I think the asymptotic safety program is fascinating but technically very difficult. The basic idea is that while vanilla quantum gravity is not renormalizable in the usual sense, it still might flow to an attractive ultra-violet fixed point as we increase the energy scale: just not a fixed point that’s a free field theory of gravitons! The problem is that people typically compute renormalization group flows using approximations (e.g. the “1-loop approximation”, “2-loop approximation” and so on, but also others). I worry whether these approximations can be trusted when looking for an ultraviolet fixed point for quantum gravity.

It’s not just nitpicking over rigor. I’m really worried that these approximations could break down and give seriously wrong results! You can see more specifics here:

• Jacques Distler, Asymptotic safety.

]]>In addition, it is known that

(1) The standard model is perturbatively renormalizable;

(2) In the IR it flows away from and not into the Gaussian fixed-point.

From all of this it follows that the UV theory – with a cutoff – lives somewhere close to the critical surface of the Gaussian fixed-point. One manifestation of the fact that the cutoff cannot be removed is the Landau pole.

The issue of fine tuning, such as there is one, is that since the Higgs mass suffers additive renormalization, the UV theory must be tuned to be agonizingly close to – but not quite on – the critical surface. Interestingly, for fermions, chiral symmetry ensures that the renormalized mass is proportional to the bare mass and so any tuning is much less severe.

]]>I’ve posted the derivation I mentioned earlier to my blog, because I have in the past wanted to have it available, for which the link is: https://quantumclassical.blogspot.com/2019/02/chemical-potential-raised-states-of_28.html.

]]>I think so, taking a Lorentz invariant convex sum of covariant pure states can be, and is in the above case, however Lorentz invariant integrals not well-defined for such cases, so IMO they should be avoided. I would for preference compute results for the state above by using the Lorentz invariant cyclic identity of the Trace and the Lorentz invariant commutator of the number operator with the creation operator (which is how I derived the expression for the generating function for this state that I give in my comment below). For a free bosonic quantum field,

can be successively manipulated using the cyclicity and a Baker-Campbell-Hausdorff identity, with a factor appearing repeatedly so that one obtains a sum in the exponent, which becomes , which becomes part of the factor. [Sorry that’s so quick, but editing equations on Azimuth is terrifying because there’s no preview.] We’re working here with unitarily inequivalent representations of the algebra, which are, loosely put, separated by thermodynamically infinite divides.

This state could be said to be in a mixed state because it’s interacting with another Lorentz invariant field (which has been traced out) as a zero temperature bath with which it exchanges particles in a Lorentz invariant way. For any algebra of observables that doesn’t include absolutely every field (including dark matter, dark energy, or anything else we haven’t found yet, …), the state over that subalgebra must be a mixed state if there’s any interaction. For the EM field in interaction with a Dirac spinor field, for example, the vacuum state for the EM field with the Dirac spinor field traced out should be a mixed deformation of the free field vacuum state, which *is* pure. Haag’s theorem, after all, insists that the interacting vacuum sector must not be unitarily equivalent to the free vacuum sector.

So the Lorentz-invariant vacuum is a mixed state? One way to think of a mixed state is as any of various pure states with various probabilities (although unlike with classical probability, the way the mixed state is divided into pure states is somewhat arbitrary). Greg argues that no possible pure state is Lorentz-invariant, and so each picks out, say, a preferred reference frame; but each reference frame is preferred with the same probability, so the overall mixed state is still Lorentz-invariant. Is this right?

]]>Perhaps inevitably, one mistake: the higher energy state should be written as, say, and similarly

Hopefully there are no others.

Greg, if a quantum field has a complex structure, it already contains a classical random field of Quantum Non-Demolition measurements, so the example I give above suffices.

We can present the algebraic structure of the measurement operators and vacuum state of a quantized bosonic field, using test functions and a pre-inner product on the test function space, as

Then we find , which is zero, satisfying microcausality, when have space-like separated supports.

If we can construct an involution for which which we can for the complex scalar field and, using the Hodge dual, for the empirically significant EM field, then we find for that

. is a classical random field, commuting with itself everywhere in space-time.

To show how this works through for the example above, using the number operator, for the vacuum state we can construct generating functions both for the quantum field and for the random field,

then for the higher energy state we just have

which is just to increase the Gaussian noise without increasing the commutator that ensures that the vacuum state is the minimum energy state in the vacuum sector. As I say above, this state, generated by using the number operator, is just baby steps. [Note that these generating functions and the commutation relations together are enough to fix the algebraic structure of the quantum and random fields completely. The pre-inner product fixes the connection, of different kinds, between the algebra and the space-time geometry.]

These are all trivial, albeit unconventional computations. It’s not a new theory, it’s just a new way of looking at what we’ve got. Still, is a true classical random field of QND measurements hidden away inside some quantum fields, for which we can construct isomorphisms between the random field Hilbert space and the quantum field Hilbert space, and, if we introduce the vacuum projection operator as an observable (which is routine in practical QFT but is less common in the algebraic QFT literature), we can construct an isomorphism between the algebras of observables. Which, I submit, is nice to have as well as the relatively undefined mess that is quantization. arXiv:1709.06711 has been recommended by a referee for publication, but I’m making some small changes, so not quite there.

Could the 25 fundamental constants be uncomputable real numbers (in the sense of slide #26 of the following paper presented at Unilog2013)?

https://www.dropbox.com/s/3uaiscas7h6tolk/42_Resolving_EPR_UNILOG_2015_Presentation.pdf?dl=0

Reflecting Mitchell Porter’s comment that the “only alternative I can see, is that quantum gravity has a very unexpected high-energy behaviour”, could the limiting behaviour of some physical processes involve phase changes (singularities due to, say, hidden variables) which do not correspond to the limits of any conceivable mathematical representations of the processes (in the sense of the gedanken in Section 15: ‘The mythical completeness of metric spaces’, of the following manuscript)?

https://www.dropbox.com/s/m2qx02zqpjom90g/16_Anand_Dogmas_Submission_tHPL_Anonymous.pdf?dl=0

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