Narrow Results

We develop the theory of the momentum map for the Maxwell–Lorentz equations for a spinning extended charged particle. The development relies on the Hamilton–Poisson structure of the system. This theory is indispensable for the study of long-time behavior and radiation of the solitons of this system, in particular for the proof of orbital stability of the corresponding solitons moving with constant speed and rotating with constant angular velocity.

We apply the theory to the translation and rotation symmetry groups of the system. The general theory of the momentum map results in formal equations for the corresponding invariants. We solve these equations obtaining expressions for the momentum and the angular momentum.

We check the conservation of these expressions and their coincidence with known classical invariants. For the first time, we specify a general class of finite energy initial states which provide the conservation of angular momentum.

Frobenius reciprocity asserts that induction from a subgroup and restriction to it are adjoint functors in categories of unitary $G$-modules. In the 1980s, Guillemin and Sternberg established a parallel property of Hamiltonian $G$-spaces, which (as we show) unfortunately fails to mirror the situation where more than one $G$-module “quantizes” a given Hamiltonian $G$-space. This paper offers evidence that the situation is remedied by working in the category of prequantum$G$-spaces, where this ambiguity disappears; there, we define induction and multiplicity spaces and establish Frobenius reciprocity as well as the “induction in stages” property.

Let $(Z,\omega )$ be a Kähler manifold and let $U$ be a compact connected Lie group with Lie algebra $𝔲$ acting on $Z$ and preserving $\omega$. We assume that the $U$-action extends holomorphically to an action of the complexified group $U^{ℂ}$ and the $U$-action on $Z$ is Hamiltonian. Then there exists a $U$-equivariant momentum map $\mu :Z\to 𝔲$. If $G\subset U^{ℂ}$ is a closed subgroup such that the Cartan decomposition $U^{ℂ}=U\text{exp}(i𝔲)$ induces a Cartan decomposition $G=K\text{exp}(𝔭),$ where $K=U\cap G$, $𝔭=𝔤\cap i𝔲$ and $𝔤=𝔨\oplus 𝔭$ is the Lie algebra of $G$, there is a corresponding gradient map ${\mu }_{𝔭}:Z\to 𝔭$. If $X$ is a $G$-invariant compact and connected real submanifold of $Z,$ we may consider ${\mu }_{𝔭}$ as a mapping ${\mu }_{𝔭}:X\to 𝔭.$ Given an $\mathrm{\text{Ad}}(K)$-invariant scalar product on $𝔭$, we obtain a Morse like function $f=\frac{1}{2}\parallel {\mu }_{𝔭}{\parallel }^2$ on $X$. We point out that, without the assumption that $X$ is a real analytic manifold, the Lojasiewicz gradient inequality holds for $f$. Therefore, the limit of the negative gradient flow of $f$ exists and it is unique. Moreover, we prove that any $G$-orbit collapses to a single $K$-orbit and two critical points of $f$ which are in the same $G$-orbit belong to the same $K$-orbit. We also investigate convexity properties of the gradient map ${\mu }_{𝔭}$ in the Abelian case. In particular, we study two-orbit variety $X$ and we investigate topological and cohomological properties of $X$.

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrödinger framework from this perspective and provide a description of the Weyl–Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.

This paper concerns the Routh reduction procedure for Lagrangians systems with symmetry. It differs from the existing results on geometric Routh reduction in the fact that no regularity conditions on either the Lagrangian L or the momentum map J_L are required apart from the momentum being a regular value of J_L. The main results of this paper are: the description of a general Routh reduction procedure that preserves the Euler–Lagrange nature of the original system and the presentation of a presymplectic framework for Routh reduced systems. In addition, we provide a detailed description and interpretation of the Euler–Lagrange equations for the reduced system. The proposed procedure includes Lagrangian systems with a non-positively definite kinetic energy metric.

