This site documents the Divine Mind:
the Universe that enables us to create ourselves
page 13: Is Hilbert space independent of Minkowski space?
13.1 Do Lorentz transformations affect Hilbert space?13.2 Entanglement: infinite velocity or no space-time?
13.3 The Lagrangian view of quantum mechanics
13.4: Quantum non-locality
13.1: Do Lorentz transformations affect Hilbert space?
Since the advent of special relativity, classical physical field theories are usually described in flat Minkowski space-time. The special principle of relativity holds that every observer sees the same laws of physics, including the same speed of light, in their own rest frame. This defines the Lorentz transformation which enables each observer to compute their local equivalent of the spacetime intervals observed between distant events in moving frames. This transformation is expressed succinctly by the 1, 1, 1, -1 metric of Minkowski space so that if we set the speed of light c to 1, all observers see an invariant interval ds2 = dx2 + dy2 + dz2 - dt2. Minkowski space - Wikipedia, Tests of special relativity - Wikipedia
It seems to be generally accepted in quantum field theory that the Lorentz transformation applies equally to states in Hilbert space and to particles in Minkowski space. This implies that the domain of Hilbert space is Minkowski space. Martinus Veltman (1994): Diagrammatica: The Path to the Feynman Rules page 20.
What is the reason for this? Perhaps it is just familiarity. Physics has always been a matter of space and time. Why would it ever be different? There may also be Newton's theological feeling that space and time were part of God's creation. In the General Scholium he writes:
And from his true dominion it follows that the true God is a Living, Intelligent, and Powerful Being; . . . He endures forever, and is every where present; and, by existing always and every where, he constitutes Duration and Space. Since every particle of Space is always, and every indivisible moment of Duration is every where, certainly the Maker and Lord of all things cannot be never and no where.
It would be hard for a Newtonian Christian to think otherwise. On this site we have replaced the well known God of Aquinas and Newton with an initial singularity which is formally identical to this God, and may be the creator and sustainer of Hilbert space as described on pages 8: The theology of the Trinity and 9: The active creation of Hilbert space. With this identification, it is natural to think of spacetime as a consequence of quantum processes made possible by a dynamic initial singularity rather than as Newton's divinely established domain for classical cosmology. Isaac Newton (1726): General Scholium
13.2: Entanglement: infinite velocity or no space-time?
If the quantum world constitutes a layer of the Universe built on the initial singularity before the emergence of observable energy, time, space and momentum, the situation would be reversed, and Hilbert space would become the domain of Minkowski space. The phenomenon of entanglement suggests that the Hilbert quantum world exists prior to and independent of the Minkowski classical world. It seems more reasonable to attribute the apparently instantaneous long distance propagation of correlations associated with entanglement to the absence of space rather than to the propagation of these correlations at infinite velocity.
If this is the case, there opens up a new degree of freedom in the relationship between quantum and classical dynamics which may make it possible to remove some of the confusion in quantum field theory noted by Kuhlmann in the epigraph to this site on page 1: Abstract. Meinard Kuhlmann (Stanford Encyclopedia of Philosophy): Quantum Field TheoryEinstein appears never to have been truly happy with quantum mechanics, and often sought to demonstrate its weaknesses. This may have been because he felt that nature should go its own way independently of any observers. In the quantum world, however, observation is part of the physics. Although some have felt that 'observer' implied a conscious being, we can equally well imagine the universe observing itself to create real events. In the course of a paper intended to show that quantum mechanics is incomplete, the authors (E, P & R) identified 'spooky action at a distance' which is a now seen as a consequence of entanglement. Einstein, Podolsky & Rosen: Can the Quantum Mechanical Description of Physical Reality be Considered Complete?, Quantum entanglement - Wikipedia, Gabriel Popkin (2018): Einstein's 'spooky action at a distance' spotted in objects almost big enough to see
EPR equate reality with predictability: If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.
