Quantum of the bare cosmological constant

We show that there exist scalar field theories with well-defined one-particle states in general D dimensional nonstationary curved spacetimes whose propagating modes are localized on d ≤ D dimensional hypersurfaces, and the corresponding stress tensor resembles the bare cosmological constant λB in the D dimensional bulk. We show that nontrivial d = 1 dimensional solutions correspond to λB < 0. Considering free scalar theories we find that for d = 2 the symmetry of the parameter space of classical solutions corresponding to λB 6= 0 is O(1, 1) which enhances to Z2×Diff(R 1) at λB = 0. For d > 2 we obtain O(d−1, 1), O(d−1)×Diff(R 1) and O(d−1, 1)×O(d−2)×Diff (R1) corresponding to, respectively, λB < 0, λB = 0 and λB > 0.


Introduction
In this paper we propose novel scalar field theories, the scalar creepers, in general nonstationary curved spacetime whose stress tensor resemble a bare cosmological constant and the corresponding one-particle states are localized on lower dimensional hypersurfaces.
The particle interpretation of states of ordinary quantum field theory (QFT) in Minkowski spacetime can be extended to QFT in stationary curved spacetime. However, there is a lore that a physically meaningful notion of particles do not exist for QFT in a general nonstationary curved spacetime except in an approximate or asymptotic sense. So, some authors argue that the particle interpretation of states should not be considered as an essential feature of QFT [1]. The scalar creepers do not describe ordinary matter field but they have well-defined one particle states in general.
Another question in QFT is the existence of matter fields whose propagating modes are localized on d < D dimensional hypersurfaces of the D-dimensional spacetime [2,3]. We show that the propagator of the one-particle states of the scalar creepers are delta-function localized on d dimensional hypersurfaces. In general d ≤ d * , where d * is the number of linearly independent nowhere-zero vector fields in the spacetime which can be computed in the homotopy theory [4].
The scalar creepers are not ordinary matter fields. Their stress tensor resembles a bare cosmological constant, i.e., they all act like perfect fluid with equation of state w = −1. An unsettled issue in QFT in curved spacetime is the cosmological constant problem [5]. Considering a general four dimensional nonstationary curved spacetime (M 4 , g) 1 local Lorentz symmetry implies that the expectation value of the quantum vacuum stress-energy tensor of ordinary fields T µν = − ρ g µν where ρ ∼ Λ 4 is the vacuum energy density and Λ is the high energy cutoff of the ordinary QFT. So T µν adds 8πG ρ ∼ M −2 Pl Λ 4 to the effective cosmological constant whose observed value is λ eff ∼ 10 −122 M 2 Pl . Since λ eff = λ B + 8πG ρ this requires an incredible fine-tuning of λ B . Recently Wang and Unruh have shown that the cosmological constant problem can be resolved if fluctuations of ρ are taken into account and λ B has taken a large negative value −λ B ≫ Λ 2 [6,7]. We show the stress tensor of d = 1 dimensional creepers resembles λ B < 0, and in the simplest d ≥ 2 dimensional models we find that the symmetry of the parameter space of classical solutions corresponding to λ B < 0 is O(d −1, 1) which enhances to O(d −1) ×Diff(R 1 ) at λ B = 0, indicating a phase transition.
In summary, the scalar creepers have the following properties: 1. Similarly to ordinary scalars, they are natural extensions of scalars in Minkowski spacetime to curved spacetime. Their actions are diffeomorphism invariant.
2. They have a well-defined notion of one-particle states in nonstationary curved spacetimes, localized to d ≤ D dimensional hypersurfaces without using warp factors or potential wells, hence the moniker.
3. Their stress tensor resembles a bare cosmological constant, i.e., they all act like perfect fluid with equation of state w = −1. So they do not describe ordinary matter field.
In section 2, we define scalar creepers as scalar fields whose quanta are confined to d ≤ D dimensional flat hypersurfaces of (M D , η) and examine their existence in (M D , g) in section 3. We discuss their application to cosmology in section 4 and recapitulate the main results in section 5.

