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.

I was quite confused by force description by second derivative versus force generation, where in Klein-Gorden equation, there is also second derivative terms, but no interaction, force is generated as the equation is Linear. Hence dig deep to clarify the two. In General Relativity, this idea is taken even further: Gravity is curvature of spacetime. … Continue reading Clarify Force Description versus Force Generation

Here’s Stokes’ theorem in its most general and elegant form from differential geometry, using the beautiful, rigorous differential geometry terminology: Now knowing the field A and field strength FμνF_{\mu\nu}Fμν are 1-form and 2-forms: The field theory become easier.

It's the same object written in two different math dialects: differential form language and index (tensor) language. Now let's upgrade from electromagnetism to full Yang–Mills: Now let's deduce the A wedge A term: Here I need to derive [Ta,Tb]=ifabcTc too:

It is deeply relaxing and joyful to arrive at Yang–Mills theory and suddenly realize that Maxwell’s equations appear as the special case of a Yang–Mills theory with an abelian gauge group. Dive down to detailed form:

A k-form assigns a (multi)linear, antisymmetric “density” to each point that can be evaluated on k tangent vectors. Density here is a bit abstract, but it  is not a “density” in the physics sense (mass per volume), but you can picture it as an oriented, multilinear measuring device that eats kk tangent vectors and spits out a number telling you “how much kk-dimensional … Continue reading Revisit Differential Forms by Taking Exterior Derivative of 1 form and 2 form

write out the narrative, create a Manim animation for it following the same style as the Yang-Mills video - with sections for: Outro Title/intro Equation breakdown (showing each term appearing) Visual explanations (matrices, commutators, comparing Abelian vs non-Abelian) Physical interpretations then i will record the voice to mp3. combine them with ffmpeg, the final output … Continue reading Learn 3Blue1Brown’s Workflow to Create High Quality Video

Let's start from global U(1) symmetry, A quantum wavefunction for a charged particle is ψ(x)\psi(x) An extra term pops out A miu is not physically observable, so we need to so some math tricks to surface it out In this case, i.e. electromagnetism, the phase change is a simple U(1) group, to extend U(1) to … Continue reading Derive electromagnetism’s Field Strength From Scratch

OpenCode is completely open-sourced and offers several free LLMs, including GLM4.7, which is reported to be quite impressive and comparable to Claude's Sonnet 4.5. It also performs well in math tasks. I'm eager to explore it further and see what other savvy AI users have shared about its use cases. This is provided by David … Continue reading OpenCode Use Cases