Archives for Mathematical Physics
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(…
Temporal Green’s Functions
Temporal Green’s Function Temporal Green’s Function The temporal Green’s function is the Green’s function that solves a differential equation involving time — typically the evolution equation of a dynamical system…
Momentum Space Representation of the 1\r Operator
Momentum Space Representation of the \( \frac{1}{r} \) Operator The operator \( \frac{1}{r}, \quad r = |\mathbf{x}| \) plays a central role in quantum mechanics, especially in Coulomb potentials,…
Worked examples Kac Algebras, Lie Groups, Lie Algebras
1️⃣ Lie Group → Lie Algebra: Example with \( \mathrm{SO}(3) \) Step 1: Define the Lie group \( \mathrm{SO}(3) \) = Special Orthogonal Group in 3D Set of all \(…
Lie Group Algebras
Overview Lie groups and Lie algebras are two deeply connected mathematical structures used to study symmetries in mathematics and physics. They are foundational in areas such as particle physics,…
Abelain Group
Z(p∞) = { z ∈ ℂ | zpk = 1 for some integer k ≥ 1 } Proof that Z(p∞) is an Abelian Group We define the set: Z(p∞) =…
Examples of Taylor SEries versus Fourier Series
Intro Which works better for a given function - a Taylor expansion or a Fourier Expansion? This post explores the pros and cons of each, using specific examples. Examples of…
Taylor Series versus Fourier Series for a function
. Domain of Representation Taylor Series: Works best for local approximations around a single point (Maclaurin series if centered at zero). Fourier Series: Represents a function over an entire interval…
Functions ONLY definable by their integrals – with applications
Functions ONLY Defined by Their Integrals 1. The Gamma Function \( \Gamma(x) \) \( \Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \, dt \), for \( x > 0 \). Applications: Generalization…
Functions Defined by Their Integrals – with applications
Functions Defined by Their Integrals Functions that are defined by their integrals often arise in fields like physics, probability theory, and engineering, where the direct formulation of a function may…