Four-mode squeezed states: two-field quantum systems and the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$

We construct the four-mode squeezed states and study their physical properties. These states describe two linearly-coupled quantum scalar fields, which makes them physically relevant in various contexts such as cosmology. They are shown to generalise the usual two-mode squeezed states of single-field systems, with additional transfers of quanta between the fields. To build them in the Fock space, we use the symplectic structure of the phase space. For this reason, we first present a pedagogical analysis of the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$ Sp ( 4 , R ) and its Lie algebra, from which we construct the four-mode squeezed states and discuss their structure. We also study the reduced single-field system obtained by tracing out one of the two fields. This procedure being easier in the phase space, it motivates the use of the Wigner function which we introduce as an alternative description of the state. It allows us to discuss environmental effects in the case of linear interactions. In particular, we find that there is always a range of interaction coupling for which decoherence occurs without substantially affecting the power spectra (hence the observables) of the system.

Generalized punctual Hilbert schemes and 𝔤-complex structures

We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple properties with Hitchin components, and which are conjecturally homeomorphic to them. For simple complex Lie algebras, this generalizes the higher complex structure. For real Lie algebras, this should give an alternative description of the Hitchin–Kostant–Rallis section.

ON AN INTEGRAL OF -BESSEL FUNCTIONS AND ITS APPLICATION TO MAHLER MEASURE

Cogdell et al. [‘Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space’, Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density $a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$ , where $J_0$ is the order-zero Bessel function of the first kind and a and $r_m$ are positive real numbers. To facilitate our proof we develop an alternative description of the integral’s asymptotic behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix.

Remembering

This chapter makes the case for understanding memory in terms of qualitatively different codes represented by active processes, as opposed to such memory systems as episodic and semantic memory. A distinction is made between primary memory (PM), viewed as active conscious processing, and secondary memory (SM), viewed as the long-term representations of events and knowledge. The notion that PM involves attention paid to the information held in mind is discussed in light of current views of working memory. SM is described in terms of a hierarchically organized set of analytic representations running from specific episodes to context-free knowledge, as an alternative description to Tulving’s account in terms of separate episodic and semantic systems. The role of the external context in supporting retrieval is emphasized, and also the role of executive processes in enabling self-initiated activities when such environmental support is absent. Following previous researchers, the chapter endorses the notion of remembering as a set of active analytic operations, and stresses the similarity between the processes of perceiving and remembering. These ideas are illustrated by empirical examples.

SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems

The Chemical Master Equation is a standard approach to model biochemical reaction networks. It consists of a system of linear differential equations, in which each state corresponds to a possible configuration of the reaction system, and the solution describes a time-dependent probability distribution over all configurations. The Stochastic Simulation Algorithm (SSA) is a method to simulate sample paths from this stochastic process. Both approaches are only applicable for small systems, characterized by few reactions and small numbers of molecules. For larger systems, the CME is computationally intractable due to a large number of possible configurations, and the SSA suffers from large reaction propensities. In our study, we focus on catalytic reaction systems, in which substrates are converted by catalytic molecules. We present an alternative description of these systems, called SiCaSMA, in which the full system is subdivided into smaller subsystems with one catalyst molecule each. These single catalyst subsystems can be analyzed individually, and their solutions are concatenated to give the solution of the full system. We show the validity of our approach by applying it to two test-bed reaction systems, a reversible switch of a molecule and methyltransferase-mediated DNA methylation.

The de Broglie–Bohm Quantum Theory and Its Application to Quantum Cosmology

We review the de Broglie–Bohm quantum theory. It is an alternative description of quantum phenomena in accordance with all the quantum experiments already performed. Essentially, it is a dynamical theory about objectively real trajectories in the configuration space of the physical system under investigation. Hence, it is not necessarily probabilistic, and it dispenses with the collapse postulate, making it suitable to be applied to cosmology. The emerging cosmological models are usually free of singularities, with a bounce connecting a contracting era with an expanding phase, which we are now observing. A theory of cosmological perturbations can also be constructed under this framework, which can be successfully confronted with current observations, and can complement inflation or even be an alternative to it.

