Why Higher Geometric Quantization

The largest single presentation was a pair of talks on “The Motivation for Higher Geometric Quantum Field Theory” by Urs Schreiber, running to about two and a half hours, based on these notes. This was probably the clearest introduction I’ve seen so far to the motivation for the program he’s been developing for several years. Broadly, the idea is to develop a higher-categorical analog of geometric quantization (GQ for short).

One guiding idea behind this is that we should really be interested in quantization over (higher) stacks, rather than merely spaces. This leads inexorably to a higher-categorical version of GQ itself. The starting point, though, is that the defining features of stacks capture two crucial principles from physics: the gauge principle, and locality. The gauge principle means that we need to keep track not just of connections, but gauge transformations, which form respectively the objects and morphisms of a groupoid. “Locality” means that these groupoids of configurations of a physical field on spacetime is determined by its local configuration on regions as small as you like (together with information about how to glue together the data on small regions into larger regions).

Some particularly simple cases can be described globally: a scalar field gives the space of all scalar functions, namely maps into \mathbb{C}; sigma models generalise this to the space of maps \Sigma \rightarrow M for some other target space. These are determined by their values pointwise, so of course are local.

More generally, physicists think of a field theory as given by a fibre bundle V \rightarrow \Sigma (the previous examples being described by trivial bundles \pi : M \times \Sigma \rightarrow \Sigma), where the fields are sections of the bundle. Lagrangian physics is then described by a form on the jet bundle of V, i.e. the bundle whose fibre over p \in \Sigma consists of the space describing the possible first k derivatives of a section over that point.

More generally, a field theory gives a procedure F for taking some space with structure – say a (pseudo-)Riemannian manifold \Sigma – and produce a moduli space X = F(\Sigma) of fields. The Sigma models happen to be representable functors: F(\Sigma) = Maps(\Sigma,M) for some M, the representing object. A prestack is just any functor taking \Sigma to a moduli space of fields. A stack is one which has a “descent condition”, which amounts to the condition of locality: knowing values on small neighbourhoods and how to glue them together determines values on larger neighborhoods.

The Yoneda lemma says that, for reasonable notions of “space”, the category \mathbf{Spc} from which we picked target spaces M embeds into the category of stacks over \mathbf{Spc} (Riemannian manifolds, for instance) and that the embedding is faithful – so we should just think of this as a generalization of space. However, it’s a generalization we need, because gauge theories determine non-representable stacks. What’s more, the “space” of sections of one of these fibred stacks is also a stack, and this is what plays the role of the moduli space for gauge theory! For higher gauge theories, we will need higher stacks.

All of the above is the classical situation: the next issue is how to quantize such a theory. It involves a generalization of Geometric Quantization (GQ for short). Now a physicist who actually uses GQ will find this perspective weird, but it flows from just the same logic as the usual method.

In ordinary GQ, you have some classical system described by a phase space, a manifold X equipped with a pre-symplectic 2-form \omega \in \Omega^2(X). Intuitively, \omega describes how the space, locally, can be split into conjugate variables. In the phase space for a particle in n-space, these “position” and “momentum” variables, and \omega = \sum_x dx^i \wedge dp^i; many other systems have analogous conjugate variables. But what really matters is the form \omega itself, or rather its cohomology class.

Then one wants to build a Hilbert space describing the quantum analog of the system, but in fact, you need a little more than (X,\omega) to do this. The Hilbert space is a space of sections of some bundle whose sections look like copies of the complex numbers, called the “prequantum line bundle“. It needs to be equipped with a connection, whose curvature is a 2-form in the class of \omega: in general, . (If \omega is not symplectic, i.e. is degenerate, this implies there’s some symmetry on X, in which case the line bundle had better be equivariant so that physically equivalent situations correspond to the same state). The easy case is the trivial bundle, so that we get a space of functions, like L^2(X) (for some measure compatible with \omega). In general, though, this function-space picture only makes sense locally in X: this is why the choice of prequantum line bundle is important to the interpretation of the quantized theory.

Since the crucial geometric thing here is a bundle over the moduli space, when the space is a stack, and in the context of higher gauge theory, it’s natural to seek analogous constructions using higher bundles. This would involve, instead of a (pre-)symplectic 2-form \omega, an (n+1)-form called a (pre-)n-plectic form (for an introductory look at this, see Chris Rogers’ paper on the case n=2 over manifolds). This will give a higher analog of the Hilbert space.