Scalar creepers in Minkowski spacetime
In this section we define the scalar creepers in (M D , η) (defined in footnote 1) by giving their classical action. For quantization, we compute the propagator of non-interacting creepers by path integral and show that the propagating modes are localized on d ≤ D dimensional subspaces.
The action of d dimensional scalar creepers in (M D , η) is obtained by removing (D − d) terms corresponding to directions perpendicular to a d dimensional hypersurface from the kinetic term of the action of ordinary scalar field theory. It is given by where m 2 is a constant, and V int (φ) gives self-interaction. Although we have dropped kinetic terms corresponding to directions perpendicular to the hypersurface, we are still considering the scalar field as a function of all spacetime coordinates, i.e., where x a := x a for a = 0, · · · , d − 1, and x a ⊥ := x a for a = d, · · · , D − 1. Henceforth we study the non-interacting fields and set V int (φ) = 0. The classical field equation is given by δS/δφ = 0, in which, and For d = D, Eq.(1) is the ordinary scalar field theory in Minkowski spacetime. For d < D, the classical field equation (d) − m 2 φ = 0 is not deterministic if not meaningless altogether because it is silent about the behaviour of the classical field in directions x ⊥ perpendicular to the hypersurface. But the classical fields do not participate in particle physics. The particle interpretation of physical states comes from quantum fields whose correlation function is given by the path integral [8], where Z := Dφ e iS is the partition function. Eq.(4) implies that Since whose solution is where D (d) Eq. (9) shows that the path-integral and the correlation function (6) are well-defined although the action (1) is not classically viable since it does not include kinetic terms corresponding to directions normal to the d dimensional hypersurface. So the action (1) together with the path-integral (6) describe scalar fields whose one-particle states are delta-function localized on a d dimensional subspace.

Scalar creepers in general spacetimes
In this section we define scalar creepers in general spacetimes and show that, in addition to their ability to stick to lower dimensional hypersurfaces, they hold a well-defined notion of vacuum state and one-particle states in general nonstationary curved spacetimes, in contrast to ordinary scalar field theories [1]. Thus, before discussing the scalar creepers we review the ordinary scalar field theory briefly in order to identify the main obstacles in computing their propagator via path integral in general spacetimes.
To see the problem with ordinary scalar field theory, consider the minimally coupled massless scalar field theory in (M D , g) (defined in footnote 1) whose action is given by where e := |det g| and ∂ µ := ∂ ∂y µ . By introducing the tetrad e µ a satisfying η ab e µ a e ν b = g µν and the vector fields e a := e µ a ∂ µ , the action (10) can be written as where e a φ := e µ a ∂ µ φ is diffeomorphism invariant. Although the action (11) is similar to Eq.(1) we cannot, in general, compute the corresponding path integral and obtain the correlation function explicitly. The difficulty can be seen from the expression where ∇ µ denotes the Levi-Civita connection, and we have used the identity Comparing Eq.(12) with equations (4) and (5) reveals the roots of the difficulty: the vector fields e a are not necessarily divergence free and they do not commute with each other in general.
In order to define the creepers in general nonstationary curved spacetimes we need to replace the tetrad e a in Eq.(11) with a set of nowhere-zero vector fields v a := v a µ ∂ µ such that and In subsection 3.1 we show that the vector fields v a exist locally though they do not exist globally. We introduce the scalar creepers in subsection 3.2 and define their one-particle states in subsection 3.3.

