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Given a Cauchy surface in a curved spacetime and a suitably defined quantum state on the CCR algebra of the Klein Gordon quantum field on that surface, we show, by expanding the squared spacetime geodesic distance and the '$U$' and '$V$' Hadamard coefficients (and suitable derivatives thereof) in sufficiently accurate covariant Taylor expansions on the surface that the renormalized expectation value of the quantum stress-energy tensor on the surface is determined by the geometry of the surface and the first 4 time derivatives of the metric off the surface, in addition to the Cauchy data for the field's two-point function. This result has been anticipated in and is motivated by a previous investigation by the authors on the initial value problem in semiclassical gravity, for which the geometric initial data corresponds {\it a priori} to the metric on the surface and up to 3 time derivatives off the surface, but where it was argued that the fourth derivative can be obtained with aid of the field equations on the initial surface., Comment: 65 pages, discussion improved

Defects in crystals can have a transformative effect on the properties and functionalities of solid-state systems. Dopants in semiconductors are core components in electronic and optoelectronic devices. The control of single color centers is at the basis of advanced applications for quantum technologies. Unintentional defects can also be detrimental to the crystalline structure and hinder the development of novel materials. Whatever the research perspective, the identification of defects is a key but complicated, and often long-standing issue. Here, we present a general methodology to identify point defects by combining isotope substitution and polytype control, with a systematic comparison between experiments and first-principles calculations. We apply this methodology to hexagonal boron nitride (hBN) and its ubiquitous color center emitting in the ultraviolet spectral range. From isotopic purification of the host hBN matrix, a local vibrational mode of the defect is uncovered, and isotope-selective carbon doping proves that this mode belongs to a carbon-based center. Then, by varying the stacking sequence of the host hBN matrix, we unveil different optical responses to hydrostatic pressure for the non-equivalent configurations of this ultraviolet color center. We conclude that this defect is a carbon dimer in the honeycomb lattice of hBN. Our results show that tuning the stacking sequence in different polytypes of a given crystal provides unique fingerprints contributing to the identification of defects in 2D materials.

The 2023 second edition of a 2006 encyclopedia article on mathematical aspects of quantum field theory in curved spacetimes (QFTCST). Section-titles (with new sections indicated with stars) are: Introduction and preliminaries, Construction of a $*$-algebra for a real linear scalar field on globally hyperbolic spacetimes and some general theorems, *More about (quasifree) Hadamard states, Particle creation and the limitations of the particle concept, Theory of the stress-energy tensor, *More about the intersection of QFTCST with AQFT and the Fewster-Verch No-Go Theorem, Hawking and Unruh effects, *More about (classical and) quantum fields on black hole backgrounds, Non-globally hyperbolic spacetimes and the time-machine question, *More about QFT on non-globally hyperbolic spacetimes, Other related topics and some warnings. The article contains many references. It also includes a review of, and also compares and contrasts, recent results on the implications of QFTCST for the question of the instability of three sorts of Cauchy horizon -- first those inside black holes such as especially Reissner-Nordstr\"om-de Sitter and Kerr-de Sitter, second the compactly generated Cauchy horizons of spacetimes in which time-machines get manufactured, and third the Cauchy horizon of the spacetime which is believed to describe evaporating black holes and which underlies (one version of) the black hole information-loss puzzle., Comment: 36 pages, 7 figures, article prepared for the second edition of the Encyclopaedia of Mathematical Physics, edited by M. Bojowald and R.J. Szabo, to be published by Elsevier. V2: small improvements and additions, 3 figures redrawn, subsections renamed as sections, dedications added, acknowledgments added, 6 references added. arXiv admin note: text overlap with arXiv:gr-qc/0601008

Hawking showed that a black hole formed by collapse will emit radiation and eventually disappear. We address the challenge to define an objective notion of physical entropy which increases throughout this process in a way consistent with unitarity. We have suggested that (instead of coarse-grained entropy) physical entropy is matter-gravity entanglement entropy and that this may offer an explanation of entropy increase both for the black hole collapse and evaporation system and also for other closed unitarily evolving systems. For this to work, the matter-gravity entanglement entropy of the late-time state of black hole evaporation would have to be larger than the entropy of the freshly formed black hole. We argue that this may possibly be the case due to (usually neglected) photon-graviton interactions., Comment: 14 pages, 3 figures. (Tiny revision of) essay written for the Gravity Research Foundation 2023 Awards for Essays on Gravitation; received an honorable mention. arXiv admin note: substantial text overlap with arXiv:2206.07445