Now, maps between Hilbert spaces in QG come from Lagrangian correspondences – these might be maps of moduli spaces, but in general they consist of a “space of trajectories” equipped with maps into a space of incoming and outgoing configurations. This is a span of pre-symplectic spaces (equipped with pre-quantum line bundles) that satisfies some nice geometric conditions which make it possible to push a section of said line bundle through the correspondence. Since each prequantum line bundle can be seen as maps out of the configuration space into a classifying space (for U(1), or in general an n-group of phases), we get a square. The action functional is a cell that fills this square (see the end of 2.1.3 in Urs’ notes). This is a diagrammatic way to describe the usual GQ construction: the advantage is that it can then be repeated in the more general setting without much change.

This much is about as far as Urs got in his talk, but the notes go further, talking about how to extend this to infinity-stacks, and how the Dold-Kan correspondence tells us nicer descriptions of what we get when linearizing – since quantization puts us into an Abelian category.

I enjoyed these talks, although they were long and Urs came out looking pretty exhausted, because while I’ve seen several others on this program, this was the first time I’ve seen it discussed from the beginning, with a lot of motivation. This was presumably because we had a physically-minded part of the audience, whereas I’ve mostly seen these for mathematicians, and usually they come in somewhere in the middle and being more time-limited miss out some of the details and the motivation. The end result made it quite a natural development. Overall, very helpful!


Dirac Sigma Models










Supersymmetric Field Theories













To continue from the previous post

Twisted Differential Cohomology





Fusion Categories






Quantizing with Higher Categories





A \leftarrow X \rightarrow B








This entry is a by-special-request blog, which Derek Wise invited me to write for the blog associated with the International Loop Quantum Gravity Seminar, and it will appear over there as well.  The ILQGS is a long-running regular seminar which runs as a teleconference, with people joining in from various countries, on various topics which are more or less closely related to Loop Quantum Gravity and the interests of people who work on it.  The custom is that when someone gives a talk, someone else writes up a description of the talk for the ILQGS blog, and Derek invited me to write up a description of his talk.  The audio file of the talk itself is available in .aiff and .wav formats, and the slides are here.

The talk that Derek gave was based on a project of his and Steffen Gielen’s, which has taken written form in a few papers (two shorter ones, “Spontaneously broken Lorentz symmetry for Hamiltonian gravity“, “Linking Covariant and Canonical General Relativity via Local Observers“, and a new, longer one called “Lifting General Relativity to Observer Space“).

The key idea behind this project is the notion of “observer space”, which is exactly what it sounds like: a space of all observers in a given universe.  This is easiest to picture when one has a spacetime – a manifold with a Lorentzian metric, (M,g) – to begin with.  Then an observer can be specified by choosing a particular point (x_0,x_1,x_2,x_3) = \mathbf{x} in spacetime, as well as a unit future-directed timelike vector v.  This vector is a tangent to the observer’s worldline at \mathbf{x}.  The observer space is therefore a bundle over M, the “future unit tangent bundle”.  However, using the notion of a “Cartan geometry”, one can give a general definition of observer space which makes sense even when there is no underlying (M,g).

The result is a surprising, relatively new physical intuition is that “spacetime” is a local and observer-dependent notion, which in some special cases can be extended so that all observers see the same spacetime.  This is somewhat related to the relativity of locality, which I’ve blogged about previously.  Geometrically, it is similar to the fact that a slicing of spacetime into space and time is not unique, and not respected by the full symmetries of the theory of Relativity, even for flat spacetime (much less for the case of General Relativity).  Similarly, we will see a notion of “observer space”, which can sometimes be turned into a bundle over an objective spacetime M, but not in all cases.

So, how is this described mathematically?  In particular, what did I mean up there by saying that spacetime becomes observer-dependent?

Cartan Geometry

The answer uses Cartan geometry, which is a framework for differential geometry that is slightly broader than what is commonly used in physics.  Roughly, one can say “Cartan geometry is to Klein geometry as Riemannian geometry is to Euclidean geometry”.  The more familiar direction of generalization here is the fact that, like Riemannian geometry, Cartan is concerned with manifolds which have local models in terms of simple, “flat” geometries, but which have curvature, and fail to be homogeneous.  First let’s remember how Klein geometry works.

