QUANTUM ASPECTS

A. Weinberg-Witten theorems, emergent gravity and spin-1 gravitons

In [110], Weinberg and Witten established two very powerful theorems in QFT. Essentially, they forbid: (i) massless charged states with helicityj >1/2 which have a conserved Lorentz-covariant current and (ii) massless states with helicityj >1 which have conserved Lorentz-covariant energy-momentum tensor.

47The formal aspects of the mapping between a gauge theory in Euclidean spacetime and a gravity theory can be discussed in terms of fiber bundle theory. The details can be found in [39, 40, 107]. The result was discussed for general manifolds withdarbitrary dimensions.

Let the original manifold have metric tensorgµν(x) while the effective metric tensor is defined asegµν(X). Thus, the matrix that defines the mapx7−→X for all points in the effective manifold is given by

L^ν_µ= ˜g

g 1/2d

˜

g^ναgαµ. (162)

Its inverse is given by

L⁻¹ν µ=

g

˜ g

1/2d

g^ναg˜αµ. (163)

In (162) and (163),g= detgµν andeg= detegµν are taken as non-vanishing quantities. Thus, ambiguities are absent in this mapping.

DRAFT

At first sight, the first theorem would forbid gauge theories to exist because vector bosons and gluons are charged massless states which carry helicity j = 1. However, the conserved gauge current associated with these states are not Lorentz covariant [110]. Similarly, the second theorem forbids spin-2 states with conserved Lorentz-covariant energy-momentum tensor to be defined. In traditional GR the metric tensor g_µν is not a Lorentz covariant tensor since it is constructed over a Riemannian manifold. The linearized version of GR, on the other hand, is also a gauge theory and the same principles of the first case apply [110]. Moreover, the energy-momentum tensor of the graviton field identically vanishes.

On the other hand, the Weinberg-Witten theorem (ii) applies to emergent gravity theories for which massless spin-2 states are generated as effective or composite states that could be associated with graviton excitations. Nevertheless, one may argue that the Weinberg-Witten theorem (ii) applies to the present mechanism. However, this is not the case here. First, the theory has a few mass parameters and the theorem holds only for massless states. Second, and more important, there are no spin-2 states in this model. The fields are identified with geometry andnot with spin-2 composite fields. Gravity emerges as geometrodynamics and not as a field theory for spin-2 particles in flat space.

Nevertheless, renormalizability establishes that the quantum action has the same form of its classical version. The difference between the classical and quantum actions lies on the fields and parameters, which are their respective renormalized versions. Hence, although each configuration (A, θ) can define a geometry (ω, e), the mapping should not be applied to each configuration in the path integral, but at the quantum action itself. In this way, the resulting geometrodynamical theory is obtained from the full dynamical content of the original gauge theory. The conclusion is that there is no violation of the Weinberg-Witten theorem.

Another question may raise at this point: Where are the spin-2 excitations of the graviton? And it is an important question. The main point here was already discussed: The present approach is not a quantization of gravity geometric variables. Inhere, gravity is an effective manifestation of another theory, i.e. geometrodynamics is a classical limit of a quantum gauge theory of spin-1 excitations. On the other hand, if our SO(5) gauge theory is in fact quantum gravity and the graviton is the mediator of gravity at quantum level, then, in the present theory, the graviton has spin-1. Hence, this definition differs from the usual graviton definition which is associated with massless spin-2 states.

Nevertheless, after the emergence of gravity as geometrodynamics, linearization is allowed for weak gravitation and the spin-2 states might also be considered. In that case, these states are classical states associated with propagating fields and not quantum states.

Another way to think of the spin counting difference lies on the map R⁴ 7−→ M⁴ where the local gauge group is identified with the isometries of the tangent spaces of M⁴. In this process, the group indexes are interpreted as spacetime tangent indexes (an association prevent at high energies due to the massless character of Yang-Mills action).

Thus, after the map, each field gains an extra index and thus, extra spin degrees of freedom.

