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4.2 Geometries at different resolution scales

In this section we elaborate on the interpretation of the effective average action formalism in a gravitational context. Specifically we argue that Γ k encodes information about ‘quantum geometries’ at different resolution scales.

Recall that the effective average action Γ k may be regarded as the standard effective action where the bare action has been modified by the addition of the mode cutoff term Ck. Every given (exact or truncated) renormalization group (RG) trajectory can be viewed as a collection of effective field theories {Γ , 0 ≤ k ≤ ∞ } k. In this sense a single fundamental theory gives rise to a double infinity of effective theories – one for each trajectory and one for each value of k. As explained in Appendix C, the motivation for this construction is that one would like to be able to ‘read off’ part of the physics contents of the theory simply by inspecting the effective action relevant to the problem under consideration. If the problem has only one scale k, the values of the running couplings and masses in Γ k may be treated approximately as classical parameters.

In a gravitational context this construction has been described in Section 4.1. The above effective field theory aspect has also been used in Equation (2.62View Equation) of Section 2.4. Here we elaborate on the significance of the solutions of the effective field equations that come with it. The two distinct versions have been discussed in Item 6 of the previous Section 4.1.1; here we consider

δ¯Γ-k[ˇgk] = 0. (4.26 ) δgαβ
Formally these stationarity equations can be viewed as a 1-parametric family of in general nonlocal generalizations of the Einstein field equations. The principle of generating or selecting physical solutions via an underlying physically acceptable state has already been described in Section 2.3. We resume this discussion below. The solutions ˇgk of Equation (4.26View Equation) are k-dependent simply because the equations are. The problem of identifying ‘the same’ solution for different k can in principle be addressed by introducing an evolution equation for ˇgk, schematically obtained by differentiating Equation (4.26View Equation) with respect to k.

We now select a state which favors geometries that are smooth and almost flat on large scales as in Section 2.3. We can think of this state as a background dependent expectation functional 𝒪 ↦→ ⟨𝒪 ⟩¯g where the background has been self-consistently adjusted through the condition ⟨gαβ⟩¯g [g] = g¯αβ ∗ (see Equation (2.48View Equation)). This switches off the source, and any fixed point of the map g ↦→ ¯g∗[g] gives a particular solution to Equation (4.26View Equation) at k = 0, implicitly referring to the underlying state [156Jump To The Next Citation Point].

In terms of the effective average action ¯Γ k the state should implicitly determine a family of solutions {ˇgk,0 ≤ k ≤ ∞ }. Structures on a scale k1 are best described by Γ k1. In principle one could also use Γ k2 with k2 ⁄= k1 to describe structures at scale k1 but then a further functional integration would be needed. It is natural to think of the family { ˇgk,0 ≤ k ≤ ∞ } as describing aspects of a “quantum spacetime”. By a “quantum spacetime” we mean a manifold equipped with infinitely many metrics; in general none of them will be a solution of the Einstein field equations. One should keep in mind however that the quantum counterpart of a classical spacetime is characterized by many more data than the metric expectation values ˇgk alone, in particular by all the higher functional derivatives of Γ k evaluated at ˇgk. The second derivative for instance evaluated at ˇgk is the inverse graviton propagator in the background ˇg k. Note that all these higher multi-point functions probe aspects of the underlying quantum state.

By virtue of the effective field theory properties of Γ k the interpretation of the metrics ˇgk is as follows. Features involving a typical scale k1 are best described by Γ k1. Hence ˇgk1 is the average metric detected in a (hypothetical) experiment which probes aspects of the quantum spacetimes with typical momenta k1. In more figurative terms one can think of ˇgk as a ‘microscope’ whose variable ‘resolving power’ is given by the energy scale k.

This picture underlies the discussions in [135Jump To The Next Citation Point134Jump To The Next Citation Point] where the quantum spacetimes are viewed as fractal-like and the qualitative properties of the spectral dimension (2.53View Equation) have been derived. We refer to Section 2.4 and [135Jump To The Next Citation Point134Jump To The Next Citation Point] for detailed expositions. The fractal aspects here refer to the generalized ‘scale’ transformations k → 2k, say. Moreover a scale dependent metric ˇgk associates a resolution dependent proper length to any (k-independent) curve. The k-dependence of this proper length can be thought of as analogous to the well-known example that the length of the coast line of England depends on the size of the yardstick used.

