Archives for Quantum Field Theory
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…
Lorentz Invariance of Scalar Fields
This is one of the foundational derivations in relativistic quantum field theory: showing that the Klein–Gordon scalar field transforms consistently under Lorentz transformations and that the theory is Lorentz invariant.…
Treating a bipartite Hamiltonian relativistically
For a relativistic bipartite system, you usually do not start with a simple Hamiltonian likeH=HA⊗IB+IA⊗HB+HintH = H_A \otimes I_B + I_A \otimes H_B + H_{\text{int}}H=HA⊗IB+IA⊗HB+Hint unless you are in a…
Why there are no macroscopic Spinor Fields?
1. What Is a Spinor Field? A spinor field is a field describing spin-12\frac1221 particles: electrons quarks neutrinos The Dirac field is the standard example:ψ(x)\psi(x)ψ(x) Unlike scalar or vector fields,…
What is the difference between δ and ∂ in this derivation
For the derivation of the Least Action Principle in classical field theory , what is the difference between δ and ∂? This is one of the most important conceptual distinctions…
The principle of stationary action derivation
Start with the actionS=∫d4x L(ϕ,∂μϕ)S=\int d^4x \, \mathcal{L}(\phi,\partial_\mu \phi)S=∫d4xL(ϕ,∂μϕ) The physical field configuration is the one for which a small variation of the field,ϕ(x)→ϕ(x)+δϕ(x)\phi(x)\rightarrow \phi(x)+\delta\phi(x)ϕ(x)→ϕ(x)+δϕ(x) does not change the action to…