Klein’s Erlangen Program, carried out in the mid-19th-century, systematically brought abstract algebra, and specifically the theory of Lie groups, into geometry, by placing the idea of symmetry in the leading role.  It describes “homogeneous spaces”, which are geometries in which every point is indistinguishable from every other point.  This is expressed by the existence of a transitive action of some Lie group G of all symmetries on an underlying space.  Any given point x will be fixed by some symmetries, and not others, so one also has a subgroup H = Stab(x) \subset G.  This is the “stabilizer subgroup”, consisting of all symmetries which fix x.  That the space is homogeneous means that for any two points x,y, the subgroups Stab(x) and Stab(y) are conjugate (by a symmetry taking x to y).  Then the homogeneous space, or Klein geometry, associated to (G,H) is, up to isomorphism, just the same as the quotient space G/H of the obvious action of H on G.

The advantage of this program is that it has a great many examples, but the most relevant ones for now are:

  • n-dimensional Euclidean space. the Euclidean group ISO(n) = SO(n) \ltimes \mathbb{R}^n is precisely the group of transformations that leave the data of Euclidean geometry, lengths and angles, invariant.  It acts transitively on \mathbb{R}^n.  Any point will be fixed by the group of rotations centred at that point, which is a subgroup of ISO(n) isomorphic to SO(n).  Klein’s insight is to reverse this: we may define Euclidean space by R^n \cong ISO(n)/SO(n).
  • n-dimensional Minkowski space.  Similarly, we can define this space to be ISO(n-1,1)/SO(n-1,1).  The Euclidean group has been replaced by the Poincaré group, and rotations by the Lorentz group (of rotations and boosts), but otherwise the situation is essentially the same.
  • de Sitter space.  As a Klein geometry, this is the quotient SO(4,1)/SO(3,1).  That is, the stabilizer of any point is the Lorentz group – so things look locally rather similar to Minkowski space around any given point.  But the global symmetries of de Sitter space are different.  Even more, it looks like Minkowski space locally in the sense that the Lie algebras give representations so(4,1)/so(3,1) and iso(3,1)/so(3,1) are identical, seen as representations of SO(3,1).  It’s natural to identify them with the tangent space at a point.  de Sitter space as a whole is easiest to visualize as a 4D hyperboloid in \mathbb{R}^5.  This is supposed to be seen as a local model of spacetime in a theory in which there is a cosmological constant that gives empty space a constant negative curvature.
  • anti-de Sitter space. This is similar, but now the quotient is SO(3,2)/SO(3,1) – in fact, this whole theory goes through for any of the last three examples: Minkowski; de Sitter; and anti-de Sitter, each of which acts as a “local model” for spacetime in General Relativity with the cosmological constant, respectively: zero; positive; and negative.

Now, what does it mean to say that a Cartan geometry has a local model?  Well, just as a Lorentzian or Riemannian manifold is “locally modelled” by Minkowski or Euclidean space, a Cartan geometry is locally modelled by some Klein geometry.  This is best described in terms of a connection on a principal G-bundle, and the associated G/H-bundle, over some manifold M.  The crucial bundle in a Riemannian or Lorenztian geometry is the frame bundle: the fibre over each point consists of all the ways to isometrically embed a standard Euclidean or Minkowski space into the tangent space.  A connection on this bundle specifies how this embedding should transform as one moves along a path.  It’s determined by a 1-form on M, valued in the Lie algebra of G.

Given a parametrized path, one can apply this form to the tangent vector at each point, and get a Lie algebra-valued answer.  Integrating along the path, we get a path in the Lie group G (which is independent of the parametrization).  This is called a “development” of the path, and by applying the G-values to the model space G/H, we see that the connection tells us how to move through a copy of G/H as we move along the path.  The image this suggests is of “rolling without slipping” – think of the case where the model space is a sphere.  The connection describes how the model space “rolls” over the surface of the manifold M.  Curvature of the connection measures the failure to commute of the processes of rolling in two different directions.  A connection with zero curvature describes a space which (locally at least) looks exactly like the model space: picture a sphere rolling against its mirror image.  Transporting the sphere-shaped fibre around any closed curve always brings it back to its starting position. Now, curvature is defined in terms of transports of these Klein-geometry fibres.  If curvature is measured by the development of curves, we can think of each homogeneous space as a flat Cartan geometry with itself as a local model.

