Relative Quantum Time

The need for a time-shift invariant formulation of quantum theory arises from fundamental symmetry principles as well as heuristic cosmological considerations. Such a description then leaves open the question of how to reconcile global invariance with the perception of change, locally. By introducing relative time observables, we are able to make rigorous the Page-Wootters conditional probability formalism to show how local Heisenberg evolution is compatible with global invariance

A relational perspective on the Wigner-Araki-Yanase theorem

We present a novel interpretation of the Wigner-Araki-Yanase (WAY) theorem based on a relational view of quantum mechanics. Several models are analysed in detail, backed up by general considerations, which serve to illustrate that the moral of the WAY theorem may be that in the presence of symmetry, a measuring apparatus must fulfil the dual purpose of both reflecting the statistical behaviour of the system under investigation, and acting as a physical reference system serving to define those quantities which must be understood as relative.Comment: Version 2 contains some corrections and improvements suggested by an anonymous refere

A quantum reference frame size-accuracy trade-off for quantum channels

The imposition of symmetry upon the nature and structure of quantum observables has recently been extensively studied, with quantum reference frames playing a crucial role. In this paper, we extend this work to quantum transformations, giving quantitative results showing, in direct analogy to the case of observables, that a "large" reference frame is required for non-covariant channels to be well approximated by covariant ones. We apply our findings to the concrete setting of SU(2) symmetry

Position Measurements Obeying Momentum Conservation

We present a hitherto unknown fundamental limitation to a basic measurement: that of the position of a quantum object when the total momentum of the object and apparatus is conserved. This result extends the famous Wigner-Araki-Yanase (WAY) theorem, and shows that accurate position measurements are only practically feasible if there is a large momentum uncertainty in the apparatus

Skew Hecke Algebras

Let $G$ be a finite group, $H \le G$ a subgroup, $R$ a commutative ring, $A$ an $R$-algebra, and $\alpha$ an action of $G$ on $A$ by $R$-algebra automorphisms. Following Baker, we associate to this data the \emph{skew Hecke algebra} $\mathcal{H}_{R}(G,H,A,\alpha)$, which is the convolution algebra of $H$-invariant functions from $G/H$ to $A$. In this paper we study the basic structure of these algebras, proving for skew Hecke algebras a number of common generalisations of results about skew group algebras and results about Hecke algebras of finite groups. We show that skew Hecke algebras admit a certain double coset decomposition. We construct an isomorphism from $\mathcal{H}_{R}(G,H,A,\alpha)$ to the algebra of $G$-invariants in the tensor product $A \otimes \mathrm{End}_{R} ( \mathrm{Ind}_{H}^{G} R )$. We show that if $|G|$ is a unit in $A$, then $\mathcal{H}_{R}(G,H,A,\alpha)$ is isomorphic to a corner ring inside the skew group algebra $A \rtimes G$. Alongside our main results, we show that the construction of skew Hecke algebras is compatible with certain group-theoretic operations, restriction and extension of scalars, certain cocycle perturbations of the action, gradings and filtrations, and the formation of opposite algebras. The main results are illustrated in the case where $G = S_3$, $H = S_2$, and $\alpha$ is the natural permutation action of $S_3$ on the polynomial algebra $R[x_1,x_2,x_3]$.Comment: 35 page

Approximating relational observables by absolute quantities : a quantum accuracy-size trade-off

The notion that any physical quantity is defined and measured relative to a reference frame is traditionally not explicitly reflected in the theoretical description of physical experiments where, instead, the relevant observables are typically represented as 'absolute' quantities. However, the emergence of the resource theory of quantum reference frames as a new branch of quantum information science in recent years has highlighted the need to identify the physical conditions under which a quantum system can serve as a good reference. Here we investigate the conditions under which, in quantum theory, an account in terms of absolute quantities can provide a good approximation of relative quantities. We find that this requires the reference system to be large in a suitable sense

Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics

We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox "absolute" quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject

Evidence for widespread hydrated minerals on asteroid (101955) Bennu

Early spectral data from the Origins, Spectral Interpretation, Resource Identification, and Security-Regolith Explorer (OSIRIS-REx) mission reveal evidence for abundant hydrated minerals on the surface of near-Earth asteroid (101955) Bennu in the form of a near-infrared absorption near 2.7 µm and thermal infrared spectral features that are most similar to those of aqueously altered CM-type carbonaceous chondrites. We observe these spectral features across the surface of Bennu, and there is no evidence of substantial rotational variability at the spatial scales of tens to hundreds of metres observed to date. In the visible and near-infrared (0.4 to 2.4 µm) Bennu’s spectrum appears featureless and with a blue (negative) slope, confirming previous ground-based observations. Bennu may represent a class of objects that could have brought volatiles and organic chemistry to Earth