EPR argue from the quantum formalism that a measurement on one part of an entangled system enables us to predict with certainty the outcome of a measurement on the other system even though they are spacelike separated. They concluded that 'no reasonable definition of reality could be expected to permit this.' Many authors feel that the quantum mechanical description has become established a new definition of reality. Problems remain, however, which we discussion more detail on page 14: "Measurement": the interface between Hilbert and Minkowski spaces
Here we may exemplify entanglement and its consequences using a simple two state system. Electrons have two spin states, often called up and down. In the singlet state, one electron has spin up, the other down, so that the total spin of the singlet is zero. Singlet - Wikipedia
Entanglement establishes that when these two electrons spatially separated, they behave, when observed, as though they are still in contact. If one electron is observed to be spin up, the other will be observed to be spin down no matter how far apart they are. This is 'spooky action at a distance', but the fact that this correlation appears to be instantaneous suggests that although the electrons are distant in Minkowski space, they are still in contact in Hilbert space. If this is the case, it is a major break from conventional wisdom and opens the way for an interesting new approach to theology, spirituality and quantum field theory.Although the traditional observers Alice and Bob can communicate the fact of their observations via entanglement immediately and definitely, it cannot be used to communicate information faster than the speed of light. This is because Alice cannot control what she is going to observe, and therefore control what Bob receives, even though it is certain, in the binary case, that if Alice observes 0 Bob will observe 1. In other words, the unobserved quantum world is, as Einstein felt, incomplete. It is completed by observation or measurement which is, of course, part of the Minkowski world.
Although the biggest shock that came with quantum mechanics is the inability to predict the precise timing of events, entanglement gives us a very definite method predict both the nature and timing of entangled events, since the observation of one part of an entangled system appears to have an immediate consequence for the other part.
Experimental tests of entanglement have gradually improved until now it is widely believed that spooky action at a distance is real. It has been shown that this correlation at a distance operates at many times the velocity of light. Pan, Bouwmeester, Daniell, Weinfurter & Zeilinger (2000): Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement, Salart, Baas, Branciard, Gisin & Zbinden (2008): Testing the speed of 'spooky action at a distance', Juan Yin et al: Lower Bound on the Speed of Nonlocal Correlations without Locality and Measurement Choice Loopholes
EPR established that from the point of view of classical determinism quantum theory is incomplete. We might therefore think of the Universe at the quantum Hilbert space level as being a work in progress. As we have already noted, the uncertainty of quantum behaviour makes the evolution of the Universe possible by providing the variation necessary for creative evolution (see page 10: The emergence of quantum mechanics)
13.3: The Lagrangian view of quantum mechanics
I am trying to preserve as much as possible of the old theology of Aristotle and Aquinas as well as modern physics as I follow my dream of showing that the Universe fulfills all the roles attributed to the traditional God. Aquinas defined God as pure action, a term he inherited from Aristotle. On page 3: Action from the unmoved mover to the quantum I introduced the modern definition of action developed by Joseph-Louis Lagrange. Lagrange's new formulation of Newtonian mechanics make it easier to deal with many body problems like the whole solar system. Lagrangian mechanics - Wikipedia
In 1933, Paul Dirac felt that it would be advantageous if Hamilton's principle could also be applied to quantum mechanics. As Dirac notes, one of the features of the Lagrangian method is that it allows one to collect together all the equations of motion and express then as the stationary property of a certain action function. The principal role of quantum mechanics is to solve the eigenvalue problem to find stationary points in the spectrum of a hermitian operator acting on a Hilbert space. The second desirable feature is that the action function is a relativistic invariant. This is consistent with the idea that action in quantum mechanics is prior to the emergence of spacetime, so that relativistic transformations do not affect it. Hamilton's principle - Wikipedia, Paul Dirac (1933): The Lagrangian in Quantum Mechanics
At first sight it did not seem clear to Dirac how the Lagrangian could be introduced to quantum mechanics. The natural home of classical relativistic Lagrangian mechanics is Minkowski space: The equations involve partial derivatives of the spacetime coordinates and velocities, and no meaning can be given to such derivatives in quantum mechanics. So we must try to take over the ideas of the classical Lagrangian theory, not the equations of the classical Lagrangian theory. What Dirac discovered is that from the Lagrangian point of view quantum mechanics deals only with the time and phase element of events in spacetime and Hamilton's principle is equivalent to the quantum mechanical search for stationary phases which are integral multiples of the quantum of action which is in Minkowski space the quantum of angular momentum.