Since Hawking's 1974 discovery, we expect that a black hole formed by collapse will emit radiation and eventually disappear. Closely related to the information loss puzzle is the challenge to define an objective notion of physical entropy which increases throughout this process in a way consistent with unitarity. In recent years, this has been addressed with certain notions of coarse grained entropy. We have suggested instead that physical entropy should be identified with matter-gravity entanglement entropy and that this may offer an explanation of entropy increase both for the black hole collapse and evaporation system and also for other closed unitarily evolving systems, notably the universe as a whole. For this to work, it would have to be that the matter-gravity entanglement entropy of the late-time state of black hole evaporation is larger than the entropy of the freshly formed black hole. We argue here that this is possibly the case due to (usually neglected) photon-graviton interactions. If black hole evaporation is slowed down by putting the black hole in a slightly permeable box, we give plausibility arguments that the radiation remaining after a large black hole has evaporated will (be pure and) mainly consist of roughly equal numbers of photons and gravitons entangled with one another -- with a photon-graviton entanglement entropy possibly greater than the entropy of the freshly formed black hole. It also seems possible that, even in the absence of such a box, the matter-gravity entanglement entropy might still increase and the late-time state again be a pure state of (predominantly) photons highly entangled with soft gravitons that the Hawking-emitted photons themselves had radiated. More work is needed to find out if it is indeed so., Comment: 13 pages, 3 figures, references added, minor clarifications and improvements, more cautiously worded

We explore how the initial value problem may be formulated for globally hyperbolic, Hadamard, solutions of the semiclassical Einstein-Klein-Gordon equations. Given a set of data on an initial 3-surface, consisting of the values on the surface of a spacetime metric and its first 3 time derivatives off the surface, we introduce a notion of 'surface Hadamard' state on the CCR algebra of the surface. We conjecture that, for a given such set of classical Cauchy data with a surface Hadamard state satisfying the semiclassical constraint equations, the initial value problem will be well posed. We present similar conjectures for a semiclassical scalars model and semiclassical electrodynamics. Moreover, partly inspired by work of Parker and Simon in 1993, we define semiclassical gravity `physical solutions' to be those that are (jointly smooth) functions of $\hbar$ and of coordinates continuous in $\hbar$ at $\hbar =0$. We conjecture that for such solutions the second and third time derivatives of the metric off the surface need not be specified, but rather will be determined by that continuity condition. Assuming the initial value conjecture for such physical solutions holds, and that a stochastic rule were available which leads to quantum state collapses occurring on (non-intersecting) random Cauchy surfaces, we discuss the well-posedness of semiclassical gravity with stochastic quantum state collapses. We also discuss two notions of approximate physical semiclassical solutions (both with and without collapses): Namely solutions to order $\hbar$ (first discussed by Parker and Simon in 1993) and solutions to order $\hbar^0$. We point out that the latter do not require higher derivative terms or Hadamard subtractions, but that nevertheless order $\hbar^0$ semiclassical gravity is a distinct theory from classical general relativity capable of incorporating quantum interference phenomena., Comment: 47 pages. Corrections implemented, notably Def 11 (preliminary surface Hadamard condition) and certain equations on p. 33. Explanations improved and typos fixed. This version corresponds largely to the published version in JCAP

We present a short account of our work to provide quantum electrodynamics with a 'product picture'. It aims to complement the longer exposition in a recent paper in 'Foundations of Physics' and to help to make that work more accessible. The product picture is a formulation of QED, equivalent to standard Coulomb gauge QED, but in which the Hilbert space arises as (a certain physical subspace of) a product of a Hilbert space for the electromagnetic field and a Hilbert space for charged matter (i.e. the Dirac field) and in which the Hamiltonian arises as the sum of an electromagnetic Hamiltonian and a charged matter Hamiltonian and an interaction term. (The Coulomb gauge formulation of QED is not a product picture because, in it, the longitudinal part of the electromagnetic field is made out of charged matter operators.) We also recall a 'Contradictory Commutator Theorem' for QED which exposes flaws in previous attempts at temporal gauge quantization of QED and we explain how our product picture appears to offer a way to overcome those flaws. Additionally, we discuss the extent to which that theorem generalizes to Yang-Mills fields. We also develop a product picture for nonrelativistic charged particles in interaction with the electromagnetic field and point out how this leads to a novel way of thinking about the theory of many nonrelativistic electrically charged particles with Coulomb interactions. We explain how the provision of a product picture for QED gives hope that one will be able likewise to have a product picture for (Yang Mills and) quantum gravity -- the latter being needed to make sense of the author's 'matter-gravity entanglement hypothesis'. In an afterword, we briefly discuss similarities and differences between that hypothesis and its predictions and ideas of Roger Penrose related to a possible role of gravity in quantum state reduction and to cosmological entropy., Comment: 15 pages, no figures. V2: Subsection on Yang-Mills corrected and rewritten, minor improvements elsewhere. V3: minor corrections. V4: A few typos corrected. Version to appear in the special collection "Celebrating Sir Roger Penrose's Nobel Prize" [AVS Quantum Science, Guest Editors: Ivette Fuentes and Hendrik Ulbricht]

We bring together two topics (quantum mechanics and time passage) with the goal of clarifying questions about each. Specifically, we claim that the formalism of quantum mechanics provides an answer to the question: "What is time passage?".

