. Thus, there is a reality experienced by chess players as they experience the restrictions of chess while playing, in some sense ‘created’ or carved out by those restrictions, but at the same time the rules of chess are not arbitrary and constitute an ‘empirical fact’ although at a higher level as an element of the reality, of possible board games as experienced by designers of board games [1]. This gives a hierarchical ‘room within rooms’ structure to our experience of reality a little different from Hilbert’s analogy. Exiting a room by dropping an axiom takes us into a bigger room in which the door we just came out of is just one of the points of interest [8].

In summary, Hilbert might well have agreed with the starting point of relative realism but would have taken a more absolutist view of reality rather than subscribing to the ‘relative’ side of the thesis. My view is that one does not have to swallow the philosophy for our approach to be useful. As with the philosophy of quantum mechanics, we do not need to get bogged down with what ‘real’ actually means as long as our model explains the perception of reality and key features of this perception at an operational level, which for us means principally its hierarchical structure whereby to some degree what we experience is determined by what assumptions we are working within.

It also seems likely that Hilbert would not have agreed with the use to which we put this philosophical position. Where Hilbert might have wanted to build on empirical facts to find a complete set of precise axioms to shore up a theory more or less found by physicists, we are doing the opposite. In [1], we used our philosophical position to propose what we believe to be a single key principle or ‘crude axiom’ (rather than a complete set of precise axioms) to help determine the mathematical structure of the not-yet-known theory of quantum gravity. Back in 1988 when I was finishing my thesis, this would have been a necessity—there was seemingly no real prospect of empirical tests for any theory of quantum gravity due to its energy scale being a factor of 10^{16} out of reach (a situation that is now seen very differently). Secondly, theoretical physics back then was in my opinion still stumbling from structure to structure like Richard Feynman’s drunk looking for their keys in the light of a street lamp not because this is where he really thought he lost his keys but because this was the easiest place to look. Given our view that physics is a subset of mathematics, we apply model-building about the nature of physics to search for its mathematical structure a little more systematically in the space of mathematical structures. This is necessary because, in my opinion, Nature is unlikely to use in the correct formulation of any ultimate theory only the mathematics already in maths books (this being finite and limited by history and our collective imagination). Hence the search for the axioms of physics cannot be divorced from the discovery and structure of mathematics itself and needs to be reasonably consistent with broad features of the latter. We now turn to what we proposed in [1] for this ‘axiom’. As it is an axiom about the desired mathematical structure of physics, it is in some sense an axiom about the axioms of physics rather than an actual physical theory. Moreover, as we think of reality as having a kind of hierarchical or fractal-like structure, it can apply at several different levels.

## 2. Duality principle aka quantum Born reciprocity

If one steps back and looks at some general features of mathematics, one feature that stands out is the idea of a map or a function *f*(*x*) being evaluated on an element *x* in some space *X* (the domain of *f*). This does not cover all of mathematics but is common enough. If the space where the function takes values is something concrete and sufficiently self-evident that we think of it as directly observable (such as an integer or fraction or the real numbers obtained by completing fractions), then we can say that *f* has a value at *x* as some kind of measurement. Now the thing is that *f* is also an element of a space of maps from *X*, let us call this , and who is to say that the number *f*(*x*) is not actually *x*(*f*), the value of *x* at this point *f* in ? Mathematics itself has this striking duality between observer and observed running through it. Usually *X* has some structure and we want *f* to respect it, which in turn gives some structure. In the dual point of view, we think of *X* as and, in the nicest cases, this could be an identification of the two. The idea then is that, if physics is a branch of mathematics, its central axiom should relate to the nature of what actually physics is. Physics is nothing if it is not the assumption that some structure is ‘out there’ and that one can do an experiment to verify it. An experiment of course involves measuring or observing something, but actually what is an experimental fact is not something that really exists in isolation. Any experiment of any complexity usually involves a theory or some abstract relationship predicted by theory, and from a theorist’s point of view an experiment ‘maps out’ or confirms the assumed theory. In this sense, an experiment represents a theoretical structure. From a dual point of view, however, an experimentalist might consider that their observations are self-evident facts and a theory merely compactly represents these data. For example, data points fall approximately on a line and the theorist represents this as a linear relationship. This suggests a kind of dualism in which either point of view should be equally correct as to which is the ‘real thing’ and which is its set of representations. Clearly this is an idealization or crude model of a much richer relationship but we see it as a key feature, as well as providing a particular answer to the question posed by Plato’s cave [3].