Geometry
A straightforward approach to obtain the vector fields v a satisfying equations (14) and (15) is to work with coordinate systems x µ used in unimodular gravity [9,10] in which e = 1. In these coordinates v a µ = δ a µ where δ denotes the Kronecker delta, i.e., To confirm this proposition we only need to use the identities Henceforth by x-coordinates we mean a coordinate system in which e = 1. In order to find such coordinates explicitly we rearrange the coordinates y µ as (y 0 , y) and suppose that x := y and which implies that the determinant of the metric in the x-coordinates equals -1. Furthermore ∂ ∂x 0 is timelike as long as ∂ 0 is timelike because As an example consider the four dimensional Schwarzschild spacetime in Kruskal-Szekeres coordinates where dΩ 2 := dc 2 1−c 2 + (1 − c 2 )dϕ 2 , c := cos ϑ, 2 and r = 1 + W , in which W := W (−uve −1 ) denotes the Lambert W function. Thus, uv = e r (1 − r). Eq.(19) (with u playing the role of y 0 ) reads where we have supposed that y 0 ref ( y) = 0, i.e., the reference point is located on the event horizon r = 1. W (−uve −1 ) is real-valued for uv ≤ 1 or equivalently for r ≥ 0.

Closed FRW universe
So far we have shown that in a D dimensional spacetime there always exist D vector fields satisfying equations (14) and (15) locally. For defining creepers we need to know how many of them exist globally. The existence of nowhere-zero vector fields is a question in homotopy theory. Take n dimensional spheres S n for instance. We know that for n = 1, 3 there exist exactly n nowhere-zero vector fields while there is no such vector fields on S 2 [4]. In this subsection we show that in a D = 4 dimensional FRW universe with closed spatial sections only two nowhere-zero vector fields satisfying equations (14) and (15) coexist globally.
Suppose that where Σ, whose line element ds 2 Σ is t-independent, is diffeomorphic to a three sphere S 3 [11]. The x 0 coordinate is given by Eq.(19) and The vector field v 0 exists and its divergence is zero except for the beginning and the end when ω(t) = 0. Now we focus on the space section Σ, which we model by S 3 , embedded in (R 4 , δ). S 3 is parallelizable, i.e., there exist three independent vector fields on S 3 , The divergence of v i is zero but Thus, in an FRW universe with closed spatial sections there exist at most two vector fields, e.g., v 0 and v 1 satisfying equations (14) and (15) simultaneously. In the coordinate system (θ, φ 1 , φ 2 ) given by where I = 1, 2, and r 1 := cos θ and r 2 := sin θ, we have and

Action
In a D dimensional spacetime (M D , g) (defined in footnote 1) with d * ≤ D linearly independent nowhere-zero vector fields v a satisfying equations (14) and (15), we give the d ≤ d * dimensional creepers' action by in which the creeper φ = φ(y 0 , · · · , y D−1 ) is a scalar field, e := |det g|, and the potential V (φ) is given by Eq. (2). Following the definition of vector fields, v a φ := v a µ ∂ µ φ is a scalar, therefore the Lagrangian density L(φ; v a ) is also a scalar though it is independent of the spacetime metric g and S is diffeomorphism invariant off-shell. In the x-coordinates the action (35) is given by resembling the action (1). Using equations (15) and (18) one verifies that the classical equation of motion is given by Later in subsection 3.3 we use this result to compute the correlation function by path integral.

