Brazilian Journal of Physics, vol. 35, no. 2B, June, 2005 481 The Feynman Propagator for Quantum Gravity: Spin Foams, Proper Time, Orientation, Causality and Timeless-Ordering D. Oriti Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK, EU Received on 19 December, 2004 We discuss the notion of causality in Quantum Gravity in the context of sum-over-histories approaches, in the absence therefore of any background time parameter. In the spin foam formulation of Quantum Gravity, we identify the appropriate causal structure in the orientation of the spin foam 2-complex and the data that characterize it; we construct a generalised version of spin foam models introducing an extra variable with the interpretation of proper time and show that different ranges of integration for this proper time give two separate classes of spin foam models: one corresponds to the spin foam models currently studied, that are independent of the underlying orientation/causal structure and are therefore interpreted as a-causal transition amplitudes; the second corresponds to a general definition of causal or orientation dependent spin foam models, interpreted as causal transition amplitudes or as the Quantum Gravity analogue of the Feynman propagator of field theory, implying a notion of ”timeless ordering”. 1 Introduction emerge only in a semiclassical approximation [3]. Regarding the issue of causality, there seem to be two basic attitude one This paper discusses the issue of causality in non-perturbative can take: one can hold that we cannot speak of causality in quantum gravity, and more precisely in the spin foam formu- absence of time and therefore just as time itself also a notion lation of the theory. This has implications for the broader causality should emerge and be applicable only in a semi- issue of time in a background independent quantum theory classical limit; this is actually true without argument, actually as well as rather more technical aspects related to the spe- a truism, if we stick to the conventional notion of causality in cific approach we deal with. We will try to highlight both terms of a given spacetime geometry, and of lightcone struc- these conceptual issues and the ideas on which our results are tures in a continuous manifold; however one can argue that based and what these results actually are. These results were the notion of causality is actually more primitive than that of presented in [1], and their presentation in section 3, 4 and 5 time, more fundamental, being already present in the notion is in fact based on [1], and will be analysed and discussed in of ordering between events, of a fundamental directionality more details in [2]. in spacetime, and that as such can be present, as a seed of what will then be, in a semiclassical approximation, the usual notion of causality, even if discrete or combinatorial or al- 1.1 Time and causality in Quantum Gravity gebraic structures are used to encode gravitational degrees of freedom and thus spacetime geometry in a formulation of In quantum gravity, because of background independence Quantum Gravity. We agree with this latter point of view. and diffeomorphism invariance, neither phsyical observables nor physical states, i.e. none of the basic elements of any canonical formulation of the theory, can depend on an exter- 1.2 Causality as ordering in Quantum Gravity nal time parameter as is the case in ordinary quantum me- chanics, but time itself becomes a quantum variable insofar Indeed, certain approaches to Quantum Gravity even take this as the metric itself is a quantum variable; this is true also in a notion of causality as order as the basic ingredient for their covariant or sum-over-histories formulation, because, even if constructions, the keystone on which to found the theory; it the boundary may be assigned timelike data, still the space- is the case of the causal set approach [4] where spacetime at time geometry cannot be used to define any unique notion of the most fundamental level is taken to be a set of fundamental time in the interior of the manifold as it represents the quan- events endowed with an ordering representing their causal re- tum field we are summing over to define our theory. This lations and the task is that of constructing a suitable quantum led to the conclusion that time can not be a basic ingredient amplitude for each configuration so specified, this amplitude in a proper formulation of Quantum Gravity, or that the very being the main ingredient ina sum-over-histories formulation concept of time is not fundamantal at all and should instead of quantum gravity as a sum over causal sets, i.e. a sum over 482 D. Oriti causally ordered or oriented, discrete sets of events. tum field theory, that is of course unavailable in this back- This idea of causality as ordering as a basic ingredient