B. Cosmological constant problem

A remarkable feature of the present theory is that Newton and cosmological constants are related through Λ² = κ²/16πG. From asymptotic freedom, κ² is a big quantity at low energies. And, by assumption, G is small. Thus, Λ should be very big. In fact, if this is true, we can make two important remarks: (i) The first term in (164) can be safely neglected. Moreover, due to the absence of matter fields, torsion can be taken as very small. The resulting theory is then, the usual Einstein-Hilbert theory with (a very big) cosmological constant (see Sec. XI A); (ii) although astrophysical predictions [111, 112] determine that Λ²_obs is very small, quantum field theory predicts [113–115] a very large Λ²_{qf t}. Thus, the contribution of a pure gravitational cosmological constant, which is big in our case, can drive the cosmological puzzle to a final consistent answer. In fact, following [39, 116], the renormalized cosmological constant of our model could be determined through Λ²_obs = Λ²_ren+ Λ²_{qf t} ≡ Λe². This means that, by summing all vacuum contributions to the last term in (164), it should be replaced by Λe². We will in fact assume this at Secs. XI C and XI B.

C. Quantum predictions

It is possible to perform actual explicit computations within a semi-perturbative approach, see [40] for the first 1-loop estimates and [117] for 1 and 2-loop improved computations. In these notes, without loss of generality, we confine ourselves to 1-loop results.

The important thing at this point is to wisely find the correct mass parameters to account for m. It is intuitive that the Gribov parameter is the most probable candidate because it is a genuine non-perturbative and inevitable parameter and it is gauge invariant [49, 50].

DRAFT

1. Running parameters

In perturbative QFT, the running of the parameters can be determined by the renormalization factors of fields and parameters. In the case of the Gribov parameter, it is determined by a gap equation obtained from the minimization of the quantum action with respect toγ², see Sec. III. At 1-loop, the gap equation is⁴⁸[14, 46]

3N κ² 4

Z d⁴p (2π)⁴

1

p⁴+γ⁴ = 1. (166)

where N = 5 and pis the internal momentum. Following [46], by employing the MS renormalization scheme at the gap equation (166), it is not difficult to find (Exercise 3.13)

N κ² 16π²

5 8 −3

8ln N γ⁴

µ⁴

= 1, (167)

whereµis the energy scale and the trivial solutionγ²= 0 was excluded. Equation (167) provides (Exercise 3.14) γ²= e⁵⁶

√Nµ²e⁻⁴³

16π2 N κ2

. (168)

On the other hand, recalling that the coupling parameter is determined through the renormalization group equations [66, 67], we can write

N κ²

16π² = 1

11 3 ln^µ2

Λ²

, (169)

where Λ is the renormalization group cutoff. Hence, combining (168) and (169), it is found (Exercise 3.15) γ²= e⁵⁶

√5Λ² µ²

Λ^{2 −35/9}

. (170)

We can see from (170) and Fig. 4 that, as higher the energy is, as smaller the Gribov parameter is. This is the expected behavior [14, 46, 48] ofγ² because at high energies perturbative massless Yang-Mills theory must be recovered. At the deep infrared region, however, it seems to diverge. This behavior is an evidence that the semi-perturbative approximation needs improvements. In fact, it is known from lattice simulations for unitary groups that the coupling parameter is finite at the origin [118, 119]. This kind of improvement could affect the infrared sector of the Gribov parameter in such a way that it could also be finite.

FIG. 1: Gribov parameter as function of energy scale. The energy is in units of Λ and the Gribov parameter in units of ^e

5

√6 5Λ².

48It is important to notice that, because we are dealing with Yang-Mills theories, the following computations can be derived in the same manner that of [46], only keeping in mind that the fundamental Casimir here is 5 and the group dimension is 10.

