A way to get rid of cosmological constant and zero point energy problems of quantum fields through metric reversal symmetry

In this paper a framework is introduced to remove the huge discrepancy between the empirical value of the cosmological constant and the contribution to the cosmological constant predicted from the vacuum energy of quantum fields. An extra dimensional space with metric reversal symmetry and $R^2$ gravity is considered to this end. The resulting 4-dimensional energy-momentum tensor (obtained after integration over extra dimensions) consists of terms that contain off-diagonally coupled pair of Kaluza-Klein modes. This, in turn, generically results in vanishing of the vacuum expectation value of the energy-momentum tensor for quantum fields, and offers a way to solve the problem of huge contribution of quantum fields to the vacuum energy density.


I. INTRODUCTION
The observation of the accelerated expansion of the universe [1] boosted the studies on an old cosmological problem, namely, cosmological constant problem [2]. The standard explanation for the accelerated expansion of the universe is a positive definite cosmological constant in Einstein field equations [3,4]. A cosmological constant (CC) may be considered either as a geometrical object ( e.g. as the part of the curvature scalar that depends only on extra dimensions in a higher dimensional space) or as the energy density of a perfect fluid with negative pressure or a combination of both. (Although these two attributions may seem to be really two different manifestations of the same thing this distinction enables a more definite discussion of the problem as we shall see.) The vacuum expectation values of the energy-momentum tensors of quantum fields (i.e. the energy-momentum tensor due to zero modes of quantum fields) induce energy-momentum tensors that has the form of the CC term in Einstein field equations. This identification is the main origin of the two (probably related ) most important cosmological constant problems; 1-why is the energy density ( ∼ (10 −3 eV ) 4 [5] derived from the measurements of acceleration of the universe is so small compared to the energy scales associated with quantum phenomena ( that is, why is CC so small? ), 2-why does the zero modes of quantum fields contribute to the accelerated expansion of the universe so less than the expected?.
There are many attempts, at least partially to answer these questions, namely; symmetry principles, anthropic considerations, adjustment mechanisms, quantum cosmology and string landscape etc. [2,6]. None of these attempts have been wholly satisfactory. One of the main ideas proposed towards the solution of the problem is the use of symmetries such as supersymmetry and supergravity. However these symmetries are badly broken in nature. So it seems that they do not offer a viable solution. Recently a symmetry principle that does not suffer from such a phenomenological restriction was introduced [7, 10,11].
This symmetry amounts to invariance under the reversal of the sign of the metric and it has two different realizations. The first realization is implemented through the requirement of the invariance of physics under the multiplication of the coordinates by the imaginary number i [7,8,9]. The second realization corresponds to invariance under signature reversal [10,12,13] and may be realized through extra dimensional reflections [10]. In this paper both realizations of the symmetry are named by a common name, "metric reversal symmetry".
In the previous studies the symmetry is implemented for a cosmological constant that is geometrical in origin e.g. a bulk CC or a CC that is induced by the part of the curvature scalar that depends on the extra dimensions only. The aim of the present paper is to extend this symmetry to a possible contribution to CC induced by the vacuum expectation value of the energy-momentum tensor of quantum fields (i.e. quantum zero modes). The main difficulty in applying the symmetry to the contribution of the quantum zero modes is that, in the simple setting considered in the previous studies, it is not possible to impose it so that the matter Lagrangian corresponding to a field is non-vanishing after integration over extra dimensions (i.e. so that the field is observable at the usual 4-dimensions at the current accessible energies) while the quantum vacuum contributions of the fields are forbidden. This point will be mentioned in more detail in the following section. To this end, in this paper the space is taken to be a union of two 2(2n + 1) dimensional spaces and the gravitational Lagrangian is taken to be R 2 where R is the curvature scalar. Robertson-Walker metric is embedded in one of these 2(2n + 1) dimensional spaces. Both realizations of the metric reversal symmetry are imposed. The 4-dimensional Robertson-Walker metric reduces to the Minkowski metric after the symmetry imposed and the action corresponding to matter Lagrangian is forbidden by the requirement of the invariance under x A → ix A . The requirement of the implementation of (either realization of) the symmetry on each space separately restricts the form of the gravitational action and only some part of the gravitational action survives and it can be identified by the usual Einstein-Hilbert action after integration over extra dimensions. After breaking the x A → ix A symmetry (while preserving the signature reversal symmetry) the Minkowski metric converts to the Robertson-Walker metric (with a slowly varying Hubble constant), and results in a small non-vanishing matter Lagrangian (and action). The unbroken signature reversal symmetry imposes the resulting matter Lagrangian generically contain at least one pair of off-diagonally coupled Kaluza-Klein modes in each homogeneous term and hence necessarily contains mixture of different Kaluza-Klein modes. This, in turn, causes the vacuum expectation value of energy-momentum tensor be zero as we shall see. Then the accelerated expansion of the universe may be attributed to some alternative methods such as quintessence [14,16], phantoms [15,16] etc. or a small CC may be induced classically after breaking of the x A → ix A symmetry as we shall see.

