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

Representation theory is the study of how abstract algebraic structures (like groups, algebras, Lie groups, Lie algebras) can be represented concretely as linear transformations (matrices) acting on vector spaces. Concisely, homomorphism into linear transformations! Let's talk about Lorentz representation, Spinor representation and Lie Algebra version to grasp this concept. there are three rotations Ji and … Continue reading Representation and Gauge Theory

Let's go step by step to see exactly why AμA_\mu​ must appear, and why that object is the photon.

Physics asks: how does something change as you move? Geometry asks: what does “moving” even mean if space has structure? Gauge theory answers both by saying: Forces arise because “comparison across space” is not trivial. ConceptPhysical MeaningGeometric MeaningGauge Potential ($A_\mu$)The Photon/Gluon fieldThe Connection (how to compare fibers)Field Strength ($F_{\mu\nu}$)The Electric/Magnetic ForceThe Curvature (how much the … Continue reading Force as Curvature

Math and physics are not about calculation first; they are about insight. Their purpose is to discover deep and abstract connections beneath complex phenomena, where many seemingly different things turn out to be the same at a structural level. Once such an insight is found, symbols appear—not as decoration, but as compression. Symbols allow complex … Continue reading Math and Physics: The Art of Seeing Deep Connections

On a smooth manifold, vector lives in the tangent vector space, while 1-form lives in the cotangent space. Note lot of confusion comes from the understanding of the basis vector and basis covetor. Vectors transform with Jacobian, 1-forms transform with the inverse transpose Jacobian. On a non-flat surface (a manifold), basis vectors are no longer … Continue reading Vectors vs. 1-forms: Two Different Spaces