Although we do not attempt to model the precise nature of theory and experiment across the physical sciences, one can argue that something so basic *should* be reflected in the structure of any ultimate theory and, meanwhile, we can see elements of it in particular contexts. We have written about this extensively elsewhere, so here we give only a concise overview. First of all, we organize mathematics, in general, according to abstract concepts and their representations, as shown in figure 1 taken from Majid [1,9]. The arrows here are meant to be inclusion functors between categories of structures, loosely interpreted. The familiar case here is that of an Abelian group *G*. Its set of representations itself forms a group and says that from a mathematical point of view one is free to reverse which is the abstract structure and which is its representation. For example, in physics, *G* could be position space ; then in a suitable setting would be momentum space , a self-dual example in the self-dual category. The *principle of representation-theoretic self-duality* [1] or ‘generalized Mach principle’ is the idea that physics should admit a reversal of which parts are structure and which parts are representation, for example which is position and which is momentum. This need not result in the same theory but merely a dual theory. The strong version is that the dual theory should have the same form but possibly with different values of parameters. From this point of view, Boolean algebra with its de Morgan duality is arguably the ‘birth’ of physics [9], while the next self-dual category beyond Abelian groups is Hopf algebras or ‘quantum groups’. Thus I argued in my 1988 PhD thesis that constructing non-commutative, non-cocommutative Hopf algebras could be seen as a toy model of constructing elements of quantum gravity, and used this to obtain one of the two main classes of such true quantum groups at the time when these were first being introduced, the *bicrossproduct* ones associated with local Lie group factorizations. This was around the same time as V.G. Drinfeld introduced the other (and more famous) class of *q*-deformed quantum groups coming from quantum integrable systems. I will say more about bicrossproducts shortly.

Quantum groups here are a big enough category to include non-Abelian groups and their Fourier duality. If *G* is a compact Lie group, say, its function algebra *C*(*G*) and its group convolution algebra *C**(*G*) can be completed to mutually dual Kac or Hopf–von Neumann algebras. At the algebraic level, we have coordinate algebras and enveloping algebras as essentially dual. Traditionally, one has to do non-Abelian Fourier transform categorically but in the language of Hopf algebras it becomes quantum Fourier transform, for example (indicating a suitable completion that includes exponentials). Here is regarded as a ‘coordinate algebra’ of a non-commutative space. We will come to the physics of this shortly, but for the moment we continue along the self-dual axis in figure 1. Here in the search for the ‘next’ self-dual category, I found in 1990 the following duality construction for functors between monoidal categories [9,10]. Here a monoidal category means there is a ⊗ product which is associative up to an associator cocycle and another one, for example the category of vector spaces, in which we construct our representations. The objects of are pairs (*V*,λ_{V}), where *V* is an object of and λ_{V}∈Nat(*V* ⊗*F*,*F*⊗*V*) is a natural transformation such that the diagram in figure 2 commutes. Here λ_{V} is a collection of morphisms (λ_{V})_{X}:*V* ⊗*F*(*X*)→*F*(*X*)⊗*V* for all which are functorial in the sense of compatible with any morphisms *X*→*Y* in , and the condition in the figure says that it ‘represents’ the tensor product of as composition in . Note that the monoidal functor *F* comes equipped with an associated natural isomorphism *f* in the sense of functorial isomorphisms *f*_{X,Y}:*F*(*X*)⊗*F*(*Y*)→*F*(*X*⊗*Y*) for all objects *X*,*Y* in , which we use. One has and the construction generalizes both group and Hopf algebra duality. The tensor product of two ‘representations’ is just
where we move *W* past *F*(*X*), then *V* past *F*(*X*). By a theorem of Mac Lane for monoidal categories, we suppress the associator between tensor products as these can be inserted afterwards.

### Example 2.1.

If *G* is a finite group and is the category of *G*-graded vector spaces, we can tensor product such spaces by the product of gradings in the group, obtaining a monoidal category. We take *F* the functor that forgets the grading; then has as objects vector spaces *V* equipped with natural isomorphisms (λ_{V})_{X}:*V* ⊗*F*(*X*)→*F*(*X*)⊗*V* sending (λ_{V})_{X}(*v*⊗*x*_{g})=*x*_{g}⊗*v*⊳*g* for some right action of *G* on *V* . One can check that this meets the requirements above. Thus, is essentially the category of representations of *G*.

This connects our monoidal duality to non-Abelian group duality. The latter also includes Hopf algebra duality when appropriately formulated. A genuinely new example of considerable interest these days in topological quantum field theory is the following.

### Example 2.2 (Drinfeld–Majid centre).

A special case of the above construction is when and *F*=id. This case was found independently by Drinfeld according to a private communication cited in [10] but the definitions and proofs are identical to the construction, just leaving out the symbol *F*. In this case, there is a tautological braiding if we assume the λ are isomorphisms.

Drinfeld’s private communication here was a letter to me after he came across the preprint version of [10]. Hence it is not the case that was obtained as a generalization of Drinfeld’s work as sometimes assumed; we simply came to essentially the same construction for different reasons. My reason was the principle of representation-theoretic self-duality, while Drinfeld’s was to generalize his famous double construction for quantum groups in [11]. Some of my follow-up work was [9].

How is this dualism reflected in physics? One setting already alluded to, and which we have called *quantum Born reciprocity (QBR)*, is Fourier duality between position and momentum space and its generalizations. In the Abelian group case this is just wave particle duality, but it also works in the non-Abelian case. If the universe is spatially *S*^{3}=*SU*_{2} (and it might be), then spatial momentum is the representation of this. These form a category but, as noted, one can also see Fourier transform at the Hopf algebra level from to (a completion of) *U*(*su*_{2}), where the latter is regarded as a quantum momentum space . Here *l*_{c} is the cosmological curvature scale and *p*_{i} are left-invariant vector fields. Dually, if the momentum space of some system were to be the non-Abelian group *SU*_{2}, then the Fourier dual would be the quantum space–time with relations [*x*_{i},*x*_{j}]=ıλ_{P}*ϵ*_{ijk}*x*_{k}, where λ_{P} is a length scale and the *U*(*su*_{2}) generators are now regarded as position coordinates. This, as we will see shortly, is thought to be the case in some models of three-dimensional quantum gravity.

In my PhD thesis [12,13], I took this point of view and the above self-duality principle as a motivation to look for self-dual-type Hopf algebras, and constructed these in the ‘bicrossproduct’ form with dual associated with any local factorization of a Lie group *X*=*G*⋈*M*. These were originally thought of as quantum phase space, but since 1994 in [14] I have also thought of them as quantum Poincaré groups acting on and respectively, as auxiliary quantum space–times. In fact there are different covariant systems with equivalent data related by *semidualization* [12,13] (where one Hopf algebra is systematically replaced by its dual),
where is the Lie algebra of *X* and in each pair we give the (possibly quantum) symmetry group and the (possibly quantum) space–time algebra on which it acts. The relevant factorization for three-dimensional quantum gravity is , where is the group of upper-triangular matrices in the Iwasawa decomposition. Focusing on the first two pairs, we have the top line of figure 3 where on the top left *U*(*h*_{3}) is the quantum space–time
2.1for *i*=1,2, which is the three-dimensional version of the Majid–Ruegg quantum space–time [14]. In its Poincaré quantum group, the momentum is commutative because its ‘enveloping algebra’ is the commutative Hopf algebra but curved as *H*_{3} is non-Abelian. The semidual of this on the top right is a classical model of a particle on *H*_{3} as curved position space with its classical *U*(*so*_{1,3})=*U*(*su*_{2})⋈*U*(*h*_{3}) symmetry containing *U*(*h*_{3}) as the translational momentum. So the roles of position and momentum are swapped between the two models—an example of QBR. It is also striking that the model on the right is classical (a particle on a curved space *H*_{3}), while the other model is quantum, so QBR interchanges classical and quantum.

In fact this picture *q*-deforms [15] as we show on the bottom line of figure 3, where the model on the bottom right is thought to encode quantum gravity with cosmological constant via an expression of the form *q*=*e*^{−λP/lc}. Its QBR-dual model on the bottom left when *q*≠1 is at some algebraic level isomorphic to two copies of acting on up to some technicalities, i.e. a classical but *q*-deformed particle on a three-sphere, and this is related by a categorical equivalence (a Drinfeld twist) to the model on the right. In other words, three-dimensional quantum gravity with cosmological constant is in some sense *self-dual* under QBR. Finally, we can take *q*→1 in different ways and the first one, on the outer right, is λ_{P}→0 (hence a classical but curved position space). Alternatively, we can send and this is the model in the centre of the figure encoding three-dimensional quantum gravity without cosmological constant (to see this one should write up to some technicalities, and then take the limit). The diagonal twist equivalence between this conventional version of three-dimensional quantum gravity with *U*(*su*_{2}) quantum space–time and the one we began with (on top left) was recently shown by P. Osei and the author [16]. More details of the point of view for three-dimensional quantum gravity are in [15].

If we leave the self-dual axis, then the dual structure is not of the same type but is still a structure. If we take a more categorical view of group duals, then the dual of a non-Abelian Lie group generally comes down to quantization, such as the method of co-adjoint orbits or the construction of the relevant representations of the Poincaré group in flat space–time by particle waves of different mass and spin. The dual object consists of all of these considered together. Geometrically, we are diagonalizing natural Laplacians or wave operators. The same principle applies when the space–time is a compact Lie group, the particle content is closely related to its representation theory. Compact Lie groups, on the other hand, are the simplest examples of Riemannian manifolds and we can similarly, albeit more loosely, think of the ‘dual’ of the latter as coming down to quantum mechanics or wave operators of different types. Recall that quantum mechanics in nice cases can be seen as the non-relativistic limit of the Klein–Gordon space–time Laplacian for fields where is factored out. In fact a Riemannian or pseudo-Riemannian manifold cannot be totally reconstructed from the scalar Laplacian alone, but if we use the Dirac operator, then one has Connes’ reconstruction of a spin manifold from an abstractly defined ‘spectral triple’ in the commutative case [17]. Either way, if one extends these ideas from classical to quantum field theory, then logically an aspect of this should be that it corresponds to *quantum Riemannian geometry*, where space–time coordinates become non-commutative. This should then be taking us towards the self-dual axis as shown in figure 1. Interestingly, [18] have now constructed quantum field theories on curved space–times as functors from the monoidal category of globally hyperbolic space–times to non-commutative *C**-algebras with tensor product, which is some kind of arrow from QFT to monoidal functors in figure 1. Such functors are not self-dual under ( )^{°}, but then this is not yet quantum gravity. At any rate, one could speculate in this context that Einstein’s equation might eventually emerge as the classical limit of a self-duality condition as it equates the Einstein tensor from the geometry side to the expectation value of the stress energy tensor from the quantum field theory side, probably requiring both to be generalized so as to be in a self-dual setting. I do not know the final framework for this, but the monoidal category dual may be a step in the right direction. In physical terms, the self-dual nature of quantum gravity as we see it is in line with the fact that the metric is the background geometry, while its fluctuations are spin-2 fields and hence among the ‘representations’ of the background geometry in the loose sense discussed above.

In summary, we were led on philosophical grounds to the view that space–time should be both curved and ‘quantum’ in the sense of a non-commutative coordinate algebra, as a consequence of a deep self-duality principle for quantum gravity [1–3]. We call this aspect the *quantum space–time hypothesis*. It is a prediction which, if confirmed, would be on a par with the discovery of gravity and indeed dual to it. What it could entail mathematically is our next topic.

## 3. Axioms of quantum Riemannian geometry

I will recap a constructive approach to this from my own work in the last decade (much of it with Edwin Beggs) rather than the better-known ‘Dirac operator’ approach of Connes [19] expressed in the axioms of a spectral triple. The two approaches have recently begun to come together with our programme of *geometric realization* of Connes’ spectral triple [20].

In fact all main approaches have in common (in our case as the starting point) the notion of differential forms (*Ω*,d) over a possibly non-commutative algebra *A* as a differential graded algebra. This means
where *Ω*^{0}=*A* and d obeys a graded-Leibniz rule with respect to the graded product ∧. We assume that *Ω* is generated by *A*,*Ω*^{1}, in which case one may focus on (*Ω*^{1},d) first and construct higher differential forms as a quotient of the tensor algebra of this over *A*. Here
where *Ω*^{1} has an associative multiplication from the left and the right by *A* (one says that *Ω*^{1} is an *A*-bimodule) and d is a derivation.

The next ingredient is a left connection,
which is a *bimodule connection* [21,22] if there also exists a bimodule map *σ* such that
The map *σ* if it exists is unique, so this is not additional data but a property that some left connections have. In the classical case where , if *X* is a vector field, then a connection ∇ defines a covariant derivative ∇_{X}:*Ω*^{1}→*Ω*^{1} by evaluating *X* against the left output of ∇ (this also works with care in the quantum case). However, we consider all such covariant derivatives together by leaving ∇ as a 1-form-valued operator on 1-forms. The curvature and torsion of a left connection (see, for example, [23]) are

Incidentally, all the same ideas except for the torsion hold for any vector bundle, which we axiomatize via its space of sections *E* as a left module over *A*, and ∇_{E}:*E*→*Ω*^{1}⊗_{A}*E*. In the bimodule case, if *E*,*F* are bimodules and ∇_{E},∇_{F} are bimodule connections, then the tensor product *E*⊗_{A}*F* has a bimodule connection
and a certain *σ*_{E⊗F}. This makes the collection of such pairs (*E*,∇_{E}) into a monoidal category with morphisms usually taken as bimodule maps that intertwine the connections. There is a forgetful functor from this to the category of bimodules over *A*, so this is an example of a monoidal functor (in the sense of figure 1) associated with any manifold and with any algebra more generally.

Next we consider a Riemannian metric *g*=*g*^{1}⊗_{A}*g*^{2}∈*Ω*^{1}⊗_{A}*Ω*^{1} (sum of such terms understood). We want this to be non-degenerate in the sense that there exists a bimodule map that is inverse, (*ω*,*g*^{1})*g*^{2}=*ω*=*g*^{1}(*g*^{2},*ω*) for all *ω*∈*Ω*^{1}. In this case, (*ω*,*ag*^{1})*g*^{2}=(*ωa*,*g*^{1})*g*^{2}=*ωa*=(*ω*,*g*^{1})*g*^{2}*a* tells us that [*a*,*g*]=0 for all *a*, i.e. *g* has to be central [24]. So even though we are doing non-commutative geometry and do not assume that 1-forms commute with functions, we will need the metric to be central. This is a significant constraint on quantum space–time in the non-commutative case, which is invisible classically. We also usually want the metric to be quantum symmetric in the sense ∧(*g*)=0.

Finally, we want ∇ to be metric compatible. There is a weak notion that makes sense for any left connection, namely it is ‘weak quantum Levi-Civita’ if it is torsion-free and
vanishes. This ‘cotorsion’ tensor was introduced in [23] and classically the condition says that ∇_{μ}*g*_{νρ}−∇_{ν}*g*_{μρ}=0. In the case of a bimodule connection, we can do better and we say this is *quantum Levi-Civita (QLC)* if it is torsion-free and ∇*g*=0 where ∇ extends to *Ω*^{1}⊗_{A}*Ω*^{1} by the tensor product formula.

Usually, one wants *A* to be a *-algebra and for * to be extended as a graded involution to *Ω* commuting with d, and for *g*^{†}=*g*, ∇°*=*σ*°†°∇ where (*ω*⊗_{A}*η*)^{†}=*η**⊗_{A}*ω**. These reality conditions in a self-adjoint basis (if one exists) would ensure that the metric and connection coefficients are real at least in the classical limit. This completes our lightning review.

By now there are many specific quantum Riemannian geometries constructed, for example, on the quantum sphere (see [25]), on the quantum space–time (2.1) (see [24]), on the functions on the permutation group , and on its dual (see [26]) each with natural differential structure, quantum metric and QLCs or weak QLCs according to the model. The Ricci tensor is only partially understood because, to follow the usual trace contraction, one would need a lifting map *i*:*Ω*^{2}→*Ω*^{1}⊗_{A}*Ω*^{1}, which is an additional datum. The Dirac operator is also only partially understood, needing both a ‘spinor’ bundle with connection compatible with ∇ and a ‘Clifford action’. At least for one can come close to the axioms of a Connes’ spectral triple at least at an algebraic level before any functional analysis completion [20].

## 4. Poisson–Riemannian geometry

Our motivation has been that quantum geometry deforms classical geometry by order λ_{P} corrections, as a measure of some quantum gravity effects. The semiclassical theory of which the above is a quantization was recently worked out by Beggs and the author in [27]. This theory is to aspects of quantum gravity as classical mechanics is to quantum mechanics, except that the deformation parameter is not , hence a kind of ‘classical quantum gravity’. One could imagine other applications, including to quantum mechanics if the phase space also has a natural Riemannian structure, so we will just call the parameter λ (and take conventions where it is imaginary).

The first layer of this is of course the Poisson structure, a tenet of mathematical physics since the early days of quantum mechanics being to ‘quantize’ functions on a manifold to a non-commutative algebra *A*. We suppose that
where we denote the product by juxtaposition and the *A* product by •. We assume all expressions can be expanded in λ and equated order by order. In this case,
defines a map { , } and the assumption of an associative algebra quickly leads to the necessary feature that this is a Lie bracket (i.e. antisymmetric and satisfies the Jacobi identity, making into a Lie algebra) and is a (Hamiltonian) vector field associated with a function *a*. It is known that every such Poisson bracket can be quantized to an associative algebra at least at some formal level [28]. The second layer is to find a differential structure *Ω*^{1} deforming the classical *Ω*^{1}(*M*). One can similarly analyse the data for this by defining a map ∇ by
The assumption of a left action and the Leibniz rule for d requires at order λ that
4.1(these follow easily from [*a*,*b*•d*c*]=[*a*,*b*]•d*c*+*b*•[*a*,d*c*] and d[*a*,*b*]=[d*a*,*b*]+[*a*,d*b*]). The first condition of (4.1) says that ∇ is a covariant derivative along Hamiltonian vector fields and the second is an additional ‘Poisson-compatibility’. The first part of (4.1) applies similarly for any bundle and can be formulated as ∇ a contravariant or Lie–Rinehart connection [29], while the second part was observed in [30,31]. Finally, the associativity of left and right actions on a bimodule gives
(this follows from the Jacobi identity [*a*,[d*b*,*c*]]+[d*b*,[*c*,*a*]]+[*c*,[*a*,d*b*]]=0). So a zero-curvature Poisson-compatible partially defined connection is what we strictly need.

