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.

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: