Author Archives: anuj
Product States – what does it mean to multiply the wavefunctions of two separate particles that are interacting?
A useful distinction is: Product state: We can assign each particle its own wavefunction and multiply them to get the joint amplitude. Entangled state: Only the combined system has a…
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…
Vacuum energy of particle antiparticle pairs may be responsible for the value of cosmological constant?
In short, No. There is an enormous discrepancy. However, this did not strictly disprove the idea that vacuum energy contributes to the cosmological constant. It showed that the simplest quantum-field-theory…
Spaces that narrowly miss becoming Hilbert Spaces
A Hilbert space is a vector space with: an inner product, which defines lengths and angles; and completeness—every Cauchy sequence converges to an element still inside the space. Several important…
A perfect nowhere dense set
A perfect nowhere dense set is a subset of a topological space (usually the real numbers R\mathbb{R}R) that satisfies two seemingly contradictory properties:Perfect: The set is closed and has no…
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…