Musings

There are two broad classes of lattice models: those which use a fixed triangulation, and those which use random triangulations. Most lattice field theories fall into the former sort, including the Regge-Calculus approach to quantum gravity (in which the edge-lengths of the bonds are the dynamical variables). But random lattice models have been proposed as a solution to the Fermion Doubling problem. And, of course, they’ve been used in the Dynamical Triangulations approach to quantum gravity, the precursor to CDT.

Whether you’re working with a fixed or a random lattice, the real problem is not formulating a suitable lattice model (lattice models are a dime-a-dozen). The real problem is finding/proving the existence of a continuum limit.

There are many, many more lattice theories than there are continuum field theories. Think about lattice Yang-Mills theory in arbitrary dimension, $d$. Only for $d\leq 4$ does a continuum theory exist. It is only in the continuum limit that the (infinite) details of the latticization drop out¹, and the resulting continuum theory depends on only a finite number of parameters (and hence is predictive). Continuum theories are relevant (infrared repulsive) deformations of Renormalization Group fixed points, and not every lattice model has such a suitable fixed point.

Conversely, just because you can write down a latticization of some QFT doesn’t mean that lattice theory has anything to do with the behaviour of the continuum theory you would like to describe. Consider the case of QED. It is absolutely straightforward to write down a lattice version of QED. However, 4d Lattice QED does not have a continuum limit. If you hold the low-energy effective gauge coupling (the fine-structure constant) fixed, then the lattice coupling diverges at a finite value of the lattice spacing. You cannot send the lattice spacing to zero.

This is not a big deal because:

1. We know that QED is just a low-energy effective theory, which is embedded in some larger theory at short distances. No one is wedded to the idea of putting QED itself on the lattice, as it’s not really fundamental.
2. The lattice spacing at which Lattice QED breaks down is exponentially smaller than the electron Compton wavelength (the length scale at which QED effects become important). This is many, many orders of magnitude shorter than the length scale at which QED is subsumed into the aforementioned larger theory.

In the case of lattice gravity, the challenge of finding a continuum limit is to be able to hold the effective Newton’s constant, $G_N$ and cosmological constant, $\Lambda$, fixed, while sending the lattice spacing to zero. The problem is more acute than in the QED case because, in the absence of a suitable RG fixed point, the length scale at which quantum gravity effects become important is the same as the lattice spacing, rather than being exponentially longer.

The precursor to CDT was the Dynamically Triangulated Random Surface model, a model of Euclidean 2d quantum gravity (coupled to matter). This model garnered a lot of interest in the early 1990s, because of its relationship to $c\lt 1$ noncritical string theories which were being actively studied in those days. But attempts to add more matter ($c\gt 1$) or to go to higher dimensions ($d\gt 2$) ran into the basic problem of the unboundedness from below of the Euclidean Einstein action. The symptom, in the lattice model, is that the triangulations which dominate the functional integral represent a branched polymer or a crumpled phase, rather than anything resembling a smooth manifold.

CDT represents an important technical advance. Rather than Euclidean gravity, it is a discretization of Wick-rotated Minkowski gravity, and does not have the wrong-signature mode that leads to trouble in the Euclidean theory. To do this, requires a specialization to manifolds which are globally-foliated by spacelike hypersurfaces of a fixed topology (usually, $S^{d-1}$). This is a highly problematic restriction and, if I were inclined to write a second post, it would be largely devoted to why that restriction is problematic. But, for present purposes, this stratagem does avoid the aforementioned sickness of the Euclidean theory. For $d=3$ (and, possibly, $d=4$), the theory avoids degenerating into a branched polymer or crumpled phase.

That’s rather nice, and an important step towards addressing the real issue, which is whether the theory has a continuum limit. In particular, we would to send $(G_N)_{\text{eff}}$ and $\Lambda_{\text{eff}}$ to zero in lattice units, as we send that lattice spacing $a\to 0$ (with $\Lambda_{\text{eff}}(G_N)_{\text{eff}}^3 \ll 1$).

In that limit (if it exists), you know what configurations should dominate the functional integral: those configurations which are “close” to a classical solution¹ of the (Wick-rotated) vacuum Einstein equations with cosmological constant. For positive $\Lambda$, these are solutions in which the 3-geometry expands and then recollapses.

