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Bias in Numbers Part 2: The Mathematics of Superposition

Allan Porter

Published Apr 21, 2022• 0 citations
Publisher

Abstract

1, 0 and infinity are all special in that they represent a concept rather than simply a quantity (something, nothing and everything). However, are there more of these concept numbers that we can fathom? How twentieth century breakthroughs in quantum mechanics sheds could be used to shed light on more.
DOI
10.5281/zenodo.6474347

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In physics, we construct idealized mathematical models in order to explain various phenomena which we observe or create in our laboratories. In this article, I recall how sophisticated mathematical models evolved from the concept of a number created thousands of years ago, and I discuss some challenges and open questions in quantum foundations and in the Standard Model. We liberated nuclear energy, landed on the Moon and built 'quantum computers'. Encouraged by these successes, many believe that when we reconcile general relativity with quantum theory we will have the correct theory of everything. Perhaps we should be much humbler. Our perceptions of reality are biased by our senses and by our brain, bending them to meet our priors and expectations. Our abstract mathematical models describe only in an approximate way different layers of physical reality. To describe the motion of a meteorite, we can use a concept of a material point, but the point-like approximation breaks completely when the meteorite hits the Earth. Similarly, thermodynamic, chemical, molecular, atomic, nuclear and elementary particle layers of physical reality are described using specific abstract mathematical models and approximations. In my opinion, the theory of everything does not exist.
The early twentieth century was a revolutionary time in the history of physics. People often think of Einstein's special theory of relativity of 1905, which changed our conceptions of time and space. But among physicists, quantum mechanics is usually regarded as an even more radical change in our thinking about the physical world. Quantum mechanics, which was developed between 1900 and 1926, began as a theory of atoms and light but has now become the framework in terms of which all basic physical theories are expected to be cast. We need quantum ideas not only to understand atoms, molecules, and elementary particles, but also to understand the electronic properties of solids and even certain astronomical phenomena such as the stability of white dwarf stars. The theory was radical in part because it introduced probabilistic behavior as a fundamental aspect of the world, but even more because it seems to allow mutually exclusive situations to exist simultaneously in a “quantum superposition.” We will see later that the possibility of quantum superposition is largely responsible for a quantum computer's distinctive advantage over an ordinary computer. The present chapter is devoted to introducing the basic principles of quantum mechanics. There are essentially four components of the mathematical structure of quantum mechanics. We need to know how to represent (i) states, (ii) measurements, (iii) reversible transformations, and (iv) composite systems. We will develop the first three in stages, starting with a very simple case – linear polarization of photons – and working toward the most general case.
In the early seventeenth century Galileo broke free from the hold of ancient Platonic and Aristotelian philosophy. He drastically changed the framework through which we view the natural world when he asserted that we should base our theory of reality on what we can observe rather than pure thought. In the process, he invented what we would come to call science. This set the stage for all the breakthroughs that followed--from Kepler to Newton to Einstein. But in the early twentieth century when quantum physics, with its deeply complex mathematics, entered into the picture, something began to change. Many physicists began looking to the equations first and physical reality second. As we investigate realms further and further from what we can see and what we can test, we must look to elegant, aesthetically pleasing equations to develop our conception of what reality is. As a result, much of theoretical physics today is something more akin to the philosophy of Plato than the science to which the physicists are heirs. In The Dream Universe, Lindley asks what is science when it becomes completely untethered from measurable phenomena?
The twentieth century has seen two major revolutions in our theories of physics concerning nature, and these have made us change many of our concepts about the terms in which nature can be described. The new theories born in these revolutions are the theory of relativity and of quantum mechanics. The biological sciences had their revolutions in the nineteenth century, and while remarkable progress has been made since, nothing comparable to that upheaval has occurred in this century. Of the two massive changes in the concepts of the physical sciences, we can discuss but one here. [excerpt]