On Wed, Jun 10, 2020 at 11:48 PM smitra <smi...@zonnet.nl> wrote: > On 09-06-2020 01:32, Bruce Kellett wrote: > > On Mon, Jun 8, 2020 at 10:33 PM smitra <smi...@zonnet.nl> wrote: > >> > >> You are confusing the nontechnical introduction for the rigorous > >> content that comes later. > > > > Where later? The only justification offered for the addition of a > > random "fluctuation" field to the classical background inflaton field > > is the hand-waving heuristics of the introduction. Sure, he is > > reasonably rigorous in his quantization of this added "fluctuation" > > field, but that does not justify it in the first place. > > It is mentioned that the field can be treated in a classical way with > refs to the literature. You then only need a quantum description for the > fluctuations, so the classical field is treated as effective background > field. >
I am aware of this. This was, in fact, the problem that I pointed to. This addition of "fluctuations", even if they are quantized rigorously, is still unjustified and ad hoc. >> > >> They do proceed in a heuristic way, but this is not unjustified. > >> Your arguments against it based on energy conservation are not valid. > > > > Oh! Where do my arguments based on energy conservation fail? > > Your arguments fail because you are considering the total energy of the > entire universe. If you consider the energy of a field in a box and > impose boundaryy conditions then you have closed system and you can > consider the system to be in an eigenstate of the Hamiltonian where the > total energy and the square of the energy are well defined and the > expectation value of the latter is then equal to the square of the > former, so no fluctuations. > > But the problem at hand is to consider the local energy density at some > position. This is not conserved! > Energy and momentum are conserved locally, even in GR where global energy conservation fails in a non-static universe. > >> And if > >> it were as simple as that then no one in that field who are all big > >> experts in QFT would write articles saying that quantum fluctuations > >> are a source of the density fluctuations. > > > > That is just an argument from authority -- which justifies nothing. > > After all, there was a time when all the authorities thought that the > > stars were attached to a crystalline "celestial sphere", and that the > > earth was the centre of the universe (and flat!). > > What I'm saying is not that we just need to blindly trust the experts, > but rather that it's not plausible that with their level of expertise > they could have overlooked a counterargument based on elementary quantum > mechanics. > What we need is a coherent argument. As pointed out, experts are frequently wrong, and science is not decided by consensus. >> The energy density of a field does > >> have a variance just like the field strength itself has, and this > >> then does couple to gravity. > > > > In quantum mechanics, all that can have variances are superpositions > > of eigenstates. Conservation laws forbid variations of energy (or > > other conserved quantities) in eigenstates. The vacuum is, by > > definition, an energy eigenstate (the lowest possible energy state), > > so its energy cannot fluctuate, and does not have a variance. > > Similarly for a simple harmonic oscillator, and the SHO is a model for > > the modes (energy eigenstates) that make up a general quantum field. > > > > The vacuum energy from zero point energies of quantum fields does not > > couple to gravity -- that is the 120 orders of magnitude mistake about > > the origin of the cosmological constant. The non-connected vacuum > > loops of perturbation theory are all of strictly zero energy, and they > > do not couple to gravity. If they did, they would no longer be > > non-connected, and would merely form standard radiative corrections to > > propagators or vertex functions. > > > > You only have decoupled SHO in momentum space, assuming that there are > no couplings to other fields or a phi^4 self-interaction. Do you not know what 'decoupled' means? It means that although fields may self-interact (through phi^4 terms, for instance), and may interact with the other fields in the theory, there are no external legs, or couplings to external systems. Such decoupled closed loops in QFT do not contribute to the physics, and they are strictly of zero energy. In real space > the SHO are coupled via the 1/2 (nabla psi)^2 term. I think that in inflation theory, gradient terms in the fields are generally neglected as irrelevant compared to the time variation. Besides, such non-local effects are irrelevant for local energy conservation. So, the local energy > in a small volume is not contained in a set of SHO that are decoupled > from the other oscillators. The coupling is trivial in the sense that > one can decouple the oscillators by performing a Fourier-transform, but > you are then working with linear combinations of the SHO in real space. > Maybe that is what I mean when I say that only in superpositions of energy eigenstates can you have fluctuations, or a variance of the expectation value over repeated measurements. The simplest analogue is to consider a system of two SHOs: > > H = 1/(2 m) (p1^2 + p2^2) + m/2 omega^2 (x1^2 + x2^2) + g (x1-x2)^2 > > If g = 0 then we have two independent oscillators with angular frequency > omega. But g is not zero, this is analogous to the squared gradient term > in field theory. If you couple the oscillators, they are not independent. Duh! Just like in that case the coupling can be eliminated > by a transformation. You then get two independent oscillators, but they > are now not localized in the old coordinates. You need to consider the > energy of not the new oscillators, but the energy contained in each of > the original oscillator: > > H1 = p1^2/(2m) + (m/2 omega^2 +g)x1^2 -g x1 x2 > > H2 = p2^2/(2m) + (m/2 omega^2 +g)x2^2 -g x1 x2 > > > The expectation value of these energies do fluctuate. > You can introduce coupled harmonic oscillators, but that is not how you form a quantized field theory. Such fluctuations arise from non-local couplings -- they are not fluctuations of the original quantum field. Energy-momentum is locally conserved, even in GR and an expanding universe. Bruce -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. 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