This idea, that the curvature of a manifold depends on the model geometry being used to measure it, shows up in the way we apply this geometry to physics.

Gravity and Cartan Geometry

MacDowell-Mansouri gravity can be understood as a theory in which General Relativity is modelled by a Cartan geometry.  Of course, a standard way of presenting GR is in terms of the geometry of a Lorentzian manifold.  In the Palatini formalism, the basic fields are a connection A and a vierbein (coframe field) called e, with dynamics encoded in the Palatini action, which is the integral over M of R[\omega] \wedge e \wedge e, where R is the curvature 2-form for \omega.

This can be derived from a Cartan geometry, whose model geometry is de Sitter space SO(4,1)/SO(3,1).   Then MacDowell-Mansouri gravity gets \omega and e by splitting the Lie algebra as so(4,1) = so(3,1) \oplus \mathbb{R^4}.  This “breaks the full symmetry” at each point.  Then one has a fairly natural action on the so(4,1)-connection:

\int_M tr(F_h \wedge \star F_h)

Here, F_h is the so(3,1) part of the curvature of the big connection.  The splitting of the connection means that F_h = R + e \wedge e, and the action above is rewritten, up to a normalization, as the Palatini action for General Relativity (plus a topological term, which has no effect on the equations of motion we get from the action).  So General Relativity can be written as the theory of a Cartan geometry modelled on de Sitter space.

The cosmological constant in GR shows up because a “flat” connection for a Cartan geometry based on de Sitter space will look (if measured by Minkowski space) as if it has constant curvature which is exactly that of the model Klein geometry.  The way to think of this is to take the fibre bundle of homogeneous model spaces as a replacement for the tangent bundle to the manifold.  The fibre at each point describes the local appearance of spacetime.  If empty spacetime is flat, this local model is Minkowski space, ISO(3,1)/SO(3,1), and one can really speak of tangent “vectors”.  The tangent homogeneous space is not linear.  In these first cases, the fibres are not vector spaces, precisely because the large group of symmetries doesn’t contain a group of translations, but they are Klein geometries constructed in just the same way as Minkowski space. Thus, the local description of the connection in terms of Lie(G)-valued forms can be treated in the same way, regardless of which Klein geometry G/H occurs in the fibres.  In particular, General Relativity, formulated in terms of Cartan geometry, always says that, in the absence of matter, the geometry of space is flat, and the cosmological constant is included naturally by the choice of which Klein geometry is the local model of spacetime.

Observer Space

The idea in defining an observer space is to combine two symmetry reductions into one.  The reduction from SO(4,1) to SO(3,1) gives de Sitter space, SO(4,1)/SO(3,1) as a model Klein geometry, which reflects the “symmetry breaking” that happens when choosing one particular point in spacetime, or event.  Then, the reduction of SO(3,1) to SO(3) similarly reflects the symmetry breaking that occurs when one chooses a specific time direction (a future-directed unit timelike vector).  These are the tangent vectors to the worldline of an observer at the chosen point, so SO(3,1)/SO(3) the model Klein geometry, is the space of such possible observers.  The stabilizer subgroup for a point in this space consists of just the rotations of space around the corresponding observer – the boosts in SO(3,1) translate between observers.  So locally, choosing an observer amounts to a splitting of the model spacetime at the point into a product of space and time. If we combine both reductions at once, we get the 7-dimensional Klein geometry SO(4,1)/SO(3).  This is just the future unit tangent bundle of de Sitter space, which we think of as a homogeneous model for the “space of observers”

A general observer space O, however, is just a Cartan geometry modelled on SO(4,1)/SO(3).  This is a 7-dimensional manifold, equipped with the structure of a Cartan geometry.  One class of examples are exactly the future unit tangent bundles to 4-dimensional Lorentzian spacetimes.  In these cases, observer space is naturally a contact manifold: that is, it’s an odd-dimensional manifold equipped with a 1-form \alpha, the contact form, which is such that the top-dimensional form \alpha \wedge d \alpha \wedge \dots \wedge d \alpha is nowhere zero.  This is the odd-dimensional analog of a symplectic manifold.  Contact manifolds are, intuitively, configuration spaces of systems which involve “rolling without slipping” – for instance, a sphere rolling on a plane.  In this case, it’s better to think of the local space of observers which “rolls without slipping” on a spacetime manifold M.

