Abstract We argue that there is a simple, unique, reason for all quantum paradoxes, and that such a reason is not uniquely related to quantum theory. It is rather a mathematical question that arises at the intersection of logic, probability, and computation. We give our ‘weirdness theorem’ that characterises the conditions under which the weirdness will show up. It shows that whenever logic has bounds due to the algorithmic nature of its tasks, then weirdness arises in the special form of negative probabilities or non-classical evaluation functionals. Weirdness is not logical inconsistency, however. It is only the expression of the clash between an unbounded and a bounded view of computation in logic. We discuss the implication of these results for quantum mechanics, arguing in particular that its interpretation should ultimately be computational rather than exclusively physical. We develop in addition a probabilistic theory in the real numbers that exhibits the phenomenon of entanglement, thus concretely showing that the latter is not specific to quantum mechanics.
Abstract We argue that there is a simple, unique, reason for all quantum paradoxes, and that such a reason is not uniquely related to quantum theory. It is rather a mathematical question that arises at the intersection of logic, probability, and computation. We give our ‘weirdness theorem’ that characterises the conditions under which the weirdness will show up. It shows that whenever logic has bounds due to the algorithmic nature of its tasks, then weirdness arises in the special form of negative probabilities or non-classical evaluation functionals. Weirdness is not logical inconsistency, however. It is only the expression of the clash between an unbounded and a bounded view of computation in logic. We discuss the implication of these results for quantum mechanics, arguing in particular that its interpretation should ultimately be computational rather than exclusively physical. We develop in addition a probabilistic theory in the real numbers that exhibits the phenomenon of entanglement, thus concretely showing that the latter is not specific to quantum mechanics.
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by being based on two main postulates: 1. the theory should be logically consistent; 2. inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide between QT, seen as a generalised theory of probability, and classical probability. All quantum paradoxes, and entanglement in particular, arise from the clash of trying to reconcile a computationally intractable, somewhat idealised, theory (classical physics) with a computationally tractable theory (QT) or, in other words, from regarding physics as fundamental rather than computation.
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from two main postulates (i) the theory should be logically consistent; (ii) inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide between QT, seen as a generalised theory of probability, and classical probability. All quantum paradoxes, and entanglement in particular, arise from the clash of trying to reconcile a computationally intractable, somewhat idealised, theory (classical physics) with a computationally tractable theory (QT) or, in other words, from regarding physics as fundamental rather than computation.
Logical foundation for quantum theory is considered. I claim that quantum theory correctly represents Nature when mathematical physics embraces and indeed features, logical anomalies inherent in pure mathematics. This approach links undecidability in arithmetic with the logic of quantum experiments. The undecidablity occupies an algebraic environment which is the missing foundation for the 3-valued logic predicted by Hans Reichenbach, shown by him to resolve `causal anomalies' of quantum mechanics, such as: inconsistency between prepared and measured states, complementarity between pairs of observables, and the EPR-paradox of action at a distance. Arithmetic basic to mathematical physics, is presented formally as a logical system consisting of axioms and propositions. Of special interest are all propositions asserting the existence of particular numbers. All numbers satisfying the axioms permeate the arithmetic indistinguishably, but these logically partition into two distinct sets: numbers whose existence the axioms determine by proof, and numbers whose existence axioms cannot determine, being neither provable nor negatable. Failure of mathematical physics to incorporate this logical distinction is seen as reason for quantum theory being logically at odds with quantum experiments. Nature is interpreted as having rules isomorphic to the abovementioned axioms with these governing arithmetical combinations of necessary and possible values or effects in experiments. Soundness and Completeness theorems from mathematical logic emerge as profoundly fundamental principles for quantum theory, making good intuitive sense of the subject.
Quantum mechanics is empirically unrivalled, yet the origin of its distinctive non-classical features remains a foundational question. This paper proves, within the unchanged standard quantum formalism, that these features—collectively termed “quantumness”—arise if and only if certain configurations of the system are indistinguishable in principle. Quantumness is defined as any deviation from classical probability distributions, i.e., probabilities that require coherent amplitude addition rather than incoherent probability addition. Three theorems are established using only the standard Hilbert-space formalism and unitary evolution: 1. Indistinguishability of configurations is necessary and sufficient for quantumness.2. Indistinguishability of identical particles is neither necessary nor sufficient.3. An equivalent restatement in precise operational terms: quantumness occurs exactly when a variable lacks physical definition (distinguishability in principle by macroscopic outcomes). These results are presented and interpreted within the Logical Interpretation of quantum mechanics, a minimalist framework that adds no new postulates and applies classical logic strictly to physically defined facts. The interpretation provides the conceptual tools—particularly the precise notion of “indistinguishability in principle”—that make the theorems sharp and unified. This work unifies the origin of quantumness and serves as the conceptual prequel to the full Logical Interpretation developed in the author’s companion paper.