We construct a dual pair associated to the Hamiltonian geometric formulation of perfect fluids with free boundaries. This dual pair is defined on the cotangent bundle of the space of volume preserving embeddings of a manifold with boundary into a boundaryless manifold of the same dimension. The dual pair properties are rigorously verified in the infinite-dimensional Fréchet manifold setting. It provides an example of a dual pair associated to actions that are not completely mutually orthogonal.

The energy–momentum tensor for a particular matter component summarises its local energy–momentum distribution in terms of densities and current densities. We re-investigate under what conditions these local distributions can be integrated to meaningful global quantities. This leads us directly to a classic theorem by Max von Laue concerning integrals of components of the energy–momentum tensor, whose statement and proof we recall. In the first half of this paper, we do this within the realm of Special Relativity (SR) and in the traditional mathematical language using components with respect to affine charts, thereby focusing on the intended physical content and interpretation. In the second half, we show how to do all this in a proper differential-geometric fashion and on arbitrary spacetime manifolds, this time focusing on the group-theoretic and geometric hypotheses underlying these results. Based on this we give a proper geometric statement and proof of Laue’s theorem, which is shown to generalise from Minkowski space (which has the maximal number of isometries) to spacetimes with significantly less symmetries. This result, which seems to be new, not only generalizes but also clarifies the geometric content and hypotheses of Laue’s theorem. A series of three appendices lists our conventions and notation and summarises some of the conceptual and mathematical background needed in the main text.

The influence of a curved spacetime $M$ on the physical behavior of an ideal gas of $N$ particles is analyzed by considering the phase space of the system as a region of the cotangent bundle $T^{\ast }{M}^N$ and using Souriau’s Lie group thermodynamics to define the corresponding probability distribution function. While the construction of the phase space respects the separability of the system, by forcing each particle to satisfy the so-called mass-shell constraint, the probability distribution is constructed by mixing the natural symplectic structure of the cotangent bundle with a Hamiltonian description of the system. In this way, the spacetime is introduced into the statistics and its isometries turn out to be of special interest because the distributions are parametrized by the elements of the Lie algebra of the isometry group, through the momentum map of the action of the isometries in $T^{\ast }{M}^N$. We find the Gibbs distribution that, in the simplest case of a flat spacetime, reduces to the so-called modified Jüttner distribution, used to describe ideal gases in the regime of special relativity. We also define a temperature-like function using the norm of a Killing vector, which allows us to recover the so-called Tolman–Ehrenfest effect. As a particular example, we study the outer region of a Schwarzschild black hole, for which a power series expansion of the Schwarzschild radius allows us to represent the partition function and the Gibbs distribution in terms of the corresponding quantities of the Minkowski spacetime.

A unitary representation of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators idπ(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra 𝔤. In the first part of the present paper we explain how this concept leads to a fruitful interaction between the areas of infinite dimensional convexity, Lie theory, symplectic geometry (momentum maps) and complex analysis. Here open invariant cones in Lie algebras play a central role and semibounded representations have interesting connections to C*-algebras which are quite different from the classical use of the group C*-algebra of a finite dimensional Lie group. The second half is devoted to a detailed discussion of semibounded representations of the diffeomorphism group of the circle, the Virasoro group, the metaplectic representation on the bosonic Fock space and the spin representation on fermionic Fock space.

In this note we introduce the concept of a semi-bounded unitary representations of an infinite-dimensional Lie group G. Semi-boundedness is defined in terms of the corresponding momentum set in the dual 𝔤' of the Lie algebra 𝔤 of G. After dealing with some functional analytic issues concerning certain weak-*-locally compact subsets of dual spaces, called semi-equicontinuous, we characterize unitary representations which are bounded in the sense that their momentum set is equicontinuous, we characterize semi-bounded representations of locally convex spaces in terms of spectral measures, and we also describe a method to compute momentum sets of unitary representations of reproducing kernel Hilbert spaces of holomorphic functions.