At higher energies ground based gamma ray telescopes have set limits on the annihilation of dark matter in dwarf spheroidal galaxies and in clusters of galaxies The PAMELA experiment launched has detected a larger number of positrons than expected These extra positrons could be produced by dark matter annihilation but may also come from pulsars No excess of anti protons has been observed The Alpha Magnetic Spectrometer on the International Space Station is designed to directly measure the fraction of cosmic rays which are positrons The first results published in April indicate an excess of high energy cosmic rays which could potentially be due to annihilation of dark matter A few of the WIMPs passing through the Sun or Earth may scatter off atoms and lose energy This way a large population of WIMPs may accumulate at the center of these bodies increasing the chance that two will collide and annihilate This could produce a distinctive signal in the form of high energy neutrinos originating from the center of the Sun or Earth It is generally considered that the detection of such a signal would be the strongest indirect proof of WIMP dark matter High energy neutrino telescopes such as AMANDA IceCube and ANTARES are searching for this signal WIMP annihilation from the Milky Way Galaxy as a whole may also be detected in the form of various annihilation products The Galactic Center is a particularly good place to look because the density of dark matter may be very high there Alternative theories edit Mass in extra dimensions edit In some multidimensional theories the force of gravity is the unique force able to have an effect across all the various extra dimensions which would explain the relative weakness of the force of gravity compared to the other known forces of nature that would not be able to cross into extra dimensions electromagnetism strong interaction and weak interaction

In that case dark matter would be a perfect candidate for matter that would exist in other dimensions and that could only interact with the matter on our dimensions through gravity That dark matter located on different dimensions could potentially aggregate in the same way as the matter in our visible universe does forming exotic galaxies Topological defects edit Dark matter could consist of primordial defects defects originating with the birth of the universe in the topology of quantum fields which would contain energy and therefore gravitate This possibility may be investigated by the use of an orbital network of atomic clocks which would register the passage of topological defects by monitoring the synchronization of the clocks The Global Positioning System may be able to operate as such a network Modified gravity edit Numerous alternative theories have been proposed to explain these observations without the need for a large amount of undetected matter Most of these theories modify the laws of gravity established by Newton and Einstein The earliest modified gravity model to emerge was Mordehai Milgrom s Modified Newtonian Dynamics MOND in which adjusts Newton s laws to create a stronger gravitational field when gravitational acceleration levels become tiny such as near the rim of a galaxy It had some success explaining galactic scale features such as rotational velocity curves of elliptical galaxies and dwarf elliptical galaxies but did not successfully explain galaxy cluster gravitational lensing However MOND was not relativistic since it was just a straight adjustment of the older Newtonian account of gravitation not of the newer account in Einstein s general relativity Soon after attempts were made to bring MOND into conformity with general relativity this is an ongoing process and many competing hypotheses have emerged based around the original MOND model—including TeVeS MOG or STV gravity and phenomenological covariant approach among others In John W Moffat proposed a modified gravity hypothesis based on the nonsymmetric gravitational theory NGT that claims to account for the behavior of colliding galaxies This model requires the presence of non relativistic neutrinos or other candidates for cold dark matter to work Another proposal uses a gravitational backreaction in an emerging theoretical field that seeks to explain gravity between objects as an action a reaction and then a back reaction Simply an object A affects an object B and the object B then re affects object A and so on creating a sort of feedback loop that strengthens gravity In another group has proposed a modification of large scale gravity in a hypothesis named dark fluid In this formulation the attractive gravitational effects attributed to dark matter are instead a side effect of dark energy Dark fluid combines dark matter and dark energy in a single energy field that produces different effects at different scales This treatment is a simplified approach to a previous fluid like model called the generalized Chaplygin gas model where the whole of spacetime is a compressible gas Dark fluid can be compared to an atmospheric system Atmospheric pressure causes air to expand but part of the air can collapse to form clouds In the same way the dark fluid might generally expand but it also could collect around galaxies to help hold them together Another set of proposals is based on the possibility of a double metric tensor for space time It has been argued that time reversed solutions in general relativity require such double metric for consistency and that both dark matter and dark energy can be understood in terms of time reversed solutions of general relativity Fractality of Spacetime edit Applying relativity to fractal non differentiable spacetime Laurent Nottale in his Scale Relativity theory suggests that potential energy arises due to the fractality