He found that the Lagrangian could be applied to quantum theory if he were is restrict the treatment of classical action in 4-dimensional spacetime to the single dimension like time which correlates with the evolution of phase in quantum mechanics. The Lagrangian then appears as the exponent in a complex representation of the evolution of the phase of a quantum system. Stationary action in the classical sense appears as fixed phase in the quantum regime. This seems to connect with the role of Hermitian operators in quantum theory which select stationary eigenvectors out of the ceaseless motion of quantum systems described by the Schrödinger equation. One of the important symmetry principles in physics is known as local symmetry. All the modern field theories for the fundamental interactions use this symmetry. Local symmetry is called gauge invariance, but as Chen Ning Yang points out, if we were to rename it today we would call it phase invariance and gauge fields would be called phase fields (Auyang page 44). Hermitian adjoint - Wikipedia, Schrödinger equation - Wikipedia, Sunny Auyang (1995): How is Quantum Field Theory Possible?Feynman used this idea to create his path integral representation of quantum mechanics. He was able to show this to be equivalent to the Schrödinger and Heisenberg representations of the theory. The path integral has since become the representation of choice in the introduction of quantum theory into quantum field theory and attempts to produce a quantum theory of gravitation. Anthony Zee (2010): Quantum Field Theory in a Nutshell, Michio Kaku (1998): Introduction to Superstrings and M-Theory
Feynman's representation also supports the idea, proposed here, that Hilbert space is prior to and independent of Minkowski space. The path integral approach assumes that a particle moving from a to b follows every possible path between these two points. Its quantum phase is integrated along every one of these paths and the sum of these integrals computed to produce an amplitude for the transition from a to b. The absolute square of this amplitude yields the probability of the transition. Given the size of the space-time universe and the relativistic restriction on the speed of particle motion, this computation can only make sense in a Hilbert space where spacetime is absent. Although the existence of null geodesics is made possible by the metric of Minkowski space, the transit of a particle on a null geodesic across a light year of spacetime still takes a year. Feynman & Hibbs (1965): Quantum Mechanics and Path Integrals
We can imagine that the path integral approach selects paths which, like a Bohr orbit in an atom, involve a stationary integral number of quanta of action. The overall program on this site is that we should replace continuous arithmetic (which leads to infinities and other problems) with logic and interpret the quantum of a action as a logical operator. Such operators are discrete rather than continuous, which is consistent with the quantum nature of the world. Bohr model - Wikipedia Finally, the independence of Hilbert and Minkowski space is supported by the radical differences in the spaces which is rooted in the meaning of their metrics, space-time distance in the case of Minkowski space, phase or angle or rate of change of phase in Hilbert space. This appears in the infinitesimals ds2 = dx2+ dy2 + dz2 − dt2 and p = i h dφ/dx.13.4: Quantum non-locality
The experiments referred to above have demonstrated that entangled particles can act upon one another at a distance even though their separation is spacelike. This is called quantum non-locality. Quantum nonlocality - Wikipedia, Spacetime - Wikipedia
Classical physics is founded on a belief in local realismwhich has three features:
1. regularities in observed phenomena point to the existence of physical reality independent of human observers;
2. consistent sets of observations underlie 'inductive inference', the notion that we can use them to devise models of what is going on behind the scenes; and
3. causal influences cannot travel faster than the velocity of light.
Long experience and detailed argument as shown that quantum mechanics is not a local realistic theory. Bernard d'Espagnat (1979): The Quantum Theory and Reality
The EPR argument was perhaps the first hint that local realism is false. John Bell studied EPR and formulated a first version of Bell's theorem which would show that quantum mechanics was not a local realistic theory. The phenomena and theory both appear to point to the fact that the quantum Hilbert world is prior to and independent of the relativistic Minkowski world. John Bell (1987): Speakable and Unspeakable in Quantum Mechanics, Myrvold, Genovese & Shimony (Stanford Encyclopedia of Philosophy): Bell's Thorem
Research continues into the application of entanglement to quantum error correction. Nielsen & Chuang (2016): Quantum Computation and Quantum Information
(Revised Wednesday 7 August 2024)
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Notes and references
Further reading
Books
Auyang (1995), Sunny Y., How is Quantum Field Theory Possible?, Oxford University Press 1995 Jacket: 'Quantum field theory (QFT) combines quantum mechanics with Einstein's special theory of relativity and underlies elementary particle physics. This book presents a philosophical analysis of QFT. It is the first treatise in which the philosophies of space-time, quantum phenomena and particle interactions are encompassed in a unified framework.'
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Bell (1987), John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987 Jacket: JB ... is particularly famous for his discovery of a crucial difference between the predictions of conventional quantum mechanics and the implications of local causality . . . . This work has played a major role in the development of our current understanding of the profound nature of quantum concepts and of the fundamental limitations they impose on the applicability of classical ideas of space, time and locality.