DRAFT

Now, the ratioγ²/κ² is easily determined from (169) and (170), γ²

κ² = 55 48π²

e⁵⁶

√5

! Λ²

µ² Λ²

^−35/9 ln

µ² Λ²

. (171)

The very interesting behavior of γ²/κ² is plotted in Fig. 2 and the inverse ratio is displayed in Fig. 5. As expected, at high energies,γ²/κ²asymptotically vanishes. Then, in a scale right above Λ,γ²/κ²achieves a maximum. At this region, BRST soft breaking dominates and the rescaling of the fields (157) is allowed. After that, it fastly drops to zero atµ= Λ. Is exactly at this point that the theory suffers the In¨on¨u-Wigner deformation which induces the breaking to theSO(4) theory and the geometric phase starts over. This point is also recognized as the point of phase transition in non-Abelian gauge theories. Below this point, another theory takes place. In the case of QCD, ΛQCD ≈237M eV, the Yang-Mills action should be replaced by an action based on hadrons and glueballs excitations. In the case of gravity, the Yang-Mills action must be substituted by a geometrodynamical action and the phase transition scale is expected to be around Planck scale.

FIG. 2: The ratio γ²/κ² as function of energy scale. The energy is in units of Λ and the Gribov parameter in units of

55 48π²

e⁵6

√ 5

Λ².

Below the transition point Λ, where the coupling parameter diverges, the squared coupling parameter acquires negative values. This indicates that below Λ the perturbative predictions are actually meaningless. However, from lattice predictions [119, 120], there are strong evidences that the non-perturbative coupling is actually finite at the origin and presents no divergence at the transition scale. On the other hand, we can argue that the divergence is a strong signal that there is a phase transition at that point. Thus, neither way, the coupling parameter should drop out in favor of an effective coupling. In the case of QCD, it is not known how to obtain this parameter from the dynamics of Yang-Mills theories. However, in the present case, we can interpret the ratioγ²/κ²as the effective coupling, which is related to Newton‘s constant.

The analysis for the cosmological constant is easier because Λ ∼ γ². Thus, it should be very large indeed. As discussed in Sec. X B, this is actually a very welcome feature because it may compensate the QFT predictions and, eventually, it can provide a small effective value [116].

2. Numerical estimates

There are two main ways to calculate the parameters of interest. The usual way [6, 68] is: i) to fix the renormalization group parameter at the transition energy. Its value is a phenomenological quantity and cannot be derived by theory. In our case is Planck energy; ii) chose a energy scale in such a way thatN κ²/16π²and ln(µ²/Λ²) are as small as possible;

iii) compute the physical parameters you want. Another way is: i) Fix a physical parameter to its experimental value;

ii) chose a energy scale in such a way thatN κ²/16π² and ln(µ²/Λ²) are as small as possible; iii) Compute the other parameters as well as the transition scale. We choose to discuss here the second way (both ways are discussed in detail in [117] and there are no significant difference between them).

DRAFT

FIG. 3: The ratio κ²/γ² as function of energy scale. The energy is in units of Λ and the Gribov parameter in units of

48π² 55

_√

5 e⁵6

Λ⁻²2.

Following the second approach, the scalesµand Λ can be estimated by fixing the current value of Newton’s constant toG⁻¹≈1.491×10³²T eV². Thus, combining (165) and (171), one easily achieves

7.323×10³¹T eV²= Λ² µ

Λ −70/9

ln µ²

Λ²

. (172)

We have little freedom to choose µas long as N κ²/16π² <1. Let us work atµ² = 2Λ². Thus, N κ²/16π²≈0.393.

This provides ln(µ²/Λ²)≈0.693. Accepting these as reasonable values, we achieve for the renormalization group scale

Λ²≈2.122×10³³T eV², (173)

which is quite close (for a 1-loop semi-perturbative estimate) to Planck energyE_p²= 1.491×10³²T eV². Although some approximations and extrapolations have been considered, we can interpret these values as a good result, indicating that the geometric phase of gravity appeared right before Planck scale. This means that right above Planck scale, where quantum mechanics starts to make sense, gravity is already in its geometric phase.

We can also estimate the value of the cosmological constant for the chosen scales. From (165) and (170), it provides

Λ²≈1.106×10³² T eV². (174)

This is a huge amount of energy and is three orders of magnitude greater than quantum field predictions [121] of Λ²_{qf t} ∼ −3.71×10²⁸T eV²and would not cancel it to provide the observational data value Λ²_obs∼1.686×10⁻⁶⁸eV². However, for 1-loop approximation at zero temperature, it is quite remarkable that a solution accommodating Newton constant and Planck energy scale could be found.

XI. CLASSICAL ASPECTS