We get it naturally that QRQ-1 is rotating R, but think deeper, isn't QR already complete the rotation action?? Yes, it Q rotates R, but it does not rotate axis of R! But QRQ-1 is not simple composition, it represents rotation of R in a different, rotated frame! Another angle to understand QRQ-1 is let … Continue reading Differentiate QR and QRQ-1 Where Q,R∈SO(3)

Lie Bracket is the Derivative of Conjugation, Hence Measuring How Rotation Fails to Commute! To illustrate this point, I'd like to start from the definition of Lie Algebra and Lie Bracket. A Lie algebra is a vector space closed under lie bracket. Its elements represent infinitesimal motions, and whose bracket measures how these motions fail … Continue reading Lie Bracket is the Derivative of Conjugation, Hence Measuring How Rotation Fails to Commute

It is very misleading that we are taught light travels at the speed of light, which is a constant value, and that it takes about 8 minutes for light from the Sun to reach the Earth. We are also told that photons, the particles of light, are massless. So how can a massless entity travel? … Continue reading Massless Photo Doesn’t Travel at Speed of Light

Context & Output Capacity: it has 1 million token context window -- the first Opus model to support this (in beta via the developer platform); and 128K token output -- a significant bump in how much content can be generated in a single response. Agent Teams: a new capability where teams of agents can split … Continue reading What’s in Opus4.6 that 4.5 Doesn’t Have

Nebla is ubiquitous in mathematics; the Nabla dot and Nabla cross product are concepts we learn in middle school, but revisiting them today through the lens of differential forms opens up a new door and provides a fresh perspective that's truly mind-opening. calculus is quite limited and going further could be confusing, for example when … Continue reading Dot and Cross Product in Differential Form Language

It's actually quite simple once you know the vector and covector spaces and in both spaces there are two components. It is change of the covector component AνA_\nuAν​ in direction xμx^\muxμ. What's the sequence of applying vectors to (0,n) tensor?

Let's derive the Lagrangian for the photon field (the electromagnetic field). in the end, want to clarify the math symbols: Fμν​ (Fμν​=∂μ​Aν​−∂ν​Aμ​) is simply the components of the 2-form F=dAF = dAF=dA:F=dA=12Fμνdxμ∧dxνF = dA = \frac12 F_{\mu\nu} dx^\mu \wedge dx^\nuF=dA=21​Fμν​dxμ∧dxν

Lagrangian of the electron field is the heart of how electrons behave in quantum field theory. There are two layers to this: Electron interacting with electromagnetism (real world) Electron by itself (free field) Electrons aren’t isolated — they interact with the electromagnetic fieldAμ(x)A_\mu(x)Aμ​(x). This term produces Coulomb force, magnetic force, photon emission, photon absorption and … Continue reading Lagrangian of the Electron Field

Here is a high-level derivation how charge generates forces in fields that live on spacetime, I will dive deeper later on. Force = exchange of gauge field quanta required by local symmetry, it is the price nature pays to keep symmetry local. It naturally makes me connect charge to energy, if energy is generator of … Continue reading Charge Generates Forces in Fields That Live on Spacetime; Energy Generates Force by Curving Spacetime Itself

What does “generator” mean? A generator is something that produces a transformation. Now let's start from deducing Energy is the generator of time shift first, then move to the understanding of "charge"! Now we do Hence, Electric charge is literally: Momentum in the direction of phase, or motion in the internal symmetry space.