The concept of potentia as proposed by Heisenberg to understand the structure of quantum mechanics, has just remained a fanciful speculation as of now. In this paper we provide a physically consistent and a mathematically justified ontology of this model based on a fundamental role played by the discrete subgroups of the relevant Lie groups. We show that as such, the space of potentia arises as a coexisting dual space to the real three dimensional space, while these two sit piggyback on each other, such that the collapse of wave function can be understood in a natural manner. Quantum nonlocality and quantum jumps arise as a natural consequence of this model.
The concept of potentia as proposed by Heisenberg to understand the structure of quantum mechanics, has just remained a fanciful speculation as of now. In this paper we provide a physically consistent and a mathematically justified ontology of this model based on a fundamental role played by the discrete subgroups of the relevant Lie groups. We show that as such, the space of potentia arises as a coexisting dual space to the real three dimensional space, while these two sit piggyback on each other, such that the collapse of wave function can be understood in a natural manner. Quantum nonlocality and quantum jumps arise as a natural consequence of this model.
It is argued that quantum theory is best understood as requiring an ontological duality of res extensa and res potentia, where the latter is understood per Heisenberg's original proposal, and the former is roughly equivalent to Descartes' 'extended substance.' However, this is not a dualism of mutually exclusive substances in the classical Cartesian sense, and therefore does not inherit the infamous 'mind-body' problem. Rather, res potentia and res extensa are proposed as mutually implicative ontological extants that serve to explain the key conceptual challenges of quantum theory; in particular, nonlocality, entanglement, null measurements, and wave function collapse. It is shown that a natural account of these quantum perplexities emerges, along with a need to reassess our usual ontological commitments involving the nature of space and time.
Abstract Heisenberg was a very dedicated Kantian, and viewed the discoveries of early twentieth century physics as disqualifying of the Kantian metaphysical worldview. He made an effort – not an especially successful one – to reconfigure the Kantian critique by replacing noumena with the quantum-level reality, and proposed a new entity as mediative between the quantum realm and the subject. The new entity, potentia (borrowed from Aristotle, and altered to serve Heisenberg’s purpose) fulfills that role by means of its hybrid nature: it’s partly mental, partly a reification of probabilities. This dual nature allows epistemic access to the probabilistic nature of quantum-level interactions, out of which our phenomenal world is realized when a measurement is made. How well this notion serves the purpose of rehabilitating the Kantian critique is questionable, as Heisenberg’s conception is problematic and, perhaps, internally inconsistent. My dissertation follows Heisenberg’s development of potentia, examines how well it addresses the failure of the Kantian problem, and asks whether his proposal is an obscurum per obscurius. Thereafter I introduce a number of other, similar proposals to potentia, consider them on their own merits, compare them to Heisenberg’s conception, and use this conjunction of potentia proposals to made a judgment regarding whether the use of potentia is a philosophically defensible way of addressing the recondite problems of quantum mechanics.
I build upon Heisenberg’s proposal in 1958 that quantum states are ontologically real “potentia.” On this view reality consists in ontologically real Possibles and ontologically real Actuals. Then measurement converts Possibles to Actuals. This ontological hypothesis, explicit in Heisenberg’s proposal, suggests a natural place for “mind.” Mind converts Possibles to Actuals. This role for mind is a scientific hypothesis open to test. Modest evidence now supports the hypothesis. If very well confirmed, such results alter the foundations of quantum mechanics and afford a role of mind in the becoming of the universe. More, the ontological proposal is not a substance dualism, for potentia are not substances. Thus, this proposal for the role of mind does not seem to inherit many of the problems raised in a substance dualism such as Descartes’ Res cogitans and Res extensa, (Descartes), or in a substance neutral monism such as Russell, (Russell). The proposal is radical. Far more evidence is needed.
In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms. Unitary representations of this group describe a wide variety of quantum systems, and have predicted previously unsuspected possibilities – notably, anyons and nonabelian anyons in two space dimensions. We present here foundational reasons why this semidirect product group serves as a universal kinematical group for quantum mechanics. We obtain thus a unified account of all possible quantum kinematics for systems with mass in an arbitrary physical space, and clarify the role played by topology in quantum mechanics. Our development does not require quantization of classical phase space; rather, the classical limit follows from the quantum mechanics. We also consider the relationship of our development to Heisenberg quantization.
IF, AS HEISENBERG SUGGESTED, potentia indeed constitute an ontologically fundamental species of reality, then the mathematical representation of the quantum mechanical evolution of the matrix of potentia to the matrix of probabilities-that is, the valuation of potentia as probabilities between zero and one-ought to be heuristically useful in understanding the ontological implications of this evolution.The mathematical representation should prove helpful in visualizing not only the process described by quantum mechanics but also, as we shall see in the following chapters, the process described by Whitehead.The basic formalism introduced and explored in this chapter is intended to be comprehensible to readers unfamiliar with mathematics.Some familiarity with the Pythagorean theorem and the addition of vectors will be useful, but not necessary.
Heisenberg offered an interpretation of the quantum state which made use of a quantitative version of an earlier notion, , of Aristotle by both referring to it using its Latin name,potentia, and identifying its qualitative aspect with . The relationship between this use and Aristotle's notion was not made by Heisenberg in full detail, beyond noting their common character: that of signifying the system's objective capacity to be found later to possess a property in actuality. For such actualization, Heisenberg requiredmeasurementto have taken place, an interaction with external systems that disrupts the otherwise independent, natural evolution of the quantum system. The notion of state actualization was later taken up by others, including Shimony, in the search for a law-like measurement process. Yet, the relation of quantum potentiality to Aristotle's original notion has been viewed as mainly terminological, even by those who used it thus. Here, I reconsider the relation of Heisenberg's notion to Aristotle's and show that it can be explicated in greater specificity than Heisenberg did. This is accomplished through the careful consideration of the role of potentia in physical causation and explanation, and done in order to provide a fuller understanding of this aspect of Heisenberg's approach to quantum mechanics. Most importantly, it is pointed out that Heisenberg's requirement of an external intervention during measurement that disrupts the otherwise independent, natural evolution of the quantum system is in accord with Aristotle's characterization of spontaneous causation. Thus, the need for a teleological understanding of the actualization of potentia, an often assumed requirement that has left this fundamental notion neglected, is seen to be spurious. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.