List of Footnotes

1		There can be circumstances for which energy is not the criterion used to define the effective theory, and for which $ℒeff$ is not real. The resulting failure of unitarity in the effective theory reflects the possibility in these theories of having states in the effective theory converting into states that have been removed in its definition.
2		We return below to a discussion of how effective Lagrangians can be defined using dimensional regularization.
3		Examples where $H$ is not bounded from below can arise, such as for charged particles in a sufficiently strong background electric field [133 ]. In such situations the runaway descent of the system to arbitrarily low energies is interpreted as being due to continual particle pair production by the background field.
4		A Killing vector field satisfies the condition $ˆ ˆ ˆ ˆ ∇ μK ν + ∇ νKμ = 0$ , which is the coordinate-invariant expression of the existence of a time-translation invariance of the background metric. The carets indicate that the derivatives are defined, and the indices are lowered by the background metric $ˆgμν$ .
5		For example, this could happen for a charged particle in a decreasing magnetic field if the effective theory is set up so that the dividing energy $Λ(t)$ is not time dependent. Then Landau levels continuously enter the low-energy theory as the magnetic field strength wanes.
6		These authors have slightly different spins on the more philosophical question of whether trans-Planckian physics is likely to be found to be non-adiabatic.
7		In the inflationary context we take ‘adiabatic vacuum’ to mean the Bunch–Davies vacuum [26 ]. See, however, [45 , 46 , 17 , 64 , 65 , 76 , 77 , 39 , 40 ] for arguments against the use of non-standard vacua in de Sitter space.
8		The point of the non-relativistic power-counting of the previous section is to show that the third, large, $r$ -independent dimensionless quantity $GMiMj ∕ℏc$ does not appear in the interaction energy.
9		Notice that the curvature-squared terms can no longer be eliminated by performing field redefinitions once classical sources are included. Instead they can only be converted into the direct source-source interactions in which we are interested.
10		The necessity for renormalizing $a$ and $b$ in addition to Newton’s constant at one loop reflects the fact that general relativity is not renormalizable. Still higher-curvature terms would be required to absorb the divergences at two loops and beyond.