Quantum theory in the reference frame of a physical reference system

The standard formalism of quantum mechanics is extended to describe a total system including the reference system (RS), with respect to which the total system is described. The RS is assumed to be able to act as a measuring apparatus, with measurement records given by the values of some reference properties of the RS. In order to describe the total system, we define a frame of reference (FR) as a set of states that can be used to express all other states of the total system. The theory is based on four basic postulates, which have, loosely speaking, the following contents. (i) A reference property of a RS has a definite value and is sufficiently stable in the FR directly related to the reference property. (ii) States of the total system are associated with vectors in the Hilbert space. (iii) Schrödinger equation is the dynamical law in each valid FR. (iv) Under certain condition a property of a system can be regarded as a reference property; vector descriptions of the total system given in different FRs of the same RS may have a probabilistic relationship like in Born's rule.The standard formalism of quantum mechanics is extended to describe a total system including the reference system (RS), with respect to which the total system is described. The RS is assumed to be able to act as a measuring apparatus, with measurement records given by the values of some reference properties of the RS. In order to describe the total system, we define a frame of reference (FR) as a set of states that can be used to express all other states of the total system. The theory is based on four basic postulates, which have, loosely speaking, the following contents. (i) A reference property of a RS has a definite value and is sufficiently stable in the FR directly related to the reference property. (ii) States of the total system are associated with vectors in the Hilbert space. (iii) Schrödinger equation is the dynamical law in each valid FR. (iv) Under certain condition a property of a system can be regarded as a reference property; vector descriptions of the total system given in different FRs of the same RS may have a probabilistic relationship like in Born's rule.

arXiv:0806.4843Jun 30, 20082 citations

Quantum Reference Frames and The Poincaré Symmetry

Oleg Kabernik

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.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.

Sep 19, 20142 citations

Structure of Physical Theories

K. Kong Wan

This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).

Aug 27, 20206 citations

Quantum mechanics from two physical postulates

Mehrafarin, M

For an arbitrary preparation, quantum mechanical descriptions refer to the complementary contexts set by incompatible measurements. We argue that an arbitrary preparation, therefore, should be described with respect to such a context by its degrees of disturbance (represented by real numbers) and their probability distribution (postulate 1). Measurement contexts thus provide reference frames for the preparation space of a physical system; a preparation being described by a point in this space with the aforementioned as its coordinates relative to a given measurement apparatus. However, all measurement contexts are equivalent with regard to the description of a given preparation; there is no preferred measurement (postulate 2). In the framework provided by the preparation space, we show that quantum mechanics emerges naturally from the above postulates in a new formulation which is manifestly canonical; provided the degrees of disturbance are identified with the quantum phases of the preparation with respect to (the basis furnished by) the measurement apparatus.For an arbitrary preparation, quantum mechanical descriptions refer to the complementary contexts set by incompatible measurements. We argue that an arbitrary preparation, therefore, should be described with respect to such a context by its degrees of disturbance (represented by real numbers) and their probability distribution (postulate 1). Measurement contexts thus provide reference frames for the preparation space of a physical system; a preparation being described by a point in this space with the aforementioned as its coordinates relative to a given measurement apparatus. However, all measurement contexts are equivalent with regard to the description of a given preparation; there is no preferred measurement (postulate 2). In the framework provided by the preparation space, we show that quantum mechanics emerges naturally from the above postulates in a new formulation which is manifestly canonical; provided the degrees of disturbance are identified with the quantum phases of the preparation with respect to (the basis furnished by) the measurement apparatus.

Feb 20, 20046 citations

Mathematical formalism of many-worlds quantum mechanics

Zeqian Chen

We combine the ideas of Dirac's orthonormal representation, Everett's relative state, and 't Hooft's ontological basis to define the notion of a world for quantum mechanics. Mathematically, for a quantum system $\mathcal{Q}$ with an associated Hilbert space $\mathbb{H},$ a world of $\mathcal{Q}$ is defined to be an orthonormal basis of $\mathbb{H}.$ The evolution of the system is governed by Schrödinger's equation for the worlds of it. An observable in a certain world is a self-adjoint operator diagonal under the corresponding basis. Moreover, a state is defined in an associated world but can be uniquely extended to the whole system as proved recently by Marcus, Spielman, and Srivastava. Although the states described by unit vectors in $\mathbb{H}$ may be determined in different worlds, there are the so-called topology-compact states which must be determined by the totality of a world. We can apply the Copenhagen interpretation to a world for regarding a quantum state as an external observation, and obtain the Born rule of random outcomes. Therefore, we present a mathematical formalism of quantum mechanics based on the notion of a world instead of a quantum state.We combine the ideas of Dirac's orthonormal representation, Everett's relative state, and 't Hooft's ontological basis to define the notion of a world for quantum mechanics. Mathematically, for a quantum system $\mathcal{Q}$ with an associated Hilbert space $\mathbb{H},$ a world of $\mathcal{Q}$ is defined to be an orthonormal basis of $\mathbb{H}.$ The evolution of the system is governed by Schrödinger's equation for the worlds of it. An observable in a certain world is a self-adjoint operator diagonal under the corresponding basis. Moreover, a state is defined in an associated world but can be uniquely extended to the whole system as proved recently by Marcus, Spielman, and Srivastava. Although the states described by unit vectors in $\mathbb{H}$ may be determined in different worlds, there are the so-called topology-compact states which must be determined by the totality of a world. We can apply the Copenhagen interpretation to a world for regarding a quantum state as an external observation, and obtain the Born rule of random outcomes. Therefore, we present a mathematical formalism of quantum mechanics based on the notion of a world instead of a quantum state.

arXiv:1504.08059Apr 30, 20150 citations

Quantum mechanics and the covariance of physical laws in quantum reference frames

Flaminia Giacomini, Esteban Castro-Ruiz, Časlav Brukner

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.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.

arXiv:1712.07207Nature CommunicationsJan 30, 2019297 citations

\mathcal{Q}

with an associated Hilbert space

\mathbb{H},

a world of

\mathcal{Q}

is defined to be an orthonormal basis of

\mathbb{H}.

The evolution of the system is governed by Schrödinger's equation for the worlds of it. An observable in a certain world is a self-adjoint operator diagonal under the corresponding basis. Moreover, a state is defined in an associated world but can be uniquely extended to the whole system as proved recently by Marcus, Spielman, and Srivastava. Although the states described by unit vectors in

\mathbb{H}

may be determined in different worlds, there are the so-called topology-compact states which must be determined by the totality of a world. We can apply the Copenhagen interpretation to a world for regarding a quantum state as an external observation, and obtain the Born rule of random outcomes. Therefore, we present a mathematical formalism of quantum mechanics based on the notion of a world instead of a quantum state.