of space and accounts for the missing mass energy observed at cosmological scales citation needed Popular culture edit Main article Dark matter in fiction Mention of dark matter is made in some video games and other works of fiction In such cases it is usually attributed extraordinary physical or magical properties Such descriptions are often inconsistent with the properties of dark matter proposed in physics and cosmology See also edit Physics portal Cosmology portal Chameleon particle Conformal gravity General Antiparticle Spectrometer Illustris project Light dark matter Mirror matter Multidark research program Conformal gravity are gravity theories that are invariant under conformal transformations in the Riemannian geometry sense more accurately they are invariant under Weyl transformations where is the metric tensor and is a function on spacetime Contents Weyl squared theories Four derivative theories Conformal unification to the Standard Model See also Notes References See also Wigner–Weyl transform for another definition of the Weyl transform In theoretical physics the Weyl transformation named after Hermann Weyl is a local rescaling of the metric tensor which produces another metric in the same conformal class A theory or an expression invariant under this transformation is called conformally invariant or is said to possess Weyl symmetry The Weyl symmetry is an important symmetry in conformal field theory It is for example a symmetry of the Polyakov action The ordinary Levi Civita connection and associated spin connections are not invariant under Weyl transformations An appropriately invariant notion is the Weyl connection which is one way of specifying the structure of a conformal connection A quantity f has conformal weight k if under the Weyl transformation it transforms via Thus conformally weighted quantities belong to certain density bundles see also conformal dimension Let Aµ be the connection one form associated to the Levi Civita connection of g Introduce a connection that depends also on an initial one form via Then is covariant and has conformal weight External links Weyl squared theories edit The simplest theory in this category has the square of the Weyl tensor as the Lagrangian where is the Weyl tensor This is to be contrasted with the usual Einstein–Hilbert action where the Lagrangian is just the Ricci scalar The equation of motion upon varying the metric is called the Bach equation where is the Ricci tensor Conformally flat metrics are solutions of this equation Since these theories lead to fourth order equations for the fluctuations around a fixed background they are not manifestly unitary It has therefore been generally believed that they could not be consistently quantized This is now disputed Four derivative theories edit Conformal gravity is an example of a derivative theory This means that each term in the wave equation can contain up to derivatives There are pros and cons of derivative theories The pros are that the quantized version of the theory is more convergent and renormalisable The cons are that there may be issues with causality A simpler example of a derivative wave equation is the scalar derivative wave equation The solution for this in a central field of force is The first two terms are the same as a normal wave equation Because this equation is a simpler approximation to conformal gravity m corresponds to the mass of the central source The last two terms are unique to derivative wave equations It has been suggested that small values be assigned to them to account for the galactic acceleration constant also known as dark matter and the dark energy constant The solution equivalent to the Schwarzschild solution in general relativity for a spherical source for conformal gravity has a metric with to show the difference between general relativity bc is very small so can be ignored The problem is that now c is the total mass energy of the source b is the integral of density times distance to source squared So this is a completely different potential to general relativity and not just a small modification The main issue with conformal gravity theories as well as any theory with higher derivatives is the typical presence of ghosts which point to instabilities of the quantum version of the theory although there might be a solution to the ghost problem An alternative approach is to consider the gravitational constant as a symmetry broken scalar field in which case you would consider a small correction to Newtonian gravity like this where we consider to be a small correction in which case the general solution is the same as the Newtonian case except there can be an additional term where there is an additional component varying sinusoidally over space The wavelength of this variation could be quite large such as an atomic width Thus there appears to be several stable potentials around a gravitational force in this model Conformal unification to the Standard Model edit By adding a suitable gravitational term to the standard model action with gravitational coupling the theory develops a local conformal Weyl invariance in the unitary gauge for the local SU The gauge is fixed by requiring the Higgs scalar to be a constant This mechanism generates the masses for the vector bosons and matter fields with no physical degrees of freedom for the Higgs See also edit Conformal supergravity Scalar field dark matter Self interacting dark matter Theoretical physics is a branch of physics which employs mathematical models and abstractions of physical objects and systems to rationalize explain and predict natural phenomena This is in contrast to experimental physics which uses experimental tools to probe these phenomena The advancement of science depends in general on the interplay