II. A BRIEF OVERVIEW OF METRIC REVERSAL SYMMETRY
We consider two different realizations of a symmetry that reverses the sign of the metric and leaves the gravitational action invariant, where S and g denote the number of space-like dimensions and determinant of the metric tensor, respectively. I call this symmetry, metric reversal symmetry.
The first realization of the symmetry [7]is generated by the transformations that multiply all coordinates by the imaginary number i The second realization [10] is generated by the signature reversal The requirement of the invariance of Eq.(1) under either of the realizations, Eq.(3) and Eq.(4) sets the dimension of the space D to D = 2(2n + 1) , n = 0, 1, 2, 3, .... .
Hence both realizations forbid a bulk cosmological constant (CC) term (provided that S G remains invariant) where Λ is the bulk CC.
In fact these conclusions are valid for signature reversal symmetry in a more general setting where the whole space consists of a 2(2n + 1) dimensional subspace whose metric transforms like (4) and the metric tensor for the rest of the space is even under the symmetry.
A higher dimensional metric with local Poincaré invariance may be written as [17] where x and µ ν = 0, 1, 2, 3 denote the usual 4-dimensional coordinates and indices; y denotes extra dimensional coordinates, andã,b=4, 5, ...2(2n + 1), e ′ , d ′ =2(2n + 1), ...., D denote the extra dimensional indices. We let, We take the underlying symmetry that induces (10) be an extra dimensional reflection symmetry. For example one may take Ω(y c ) = cos k y y = y D where k is some constant and take the symmetry transformation be a reflection about kz = π 2 given by There is a small yet important difference between simply postulating a signature reversal symmetry or realizing it through (9) and (11) although both forbid a cosmological constant (CC). In the case of (9) and (11), one may take a non-vanishing CC from the beginning and it cancels out after integration over extra dimensions while this is not possible if one simply postulates the metric reversal symmetry.
The action functional corresponding to the matter sector is where L M is the Lagrangian for a matter field. If the symmetry is applicable to the matter sector then the symmetry must leave S M invariant. One may take the dimension where the field propagates as D = 2(2n + 1) so that (at least) the kinetic part of S M is invariant under the symmetry transformations. For example the kinetic part of the Lagrangian of a scalar field φ transforms like R under the transformations, (3) and/or (4) so that S M is invariant under the symmetry if φ propagates in a 2(2n + 1) dimensional space and φ → ±φ under the symmetry transformation. Meanwhile this allows non-zero contributions to the CC through the vacuum expectation of energy-momentum tensor of quantum fields. The 4-dimensional energy-momentum tensor for (14) at low energies, T ν µ , is where we employed the metric (9), andg and g e denote the determinants of (gãb) and (g e ′ d ′ ), and δ ν µ denotes the Kronecker delta. If the signature reversal symmetry is imposed through an extra dimensional reflection, for example, by (11) and (12) then the last term in (15) cancels out while the other terms survive after the integration over the extra dimensions.
So the 4-dimensional energy-momentum tensor in general gives non-zero contribution to vacuum energy density through its vacuum expectation value after quantization. One may allow L φ k by letting φ propagates in a 4n dimensional but this would allow a bulk CC. In other words one may adjust the dimension of the space where the field propagates so that (13) is allowed and hence the symmetry is true for matter sector but this allows either a bulk CC or the contribution of quantum zero modes. The situation is the same for gauge fields and fermions. So one should consider this as a classical symmetry [8] or one should construct a more sophisticated framework where the symmetry applies both at classical and quantum levels. Constructing such a model will be the aim of the following sections.
This together with the requirement that after integration over extra dimensions it should correspond to the solution of the 4-dimensional Einstein equations with a cosmological constant (as the only source) implies that In other words the first realization of the symmetry, Eq.(17) requires the 4-dimensional part of the metric be the usual Minkowski metric, that is, Eq. renormalized value of CC is proportional to the particle masses [18]. So even a free electron contributes to CC by an amount that is ∼ 10 33 times larger than the observational value of CC. Therefore the first realization of metric reversal symmetry by itself can not be used to make CC vanish (or tiny). In the next section we will see how the signature reversal symmetry (realized through extra dimensional reflections) can be used to make the contribution of the quantum zero modes vanish. However the first realization has an advantage over the second one especially when the second realization is considered to be an extra dimensional reflection of the form of (12). Extra dimensional reflections do not act on the 4-dimensional coordinates so they can not forbid a contribution from the 4-dimensional part of the metric, for example through a(t) while the first realization always does by setting it to zero as we have seen. So in the next section we will employ both realizations of the symmetry. The second realization through extra dimensional reflections will cancel the contributions to CC while the first one will allow a small CC after it is broken by a small amount.
Next see what is the form of the conformal factor Ω when both realizations of the symmetry are imposed. We have obtained in (20) the form of the metric after the first realization of the symmetry is imposed. Eqs. (17,18) set the form of the conformal factor Ω in (16) to one of the followings Ω(y) = Ω(|y|) or Ω(y) = f (y)f (iy) (e.g. cos ky cosh ky) (21) where f (y) is an even function in y i.e. f (−y) = f (y). Next apply (12) to (21) and require (10) and take the extra dimension y be an S 1 /Z 2 interval. This restricts the form of Ω to Ω(y) = cos k|y| or Ω(y) = tan k|y| (22) where cot k|z| has been excluded because it blows out at the location of the branes at k|y| = 0 and k|y| = π. For simplicity I take Ω(y) = cos k|y| (23) in the next section whenever necessary.

