vanhees71 said:
The divergences of higher-order perturbative contributions to the S-matrix elements in relativistic QFT come from a too sloppy use of the field operators, which are operator-valued distributions and thus cannot be so easily multiplied at the same space-time argument. This is however just sloppyness of the physicists and can be cured by using "smeared operators" as in the Epstein-Glaser approach. Then everything is finite right from the beginning, but that's not the point concerning the dark-energy/cosmological-constant problem.
Absolutely: it is much too often not mentioned that field operators are operator-valued singular distributions, hence the infinities occurring in QFT are the result of exactly this "sloppy" use and must be dealt with by the renormalization programme. However, as dirty as it might look, it works, and at least eventually remedies the mathematically ill-defined calculational approach in the end, with spectacular results.
By the way, I am not overly knowledgeable on the Epstein--Glaser approach, but I have intended to give the books by Günter Scharf a read some time.
And also yes: I don't think it has anything to do with the cosmological constant problem.
Even when everything is finite, which you can as well achieve with the more handwavy approach using some other regularization than "smearing" like dim. reg. or just at any stage calculate everything with physical quantities and add appropriate counterterms to the self-energy, proper-vertex functions, and particularly also our case of 1PI vacuum diagrams as in the (generalized) BPHZ approach. The problem is that you have to choose a renormalization scale and a renormalization prescription and at each (loop) order perturbation theory fit the finite parameters (wave-function normalization factors, masses, coupling constants) to some measured cross sections. The quantities like physical masses and couplings are then defined by the renormalized, connected n-point Green's functions. E.g., the masses of particles or masses and widths of resonances are given by the corresponding poles of the one-particle Green's functions. In this way, to the order the perturbation theory is evaluated, the S-matrix elements (cross sections) are invariant under the choice of the energy-momentum scale and the renormalization scheme,
I do agree so far. But all of the above applies to QFT in general, in simple Minkowski space. One would think that it would also apply to QFT defined in curved spacetimes, but here's where additional issues come in.
but changing the renormalization scheme and/or the energy-momentum scale changes the renormalized parameters including the "zero-point energy", and this "running of parameters", determined by the renormalization-group equations, is the problem. Adapting everything at the low-energy scales needed to describe the outcome of our experiments with accelerators, implies a huge value of the "zero-point energy" when using the RG equations to calculate this parameter at higher energy-momentum scales like the GUT scale or even the Planck-mass scale. Here a tremendous finetuning is needed, and that's considered unnatural.
This is where I am not really able to follow. If am I am getting you wrong, my apologies, but are you hinting at the diffculty of extrapolating a QFT to regimes way beyond their scope of applicability? Again, this is an issue not of QFTs in curved spacetimes, but of QFTs in general, and is well-accepted by the philosophy of effective theory. Let alone the fact that no viable candidate of a GUT exists nowadays that is physically sufficiently predictive, or even mathematically well-defined.
Also, one general statement often mentioned is that in a general curved spacetime, normal ordering does not work anymore, as re-defining energy levels contradicts the fact that absolute energy levels count in GR. I agree to the latter, but not to the former per se. Because hang on, here's my first objection: we've just said that all along we are using a mathematically ill-defined procedure to handle infinities, and the logic is they are unphysical artifacts of the sloppy math applied, but now all of a sudden these mathematically ill-defined entities are to become physically meaningful?
What I do have understood though, and this seems quite natural using some basic notion of QFT, is the problem of being able to define a vacuum state in general. Lacking Poincare invariance in a general spacetime, it is simply not possible to have a notion of a "ground state" as being the one having lowest energy. The existence of a vacuum state, however, is necessary for defining a Fock space as we know it, and the vacuum is then unique. Stationary spacetimes, permitting a timelike Killing field, belong to the class that allow for this. Hence as I see it, standard QFT approach is in general not applicable anyway.
Now, as mentioned in a previous post, Birrell/Davies (
https://www.cambridge.org/core/books/quantum-fields-in-curved-space/95376B0CAD78EE767FCD6205F8327F4C) seems to be quite a good starting point for understanding the issues. However, repeating and understanding the calculations therein requires much more time and effort than I have been able to put in so far.
They devote a whole chapter §6 on the renormalization of the stress-energy tensor ##\langle T^μ{}_ν\rangle##. As said, without having reproduced any calculation in detail, there are indeed renormalization schemes, but they are more complicated than in flat spacetime. But still there are some highly symmetrical spacetimes that allow the calculation of ##\langle T^μ{}_ν\rangle_\textrm{ren}##. A massive, free scalar field on a de Sitter space, for example, leads to a renormalized stress-energy tensor as given in eq. (6.183), which also correctly implies the trace anomaly present in that case. The same scalar field defined on an Einstein static spacetime delivers an energy density of (eq. (6.190))
$$ρ=\frac{1}{480π^2 a^4},$$
where ##a## is the radius of the space 3-sphere. This would then lead to (##κ## is the Einstein gravitational constant)
$$Λ=\frac{1}{a^2}=\frac{κρ}{2},$$
and hence
$$Λ=\frac{κ}{960π^2 a^4}.$$
Now, as I must admit, I need more time to understand renormalization techniques in curved spacetimes. However, there seem to be very few explicit examples and even less mathematically strict results. My main critique is that all over the literature, seemingly simple results are shown, and eq. (3.5) in above-cited Weinberg's review is no exception. There is no way this equation is physically meaningful in any sense if the baseline for doing the respective calculations is as given in Birrell/Davies. For general spacetimes, a Fourier mode decomposition of a field is not even possible. How then is a quantity like ##\rho## even calculated in general?
Quantum effects in general and QFT on a curved spacetime in specific still seems to be a poorly understood concept, and I am wondering if we are actually in a position already to assess whether there is a cosmological constant problem or not.
