We hear perturbation in QFT a lot, but what is it? It's Perturbing the free (non-interacting) theory by turning on interactions. The Lagrangian density of two fields consist the free part and interaction part. The latter, full interacting theory is generally impossible to solve exactly. So Treat the interaction, i.e. the time evolution / amplitudes … Continue reading Perturbation in QFT

According to Born’s rule, the probability is obtained from the square of the wavefunction. More generally, this is written as the product of the complex conjugate of the wavefunction and the wavefunction itself, ψ∗ψor(𝜓ˉ𝜓)ψ ∗ ψ or( 𝜓 ˉ 𝜓 ) to ensure the probability density is real and non-negative. Now focus on the Dirac … Continue reading Need Dirac Adjoint to Describe 1/2 Spinor’s Lorentz Transformation

It's originally invented by Green to solve difficult PDE problems, let's reduce it to simple matrix form first Now let's solve a real, simple PDE problem: If we increase the number of grid points: The formula turns into Green's function A Green's function is literally the inverse of a differential operator, just like: A−1

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