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$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, …

Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$. The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition $$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$ where $\{A,B\} …

EDIT: Let $\Omega\subset \mathbb{R}^3$ be a bounded domain with smooth connected boundary. Let $f\colon \mathbb{R}^3\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\m …

Consider the (quantum) Hamiltonian on the real line $$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$ Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate min …

It is well known how to present solutions on the heat equation using the path integral (including the case of Riemannian manifold). Is there a way to present solutions of the Dirac equation using path …

In many courses in theoretical classical mechanics Hamiltonian formalism takes an important place. However I did not see it applied to problems of classical mechanics (unless one expands the scope of …

I am looking for an explicit (preferably simple) example of an ODE with time-independent coefficients in $\mathbb{R}^3$ such that there does not exist an Euler-Lagrange equation $$\frac{\partial L}{\p …

In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) se …

EDIT: In general relativity given a manifold $M$ one can consider a functional on (pseudo-) Riemannian metrics $g$ $$\int_M R\,\, dvol_g,$$ where $R$ is the scalar curvature and $vol_g$ is the (pseudo …

It is claimed that classical mechanics motivates introduction of symplectic manifolds. This is due to the theorem that the Hamiltonian flow preserves the symplectic form on the phase space. I am won …

The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order …

In (mathematical) physics the equations of motion of a system of particles are often interpreted as Euler-Lagrange equations for appropriate Lagrangian $L=L(x,\dot x,t)$ where $x$ is a collection of v …

This is a continuation of Classification of conformal diffeomorphisms of Minkowski space Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric: $$g=(x^0)^2-(x^1)^2-\dots -(x …

Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric: $$g=(x^0)^2-(x^1)^2-\dots -(x^n)^2.$$ Is there a classification of diffeomorphisms $F\colon \mathbb{R}^{n+1}\tilde\to …

EDIT: Given a system of $N\geq 3$ charged point particles in $\mathbb{R}^3$ of the same charge which interact according to Coulomb law (thus they repell one from each other). Is it possible that t …