Brilliant anaylsis. The "graduate student arc" framing is spot-on and more revealing than most AI capability discussions. AI can follow directed prompts and execute compuations, but lacks the broader context awarenes to spot foundational issues like Torre-Varadarajan. This gap between local competence and global understanding is exacty what makes careful human oversight still essential, otherwise we risk flooding literature with plausibly-writen but fundementally flawed work.
Thanks for your interest in my paper and for pointing me to the Torre-Varadarajan work.
TV identify a global obstruction: even though each infinitesimal Tomonaga–Schwinger deformation is locally well-defined, the accumulated evolution from one Cauchy surface to a distant one generally fails to be represented by a unitary operator on the original Fock space.
Local TS integrability still guarantees an algebraic map between initial and final hypersurfaces, but TV show that this map need not admit a unitary implementation in a single Hilbert representation.
Thus, the TV obstruction concerns global realizability, and is logically distinct from the possibility that the infinitesimal TS conditions themselves might break down in a modified (e.g., state-dependent) quantum theory. In this case, even small, spacelike-separated deformations would fail to commute.
Thanks for your response. I checked with Varadarajan and he responded "whole point is that infinitesimal deformations on the wave function are ill-defined".
I think this is because the obstruction comes from UV (the Boguliubov beta coefficient goes as 1/k) and so even for generic small deformations there is no bounded map. But it's been a while since I read Varadarajan-Torre.
Take my comment with a grain of salt, because I should also look more carefully at their paper.
But the fact that their result depends on UV effects suggests there is a version of the TS conditions that holds for a coarse grained (or suitably regulated) theory. It seems like the problem they identify only emerges when the number of degrees of freedom is strictly infinite.
Should the commutator following the below sentence "This is expressed by following condition of foliation independence" should have sigma(y) for the second operator?
Brilliant anaylsis. The "graduate student arc" framing is spot-on and more revealing than most AI capability discussions. AI can follow directed prompts and execute compuations, but lacks the broader context awarenes to spot foundational issues like Torre-Varadarajan. This gap between local competence and global understanding is exacty what makes careful human oversight still essential, otherwise we risk flooding literature with plausibly-writen but fundementally flawed work.
Thanks for your interest in my paper and for pointing me to the Torre-Varadarajan work.
TV identify a global obstruction: even though each infinitesimal Tomonaga–Schwinger deformation is locally well-defined, the accumulated evolution from one Cauchy surface to a distant one generally fails to be represented by a unitary operator on the original Fock space.
Local TS integrability still guarantees an algebraic map between initial and final hypersurfaces, but TV show that this map need not admit a unitary implementation in a single Hilbert representation.
Thus, the TV obstruction concerns global realizability, and is logically distinct from the possibility that the infinitesimal TS conditions themselves might break down in a modified (e.g., state-dependent) quantum theory. In this case, even small, spacelike-separated deformations would fail to commute.
Steve
Hi Steve,
Thanks for your response. I checked with Varadarajan and he responded "whole point is that infinitesimal deformations on the wave function are ill-defined".
I think this is because the obstruction comes from UV (the Boguliubov beta coefficient goes as 1/k) and so even for generic small deformations there is no bounded map. But it's been a while since I read Varadarajan-Torre.
Best,
Nirmalya
Take my comment with a grain of salt, because I should also look more carefully at their paper.
But the fact that their result depends on UV effects suggests there is a version of the TS conditions that holds for a coarse grained (or suitably regulated) theory. It seems like the problem they identify only emerges when the number of degrees of freedom is strictly infinite.
That's not implausible.
Minor typo:
Should the commutator following the below sentence "This is expressed by following condition of foliation independence" should have sigma(y) for the second operator?
Good catch, thank you! Corrected.