In history, Dirac derived Dirac Equation by combining Quantum Mechanics and Special Relativity, led the equation, Later: Lagrangian is constructed to reproduce the equation In this post, Let’s derive the Dirac equation from the Dirac Lagrangian step-by-step using the Euler–Lagrange equation.

Noether's theorem states that For every continuous symmetry of the action, there is a corresponding conserved quantity. Continuous symmetry → a transformation that can be done smoothly, not discrete. Action → integral of the Lagrangian over time. SymmetryConserved quantityTime translation (Lagrangian doesn’t depend explicitly on time)EnergySpace translation (Lagrangian doesn’t depend on position)MomentumRotational symmetry (Lagrangian doesn’t … Continue reading Conservation of Charge

Classic Field Theory: Before quantization, understand classical fields. Topics: Action principle Euler–Lagrange equations Lagrangian density Noether's theorem Important fields: Maxwell's Equations Scalar field theory Vector fields Key idea:Fields are the fundamental objects, not particles. Relativistic Wave Equations: Next learn equations describing relativistic particles. Important equations: Klein–Gordon Equation (scalar particles) Dirac Equation (spin-½ particles) Key concepts: … Continue reading Roadmap to Learn Quantum Electrodynamics (QED)

Knowing Minkowski's invariance ( p^2 = m^2 ), we can then deduce the propagator in the Klein-Gordon equation, that is, scalar field, then pole of the propagator in Quantum Field Theory. This is one of the clever conceptual insights physicists discovered. The reasoning is: Fields have wave equations. Wave equations allow plane waves with p2=m2p^2=m^2. … Continue reading Physicists Identify Particles by Looking for Poles in Correlation Functions

Suppose a signal function f(t), the Fourier transform rewrites it as a sum of oscillations: It's frequency description is Take a simple one frequency wave function as example to visualize: This is essential because field is just a 4-D Fourier transform.

For a deep understanding of Green's function, we need to refresh knowledge on probability, starting from two variables x1 and x2: Given the coupling effect here due to the matrix A is not 1, 0 and 0, 1, large values of x1,x2x_1,x_2x1​,x2​ are strongly suppressed. This coupling is exactly why correlations appear, and those correlations … Continue reading Green’s Function in QFT 0

It's fascinating to see that Green's function, which I once thought was merely a trick to solve challenging PDE problems, is inherently present in QFT, especially when a genius like Richard Feynman attempts to develop the algebra needed to compute field interactions. In this blog, I want to lay down the fundamentals and then progress … Continue reading Green’s Function in QFT 1: Basics

Deriving this Lagrangian in scalar field: the key is to Replace particle with a field. If the field interacts with itself, the simplest form, the Lagrangian is

Physics is described by the Lagrangian: L = T - V where T is the kinetic energy and V is the potential energy. For fields,

Imagine a particle moving from point A to point B. In quantum mechanics, it doesn’t just take a single path — it explores all possible paths. Each path has a complex amplitude (or wave function) with a phase determined by the action along that path. The sum of all these amplitudes determines the probability for … Continue reading Feynman Paths to Feynman Diagrams: A Journey Through Quantum Arrows