Relativistic Quantum Mechanics and Related Topics

Jonathan Dimock

Explaining the concepts of quantum mechanics and quantum field theory in a precise mathematical language, this textbook is an ideal introduction for graduate students in mathematics, helping to prepare them for further studies in quantum physics. The textbook covers topics that are central to quantum physics: non-relativistic quantum mechanics, quantum statistical mechanics, relativistic quantum mechanics and quantum field theory. There is also background material on analysis, classical mechanics, relativity and probability. Each topic is explored through a statement of basic principles followed by simple examples. Around 100 problems throughout the textbook help readers develop their understanding.

Ph. Droz-Vincent

Stephen L. Adler

Abstract We turn in this chapter and the next to a discussion of topics relating to relativistic single-particle equations in quaternionic quantum mechanics. In Secs. 11.1 through 11.3 we develop the theory of the quaternionic Klein-Gordon equation, and then proceed in Secs. 11.4 and 11.5 to develop the theory of the quaternionic Dirac equation.

Maki Takahashi

This thesis explores a broad range of topics in the foundations of relativity and quantum information theory. The first and main topic of this thesis is on relativistic quantum information theory. Here we construct a reformulation of quantum information, which is consistent with relativity theory. We will see that by providing a rigorous formulation starting with the field equations for a massive fermion and a photon we can construct a theory for relativistic quantum information. In particular we provide a measurement formalism, a transport equation which describes the unitary evolution of a state through spacetime as well as how to extend this to multipartite systems. The second topic concerns the nature of time, duration and clocks in current physical theories and in particular for Newtonian mechanics. We analyse the relationship between the readings of clocks in Newtonian mechanics with absolute time. We will see that in order to answer this question we must provide not only a model for a clock but also solve what is referred to as Newton’s Scholium problem. We then compare this with other dynamical theories in particular quantum mechanics and general relativity where the treatment of time is quite different from Newtonian mechanics. The final topic is rather different from the first two. In this chapter we investigate a range of methods to perform tomography in a solid-state qubit device, for which a priori initialization and measurement of the qubit is restricted to a single basis of the Bloch sphere. We explore and compare several methods to acquire precise descriptions of additional states and measurements, quantifying both stochastic and systematic errors, ultimately leading to a tomographically-complete set that can be subsequently used in process tomography. We focus in detail on the example of a spin qubit formed by the singlet-triplet subspace of two electron spins in a GaAs double quantum dot, although our approach is quite general.

Armin Wachter