The quantum cosmological constant

We present an extension of general relativity in which the cosmological constant becomes dynamical and turns out to be conjugate to the Chern-Simons invariant of the Ashtekar connection on a spatial slicing. The latter has been proposed in \cite{Chopin-Lee} as a time variable for quantum gravity: the Chern-Simons time. In the quantum theory the inverse cosmological constant and Chern-Simons time will then become conjugate operators. The"Kodama state"gets a new interpretation as a family of transition functions. These results imply an uncertainty relation between $\Lambda$ and Chern-Simons time; the consequences of which will be discussed elsewhere.


Introduction
The long standing cosmological constant problem comes in many guises. Before the observation of cosmic acceleration, the search had been to seek a theory where all contributions to the cosmological constant summed up to zero. If the data on acceleration continues to be consistent with a plain cosmological constant, rather than a more general form of dark energy, then the new cosmological constant problem becomes a question of extreme fine tuning. In this work we take an agnostic position on this matter and are instead motivated by insights from non-perturbative quantum gravity.
In quantum mechanics, while perturbation theory is a powerful tool in unveiling numerous physical observables, it can often times obscure or be blind to important nonperturbative information. For example, quantum tunnelling is a non-perturbative process and any perturbative expansion in the vicinity of the tunnelling barrier would be blind to tunnelling. It is in this context we would like to address the issue of both the IR and UV versions of the cosmological constant problems 1 . In both cases, the evaluation of the vacuum energy arises from choosing a classical space time background and calculating the relevant perturbative vacuum diagrams, which are divergent and, after imposing regurlarization, deemed to be highly fine tuned.
Given the difficulty of the problem, perhaps a new approach is needed. We explore the idea that the cosmological constant is a dynamical variable, not in the sense of a dynamical field, as in quintessence models, but as a degree of freedom for the entire spatial or spacetime manifold. Even though similar ideas have been explored before [2,4,5], the novelty of this paper is that it examines the issue from the unique vantage point of the Ashtekar variables [6] and its associated quantization [7]. In this opening article we focus on the classical formulation of the theory. This will give us guidance for turning the cosmological constant into an operator in the quantum theory, which we develop in later papers in this series.
In this work it is convenient to formulate general relativity by gauging the complexified Lorentz group, SL(2, C) C on a four dimensional manifold M. The space-time connection A ab = −A ba is a one form, valued in sl(2, C) C the Lie algebra of SL(2, C) C . This Lie algebra is represented by complex antisymmetric, 4 × 4 matrices, M ab = −M ba , where a, b, c = 0, 1, 2, 3 are internal Lorentz indices. The resulting gravitational dynamics is determined by a connection A ab , a two form Σ ab , valued in the Lie algebra of SL(2, C) and a scalar field which provides a map Ψ : sl(2, C) C → sl(2, C) C , which is written as Ψ abcd with the following symmetries and constraints, It is then convenient to change to two component spinor indices where A, B = 0, 1 are left handed spinor indices while A ′ , B ′ = 0 ′ , 1 ′ are right handed spinor indices. The connection then decomposes into and the two form Σ ab similarly decompose. Very importantly, the curvature two form decomposes the same way where is the left handed part of the curvature tensor 2 The scalar fields Ψ abcd decompose into pure spin two fields represented by Ψ ABCD and Ψ A ′ B ′ C ′ D ′ , both totally symmetric, and mixed components Ψ ABA ′ B ′ on symmetric pairs of indices. Thus, and the same for primed indices represents the spin two field. In our work we formulate an extension of general relativity in which the cosmological constant, Λ, varies 3 , and is determined by the solution of an equation of motion. The theory has contributions proportional to both Λ and 1 Λ so the process that determines its value is non-perturbative in Λ. The term in Λ is of course just the spacetime volume: 2 In much of the literature on LQG, R AB is denoted F AB . 3 Prior works in which Λ varies have included [8,2,28,29,5].
where Σ AB = e A ′ A ∧ e B A ′ is the self-dual two form of the spacetime metric, e B A ′ . We propose a novel coupling of 1 Λ to the topological invariant M R AB ∧ R AB , where R AB is the left handed part of the curvature tensor. In the Euclidean theory which, for simplicity, we will be studying in this paper, this reads This is motivated by a key property of deSitter spacetime, which is that it is a self-dual solution, in the sense that Thus, a formal duality transformation that has deSitter spacetime as a fixed point, must have symmetry: and therefore it must takes the usual cosmological constant term, (6), to its dual, which is S new , given by Eq. (7). Right at the start we find an interesting quantum implication. A reflection of this duality is that the solution to the Hamilton-Jacobi equation that corresponds to deSitter spacetime is [12] where Y CS (A) is the Chern-Simons invariant of the Ashtekar connection. This is of course closely related to S new , as we will discuss below. It leads to to a semiclassical state, which is called the Kodama state [9]. Remarkably in some regularizations and ordering prescriptions this is an exact solution to all the constraints of quantum gravity [12]. There have, however, been issues concerning the physical adequacy and interpretation of the Kodama state [15]. Below we propose a new interpretation for this state, stemming from our proposal. The form of the new term (7), and particularly that it is CP odd, suggests an anology between 3 Λ and the theta angle in QCD [10]. Relating the cosmological constant problem to the theta vacuum was considered in the works of [11,13,14]. This further suggests treating Λ as a dynamical degree of freedom. We explore three versions of this idea, in which Λ is chosen to be a dynamical field, or a function of time in a preferred 3 + 1 slicing, or a single variable for all spacetime 4 . We discover in each case that the Λ equation of motion determines its value.
Finally, in one case for the realization of this theory, we show how the Hamiltonian formalism might be set up and lead to the canonical quantization of the theory. Even ignoring details of the dynamics at the classical level, we can see that a quantum uncertainty principle would always be in action, rendering Lambda and Chern-Simons time [1], complementary variables. This may possibly leads to deep implications for quantum cosmology and quantum gravity, which we outline in the concluding Section, and take up again elsewhere.
Before beginning, we note that the idea that the cosmological constant is conjugate to a measure of time has appeared before, in the context of unimodular gravity [28,29]. There the conjugate measure of time is four volume to the past of a three slice.

