An enlightening case: the anomaly-induced effective action

was originated in the high-energy regime, i.e. when the electrons are light compared to the energy scale in consideration. On the contrary, in the case of heavy electrons the result is the decoupling of the electrons and only local operators appear.

1.2.4 An enlightening case: the anomaly-induced effective action

The construction of the vacuum QEA for gravity illustrated in Section1.2.1can be carried out exactly for a theory with massless, conformally-coupled matter fields, inD = 2 space-time dimensions, by integrating the conformal anomaly. The same procedure can be used inD=4 to obtain exactly the part of the QEA depending on the conformal mode of the metric. We follow the discussion in Section 3 of [29], see also [36,48,49,52,53,54] for reviews.

The anomaly-induced effective action is an explicit example of an exact QEA derivation for gravity starting from the fundamental action of the theory. Knowing both the starting point (a healthy classical theory) and the final point (the anomaly-induced QEA) will allow us to discuss some conceptual points and avoid possible misunderstandings about the degrees of freedom appearing in a QEA. It will be

quite simple and very useful to adapt those discussions to a correct understanding of nonlocal gravity models, considering their lack of a fundamental derivation.

Let us consider 2D gravity, including also a cosmological constantλ, coupled to Nmassless matter fields,

S=

Z

d²xp

−^g(κR−^λ) +S_m, (1.2.14) and let us suppose that theN =N_s+N_f matter fields included inS_mare given byN_s conformally-coupled massless scalars⁴andN_f conformally-coupled massless Dirac fermions. In D = 2 we use a = 0, 1 as Lorentz indices, and the metric signature isη_ab = (−^,+). For conformal matter fields, classically the traceT_a^a of the energy-momentum tensor vanishes. However, at the quantum level the vacuum expectation value ofT_a^a is non-zero, and is given by

h⁰|^Ta^a|⁰i= ^N

24πR, (1.2.15)

where N = Ns+N_f. The result in eq. (1.2.15) is the trace anomaly (or conformal anomaly) and its derivation for a conformally-coupled scalar field can be found in Section 14.3 of [53]. Its peculiarity is that, even if it can be obtained with a one-loop computation, it is actually exact at all perturbative orders, see again Section 14.3 of [53] for a proof. In other words, no contribution to the trace anomaly comes from higher loops. Using eq. (1.2.9) we can deduce the QEA corresponding to the vacuum expectation value in eq. (1.2.15). To perform the calculation, let us write

g_ab =e^2σg¯_ab, (1.2.16)

where ¯g_ab is a fixed reference metric. The functionσ(x)of spacetime coordinates is known as the conformal mode of the metric. Correspondingly, the Ricci scalar can be written as

R=e^−2σ(R−^2σ), (1.2.17) where the overbars denote the quantities computed with the metric ¯g_ab. InD=2, we can always find a local coordinate transformation such that the reference metric is

¯

g_ab= η_aband henceg_ab =e^2ση_ab. Then the Ricci scalar simplifies toR= −^{2e−2σ2ησ} where2η is the flat-space d’Alembertian, while the determinant of the metric gives

√−^g =e^2σ. From eq. (1.2.9) it follows that δΓm = ¹

2 Z

d²xp

−^g h⁰|^Tab|⁰i^δgab =

Z

d²xp

−^g h⁰|^Tab|⁰i^gabδσ, (1.2.18) and we can express the functional derivative ofΓ[σ]with respect toσ(x)as

δΓm

δσ =^p−^gh⁰|^Ta^a|⁰i=− ^N

12π2ησ. (1.2.19)

Finally, the integration of eq. (1.2.19) gives Γm[σ]−Γm[0] =− ^N

24π Z

d²xσ2ησ. (1.2.20)

4For reference, we recall that Ss = −¹₂R d⁴x√

−^ggµν∂µϕ∂_νϕ+¹₆Rϕ²

is the action for a conformally-coupled scalar fieldϕin D = 4 spacetime dimensions. The action is invariant under conformal transformationsgµν→^g˜µν=Ω²(x)gµν, for any arbitrary function of spacetimeΩ²(x).

Since we have setD = 2 we can say thatΓm[0] = 0, because whenσ = 0 we have locallyg_ab = η_ab and there cannot be any curvature determining a non-zero value of Γm[0]. The exactness of the trace anomaly in eq. (1.2.15) also implies that this derivation of the anomaly-induced QEA inD=2 is exact at all perturbative orders.

The result can also be rewritten in a generally-covariant form by using nonlocal operators. To achieve such a form we observe that2g = e^−2σ2η, where 2g is the d’Alembertian computed with the full metricg_ab = e^2ση_ab. Then, the equationR =

−^2e−2σ2ησ gives R = −22gσ, which can be inverted as σ = −(1/2)2⁻_g¹R. The inverse d’Alembertian2⁻_g¹is the nonlocal operator that allows us to write the QEA in a manifestly covariant way, known as the Polyakov QEA:

Γm[gµν] =−_24π^N As we said at the end of Section1.2.1, a study of the full quantum gravitational dynamics also requires to integrate over the quantum fluctuations of the metric.