in ground independent context. The fact that this is possible a sum-over-histories formulation of the theory has a formal or not, thus the exact way the Quantum Gravity transition implementation also in the analytic path integral approach to amplitudes are defined allows a distinction between differ- quantum gravity as developed by Teitelboim [5], so not as- ent kinds of transition amplitudes, causal and a-causal ones. suming any discrete substratum for spacetime, and his con- More precisely [5], the distinction between the different tran- struction can be seen as a formal realization of what seems sition amplitudes in registering initial and final 3-geometry to be a general principle in any sum-over-histories quantum (h1; h2) in the boundary data of the path integral is obtained gravity theory: implement the causality principle by requir- by choosing different range of integration in the proper time ing that each history summed over, i.e. each spacetime geom- formulation of the theory; using a canonical decomposition etry, is oriented, that this orientation matches that of the of the metric variables, the role of proper time is played by boundary states/data and is registered in the quantum ampli- the lapse function; an unrestricted range of integration leads tude assigned to each history, such that, when a canonical to the (quantum gravity analogue of the) Hadamard function, formulation is available, the two boundary states can be dis- while a restriction to positive lapses leads to the (analogue of tinguished as either ’in-state’ or ’out-state’; this orientation the) Feynman propagator. The expressions for the two quan- dependence replaces the notion of ’time-ordering’ of quan- tities have just a formal meaning, but read: c Z +1 h R ³ ´ i 2 1 i 3 ij _ i GH (h ; h ) = DN e M d xdt ¼ hij ¡ NH¡ N Hi D¼Dh ¡1 Z +1 h R i 3 ij _ i 2 1 i d xdt(¼ hij ¡NH¡N Hi) GF (h ; h ) = DN e M D¼Dh : 0 d The idea behind this construction is best understood recall- The problem we address in this work is how to construct the ing basic facts from the theory of a quantum relativistic analogue quantities in a spin foam context, i.e. in a purely particle (that cna be thought indeed as quantum gravity in combinatorial, algebraic and group theoretic way, in absence 0 + 1 dimensions. There [6] the main difference between of any smooth manifold structure and any metric field. We the different transition amplitudes (and 2-point functions) one will see that this can be achieved in full generality and in a can construct, in particular the Feynman propagator and the very natural way. Hadamard function, is exactly the way they encode (or fail to encode) causality restrictions, in the fact that the amplitude registers or not that one of the two is in the causal future of the other; this order can be imposed in each of the histories 2 Spin Foam models of Quantum summed over in a clear way using a proper time formula- Gravity tion: starting from the same proper time dependent expres- sion g(x ; x ;T ), which can be given a sum-over-histories 1 2 Spin foam models [7, 8] are currently being studied as a new form, the Hadamard function is obtained eliminating this de- more rigorous implementation of the path integral approach pendence by integrating over the proper time variable with an to quantum gravity. As such they are constructed by a defi- infinite range over both positive and negative values; the re- nition of histories of the gravitational field, interpreted as 4- sulting amplitude is then a-causal, and real, and does not reg- dimensional geometries for a given spacetime manifold, and ister any ordering between its arguments; the Feynman propa- an assignment of quantum amplitudes to these geometries, gator is instead obtained restricting this integration to positive i.e. suitable complex functions of the geometric data char- or negative proper times only, and this ordering is precisely acterizing each history. Different models have been proposed what makes it causal; denoting G§ (x ; x ) the positive and W 1 2 and derived from many different points of view, including lat- negative Wightman function, the result is: tice gauge theory type derivations [9] and group field theory G (x ; x ) = G+ (x ; x ) + G¡ (x ; x ) = formulations [10], for both the Riemannian and Lorentzian H 1 2 W 1 2 W 1 2 signatures, and in different dimensions, which counts as one Z +1 of the attractive features of this approach. The peculiarity = dT g(x1; x2;T ) ¡1 of the spin foam framework, as compared with the tradi- tional path integral for gravity, is that the spacetime mani- 0 0 + GF (x1; x2) = θ(x2 ¡ x1) GW (x1; x2) + folds on which the gravitational data are given are combina- Z +1 torial and discrete ones, and specifically are given by combi- 0 0 ¡ +θ(x1 ¡ x2) GW (x1; x2) = dT g(x1; x2;T ) natorial 2-complexes, i.e.