Usually the resolving power of a microscope is characterized by a length scale ℓ defined as the smallest distance of two points the microscope can distinguish. In the above analogy between the effective average action and a “microscope” the resolving power is implicitly given by the mass scale k and it is not immediate how k relates to a distance. One would like to know the minimum proper distance ℓ(k1) of two points which can be distinguished in a hypothetical experiment with a probe of momentum k1, effectively described by the action Γ k1. Conversely, if one wants to ‘focus’ the microscope on structures of a given proper length ℓ one must know the k-value corresponding to this particular value of ℓ. For non-gravitational theories in flat Euclidean space one has ℓ(k ) ≈ π∕k, but in quantum gravity the relation is more complicated.

Given a family of solutions {ˇgk, 0 ≤ k ≤ ∞ } with the above interpretation the construction of a candidate for ℓ(k) proceeds as follows [135Jump To The Next Citation Point134Jump To The Next Citation Point184Jump To The Next Citation Point]: One considers the spectral problem of the (tensor) Laplacian − Δ ˇgk associated with ˇgk. To avoid technicalities inessential for the discussion we assume that all geometries in the family {ˇgk} are compact and closed. The spectrum of − Δ ˇgk will then be discrete; we write − Δ φ (ˇg ) = ℰ (ˇg )φ (ˇg ) ˇgk n k n k n k, n ∈ ℕ, for the spectral problem. As indicated, both the spectral values ℰn(ˇgk) and the eigenfunctions φn(ˇgk) will now depend on k.

The collection of eigenfunctions

{ 2} φn(ˇgk) with n such that ℰn(ˇgk) ≈ k , (4.27 )
will be referred to as “cutoff modes at scale k”. The significance of these modes can be understood by returning to the original functional integral (4.3View Equation), which one can think of being performed over the (tensorial) eigenmodes of the k-independent Laplacian − Δ ¯g of the background metric considered. Schematically Γ¯k[¯g ] contains information about the functional integral (4.3View Equation) where all the eigenmodes with spectral values ℰn (g¯) obeying ℰn(¯g) ≥ k2 have been integrated out; the modes with ℰn (¯g) ≈ k2 are ‘the last’ to be integrated out. If one now takes for the background metric ¯g the itself k-dependent solution ˇg k, the equation ℰ (ˇg ) ≈ k2 n k implicitly defines n and hence selects the “cutoff modes at scale k”. Note that at the level of ¯ Γ k the two metrics ⟨gαβ⟩ and ¯gαβ are already identified.

Given a wave function in Equation (4.27View Equation) one can now ask what a typical coordinate distance Δkx α is over which it varies. Converting this into a proper distance with the metric ˇgk defines the proposed resolving power

[ α β]1∕2 ℓ(k;x) := (ˇg(x )k)αβ Δkx Δkx . (4.28 )
The definition (4.28View Equation) is motivated by the fact that the “last set of modes integrated out” should set the length scale over which the (covariant version) of the averaging has been performed. In this sense ℓ(k;x) is a substitute for the “blocking size” in the spirit of Kadanoff–Wilson. The scale ℓ(k;x) may vary with the point x on the manifold considered. A case study of the relation between ℓ and k in a simple situation can be found in [184].

Since ¯ Γ k[g ] depends on the choice of the mode-cutoff scheme so will the solutions of Equation (4.26View Equation), and hence the resolving power ℓ(k;x ). It can thus not be identified with the resolving power of an actual experimental set up, but is only meant to provide an order of magnitude estimate. The scheme independence of the resolution which can be achieved in an actual experimental set up would in this picture arise because the scheme dependence in the trajectory cancels against that in the ℓ versus k relation.

This concludes our presentation of the effective average action formalism for gravity. In the next Section 4.3 we will use the FRGE for Γ k as a tool to gain insight into the gravitational renormalization flow.

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