On a Tangent [Frame]
Spacetime is like a sweet sixteen party. Fermions are the teenagers who get to propagate and interact on this spacetime. Based on what little I know about sweet sixteen parties, they involve a teenager dancing around, then changing into a different dress, and then dancing around some more. The tangent space is like the dressing room in the back of the party, where fermions go to transform into the appropriate dress so they can interact more with attractive gauge bosons on the spacetime. So how does the fermion go from the spacetime ‘upstairs’ to the tangent space? They take the elevator called the vielbein.
(Sometimes the teenagers on those `sweet sixteen’ reality TV shows get upset and throw a tantum because of some adolescent love triangle. That’s a chiral anomaly.)
Spacetime is a semi-Riemannian manifold with a Minkowskian tangent space. In general relativity, the spacetime is invariant under the diffeomorphism group, that is the group of coordinate transformations. This is just saying that the underlying physics is independent of how we choose to draw coordinate charts.Fields, such as the electron or photon field, live on this spacetime. The spacetime is where they propagate and where they interact locally with other fields to produce interesting physics. However, the spacetime isn’t necessarily where the fields transform. In particular, fermions transform in the tangent space. This is because fermions are represented by spinor fields, which are spin 1/2 representations of the Lorentz group. The Lorentz group acts on Minkowski space, i.e. the tangent space at a point of spacetime. Note that the general linear group does not admit any such spin 1/2 representations.
Hence on a curved spacetime, we need to introduce an `elevator’ so that our fermions can `go upstairs’ to the tangent frame where they can transform under the spin-1/2 representation of the Lorentz group. This elevator is the vielbein.
A quick word on nomenclature: vielbein (rhymes with ‘Einstein’) means “many legs.” In four dimensions we would call this object a vierbein, meaning “four-legs.” There are associated einbeins, dreibeins, and so forth, but vielbein is the general term.
By the equivalence principle, at any point of our spacetime we can choose our coordinates to be Minkowskian. However, this is only true at that given point. The idea of the vierbein is to define a tangent space index structure based on the fact that one can make this transformation at any point. Let us denote as the coordinate transformation from point to a reference epoint .
Then let us define the vielbein by . Here is a tangent space index and is a spacetime index. Then we can write our spacetime metric as:
above is just the Minkowski metric. And so the vielbein is, in some sense, the `square root’ of the metric. It acts to convert spacetime coordinates into tangent space coordinates.
There are lots of useful properties one can deduce, but I’ll leave that to you to discover. (A useful reference is chapter 12 of Anomalies in Quantum Field Theory by Bertlmann, though people will certainly have their own favorites. )
What I’d like to get back to is the idea that fermions transform under the Lorentz group, and not the diffeomorphism group. The -matrices which are used to define the Dirac operator are defined on the tangent space. In order to connect it to the spacetime, we need to use the vielbein:
Here on the right hand side lives in the tangent frame and is a constant matrix satisfying the Clifford algebra. The on the left hand side is a function of spacetime and that contracts with the covariant derivative. By the way, why didn’t you have to worry about this `position dependent’ matrix when you were doing quantum field theory the first time around? Because you were working in flat space where the spacetime and tangent space are the same, hence the vierbein is trivial.
It is worth noting that one can do a lot of differential geometry from the tangent frame. In fact, one can define a `spin connection’ based on the vierbeins that takes the place of the Christoffel connection that one is used to from a first general relativity course, but this is another subject.
Anyway, there’s much more to this story but I hope I was able to describe a small facet of the big picture. My big picture, by the way, is using this to play with models of bulk fermions in warped space. A particular application that has caught my attention: neutrino masses in RS1 (Grossman and Neubert, Phys. Lett. B464 (2000) 361).
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