In [27], we make two convenient variations. First of all, we are *not* going to require zero curvature because the effect of curvature is visible only at order λ^{2}, so we do not really need this in the order λ theory. If there is curvature, then it will not be possible to have an associative differential calculus of classical dimension on *A*, but this is actually a situation that one frequently encounters in non-commutative geometry. We can either absorb this in a higher-dimensional associative differential structure or we can live with non-associative differentials at order λ^{2}. Strictly speaking, the same applies to the Poisson bracket obeying the Jacobi identity not being strictly needed at order λ^{2}, in which case we would have *A* itself being non-associative. Secondly, for simplicity, we are going to make the assumption that is indeed the restriction of an actual connection ∇. This will allow us to speak more freely of geometric concepts such as the contorsion tensor. In fact this assumption is not critical; if the Poisson tensor in these coordinates is *ω*^{μν}, then we are in most formulae making use only of the combination ∇^{μ}:=*ω*^{μν}∇_{μ} rather than the full covariant derivatives ∇_{μ} themselves. It means that our data have redundant ‘auxiliary modes’ that do not affect the quantum differential structure at order λ, a situation not unfamiliar from other situations such as gauge theory. There is also the matter of extending from *Ω*^{1} to forms of all degrees but this turns out [27] to impose no further conditions.

The third layer is the construction of a quantum metric and the natural data for this will be a classical metric *g* on *M*. As one might guess, the metric compatibility of ∇ is just that ∇*g*=0. To avoid confusion, we will write for the classical Levi-Civita connection of *g* and we let *S* be the cotorsion tensor of ∇ whereby . It is well known, in general relativity, that a metric-compatible connection is determined by its torsion tensor *T* or equivalently a cotorsion tensor *S* antisymmetric in its outer indices when all indices are lowered. Hence under our simplifying assumption the data for ∇ can be thought of as *T* or *S*. In this case, Poisson compatibility of ∇ can be written as [27]
4.2The fourth layer is more specialized as it is specifically the quantization data for a bimodule quantum Levi-Civita connection (one could be happy with something weaker) and comes down to the identity
4.3where the curvature *R* of ∇ combines with the contorsion to define
4.4The latter is called the *generalized Ricci 2-form* associated with our classical data. In summary, the field equations of Poisson–Riemannian geometry come down to [27]:

(0) A metric

*g*_{μν}and an antisymmetric bivector*ω*^{μν}typically obeying the Poisson bracket Jacobi identity.(1) A metric-compatible connection ∇ at least along Hamiltonian vector fields.

(2) Poisson-compatibility of ∇ given in the fully defined case by (4.2).

(3) The optional quantum Levi-Civita condition (4.3).

These equations can be quite restrictive, particularly if one also wants to preserve a symmetry.

### Example 4.1 (quantizing the Schwarzschild black hole [27]).

Solving the above equations for the Schwarzschild metric in polar coordinates *t*,*r*,*θ*,*ϕ* and asking to preserve rotational symmetry leads to a unique Poisson tensor *ω* and unique ∇ up to auxiliary modes. This leads to *r*,*t*,d*r*,d*t* central (unquantized) and for each *r*,*t* one has a radius *r* ‘non-associative fuzzy sphere’
to order λ in coordinates where . Here ∇ on *S*^{2} is the Levi-Civita connection with constant curvature, hence *Ω*^{1} is not associative at order λ^{2}.

This uniqueness result was extended to generic static spherically symmetric space–times in [32].

## 5. Quantum gravity on a square graph

The mathematics of quantum Riemannian geometry is simply more general than classical Riemannian geometry and includes discrete [33] as well as deformation examples. What is significant in this section is that whatever we find *emerges* from little else but the axioms applied to a square graph as ‘manifold’.

