G. Esposito

1 - Historical background1 - Historical background

2 - GR singularities in the sixties2 - GR singularities in the sixties

3 - Feynman quantization of GR3 - Feynman quantization of GR

4 - Quantum cosmology4 - Quantum cosmology

5 - Singularity avoidance?5 - Singularity avoidance?

6 - Assessment6 - Assessment

• Stephen Hawking, a Ph.D. student of Dennis Sciama, is diagnosed with ALS on his way back from a trip with his father.

• Dennis Sciama, worried by the slow progress of his student, suggests looking at a paper by Penrose in Phys. Rev. Lett., which applies global methods to gravitational collapse of stars.

• Hawking understands how to apply such techniques to the early Universe.

• Spacetime is singularity-free if it is timelike and null geodesically complete (definition used to prove theorems by Hawking, Penrose and Geroch).

• The Hawking-Penrose theorem: spacetime singularities are a generic property of GR, no matter how symmetric or asymmetric is our model of the Universe.

3.1 Fully gauge-invariant boundary conditions in QFT

• Both (1) and (2) are preserved under unfinitesimal gauge transformations provided that the ghost obeys homogeneous Dirichlet conditions as in (3). For gravity, we choose Ф so as to have an operator Ρ of Laplace type in the Euclidean theory.

• On the Euclidean 4-ball, we expand metric perturbations һμν in terms of scalar, transverse vector, transverse-traceless tensor harmoncs on S³. For vector, tensor and ghost modes, boundary conditions reduce to Dirichlet or Robin. For scalar modes, one finds eventually the eigenvalues E ═ X² from the roots X of

(4) Jיn (x)± n Jn (x) ═ 0,

(5) Jיn (x)+(- ± ) Jn (x) ═ 0.

5.1 Eigenvalue condition for scalar modes

_x

x_2

_nx

Note that both x and –x solve the same equation.

5.2 Four spectral zeta-functions for scalar modes

Four spectral zeta-functions

5.3 Regularity at the origin of ζB+

5.3 Regularity at the origin of ζB+

5.3 Regularity at the origin of ζB+

5.3 Regularity at the origin of ζB+

5.3 Regularity at the origin of ζB+

5.3 Regularity at the origin of ζB+

• From: gts@pku.edu.cnSubject: »Ø¸´: My opinion.Date: Wed, November 2, 2005 8:38 pmTo: "Esposito Giampiero" gesposit@na.infn.it

• Greetings Giampiero, Thank you for offering your opinion on the issue which I raised in my critique of your work. However, I am still not sure whether I understand your point. You write that only fully diffeomorphism-invariant boundary conditions lead to a positive zeta(0) value for pure gravity on the 4-ball. Logically then, only fully diffeomorphism-invariant boundary conditions lead to a vanishing cosmological wave function for vanishing three geometries at one-loop level at least on the Euclidean 4-ball.

• Yet, this is not what you write. You write, instead, that your finite one-loop wave function for pure gravity in the presence of a 3-sphere boundary is not recovered within other schemes. The issue which I raised related to a vanishing wave-function for vanishing three geometries, not simply to a finite one. In hep- th/0507264 you did not simply arrive at a finite and non-vanishing value for that wave-function at one-loop but at a vanishing value instead. Then you interpreted that result as a quantum avoidance of the cosmological singularity, which was a point on which I raised an objection because any finite value results in such a quantum avoidance. In addition, even if in other schemes the cosmological wave function for pure gravity turns out to be divergent at one-loop, as you write, the procedure of perturbative renormalisation remains

• always valid at the scales involved and its resulting finite value would always be the physically acceptable value for the cosmological wave function for pure gravity at one-loop order. Finally, there is the fundamental issue raised by the presence of a non-trivial boundary in Euclidean Quantum Cosmology. As you write, your results do not take it into consideration. As I have shown in my work on such bounded manifolds the boundary will not affect any ultraviolet divergence and, for that matter, the cut-off scale behaviour of the theory. However, it will certainly affect the finite part of the associated amplitude and, for that matter, the effective action and the cosmological wave function at one loop. Two serious implications follow from this fact. First, even if - through your scheme - you obtain a vanishing one-loop wave function of the Universe at the limit of small three-geometries on the unbounded 4-ball (S_4) I would certainly expect boundary contributions to alter that zero value on the bounded C_4. Second (and more physically important), the one-parameter ambiguity of renormalisation will inevitably drive the one-loop cosmological wave function away from its zero value which you claim. In fact, this result is as relevant on S_4 as it is on C_4.

• Your calculation is mathematically important and physically significant. However, with all due respect, on account of the above- stated reasons I am not yet convinced that you have achieved a breakthrough in the current understanding of the issue of the big-bang singularity. I share your enthusiasm for Quantum Cosmology. Higher loop contributions to the cosmological wave-function are, indeed, a real challenge. I have actively pursued them myself and I shall return to them when I have developed a better intuitive feeling as to the manner in which the quantum Universe avoids the classical "big-bang" singularity. Hopefully very soon. Best regards, George

• R. Penrose and S.W. Hawking, Proc. R. Soc. Lond. A314, 529 (1970).

• J.B. Hartle and S.W. Hawking, Phys. Rev. D28, 2960 (1983).

• S.W. Hawking, Nucl. Phys. B239, 257 (1984).

• G. Esposito, G. Fucci, A.Y. Kamenshchik and K. Kirsten, Class. Quantum Grav. 22, 957 (2005); JHEP 0509, 063 (2005).

Thank you