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The Hartle-Hawking wave function in 2d causal set quantum gravity

We define the Hartle-Hawking no-boundary wave function for causal set theory (CST) over the discrete analogs of spacelike hypersurfaces. Using Markov Chain Monte Carlo and numerical integration methods we analyse the wave function in non- perturbative 2d CST. We find that in the low temperature regime it is dominated by causal sets which have no continuum counterparts but possess physically interesting geometric properties. Not only do they exhibit a rapid spatial expansion with respect to the discrete proper time but also a high degree of spatial homogeneity. The latter is due to the extensive overlap of the causal pasts of the elements in the final discrete hypersurface and corresponds to high graph connectivity. Our results thus suggest new possibilities for the role of quantum gravity in the observable universe.

The Hartle-Hawking (HH) prescription for the ground state wave function over spatial closed 3-geometries (Σ, h) is the Euclidean functional integral over 4-geometries (M, g) where ∂M = Σ, g| Σ = h, I E (g) is the Euclidean Einstein action and A is a normalisation constant [1]. This ground state is meant to represent the initial and only boundary condition of the universe from which further evolution of Ψ can be (uniquely) determined. The use of the Euclidean path integral is motivated by direct analogy with quantum field theory on a fixed background.
While the simplicity and ingenuity of this proposal is undeniable, the continuum path integral is notoriously ambiguous. It is therefore desirable to find a framework in which it can be regulated in some fashion. In causal set theory (CST) the spacetime continuum is replaced by an underlying discrete substructure given by a locally finite partially ordered set or causal set [2,3,4]. Here the path integral is replaced by a sum over all possible discrete geometries, or causal sets, thus providing a concrete framework for examining the HH wave function in a fully non-perturbative setting.
The closest analog of a Cauchy hypersurface in a causal set C is an inextendable antichain [5] which is a maximal subset of unrelated elements in C. Thus, a final spatial boundary of a spacetime region is represented in C by a future-most inextendable antichain A f in C and an initial spatial boundary by a past-most inextendable antichain A i . The main characteristic of such an antichain is its cardinality although its connectivity to the bulk elements also contains significant geometric information. In order to implement the no boundary proposal in CST, we restrict our sum to those causal sets with a future-most antichain A f of fixed but arbitrary cardinality and a single minimal past element A i . This latter condition is the exact discrete analog of what the authors of [1] refer to as an initial spatial "zero" geometry, a single point, which captures the idea of a universe emerging from nothing.
In the continuum the Euclidean path integral serves two purposes. First, the no-boundary continuum topology does not support a singularity free causal Lorentzian geometry [6,7], and second, Euclideanisation gives a well-defined measure theoretic interpretation of the probability without having to resort to the machinery of measurement theory and external observers. In CST the first concern is no longer valid since every causal set is causal and "nonsingular" while remaining "Lorentzian". Additionally, the Euclideanisation of the measure can be achieved without relinquishing the Lorentzian nature of the causal set, by introducing a supplementary variable, the "inverse temperature" β [8,9]. Thus, the sum is one over Lorentzian discrete geometries rather than Euclidean ones. We may therefore define the HH wave function in causal set theory as (2) where Ω N is the space of N element "no-boundary" causal sets i.e., with N i ≡ |A i | = 1, and S(C) is an action. Keeping N fixed corresponds in the continuum to keeping the spacetime volume V fixed as one does in unimodular gravity. N thus acts as a proxy "time" label, which needs interpretation in a fully covariant setting. We will attend to this question at the end of this paper. When β = 0, Ψ When there is no restriction on N i or N f this distribution is dominated in the asymptotic limit by the Kleitman-Rothschild posets [10] and it is an interesting question how a restriction on N f affects this result.