We provide a theoretical foundation for the notion of the quantum coherent state of the electrostatic field of a static external charge distribution introduced in a 1998 paper and rederive formulae there for the inner products of a pair of such states. Contrary to what one might expect, these inner products are non-zero whenever the total charges of the two charge distributions are equal, even if the charge distributions themselves differ. We actually display two different frameworks for these same coherent states, in the second of which Gauss's law only holds in expectation value. We propose an experiment capable of ruling that out. The first framework leads to a 'product picture' for full QED -- i.e. a reformulation of standard QED in which it has a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free charged-matter Hamiltonian and an interaction term, acting on a 'physical subspace' of the full tensor product of charged-matter and electromagnetic-field Hilbert spaces. (The traditional Coulomb gauge formulation of QED isn't a product picture because, in it, the longitudinal part of the electric field is a function of the charged matter operators.) We do this for both Maxwell-Dirac and Maxwell-Schr\"odinger QED. For all states in the physical subspace of each of these systems, the charged matter is entangled with longitudinal photons and Gauss's law holds on the physical subspace as an operator equation; albeit the electric field operator and the Hamiltonian, while self-adjoint on the physical subspace, fail to be self-adjoint on the full tensor-product Hilbert space. Analogues of our coherent state inner products and of the product picture play a role in the author's matter-gravity entanglement hypothesis. Also, the product picture amounts to a temporal gauge quantization of QED which appears to be free from the difficulties of previous versions., Comment: 63 pages, 3 figures. v2: title changed, substantial revision, new material added, v3: a few corrections and improvements, v4: more corrections and improvements, improved discussion of how the product picture evades the Contradictory Commutator Theorem; to appear in Foundations of Physics

I recall my 'matter-gravity entanglement hypothesis' and briefly review the evidence for it, based partly on its seeming ability to resolve a number of puzzles related to quantum black holes including the black hole information loss puzzle. I point out that, according to this hypothesis, there is a quantity, i.e. the universe's 'matter-gravity entanglement entropy' -- which deserves to be considered the 'entropy of the universe' and which, with suitable initial conditions, will plausibly increase monotonically with cosmological time. In the last section, which is more tentative and raises a number of further puzzles and open questions, I discuss the prospects for a notion of 'events' which 'happen' whose statistical properties are described by this entropy of the universe. It is hoped that such a theory of events may be a step on the way towards explaining how initial quantum fluctuations convert themselves into inhomogeneities in a seemingly classical universe., Comment: 17 pages, 7 figures. Invited contribution to the proceedings of the conference "Progress and Visions in Quantum Theory in View of Gravity", Leipzig, Germany, October 2018, based on part of a talk at the workshop "The Mysterious Universe: Dark Matter -- Dark Energy -- Cosmic Magnetic Fields" MITP, Mainz, Germany, June 4, 2019. (v2 No change) v3: References added v4: typos etc. corrected

Editorial Note with a mathematical and historical introduction to a 1932 paper by Erwin Schr\"odinger on the generalization of the Dirac equation to a curved spacetime -- to appear in the 'Golden Oldie' section of the Journal of General Relativity and Gravitation alongside an English translation of that paper. The Schr\"odinger paper is of interest as the first place that the well-known formula $g^{\mu\nu}\nabla_\mu\nabla_\nu + m^2 + \frac{R}{4}$ was obtained for the 'square' of the Dirac operator in curved spacetime. This formula is known by a number of names and we explain why we favour the name 'Schr\"odinger-Lichnerowicz formula'. We also aim to explain how the modern notion of `spin connection' emerged from a debate in the physics journals in the period 1929-1933. We discuss the key contributions of Weyl, Fock and Cartan and explain how and why they were partly in conflict with the approaches of Schr\"odinger and several other authors. We reference and comment on some previous historical accounts of this topic., Comment: 15 pages, no figures. Addendum added mainly to mention that, in addition to the rediscovery of Schr\"odinger's 1932 formula for the 'square' of the curved spacetime Dirac operator by Andr\'e Lichnerowicz in 1962/3, it was also rediscovered in 1963 by Asher Peres -- and to discuss again the naming of that formula