The search is on for the abstract formalization of the process of quantization – the process that reads in a “classical field theory” – for instance presented in form of a gauge theory or in form of a σ-model background field data – and spits out the corresponding quantum field theory.
There is an open problem of mathematical (-physics) model building:
While for various aspects and facets of this question there are well-developed formalisms – such as geometric quantization or deformation quantization or BV theory – a full answer is certainly still missing, not the least because the full formalization of the question itself has still to be established.
Considerable progress on this formulation of the question has been achieved with the formalization and proof of the cobordism hypothesis in On the Classification of Topological Field Theories by Jacob Lurie. This at least indicates what the result of any full quantization procedure should be in that it clarifies what exactly a topological quantum field theory is: a morphism from the (∞,n)-category of cobordisms .
In On the Classification of Topological Field Theories Jacob Lurie indicates some first steps towards finding a similar formalization of “classical field theory” (in terms of his -categories of “families”) and a systematic procedure for turning the classical theory into the quantum theory. These thoughts were further developed in the article
(See there also for some comments along the lines here.)
At the same time Stephan Stolz and Peter Teichner in their work on geometric models for elliptic cohomology? match QFTs to cohomology theories and are seeing there the analogy between pushforward in cohomology theory and the quantization process from classical field theories to quantm field theories.
For the purpose of the present entry this indication of a quantizaton proposal by Lurie et al. and Stolz-Teichner mainly serves as a reference for the idea itself that a formalization of something interesting is to be sought here, and of the kind of abstract nonsense answer one hopes to find. We will however discuss a somewhat different-looking approach. It may well be related to the Lurie-et al proposal in the end, but for the time being we shall not concentrate on that relation.
Rather, the approach for a formalization of the quantization procedure that shall be discussed at this entry here is more in the spirit as indicated to some extent in the work of David Ben-Zvi et. al, which is described in some detail at geometric ∞-function theory. This is essentially based on the idea that a quantum field theory is obtained from a σ-model target space object by homming extended cobordism cospans into the target object and then pull-pushing geometric function objects through the resulting spans of configuration space objects .
The main result of David Ben-Zvi et. al.‘s work on this approach is that they point out that as soon as the geometric function object one uses satisfies the two fundamental theorems of geometric infinity-function theory, a considerable amount of rich structure – that has in parts been known by itself – gets unified into one coherent elegant story: the nature of partition functions (i.e. traces), of centers, of Hochschild (co)homology, Deligne-Kontsevich-statements, etc. all are understood by means of a suitable geometric function theory as induced from the underlying geometry of configuration space objects as well as the loop space objects of .
But more precisely, the premise of the discussion here is that a fundamental understanding of the situation is obtained after regarding seriously the fact that
This means that we shall take the classical structure which the sought-after quantization procedure is supposed to send to an FQFT to be, essentially, a functor
from some path n-groupoid of to some coefficient object .
Here the k-morphisms in are generated from the existing -morphisms in as well as the -dimensional smooth paths in . The functor on this encodes the parallel transport of a connection on a principal ∞-bundle over : this assignment of k-morphisms in to -dimensional paths in is the assignment of the classical action to trajectories in .
More precisely, this is the classical action assigned to topologically disk-shaped -paths. The full classical action of a classical σ-model quantum field theory is an extension of
from disk-shaped cobordisms to all cobordisms in (possibly and usually after an extension of the codomain to some new codomain ). This extension knows not just the parallel transport, but also the holonomy of the classical background field.
The question to be analyzed here is how this classical action transmutes systematically to an FQFT on abstract bordisms (possibly with extra structure, if we pass from TQFT to richer theories like CFT?). This step should be the path integral:
in some way the assignment of to a cobordism is expected to be obtained by “summing” in some way the value of on elements in the fiber of the forgetful functor over , i.e. over all possible ways to map into – the sum of over all paths or field configurations .
This section describes the general formalism that we want to study as a candidate for (aspects of) a fundamental abstract-nonsense? description of quantization?. The next section then picks special (simple) cases and works out the action of the abstract machinery on these.
In rough outline the procedure is this:
In a context of sufficiently general spaces;
we extend the background field parallel transport to an action functional .
the points of are the Dirac--distributional sections or quantum states
through the corresponding span? of total spaces.
The definition of quantum propation defined by a classical background field that we describe below is quite simple once we suitably organize the material by establishing a bit of terminology.
We need to talk about various spaces of sorts:
a space that is the target space – often, but not always, to be interpreted as physical spacetime – in which the -dimensional object propagates – for for instance a point particle such as an electron – whose dynamics we want to encode.
Moreover, for each -dimensional manifold? we need a path space or configuration space of maps from into .
The general abstract nonsence? for dealing with general spaces of this sort is that of space and quantity?. As described at motivation for sheaves, cohomology and higher stacks, this leads one to describe a space as an ∞-stack? on some site? .
In the simplest case, which the reader should keep in mind, the site? in question is the point? and an ∞-stack? on it is just an ∞-groupoid?. In turn in simple cases of this, which the reader should still keep in mind, this ∞-groupoid? will be just a groupoid? or at most a 2-groupoid?.