A Plebanski formulation for our poposal
We work first in the Euclidean case. We fix a topology M = I × Σ, where I is the interval and we take Σ compact. We start with an action for general relativity coupled to N chiral fermion fields, all expressed in terms 5 of Ashtekar variables [18,19], but with the addition of a new, third term, which will suffice to make Λ dynamical, as we shall presently see. We divide the action by the Planck's constant for reasons to become apparent below. We then rewrite this action, with its new term, in the Plebanski formulation [20,21,16]: The new term can be rewritten as: 5 Where P abcd + is the projection operator onto self-dual two forms.
We note that if we exponentiate this action (divided by ), the first and second terms gives a generalization of the Kodama state on the initial and final slice. The last term can be written as: and vanishes if Lambda is forced to be a constant. The field equations for our theory, in the absence of matter, are: with The solution to (17) is that there exists a frame field e AA ′ such that, is the self-dual two form of the metric made from e AA ′ . We also note that if is the torsion, then A new feature is then a contribution to the torsion (20) related to the derivative of the cosmological constant.

The underlying duality
We can see that the terms that involve Λ: have an interesting structure: they have a formal duality symmetry under: The self-dual solutions, including deSitter spacetime, are the self-dual points: where Φ ABCD is a Lagrange multiplier. We say this symmetry is formal because on shell Σ AB satisfies (17), which is typically not satisfied by R AB .
Thus, we can extend the theory to one which has (25) as an exact symmetry: Instead of (21) this says that there is a frame field such that When we write the action and field equations in terms of this new e AA ′ we find this yields again an action for the Einstein equations.

Three cases for the realization of the theory
The duality (25) results in the determination of Λ as a function of the other fields. To see this we study the field equations for varying Λ. There are three cases, depending on what we choose Λ to be a function of.
• Case I: Λ(x µ ) is a function of space and time.
Varying by Λ(x µ ) we find an equation for Λ(x µ ): Plugging this back into the action, we find This gives an interesting set of equations, the question is whether they are consistent. Further study of this case is left to a future publication.
• Case II: Λ is a function of time on some preferred 3 + 1 slicing.
We fix an explicit slicing such as constant mean curvature slicing. This gives a time coordinate t. We also define Chern-Simons time by an integral over this slicing, leading to τ CS (t). We fix Λ to be a function of the slicing.
Varying by Λ(t) we find an equation for Λ(t): Plugging this back into the action, we find This is similar to the theory described in [5], only rather than being conjugate to Newton's constant, G, it appears the cosmological constant is conjugate to the Chern-Simons time τ CS in the preferred slicing.

The equation of motion (18) becomes non-local
This theory is also under investigation.
We can check the homogeneous solutions, with 6 and the torsion (20) It is an important open question whether there are non-trivial solutions where Λ(t) varies, with matter or non-zero Weyl tensor, which are not equivalent to deSitter spacetime.
• Case III: Λ is one variable over all of spacetime [2].
Varying by Λ we find an equation for Λ: 6 Note that the covariant curl of (18) vanishes because the torsion is (20).
Plugging this back into the action, we find This is a version of the Kalapor-Padilla theory [2] We can write out all the components of R AB as where Φ AA ′ BB ′ and R are, respectively, the trace-free part of the Ricci tensor and its trace. Then, the Poyntriagin density is It is interesting to note that in all cases the cosmological constant is proportional to the square root Pontryagin density. It was shown that in metric variables the Pontryagin density vanishes in most cosmological space-times. However in approximate de-Sitter spacetimes, while the background Pontryagin density is zero, backreaceted chiral gravitational waves are sourced by a dynamical field that is coupled to the density and this will yield a non-vanishing Pontryagin density [23,24]. It is interesting to expect that such a mechanism could generate a small cosmological constant today and we leave this for a future work [25].
For the remainder of this paper we study Case II.

