Physics is not just a subject — it is a language. One written in the ink of mathematics, spoken through the geometry of spacetime, and heard in the hum of a quantum field. think.ai is a space to explore that language, one equation at a time.

“The most incomprehensible thing about the universe is that it is comprehensible.” — Albert Einstein

From the Schrödinger equation $i\hbar\, \frac{\partial \psi}{\partial t} = \hat{H}\psi$ governing the evolution of quantum states, to the Einstein field equations $R_{\mu\nu} - \tfrac{1}{2}Rg_{\mu\nu} = 8\pi G\, T_{\mu\nu}$ curving the fabric of spacetime — every post here is an attempt to understand these deep structures from first principles.

What This Blog Is About

This is a research and learning blog focused on the mathematical foundations of modern physics. The content ranges from rigorous derivations to conceptual discussions, always anchored in the formalism:

  • Quantum mechanics — operator algebra, Hilbert spaces, measurement theory, path integrals
  • Quantum field theory — canonical quantisation, Feynman diagrams, renormalisation
  • Attosecond physics — ultrafast laser dynamics, electron wavepacket evolution
  • Mathematical methods — differential geometry, Lie groups, functional analysis, and their role in physics
  • Computational physics — numerical approaches to problems that resist closed-form solutions

Every topic will be treated with mathematical honesty. No hand-waving where derivations are possible.

Why think.ai?

The name reflects a simple philosophy: good physics requires active, careful thinking. Modern AI tools can assist in computation and pattern recognition, but the deep conceptual work — understanding why $\Delta x\, \Delta p \geq \hbar/2$ is not just a measurement limitation but a fundamental property of non-commuting observables — still belongs to the human mind working carefully through the mathematics.

This blog sits at that intersection: rigorous theory, modern tools, and the kind of slow thinking that physics demands.

Current Research Threads

The areas I am actively working through and writing about:

  1. Hydrogen atom under attosecond pulses — time-dependent perturbation theory and ionisation dynamics
  2. Canonical commutation relations — the algebra $[\hat{x},\, \hat{p}] = i\hbar$ and its consequences for uncertainty, coherent states, and the harmonic oscillator
  3. Amplituhedron geometry — how scattering amplitudes in $\mathcal{N}=4$ SYM can be computed from the volume of a geometric object, bypassing Feynman diagrams
  4. Quantum systems in confined geometries — boundary conditions, discrete spectra, and topological effects

What to Expect

Posts will be of two kinds:

Derivation posts walk through a result step by step, starting from axioms or first principles and arriving at a non-trivial conclusion. These are the core of the blog.

Concept posts discuss the physical meaning and context of a result — what it tells us about nature, where it comes from historically, and where it leads.

All mathematical notation is rendered with KaTeX, so inline expressions like $E = \hbar\omega$ and display equations like

\[\mathcal{L} = \bar{\psi}(i\gamma^\mu \partial_\mu - m)\psi\]

will appear cleanly throughout.

The first technical post is already up: a derivation of the Heisenberg uncertainty principle from canonical commutation relations. More to follow.