Stress tensor
Now we couple the scalar creepers to Einstein's gravity. For this purpose we first recall the definition of the stress tensor and the consequences of the off-shell diffeomorphism invariance in ordinary matter field theory. In a theory whose action is S = d D y eL the stress tensor is defined by A diffeomorphism is given by the map y → y ξ such that δ ξ y := y ξ − y = ξ(y). We suppose that the vector field ξ(y) is zero on the boundary and drop all the boundary terms subsequently. The corresponding variation of a scalar φ, a vector v, the one-form dφ, the tensor g and the volume element e are given by the Lie derivatives Since L is a scalar, similarly to Eq.(41) we have δ ξ L = −ξ µ ∂ µ L which together with Eq.(45) results in the off-shell diffeomorphism invariance δ ξ S = 0. For ordinary matter fields we have Using equations (41) and (42) together with the classical field equation one verifies that the second term on the right hand side of Eq.(46) vanishes. Furthermore, by using Eq.(44) and integration by parts we obtain Thus in ordinary scalar theory, the off-shell diffeomorphism invariance together with the classical field equation imply that the stress tensor is conserved on-shell, i.e., Similarly, the Einstein-Hilbert action S EH (in the metric formulation) depends on the spacetime metric only and Defining the Einstein's tensor by and using Eq.(44) and δ ξ S EH = 0, we conclude that ∇ µ G µν = 0. On the other hand, Einstein's field equation given by reads G µν = T µν . Thus, Eq.(49) shows that ordinary scalar theories whose actions are diffeomorphism invariant off-shell can be coupled to Einstein's gravity on-shell. In the case of scalar creepers, L(φ; v a ) is independent of the metric g µν and Eq.(35) together with the identity δe = 1 2 eg µν δg µν imply that the creepers' stress tensor is given by which resembles a perfect fluid with equation of state w = −1. Coupling the scalar creepers to Einstein's gravity results in the field equation which gives G µν = T µν . Consistency of this equation with the identity ∇ µ G µν = 0 requires that Therefore a consistent coupling to gravity requires In other words, the scalar creepers couple to gravity similarly to a cosmological constant term. We study this phenomenon in section 4. We should not have expected to obtain Eq.(55) as a direct consequence of the off-shell diffeomorphism invariance of S and the classical field equation (38) similarly to Eq.(49), because there exist solutions to the classical field equation for which the on-shell value of L(φ; v a ) is not constant. In fact we can not obtain Eq.(55) by following those steps that led us to Eq.(49) although L(φ; v a ) is a scalar and consequently δ ξ S = 0 off-shell. This can be verified by noting that where we have used equations (43) to obtain the last term which is new compared to Eq.(46). By integration by parts and using equations (15), (38), (39) and (41) we obtain So the third term in Eq.(57) cancels the δS/δφ term therein and by inserting Eq.(58) in Eq.(57) we obtain Thus there is no contribution from the classical field equation. After using equations (53), (44) and (41) in Eq.(59) we simply end up with δ ξ S = 0. In summary, a scalar creeper on a d ≤ D dimensional hypersurface is defined by equations (35) and (36). The creepers are perfect fluids with equation of state w = −1 and consequently they do not correspond to ordinary matter field. In order to couple the scalar creepers to gravity their Lagrangian should be constant on-shell.

Quantization
The propagator of the d dimensional scalar creepers in a (M D , g) can be computed by using the classical action in the path integral. For V int (φ) = 0 the path integral together with Eq.(38) give in spite of the fact that we have not defined the path integral, especially the temporal (Feynman) boundary conditions yet. We have only assumed that the path integral exists and the integration by parts is applicable.
To compute D F (y, y ′ ) we note that the creepers' action in the x-coordinate system given by Eq.(37) is identical to Eq.(1), the operator (d) defined in Eq.(39) is given by similarly to Eq.(5), and 1 e δ D (y−y ′ ) = δ D (x−x ′ ). Therefore Eq.(61) in the x-coordinates is equivalent to Eq. (8). Consequently the correlation function is given by Eq.(9), and the corresponding particle description is also applicable here, though the d ≤ D dimensional subspace is embedded in a nonstationary spacetime. Precisely we can consider an auxiliary action whose second quantization indicates that there exist a vacuum state |0 with respect to the "time coordinate" x 0 (cf. Eq.(20)) and a set of creation and annihilation operators a † (p) and a(p) with commutation relations such that a(p) |0 = 0 [8]. The field operator is given by where x := (x 0 , x), p := (E(p), p), E(p) := p · p + m 2 , p · x := E(p)x 0 − p · x, and the inner product We postulate, and define the path integral accordingly, that D (d) F in Eq.(9) equals the corresponding Feynman propagator given by 3 which gives the amplitude for particles with "positive frequency" to propagate from x ′ to x and from x to x ′ for x 0 > x ′ 0 and x 0 < x ′ 0 respectively [8]. Consequently, we have a notion of vacuum state, creation and annihilation operators and "time-ordering" of n-point functions with respect to the "time-coordinate" x 0 , although Eq.(20) demonstrates that the vector field v 0 is not necessarily timelike everywhere. In particular, for d = 2 and m = 0, Eq.(37) reads in which ∂ ± := 1 2 ∂ ∂x 0 ± ∂ ∂x 1 . This theory can be interpreted as a c = 1 conformal field theory [12] embedded in (M D , g).