Let *X* be a discrete set and be the usual commutative algebra of complex functions on it as our ‘space–time algebra’. It is an old result that its possible 1-forms and differential (*Ω*^{1},d) are in 1–1 correspondence with directed graphs with *X* as the set of vertices. Here *Ω*^{1} has basis {*ω*_{x→y}} over labelled by the arrows of the graph and differential . In this context, a quantum metric
requires weights for every edge and for every edge to be bi-directed (so there is an arrow in both directions). Taking all weights to be 1 is the so-called ‘Euclidean metric’ [33]. The calculus over is compatible with complex conjugation on functions and . Finding a QLC for a metric depends on how *Ω*^{2} is defined and one case where there is a canonical choice of this is *X* a group and the Cayley graph generated by right translation by a set of generators. Previously, a QLC was found for some specific groups and the ‘Euclidean metric’ but here we give a first calculation for a reasonably general class of metrics.

We take with its canonical two-dimensional calculus given by a square graph with vertices 00,01,10,11 in an abbreviated notation as shown in figure 4. The graph is regular and there are correspondingly two basic 1-forms
with relations and differential
where *R*_{1}*f* shifts by 1 mod 2 (i.e. takes the other point) in the first coordinate, similarly for *R*_{2}, and ∂^{i}=*R*_{i}−id. The exterior algebra is the usual Grassmann algebra on the *e*_{i} (they anticommute). The general form of a quantum metric and its inverse are
where the *a*,*b* are non-vanishing functions. With the standard * structure , the metric obeys the reality condition in §3 if *a*,*b* are real-valued. In terms of the graph their eight values are equivalent to the values of *g* on the eight arrows as shown in figure 4. It is natural here to focus on the symmetric case where the metric weight assigned to an edge does not depend on the direction of the arrow. This means ∂^{1}*a*=∂^{2}*b*=0 and we assume this now for simplicity. In this case, we find a one-parameter family of torsion-free metric-compatible or ‘quantum Levi-Civita’ connections:
where *Q*,*α*,*β* are functions on the group defined as
when we list the values on the points in the same binary sequence as above. Here *q* is a free parameter and *χ*(*i*,*j*)=(−1)^{i+j}=(1,−1,−1,1) is a function on . If we write *σ* as a matrix *σ*^{i1i2}_{j1j2}, where the multi-indices are in order 11,12,21,22, we have
5.1What we have coming out of the axioms is a field of such ‘generalized braiding’ matrices because the entries here are functions on the group. The eigenvalues are −1,*αβ*,−*Q*^{−1},*Q* as functions on the group. Note that these ‘generalized braidings’ have a broadly eight-vertex form normally associated with quantum integrable systems but here arise out of nothing but the requirements of quantum Riemannian geometry applied to a square graph.

The Laplacians for the above QLCs are computed as
using our formula for ∇, the connection property, and . The curvatures are given by
where Vol=*e*_{1}∧*e*_{2}, and a similar formula for *R*_{∇}*e*_{2} interchanging *e*_{1},*e*_{2}; *R*_{1},*R*_{2}; *α*,*β*; *a*,*b* and *Q*,−*Q*^{−1} (so that Vol also changes sign). One can discern contributions from *q*≠1 and from *a*,*b* non-constant. The connection reality condition comes down to
5.2so that in particular the function *Q*−*Q*^{−1} is pointwise imaginary.

Next we find the Ricci tensor defined by a lifting map *i*, for which, in our case, there is a canonical choice . If we write *R*_{∇}*e*_{i}=*ρ*_{ij} Vol⊗*e*_{j}, then
as the matrix of coefficients on the left in our tensor product basis. Applying ( , ) again, we have scalar curvature
which is invariant under the interchange above. For the ‘purely quantum’ case where *q*≠1 and *a*,*b* are constant, the QLCs and their curvature reduce to
as the intrinsic quantum Riemannian geometry of with its square metric. This has
which we see is quantum symmetric but does not obey the same reality condition as the metric if we impose (5.2) needed for the connection to obey its ‘reality’ condition. This is a purely quantum effect because classically there would be no curvature when *a*,*b* are constant.

The general Ricci curvature is quite complicated but, for *q*=1, say, it has values
5.3for the matrix of coefficients. This is not quantum symmetric or real in the sense of the metric. For the scaler curvature the general formula is
Finally, it is not obvious what measure we should use to integrate either of these but if we take measure *μ*=|*ab*|=*ab* (we assume for now the *a*,*b* are positive edge lengths, i.e. the theory has Euclidean signature) and sum over then we have
5.4independently of *q*. We consider this action as some kind of energy of the metric configuration. If we took other measures such as *μ*=1 or , then we would not have invariance under *q*, so the action would not depend only on the metric but on the choice of ∇.

