1,721,004 research outputs found
A solution of the paradox of the double-slit experiment
No full textWe argue that the double-slit experiment can be understood much better by considering it as an experiment wherebyone 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 how the concept of undecidability can be used in an intuitive way to make sense of the double-slit experimentand the quantum rules for calculating coherent and incoherent probabilities. We meet here a situation where the electrons always behave in a fully deterministic way (following Einstein's conception of reality), while the detailed design of the set-up may render the question about the way they move through the set-up experimentally undecidable (which follows more Bohr's conception of reality).We show that the expression 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 , which has the same numerical value as , such that , but with and . Here and are the correct contributions from the slits to the total wave function . We have then 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 and would be the true physical contributions to like in the case of waves in a water tank. The solution proposed here is not 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 S or going through slit S) 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
What is the reason for the asymmetry between the twins in the twin paradox?
No full textLargely improved presentationThe true difficulty of the twin paradox does not reside in the algebra that shows that the traveling twin ages less than the twin who stays at home. The truly startling part of the paradox resides in the much more difficult question why the argument cannot be reversed by symmetry, because there is no such thing as a preferred reference frame,and motion ought to be relative.Can the traveling twin not claim with equal rights to have stayed at home while the other twin has made the journey?The answer to this question is not what anyone has thought. Most of the time text books invoke the accelerations intervening in the trip to explain the asymmetry.We will show that one can formulate and solve the paradox without making any reference to accelerations. There is actually something very simple that has been overlooked.In drafting the protocol which defines the journey, we unwittingly pick a preferred reference frame, because we define the protocol with respectto a given frame, which thereby becomes special. It is this selection of a special reference frame which introduces the asymmetry. Hence, the reference frame wherein we define the protocol for the journey will act like an absolute frame and it is its unavoidable introduction which breaks the symmetry between the twins.There is an infinity of protocols that can be selected to define a trip and each of these trips leads to its own corresponding twin paradox, with its own outcomeas to which twin will age less.Whereas the individual trips of the two twins within a given protocol are asymmetrical, the set of all possible trips is symmetrical, such that the symmetry of the Lorentz group is indeed respected
A Lorentz boost as a product of two space-time reflections and some additional results about Clifford algebra
No full textThis is a technical clarifying note consisting of two parts. In the first part we derive the expression for a boost in two representations of the homogeneous Lorentz group, viz. the two-dimensional representation SL(2,C) and the four-dimensional Dirac representation in its Cartan-Weyl form. The derivation is purely algebraic. It uses the development of a Clifford algebra for a group of isometries of a vector space, whereby the group is generated by reflections. We prove that a boost can be obtained as a product of two space-time reflections, in perfect analogy with the way a rotation in R 3 can be obtained as a product of two reflections. The derivation does therefore not rely on physical considerations as in Einstein's approach. It is purely based on symmetry arguments. The second part deals with the justification of the definition of a Clifford algebra given in certain mathematical textbooks, which immediately introduce a basis of multi-vectors 1, ej, ej 1 ∧ ej 2 , ej 1 ∧ ej 2 ∧ ej 3 , • • • for this algebra. Rather than as a bemusing "postulate" that descends from heaven, we will present the introduction of this basis as an obvious result of a logical construction of the group representation theory. This will provide the reader with a much better understanding of what is going on behind the scenes of the formalism. We prove that this basis of multi-vectors 1, ej, ej 1 ∧ ej 2 , ej 1 ∧ ej 2 ∧ ej 3 , • • • is orthogonal in terms of a scalar product whose use is very natural in vector spaces of matrices
Normalization errors in CHSH Bell inequalities
No full textExplanations added to make the paper self-contained.We consider two different definitions of determinism, the traditional one and a new alternative one. The alternative definition introduces a new description of what happens during a photon correlation measurement. The CHSH Bell-type inequality is derived according to the traditional definition of determinism. It can then not be applied in the usual way to the experiments it has been designed for, when in the analysis of the experiments we follow the alternative type of definition. In fact, the usual procedure introduces then a normalization error with a confusion between absolute and conditional probabilities. Within the frame work of the alternative definition the inequality is no longer violated and determinism is not defeated
Exact theory of the Stern-Gerlach experiment - extended version