We now examine the above proposal in detail by restricting to 2d CST which has proven to be a non-trivial testing ground for the full theory [9,11]. In 2d CST Ω N is restricted to the set Ω 2d of N -element 2d orders defined as follows. Figure 1: An illustration of a 2d order in Ω N with N i = 1 and arbitrary N f 2d order can be embedded into 2d Minkowski spacetime using lightcone coordinates, not every 2d order admits a continuum approximation as defined in CST [2,3,4]. An important example of one that does is the 2d random order where the elements of U and V are chosen at random and independently from S. A typical 2d order can be shown to be approximated by a region in 2d Minkowski spacetime [11]. A natural choice of action for the 2d order is the discrete analog of the 2d Einstein-Hilbert action for a causal set C [12,13] where N i is the number of (i−1)-element intervals [12] 1 , = lp l is a measure of the non-locality scale l > l p and f (i + 1, [14,12]. The uniform distribution over Ω 2d without restrictions on N f is dominated by random 2d orders which are approximated by 2d Minkowski spacetime [11] and we will see in what follows that fixing N f changes the associated partition function as a function of N f . It is in fact possible to calculate Ψ 0 (N f , β) explicitly (up to normalisation) for large N f and we show now how to do this for N f = N − 1, N − 2. Let e 0 ∈ A i be the initial element to the past of all other elements (see Figure 1). We will refer to elements not in e 0 ∪ A f as "bulk elements". For N f = N − 1 the sum consists of a single term since there are no bulk elements. Here, the only non-vanishing i-element interval is where R = 2 N (1 − 2 ) + 4 2 (we suppress the β, dependence in Ψ 0 for ease of notation). Note that any relabelling of the elements of A f is an automorphism. For N f = N − 2 there is a single bulk element e 1 . Let F be the set of its future elements in A f , with l ≡ |F |. l can therefore vary from 1 to N f . For a given l, N 1 = N − 1 while N 2 = l and N i = 0 otherwise. In addition there is a multiplicity µ associated to every l corresponding to the distinct labelled 2d orders. It is calculated as follows. For every using the ratio of the probabilities shows an interesting dependence on the parameters , β as shown in Figure 2.
As is already evident, analytic calculations for smaller N f values rapidly become more complicated and eventually run into the problem of the enumeration of finite element causal sets, an open question in combinatorics. Thus the only feasible method of evaluating Ψ 0 (N f ) for large enough N and arbitrary choice of N f is via a numerical computation. In [9] the expectation values of various observables for 2d causal set quantum gravity were obtained using Markov Chain Monte Carlo(MCMC) methods for the unrestricted partition function, i.e., with no constraints on N f . The Markov Chain move is: (i) pick a pair (u i , u j ) or (v i , v j ) at random i = j and (ii) exchange them. This move satisfies detailed balance. As β changes, the system undergoes a phase transition from a continuum phase at high temperatures to a crystalline phase at low temperatures. In this work we use these same methods to first evaluate S 2d β (N f ). As we describe in detail below, the phase transition shows a nontrivial dependence on N f which one could not have guessed from the unrestricted partition function. Namely, there is a turning point value of N f at which the inverse temperature of the phase transition β c takes on a minimum value while for some large enough N f the transition disappears altogether.
In order to obtain where Z β (N f ) is the 2d partition function for a given N f , we have to (a) numerically evaluate the integral in the exponential and (b) evaluate Z 0 (N f ). While the numerical integration can be performed using relatively straightforward best fit interpolating functions, calculating Z 0 (N f ) is more tricky. We adopt the following route. The partition function Z 0 (free) without any restrictions on N f can be expressed as a sum over the restricted partition functions Since each Z 0 (N f ) is itself a uniform measure, counting the contribution to Z 0 (free) from 2d orders which have N f maximal elements is sufficient. Moreover, since we know that Z 0 (free) is dominated by 2d random orders, this is equivalent to generating an ensemble of 2d random orders and finding the frequency of those with a given N f . An alternative but equivalent route is to perform the MCMC simulation at β = 0 and again obtain the frequency for the causal sets with a given N f . This gives us Ψ To implement this proposal we have run extensive MCMC simulations for an ensemble of 50 element 2d orders for = 0.12, 0.5, 1.0 and over several values of β. In order to compare with the best data in [9] we focused our attention on the = 0.12 trials with the largest number of β values, while there were fewer β values for the = 0.5, 1 trials. In all three cases, the qualitative features observed remain the same. However, since thermalisation problems set in at different values of β for different , we have we have more extensive data in a sense for = 0.5, 1. Our conclusions will take all this data into account.
Defining a sweep to be N (N − 1)/2 moves we perform 10, 000 sweeps, with each sweep having 1225 moves, and let N f = 1, . . . 47, β = 0.1, . . . 8. For N f = 48, 49 we use analytic results. Thermalisation typically occurs very quickly, an example of which is given in Figure 3 until around N f = 46. The thermalisation is poorer for N f = 47, but gives results consistent with the other values of N f .
The N f dependence of the phase transition is shown in Figure 4 and bears strong similarities to the phase behaviour for the unrestricted boundary [9], but with important differences. As expected, for small values of N f ∼ 1, the similarity to [9] is strong and one has the continuum phase for small β which transitions to a crystalline phase for large β. As N f increases, while this feature remains, the critical point β c first begins to decrease achieving a minimum value β min c around N f ∼ 30 after which it begins to increase again. For N f > 40, the nature of the transition changes since a reduced bulk makes the two phases less distinguishable. The rich phase structure exhibited in this Figure contains in it the essence of what we will eke out with more care below. In Figure 5 we show the best fit interpolating functions we have obtained using Mathematica. We begin by approximating S N (N f , ζ) by an integrable function. A reasonable guess for S N (N f , ζ) shown in Figure 5 with error bars for different N f is S 2d (β) = (a 1 + a 2 β) tanh (a 3 β + a 4 ) + a 5 + a 6 β , which we then fit to S N (N f , ζ) using Mathematica to determine best estimates for all six a i using the inverse square of the measurement error on S N (N f , ζ) as weights. Figure 5 also shows the errors   in these fits which come from the approximating function. Due to the lower quality of the data for = 0.5 this guess does not work well, and instead we use the interpolation function Step [x] is an approximation of the Heaviside step function (see Figure 8).
