I am greatly interested in learning how spinors, antimatters surface out after solving Dirac Equation. Let's use the simplest case, rest frame to start. The full positive-energy solution is So the final complete wavefunction solution is It contains the 4 dimensions to describe the quantum state of the particle/wave/quantum and turns out it's a spinor … Continue reading Dirac Equation Solution

Why Dirac wanted to create the Dirac equation and solve it? Because he wants an equation that describes matter in a world where: Probability is conserved Spacetime is Lorentzian (special relativity is non-negotiable) States superpose (quantum mechanics) Time evolution is causal and first-order Special Relativity wants E2=p2+m2E^2 = p^2 + m^2 and Quantum mechanics wants … Continue reading Dirac Equation

It's because of how identical particles and spacetime symmetry work — let's unpack it. In quantum physics: identical particles are indistinguishable exchanging two identical particles cannot produce a new physical state So the state must be: unchanged, or changed only by a phase Fermions are antisymmetry: You may ask why fermions are antisymmetry, it relates … Continue reading Why Two Fermions Are Exclusive

When we think about fields, it always means something broad, wavy and unreal, it's such a big stretch to connect it to form the rich world we live in where concrete objects are everywhere. We need to know that Concrete objects are stable field configurations. There are three fundamental reasons: First, Fields are quantized, so … Continue reading If Everything is just fields, why does anything become solid, stable, and concrete — including our own bodies?

What is a generator? A generator is the object that tells you how something changes under an infinitesimal symmetry transformation. We need to entirely understand this via the lens of Lie Group and Lie Algebra. generator lives in the tangent vector space and is making the rotation on infinitesimal scale. The laws of nature are … Continue reading Energy is the Generator of Time Change

Recent news comes out that Meta is acquiring Manus for approximately $2 billion, a move that appears to lack technical rationale. What Manus really has is not some breakthrough AI. It is mostly a large amount of prompting and detailed instruction layered on top of the same commodity LLMs everyone else uses. They are good … Continue reading Meta Acquiring Manus for Billions Seems Like a Joke

What is representation theory? A group G is an abstract symmetry such as rotations, boots, gauge transformations, phase rotations and diffeomorphisms, etc. but physics cannot act with abstract symbols, it needs objects that transform, so we need to use representation. A representation is a rule that assigns to each group element ggg a linear transformation … Continue reading A Particle is a Representation of the Symmetry Group

Penrose said "spinors are the square roots of vectors", it reveals deep insight of how the nature works. Where we can see a three-component vector can be expressed by 2x2 complex-element matrix. But the key idea of spinors are square roots of vectors go further: Now illustrate with very concrete example How to connect the … Continue reading Spinors Are the Square Roots of Vectors

We know Lorentz group SO(1,3) in transforming the 4-component space-time vector very well. A quick recap on how it's deduced: But to describe mathematically spinor, which has “topological twists” and do not respond to 4-vectors, we need something more complex then simple Lorentz transform. We need its double cover SL(2, C). Chirality is a property … Continue reading What Is Weyl Spinor and Chilarity

What's adjoint representation? We also call such group abelian, meaning Tg and gT commute, and if it doesn't transform in such a way, it's non-abelian. It's hard to understand the Einstein symbols in this representation formula, let's use a concrete example to illustrate: This leads to deep understanding that Group typeWhat happensAbelian (U(1))(gTg^{-1}=T) → no … Continue reading Adjoint Representation