IV. THE MODEL: CLASSICAL ASPECTS
In this section we employ both realizations of the metric reversal symmetry in a space that is the sum of two 2(2 + 1) dimensional spaces (where the usual 4-dimensional is embedded in one of them) and modify the curvature term S G so that the metric reversal symmetry becomes a good candidate to explain the huge discrepancy between the observed value of cosmological constant (CC) and the theoretically expected contribution to it through quantum zero modes. In this study I adopt the view that the symmetry forbids both the geometrical and the vacuum energy density contributions to CC. Hence CC is forced to be zero when the symmetry is manifest, and it is tiny when the symmetry is broken by a tiny amount (instead of seeking a solution where both contributions cancel each other up to a very big precession to explain the observed value of CC). In this section the main classical aspects of a framework to this end are introduced.
Consider the whole space be a sum of two 2(2n + 1) dimensional spaces with the metric Ω y (y) = cos k|y|) , Ω z (z) = cos k ′ |z| (25) The usual four dimensional space is embedded in the first space g AB dx A dx B as it is evident from (24). We take the action be invariant under both realizations of metric reversal symmetry, that is, and As in (20) and (23)

A. Curvature Sector
We replace the gravitational action in (2) by an R 2 action where the unprimed quantities denote those corresponding to the N = 2(2n + 1) dimensional space, and the primed quantities denote those corresponding to the N ′ = 2(2m + 1) dimensional space. Under the transformations (28,29) We observe that under the action of the symmetry transformations to only one of the spaces, the unprimed or the primed spaces. So, only the cross terms RR ′ are allowed. In other words only these terms may survive after integration over extra dimensions. In fact it is obvious from the above transformation rules that an Einstein-Hilbert type of action is not allowed directly because each piece R and R ′ inR is odd while dV is even under a transformation applied to both subspaces, the unprimed and the primed subspaces. Since only RR ′ terms are allowed (30) becomes andG is a dimensionless constant. In other words in the usual 4-dimensions at low energies

B. Matter Sector
In this subsection we consider the matter action and we consider the 4-dimensional form of S M after integration over extra dimensional spaces. Then we study the vacuum expectation value of the energy-momentum tensor induced by the corresponding Lagrangian in the section after the next section.
It is evident that under the first realization of the symmetry for a space consisting of the sum of two 2(2n + 1) dimensional spaces as in (24). The for the usual fields [8]. So S M is not invariant under the symmetry generated by In other words the first realization of the metric reversal symmetry is maximally broken in the matter sector (and hence the scale factor a(t) in the Robertson-Walker metric may be time dependent). On the other hand I take a higher dimensional version of the P T symmetry be almost exact and broken by a tiny amount. In other words I adopt which is a subgroup of the group generated by The symmetries in (43) are imposed on each subspace separately. Next I impose an additional 4-dimensional PT symmetry generated by In the case of fermions the integers n, m in (46,49) should be replaced by 1 2 n, 1 2 m, respectively. One observes that since n(m) are the eigenvalues of ∂ ∂y ( ∂ ∂z ) i.e. they are the momenta corresponding to the directions y and z. There are two eigenvalues i.e. ± 1 of the each transformation in (50) since application of the transformations twice results in the identity transformation. Now we show that the fields (46,49) are the eigenstates of the transformations (50). First consider (46). Applying the transformation (43) and using (50), φ AA in (46) transforms to There will be no mixture of the eigenstates of (43) in the Lagrangian because the Lagrangian is invariant under (43). So φ AA is either odd or even under (43). In the light of (50,52) the eigenstates of φ AA under the transformation are determined by φ AA n,m (x). The same conclusion is true for all φ's (46,49). So, for all φ's (46,49) we have two cases for each symmetry in (50) Meanwhile one may write (46,49) in the following form as well

Scalar Field
First consider L φk , the kinetic part of the Lagrangian L M k for a scalar field (in the space given in (24)) The expressions for φ AS , φ SA , φ AA are the same as (A3) up to minus and pluses in front of the φ mn terms. Hence the expressions for φ AS , φ SA , φ AA are the same as (61)  A more detailed analysis of Eq.(61) and these points will be given in the next section.
Next consider a bulk mass term (for φ SS ) dy cos ky cos (n k|y|) cos (r k|y|) The common aspect of the equations (61)  In the next subsection we consider one additional example, that is, the kinetic term for fermions because it is not a straightforward generalization of the scalar case. We will see that the same conclusion also holds in that case as expected.

V. THE RELATION TO LINDE'S MODEL
It is evident from (61) that the 4-dimensional kinetic term contains the zero mode φ 00 while the other terms i.e the mass terms do not contain the zero mode. This implies that there is a zero mass eigenstate that contains φ 00 . However the form of (61) is rather involved since it involves, in general, mixing of all Kaluza-Klein modes. An important aspect of this mixing is the absence of diagonal terms in the mixing terms. We will see in the next section how this plays a crucial role in making the vacuum expectation value of energymomentum tensor zero. Before passing to this issue, first we should make the form of (61) more manageable. In any case one should diagonalize (61) so that, at least, the fields in the 4-dimensional kinetic term couple to each other diagonally i.e. we should pass to the interaction basis. One observes due to the signature reversal symmetry (induced through extra dimensional reflections) that all the terms in the 4-dimensional kinetic term in (61) are mixed so that the terms with odd n's mix with the even n's, and the odd m's with odd m's, the even m's with even m's. There is the same behavior for the terms with the coefficient k 2 , and a similar behavior for the terms with the coefficient k ′2 (the odd n's mix with the odd n's, the even n's mix with the even n's, and the odd m's mix with the even m's and vice versa). So the form given by the 4-dimensional part of (61) may be only induced by the mixture of either of The each sum may be an infinite series if all modes are mixed or it may correspond to a set of finite sums if the modes mix with each other in a set of subsets of r and s in (61).
In the expansion of φ EE SS the sum over j starts from one because we take the zero mode φ 00 in a different eigenstate as we will see. The requirement that the internal symmetries that may be induced by extra dimensional symmetries and the usual space-time symmetries are independent requires the whole space be a direct product of the 4-dimensional space with the extra dimensional space. This, in turn, requires all φ n,m (x)'s in the above equations be the same up to constant coefficients, that is, |2j − 2| |2l − 1| ((2j) 2 + 1)((2l + 1) 2 + 1) cos (2j)ky cos (2l where the SS indices are suppressed. In the light of (74,75) Eq.(61) becomes where the form of the coefficients C i , i = 1, 2, 3, ..., 12 are given in Appendix C. The diagonalization of (76) results in where It is evident from (77) that the scalar kinetic Lagrangian (61) is equivalent to a Lagrangian that consists of a set of usual scalars and a set of ghost scalars. In fact this conclusion is valid for all quadratic terms for all fields e.g.ψ n,m ψ r,s where n = r and/or m = s due to the symmetry and this term is equivalent to 1 2 (ψ 1 ψ 1 −ψ 2 ψ 2 ) whereψ 1 = ψ n,m + ψ r,s , ψ 2 = ψ n,m − ψ r,s . This setting is similar to Linde's model [19] and its variants [20]. Only mixing between the usual particles and ghost sector may be induced through quartic and higher order terms. A detailed analysis of such possible mixings and suppressing these couplings needs a separate study by its own.