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Feynman (1965), Richard P, and Albert P Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill 1965 Preface: 'The fundamental physical and mathematical concepts which underlie the path integral approach were first developed by R P Feynman in the course of his graduate studies at Princeton, ... . These early inquiries were involved with the problem of the infinite self-energy of the electron. In working on that problem, a "least action" principle was discovered [which] could deal successfully with the infinity arising in the application of classical electrodynamics.' As described in this book. Feynman, inspired by Dirac, went on the develop this insight into a fruitful source of solutions to many quantum mechanical problems.
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Galilei (1610, 1957), Galileo, and Stillman Drake (translator), Discoveries and Opinions of Galileo: Including the Starry Messenger (1610 Letter to the Grand Duchess Christina), Doubleday Anchor 1957 Amazon: 'Although the introductory sections are a bit dated, this book contains some of the best translations available of Galileo's works in English. It includes a broad range of his theories (both those we recognize as "correct" and those in which he was "in error"). Both types indicate his creativity. The reproductions of his sketches of the moons of Jupiter (in "The Starry Messenger") are accurate enough to match to modern computer programs which show the positions of the moons for any date in history. The appendix with a chronological summary of Galileo's life is very useful in placing the readings in context.' A Reader.
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Kaku (1998), Michio, Introduction to Superstrings and M-Theory (Graduate Texts in Contemporary Physics), Springer 1998 ' Called by some "the theory of everything," superstrings may solve a problem which has eluded physicists for the past 50 years -- the final unification of the two great theories of the twentieth century, general relativity and quantum field theory. This is a course-tested comprehensive introductory graduate text on superstrings which stresses the most current areas of interest, not covered in other presentation, including: string field theory, multi loops, Teichmueller spaces, conformal field theory, and four-dimensional strings. The book begins with a simple discussion of point particle theory, and uses the Feynman path integral technique to unify the presentation of superstrings. Prerequisites are an acquaintance with quantum mechanics and relativity. This second edition has been revised and updated throughout.'
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Nielsen (2016), Michael A., and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2016 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002.
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Veltman (1994), Martinus, Diagrammatica: The Path to the Feynman Rules, Cambridge University Press 1994 Jacket: 'This book provides an easily accessible introduction to quantum field theory via Feynman rules and calculations in particle physics. The aim is to make clear what the physical foundations of present-day field theory are, to clarify the physical content of Feynman rules, and to outline their domain of applicability. ... The book includes valuable appendices that review some essential mathematics, including complex spaces, matrices, the CBH equation, traces and dimensional regularization. . . .'
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Zee (2010), Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2010 ' Since it was first published, Quantum Field Theory in a Nutshell has quickly established itself as the most accessible and comprehensive introduction to this profound and deeply fascinating area of theoretical physics. Now in this fully revised and expanded edition, A. Zee covers the latest advances while providing a solid conceptual foundation for students to build on, making this the most up-to-date and modern textbook on quantum field theory available. This expanded edition features several additional chapters, as well as an entirely new section describing recent developments in quantum field theory such as gravitational waves, the helicity spinor formalism, on-shell gluon scattering, recursion relations for amplitudes with complex momenta, and the hidden connection between Yang-Mills theory and Einstein gravity. Zee also provides added exercises, explanations, and examples, as well as detailed appendices, solutions to selected exercises, and suggestions for further reading.'
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Links
Bernard d'Espagnat (1979), The Quantum Theory and Reality, 'The doctrine that the world is made up of objects whose existence is independent of human consciousness turns out to be in conflict with quantum mechanics and with facts established by experiment' |
Bohr model - Wikipedia, Bohr model - Wikipedia, the free encyclopedia, 'In atomic physics, the Rutherford–Bohr model or Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with attraction provided by electrostatic forces rather than gravity.' back |
Einstein, Podolsky & Rosen (1935), Can the Quantum Mechanical Description of Physical Reality be Considered Complete?, A PDF of the classic paper. 'In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false, One is thus led to conclude that the description of reality given by the wave function is not complete.' back |
Gabriel Popkin (2018), Einstein's 'spooky action at a distance' spotted in objects almost big enough to see, ' One of the strangest aspects of quantum physics is entanglement: If you observe a particle in one place, another particle—even one light-years away—will instantly change its properties, as if the two are connected by a mysterious communication channel. Scientists have observed this phenomenon in tiny objects such as atoms and electrons. But in two new studies, researchers report seeing entanglement in devices nearly visible to the naked eye.' back |
