Origen, toxicidad y legislación ambiental 74

A natural first instinct when attempting to solve the cosmological constant problem is to play with the idea that there exists some set of fields_{φi} whose dynamics will generically

drive Λ_→0. As Weinberg says, this idea has been “tried by virtually everyone” [115], but without much success.

There are quite general arguments that indicate that this route is a difficult one, in particular the “no-go” theorem of Weinberg himself [115], which we review here. The key idea is that a natural solution of the CC problem would require the fact Λ _≈ 0 to follow entirely from the equations of motion for the fields_{φi}. For instance, an unnatural solution

would be to simply assume that the potential for these additional fieldsV(_{φi}) simply takes

on exactly the right value to cancel off the true value of Λ. Such a solution would be nothing but a reshuffling of where the fine tuning is taking place. Rather than small Λ originating from a fine tuning in the fundamental UV physics, it would originate from fine tuning within

V(_{φi}), rendering the idea unpalatable.

Instead, it would be much more natural if the equations of motion for the fields _{φi}

were proportional to Tµµ. If this were the case, then the dynamics of {φi} driveTµµ→ 0

and hence drive spacetime to be approximately flat, as Einsteins equations tell usR _∝Tµµ.

Let us attempt to construct such a scenario in the restricted case where the fields_{φi} and

are constant throughout spacetime, i.e. they preserve Poincar´e invariance, and we want the

{φi} to drive us to a scenario in which the metric gµν is also spacetime independent. In

terms of the lagrangianL, we are in effect requiring the statement

X i fi ∂L ∂φi ∝ Tµµ, (1.9)

where_{fi} are generic functions of the{φi}. If we can find a system which obeys a form of

(1.9), thenTµµvanishes as the fields{φi}settle into their minima and we have accomplished

our goal.

The condition (1.9) turns out to be a quite severe restriction, however. From the as- sumption thatgµν is spacetime independent, we can write writeTµµ= √2_−ggµν_∂g∂L_µν and so

(1.9) becomes X i fi ∂L ∂φi =gµν ∂L ∂gµν . (1.10) This (1.10) is actually a statement of the symmetries of the fields; it expresses that the lagrangian must be symmetric under the combined transformations

δφi =−ǫfi, δgµν =ǫgµν . (1.11)

In order to see why this is problematic, it is useful to redefine our fields to a new diagonal basis, _{φi} → {ϕ, σa}, which is chosen such that the symmetry in terms of {ϕ, σa}is

δϕ=₋ǫ , δσa= 0, δgµν =ǫgµν . (1.12)

Any non-derivative interaction which obeys this symmetry must then be built from the in- variant effective metriceϕ_g

µν and general coordinate invariance further dictates that the in-

teraction must be_∝p₋deteϕ_g

µν =e2ϕ√−g. The most general, symmetric, non-derivative

interaction that will involve all the fields _{gµν, ϕ, σa} is then L=√−g e2φL′({σa}) where

L′(_{σa}) only involves the fields {σa} and would generally include a cosmological constant

piece L′(_{σa})⊃Mpl2Λ.

The problem we will find is that while we were explicitly looking for interactions which would naturally set Tµµ → 0, the lagrangianL=√−g e2φL′({σa}) will only set Tµµ→ 0

for a precise tuning of the parameters inL′(_{σa}), which is exactly what we were trying to

avoid. Specifically, the equations of motion for ϕand σa are

∂L ∂ϕ = 2 √ −g e2φL′(_{σa}), ∂L ∂σa =√₋g e2φ∂L′({σa}) ∂σa . (1.13)

The σa equations simply define the equilibrium values of {σa} through the conditions ∂L′₍

{σa})

∂σb = 0. On the other hand, it is the ϕ equation which sets the trace of the en-

ergy momentum tensor,

∂L ∂ϕ = 2 √ −g e2ϕL′(_{σa}) =gµν ∂L ∂gµν = 1 2 √ −gTµµ . (1.14)

The only way to get a stationary configuration for ϕ, and hence set Tµµ → 0, is to then

enforce that the_{σa}equilibrium additionally setsL′({σa}) = 0, a condition which can only

arise through fine tuning. That is, to satisfy this condition we must balance all couplings inL′(_{σa}) against one another, and against the cosmological constant that might appear

inL′(_{σa}), so that they precisely cancel.

Therefore, the simplest attempts at adjustment mechanisms will generically suffer from fine tuning issues themselves and do not represent improvements of the problem. It is possible to evade this conclusion by bypassing the assumptions made above, for example by exploring scenarios in which the _{φi} are not spacetime independent. These are active

lines of research [26, 27], but we will not discuss them in detail here.