Approaches to Quantum Gravity

Does locality fail at intermediate length scales? 39(and flat) space would show no differences at all. (The massive case might tell adifferent story, though.)3.3.1 Fourier transform methods more generallyWhat we’ve just said is essentially that the Fourier transform of vanishesKnowhere in the complex z-plane (z ≡ k · k), except at the origin. But this drawsour attention to the Fourier transform as yet another route for arriving at a nonlocalD’Alembertian. Indeed, most people investigating deformations of seemto have thought of them in this way, including for example [9; 10]. They havewritten down expressions like exp( /K ), but without seeming to pay muchattention to whether such an expression makes sense in a spacetime whose signaturehas not been Wick rotated to (+ +++). In contrast, the operator of thisKpaper was defined directly in “position space” as an integral kernel, not as a formalfunction of . Moreover, because it is retarded, its Fourier transform is ratherspecial .... By continuing in this vein, one can come up with a third derivation ofas (apparently) the simplest operator whose Fourier transform obeys the analyticityand boundedness conditions required in order that be well-defined andKKretarded.The Fourier transform itself can be given in many forms, but the following isamong the simplest:Ke ik·x | x=0= 2zi∫ ∞0dte itz/K(t − i) 2 (3.11)where here, z =−k · k/2.It would be interesting to learn what operator would result if one imposed “Feynmanboundary conditions” on the inverse Fourier transform of this function, insteadof “causal” ones.3.4 What next?Equations (3.7) and(3.6) offer us two distinct, but closely related, versions of ,one suited to a causet and the other being an effective continuum operator arisingas an average or limit of the first. Both are retarded and each is Lorentz invariant inthe relevant sense. How can we use them? First of all, we can take up the questionsabout wave-propagation raised in the introduction, looking in particular for deviationsfrom the simplified model of [15] based on “direct transmission” from sourceto sink (a model that has much in common with the approach discussed aboveunder the heading “First approach through the Green function”). Equation (3.7),