Unlike our basic theories of space and time, quantum mechanics is not a locally causal theory. Moreover, it is widely believed that any hopes of restoring local causality within any realistic theory have been undermined by Bell's theorem and the experimental investigations it has inspired. In this pedagogical paper, John S. Bell's amusing example of Dr. Bertlmann's socks to illustrate the results of these experiments is reconsidered, first within a toy model of a two-dimensional one-sided world of a non-orientable Möbius strip, and then within a real world of three-dimensional quaternionic sphere, S3, which results from an addition of a single point to IR3at infinity. In the latter quaternionic world, which happens to be the spatial part of a solution of Einstein's field equations of general relativity, the singlet correlations between a pair of entangled fermions can be understood as classically as those between Dr. Bertlmann's colorful socks.
Unlike our basic theories of space and time, quantum mechanics is not a locally causal theory. Moreover, it is widely believed that any hopes of restoring local causality within any realistic theory have been undermined by Bell's theorem and the experimental investigations it has inspired. In this pedagogical paper, John S. Bell's amusing example of Dr. Bertlmann's socks to illustrate the results of these experiments is reconsidered, first within a toy model of a two-dimensional one-sided world of a non-orientable Möbius strip, and then within a real world of three-dimensional quaternionic sphere, S3, which results from an addition of a single point to IR3at infinity. In the latter quaternionic world, which happens to be the spatial part of a solution of Einstein's field equations of general relativity, the singlet correlations between a pair of entangled fermions can be understood as classically as those between Dr. Bertlmann's colorful socks.
I point out critical errors in the article “Dr. Bertlmann's Socks in a Quaternionic World of Ambidextral Reality” by J. Christian, published in IEEE ACCESS. Christian's model does not generate the singlet correlations but in fact simply reproduces the Bertlmann effect. John Bell's colleague Reinhold Bertlmann of CERN, in his younger days, always wore one pink and one blue sock, at random. The moment you saw his left foot, you knew what colour sock would be on his right foot. Action at a distance? As John Bell liked to explain, quantum entanglement cannot be explained away in such an easy way. Yet Christian's model assigns the two particles of the EPR-B experiment an equal and opposite spin at the source, the choice being determined by a fair coin toss. However they are measured, these spins are recovered. Christian's computer simulation works by not actually simulating his model at all but by almost directly tracing the negative cosine built into his computer algebra package. Bell's theorem has not been disproved. Debate as to what it means for the foundations of physics as well as for quantum information engineering (quantum communication, computation) is more lively today than ever before. A possible role for Geometric Algebra is still wide open and deserves further investigation, informed by a proper understanding of the mathematical content of Bell's theorem.
In this paper, I respond to a critique of one of my papers previously published in this journal, entitled ‘Dr. Bertlmann’s socks in a quaternionic world of ambidextral reality.’ The geometrical framework presented in my paper is based on a quaternionic 3-sphere, orS3, taken as a model of the physical space in which we are inescapably confined to perform all our experiments. The framework intrinsically circumvents Bell’s theorem by reproducing the singlet correlations local-realistically, without resorting to backward causation, superdeterminism, or any other conspiracy loophole. In this response, I demonstrate point by point that, contrary to its claims, the critique has not found any mistakes in my paper, either in the analytical model of the singlet correlations or in its event-by-event numerical simulation based on Geometric Algebra.
Unlike our basic theories of space and time, quantum mechanics is not a locally causal theory. Moreover, it is widely believed that any hopes of restoring local causality within a realistic theory have been undermined by Bell's theorem and its supporting experiments. By contrast, we provide a strictly local, deterministic, and realistic explanation for the correlations observed in two such supporting experiments, performed independently at Orsay and Innsbruck. To this end, a pair of local variables is constructed to simulate detections of photon polarizations at various angles, chosen freely by Alice and Bob. These generate purely random outcomes, A = +1 or -1 and B = +1 or -1, occurring within a parallelized 3-sphere. These outcomes do not depend on the contexts of measurements, but are determined entirely by the randomly chosen initial orientation of the 3-sphere. When they are compared, however, the correlation between them turn out to be exactly equal to -cos2(alpha - beta), with the corresponding CHSH inequality violated for the polarization angles alpha, alpha', beta, and beta' in precisely the manner predicted by quantum mechanics. The key ingredient in our explanation is the topology of the 3-sphere, which remains closed under multiplication, thus preserving the locality condition of Bell. It allows us to model the physical space as a 3-sphere, and reveals that the illusion of quantum nonlocality in the present case stems from a twist in the Hopf fibration of the 3-sphere.
Introduction The philosopher in the street, who has not suffered a course in quantum mechanics, is quite unimpressed by Einstein–Podolsky–Rosen correlations. He can point to many examples of similar correlations in everyday life. The case of Bertlmann's socks is often cited. Dr. Bertlmann likes to wear two socks of different colours. Which colour he will have on a given foot on a given day is quite unpredictable. But when you see (Fig. 1) that the first sock is pink you can be already sure that the second sock will not be pink. Observation of the first, and experience of Bertlmann, gives immediate information about the second. There is no accounting for tastes, but apart from that there is no mystery here. And is not the EPR business just the same? Consider for example the particular EPR gedanken experiment of Bohm (Fig. 2). Two suitable particles, suitably prepared (in the ‘singlet spin state’), are directed from a common source towards two widely separated magnets followed by detecting screens. Each time the experiment is performed each of the two particles is deflected either up or down at the corresponding magnet. Whether either particle separately goes up or down on a given occasion is quite unpredictable. But when one particle goes up the other always goes down and vice-versa. After a little experience it is enough to look at one side to know also about the other.
It is well known that correlations produced by common causes in the past cannot violate Bell's inequalities. This was emphasized by Bell in his celebrated example of Bertlmann's socks. However, if common causes are induced by the very measurement process i.e., actualized at each run of a joint measurement, in a way that depends on the type of joint measurement that is being executed (hence, the common causes are contextually actualized), the resulting correlations are able to violate Bell's inequalities, thus providing a simple and general explanation for the origin of quantum correlations. We illustrate this mechanism by revisiting Bertlmann's socks example. In doing so, we also emphasize that Bell's inequalities, in their essence, are about demarcating 'non-induced by measurements' (non-contextual) from 'induced by measurements' (contextual) common causes, where the latter would operate at a non-spatial level of our physical reality, when the jointly measured entangled entities are microscopic in nature.