An ambient approach to conformal geodesics

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third-order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article, we give a third description using ideas from holography. A conformal [Formula: see text]-manifold [Formula: see text] can be seen (formally at least) as the asymptotic boundary of a Poincaré–Einstein [Formula: see text]-manifold [Formula: see text]. We show that any curve [Formula: see text] in [Formula: see text] has a uniquely determined extension to a surface [Formula: see text] in [Formula: see text], which we call the ambient surface of [Formula: see text]. This surface meets the boundary [Formula: see text] in right angles along [Formula: see text] and is singled out by the requirement that it be a critical point of renormalized area. The conformal geometry of [Formula: see text] is encoded in the Riemannian geometry of [Formula: see text]. In particular, [Formula: see text] is a conformal geodesic precisely when [Formula: see text] is asymptotically totally geodesic, i.e. its second fundamental form vanishes to one order higher than expected. We also relate this construction to tractors and the ambient metric construction of Fefferman and Graham. In the [Formula: see text]-dimensional ambient manifold, the ambient surface is a graph over the bundle of scales. The tractor calculus then identifies with the usual tensor calculus along this surface. This gives an alternative compact proof of our holographic characterization of conformal geodesics.

A Probabilistic Approach to Extended Finite State Mean Field Games

We develop a probabilistic approach to continuous-time finite state mean field games. Based on an alternative description of continuous-time Markov chains by means of semimartingales and the weak formulation of stochastic optimal control, our approach not only allows us to tackle the mean field of states and the mean field of control at the same time, but also extends the strategy set of players from Markov strategies to closed-loop strategies. We show the existence and uniqueness of Nash equilibrium for the mean field game as well as how the equilibrium of a mean field game consists of an approximative Nash equilibrium for the game with a finite number of players under different assumptions of structure and regularity on the cost functions and transition rate between states.

Descriptions and the materiality of texts

This article builds on the notions of thick and thin description elaborated by Geertz and looks at what descriptive methods have been used in the field of papyrology, a sub-discipline of classics that studies ancient manuscripts on papyrus fragments recovered through legal and illegal excavations in Egypt from the 19th century. Past generations of papyrologists have described papyri merely as resources to retrieve ancient ‘texts’. In the article I argue these descriptions have had negative effects in the way this ancient material has been studied, preserved, and also exchanged through the antiquities market. Through a series of case studies, I offer an alternative description of papyrus fragments as things, which have a power that can be activated under specific circumstances or entanglements. In demonstrating papyrus manuscripts’ unstable nature and shifting meanings, which are contingent on such entanglements, the article calls for a new politics and ethics concerning their preservation and exchange.

The unity of the divine and the human in Christ: The Chalcedonian definition and Paul Tillich’s Spirit Christology

The Christological disputes of the 6th–7th centuries (the polemics of Leontius of Byzantium with the Nestorians and Eutychians, and Maximus the Confessor with the monoenergistes/monothelites) showed that the Chalcedonian definition gives rise to a number of problems that cannot be solved within the framework of traditional theology: the unclear ontological status of human nature without a human hypostasis; the inconsistency of the ontological models underlying trinitology and Christology; the need to resort to an artificial interpretation of the gospel testimonies about Christ. However, the Chalcedonian definition is only one possible way to describe the unity of the divine and the human in Christ. The Christology of Paul Tillich is considered as an example of an alternative description in which the above problems do not arise. Tillich’s idea is to replace the traditional concept of the Logos incarnated in man with the concept of the Spirit of God transforming man. According to this view, God does not act on human nature without hypostasis, but on the hypostasis of man through its unifying center. During the earthly life of Christ, this effect occurred only in the hypostasis of Christ as man. And after (and thanks to) the death on the cross and the resurrection of Christ, it extends to all people.