Archives for Quantum Field Theory
Scalar Lagrangian Full Expansion
For a real scalar field \(\phi(x)\), the standard interacting Lagrangian density is\ Using natural units, \(\hbar=c=1\), and the metric \(\eta_{\mu\nu}=\operatorname{diag}(+1,-1,-1,-1)\), let us expand each part. 1. Expand the spacetime derivatives…
Renormalization in QFT with a few examples
Renormalization is the process of relating the parameters in a quantum field theory to quantities we actually measure. It also explains why the effective strength of an interaction changes with…
Relativistic causality does not require every two-point function to vanish outside the light cone. It requires local observables to commute there = [ϕ(x),ϕ(y)]=0for spacelike separation.
The key distinction is between:correlation\text{correlation} andcausal influence.\text{causal influence}. A two-point function measures correlation. A commutator measures whether one local operation can affect another. 1. The two-point function can be nonzero For…
Feynman’s 1949 paper – Space-Time Approach to Quantum Electrodynamics
Feynman’s Spacetime Approach to QED: Opening Pages Explained These opening two pages are Feynman’s roadmap for a new way of calculating quantum electrodynamics. He is not yet deriving the machinery;…
Peskin and Schroeder’s Problem 2.1 is really about a subtle question…
If Maxwell’s equations follow from a perfectly Lorentz- and gauge-invariant Lagrangian, why does the straightforward Noether energy–momentum tensor look neither symmetric nor gauge invariant? The solution has two stages: derive…
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…
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…