Now, Minkowski space has a slicing into space and time – in fact, one for each observer, who defines the time direction, but the time coordinate does not transform in any meaningful way under the symmetries of the theory, and different observers will choose different ones.  In just the same way, the homogeneous model of observer space can naturally be written as a bundle SO(4,1)/SO(3) \rightarrow SO(4,1)/SO(3,1).  But a general observer space O may or may not be a bundle over an ordinary spacetime manifold, O \rightarrow M.  Every Cartan geometry M gives rise to an observer space O as the bundle of future-directed timelike vectors, but not every Cartan geometry O is of this form, in any natural way. Indeed, without a further condition, we can’t even reconstruct observer space as such a bundle in an open neighborhood of a given observer.

This may be intuitively surprising: it gives a perfectly concrete geometric model in which “spacetime” is relative and observer-dependent, and perhaps only locally meaningful, in just the same way as the distinction between “space” and “time” in General Relativity. It may be impossible, that is, to determine objectively whether two observers are located at the same base event or not. This is a kind of “Relativity of Locality” which is geometrically much like the by-now more familiar Relativity of Simultaneity. Each observer will reach certain conclusions as to which observers share the same base event, but different observers may not agree.  The coincident observers according to a given observer are those reached by a good class of geodesics in O moving only in directions that observer sees as boosts.

When one can reconstruct O \rightarrow M, two observers will agree whether or not they are coincident.  This extra condition which makes this possible is an integrability constraint on the action of the Lie algebra H (in our main example, H = SO(3,1)) on the observer space O.  In this case, the fibres of the bundle are the orbits of this action, and we have the familiar world of Relativity, where simultaneity may be relative, but locality is absolute.

Lifting Gravity to Observer Space

Apart from describing this model of relative spacetime, another motivation for describing observer space is that one can formulate canonical (Hamiltonian) GR locally near each point in such an observer space.  The goal is to make a link between covariant and canonical quantization of gravity.  Covariant quantization treats the geometry of spacetime all at once, by means of a Lagrangian action functional.  This is mathematically appealing, since it respects the symmetry of General Relativity, namely its diffeomorphism-invariance.  On the other hand, it is remote from the canonical (Hamiltonian) approach to quantization of physical systems, in which the concept of time is fundamental. In the canonical approach, one gets a Hilbert space by quantizing the space of states of a system at a given point in time, and the Hamiltonian for the theory describes its evolution.  This is problematic for diffeomorphism-, or even Lorentz-invariance, since coordinate time depends on a choice of observer.  The point of observer space is that we consider all these choices at once.  Describing GR in O is both covariant, and based on (local) choices of time direction.

This is easiest to describe in the case of a bundle O \rightarrow M.  Then a “field of observers” to be a section of the bundle: a choice, at each base event in M, of an observer based at that event.  A field of observers may or may not correspond to a particular decomposition of spacetime into space evolving in time, but locally, at each point in O, it always looks like one.  The resulting theory describes the dynamics of space-geometry over time, as seen locally by a given observer.  In this case, a Cartan connection on observer space is described by to a Lie(SO(4,1))-valued form.  This decomposes into four Lie-algebra valued forms, interpreted as infinitesimal transformations of the model observer by: (1) spatial rotations; (2) boosts; (3) spatial translations; (4) time translation.  The four-fold division is based on two distinctions: first, between the base event at which the observer lives, and the choice of observer (i.e. the reduction of SO(4,1) to SO(3,1), which symmetry breaking entails choosing a point); and second, between space and time (i.e. the reduction of SO(3,1) to SO(3), which symmetry breaking entails choosing a time direction).

This splitting, along the same lines as the one in MacDowell-Mansouri gravity described above, suggests that one could lift GR to a theory on an observer space O.  This amount to describing fields on O and an action functional, so that the splitting of the fields gives back the usual fields of GR on spacetime, and the action gives back the usual action.  This part of the project is still under development, but this lifting has been described.  In the case when there is no “objective” spacetime, the result includes some surprising new fields which it’s not clear how to deal with, but when there is an objective spacetime, the resulting theory looks just like GR.




Categories with Structures





Tensor Categories and Field Theories













State Sum Models


Higher Gauge Theory
































There is no abiding thing in what we know. We change from weaker to stronger lights, and each more powerful light pierces our hitherto opaque foundations and reveals fresh and different opacities below. We can never foretell which of our seemingly assured fundamentals the next change will not affect.

