Types of quantum field thories
In physics, the term general covariance is meant to indicate the property of a physical system or model (in theoretical physics) whose configurations, action functional and equations of motion are all equivariant under the action of the diffeomorphism group on the smooth manifold underlying the spacetime or the worldvolume of the system. Here “covariance” is as in “covariant tensor” another term for behaviour under diffeomorphisms.
The term general relativity for Einstein-gravity originates in at least closely related ideas (see History and original formulation below), and Einstein-gravity is the archtypical example of a generally covariant physical system:
here, the configuration space of physical fields over a smooth manifold is not quite the space of Riemannian metrics on itself, but is the quotient of this space by the action of the diffeomorphism group : two Riemannian metrics and on represent the same field of gravity on if there is a diffeomorphism such that .
Or rather, such a diffeomorphism is a gauge transformation between the two field configurations. The configuration space is not the naive quotient of fields by diffeomorphisms as above, but is the homotopy quotient, or action groupoid, denoted . In the physics literature this action groupoid is most familiar in its infinitesimal approximation, the corresponding Lie algebroid, whose formal dual is a BRST complex whose degree-1 elements are accordingly called the diffeomorphism ghosts (see there).
As with all gauge transformations, they relate physical configurations which may be nominally different, but equivalent. Therefore general covariance is an instance of the general principle of equivalence in mathematics which says that sensible statements about objects must respect the isomorphisms and more general equivalences between these objects.
A physical system which is not generally covariant in this sense is hence one where the smooth manifold as above, underlying spacetime/worldvolume is not regarded as modelling an absolute physical system (such as the observable universe in gravity), but a subsystem that is equipped with ambient structure that breaks the diffeomorphism symmetry. Notably systems like electromagnetism or Yang-Mills theory have traditionally been written in a non-generally covariant form describing gauge fields on a fixed gravitational background, as for instance the space inhabited by a particle accelerator. This ambient structure on the spacetime breaks its general diffeomorphism invariance and hence the effective resulting theory on this background is not generally covariant (a special case of the general phenomenon of spontaneous symmetry breaking).
On the other hand, such a model consisting of background (e.g. the particle accelerator) and quantum fields propagating in it is ultimately to be understood as an approximation to a more encompassing model in which also the background is dynamical, and which is again generally covariant. Specific for electromagnetism and Yang-Mills theory this refined generally covariant model is known as Einstein-Maxwell theory or more generally Einstein-Yang-Mills theory.
The idea of general covariance has a long and convoluted history and the literature witnesses plenty of disagreement about how to interpret and formalize it in technical detail (Norton). Already early arguments by Einstein himself (e.g. the “hole paradox” (Einstein-Grossmann)) show that the discussion has suffered from the beginning from lack of the basic category theoretic concept of isomorphism in the category Diff of smooth manifolds. Below in Formalization in homotopy type theory we indicate a formalization of general covariance that is general, fundamental, and accurately reflects the role of the term in theoretical physics.
The question of general covariance of physical theories in space and time can be traced back to the famous debate between Gottfried Wilhelm Leibniz and Samuel Clarke (the latter assisted by Sir Isaac Newton) on the ontological status of space in the years 1715–1716 (Alexander), the central question being if space exists as a substance or as an absolute being and absolute motion is present (Clarke) or if it is constituted only in relation to co-existent things allowing for relativism in motions only (Leibniz). This kind of problems also played an important role when the general theory of relativity was being developed in the years around 1910. While Albert Einstein first characterized generally covariant field equations as inadmissible since they did not determine the metric field uniquely as shown in the hole argument ( Lochbetrachtung ) in the appendix of (Einstein-Grossmann), he later accepted (Einstein 1916) that all physical laws had to be expressed by equations that are valid in all coordinate systems, i. e., which are covariant (generally covariant) under arbitrary substitutions.
Die allgemeinen Naturgesetze sind durch Gleichungen auszudrücken, die für alle Koordinatensysteme gelten, d. h. die beliebigen Substitutionen gegenüber kovariant (allgemein kovariant) sind. (Einstein 1916 p. 776)
The hole argument was dismissed by the reasoning that it is not the spacetime metric that has to be fixed uniquely by the field equations, but only the physical phenomena that occur in spacetime need to be given a unique expression with reference to any description of spacetime. All physical statements are given in terms of spacetime coincidences; measurements result in statements on meetings of material points of the measuring rods with other material points or in coincidences between watch hands and points on the clockface. The introduction of a reference system merely serves the easy description of the totality of all these coincidences (point-coincidence argument) (Einstein 1916 p. 776f).
