Abstract. Physics is real. Measurement produces real numbers. Yet quantum mechanics uses com-plex arithmetic, in which √−1 is necessary but mysteriously relates to nothing else. By applying the same sort of symmetry arguments that Cox [1, 2] used to justify probability calculus, we are now able to explain this puzzle. The dual device/object nature of observation requires us to describe the world in terms of pairs of real numbers about which we never have full knowledge. These pairs combine according to complex arithmetic, using Feynman’s rules.
Abstract. Physics is real. Measurement produces real numbers. Yet quantum mechanics uses com-plex arithmetic, in which √−1 is necessary but mysteriously relates to nothing else. By applying the same sort of symmetry arguments that Cox [1, 2] used to justify probability calculus, we are now able to explain this puzzle. The dual device/object nature of observation requires us to describe the world in terms of pairs of real numbers about which we never have full knowledge. These pairs combine according to complex arithmetic, using Feynman’s rules.
Complex numbers are an intrinsic part of the mathematical formalism of quantum theory and are perhaps its most characteristic feature. In this article, we show that the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.
Dirac called the imaginary unit in the Schrödinger equation the deepest thing in quantum mechanics, and admitted he did not understand it. This note offers a candidate: i is the complex place of a single number — Q, the real root of x⁴ − x − 1 — and the strangeness of quantum mechanics is the strangeness of that one number's arithmetic, seen from the inside. For a century quantum theory has been taught as a list of postulates — complex amplitudes, a Born rule, unitary evolution, a dagger — accepted because they work and never because they are understood. Grant a single identification, that the state space of a quantum system is the complex place of Q's field, and those postulates stop being assumptions. They become theorems. The imaginary unit is the field's complex structure; the dagger is Galois conjugation; observables are real because they are Galois-fixed; unitary evolution is the field's unit circle; and the Born rule — the squared modulus, long the least-explained law in physics — is the field's own norm, made unique by Gleason's theorem. Time reversal and charge conjugation appear as named elements of the Galois group S₄, the Weyl group of su(4). And a proved theorem locates the classical/quantum boundary precisely: in the family xⁿ − x − 1 the root is Pisot — its powers settle, its world definite — only through three dimensions, and at the fourth that settling fails. That, on the reading proposed here, is where the classical world ends and the quantum world begins. The dozen postulates of the physics textbooks are thus compressed to one here, named plainly as a posit, with every claim labeled at its evidential tier and the composition of many systems left as the open edge. It is not a metaphor and not a fit. Quantum Mechanics is the arithmetic of a single number — and, at last, a single thing to be right or wrong about.
Abstract Quantification starts with sum and product rules that express combination and partition. These rules rest on elementary symmetries that have wide applicability, which explains why arithmetical adding up and splitting into proportions are ubiquitous. Specifically, measure theory formalizes addition, and probability theory formalizes inference in terms of proportions. Quantum theory rests on the same simple symmetries, but is formalized in two dimensions, not just one, in order to track an object through its binary interactions with other objects. The symmetries still require sum and product rules (here known as the Feynman rules), but they apply to complex numbers instead of real scalars, with observable probabilities being modulus squared (known as the Born rule). The standard quantum formalism follows. There is no mystery or weirdness, just ordinary probabilistic inference.
Why quantum? Why spacetime? We find that the key idea underlying both is uncertainty. In a world lacking probes of unlimited delicacy, our knowledge of quantities is necessarily accompanied by uncertainty. Consequently, physics requires a calculus of number pairs and not only scalars for quantity alone. Basic symmetries of shuffling and sequencing dictate that pairs obey ordinary component-wise addition, but they can have three different multiplication rules. We call those rules A, B and C. “A” shows that pairs behave as complex numbers, which is why quantum theory is complex. However, consistency with the ordinary scalar rules of probability shows that the fundamental object is not a particle on its Hilbert sphere but a stream represented by a Gaussian distribution. “B” is then applied to pairs of complex numbers (qubits) and produces the Pauli matrices for which its operation defines the space of four vectors. “C” then allows integration of what can then be recognised as energy-momentum into time and space. The picture is entirely consistent. Spacetime is a construct of quantum and not a container for it.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Overview: Two Simple Rules and Two Easy Applications The Historical Route The Feynman Route Nitty-Gritty: Why Probabilities? Why Vector Spaces? Why Complex Numbers? Whence the Trace Rule? How Probabilities Depend on Measurement Outcomes How Probabilities Depend on the Times of Measurements Whence Those Simple Rules? Why Spin? Composite Systems Why Fermions? Special Relativity in a Nutshell Why Relativity? Whence the Classical Forces? Meaning of EnergyA and MomentumA Why Relativistic Quantum Mechanics? Making Sense: Why Measurements? The Importance of Unperformed Measurements The Stability of Matter A Sleight of Hand That Almost Worked A Sleight of Hand That Cannot Work Pseudo-Problems and Gratuitous Solutions A Space for the Quantum World A Particle by Itself The Emergence of Form CreationA in a Nutshell Defining MacroworldA Assigning Reality The Supervenience of MicroA on MacroA ManifestationA - A Better Paradigm? Not the Theory but the World is Incomplete Explanation or Magic Wand?