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

The reason physics and math are difficult for most people is poor teaching. Instead of revealing intuitive concepts—like how homogeneous transformations connect to physical reality or why Hermitian operators represent observables—teachers often overwhelm students with numerous quantum formulas. This makes the subject hard to follow. The right approach is to emphasize intuition and reasoning, which … Continue reading Homogeneous Transformation Making It Physical and Hermitian Operators Are Observables

Nature chooses symmetries. Making those symmetries local creates connections (fields). Different representations of those symmetries become particles. Empirically, the universe respects internal rotation symmetries:SU(3)×SU(2)×U(1)SU(3)\times SU(2)\times U(1)They are the smallest groups that match: color triplets weak doublets electric phases When symmetry is global → nothing happens.When symmetry is local → derivatives break symmetry → so nature … Continue reading Symmetries to Particles

This is to be elaborated in the future, documenting here first:

In Maxwell (electromagnetism) equation, U(1) gauge group is used, and U(1) is abelian, so it doesn't require Yang-Mills theory. However, What if the internal symmetry is non-Abelian (like SU(2)SU(2)SU(2))? The gauge field has multiple components: The curvature picks up A∧A; Gauge bosons interact with each other. This is Yang–Mills theory. Here is the Yang-Mills Equation

What is gauge field? It is a Lie Algebra Valued 1-form on space time. This 1-form eats tangent vector (direction of motion) and returns something linear, but this linear thing is not a number but an infinitesimal symmetery generator! Let's illustrate by a concrete example: SU(2) gauge field acting on a spacetime vector. Then we … Continue reading Gauge Field

Learning math is learning a language, being exact and precise is the key, which requires some efforts to fully understand and practice using them. So one needs to know basis vector, basis covector, their components to achieve math fluency! d is a map (an operator)! Here is its official definition: So make it crystal clear … Continue reading To Master Mathematical Language, Exactness Is Key