Next, we estimate Z 0 (N f ) using both numerical simulations as well as our analytic results. Simulating 1.138 · 10 10 2d random orders we find a non-trivial frequency profile for N f up to 19. In this sampling we find only 3 2d orders with N f = 19 and hence for larger N f values the frequency becomes numerically insignificant. Using a log plot and the values of Z 0 (N f ) for N f = 49, 48 from our analytic calculations, however, we can again use a best fit estimate to find Z 0 (N f ) as shown in Figure 6.
Putting these two pieces together we see an interesting struggle between the "entropic" component coming from Z 0 (N f ) and the action component played out in the HH wave function in Eqn (6). Figure 7 shows |Ψ 0 (N f )| 2 at different β values for = 0.12, 0.5. For small β, |Ψ 0 (N f )| 2 is dominated by the entropic contribution and is peaked around N f ∼ 4 at β = 0 and then shifts downward slightly as β increases. For larger β however, |Ψ 0 (N f )| 2 develops a second peak at large N f ∼ 23 which gets more and more pronounced. Unfortunately the thermalisation properties of the data begin to deteriorate beyond β ∼ 8 thus making it less reliable, though it does appear that the second peak continues to grow. This shifting of peaks also occurs for = 0.5 and = 1; for these the second peak in fact begins to dominate the first as one goes to larger β as shown in the second figure of Figure  ??. Hence it appears that as β goes well past β min c the second peak dominates the first.
An error estimate on these results can be obtained from the errors in the interpolating functions used for S 2d and Z 0 (N f ). |Ψ 0 (N f )| 2 with the error is shown in Figure 9 for   in our approximating function for S N (N f , ζ) . The green region is the difference between the lower limit of S N (N f , ζ) and the mean while the blue region marks the difference between the upper limit of S N (N f , ζ) and the mean. We can see that the second peak appears considerably earlier for the lower limit and considerably later for the upper limit. For = 0.12, the error appears to wipe out the second peak. This is due to the fact that the error becomes much larger around the phase transition but lessens as β increases, which in turn is because the lower limit on S N (N f , ζ) starts the phase transition earlier, while the upper limit starts later. For = 0.12, the appearance of the second peak around the phase transition already coincides with the thermalisation limit and one cannot proceed to larger β. On the other hand for = 0.5, the thermalisation limit occurs at a much larger β compared to the appearance of the second peak, and hence we see clearly that the errors become small enough to see the dominance of the second peak. We have obtained similar plots to examine the confidence region of our estimator for Z 0 (N f ), but found the error to be subleading compared to the uncertainty on the approximation of the average action.
While the first peak geometries clearly belong to the continuum phase, i.e., the 2d random order, the nature of the second peak geometry is of particular interest, since these geometries are truly discrete, having no continuum counterparts. Figure 10 shows a 2d order generated at the end of the MCMC trial for N f = 23, β = 7.6. This 2d order represents a random sample from the thermalised ensemble of geometries belonging to the second peak. It shows the distinctive layered structure characteristic of the crystalline phase of [9] and is clearly non-manifold-like.
After having isolated the existence of these second-peak geometries, we performed more extensive MCMC simulations for a range of observables around the first and second peak β and N f values in order to assess the nature of these geometries. These observables include the proper-time or height of the 2d order, the distribution of the i-element intervals N i as well as the ordering fraction which is the ratio of the number of actual relations in the causal set to the number of possible relations N 2 . The first peak geometries around N f ∼ 4 are predictably those which are approximated by 2d Minkowski spacetime, while the secondpeak geometries share the crystalline or layered structure noticed in [9]. These second-peak geometries are clearly non-manifold-like, but nevertheless, as we show below they possess characteristics of relevance to the observable universe. In particular, let us examine the ratio of N f to the height of the poset shown in Figure 11 for N f = 22, 23, 24, i.e. the 2d orders around the second peak at N f = 23. This ratio shows a fairly rapid expansion from a single initial element A i to a final A f with a large number of elements N f = 23. Indeed as shown in Figure 10 most of the elements in A f are just 3 time steps away from the initial element and so despite being spatially large with N f = 23, the universe is still very young. Moreover, we can quantify the homogeneity of A f by looking at the distribution of the cardinality of the past volumes of the elements in A f . Figure 12 shows this distribution for N f = 23 which is the location of the second-peak. Most elements in A f thus have 27 elements to their pasts with the distribution falling off sharply. This shows that A f has a strong degree of homogeneity. Additional tests for homogeneity are of course possible, but they are difficult to interpret for such small values of N . We conclude that not only do non-manifold contributions to the path integral play an important role in determining initial conditions but they possess certain features consistent with our observable universe, despite being non-manifold-like. This lends support to the idea that one needs quantum gravity to explain aspects of the observable universe. The most striking example of this of course is the prediction from CST of the value of the cosmological constant [15,16,17]. While the calculations presented here are far more limited by both N and the spacetime dimension, these are promising features to look for in full CST.
The role played by β in our analysis is non-trivial, but conceptually opaque. In full 2d quantum gravity β is a Wick rotation parameter with β → −iβ being the physically relevant quantum regime. Calculations with the Euclidean measure are assumed to analytically continue to this quantum regime in a manner similar to quantum field theory. In contrast, since the HH wavefunction is defined as a Euclidean path integral there is no essential need for a β different from 1 -all it provides is an overall scaling of the action. While in higher dimensions β can be absorbed into rescaling of l p this is not the case in 2d quantum gravity because of the absence of a fundamental scale. However, our analysis clearly demonstrates that β plays a physical role -tuning β shifts the peak contributions from manifold-like causal sets to non-manifold-like causal sets. Recent work on the scaling properties of 2d CST shows that both β and play a significant role in the large N behaviour of the theory, and it is plausible that an understanding may lie in this direction [18].
Additionally, the boundary term plays a crucial role in continuum formulations of quantum gravity. It would be interesting to see how our results are affected by the recent proposal for a discrete Gibbons Hawking term [19].
We now return to the question of whether there is a fully covariant interpretation of Ψ 0 (N f ). In the continuum the HH proposal gives the amplitude for a spatial initial condition with the assumption that subsequent evolution is via Hamiltonian dynamics. In a fully path integral framework without ad-hoc restrictions on topology, it is possible that initially disconnected topologies could merge, thus rendering inconsequential the intended import of an initial condition. We address this question within the CST framework for quantum gravity.
To begin with, we note that restricting our attention to finite, fixed cardinality N of the causal sets leads to a CST version of the "problem of time", since such causal sets come with a particular time-labelling. While this labelling can enter the measure, as it does in the 2d quantum gravity case we have examined above, the observables of the theory must be label invariant. Following [20], the first step is to embed the finite sample space Ω N into the space Ω of infinite element past finite causal sets as in the sequential growth models of [21]. In particular, in keeping with the initial boundary condition that N i = 1 we can restrict Ω to causal sets which are not only past finite, but originary i.e., with a single initial element.
Within such a set up, a CST dynamics is given by the measure space (Ω, A, µ), where A is the event algebra whose elements are the observables, and µ is a measure, either classical or quantum [22]. The observables are not in general label invariant but can be made covariant by quotienting the event algebra by relabellings. Alternatively, one can consider the (sigma) event algebra R over the set of unlabelled causal sets Ω cov as in [20]. Define a stem σ ⊂ C ∈ Ω cov as a past-set i.e., P ast(σ) = σ and a stem event α σ = {C ∈ Ω cov |σ is a stem in C}. The set of the stem events generate the stem sigma algebra R. Note that a stem event is also an event in the set of labelled causal sets Ω but remains covariant even so.
Next, we define the Hartle-Hawking event α N HH (A f ) as the set of originary causal sets in Ω cov for which A f is an inextendable antichain, with |P ast(A f )| = N −N f . In particular, this means that A f is dividing, i.e., every element in such a causal set lies either in A f , P ast(A f ) or F ut(A f ). We now show that α N HH (A f ) belongs to R and is therefore a covariant event. We show this using arguments similar to those in the Proposition in [5].
We begin with a finite causal set c N ∈ Ω N with A f ⊂ c N is its set of future most elements. Let Finally, consider the set S N of all possible c N ⊂ Ω N with A f is an inextendable antichain and |P ast(A f )| = N − N f . The HH event is then the set α N HH (A f ) = c N ∈S N Φ c N ∈ R. It is clear from this construction that A f is indeed an inextendable, dividing antichain in any infinite element causal set in this set, with |P ast(A f )| = N − N f , thus ensuring that no disjoint universe can "join up" at a coordinate time greater than N .
In order to complete the story, we need a quantum measure on the Hartle-Hawking observable which we take to be given by our prescription Eqn (2). Needless to say, in order to do this consistently one would begin with a measure on the stem sets and extend this to R, but this is beyond the scope of the current work.