H.G. Wells, A Modern Utopia

So there’s a recent paper by some physicists, two of whom work just across the campus from me at IST, which purports to explain the Pioneer Anomaly, ultimately using a computer graphics technique, Phong shading. The point being that they use this to model more accurately than has been done before how much infrared radiation is radiating and reflecting off various parts of the Pioneer spacecraft. They claim that with the new, more accurate model, the net force from this radiation is just enough to explain the anomalous acceleration.

Well, plainly, any one paper needs to be rechecked before you can treat it as definitive, but this sort of result looks good for conventional General Relativity, when some people had suggested the anomaly was evidence some other theory was needed.  Other anomalies in the predictions of GR – the rotational profiles of galaxies, or redshift data, have also suggested alternative theories.  In order to preserve GR exactly on large scales, you have to introduce things like Dark Matter and Dark Energy, and suppose that something like 97% of the mass-energy of the universe is otherwise invisible.  Such Dark entities might exist, of course, but I worry it’s kind of circular to postulate them on the grounds that you need them to make GR explain observations, while also claiming this makes sense because GR is so well tested.

In any case, this refined calculation about Pioneer is a reminder that usually the more conservative extension of your model is better. It’s not so obvious to me whether a modified theory of gravity, or an unknown and invisible majority of the universe is more conservative.

And that’s the best segue I can think of into this next post, which is very different from recent ones.


I was thinking recently about “fundamental” theories.  At the HGTQGR workshop we had several talks about the most popular physical ideas into which higher gauge theory and TQFT have been infiltrating themselves recently, namely string theory and (loop) quantum gravity.  These aren’t the only schools of thought about what a “quantum gravity” theory should look like – but they are two that have received a lot of attention and work.  Each has been described (occasionally) as a “fundamental” theory of physics, in the sense of one which explains everything else.  There has been a debate about this, since they are based on different principles.  The arguments against string theory are various, but a crucial one is that no existing form of string theory is “background independent” in the same way that General Relativity is. This might be because string theory came out of a community grounded in particle physics – it makes sense to perturb around some fixed background spacetime in that context, because no experiment with elementary particles is going to have a measurable effect on the universe at infinity. “M-theory” is supposed to correct this defect, but so far nobody can say just what it is.  String theorists criticize LQG on various grounds, but one of the more conceptually simple ones would be that it can’t be a unified theory of physics, since it doesn’t incorporate forces other than gravity.

There is, of course, some philosophical debate about whether either of these properties – background independence, or unification – is really crucial to a fundamental theory.   I don’t propose to answer that here (though for the record my hunch at he moment is that both of them are important and will hold up over time).  In fact, it’s “fundamental theory” itself that I’m thinking about here.

As I suggested in one of my first posts explaining the title of this blog, I expect that we’ll need lots of theories to get a grip on the world: a whole “atlas”, where each “map” is a theory, each dealing with a part of the whole picture, and overlapping somewhat with others. But theories are formal entities that involve symbols and our brain’s ability to manipulate symbols. Maybe such a construct could account for all the observable phenomena of the world – but a-priori it seems odd to assume that. The fact that they can provide various limits and approximations has made them useful, from an evolutionary point of view, and the tendency to confuse symbols and reality in some ways is a testament to that (it hasn’t hurt so much as to be selected out).

One little heuristic argument – not at all conclusive – against this idea involves Kolmogorov complexity: wanting to explain all the observed data about the universe is in some sense to “compress” the data.  If we can account for the observations – say, with a short description of some physical laws and a bunch of initial conditions, which is what a “fundamental theory” suggests – then we’ve found an upper bound on its Kolmogorov complexity.  If the universe actually contains such a description, then that must also be a lower bound on its complexity.  Thus, any complete description of the universe would have to be as big as the whole universe.

Well, as I said, this argument fails to be very convincing.  Partly because it assumes a certain form of the fundamental theory (in particular, a deterministic one), but mainly because it doesn’t rule out that there is indeed a very simple set of physical laws, but there are limits to the precision with which we could use them to simulate the whole world because we can’t encode the state of the universe perfectly.  We already knew that.  At most, that lack of precision puts some practical limits on our ability to confirm that a given set of physical laws we’ve written down is  empirically correct.  It doesn’t preclude there being one, or even our finding it (without necessarily being perfectly certain).  The way Einstein put it (in this address, by the way) was “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”  But a lack of certainty doesn’t mean they aren’t there.

