Approaches to Quantum Gravity

The spin foam representation of loop quantum gravity 273Constraint Hamiltonianvector fieldΓgauge orbitCONSTRAINTSURFACEreducingredquantizingquantizingH kinreducing(spin foam rep)H physΓredFig. 15.1. On the left: the geometry of phase space in gauge theories. On theright: the quantization path of LQG (continuous arrows).phase space variables – schematically C(p, q) = 0for(p, q) ∈ Ɣ – which arereferred to as constraints. The constraints restrict the set of possible states of thetheory by requiring them to lie on the constraint hyper-surface. In addition, throughthe Poisson bracket, the constraints generate motion associated with gauge transformationson the constraint surface (see Fig. 15.1). The set of physical states (theso called reduced phase space Ɣ red ) is isomorphic to the space of orbits, i.e. twopoints on the same gauge orbit represent the same state in Ɣ red described in differentgauges (Fig. 15.1).In general relativity the absence of a preferred notion of time implies that theHamiltonian of gravity is a linear combination of constraints. This means thatHamilton equations cannot be interpreted as time evolution and rather correspondto motion along gauge orbits of general relativity. In generally covariant systemsconventional time evolution is pure gauge: from initial data satisfying the constraintsone recovers a spacetime by selecting a particular one-parameter family ofgauge-transformations (in the standard ADM context this amounts to choosing aparticular lapse function N(t) and shift N a (t)).From this perspective the notion of spacetime becomes secondary andthe dynamical interpretation of the the theory seems problematic (in the quantumtheory this is referred to as the “problem of time”). A possible reason forthis apparent problem is the central role played by the spacetime representationof classical gravity solutions. However, the reason for this is to a large part dueto the applicability of the concept of test observers (or more generally test fields)