When integrating over those metric flucuations in the path integral, the setting of g_ab = e^2σg¯_ab, as in eq. (1.2.16), gives a gauge-fixing condition. Therefore, besides the conformal modeσ, also the Faddeev-Popov ghosts will affect the path-integral integration. It can be shown, see [60,61,62], that the Faddeev-Popov ghosts give a contribution−26 to be added to N, while the conformal factorσ gives a contri-bution +1. Therefore, neglecting the topologically-invariant Einstein-Hilbert term and including the cosmological constantλ, the exact quantum effective action of 2D gravity reads From eq. (1.2.17) and dropping terms depending only on the reference metric ¯g_ab, and not onσ, we can write a local expression forΓin terms of the conformal mode:

Γ=

It is useful to explain a possible confusion that emerges when trying to extract the spectrum of the theory from the QEA in eq. (1.2.23). We are going to see that such an operation is not allowed and only the fundamental action correctly expresses the physical content of the theory. We can also rephrase the situation by saying that the interpretation of a formal expression like that in eq. (1.2.23) changes completely in the two cases in which we take it as a fundamental action or as a quantum effec-tive action. Understanding this point is important for a correct interpretation of the phenomenological nonlocal models that we will study later.

Let us try to read the spectrum of the quantum theory from eq. (1.2.23), treating it as if it were the fundamental action of a QFT; then we would conclude that, for N 6= 25, there is one dynamical degree of freedom,σ. Recalling that our signature isη_ab = (−^,+), we would also conclude that forN>25 this degree of freedom is a ghost and forN < 25 it has the right sign for the kinetic term. For N = 25 there is no dynamics at all.

This conclusion is clearly wrong because we know that eq. (1.2.23) is the QEA of a fundamental theory which is just 2D gravity coupled to Nhealthy fields, and there cannot be any ghosts in the spectrum of the fundamental theory. If we perform

the quantization of the fundamental theory in the gauge of eq. (1.2.16), the fields in-volved are the matter fields, the Faddeev-Popov ghosts and the conformal modeσ.

Each of them has its own creation and annihilation operators, which generate the full Hilbert space of the theory. However, as always in theories with a local invari-ance (in this case diffeomorphism invariinvari-ance) the physical Hilbert space is a subset of the full Hilbert space. Similarly to the physical-state condition h^s0|^∂µA^µ|^si = 0 required in the Gupta-Bleuler quantization of electrodynamics to discard the ghost states associated toA0, here the condition on physical states|^si^and|^s0i^{can be} ob-tained from

h^s0|^Ttot^ab|^si=0 , (1.2.24) whereT_tot^ab is the sum of the energy-momentum tensors of matter, ghosts andσ. As explicitly proved in [63], this condition eliminates from the physical spectrum both the states associated with the reparametrization ghosts and the states generated by the creation operators of the conformal mode. The physical spectrum of the funda-mental theory is simply given by the quanta of theNhealthy matter fields⁵, with no ghosts in the theory. ThoseNmatter fields are not visible in the QEA of eq. (1.2.23) because we integrated them out in the path integral. We also observe that the fact that no physical quanta are associated toσdoes not mean that the fieldσitself has no physical effects. For example in QED there are no physical quanta associated toA₀, but the interaction mediated by A₀ is exactly what is responsible for the Coulomb potential in the static case. In the language of Feynaman diagrams, this is possible because the condition of having no physical states forσin gravity (or forA₀in QED) just means thatσ cannot appear in external lines, but it can still appear in internal lines and mediate interactions in that way.

In the caseD=4 the trace anomaly is de-pend on the number of massless conformally-coupled scalars, fermions and vector fields. This result for the trace anomaly is exact and receives contribution only at one loop order. When we introduce the conformal modeσby writinggµν = e^2σg¯µν, where∆4is the Paneitz operator

∆4≡2²+2R^µν∇_µ∇_ν− ²

3R2+¹

3g^µν∇_µR∇_{ν. (1.2.27)} The quantityΓanom[g¯µν], corresponding toΓanom[gµν]evaluated at σ = 0, cannot be determined from the conformal anomaly alone, because inD= 4 we can no longer

5The conformal mode is never a propagating degree of freedom and, in D=2, there are no graviton polarizations.

set ¯g_µν = η_µν. The maximal information that we can get is the dependence of the QEA on the conformal modeσ.

The covariantization of eq. (1.2.26) is not uniquely determined and a possiblity is given by the Riegert action [64]

Γanom[gµν] = Γc[gµν]− ^b3 It is interesting that, again, a covariant form requires the use of nonlocal operators.

In this case the inverse Paneitz operator is needed, while for D = 2 the inverse d’Alembertian appeared.