The purpose of this dissertation is to construct a radically new type of mathematics whose underlying logic differs from the ordinary classical logic used in standard mathematics, and which we feel may be more natural for applications in quantum mechanics. Specifically, we begin by constructing a first order quantum logic, the development of which closely parallels that of ordinary (classical) first order logic --- the essential differences are in the nature of the logical axioms, which, in our construction, are motivated by quantum theory. After showing that the axiomatic first order logic we develop is sound and complete (with respect to a particular class of models), this logic is then used as a foundation on which to build (axiomatic) mathematical systems --- and we refer to the resulting new mathematics as "quantum mathematics." As noted above, the hope is that this form of mathematics is more natural than classical mathematics for the description of quantum systems, and will enable us to address some foundational aspects of quantum theory which are still troublesome --- e.g. the measurement problem --- as well as possibly even inform our thinking about quantum gravity.\n\nAfter constructing the underlying logic, we investigate properties of several mathematical systems --- e.g. axiom systems for abstract algebras, group theory, linear algebra, etc. --- in the presence of this quantum logic. In the process, we demonstrate that the resulting quantum mathematical systems have some strange, but very interesting features, which indicates a richness in the structure of mathematics that is classically inaccessible. Moreover, some of these features do indeed suggest possible applications to foundational questions in quantum theory. \n\nWe continue our investigation of quantum mathematics by constructing an axiomatic quantum set theory, which we show satisfies certain desirable criteria. Ultimately, we hope that such a set theory will lead to a foundation for quantum mathematics in a sense which parallels the foundational role of classical set theory in classical mathematics. One immediate application of the quantum set theory we develop is to provide a foundation on which to construct quantum natural numbers, which are the quantum analog of the classical counting numbers. It turns out that in a special class of models, there exists a 1-1 correspondence between the quantum natural numbers and bounded observables in quantum theory whose eigenvalues are (ordinary) natural numbers. This 1-1 correspondence is remarkably satisfying, and not only gives us great confidence in our quantum set theory, but indicates the naturalness of such models for quantum theory itself. We go on to develop a Peano-like arithmetic for these new "numbers," as well as consider some of its consequences. Finally, we conclude by summarizing our results, and discussing directions for future work.
We argue that the double-slit experiment can be understood much better by considering it as an experiment whereby
one uses electrons to study the set-up rather than an experiment whereby we use a set-up to study the behaviour of electrons.
We also show that the concept of undecidability (like e.g. occurs in G\odel's theorem) can be used in an intuitive way to make sense of the double-slit experiment
and the quantum rules for calculating coherent and incoherent probabilities. We meet here a situation where the electrons always behave in a fully
deterministic way, while the detailed design of the set-up may render the question about the way they move through the set-up experimentally undecidable.
It is very important to make a distinction in quantum mechanics
between the determinism of nature (Einstein)
and the decidability of a question within an experimental set-up (Bohr).
The former is about the absolute truth of an answer to a yes or no question, and follows binary logic
(true or false),
the latter about what an experimental set-up
can decide and tell about the truth of that answer, and follows ternary logic (true, false or undecidable).
Binary and ternary logic are incompatible.
The viewpoints of Bohr and Einstein are thus operating on different levels and it is only by confusing these two levels
that these two viewpoints seem to be irreconcilable.
We show that the expression ψ1+ψ2 for the wave function of the double-slit experiment is numerically correct, but logically flawed. It has to be replaced in the interference region
by the logically correct expression ψ1′+ψ2′, which has the same numerical value as
ψ1+ψ2, such that ψ1′+ψ2′=ψ1+ψ2,
but with ψ1′=2ψ1+ψ2e4π=ψ1 and
ψ2′=2ψ1+ψ2e−4π=ψ2.
Here ψ1′ and ψ2′ are the correct
contributions from the slits to the total wave function ψ1′+ψ2′.
We have then $p = |\psi'_{1} + \psi'_{2}|^{2} = |\psi'_{1}|^{2} + |\psi'_{2}|^{2} = p_{1}+p_{2} $ such that the paradox
that quantum mechanics (QM) would not follow the traditional rules of probability calculus disappears.
The paradox is rooted in the wrong intuition that ψ1 and ψ2 would be the true physical contributions to
ψ1′+ψ2′=ψ1+ψ2 like in the case of waves in a water tank.
The solution proposed here is not {\em ad hoc} but based on an extensive analysis of the geometrical meaning of spinors within group representation theory
and its application to QM. Working further on the argument one can even show that an interference pattern is the only way to satisfy simultaneously
two conditions: The condition obeying binary logic (in the spirit of Einstein) that the electron has only two mutually exclusive options to get to the detector (viz. going through slit S1 or going through slit S2) and the condition obeying ternary logic (in the spirit of Bohr)
that the question which one of these two options the electron has taken is experimentally undecidable.
A very important element in the analysis is the problem of the non-existence of {\em common} probability distributions for different experimental set-ups.
And this is a recurrent theme in other situations that are felt as paradoxical.
The CHSH inequality used in the experiments of Aspect {\em et al.}
clashes with quantum mechanics because it
is based on the assumption that there would exist a {\em common} probability distribution for the
hidden variables in the different experiments that have to be made to determine the various quantities that intervene in the inequality.
That common probability distribution for measured quantities does not exist is well known from quantum mechanics itself (because the operators that come into play do not commute and therefore do not have common eigenstates, i.e.
common probability amplitudes). But this non-existence of common probability distributions applies also to certain variables in the set-ups of the different experiments and the latter fact
can also be explained by purely classical reasoning.