We discuss the modern formulation of general covariance in differential geometry.
The statement of general covariance is that the distinct configurations of the gravitational field form the set . In particular, if is compact, then the Einstein-Hilbert action functional which a priori is defined on descends to
While this captures the idea of general covariance accurately, for further development of the theory of gravity, however, the set needs to be refined. It is really equipped with the structure of a smooth space (in order to perform variational calculus and hence derive the equations of motion of the theory), and second it is to be refined to a smooth groupoid .
Finally, for setups that admit to introduce fermions/spinors into the model one needs to refine Riemannian metrics to vielbein fields/orthogonal structures. The fully refined generally covariance smooth configuration groupoid is then , discussed in more detail below.
The principle of equivalence in general relativity is formally the statement that around every point in a (pseudo-)Riemannian manifold one can find a coordinate system such that the Levi-Civita connection vanishes (“Riemann normal coordinates”), which means that to first infinitesimal order around that point particle dynamics subject to the force of gravity is equivalent to dynamics in Minkowski spacetime with vanishing field of grvity.
By the above this is a special case of the principle of general covariance: for every field configuration and every given point there is a gauge equivalent field configuration such that the “force of gravity” (the Levi-Civita connection) vanishes at that point.
It is via this relation that the physical “principle of equivalence” relates to the mathematical principle of equivalence: this says that formulations need to respect the given notion of equivalence/gauge transformation, and so
principle of equivalence in mathematics principle of general covariance principle of equivalence in physics .
We discuss here that general covariance in field theory has a natural formalization in homotopy type theory, hence internal to any (∞,1)-topos. For exposition, background and further details on the discussion of classical/quantum field theory in this fashion see (Schreiber, ESI lectures) and (Schreiber-Shulman).
Then the central idea of general covariance is that for
But with just a little bit of formalization the apparent paradox is resolved, because the above evidently just says that the “moduli space” for “subsets of spacetime” is not the manifold itself, but is rather a “moduli stack” namely the quotient stack of by the action of the diffeomorphism group.
to it are equivalent if (and only if) there is a diffeomorphism relating them, as above.
So if in a generally covariant field theory spacetime is not actually the manifold , but rather the quotient stack , then also a field in this generally covariant field theory should be a field on that quotient stack, not on itself.
From this it is clear that for a generally covariant field theory we are instead to declare that the space of field configurations is
In words, the right hand side is the time-honored answer: two fields on a spacetime manifold which are such that one goes over into the other when pulled back along a diffeomorphism are gauge equivalent. This is the statement of general covariance, derived here, formally, from just the condition that any two shapes in spacetime are to be equivalent if related by a diffeomorphism.
Here to read the above equivalence as a theorem, we have to read the left hand side, as it should, be “in the context of -actions”. Such context-dependence is precisely what dependent homotopy type theory takes care of, and this is what the following technical statement deals with.
But in homotopy type theory, types are homotopy types and, hence so are the contexts. A type in context is now in general something more refined than just a family of types. It is really a family of types equipped with equivariance structure with respect to the homotopy groups of the context type.
Specifically, if the context type is connected and pointed, then it is equivalent to the delooping of an ∞-group . One finds – this is discussed at ∞-action – that the context defined by the type is that of -equivariance: every type in the context is a type in the original context, but now equipped with a -∞-action. In a precise sense, the homotopy type theory of --actions is equivalent to plain homotopy type theory in context .
In the following we discuss this for the case that is an automorphism ∞-group of a type which is regarded as representing spacetime or a worldvolume. We show that in this context the rules of homotopy type theory automatically induce the principle of general covariance and naturally produce configurations spaces of generally covariant field theories: for a moduli object for the given fields, so that a field configuration is a function , the configuration space of covariant fields is the function type but formed in the “general covariance context” . When interpreted in smooth models, is the smooth groupoid of field configurations and diffeomorphism gauge transformations acting on them, the Lie integrations of the BRST complex whose degree-1 elements are the diffeomorphism ghosts.
More precisely, we show the following.
Consider in homotopy type theory two types , to be called spacetime and field moduli. Let
be the image of the name of , with essentially unique term
Perform the canonical context extension of and trivial context extension of to get types in context
Form then the type of field moduli ”” in this context:
whose objects are field configurations on and whose morphisms are diffeomorphism gauge transformations between them. In particular the corresponding Lie algebroid is dual to the BRST complex of fields with diffeomorphism ghosts in degree 1.