Hamilton's principle - Wikipedia, Hamilton's principle - Wikipedia, the free encyclopedia, 'In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action . . . It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it.' back |
Hermitian adjoint - Wikipedia, Hermitian adjoint - Wikipedia, the free encyclopedia, ' Hermitian operators: A bounded operator A : H → H is called Hermitian or self-adjoint if |
Isaac Newton (1726), General Scholium, 'Published for the first time as an appendix to the 2nd (1713) edition of the Principia, the General Scholium reappeared in the 3rd (1726) edition with some amendments and additions. As well as countering the natural philosophy of Leibniz and the Cartesians, the General Scholium contains an excursion into natural theology and theology proper. In this short text, Newton articulates the design argument (which he fervently believed was furthered by the contents of his Principia), but also includes an oblique argument for a unitarian conception of God and an implicit attack on the doctrine of the Trinity, which Newton saw as a post-biblical corruption. The English translation here is that of Andrew Motte (1729). Italics and orthography as in original. back |
Juan Yin et al, Lower Bound on the Speed of Nonlocal Correlations without Locality and Measurement Choice Loopholes , ' In their well-known paper, Einstein, Podolsky, and Rosen called the nonlocal correlation in quantum entanglement a “spooky action at a distance.” If the spooky action does exist, what is its speed? All previous experiments along this direction have locality and freedom-of-choice loopholes. Here, we strictly closed the loopholes by observing a 12 h continuous violation of the Bell inequality and concluded that the lower bound speed of spooky action was 4 orders of magnitude of the speed of light if Earth’s speed in any inertial reference frame was less than 10-3 time the speed of light. ' back |
Lagrangian mechanics - Wikipedia, Lagrangian mechanics - Wikipedia, the free encyclopedia, ' Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle. Given a system of point masses and a pair, t1 and t2 Lagrangian mechanics postulates that the system's trajectory (describing evolution of the system over time) . . . must be a stationary point of the action functional S = ∫ L dt. By convention, L = T − V, where T and V are the kinetic and potential energy of the system, respectively.' back |
Minkowski space - Wikipedia, Minkowski space - Wikipedia, the free encyclopedia, ' By 1908 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four-dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval x2 + y2 + z2 − c2 t2.' back |
Myrvold, Genovese & Shimony (Stanford Encyclopedia of Philosophy), Bell's Theorem, ' Beginning in the 1970s, there has been a series of experiments of increasing sophistication to test whether the Bell inequalities are satisfied. With few exceptions, the results of these experiments have confirmed the quantum mechanical predictions, violating the relevant Bell Inequalities. Until recently, however, each of these experiments was vulnerable to at least one of two loopholes, referred to as the communication, or locality loophole, and the detection loophole (see section 5). Finally, in 2015, experiments were performed that demonstrated violation of Bell inequalities with these loopholes blocked. This has consequences for our physical worldview; the conditions that entail Bell inequalities are, arguably, an integral part of the physical worldview that was accepted prior to the advent of quantum mechanics. If one accepts the lessons of the experimental results, then some one or other of these conditions must be rejected.' back |
P. A. M. Dirac (1933), The Lagrangian in Quantum Mechanics, ' . . . there is an alternative formulation [to the Hamiltonian] in classical dynamics, provided by the Lagrangian. This requires one to work in terms of coordinates and velocities instead of coordinates and momenta. The two formulation are closely related but there are reasons for believing that the Lagrangian one is more fundamental. . . . Secondly the lagrangian method can easily be expressed relativistically, on account of the action function being a relativistic invariant; . . .. ' [This article was first published in Physikalische Zeitschrift der Sowjetunion, Band 3, Heft 1 (1933), pp. 64–72.] back |
Pan, Bouwmeester, Daniell, Weinfurter & Zeilinger (2000), Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement, ' Abstract Bell's theorem states that certain statistical correlations predicted by quantum physics for measurements on two-particle systems cannot be understood within a realistic picture based on local properties of each individual particle—even if the two particles are separated by large distances. Einstein, Podolsky and Rosen first recognized the fundamental significance of these quantum correlations (termed ‘entanglement’ by Schrödinger) and the two-particle quantum predictions have found ever-increasing experimental support. A more striking conflict between quantum mechanical and local realistic predictions (for perfect correlations) has been discovered; but experimental verification has been difficult, as it requires entanglement between at least three particles. Here we report experimental confirmation of this conflict, using our recently developed method to observe three-photon entanglement, or ‘Greenberger–Horne–Zeilinger’ (GHZ) states. The results of three specific experiments, involving measurements of polarization correlations between three photons, lead to predictions for a fourth experiment; quantum physical predictions are mutually contradictory with expectations based on local realism. We find the results of the fourth experiment to be in agreement with the quantum prediction and in striking conflict with local realism.' back |