However, this got me thinking about fundamental theories from the point of view of epistemology, and how we handle knowledge.


First, there’s a practical matter. The idea of a fundamental theory is the logical limit of one version of reductionism. This is the idea that the behaviour of things should be explained in terms of smaller, simpler things. I have no problem with this notion, unless you then conclude that once you’ve found a “more fundamental” theory, the old one should be discarded.

For example: we have a “theory of chemistry”, which says that the constituents of matter are those found on the periodic table of elements.  This theory comes in various degrees of sophistication: for instance, you can start to learn the periodic table without knowing that there are often different isotopes of a given element, and only knowing the 91 naturally occurring elements (everything up to Uranium, except Technicium). This gives something like Mendeleev’s early version of the table. You could come across these later refinements by finding a gap in the theory (Technicium, say), or a disagreement with experiment (discovering isotopes by measuring atomic weights). But even a fairly naive version of the periodic table, along with some concepts about atomic bonds, gives a good explanation of a huge range of chemical reactions under normal conditions. It can’t explain, for example, how the Sun shines – but it explains a lot within its proper scope.

Where this theory fits in a fuller picture of the world has at least two directions: more fundamental, and less fundamental, theories.  What I mean by less “fundamental” is that some things are supposed to be explained by this theory of chemistry: the great abundance of proteins and other organic chemicals, say. The behaviour of the huge variety of carbon compounds predicted by basic chemistry is supposed to explain all these substances and account for how they behave.  The millions of organic compounds that show up in nature, and their complicated behaviour, is supposed to be explained in terms of just a few elements that they’re made of – mostly carbon, hydrogen, oxygen, nitrogen, sulfur, phosphorus, plus the odd trace element.

By “more fundamental”, I mean that the periodic table itself can start to seem fairly complicated, especially once you start to get more sophisticated, including transuranic elements, isotopes, radioactive decay rates, and the like. So it was explained in terms of a theory of the atom. Again, there are refinements, but the Bohr model of the atom ought to do the job: a nucleus made of protons and neutrons, and surrounded by shells of electrons.  We can add that these are governed by the Dirac equation, and then the possible states for electrons bound to a nucleus ought to explain the rows and columns of the periodic table. Better yet, they’re supposed to explain exactly the spectral lines of each element – the frequencies of light atoms absorb and emit – by the differences of energy levels between the shells.

Well, this is great, but in practice it has limits. Hardly anyone disputes that the Bohr model is approximately right, and should explain the periodic table etc. The problem is that it’s largely an intractable problem to actually solve the Schroedinger equation for the atom and use the results to predict the emission spectrum, chemical properties, melting point, etc. of, say, Vanadium…  On the other hand, it’s equally hard to use a theory of chemistry to adequately predict how proteins will fold. Protein conformation prediction is a hard problem, and while it’s chugging along and making progress, the point is a theory of chemistry alone isn’t enough: any successful method must rely on a whole extra body of knowledge.  This suggests our best bet at understanding all these phenomena is to have a whole toolbox of different theories, each one of which has its own body of relevant mathematics, its own domain-specific ontology, and some sense of how its concepts relate to those in other theories in the tookbox. (This suggests a view of how mathematics relates to the sciences which seems to me to reflect actual practice: it pervades all of them, in a different way than the way a “more fundamental” theory underlies a less fundamental one.  Which tends to spoil the otherwise funny XKCD comic on the subject…)

If one “explains” one theory in terms of another (or several others), then we may be able to put them into at least a partial order.  The mental image I have in mind is the “theoretical atlas” – a bunch of “charts” (the theories) which cover different parts of a globe (our experience, or the data we want to account for), and which overlap in places.  Some are subsets of others (are completely explained by them, in principle). Then we’d like to find a minimal (or is it maximal) element of this order: something which accounts for all the others, at least in principle.  In that mental image, it would be a map of the whole globe (or a dense subset of the surface, anyway).  Because, of course, the Bohr model, though in principle sufficient to account for chemistry, needs an explanation of its own: why are atoms made this way, instead of some other way? This ends up ramifying out into something like the Standard Model of particle physics.  Once we have that, we would still like to know why elementary particles work this way, instead of some other way…

An Explanatory Trilemma

There’s a problem here, which I think is unavoidable, and which rather ruins that nice mental image.  It has to do with a sort of explanatory version of Agrippa’s Trilemma, which is an observation in epistemology that goes back to Agrippa the Skeptic. It’s also sometimes called “Munchausen’s Trilemma”, and it was originally made about justifying beliefs.  I think a slightly different form of it can be applied to explanations, where instead of “how do I know X is true?”, the question you repeatedly ask is “why does it happen like X?”

