Ivancevic_Applied-Diff-Geom

1066 Applied Differential Geometry: A Modern Introduction(6.107). Without it, this expression is an infinite sum (since the volume canbecome arbitrarily large) of complex terms whose convergence propertieswill be very difficult to establish. In this situation, a Wick rotation is simplya technical tool which – in the best of all worlds – enables us to performthe state sum and determine its continuum limit. The end result will haveto be Wick–rotated back to Lorentzian signature.Fortunately, Lorentzian dynamical triangulations come with a naturalnotion of Wick rotation, and the strategy we just outlined can be carriedout explicitly in two space–time dimensions, leading to a unitary theory.In higher dimensions we do not yet have sufficient analytical control of thecontinuum theories to make specific statements about the inverse Wickrotation. Since we use the Wick rotation at an intermediate step, one canask whether other Wick rotations would lead to the same result. Currentlythis is a somewhat academic question, since it is in practice difficult to findsuch alternatives. In fact, it is quite miraculous we have found a singleprescription for Wick–rotating in our regularized setting, and it does notseem to have a direct continuum analogue (for more comments on this issue,see [Dasgupta and Loll (2001); Dasgupta (2002)]).Our Wick rotation W in any dimension is an injective map fromLorentzian– to Euclidean–signature simplicial space–times. Using the notationT for a simplicial manifold together with length assignments ls 2 andlt 2 to its space– and time–like links, it is defined byT lor = (T, {ls 2 = a 2 , lt 2 = −a 2 W}) ↦−→ T eu = (T, {ls 2 = a 2 , lt 2 = a 2 }).(6.108)Note that we have not touched the connectivity of the simplicial manifoldT , but only its metric properties, by mapping all time–like links of T intospace–like ones, resulting in a Euclidean ‘space–time’ of equilateral buildingblocks. It can be shown [Ambjørn et. al. (2001a)] that at the level of thecorresponding weight factors in the path integral this Wick rotation 14 hasprecisely the desired effect of rotating to the exponentiated Regge action ofthe ‘Euclideanized’ geometry,e iS(T lor )W↦−→ e−S(T eu) . (6.109)14 To get a genuine Wick rotation and not just a discrete map, one introduces a complexparameter α in l 2 t = −αa2 . The proper prescription leading to (6.109) is then an analyticcontinuation of α from 1 to -1 through the lower–half complex plane.