added more semi-publication data to this item:

- Hans-Jürgen Borchers,
*On Revolutionizing of Quantum Field Theory with Tomita’s Modular Theory*, ESI Preprint 773 (1999) [pdf]

Did Dyson need to specify ’twice-differentiable’?

This comes from the fact that the Riemann tensor is second order in derivatives of the metric, so that the minimum order differentiablitiy that one needs to require in order to preserve the Einstein-Hilbert action of pure gravity is 2. But since one will generally want to allow various matter couplings which may have higher order derivatives in them (the whole action functional being transgressed from a Lagrangian density on the jet bundle) it is natural and has become standard to require arbitrary differntiability, hence diffeomorphism invariance.

Clearly I cannot speak for Dyson, but my suggestion that the axioms of locally covariant AQFT (as in the article you now quote) are the fully satisfactory answer to the request for diffeomorphism-invariant quantum field theory on curved spacetime is well supported and widely accepted.

Of course this describes QFT on general but classical gravity backgrounds. This includes perturbative quantum gravity but likely not non-perturbative quantum gravity. The latter remains famously an open issue.

]]>And how did this prediction pan out?

If we try to replace the Poincaré group P by the Einstein group E, we have no way to define a space-like relationship between two regions, and axiom (5) becomes meaningless…an E-invariant axiom of local commutativity to replace axiom (5) will require at least some quantum-mechanical analog of Riemannian geometry.

Would he take p.11 of QFT on curved spacetimes: axiomatic framework and examples as providing the right solution?

]]>It’s called the

diffeomorphism group! :-)

Did Dyson need to specify ’twice-differentiable’?

]]>Of course, you’re suggesting that there is plenty to say, as treated at AQFT on curved spacetimes.

But is there some ’Einstein group’ at play?

]]>It’s called the *diffeomorphism group*! :-)

Wikipedia sometimes gets stuck with the weirdest of attributions, maybe because it discourages experts from editing, who might not be considered “neutral” enough.

]]>The ’E-invariance’ and ’P-invariance’ there refer to the Einstein group and the Poincaré group. Do we have the former by a different name? Dyson defines it as consisting of

all one-to-one and twice-differentiable transformations of the coordinates.

There’s a Wikipedia entry suggesting that Mendel Sachs found the group that Einstein was looking for. I see this same Sachs is generally not held in high esteem - Physics Stack Exchange.

Dyson ends that section with

The answer to my challenge will necessarily involve a delicate weaving together of concepts from differential geometry, functional analysis, and abstract algebra. With these words of warning I leave the problem to you.

With the advantage of the best part of 50 years, is there anything to say to Dyson’s challenge?

]]>added quote from Dyson 72

]]>I have tried to expand a little the opening sentences in the *Idea*-section at *Haag-Kastler axioms*.

Also I tried to clean up the list of references. There were some pointers to articles by Summer et al. dropped there which clearly belonged to other entries: I moved them to *scattering amplituce* and to *modular theory*.

At Haag-Kastler axioms I have expanded the section on Terminology and started a section on Motivation.

]]>With Todd Trimble I was discussing by email the motivation and physical meaning of the Haag-Kastler axioms. This is clearly something that should go into the $n$Lab, and so I have started expanding the entry.

So far all I did is add more motivation and explanation to the first two axioms: 1. isotonic copresheaf of algebras and 2. causal locality.

I won’t have much time to work on this these days, but maybe a little bit. I hope Todd will reply here and that in the course of further discussion we can see how the content of that entry can further be improved.

]]>Nice.

I edited a bit and added a bit.

]]>Tim van Beek has graced us with these: Haag-Kastler axioms.

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