Despite – possibly – its appearance, the reader should take this statement as suggesting an immense simplification instead of a huge complification: the language of (∞,1)-topos? is a user interface that makes pretty sophisticated and rich notions of generalized spaces have the same look-and-feel as just plain topological space?s. It’s the inner workings of the formalism that will take of things coming out right even when we talk about richer objects.
In particular, it is the machinery of models for ∞-stack (∞,1)-toposes? that provides all the tools for actually performing the operations that we shall consider. The reader unfamiliar with that will suffer little loss from concentrating his or attention on the toy examples where all “spaces” involved are finite groupoid?s and trust that everything goes through analogously also in the other examples.
But, as indicated above, it is actually, more: it is a differential cocycle on : something that depends not just on itself, but also on a notion of paths in .
To encode this we may pick a cosimplicial object?
in our -category of spaces, that encodes which object in models the standard -simplex? regarded as a space. For instance if is the (∞,1)-topos? of Lie ∞-groupoid?s modeled by ∞-stack?s on Diff?, we would take to be the standard -simplex regarded as a smooth manifold? in the standard way.
Then denote the space of disk-shaped -dimensional paths in by
These spaces glue together to the path ∞-groupoid
For instance if is a global orbifold?, the 1-morphisms in are generated from smooth paths in as well as orbifold jumps subject to the relation
More generally, let be some poset? and specify a functor
We think the image of some under as the realization in of some abstract cobordism? whose boundary component inclusions are the images of all morphisms in .
So encodes a multi-cospan? of cobordisms in . In principle will in general be supposed to range over all cobordisms in some sense, but for our discussion it will be complete sufficient to concentrate on much less. Most of our discussion concerns a handful of cobordisms, usually just two of them and their composites.
In any case, by slight abuse of notation, we shall write
for the (homotopy?) colimit? over this multi-cospan? of cobordisms and speak of the -groupoid of bordisms in . If we assume that ranges at least over the disk-shaped cobordisms we have a canonical inclusion
A background field that is flat (meaning that it has vanishing field strength?) is a morphism
The more general situation is described at …, but need not concern us here for the moment. Its quantization is structurally slightly but not, for our purposes, essentially different from that of the flat case.
where is the component of (as a morphism out of a colimit) on points in .
We are interested in forming the associated -bundle induced by a choice of representation?
on some (∞,1)-category? . See below for examples.
As a cocycle this is just the composite
This is the background field that encodes the forces acting on the objects propagating in whose quantum dynamics we seek.
We will often abuse notation and write just for , when the context is clear.
A choice of extension of the parallel transport morphism through the inclusion is the corresponding action functional
This extension amounts to choosing (possibly uniquely) traces over parallel transport.
example Let encode the parallel transport in a vector bundle? with connection. And consider the extension to with just 1-dimensional cobordisms in . Then the extension amounts to finding a value for on the circle. To be functorial and symmetric monoidal, it must be true that the value on the circle is the result of cutting the circle open at any point, evaluating on the resulting 1-disk-shaped cobordism and then applying the trace on the vector space over the endpoint.
For each cobordism the component of the action functional will define a total space of an associated bundle . These are discussed in the next section.
A state of the quantum theory over a cobordism is supposed to be a section? of the bundle over the configuration space of fields on , where we allow generalized sections that may be distributional. This we define now.
In order to define sections and states, we need to fix a point? in the coefficient object , i.e. a morphism
definition (ground monoid)
i.e. the lax pullback?
example: ordinary linear quantum mechanism
In the special case that Vect? over some ground field the canonical point is that ground field. Then the “ground monoid” is again canonically identified with the ground field: . Therefore the name “ground monoid” as a generalization of this situation.
More generally, for any other point of , we say that the lax pullback?
is the object of elements of relative to the chosen point.
example: ordinary vectors
Moreover, given the background field differential cocycle or the action functional
we have the lax pullback?
This is really the action groupoid? of the “action?” of on : it is much like the Atiyah Lie groupoid? of a principal bundle?, with the difference that its objects are elements in an associated (not a principal bundle) and its morphism connect only elements in the fibers that are related by parallel transport along the corresponding path down in .
For each component we get a corresponding lax pullback
in particular the fiber over of
is the fiber of the underlying --bundle over -
Picking a point in the total space is like picking a -distributional section of with support at and value there. Picking two points is naturally interpreted as picking the sum of two such sections.
Picking instead a map for a finite group of cardinality is like picking a -distributional section with value and dividing its value by .
Then for a cobordism, a a section of the background field or state of the quantum system over is an object
For some choices of this general notion of section may be degroupified – to yield an ordinary notion of (still possibly distributional) section.
In the case that we have that is (the discrete category? on the set underlying) the vector space that is the fiber of the background vector bundle over at .
