Digesting a Paper on Flatness (Part 4)

See Part 1 to understand what this is about.

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One of us has given a formal argument, based on Picard-Lefschetz theory, that a path integral for a quantum gravitational transition amplitude can never yield a positive semiclassical exponent [18, 19, 20]. However, for a statistical ensemble, a formal argument indicates precisely the opposite.

See Part 3 for the semiclassical exponent, iS/ħ. A transition amplitude, also known as a probability amplitude, is a weighting of the probability of a particular state (or eigenstate).  This is a complex number (with a real and an imaginary component), which reflects the wave function.

Consider, for example, the partition function Z(β) = Tr(e^-βH). The time reparameterization invariance of general relativity means that the Hamiltonian H vanishes on physical states [^{we only consider cosmologies in which space is compact}].

The Hamiltonian H is the “sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with (a) system”.  Note that it does not include the inherent energy of the particles (ie that pertaining to the mass of the particles).  Basically, all that is being said here is that if we consider a system as having no momentum and no potential, then the total energy is given only by E=mc².  Note that the Wikipedia article on partition functions states that “the dimension of e^-βH is the number of energy eigenstates of the system” (there’s a slight difference in that they give the Hamiltonian H a hat, maybe to distinguish it from the Hubble parameter, H).

Thus, Z = e^S simply counts the number of states. If Z is approximated by a saddle, the semiclassical exponent must be positive.

The equation Z = e^S is a rewording of a generalisation of the Boltzman equation (S = k_B log W generalises to S = -k_B Σp_i ln p_i – which can be rearranged to [dropping the subscripts i and B for ease of representation] S = k ln p^-Σp and so e^S = p^-Σp+k).  For any non-trivial (that is small) number of states, it’s true that S must be positive (and the larger, the more states) – if real (see Part 3), unless Z = e^S has been normalised.

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Hopefully it’s quite obvious that this series is more of an open pondering session rather than any statement of fact about what the authors of Gravitational entropy and the flatness, homogeneity and isotropy puzzles intended to convey.  If I have misinterpreted them, then I’d be happy to hear about it.