VI. VACUUM EXPECTATION VALUE OF ENERGY-MOMENTUM TENSOR IN THE PRESENCE OF METRIC REVERSAL SYMMETRY
The 4-dimensional energy momentum tensor corresponding to the action (76) is It is evident from (79) that all terms consist of off-diagonally coupled Kaluza-Klein modes.
As we have remarked before any 4-dimensonally Lagrangian term (after integration over extra dimensions) necessarily contains at least a pair of Kaluza-Klein modes that are offdiagonally coupled in the space given by (24). (As we have remarked in the previous section, this is due to the fact that if a term wholly consists of pairs of diagonally coupled Kaluza-Klein modes then that term is even under the signature reversal symmetry in contradiction with the invariance of the action under the signature reversal symmetry.) This, in turn, leads to cancellation of the vacuum expectation value of T ν µ since it is proportional to terms of the form < 0|T ν µ |0 > ∝ < 0| a n,m a † r,s |0 > = 0 , < 0| a † r,s a r,s |0 > = 0 n = r and/or m = s (because a r,s |0 > = 0, and [ a n,m , a † r,s ] = 0 for n = r and/or m = s) where a n,m , a † n,m are the creation and annihilation operators in the expansion the quantum fields (in Minkowski space) given by The same reasoning is true for all fields. Therefore the vacuum energy density of all fields in this scheme is zero.
In this scheme the Casimir effect can be seen as follows: Introduction of (metallic) boundaries into the vacuum results in a change in the vacuum configuration for the usual particles while the ghost sector vacuum remains the same. This point can be seen better when one considers the the energy momentum tensor written in terms of the usual and ghost fields by using (77) To see the situation better let us consider a simple case, for example the part of the energymomentum tensor that contains the zero mode. After introduction of the (metallic) boundary the vacuum expectation value of the corresponding part of the energy momentum tensor changes as follows where the subscript 0 denotes complete vacuum (without any boundary) and the subscript where α << 1 is a constant that reflects that L cl is small since it corresponds to the breaking For α v 1,0 v 0,1 ≃ 1 (85) results in the observed value of Λ ≃ (10 −3 eV ) 4 for L, L ′ in the millimeter scale and for α v 1,0 v 0,1 ≃ Mew M pl ≃ 10 −17 , for example, L(L ′ ) < 10 −7 m. In any case a non-zero CC if exists is a classical phenomena in this scheme. Another point is that the energy density due to CC obtained in a way similar to (85) may be argued to be in the order of matter (ie. the usual matter plus dark matter) density since both are induced by matter Lagrangian that corresponds to breaking of the However there is a difference between the two cases. The induction of S M corresponds to breaking the symmetry that corresponds to the simultaneous application of x A → ix A and

VIII. CONCLUSION
We have considered a space that is a sum of two 2(2n + 1) dimensional spaces with R 2 gravity and metric reversal symmetry. The usual 4-dimensional space is embedded in one of these subspaces. We have shown that the curvature sector reduces to the usual Einstein-Hilbert action, and the 4-dimensional energy-momentum tensor of matter fields generically mixes different Kaluza-Klein modes so that each homogeneous term contains at least one pair of off-diagonally coupled Kaluza-Klein modes. This, in turn, results in vanishing of the vacuum expectation value of the energy-momentum tensor of quantum fields. I have also shown that such a model is equivalent to a variation of Linde's model (where the universe consists of the usual universe plus a ghost one). There may be some relation between this scheme and the Pauli-Villars regularization scheme [23] ( that employs ghost-like auxiliary fields for regularization), and also between this scheme and Lee-Wick quantum theory [24].
In my opinion all these points need further and detailed studies in future.

Acknowledgments
This work was supported in part by Scientific and Technical Research Council of Turkey under grant no. 107T235.