For consider the underlying morphism of plain -groupoids and interpret this as the vector
In total the above discussion yields, starting from a background field , a multispan? of -bundles over configuration spaces .
this realizes as the universal cocone whose component under is
by lax pullback of this induces a span of total spaces
definition the “quantum propagation along ” is the functor of geometric function object?s
Reminder on sections In order to unwrap the above abstract prescription in some special cases recall how we extract from the value of a generalized section an ordinary vector in the case that is a vector bundle?:
given we extract the corresponding “degroupoidified” vector of the corresponding section over as the pullback
The background field for the electrically charged particle is the electromagnetic field?. As discussed there, is this modeled by a Deligne cocycle? in degree 2, or equivalently a line bundle with connection.
Then an electromagnetic background field over a manifold? is encoded in a morphism
See connection on a bundle? for more details on this special case.
We choose the standard representation
of on .
is the total space of the corresponding line bundle?.
for the interval? the configuration space is the path space of ;
the bundle can be thought of as having as fiber over a path the space of pairs of vectors that are related by parallel transport along , i.e. those for which .
The two span maps
project on the corresponding component:
To see how the corresponding quantum propagation acts on a section, it is sufficient to look at a single -distribution state
that hits a vector . In diagrams this means that
Then the pull-push propagation functor
sends to the composite right diagonal morphism in
We see that
is the space with points the set of pairs such that ;
hence the map
is on the level of sets the map whose fiber over the vector at is the set of paths such that .
This means that, by the above prescription, the degroupoidification of our propagated section is the section whose value over is formally
Since the cobordisms are disk-shaped, the groupoids of -principal bundles on them are trivial and we have
Hence the background field for DW theory is actually just a
This just reflects the fact that -principal bundles for discrete group have a unique and flat connection?.
By the discussion at group cohomology? this DW background field is nothing but a group cocycle defining a class in degree 3 group cohomology . This is the form in which the background data for DW theory was originally and is still usually presented.
The standard representation of the gauge group (recall that is from the sigma-model? perspective not a gauge group but defines the target space!) is on
is under the above “degroupoidification” the action of Dijkgraaf-Witten theory, where the -weight comes from the fact that the pulled back state picks up the automorphisms of by the push-forward
Actually, more precisely due to the nature of the homotopy pullback? this picks up another factor of .
background field is a morphism .
So this is just as Dijkgraaf-Witten theory, discussed above, but in one degree higher.
Accordingly, its discussion is pretty analogous to that of the Dijkgraaf-Witten model discussed above. The important difference is that now the automorphisms of a principal 2-bundle? classified by a map form a 2-group? themselves, so that the Leinster measure? coefficient in our degroupoidification formula becomes slightly more subtle.
The right factor was originally found in the literature by requiring the path integral of the Yetter model to be independent of choices made when representing -2-bundles as cocycles. The following proposition shows that the factor obtained this way is indeed nothing but the 2-groupoid Leinster measure.
this is an old version of this proof, I’ll upload a better one…
The combinatorial weighting for the “categorified DW” theory introduced by Yetter (and studied by Girelli, Pfeiffer, Popescu, Porter, Martins etc.) is indeed the Leinster measure? of the corresponding configuration space.
Recall that for ordinary DW theory for finite group config space is the groupoid , which, since is finite, may be computed by choosing a triangulation of , letting denote the free category on the corresponding graph and considering .
Since there are
many ways to assign group element to the vertices of the triangulation, there are many natural transformations starting at any functor and therefore the Leinster measure of the hom-groupoid is
Now, Yetter et al find that when target space is replaced by
then the corresponding combinatorial weight becomes
where is the number of vertices, as before, and where is the number of edges in .
(See for instance theorem 2.13 on p. 127 of Martins-Porter’s On Yetter's invariants or the discussion below theorem III.2 on p. 6 of Girelli-Pfeiffer-Popescu’s BF to BFCG (while the factor is not manifest in their expression (15))).
The claim here is that this factor is precisely the Leinster measure? on the configuration space
Here I am taking, to replace awkward pseudofunctors with nicer strict 2-functors, now to be the strict 2-groupoid generated from the triangles in .
Then the above hom-groupoid is supposed to be the 1-groupoid obtained by passing to equivalence classes of 1-morphisms in the 2-groupoid whose objects are strict 2-functors , whose morphisms are pseudonatural transformations of these, and whose 2-morphisms are modifications of those.
To prove this, we need to compute how many 1-morphisms in there start at any given object.
But that’s easy: a 1-morphism
in the 2-groupoid of strict 2-functors is a transformation whose component is an assignment of squares in to edges in :
By the general yoga of strict 2-groups, we know that the square on the right is entirely and precisely fixed by
the source 2-functor value on the top;
the two transformation elements on the two sides;
and the label of the 2-morphism filling the square.
This uniquely determines the value of the pseudofunctor which is the target of the transformation whose component is.
Therefore, clearly, there are precisely
many 1-morphisms in
starting at any given object.
But some of them are related by 2-morphisms
To figure out many, we need to compute how many modifications there are starting at any transformation .
But again the same logic applies: such a transformations is given by a component map which sends vertices in to 2-cells in :
Again one sees, by drawing the relevant naturality tin-can diagram which I won’t bother to do here in MathML, that for each such choice of assignments of elements in to vertices in there is precisely one modification .
This means that we have
The Leinster measure in this situation is
precisely the combinatorial factor which people found makes the Yetter state sum model a topological invariant.