The basis for a Hamiltonian treatment
Let us now apply a 3 + 1 decomposition to the action, which yields: As explained above, we assume that Λ is a function of time, t, alone. Then, we have, as usual, the canonical brackets: In addition Λ has a momentum P such that, and there is a new primary constraint We add W to the Hamiltonian with a new lagrange multiplier, φ.
In a future paper we will study further the algebra and other properties of this system of constraints. For the purpose of this paper it is sufficient to note that Lambda, or a function thereof, appears to be conjugate to a function of the Chern-Simons time [1] once the new primary constraint is taken into account. This suggests at once a quantum theory containing a Heisenberg uncertainty principle involve Lambda and CS time, something we start to explore here.

Towards a quantum theory
Given the classical canonical structures unveiled in Section 5 we can now lay down the basis of the quantum theory, resulting in a new interpretation for the Kodama state. Combining (42) and (43) we can infer the Poisson bracket: Obtaining this classical structure was the ultimate goal of this first paper. It suggests that in a quantum theory we can elevate Λ and τ CS = Σ Y CS (A) to operators with commutation relations, We note that the commutator of Λ is proportional to Λ 2 , so the larger Λ is in Planck units the less classical it is. Specifically, using purely kinematical arguments, we can derive an uncertainty principle of the form: If the expectation of Lambda is large in Planck units, then Lambda and CS time are complementary or incompatible variables. If it is not, as seems to be the case in the "current" Universe, they can be treated as classical variables. In our theory, the onset of classicality in cosmology is therefore related to the observed smallness of Lambda.
Note that since [q, p] = i implies [f (q), p] = if ′ (q), we can re-express the commutator (46) in the more canonical form: It is then natural to consider representations that diagonalize either1 Λ orτ CS , i.e. one of the two complementary variables. In the CS time representation we have: The Kodama state then appears as an eigenstate of1 Λ in the CS time basis: More generally, the Kodama state can be seen as a transition amplitude between eigenstates ofτ CS and those of1 Λ : Within a variable Lambda theory the Kodama state therefore receives a new interpretation as a transition amplitude. We further note that it satisfies a new operator self-dual equation with a quantum Λ operator:

Outlook
In this paper we have introduced the term S new , given by Eq. (7), aiming at introducing a variable or dynamical Λ, and we have just begun a study of the implications for quantum gravity and cosmology. At the level of the classical theory, we have left open important questions. To begin with we will want to extend these results to the Lorentzian theory, in which case the Chern-Simons time becomes the imaginary part of the Chern-Simons invariant of the Ashtekar connection [1].
In each of the three cases we have considered in Section 4 we need to establish whether or not the field equations force Λ to be constant. If the field equations allow, we will want to study classical solutions where Λ varies classically.
We note that matter couplings may play a key role, as they may introduce terms in the torsion that compensate those due to derivatives of Λ, given by eq. (20). One reason to expect this will be noted below.
But even if Λ is constrained to be constant at the classical level, there may be allowed transitions in the quantum theory in which the value of Λ changes. These would be a new kind of tunnelling, which may be important in the early universe. Naively, the amplitude for such a transition would be proportional to A quantum uncertainty principle between Λ and a measure of time could also have deep cosmological consequences. These will be the subject of separate papers [27]. Further novelties in the the quantum theory would result from a possible chiral gravitational anomaly. Let us write the quantum partition function as We conjecture that under the integral DΨDΨ for a set of chiral fermions, there will be a chiral anomaly, where J is the dual of the chiral current, J abc = ǫ abcdJ d , wherẽ Note that in ordinary, perturbative quantum gravity, there is a chiral anomaly of the form of (57) so it is natural to conjecture that the anomaly appears here as well. On the basis of this conjecture, we can write the action as Therefore, in our theory, we can write the CS phase as: We then have a torsion added to the connection This modifies the curvature tensor and the Einstein Equation. These aspects of the quantum theory will be explored in companion papers.
To conclude, we have laid down the basis for a new theory of a "quantum cosmological constant" that could address the nagging problems Λ leads to in cosmology and quantum gravity. The glimpses obtained already shed new light on outstanding issues, such as how to interpret the Kodama state in Quantum Gravity. Finally, as we will argue elsewhere, the quantum theory also hints at how a non-perturbative approach might resolve the problem of the smallness of Λ in our Universe.