So does this Minkowskian theory have a suitable RG fixed point, which would lead to a continuum limit? Reuter et al claim that Euclidean Einstein Gravity (with a cosmological constant) has an RG fixed point. But, as we’ve seen, their alleged fixed point corresponds to a solution of an essentially perturbative RG equation lying well beyond the realm of validity of perturbation theory. In other words, it is a mirage.

What one really needs is a truly nonperturbative analysis, and that’s what a lattice model affords us the possibility of attempting. No one has actually tackled the question of the existence of a fixed point head-on (through some numerical ‘block-spinning’ of an ensemble of triangulations). But one very crude test is to compare the Monte-Carlo simulations that they have done with what one might expect from a semiclassical analysis.

Ambjørn et al are rather upbeat about the agreement between their simulations and a mini-superspace analysis. But, when I stare at their results, it looks to me as if the effective cosmological constant is quite large (the comparison is insensitive to the effective value of $G_N$), and the departures from semiclassical behaviour (which one expects for small 3-geometries) set in at a scale-factor not all that far removed from the maximal size of the universe. In other words, their results don’t look terribly semiclassical at all.

Perhaps, this is because they are still far from the continuum limit (with their pitifully small average number of simplices), and larger simulations will start looking more semiclassical. That would be evidence (but, still, very weak evidence) for the existence of a continuum limit.

More likely, as I’ve argued previously, pure quantum gravity doesn’t have a suitable RG fixed point. Only by adding some as yet unknown (and, for practical purposes, unknowable) set of matter degrees of freedom would such an RG fixed point emerge. In that case, you wouldn’t even expect the lattice theory of “pure” gravity to have a continuum limit.

Still, regardless of whether CDT has a continuum limit, it is likely to be an interesting source of insights, just as lattice QED has proven to fruitful to think about, despite the fact that it doesn’t have a continuum limit, either.

Update (2/4/2006): Shellings

Bah! After further private discussions with Greg, I no longer believe this objection is pertinent. A shelling implies an actual ordering of the $d$-simplices. A discrete time function only determines a partial ordering of the $d$-simplices. For triangulations of $X=M\times\mathbb{R}$, that partial ordering both exists and is unique. On the other hand, Ambjørn et al are largely interested in $X=M\times I$ or $X=M\times S^1$, for which existence is not guaranteed.

I thought I should move one of Greg Kuperberg’s critiques of CDT from the comments below to someplace “above the fold.” As I said above, we are specializing to d-manifolds of a very special sort, $X^{(d)}= M^{(d-1)}\times \mathbb{R}$. In our lattice model, we are going to sum over piece-wise flat, globally time-oriented, Minkowski-signature metrics on $X$, corresponding to triangulations³ of $X$.

Every triangulation, $T$, of $X$ gives rise to such a metric, so one might hope for a discrete approximation to “the space of all Minkowskian metrics on $X$, modulo diffeomorphisms” by simply summing over triangulations, $T$.

However, to actually define the lattice model, one needs to do a certain “Wick-rotation.” This requires us to choose a (discrete) time-coordinate, $t$, on each triangulation, $T$, that is, an assignment of an integer, $t_i$, to each vertex, $v_i$, of the triangulation. Since we are trying to model a diffeomorphism-invariant theory, we would like this time-coordinate to be determined by the triangulation itself, rather than being an arbitrary extra piece of data.

For a single $(k,d+1-k)$-simplex, this is easy: simply assign an integer $t$ to the first $k$ vertices, and the integer $t+1$ to the remaining $d+1-k$ vertices. The question is:

• Can you do this compatibly for all of the $d$-simplices of the triangulation?
• Is the resulting assignment unique?

Yes.

But, rather trivially, “most” triangulations of $X=M\times I$ will not have that property. There will be paths with different numbers of timelike links (counted with sign) between the initial and the final hypersurface.

The restriction to triangulations (equivalently, piece-wise flat metrics) for which the above-described discrete time coordinate does exist is a very strange and highly non-local restrictions on the metrics you include in the functional integral. Essentially, you are restricting to metrics for which the geodesic distance between the initial and final slices (for all geodesics between the initial and final slice) is fixed. Even if there is a continuum limit, it’s not clear at all that the resulting continuum theory is a local, diffeomorphism-invariant theory.