Freeman Dyson's 1972 paper "Missed Opportunities"

[kith]

I just stumbled upon the well-written 1972 paper "Missed Opportunities" by Freeman Dyson. It can be found https://www.math.uh.edu/~tomforde/Articles/Missed-Opportunities-Dyson.pdf.

Dyson argues that over time, people have become worse in following the mathematical hints inherent in the structure of physical theories. As an example, he compares the developments in both mathematics and physics following Newtonian physics with the "failure" to discover special relativity based purely on the transformation properties of the Maxwell equations and other things. Both hindsight bias as well as selection bias might be involved here but I think it is an interesting thesis.

He goes on to identify three things he considers missed opportunities as of 1972 and thus proposes:
1) ... to create a mathematical structure preserving the main features of the Haag-Kastler axioms but possessing E-invariance instead of P-invariance.
2) ... to construct a conceptual scheme which will legalize the use of Feynman sums [...] with suitable Iagrangians which are not quadratic.
3) ... to try to achieve a rigorous definition of Feynman sums which are invariant under general coordinate transformation.
(Check the paper for details.)

Now I'm not very familiar with QFT and much less BtSM physics. Has there been considerable progress or effort in these directions? How do renormalization and contemporary QG efforts like String theory and LQG tie in with what he writes?

 

[Urs Schreiber]

... to create a mathematical structure preserving the main features of the Haag-Kastler axioms but possessing E-invariance instead of P-invariance.
[...] Has there been considerable progress or effort in these directions?

Around the year 2000, Brunetti and Fredenhagen generalized the Haag-Kastler axioms from Minkowski spacetime to general and all spacetimes. This has come to be known as locally covariant algebraic quantum field theory and arguably realizes what Dyson was asking for here, to the extent possible without being a full theory of quantum gravity: it describes quantum field theory on general but classical gravitational backgrounds.

How do renormalization and contemporary QG efforts like String theory and LQG tie in with what he writes?

Brunetti, Fredenhagen and others went on to also consider a variant of the Haag-Kastler axioms that applies to perturbative QFT, now called perturbative algebraic quantum field theory and they observed that the old concept of Epstein-Glaser renormalization finds its proper home here. This is discussed also in the PF-Insights on Interacting Quantum Fields.

Finally they went on to combine these two, to locally covariant perturbative algebraic quantum field theory and generalized Epstein-Glaser renormalization to a theory of renormalization on curved spacetime (starting with the seminal Brunetti-Fredenhagen 00). This is a mathematically solid context for discussion of topics such as the cosmological constant (see there) or the cosmic background radiation. See

• Thomas-Paul Hack,
  "Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes",
  Springer 2016 (arXiv:1506.01869, doi:10.1007/978-3-319-21894-6)

(Notice that LQG is not a thing.)

 

[Laurie K]

Dyson argues that over time, people have become worse in following the mathematical hints inherent in the structure of physical theories. As an example, he compares the developments in both mathematics and physics following Newtonian physics with the "failure" to discover special relativity based purely on the transformation properties of the Maxwell equations and other things. Both hindsight bias as well as selection bias might be involved here but I think it is an interesting thesis.

I'm surprised Freeman Dyson didn't mention Emmy Noether and her contribution to the conceptual structures of the mathematics in our modern physics. Nina Byers goes into this in detail in her paper "E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws" in 1998.

https://arxiv.org/abs/physics/9807044v2

Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as ‘the failure of the energy theorem’. In a correspondence with Klein [3], he asserted that this ‘failure’ is a characteristic feature of the general theory, and that instead of ‘proper energy theorems’ one had ‘improper energy theorems’ in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether.

At a conceptual structural level improper integrals in calculus are integrals, usually from +infinity to -infinity, that converge at their limits. If they don't converge then they are indefinite integrals which are an entirely different kettle of fish.

That doesn't mean that indefinite integrals don't play a part in our calculus or physics as an indefinite integral that cycles between +infinity and -infinity, that is also a sub function of a higher level function, is a valid use of indefinite integrals and is proper.

It just begs the question that when we apply the second proper format and there is not one complete cycle in the higher level function, can we actually say that the underlying calculus represent a valid symmetric mathematical construct that conforms to Emmy Noether's proofs?

 

[Laurie K]

I apologise as the question in my previous post was actually a bit of a trick question, a valid proper integral of any form is not equivalent to a valid improper integral because that is the underlying conceptual difference between classical and modern physics as discussed by Hilbert and Klein, and Emmy Noether only proved the improper energy functions symmetric and correct.

While Emmy Noether provided the conceptual symmetries of relativity, Arthur Compton provided the final piece of the relativity puzzle by experimentally and theoretically uniting the wave and particle natures of electromagnetic particles between 1922-23. He was awarded the Nobel prize in physics for his work in 1927.

The Compton wavelength and the reduced Compton wavelength are both named after him as a result and the relationship between the two forms the underlying fundamental difference between proper energy integrals and improper energy integrals i.e. the elementary conceptual difference between classical and modern physics.

Sorry for being pedantic in advance. If you have failed to understand correctly what the underlying conceptual and mathematical difference is between classical and modern or relativistic physics, and their many derivations, and you conflate the two together at the conceptual and theoretical levels and then apply the result on a universal scale, would you expect to see an artefact of the difference between the two, as discovered by Arthur Compton, in the experimental results?

I ask you, is the artefact that results when we divide the total calculated universal matter resulting from Lambda CDM, over the total ordinary matter as measured via either the WMAP or PLANCK data, evidence of this conceptual and theoretical conflation of classical and relativistic physics at the universal level?

Getting back to the OP, modern physics will only be able to move forward and reclaim its 'missed opportunities' when it resolves this conundrum.