When Dirac solved the famous Dirac equation, he was troubled by the introduction of "negative energy" due to the bottom spinor. Later, when quantum field theory emerged, this problem was addressed: there is no negative energy in our universe. The interpretation introduced both the electron and the positron. Therefore, the key here is to understand … Continue reading Interpret Dirac Equation without “Negative Energy”

Most quantum field theory books casually writex(t)∼ae−iωt+a†e+iωtx(t) \sim a e^{-i\omega t} + a^\dagger e^{+i\omega t} and move on as if this were obvious. But it hides something profound. Where did these operators aaa and a†a^\daggera† actually come from? Were they inserted by hand? Are they a quantum trick? No. They are forced by structure. Now … Continue reading From a Harmonic Oscillator to Antimatter

How is γ→e−+e+ derived? this could be a gateway to applying Feynman Diagram in QED Lagrangian.

Fundamental representation → how symmetry acts on matter Adjoint representation → how symmetry acts on itself, it is one of the most profound structural facts in physics. We know the SU(2) The symmetry rotates the components of the matter field, components include Electron spin, weak isospin doublets or any SU(2) doublet. Now in adjoint representation, … Continue reading Fundamental Representation: Symmetry Acts on Matter; Adjoint Representation: Symmetry Acts on Itself

In last blog, we discussed the SU(2) representations: Complexified Lorentz: SU(2)_L ⊕ SU(2)_R, We saw:so(1,3)C≅su(2)L⊕su(2)R\mathfrak{so}(1,3)_\mathbb{C} \cong \mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_Rso(1,3)C​≅su(2)L​⊕su(2)R​ So representations are tensor products:(jL,jR)=VjL⊗VjR(j_L, j_R) = V_{j_L} \otimes V_{j_R} Now let's build a structured and precise picture of Lorentz representations, Math is the most beautiful and rigorous language, so we need to clarify when we … Continue reading More on Representation Types

Representation: Every group element is realized as a linear transformation on some vector space VV, is a way to make an abstract symmetry concrete as matrices acting on vectors. Use SU(2) as example, So we have Trivial representation (spin 0), Fundamental representation (spin ½), Adjoint representation (spin 1) and Higher spin representations. In spin 0, … Continue reading Representation Types

We get it naturally that QRQ-1 is rotating R, but think deeper, isn't QR already complete the rotation action?? Yes, it Q rotates R, but it does not rotate axis of R! But QRQ-1 is not simple composition, it represents rotation of R in a different, rotated frame! Another angle to understand QRQ-1 is let … Continue reading Differentiate QR and QRQ-1 Where Q,R∈SO(3)

Lie Bracket is the Derivative of Conjugation, Hence Measuring How Rotation Fails to Commute! To illustrate this point, I'd like to start from the definition of Lie Algebra and Lie Bracket. A Lie algebra is a vector space closed under lie bracket. Its elements represent infinitesimal motions, and whose bracket measures how these motions fail … Continue reading Lie Bracket is the Derivative of Conjugation, Hence Measuring How Rotation Fails to Commute

It is very misleading that we are taught light travels at the speed of light, which is a constant value, and that it takes about 8 minutes for light from the Sun to reach the Earth. We are also told that photons, the particles of light, are massless. So how can a massless entity travel? … Continue reading Massless Photo Doesn’t Travel at Speed of Light

Context & Output Capacity: it has 1 million token context window -- the first Opus model to support this (in beta via the developer platform); and 128K token output -- a significant bump in how much content can be generated in a single response. Agent Teams: a new capability where teams of agents can split … Continue reading What’s in Opus4.6 that 4.5 Doesn’t Have