Quantum entanglement - Wikipedia, Quantum entanglement - Wikipedia, the free encyclopedia, 'Quantum entanglement is a physical phenomenon which occurs when pairs or groups of particles are generated, interact, or share spatial proximity in ways such that the quantum state of each particle cannot be described independently of the state of the other(s), even when the particles are separated by a large distance—instead, a quantum state must be described for the system as a whole. . . . Entanglement is considered fundamental to quantum mechanics, even though it wasn't recognized in the beginning. Quantum entanglement has been demonstrated experimentally with photons, neutrinos, electrons, molecules as large as buckyballs, and even small diamonds. The utilization of entanglement in communication and computation is a very active area of research.' back |
Quantum nonlocality - Wikipedia, Quantum nonlocality - Wikipedia, the free encyclopedia, 'Quantum nonlocality is the phenomenon by which the measurements made at a microscopic level necessarily refute one or more notions (often referred to as local realism) that are regarded as intuitively true in classical mechanics. Rigorously, quantum nonlocality refers to quantum mechanical predictions of many-system measurement correlations that cannot be simulated by any local hidden variable theory. Many entangled quantum states produce such correlations when measured, as demonstrated by Bell's theorem.' back |
Salart, Baas, Branciard, Gisin & Zbinden (2008), Testing the speed of 'spooky action at a distance, ' Correlations are generally described by one of two mechanisms: either a first event influences a second one by sending information encoded in bosons or other physical carriers, or the correlated events have some common causes in their shared history. Quantum physics predicts an entirely different kind of cause for some correlations, named entanglement. This reveals itself in correlations that violate Bell inequalities (implying that they cannot be described by common causes) between space-like separated events (implying that they cannot be described by classical communication). Many Bell tests have been performed, and loopholes related to locality and detection have been closed in several independent experiments. It is still possible that a first event could influence a second, but the speed of this hypothetical influence (Einstein's 'spooky action at a distance') would need to be defined in some universal privileged reference frame and be greater than the speed of light. Here we put stringent experimental bounds on the speed of all such hypothetical influences. We performed a Bell test over more than 24 hours between two villages separated by 18 km and approximately east-west oriented, with the source located precisely in the middle. We continuously observed two-photon interferences well above the Bell inequality threshold. Taking advantage of the Earth's rotation, the configuration of our experiment allowed us to determine, for any hypothetically privileged frame, a lower bound for the speed of the influence. For example, if such a privileged reference frame exists and is such that the Earth's speed in this frame is less than 10(-3) times that of the speed of light, then the speed of the influence would have to exceed that of light by at least four orders of magnitude.' back |
Schrödinger equation - Wikipedia, Schrödinger equation - Wikipedia, the free encyclopedia, ' In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. . . . In classical mechanics Newton's second law, (F = ma), is used to mathematically predict what a given system will do at any time after a known initial condition. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").' back |
Singlet - Wikipedia, Singlet - Wikipedia, the free encyclopedia, 'In theoretical physics, a singlet usually refers to a one-dimensional representation (e.g. a particle with vanishing spin). It may also refer to two or more particles prepared in a correlated state, such that the total angular momentum of the state is zero. Singlets frequently occur in atomic physics as one of the two ways in which the spin of two electrons can be combined; the other being a triplet. A single electron has spin 1/2, and transforms as a doublet, that is, as the fundamental representation of the rotation group SU(2). The product of two doublet representations can be decomposed into the sum of the adjoint representation (the triplet) and the trivial representation, the singlet. More prosaically, a pair of electron spins can be combined to form a state of total spin 1 and a state of spin 0. The singlet state formed from a pair of electrons has many peculiar properties, and plays a fundamental role in the EPR paradox and quantum entanglement' back |
Spacetime - Wikipedia, Spacetime - Wikipedia, the free encyclopedia, ' In physics, spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur.' back |
Tests of special relativity - Wikipedia, Tests of special relativity - Wikipedia, the free encyclopedia, ' Special relativity is a physical theory that plays a fundamental role in the description of all physical phenomena, as long as gravitation is not significant. Many experiments played (and still play) an important role in its development and justification. The strength of the theory lies in its unique ability to correctly predict to high precision the outcome of an extremely diverse range of experiments. Repeats of many of those experiments are still being conducted with steadily increased precision, with modern experiments focusing on effects such as at the Planck scale and in the neutrino sector. Their results are consistent with the predictions of special relativity.' back |