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arXiv:1204.3562

A Structurally Relativistic Quantum Theory. Part 1: Foundations

Emile Grgin

Published Apr 16, 2012• 0 citations
arXivHTMLPDF

Abstract

The apparent impossibility of extending non-relativistic quantum mechanics to a relativistic quantum theory is shown to be due to the insufficient structural richness of the field of complex numbers over which quantum mechanics is built. A new number system with the properties needed to support an inherently relativistic quantum theory is brought to light and investigated to a point sufficient for applications.

DOI
10.48550/arxiv.1204.3562

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The apparent impossibility of extending non-relativistic quantum mechanics to a relativistic quantum theory is shown to be due to the insufficient structural richness of the field of complex numbers over which quantum mechanics is built. A new number system with the properties needed to support an inherently relativistic quantum theory is brought to light and investigated to a point sufficient for applications.
The present article is the flrst in a program that aims at generalizing quantum mechanics by keeping its structure essentially intact, but constructing the Hilbert space over a new number system much richer than the fleld of complex numbers. We call this number system \the Quantionic Algebra. It is eight dimensional like the algebra of octonions, but, unlike the latter, it is associative. It is not a division algebra, but \almost one (in a sense that will be evident when we come to it). It enjoys the minimum of properties needed to construct a Hilbert space that admits quantum-mechanical interpretations (like transition probabilities), and, moreover, it contains the local Minkowski structure of space-time. Hence, a quantum theory built over the quantions is inherently relativistic. The algebra of quantions has been discovered in two steps. The flrst is a careful analysis of the abstract structure of quantum mechanics (the flrst part of the present work), the second is the classiflcation of all concrete realizations of this abstract structure (several additional articles). The classiflcation shows that there are only two realizations. One is standard quantum mechanics, the other its inherently relativistic generalization. The present article develops the abstract algebra of observables.
I n the early 20th century, the groundbreaking theories of quantum mechanics and special relativity emerged at roughly the same time.At first, both theories appeared to be entirely incompatible with each other.While the quantum equations written out by Erwin Schrödinger and Wolfgang Pauli dealt with the behaviours of electrons, atoms, and molecules moving at speeds encountered in classical physics, the rules of special relativity first imagined by Albert Einstein described the behaviour of matter moving at close to the speed of light.Initially, this apparent clash of theories presented a crisis in physics.However, in the late 1920s, further theoretical work carried out by physicists including Paul Dirac, Oskar Klein, and Walter Gordon culminated in a new set of equations, which proved definitively that quantum mechanics and special relativity are compatible with each other after all. An educational challengeYet with this groundbreaking discovery came an entirely new problem.In adding a new layer of complexity to the equations of non-relativistic quantum mechanics, the more complete theories of relativistic
Understanding the electron clock and the role of complex numbers in quantum mechanics is grounded in the geometry of spacetime, and best expressed with Spacetime Algebra (STA). The efficiency of STA is demonstrated with coordinate-free applications to both relativistic and non-relativistic QM. Insight into the structure of Dirac theory is provided by a new comprehensive analysis of local observables and conservation laws. This provides the foundation for a comparative analysis of wave and particle models for the hydrogen atom, the workshop where Quantum Mechanics was designed, built and tested.
Quantum mechanics, which describes the behaviour of matter and energy on very small scales, is one of the most successful theories in the history of science. Einstein's theory of special relativity, which describes the relationship between space and time, is likewise a highly successful and widely accepted theory. And yet there is a well-documented tension between the two theories, to the extent that it is still not clear that the two can ever be reconciled. This thesis is concerned with furthering the current understanding of the relationship between quantum mechanics and special relativity. In the first part of the thesis we study the behaviour of quantum information in relativistic spacetime. The field of quantum information arose from the realisation that quantum information has a number of crucial properties that distinguish it from classical information, such as the no-cloning property, quantum contextuality, and quantum discord. More recently, it has been realised that placing quantum information under relativistic constraints leads to the emergence of further unique features which are not exhibited by either non-relativistic quantum information or relativistic classical information; as part of this ongoing research programme we develop a new relativistic quantum `paradox' which puts pressure on conventional views about the spatiotemporal persistence of quantum states over time. We then study a new set of relativistic quantum protocols which involve the distribution of entangled states over spacetime, defining one task involving the distribution of the two halves of a known entangled state, and another task involving the distribution of the two halves of an unknown entangled state. The second part of the thesis deals with relativistic quantum cryptography, a field which first began attracting serious attention when it was realised that a cryptographic task known as `bit commitment,' can be implemented with perfect security under relativistic constraints. This result was highly significant, since it is provably impossible to implement bit commitment with perfect security in a purely classical or purely quantum context, and hence bit commitment is an ideal starting point for probing the power of relativistic quantum cryptography. In this thesis we propose several new relativistic quantum bit commitment protocols which have notable advantages over previously known protocols. We then move to a related task, a generalization of zero-knowledge proving which we refer to as knowledge-concealing evidencing of knowledge of a quantum state; we prove no-go theorems concerning the possibility of implementing this task with perfect security, and then set out a relativistic protocol for the task which is asymptotically secure as the dimension of the state in question becomes large. These results have interesting foundational significance above and beyond their applications in the field of cryptography, providing a new perspective on the connections between knowledge, realism and quantum states.