Search Results

Each of the different formalism of classical mechanics has its advantages and disadvantages. However, in the end all three frameworks tend to be equivalent, and thus the following list is very subject …

Does there exists a symplectic formulation of statistical physics? I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated sym …

Hey, the following is well-known (e. g. Ashtekar/Schilling, Brody/Hughston): A bounded self-adjoint operator $A$ on a Hilbert space $H$ induces a globally defined vector field $X$ on the projective …

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy m …

Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$. For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9 …

The classical phase space of the spin degrees of freedom is represented by the two-sphere with a symplectic form given by $s$ times the standard volume form. This is clearly no cotangent bundle to som …

Let $P \to M$ be a principal $G$-bundle. The moduli space of flat connections on $P$ is, by definition, the space $\mathcal{M} = \mathcal{C}_0 / \mathcal{G}$, where $\mathcal{C}_0$ denotes the subspac …

Smooth vector fields are in a one-to-one relationship with flows $\Phi: D \subseteq M\times \mathbb{R} \rightarrow M$, $$X_m = {\frac{d}{d t}}_{t=0} \Phi(m, t),$$ and by the symplectic form also with …

The easiest way to see that the norm of the curvature corresponds to the energy is to consider the special case of an abelian U(1)-Yang-Mills theory (i.e. electrodynamics). If you write out the norm s …

Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface. …

Hi, I am a physicist and currently doing my bachelor thesis about geometric quantization. In the book by Bates and Weinstein I encountered the Maslov index, which seems to be very important :-). But …

As spinors transform with a minus sign under a full rotation, there is no (non-trivial) lift of the action of the group of diffeomorphisms to the spinor bundle (i.e. the spinor bundle is not a natural …

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms $\Omega^{p …

Let $\Upsilon: G \times M \to M$ be a smooth action of a Lie group $G$ on a manifold $M$. Isenberg and Marsden (1982) define a slice at $m \in M$ as a submanifold $S \subseteq M$ containing $m$ such …

Ad 1.: Since every vector can be decomposed in its horizontal and vertical part. Thus it is enough to consider the case a) where all vectors are horizontal (this is trivial) and b) where at least one …