Some Notes on AI
A Stanford‑hosted Future of Mathematics symposium highlighted the accelerating role of artificial intelligence in research, sparking debate among leading scholars. Mathematicians such as Peter Scholze and Michael Harris voiced strong concerns that AI could erode the discipline’s core values and even threaten democratic norms. At the same time, experts noted that proof‑verification systems might mitigate AI’s propensity for plausible but inaccurate statements, especially in contentious areas like the Mochizuki abc conjecture. In physics, the Department of Energy’s Genesis Mission is pouring substantial funds into AI projects aimed at unifying fundamental theories, though optimism remains mixed.
Wick Rotating Spinors and Twistors
The talk in Marseille examined the longstanding difficulty of Wick rotating spinor fields between Minkowski and Euclidean spacetimes, highlighting that complex spacetime geometry simplifies spinor and twistor representations while real spacetimes introduce incompatibilities. The speaker proposed redefining the Minkowski Lorentz...
Formalization of QFT?
A new arXiv paper claims to formalize a free scalar quantum field theory in Lean/Mathlib by constructing a Euclidean measure that satisfies the Glimm‑Jaffe Osterwalder‑Schrader axioms. The work reproduces the classic proof that the two‑point Schwinger function yields a measure...
Twistors and Unification
The author proposes Penrose’s twistor theory as a chiral alternative to conventional spacetime symmetries, linking Wick rotation to a gauge choice in complex projective space. By treating PT≈CP³ with an SU(2,2) conformal action, particles become representations of a larger symmetry...
Twistors and Wick Rotation
The article explains how twistor theory provides a geometric framework for Wick rotating between Minkowski and Euclidean spacetimes. By treating spacetime points as CP^1 lines inside projective twistor space (PT=CP^3), the author shows that the Minkowski conformal group SU(2,2) and...
Weyl Spinor Fields and Right-Handed Spacetime
The article explains why a single Weyl spinor field cannot be Wick‑rotated using the conventional Euclidean continuation, highlighting a fundamental mismatch between Minkowski and Euclidean spinor representations. It proposes a new framework that employs only right‑handed Weyl spinors to encode...
Lorentz versus Euclidean Symmetry
The article explains how Wick rotation swaps the Lorentz symmetry SO(3,1) of Minkowski quantum field theory for the Euclidean rotation group SO(4), and how the reverse process is more subtle. It shows that Osterwalder‑Schrader (OS) reconstruction in Euclidean space breaks...
Osterwalder-Schrader and Euclidean Spinor Fields
The 1972 Osterwalder‑Schrader framework tackles the long‑standing problem of Wick rotating spinor fields by introducing a pair of independent fermionic variables, effectively doubling the degrees of freedom when moving from Minkowski to Euclidean space. Their construction preserves the Dirac adjoint...
Harmonic Oscillators
The post reviews the quantum harmonic oscillator in the Heisenberg picture, showing how ladder operators $a$ and $a^\dagger$ solve the equations of motion and generate the familiar energy spectrum. It then contrasts this elementary construction with the Osterwalder‑Schrader (OS) Euclidean framework, noting that the...