Abstract
In this chapter we return to a foundational issue raised in Sect. 1.3 called the reconstruction problem. In the discussions that follow we phrase all arguments in terms of a single-component scalar field so that none of the extra structure that comes along when dealing with metric degrees of freedom plays a role.
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- 1.
See also Ref. [4].
- 2.
Throughout this chapter we will often refer to \(\Gamma _k\) by its alternative name, the Legendre effective action, in accordance with Ref. [1].
- 3.
All momenta should be understood to be cutoff from above by \(\Lambda \), including that in the momentum integral implied by the space-time trace. The mass parameter M introduced in Ref. [2] will play no significant role here so will be neglected. Also in contrast to Ref. [2], we will not make the momenta discrete by compactifying on a torus.
- 4.
Here we commit a slight abuse of notation. Strictly in order for the action to reach a fixed point, we should change to the appropriate dimensionless variables. By \(\hat{S}^{*}\) we actually mean the action such that it takes the fixed point form after such a transformation.
- 5.
- 6.
This goes beyond the sum rule introduced in Ref. [1] since we now allow \(\tilde{C}^\Lambda \) to be unrelated to \(C^k\).
- 7.
Up to a constant of proportionality. We ignore these from now on.
- 8.
It can also be shown that this is consistent to one loop with the solution (2.6.4).
References
Morris TR (1994) The Exact renormalization group and approximate solutions. Int J Mod Phys A 09:2411–2450 arXiv: hep-ph/9308265
Manrique E, Reuter M (2009) Bare action and regularized functional integral of asymptotically safe quantum gravity. Phys Rev D79:025008 arXiv:0811.3888
Manrique E, Reuter M (2011) Bare versus effective fixed point action in asymptotic safety: the reconstruction problem. PoS CLAQG08:001 arXiv:0905.4220
Vacca G, Zambelli L (2011) Functional RG flow equation: regularization and coarse-graining in phase space. Phys Rev D 83:125024 arXiv:1103.2219
Ashtekar A, Lewandowski J (2004) Background independent quantum gravity: A Status report. Class Quant Grav 21:R53 arXiv: gr-qc/0404018
Thiemann T, Modern canonical quantum general relativity arXiv: gr-qc/0110034
Rovelli C (2008) Loop quantum gravity. Living Rev Rel 11:5
Ambjørn J, Goerlich A, Jurkiewicz J, Loll R (2012) Nonperturbative quantum gravity. Phys Rept 519:127–210 arXiv:1203.3591
Ashtekar A, Reuter M, Rovelli C, From general relativity to quantum gravity arXiv:1408.4336
Hamber HW (2015) Scaling exponents for lattice quantum gravity in four dimensions. Phys Rev D92(6):064017 arXiv:1506.0779
Liu RG (2016) Williams RM Regge calculus models of closed lattice universes. Phys Rev D93(2):023502 arXiv:1502.0300
Litim DF (2000) Optimization of the exact renormalization group. Phys Lett B486:92–99 arXiv: hep-th/0005245
Litim DF (2001) Optimized renormalization group flows. Phys Rev D 64:105007 arXiv: hep-th/0103195
Litim DF (2001) Mind the Gap. Int J Mod Phys A16:2081–2088 arXiv: hep-th/0104221
Morris TR, Slade ZH (2015) Solutions to the reconstruction problem in asymptotic safety. JHEP 11:094 arXiv:1507.0865
Hasenfratz P, Niedermayer F (1994) Perfect lattice action for asymptotically free theories. Nucl Phys B 414:785–814 arXiv: hep-lat/9308004
Morris TR (2005) Equivalence of local potential approximations. JHEP 0507:027 arXiv: hep-th/0503161
Hamzaan Bridle I, Morris TR (2016) Fate of nonpolynomial interactions in scalar field theory. Phys Rev D94:065040 arXiv:1605.0607
Polchinski J (1984) Renormalization and effective lagrangians. Nucl Phys B 231:269–295
Morris TR (1998) Elements of the continuous renormalization group. Prog Theor Phys Suppl 131:395–414 arXiv: hep-th/9802039
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Appendix 1: Why a UV Regulated Effective Average Action Must Depend on the UV Regulator
Appendix 1: Why a UV Regulated Effective Average Action Must Depend on the UV Regulator
It is clear that at least for a general form of UV cutoff, the effective average action \(\hat{\Gamma }_{k}^\Lambda [\varphi ]\) must depend on the UV regulator \(\Lambda \) as indicated. Indeed if we embed the UV cutoff in the free propagator as done in (2.2.9) then the Feynman diagrams that follow from its perturbative expansion will evidently have all free propagators \(1/p^2\) replaced by \(\Delta ^\Lambda _k(p)\). The fact that \(\hat{\Gamma }_{k}^\Lambda [\varphi ]\) thus depends on two scales, means that a bare action cannot be reconstructed which would directly give the continuum version \(\hat{\Gamma }_k\) in the usual way. This is the first “severe issue” outlined above (2.1.2).