Application to cosmology
In section 3 we defined the scalar creepers in (M D , g) and observed that they have well-defined one-particle states localized to d dimensional hypersurfaces of the bulk. In this section we couple the scalar creepers to gravity and investigate their contribution to the dark energy.
In subsection 3.2.1 we showed that the scalar creepers can be coupled to gravity consistently if This condition together with Eq.(53) implies that the creepers can be coupled to Einstein's gravity only if L(φ; v a ) is constant on-shell. In this way, T µν resembles the bare cosmological constant term λ B in the Einstein field equation suggesting that For the classical solution φ =φ = constant, such that we have Similarly to ordinary scalar field theories, such solutions are not interesting because they need a fine-tuned potential according to the requirements (72), (73) and (74) [13]. Thus, we dismiss such solutions and suppose that the creepers are not constant on-shell, though we seek classical solutions such that L is constant on-shell, corresponding to a bare cosmological constant. So we demand In the following we study d = 1 and d ≥ 1 creepers separately. In the one dimensional case we verify that the requirements (75) imply that V (φ) is necessarily zero and λ B < 0. For d ≥ 1 we confine our study to free creepers with V (φ) = 0 and identify the parameter space of classical solutions satisfying the requirements (75). We discover that for d = 2 the parameter space enjoys an O(1, 1) symmetry which enhances to Z 2 × Diff(R 1 ) at λ B = 0, while for d > 2 the symmetry

d = 1 dimensional scalar creepers
For d = 1 the field equation (38) together with Eq.(72) imply thaṫ where λ 0 is an integration constant, andφ is the momentum conjugate to φ. Recalling Eq.(2) we note that there exists no function V (φ) except for V (φ) = 0 satisfying (75), (76) and (77). Therefore λ 0 = −λ B , and Eq.(72) reads, To estimate the value of λ B in the D = 4 model, we note that since V (φ) = 0 the only energy scale at hand is M Pl . So we can rewrite the action (35) and the corresponding classical field equation in terms of the dimensionless fieldφ := M −1 Pl φ and the dimensionless coordinatesȳ := M Pl y as wherev 0 := M −1 Pl v 0 . The classical solutions isφ = κ(x 0 − x 0 ), wherex 0 := M Pl x 0 and x 0 ∈ R is an integration constant. So it is natural 4 to assume that |κ| ∼ 1 and consequently Thus equation (72) satisfies the conditions required in the references [6,7], i.e., λ B < 0 and where Λ ∼ 10 −14 M Pl is the high energy cutoff for the ordinary quantum field theory [14]. It is important to note that this result is independent of how we choose v 0 and x 0 . This result has two consequences. First of all we can identifyφ with a continuous and monotonic time which according to Eq.(79) does not commute with λ B [15]. Secondly, since |κ| ∼ 1 quite naturally, we conclude that the Wang-Unruh approach to the cosmological constant problem provides an anthropic explanation of the hierarchy problem: since |λ B | ∼ 1M 2 PL and λ eff ∼ 10 −122 M 2 Pl we should have Λ ≪ 1 in Planck units [6,7].

d ≥ 2 dimensional scalar creepers
Suppose V (φ) = 0. In this case the action (36) is invariant under adding an arbitrary constant φ 0 to φ. So in the following we study the equivalence classes of classical solutions defined accordingly, i.e., throughout this subsection, φ stands for the set of all fieldsφ such thatφ(y) − φ(y) is constant.
For d ≥ 2 the classical solutions to the field equation (d) φ = 0 satisfying the requirements (75) are given by in which f is a smooth function and such that where, for example, In the following we investigate the parameter space of the classical solutions (84) indicated by equations (86) for λ B < 0, λ B = 0 and λ B > 0 separately.

