Lorentz-covariant, unitary evolution of a relativistic Majorana qubit

We formulate a covariant description of a relativistic qubit identified with an irreducible set of quantum spin states of a Majorana particle with a sharp momentum. We treat the particle’s four-momentum as an external parameter. We show that it is possible to define an interesting time evolution of the spin density matrix of such a qubit. This evolution is manifestly Lorentz covariant in the bispinor representation and unitary in the spin representation. Moreover, during this evolution the Majorana particle undergoes an uniformly accelerated motion. We classify possible types of such motions, and finally we illustrate the behaviour of the polarization vector of the Majorana qubit during the evolution in some special cases

80 Years of Professor Wigner's Seminal Work "On Unitary Representations of the Inhomogeneous Lorentz Group"

This book celebrates the 80 years of the Professor Eugene P. Wigner paper “On Unitary Representations of the Inhomogeneous Lorentz Group", published in The Annals of Mathematics in 1939. We have collected several contributions divided into Research articles and Reviews. All contributions are technical, but the papers also bring a health element of didactic. Practitioners from several areas, from Gravity to Quantum Field Theory and Quantum Mechanics, as well as students, shall find a rich material in this Volume

Boosting Majorana zero modes

One-dimensional topological superconductors are known to host Majorana zero modes at domain walls terminating the topological phase. Their non-Abelian nature allows for processing quantum information by braiding operations that are insensitive to local perturbations, making Majorana zero modes a promising platform for topological quantum computation. Motivated by the ultimate goal of executing quantum-information processing on a finite time scale, we study domain walls moving at a constant velocity. We exploit an effective Lorentz invariance of the Hamiltonian to obtain an exact solution of the associated quasiparticle spectrum and wave functions for arbitrary velocities. Essential features of the solution have a natural interpretation in terms of the familiar relativistic effects of Lorentz contraction and time dilation. We find that the Majorana zero modes remain stable as long as the domain wall moves at subluminal velocities with respect to the effective speed of light of the system. However, the Majorana bound state dissolves into a continuous quasiparticle spectrum after the domain wall propagates at luminal or even superluminal velocities. This relativistic catastrophe implies that there is an upper limit for possible braiding frequencies even in a perfectly clean system with an arbitrarily large topological gap. We also exploit our exact solution to consider domain walls moving past static impurities present in the system

Relativistic Effects in Quantum Entanglement

One of the most fundamental phenomena of quantum physics is entanglement. It describes an inseparable connection between quantum systems, and properties thereof. In a quantum mechanical description even systems far apart from each other can share a common state. This entanglement of the subsystems, although arising from mathematical principles, is no mere abstract concept, but can be tested in experiment, and be utilized in modern quantum information theory procedures, such as quantum teleportation. In particular, entangled states play a crucial role in testing our understanding of reality, by violating Bell inequalities. While the role of entanglement is well studied in the realm of nonrelativistic quantum mechanics, its significance in a relativistic quantum theory is a relatively new field of interest. In this work the consequences of a relativistic description of quantum entanglement are discussed. We analyze the representations of the symmetry groups of special relativity, i.e. of the Lorentz group, and the Poincar\'e group, on the Hilbert space of states. We describe how unitary, irreducible representations of the Poincar\'e group for massive spin 1/2 particles are constructed from representations of Wigner's little group. We then proceed to investigate the role of the Wigner rotations in the transformation of quantum states under a change of inertial reference frame. Considering different partitions of the Hilbert space of 2 particles, we find that the entanglement of the quantum states appears different in different inertial frames, depending on the form of the states, the chosen inertial frames, and the particular choice of partition. It is explained, how, despite of this, the maximally possible violation of Bell inequalities is frame independent, when using appropriate spin observables, which are related to the Pauli-Ljubanski vector, a Casimir operator of the Poincar\'e group.Comment: 115 pages, 6 figures, diploma thesi

Octonions and Quantum Gravity through the Central Charge Anomaly in the Clifford Algebra

We derive a theory of quantum gravity containing an AdS$_3$ isometry/qubit duality. The theory is based on a superalgebra generalization of the enveloping algebra of the homogeneous AdS$_3$ spacetime isometry group and is isomorphic to the complexified octonion algebra through canonical quantization. Its first three quaternion generators correspond to an $\hbar$-quantized AdS$_3$ embedded spacetime and its remaining four non-quaternion generators to a $G$-quantized embedding $2+2$ Minkowski spacetime. The quaternion algebra's expression after a monomorphism into the complexified Clifford algebra produces a two-dimensional conformal operator product expansion with a central charge anomaly, which results in an area-law $\hbar G$ scaling satisfying the holographic principle and defines an "arrow of time". This relationship allows us to extend the theory through supersymmetry- and conformal-breaking $\mathcal O(G)$ transformations of the embedding to produce dS$_3$ and dS$_4$ spacetimes and derive a resolution to the black hole information paradox with an explicit mechanism. Unlike string theory, the theory is background-independent and suggests that our local dS$_4$ spacetime is the largest possible