Next we Fourier transform on to write our results in ‘momentum space’. We have
as the plane waves and, given the conditions we imposed on *a*,*b*, we can expand these in terms of four real momentum space coefficients as
Then some computation gives the scalar curvature for *q*=1 as
With measure *μ*=*ab* as above, this gives

To analyse this, we define *k*=*k*_{1}/*k*_{0} with |*k*|<1 corresponding to *a*>0 at all points and similarly for *l*=*l*_{1}/*l*_{0} and fix *k*_{0},*l*_{0}>0 as the average values of *a*,*b* so that we can focus on fluctuations about these as controlled by *k*,*l*. In this case, the action becomes
5.5This has a ‘bathtub’ shape with coupling constants *k*_{0},*l*_{0} and a minimum at *k*=*l*=0, which makes sense as a measure of the energy of the gravitational field. The *k*,*l* are not momentum variables but the relative amplitude of the unique allowed non-zero momentum in each direction.

In the Minkowski version, we require, say, *a*<0,*b*>0 everywhere. We suppose *k*_{0}<0,*l*_{0}>0 as the average values and require |*k*_{1}|<−*k*_{0}, |*l*_{1}|<*l*_{0} to maintain the sign. We define *k*,*l* as before for the relative fluctuations and regard as coupling constants. Now *μ*=|*ab*|=−*ab* for our measure, giving
In either case, if we ignore higher-order terms, then we have *S* quadratic in *k*,*l* as for a massless free field in a universe with only one momentum in each direction. The higher terms correspond to quartic and higher derivatives in the action from this point of view.

Finally, we can add matter using the Laplacian above. However, this Laplacian does depend on *q*. For example, one can check in the momentum parametrizaton that
in the sense of the same eigenvalues. In other words, the theory with *a*,*b* swapped is the same but has the negated value of *q*. These eigenvalues are mostly real when *q* is real, leading to *q*=±1 as the natural choices. We plot the three non-zero eigenvalues in figure 5 for *q*=1 and the two signatures, at a typical value *k*=0.5. The cross section passes a narrow region in the (*k*,*l*) plane where two of the eigenvalues become complex but otherwise they are positive. The remaining mainly small eigenvalue is zero at *l*=0 and *q*=1 or *k*=0 and *q*=−1 among possibly other zeros.

In principle, one can proceed to consider ‘functional integrals’ over any of our parametrizations of the metric field. Thus for quantum gravity we can consider integrals of the form
(this converges when we use the ı in the action, otherwise we have to renormalize due to divergence at the endpoints), and similar integrations against functions of *k*,*l* to extract expectation values of operators. If we add matter to the action via the Laplacian, then we will have a *q*-dependence as discussed. We should also in the full theory integrate over the *k*_{0},*l*_{0} rather than treating them as constants as we have above.

## 6. Conclusion

Sections 1 and 2 were philosophical in nature as a brief introduction to a principle of ‘representation-theoretic self-duality’ as an ‘axiom of physics’ [1–3] that has motivated many of my works. We saw how this at an abstract level was one route to the discovery of the ‘centre’ of a monoidal category, while as ‘QBR’ it led to the discovery of an early class of quantum groups. We also explained how the big picture leads one to the *quantum space–time hypothesis*.

Sections 3 and 4 were a brief outline of a formulation of such quantum space–times with curvature, using a bimodule approach developed mostly with Edwin Beggs [20,24,27,31]. Section 5 then proceeded with a new application to a discrete model, namely quantum Riemannian geometry on a square. Unlike lattice approximations used in physics, we do not consider the model as an approximation but rather as an exact finite geometry [33]. We found a one-parameter family of quantum Levi-Civita connections for every bidirectional metric and an Einstein–Hilbert action as a measure of the energy in the gravitational field and independent of the connection parameter.

We also found that the ‘generalized braiding’ *σ* [21,22] emerging in our case from nothing other than the axioms of quantum Riemannian geometry applied to a square graph has a strong resemblance to the eight-vertex solutions [34] of the Yang–Baxter equations in the theory of quantum integrable systems. Our *σ* does not obey the braid relations other than the trivial case (this is usually tied to flatness of the connection [33]) but has a similar flavour.

Note that while I have focused on my own path, reflected also in the bibliography, there are by now many other works on quantum space–time which I have not had room to cover.

## Data accessibility

This article has no additional data.

## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

## Footnotes

One contribution of 14 to a theme issue ‘Hilbert's sixth problem’.

- Accepted January 29, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.