Classical solutions corresponding to λ B = 0
In this case κ · κ = 0. Therefore in Eq.(84) we can absorb the term d−1 a=0 κ a x a to f (z), and without loss of generality claim that the classical solution is where f is a smooth function and such that K · K = 0, i.e., Thus the symmetry of the parameter space has two factors; O(d − 1) for the K-subspace and the diffeomorphism z → f (z).

Concluding Remarks
Second quantization of quantum field theory in four dimensional Minkowski spacetime has been successful in modelling particle physics. Thinking about our universe as a four dimensional spacetime conceivably embedded in a higher dimensional geometry, it is reasonable to seek a quantum field theory describing particles localized to a nonstationary curved hypersurface embedded in a nonstationary curved spacetime.
In [16] we have shown that a fermionic field theory exists in four dimensions with a consistent particle interpretation in general nonstationary curved spacetime whose one-particle states are localized on two dimensional subspaces. In this work, we have shown that there exist scalar field theories whose one-particle states are well-defined in general D dimensional nonstationary curved spacetimes and their quanta are localized on d ≤ D dimensional subspaces. Therefore, in addition to providing a notion of scalar particles in nonstationary curved spacetimes, this construction might be of some interest in the brane-world models of a four dimensional universe embedded in a higher dimensional bulk, see, e.g., [2,3] and references therein. In particular, the "massless" quantum field theories localized on two dimensional subspaces can be interpreted as c = 1 conformal field theories embedded in the bulk.
The main obstacle to the particle interpretation of states of a quantum field theory in general is the absence of a preferred notion of time translations in nonstationary spacetimes [1]. We have noted that this problem can be circumvented by distinguishing the time-ordering of field operators arising in the path integral formalism from the timelike directions of the spacetime. The so-called time-ordering occurring in the path integral can be attributed to the relative signs in the kinetic term. So, it can be separated from the spacetime geometry by using the action (35) whose field equation does not engage the spacetime metric.
The path integral implies that the correlation function satisfies Eq.(61). We have been able to solve this equation, and explicate the path integral accordingly, by using a coordinate system denoted by x, in which the volume element e d D y equals d D x. Such coordinate systems are familiar in unimodular gravity [9,10]. We observed that the Feynman propagator mimics the propagator of an auxiliary scalar field confined to a d dimensional flat hypersurface of a D dimensional Minkowski spacetime. So, we interpreted it accordingly as the propagating amplitude of one-particle states localized on the d dimensional hypersurface.
These scalars do not describe ordinary matter as can be seen from their stress tensor T µν = Lg µν in which L denotes the Lagrangian density. They all act like perfect fluid with equation of state w = −1. By coupling to gravity, conservation of the stress tensor implies that L is constant onshell. So these scalars add to the cosmological constant. We approached this problem classically and considered the on-shell value of L as the bare cosmological constant λ B .
The d = 1 case is almost unique. Requiring the existence of nontrivial solutions to the classical field equation implies that the potential term is zero. So the only freedom in writing the action is to choose a nowhere-vanishing vector field which is asymptotically timelike. In an FRW universe the vector field is timelike everywhere so the d = 1 creeper can be considered as a continuous and monotonic time which does not commute with λ B [15]. Furthermore, since in this model λ B < 0 it might be of some interest in the recent approaches to the cosmological constant problem [6,7].
The existence of higher dimensional creepers depends on the number of linearly independent nowhere-vanishing vector fields in the spacetime which is a question in homotopy theory [4]. Such creepers are not unique because we need to choose the potential term and also the (asymptotically) spacelike vector fields which together with the (asymptotically) timelike vector field give the action. For example in an FRW universe whose spatial sections are three dimensional spheres, d > 2 dimensional creepers do not exist but we can define d = 2 dimensional creepers and for that purpose we need to choose one spacelike vector field from the three dimensional tangent space. Of course, it is reasonable to use the SO(3) symmetry of the tangent space in order to consider all of these choices equivalent to each other.
We studied d ≥ 2 massless and non-interacting creepers and investigated the parameter space of the classical solutions corresponding to a bare cosmological constant. Contrary to the d