Approaches to Quantum Gravity

Quantum Regge calculus 365emanating from the origin in the positive direction join it to vertices 1, 2, 4 and 8(coordinate edges), 3, 5, 6, 9, 10 and 12 (face diagonals), 7, 11, 13 and 14 (bodydiagonals) and 15 (hyperbody diagonal). Small perturbations are then made aboutthe flat space edge lengths, so thatl (i)j= l (i)0(1 + ɛ(i)j), (19.8)where the superscript i denotes the base point, the subscript j denotes the direction(1, 2,...,15) and l (i)0is the unperturbed edge length (1, √ 2, √ 3 or 2). Thus forexample, ɛ (1)14would be the perturbation in the length of the edge from vertex 1 inthe 14-direction, i.e. to vertex 15. The ɛs are assumed to be small compared with 1.The Regge action is evaluated for the hypercube based at the origin and thenobtained for all others by translation. The lowest non-vanishing term in the totalaction is quadratic in the variations (the zeroth and first order terms vanish becausethe action is zero for flat space and also flat space is a stationary point of the actionsince it is a solution of the Regge analogue of the Einstein equations). It can bewritten symbolically asS R = ∑ ɛ † Mɛ, (19.9)where ɛ is an infinite-dimensional column vector with 15 components per point andM is an infinite-dimensional sparse matrix. Since all the entries corresponding tofluctuations of the hyperbody diagonal are zero, these form a one-parameter familyof zero eigenmodes. It can also be shown that physical translations of the verticeswhich leave the space flat form a four-parameter family of zero eigenmodes. Theseare the exact diffeomorphisms in this case.The matrix M is then block diagonalised by Fourier transformation or expansionin periodic modes. This is achieved by settingɛ (a,b,c,d)j= (ω 1 ) d (ω 2 ) c (ω 4 ) b (ω 8 ) a ɛ (0)j, (19.10)where ω k = e 2πi/n k, k = 1, 2, 4, 8. Acting on periodic modes M becomes a matrixwith 15 × 15 dimensional blocks M ω along the diagonal. This submatrix has theschematic form⎛⎞A 10 B 0M ω = ⎝ B † 18I 4 0⎠ , (19.11)0 0 0where A 10 is a 10 × 10 dimensional matrix and B is a 4 × 10 dimensional one;their entries are functions of the ωs. Then M ω itself is block diagonalised by a nonunitarybut uni-modular similarity transformation, and the diagonal blocks are Z =A 10 − 1 18 BB† ,18I 4 and 0. The 4 × 4 unit matrix block means that the fluctuationsɛ j for j = 7, 11, 13, 14 have been decoupled; they are constrained to vanish by the