Reviews in Mathematical Physics

We consider Bose gases of $N$ particles in a box of volume one, interacting through a repulsive potential with scattering length of order $N^{-1+\kappa }$, for $\kappa >0$. Such regimes interpolate between the Gross–Pitaevskii and thermodynamic limits. Assuming that $\kappa$ is sufficiently small, we determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing in the limit $N\to \infty$.

We study a class of dynamical semigroups ${({𝕃}^n)}_{n\in \mathbb{N}}$ that emerge, by a Feynman–Kac type formalism, from a random quantum dynamical system ${({{ℒ}}_{{\omega }_n}\circ \cdots \circ {{ℒ}}_{{\omega }_1}({\rho }_{{\omega }_0}))}_{n\in \mathbb{N}}$ driven by a Markov chain ${({\omega }_n)}_{n\in \mathbb{N}}$. We show that the almost sure large time behavior of the system can be extracted from the large $n$ asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator $𝕃$. As a physical application, we consider the case where the ${{ℒ}}_{\omega }$’s are the reduced dynamical maps describing the repeated interactions of a system ${𝒮}$ with thermal probes ${{ℰ}}_{\omega }$. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.

We study the behavior of Donaldson’s invariants of 4-manifolds based on the moduli space of anti-self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang–Mills theory, known today as Donaldson–Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. These methods are expected to extend to higher codimensions and thus might help getting a better understanding for fully extendable $n$-dimensional field theories (in the sense of Baez–Dolan and Lurie) in the perturbative setting, especially when $n\le 4$. Additionally, we relate these constructions to Nekrasov’s partition function by treating an equivariant version of Donaldson–Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov–Witten and Donaldson–Thomas theory and recall their correspondence conjecture for general Calabi–Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena.

After introducing the infinite Fermi $C^{\ast }$-tensor product of a single ${\mathbb{Z}}_2$-graded $C^{\ast }$-algebra as an inductive limit, we systematically study the structure of the so-called symmetric states, that is those which are invariant under the group consisting of all finite permutations of a countable set. Among the obtained results, we mention the extension of De Finetti theorem which asserts that a symmetric state is a “mixture” of product states, each of which is a product of a single even state. This result induces a canonical morphism of the simplexes made of the symmetric even states on the usual infinite $C^{\ast }$-tensor product and the symmetric states on the infinite Fermi $C^{\ast }$-tensor product. We then extend the so-called Klein transformation to the infinite Fermi $C^{\ast }$-tensor product, available when the parity automorphism is inner. In such a situation, we investigate further properties of product states, the last being the extremal symmetric states on such an infinite Fermi $C^{\ast }$-tensor product $C^{\ast }$-algebra. This paper is complemented with a finite dimensional illustrative example for which the Klein transformation is not implementable, and then the Fermi tensor product might not generate a usual tensor product. Therefore, in general, the study of the symmetric states on the Fermi algebra cannot be easily reduced to that of the corresponding symmetric states on the usual infinite tensor product, even if both share many common properties.

We consider how the outputs of the Kadison transitivity theorem and Gelfand–Naimark–Segal (GNS) construction may be obtained in families when the initial data are varied. More precisely, for the Kadison transitivity theorem, we prove that for any nonzero irreducible representation $({\mathscr{H}},\pi )$ of a $C^{\ast }$-algebra $𝔄$ and $n\in \mathbb{N}$, there exists a continuous function $A:X\to 𝔄$ such that $\pi (A(x,y))x_i=y_i$ for all $i\in \{1,\dots ,n\}$, where $X$ is the set of pairs of $n$-tuples $(x,y)\in {{\mathscr{H}}}^n\times {{\mathscr{H}}}^n$ such that the components of $x$ are linearly independent. Versions of this result where $A$ maps into the self-adjoint or unitary elements of $𝔄$ are also presented. Regarding the GNS construction, we prove that given a topological $C^{\ast }$-algebra fiber bundle $p:𝔄\to Y$, one may construct a topological fiber bundle $\mathcal{𝒫}(𝔄)\to Y$ whose fiber over $y\in Y$ is the space of pure states of ${𝔄}_y$ (with the norm topology), as well as bundles $\mathcal{\mathscr{H}}\to \mathcal{𝒫}(𝔄)$ and $\mathcal{𝒩}\to \mathcal{𝒫}(𝔄)$ whose fibers ${\mathcal{\mathscr{H}}}_{\omega }$ and ${\mathcal{𝒩}}_{\omega }$ over $\omega \in \mathcal{𝒫}(𝔄)$ are the GNS Hilbert space and closed left ideal, respectively, corresponding to $\omega$. When $p:𝔄\to Y$ is a smooth fiber bundle, we show that $\mathcal{𝒫}(𝔄)\to Y$ and $\mathcal{\mathscr{H}}\to \mathcal{𝒫}(𝔄)$ are also smooth fiber bundles; this involves proving that the group of ∗-automorphisms of a $C^{\ast }$-algebra is a Banach Lie group. In service of these results, we review the topology and geometry of the pure state space. A simple non-interacting quantum spin system is provided as an example illustrating the physical meaning of some of these results.

We study the radiation of photons from a classical charged particle. We particularly consider a situation where the particle has a constant velocity in the distant past, then is accelerated, and then has a constant velocity in the distant future. Starting with no photons in the distant past we seek to characterize the quantum state of the photon field in the distant future. Working in the Coulomb gauge and in a $C^{\ast }$-algebra formulation, we give sharp conditions on whether this state is or is not in Fock space.