Write for the ambient homotopy type theory, or rather an interpretation given by an (∞,1)-topos. For standard applications in physics we have Smooth∞Grpd or SmoothSuper∞Grpd or similar, but none of the following general discussion depends on a concrete choice for .
Pick once and for all an object
In the standard interpretation of the homotopy type theory in Smooth∞Grpd could be an ordinary smooth manifold or orbifold, in particular, and then is the diffeomorphism group of , regarded as a diffeological group object. In view of this archetypical example we will in the following often say diffeomorphism for short instead of auto-equivalence in and similarly refer to loosely as the diffeomorphism group of . But even in the specifical model Smooth∞Grpd/SmoothSuper∞Grpd, can be much more general than a smooth manifold or supermanifold or orbifold.
which we also write
Fact. is the context of general covariance with respect to .
In the precise formal sense.
In particular, itself is canonically equipped with the defining action of on it, which syntactically we may write
Here the total space
is the homotopy quotient or action groupoid of by . This is the type characterized by the fact that a function is a function to which is regarded as (gauge) equivalent to another function to if both differ by postcomposition with a diffeomorphism of .
be an object that represents the moduli ∞-stack of field configurations on for some model (in theoretical physics) to be described. For instance for an ∞-group and a cohesive homotopy type theory, we could have the moduli for a choice of -principal ∞-connection, being the moduli for -(higher)gauge fields. For general we may always regard as the target space of a sigma-model.
Then the internal hom
hence the function type
is the naive configuration space of the model. This is not generally covariant, precisely so by the above definition: it is not in the generally covariant context .
which is context enlargement by , the moduli type is freely moved to the general covariant context, where it is regarded as equipped with the trivial ∞-action. Accordingly we will write just with that trivial action understood, which is justified by the precise syntactic expression for it:
We may then form the configuration space of fields in the generally covariant context . As before, a field should be a function on with values in the moduli type of field configurations, but now we interpret this statement in the generally covariant context. Syntactically this simply means that a field is now a term in -context
and that accordingly the configuration space of fields is
The central observation now is that discussed at ∞-action – Examples – General covariance:
is the homotopy quotient of the naive fields by the action of the diffeomorphism group, exhibiting a gauge equivalence between any two field configurations that differ after pullback along a diffeomorphism.
This is precisely as it should be for configuration space of generally covariant theories. We have found:
Fact. In terms of homotopy type theory, configuration spaces of a generally covariant theory over are precisely the ordinary configuration spaces of fields, but formed in the context :
The fields of Einstein gravity are orthogonal structures (Riemannian metrics) on a smooth manifold SmoothMfd of dimension . As discussed at orthogonal structure and vielbein, we are to regard in the context of the delooping of the general linear group via its tangent bundle , by which we always mean here the -principal bundle to which the tangent bundle is associated.
By the discussion at principal ∞-bundle this is modulated by a morphism
in . (A detailed exposition of this and the following, with further pointers, is in (Schreiber, ESI lectures).)
and since is essentially unique we will notationally suppress it in the succedent on the right and just write
In view of the equivalence of (∞,1)-categories
this expresses the canonical -action on the coset (by mutliplication from the “other side”).
This almost verbatim expresses the familiar statement:
A vielbein on is a -equivariant map from to the coset .
This in turn almost verbatim expresses the familar equivalent statement
This is still the naive space of fields, not yet generally covariant. So we next pass to the general covariant -context and form the correct generally covariant space of fields, being the type in context given by
orthogonal frame transformations of the fibers of the tangent bundle;
general diffeomorphisms of the base .
We unwind this a bit more.
A slight subtlety in interpreting the above expression is that in
So the name and its pullback are given by a diagram of the form
in . Here the ∞-image is directly read off to be the factorization in the second column from the right of
where each square and hence each rectangle is an (∞,1)-pullback in . This shows that the automorphism -group of in the context of is just the absolute automorphism -group freely context extended. The categorical semantics of the dependent type
is the third column from the right in the above diagram. This means that the dependent sum in
Forms the internal hom in between the homotopy fiber of that third column formed in , which is the second column (and therefore now does rememeber the -action on ) with , rememeberting that the result has an -action by precomposition.
The pre-history of the idea of general covariance is reviewed in
The original articles by Einstein on the idea of general covariance include his Entwurf (sketch)
where it is fully embraced.
An attempt at a fairly comprehensive review of the history of the idea of general covariance and its reception up to modern days is in
A formalization in the context of AQFT is proposed and discussed in
A review is in
For more see the references at AQFT on curved spacetimes.
See also higher category theory and physics.