Following Ref. [2], a sharp UV cutoff and infrared optimised cutoff would appear to provide an exception however. With a sharp UV cutoff in place, (2.1.1) can alternatively be written
where the first space-time trace leads to an unrestricted momentum integral
and we mean that the second term, the “remainder term”, has the momentum integral defining the trace restricted to \(|p|>\Lambda \) as indicated. With the optimised IR cutoff profile we have \(\partial R_k(p)/\partial k = 2k\theta (k^2-p^2)\) and thus, since \(k\le \Lambda \), the remainder term vanishes in this case. At first sight this would appear then to allow us to consistently set \(\hat{\Gamma }_{k}^{\Lambda }[\varphi ] = \hat{\Gamma }_{k}^{}[\varphi ]\) in (2.8.1) (providing only that we restrict flows to \(k\le \Lambda \)), meaning that for these choice of cutoffs, the dependence of the effective average action on \(\Lambda \) disappears. This is not correct however as can be seen by expanding the inverse kernel. Define the full inverse propagator as
(temporarily suppressing the k and \(\Lambda \) dependence) and similarly define \(\Gamma ^{\prime }[\varphi ]\) to be the remainder after the term quadratic in the fields is removed (which thus starts at \(\mathcal{O}(\varphi ^3)\) in a field expansion). Then
The momentum q is the external momentum injected by the fields remaining in \(\Gamma '\):
where we have displayed as a simple example the 1PI three-point vertex defined as in (2.5.2). (The higher point vertices will have an integral over the field momenta with a delta-function restricting the sum to \(-q\).) With a sharp UV cutoff in place, not only are the external momenta \(|q|\le \Lambda \) restricted, but the momentum running through any internal line is also restricted, thus here we also have \(|p+q|\le \Lambda \). This is because ultimately all the free propagators come (via Wick’s theorem) from a Gaussian integral over the fields \(\phi (r)\) in the path integral whose momenta \(|r|\le \Lambda \) have been restricted by the sharp UV cutoff. Although the momentum p already has a sharp UV cutoff k provided by \(\partial R_k(p)/\partial k\) which means the overall UV cutoff \(\Lambda \) is invisible for it, this invisibility does not work for the other internal momenta, such as \(p+q\), hidden in the construction of the inverse kernel. In other words even if the argument p above is freed from its UV cutoff at \(\Lambda \), this cutoff remains inside the construction in all the internal propagators, such as displayed in (2.8.4), and thus despite appearances the first term on the right hand side of (2.8.1) actually still does depend non-trivially on \(\Lambda \), implying also that \(\hat{\Gamma }_{k}^\Lambda [\varphi ]\) is a non-trivial function of \(\Lambda \).
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Slade, Z.H. (2019). Solutions to the Reconstruction Problem. In: Fundamental Aspects of Asymptotic Safety in Quantum Gravity. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-19507-6_2
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