Post 4: Gravity as a Gauge Theory

I talked in the previous post about the description of electromagnetism as a gauge theory and its description in terms of loops. LQG was developed by first reformulating GR as a gauge theory and building up a loop description.

The breakthrough that allowed GR to be usefully described as a gauge theory was provided by Ashtekar in Phys Rev Lett 57 (1986) 2244.

The Ashtekar reformulation makes extensive use of the

tetrad formalism. By introducing an appropriate set of basis one-forms, the general metric in general relativity g

μν can be made to look locally like a Minkowski metric η

IJg

μν = e

Iμ(x)e

Jν(x)η

IJThink of the e

Iμ(x) as a set of basis vectors at the point x (for the pedants – I am blurring the distinction between covariant and contravariant vectors to keep this explanation simple). Spacetime is 4 dimensional, so I need 4 of these basis vectors (labelled by I), and each one will have 4 components (labelled by μ) in the x coordinate system. These basis vectors are chosen so that the metric components expressed in that basis are just the flat Minkowski space components η

IJ.

In post number 2, I talked about the idea of a connection coefficient. This is used to allow the definition of a parallel transport operation. The infinitesimal form of this parallel transport is a covariant differentiation operation defined on tensors which produces new tensors. Basically, the derivative of a (co)vector a

μ is

∂

μa

ν-Γ

ρνμa

ρWhen we have tetrads, we not only have the general tensor indices (Greek) which are handled by the connection coefficient, but also the Minkowski indices (capital Latin). To handle the latter, we introduce a "spin connection" ω and the covariant derivative of something with both general tensor indices and Minkowski space indices becomes

D

μe

Iν ≝ ∂

μe

Iν + ω

IμJe

Jν - Γ

ρνμe

IρThe tetrad field encodes the information in the metric g

μν, thus contains all the usual GR information (curvature etc). The well known relationships of GR can now be re-written in terms of the e and Γ fields.

To get to the Ashtekar variables, we need to use the ADM formalism of GR I described in post number 2. Basically I described there how spacetime is foliated by a bunch of spacelike hypersurfaces and time evolution is described in terms of the lapse and shift vectors.

The physics is contained in the three-metric g

ij induced on the hypersurfaces. Recall I said that there were also a set of variables π

ij “ conjugate” to the

g

ij. The set of g’s together with the π’s form “phase space”. (The easiest way to picture a phase space is to imagine the phase space of the simple harmonic oscillator. There we have a position coordinate q and a momentum coordinate p. The motion of the oscillator is an ellipse traced out in phase space. The actual ellipse depends upon the initial conditions.)

There are also 4 constraint equations which the π

ij and g

ij have to satisfy. These are split into two types:

The "Hamiltonian Constraint" - a single scalar constraint equation

The "Diffeomorphism Constraints" - a three vector's worth of constraints

Think of the constraints as defining a subset of the full space of values of { π

ij , g

ij }. The physical configurations are restricted to this subset.

These are

first class constraints (Their

Poisson brackets with each other preserve the constaint hypersurface). The diffeomorphism constraint generates

diffeomorphisms of the 3 surfaces. Combined with the Hamiltonian constraint, we get the generators of spacetime diffeomorphisms on the full spacetime.

Now we can choose a tetrad formulation of GR appropriate for our ADM decomposition (note this is sometimes called “3+1 decomposition” since the time variability is split off from the 3 space dimensions).

Note from now on I’m going to use Greek indices for 4 dimensional spacetime indices and latin lower case for three-space indices. To do this, we can keep the spatial part of the tetrad the same and redefine the time part to be a combination of the lapse function and shift vectors:

e

I0 = Nn

I + N

ie

Ii𝜹

kle

ki e

lj = g

ijNow, as with ADM, we identify our canonical variables and their conjugates, construct an

action integral from them, which is then used to derive the physics. Constraints between the set of canonical variables and their conjugates become

Lagrange Multipliers. This time, with the tetrad picture, it’s going to be a bit more complex, because we have, instead of the metric g

ij, the tetrad field and the connection as independent variables. Also, we have an extra symmetry – the invariance under local Lorentz transformations – which we would expect to give rise to new constraints.

The constraint algebra (the set of Poisson bracket relations amongst the constraints) is now

second class. However – (this was the breakthrough that enabled this whole field to develop ) – there is a specific set of variables which simplifies things. These were discovered by Abhay Ashektar and Amitaba Sen.

The variables are, firstly the "densitized triad"

E

ai = ee

ai (a,b,c are the three space indices, and i,j,k are the "group" indices labelling the basis vectors. e = det(e

ai)

and secondly the Ashtekar-Barbero connection

A

ia = γω

0ia + (1/2)ε

ijkω

jkaE

ai and A

ia transform like an SU(2) vector and SU(2) connection respectively.

In addition to the Hamiltonian and Diffeomorphism constraint I talked about previously, in the tetrad formulation 3+1 decomposition we now have an extra constaint called the Gauss constraint. This comes from the extra symmetry we've introduced - since we've defined these tetrads at each point, they can be transformed by local Lorentz transformations into other just-as-good tetrads.

Now that we have a classical phase space, consisting of dynamical variables and their conjugates, plus constraint conditions, what would normally be done is to apply

Dirac Quantization to this system. (Note this mechanism is not what you’ll find in most books on quantum field theory. They describe covariant quantization, whereas here we need a quantization to deal with a system on spacetime foliated by time slices and subject to constraints).

The original attempt (using γ=+/-i) at quantization used a wave functional Ψ[A] on the space of connections A (modulo gauge). This attempt was not very successful – it was not possible to establish semiclassical states and thus see how smooth spacetime could emerge (see

here).

The solution to this was to move away from defining wave functionals on the space of connections and instead to use

holonomies as the basic variables. I’ll describe this in the next post.

Edited to fix error in equation