Archives for June, 2026
Constant Coefficients for a Differential Equation -> means translational symmetry (and temporal symmetry).
Intro The key point is that constant coefficients mean the equation itself does not change when you shift the coordinates. Let's look at the KG equation carefully:(□+m2)ϕ(x)=0(\Box + m^2)\phi(x)=0(□+m2)ϕ(x)=0 or(∂2∂t2−∇2+m2)ϕ(x)=0.\left(…
Why look for eigenfunctions of energy and momentum (KG Equation)?
Chapter 2 of Peskin and Schroder 'An Intro to QFT' contains something like this:Just as in ordinary quantum mechanics, we look for eigenfunctions of momentum and energy:ϕ(x)=e−ip⋅x\phi(x)=e^{-ip\cdot x}ϕ(x)=e−ip⋅x wherep⋅x=pμxμ=Et−p⋅xp\cdot x…
Plane wave solutions to the Klein Gordon Equation
The Klein-Gordon (KG) equation is the relativistic wave equation for a spin-0 particle. In natural units (ℏ=c=1\hbar=c=1ℏ=c=1):(□+m2)ϕ(x)=0(\Box + m^2)\phi(x)=0(□+m2)ϕ(x)=0 where□≡∂μ∂μ=∂2∂t2−∇2.\Box \equiv \partial_\mu\partial^\mu = \frac{\partial^2}{\partial t^2} -\nabla^2.□≡∂μ∂μ=∂t2∂2−∇2. Explicitly,(∂2∂t2−∇2+m2)ϕ(x)=0.\left( \frac{\partial^2}{\partial t^2} -\nabla^2…
Negative energy states in Relativistic QM, but not in QFT
This is one of the deepest conceptual shifts from Relativistic Quantum Mechanics (RQM) to Quantum Field Theory (QFT). The short answer is: In RQM, negative-energy solutions appear because we are…
Non Commutavity of the Minkowski Group
The Minkowski transformation group (more commonly the Lorentz group and, when translations are included, the Poincaré group) has several important algebraic properties. One of the most important is that it…
Relativistic Particle versus Relativistic Field
Key Difference A relativistic particle is an object that obeys the relativistic energy-momentum relation E2=p2c2+m2c4E^2=p^2c^2+m^2c^4E2=p2c2+m2c4 A relativistic field is a quantity defined at every point in spacetime whose dynamics are…
Is QFT Linear?
Yes — but with an important distinction:The field equations are often linear for free fields, but interacting QFT is not linear.\boxed{\text{The field equations are often linear for free fields, but interacting QFT is not linear.}}The field equations are often linear for free fields, but interacting QFT is not linear. Let’s do a few concrete Weyl-vector examples. 1. Expand…
Dispersion Relation for Schrodinger’s Wave
The dispersion relation tells us how the wave frequency ω\omegaω depends on the wavenumber kkk. For a free Schrödinger particle, start with the time-dependent Schrödinger equation:iℏ∂ψ∂t=−ℏ22m∂2ψ∂x2i\hbar\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}…