So, the Agrippa Trilemma as classically expressed might lead to a sequence of questions about observation.  Q: How do we know chemical substances are made of elements? A: Because of some huge body of evidence. Q: How do we know this evidence is valid? A: Because it was confirmed by a bunch of experimental data. Q: How do we know that our experiments were done correctly? And so on. In mathematics, it might ask a series of questions about why a certain theorem is true, which we chase back through a series of lemmas, down to a bunch of basic axioms and rules of inference. We could be asked to justify these, but typically we just posit them. The Trilemma says that there are three ways this sequence of justifications can end up:

  1. we arrive at an endpoint of premises that don’t require any justification
  2. we continue indefinitely in a chain of justifications that never ends
  3. we continue in a chain of justifications that eventually becomes circular

None of these seems to be satisfactory for an experimental science, which is partly why we say that there’s no certainty about empirical knowledge. In mathematics, the first option is regarded as OK: all statements in mathematics are “really” of the form if axioms A, B, C etc. are assumed, then conclusions X, Y, Z etc. eventually follow. We might eventually find that some axioms don’t apply to the things we’re interested in, and cease to care about those statements, but they’ll remain true. They won’t be explanations of anything very much, though.  If we’re looking at reality, it’s not enough to assume axioms A, B, C… We also want to check them, test them, see if they’re true – and we can’t be completely sure with only a finite amount of evidence.

The explanatory variation on Agrippa’s Trilemma, which I have in mind, deals with a slightly different problem.  Supposing the axioms seem to be true, and accepting provisionally that they are, we also have another question, which if anything is even more basic to science: we want to know WHY they’re true – we look for an explanation.

This is about looking for coherence, rather than confidence, in our knowledge (or at any rate, theories). But a similar problem appears. Suppose that elementary chemistry has explained organic chemistry; that atomic physics has explained why chemistry is how it is; and that the Standard model explains why atomic physics is how it is.  We still want to know why the Standard Model is the way it is, and so on. Each new explanation gives an account for one phenomenon in terms of different, more basic phenomenon. The Trilemma suggests the following options:

  1. we arrive at an endpoint of premises that don’t require any explanation
  2. we continue indefinitely in a chain of explanations that never ends
  3. we continue in a chain of explanations that eventually becomes circular

Unless we accept option 1, we don’t have room for a “fundamental theory”.

Here’s the key point: this isn’t even a position about physics – it’s about epistemology, and what explanations are like, or maybe rather what our behaviour is like with regard to explanations. The standard version of Agrippa’s Trilemma is usually taken as an argument for something like fallibilism: that our knowledge is always uncertain. This variation isn’t talking about the justification of beliefs, but the sufficiency of explanation. It says that the way our mind works is such that there can’t be one final summation of the universe, one principle, which accounts for everything – because it would either be unaccounted for itself, or because it would have to account for itself by circular reasoning.

This might be a dangerous statement to make, or at least a theological one (theology isn’t as dangerous as it used to be): reasoning that things are the way they are “because God made it that way” is a traditional answer of the first type. True or not, I don’t think you can really call an “explanation”, since it would work equally well if things were some other way. In fact, it’s an anti-explanation: if you accept an uncaused-cause anywhere along the line, the whole motivation for asking after explanations unravels.  Maybe this sort of answer is a confession of humility and acceptance of limited understanding, where we draw the line and stop demanding further explanations. I don’t see that we all need to draw that line in the same place, though, so the problem hasn’t gone away.

What seems likely to me is that this problem can’t be made to go away.  That the situation we’ll actually be in is (2) on the list above.  That while there might not be any specific thing that scientific theories can’t explain, neither could there be a “fundamental theory” that will be satisfying to the curious forever.  Instead, we have an asymptotic approach to explanation, as each thing we want to explain gets picked up somewhere along the line: “We change from weaker to stronger lights, and each more powerful light pierces our hitherto opaque foundations and reveals fresh and different opacities below.”

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