Archives for Mathematical Physics
Finite Abelian Groups and Applications to Quantum Physics
Finite Abelian Groups and Applications to Quantum Physics What Are Finite Abelian Groups? A finite abelian group is a group \( G \) with the following properties: Closure: For any…
Galois Groups and Applications to Quantum Mechanics
Finite Galois Groups and Applications in Quantum Mechanics Examples of Finite Galois Groups 1. The Cyclic Group \( C_n \) Example: Consider the extension \( \mathbb{Q}(\alpha_n)/\mathbb{Q} \), where \( \alpha_n…
Borel Algebras and Applications in Physics
Borel Algebra and Applications in Physics Borel Algebra and Applications in Physics Examples of Borel Algebras Real Line (\( \mathbb{R} \)): The Borel algebra on \( \mathbb{R} \) is generated…
Analytic Functions on a Punctured Disk with Applications to Quantum Mechanics
Analytic Functions in Quantum Mechanics and Quantum Field Theory The following examples illustrate how different analytic functions defined on a punctured disk can be applied in quantum mechanics (QM) and…
Quantum Mechanics and the Transformation 1\(z-a)
Quantum Mechanics and the Transformation \( \frac{1}{z - a} \) Transformations of the form \( \frac{1}{z - a} \), especially in the context of complex analysis, appear in quantum mechanics,…
The Dirichlet problem and quantum entanglement
Dirichlet Problem and Quantum Entanglement The Dirichlet problem and quantum entanglement are concepts from different branches of mathematics and physics, respectively, but there are indirect connections through the underlying…
Contour Integration and Feynman Propagator
Contour integration is a powerful technique in complex analysis, used extensively in physics for evaluating integrals that arise in quantum field theory, especially when calculating Feynman propagators. Here's an explanation…
Exercises on Rings and Modules by T.Y. Lam
"Exercises on Rings and Modules" contains a collection of exercises designed to help students understand and practice concepts related to rings and modules in abstract algebra. Here are the key…
Examples of Hilbert Spaces
Hilbert Spaces can be FINITE dimensional or INFINITE Dimensional Finite Dimensional Hilbert Spaces n-Tuples of real numbers - $R^n$ - $R^5$ Inner Product would be just the dot product of…
Subspaces of Vector Spaces, Invariant Subspaces
The idea is to create a RESTRICTED space within the given vector space. Within R3 $\R^3$: $\{ \vec{i}, \vec{j}, \vec{k} \}$ we can define a space that is far more…