A quantum frame is defined by a material object subject to the laws of quantum mechanics. The present paper studies the relations between quantum frames, which in the classical case are described by elements of the Poincare' group. The possibility of using a suitable quantum group is examined, but some arguments are given which show that a different mathematical structure is necessary. Some simple examples in lower dimensional spacetimes are treated. They indicate the necessity of taking into account some "internal" degrees of freedom of the quantum frames, that can be disregarded in a classical treatment.
A quantum frame is defined by a material object subject to the laws of quantum mechanics. The present paper studies the relations between quantum frames, which in the classical case are described by elements of the Poincare' group. The possibility of using a suitable quantum group is examined, but some arguments are given which show that a different mathematical structure is necessary. Some simple examples in lower dimensional spacetimes are treated. They indicate the necessity of taking into account some "internal" degrees of freedom of the quantum frames, that can be disregarded in a classical treatment.
Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part of. While such transformations were shown to be symmetries of the system's Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work, we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations.
A reference frame F is described by the element g of the Poincaré group 𝒫 which connects F with a given fixed frame F0. If F is a quantum frame, defined by a physical object following the laws of quantum physics, the parameters of g have to be considered as quantum observables. However, these observables are not compatible and some of them, namely the coordinates of the origin of F, cannot be represented by self-adjoint operators. Both these difficulties can be overcome by considering a positive-operator-valued measure on 𝒫, covariant with respect to the left translations of the group, namely a covariance system. We develop a construction procedure for this kind of mathematical structure. The formalism is also used to discuss the quantum observables measured with respect to a quantum reference frame.
The concept of a reference frame has been a part of the physical formalism since the early days of physics. The reference frame was always understood as a physical system with respect to which states of other physical systems are described. With the adoption of quantum theory, our understanding of physical systems has fundamentally changed, yet we still rely on classical systems for the definition of reference frames. If we acknowledge the fact that all systems are quantum mechanical, including reference frames, then we have to be able to describe quantum systems relative to other quantum systems. \n \nIn this thesis we develop a framework for integrating quantum reference frames into the formalism of quantum mechanics. We show how the description of states and measurements has to be refined in order to account for the quantum nature of the reference frame. Initially, we study the implications of the new description with a quantum reference frame associated with a generic compact group. Then, we apply the same approach to the study of relativistic quantum reference frames associated with the Poincaré group, which is not compact. \n \nOur findings for the generic compact group include an analysis of how well a quantum system defines a frame of reference for other systems. Also in the generic case we analyze the effects of a measurement on a quantum reference frame. In the relativistic case we first find how the massive representations of the Poincaré group decompose into irreducible representations. This is a key problem in the analysis of quantum reference frames. Finally, we analyze a relativistic quantum reference frame that is used for measuring the total momentum. We will show how the relativistic measurements of total momentum should be described with respect to a quantum reference frame. We will also show how these measurements affect the quantum reference frame.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
In physics, every observation is made with respect to a frame of reference. Although reference frames are usually not considered as degrees of freedom, in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered as a reference frame and, if so, which description would it give of the world? Here, we introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a "superposition of coordinate transformations". We describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame, and find that entanglement and superposition are frame-dependent features. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, to an extension of the weak equivalence principle, and to the possibility of defining the rest frame of a quantum system.
Recent advances in our understanding of foundations of quantum mechanics have shown that information can be made objective through quantum states. Such objectification processes, predicted e.g. in a variety of quantum open systems, must accompany any realistic quantum-to-classical transition mechanism in order to reproduce the objective character of the classical limit of our world. However, so far only examples of simple, unstructured information, such as a value of an observable, have been studied. In this work we show that a more complicated form of information, given by a Cartesian reference frame, can also be made (at least partially) objective in quantum mechanics. The non-trivial internal gauge structure of reference frames, given by the transformation group, leads to a more general form of objectivity, where all observers see the same but modulo their relative orientations, like it happens in modern understanding of geometry. This opens a way to extend quantum objectivity beyond simple scenarios and possibly link it to the foundations of modern geometry.