No full textVersion élargie de l'article accepté pour publication dans Symmetry.The Stern-Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. But that theory is based on mathematical errors in the way it (mis)treats spinors and group theory.We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry.Contrary to what Pauli has been reading into the Stern-Gerlach experiment,the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured.Surprizingly this leads to only two energy states, which should be qualified as precession-up and precession-down rather thanspin-up and spin down.Indeed, despite the presence of the many different possible angles betweenthe spin axis and the magnetic field , the fermions can only have two possible energies . The values do thus not correspond to the continuum of values Einstein and Ehrenfest had conjectured. The energy term is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies , and as such not valid for individual fermions. The two fermion states with energy are not potential-energy states. We also explain themathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clearand understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle
On magnetic monopoles, the anomalous g-factor of the electron and the spin-orbit coupling in the Dirac Theory
No full text38 pagesWe discuss the algebra and the interpretation of the anomalous Zeeman effect and the spin-orbit coupling within the Dirac theory. Whereas the algebra for the anomalous Zeeman effect is impeccable and therefore in excellent agreement with experiment, the physical interpretation of that algebra uses images that are based on macroscopic intuition but do not correspond to the meaning of this algebra. The interpretation violates the Lorentz symmetry. We therefore reconsider the interpretation to see if we can render it consistent also with the symmetry. The results confirm clearly that the traditional physical interpretation of the anomalous Zeeman effect is not correct. We give an alternative intuitive description of the meaning of this effect, which respects the symmetry and is exact. It can be summarized by stating that a magnetic field makes any charged particle spin. This is even true for charged particles " without spin ". Particles " with spin " acquire additional spin in a magnetic field. This additional spin must be combined algebraically with the pre-existing spin. We show also that the traditional discussion about magnetic monopoles confuses two issues, viz. the symmetry of the Maxwell equations and the quantization of charge. These two issues define each a different concept of magnetic monopole. They cannot be merged together into a unique all-encompassing issue. We also generalize the minimal substitution for a charged particle, and provide some intuition for the magnetic vector potential. We finally explore the algebra of the spin-orbit coupling, which turns out to be badly wrong. The traditional theory that is claimed to reproduce the Thomas half is based on a number of errors. An error-free application of the Dirac theory cannot account for the Thomas precession, because it only accounts for the instantaneous local boosts, not for the rotational component of the Lorentz transformation. This runs contrary to established beliefs, but can be understood in terms of the Berry phase on a path through the Lorentz group manifold. These results clearly reveal the limitations of the prevailing working philosophy to " shut up and calculate "
Exact theory of the Stern-Gerlach experiment - extended version
No full textVersion élargie de l'article accepté pour publication dans Symmetry.The Stern-Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. But that theory is based on mathematical errors in the way it (mis)treats spinors and group theory.We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry.Contrary to what Pauli has been reading into the Stern-Gerlach experiment,the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured.Surprizingly this leads to only two energy states, which should be qualified as precession-up and precession-down rather thanspin-up and spin down.Indeed, despite the presence of the many different possible angles betweenthe spin axis and the magnetic field , the fermions can only have two possible energies . The values do thus not correspond to the continuum of values Einstein and Ehrenfest had conjectured. The energy term is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies , and as such not valid for individual fermions. The two fermion states with energy are not potential-energy states. We also explain themathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clearand understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle
The geometrical meaning of spinors as a key to make sense of quantum mechanics
No full textThis paper aims at explaining that the key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R 4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R 3. In the three-dimensional rotation group the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding acquired to derive the free-space Dirac equation from scratch proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine's statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM
Derivation of Malus' law by purely classical reasoning expressed in the language of group representation theory
No full textThis work is part of a reconstruction of quantum mechanics from scratch with the aim to understand what it means. In previous work we have shown that the Dirac equation can be derived by classical reasoning just using relativity and group theory. This implies that contrary to common belief quantum mechanics (QM) is not magical or radically different from classical mechanics. It is also not incompatible with the theory of relativity but completely part of it. As the experimental violations of the Bell inequalities seem to take exception with this general scheme for making sense of QM we have scrutinized the derivation of the Bell inequalities to figure out the limitations of our approach. We were able to show that the derivation contains a logical error based on wrong modelling. In the present paper we complete this investigation by showing that Malus' law can be derived by classical reasoning using group representation theory, both for electrons and photons. This is a further, be it somewhat more indirect proof that the derivation of the Bell inequalities contains an error. But more importantly it shows that the philosophy of our approach remains intact, by proving that we can also understand Malus' law with our methods and thereby validating our approach also for this physical phenomenon
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