# Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elementary physical questions. This tag is intended for questions on modern mathematical methods used in quantum theory, general relativity, string theory, integrable system etc at an advanced undergraduate or graduate level.

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### Finiteness of results in Connes-Kreimer approach

Remark: Although this is technically a physics-related post, the content heavily relies on pure mathematics, so I deemed it more appropriate here. When reading the papers by Connes and Kreimer (e.g. [...
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### How to expand $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ in $SU(2)$

I would like to calculate the following expression: $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ where $D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$ and $A_\mu^a$ are the components of a real $SU(2)$ ...
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### Decomposition of function into products

Given a single variable function $f(x)$, is there a way of decomposing it into the product of a family of function. Something similar to, $$f(x) = \prod_n p^{a_n}_n(x)$$ I am trying to find the ...
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### Question about Cartan's Theory of Spinors, Section 53 a spinor is a Euclidean tensor

Context I'm studying spinors in detail as part of research project. I'm working through Cartan's Theory of Spinors [1]. In section 53, A spinor is a Euclidean tensor, Cartan asks us to, "Consider ...
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### How to evaluate this definite integral in terms of Bessel functions.

In the context of Green's functions for the Free Klein-Gordon field, the following integral occurs: $$\int_m^{\infty}{\rho e^{-\rho r}\over\sqrt{\rho^2-m^2}}\; d\rho.$$ Here $m$, is a positive ...
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### How to find Riemann invariants of a system

I'm trying to find the Riemann invariants of the system \begin{array}{l} \frac{\partial \alpha }{\partial t} +( \alpha +V)\frac{\partial \alpha }{\partial x} -\alpha \frac{\partial V}{\partial x} =0\\ ...
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### What does it mean for a spring to have negative stiffness? [closed]

When working with 2nd order, linear, homogeneous, constant coefficient ODEs of the form my'' + by' + ky = 0, k is indicative of spring stiffness. The stiffer the spring, the more force it exerts ...
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### Properties of Harmonic Functions and Monotonicity

I have a general question surrounding certain harmonic functions. I was able to solve the Laplace equation $\Delta f$ = 0 in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, ...
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### Uniqueness and stability of equilibrium

Two points $P_1,P_2$ of mass $m$ move constrained respectively to $y=x^2, y=x^2-1$ in a vertical plane $xy$. Points are connected with a spring with constant $k>0$, and rest's length equal zero. ...
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### Green's Formula for vector fields in the Navier Stokes Weak Formulation

I am currently studying the weak formulation of the Navier-Stokes equations and came across the following equation: \int_{\Omega} \mathbf{v} \cdot \Delta \mathbf{u} \, dx = -\int_{\...
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