between experimental studies and theory In some cases theoretical physics adheres to standards of mathematical rigor while giving little weight to experiments and observations a For example while developing special relativity Albert Einstein was concerned with the Lorentz transformation which left Maxwell s equations invariant but was apparently uninterested in the Michelson–Morley experiment on Earth s drift through a luminiferous ether citation needed Conversely Einstein was awarded the Nobel Prize for explaining the photoelectric effect previously an experimental result lacking a theoretical formulation Contents Overview History Mainstream theories Examples Proposed theories Examples Fringe theories Examples Thought experiments vs real experiments See also Notes References Suggested Reading List External links OverviewA physical theory is a model of physical events It is judged by the extent to which its predictions agree with empirical observations The quality of a physical theory is also judged on its ability to make new predictions which can be verified by new observations A physical theory differs from a mathematical theorem in that while both are based on some form of axioms judgment of mathematical applicability is not based on agreement with any experimental results A physical theory similarly differs from a mathematical theory in the sense that the word theory has a different meaning in mathematical terms b The equations for an Einstein manifold used in general relativity to describe the curvature of spacetime A physical theory involves one or more relationships between various measurable quantities Archimedes realized that a ship floats by displacing its mass of water Pythagoras understood the relation between the length of a vibrating string and the musical tone it produces Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles and the quantum mechanical idea that action and energy are not continuously variable Theoretical physics consists of several different approaches In this regard theoretical particle physics forms a good example For instance phenomenologists might employ semi empirical formulas to agree with experimental results often without deep physical understanding c Modelers also called model builders often appear much like phenomenologists but try to model speculative theories that have certain desirable features rather than on experimental data or apply the techniques of mathematical modeling to physics problems d Some attempt to create approximate theories called effective theories because fully developed theories may be regarded as unsolvable or too complicated Other theorists may try to unify formalise reinterpret or generalise extant theories or create completely new ones altogether e Sometimes the vision provided by pure mathematical systems can provide clues to how a physical system might be modeled f e g the notion due to Riemann and others that space itself might be curved Theoretical problems that need computational investigation are often the concern of computational physics Theoretical advances may consist in setting aside old incorrect paradigms e g aether theory of light propagation caloric theory of heat burning consisting of evolving phlogiston or astronomical bodies revolving around the Earth or may be an alternative model that provides answers that are more accurate or that can be more widely applied In the latter case a correspondence principle will be required to recover the previously known result Sometimes though advances may proceed along different paths For example an essentially correct theory may need some conceptual or factual revisions atomic theory first postulated millennia ago by several thinkers in Greece and India and the two fluid theory of electricity are two cases in this point However an exception to all the above is the wave particle duality a theory combining aspects of different opposing models via the Bohr complementarity principle Relationship between mathematics and physicsPhysical theories become accepted if they are able to make correct predictions and no or few incorrect ones The theory should have at least as a secondary objective a certain economy and elegance compare to mathematical beauty a notion sometimes called Occam s razor after the th century English philosopher William of Occam or Ockham in which the simpler of two theories that describe the same matter just as adequately is preferred but conceptual simplicity may mean mathematical complexity They are also more likely to be accepted if they connect a wide range of phenomena Testing the consequences of a theory is part of the scientific method Physical theories can be grouped into three categories mainstream theories proposed theories and fringe theories HistoryFor more details on this topic see History of physics Theoretical physics began at least years ago under the Pre socratic philosophy and continued by Plato and Aristotle whose views held sway for a millennium During the rise of medieval universities the only acknowledged intellectual disciplines were the seven liberal arts of the Trivium like grammar logic and rhetoric and of the Quadrivium like arithmetic geometry music and astronomy During the Middle Ages and Renaissance the concept of experimental science the counterpoint to theory began with scholars such as Ibn al Haytham and Francis Bacon As the Scientific Revolution gathered pace the concepts of matter energy space time and causality slowly began to acquire the form we know today and other sciences spun off from the rubric of natural philosophy Thus began the modern era of theory with the Copernican paradigm shift in astronomy soon followed by Johannes Kepler s expressions for planetary orbits which summarized the meticulous observations of Tycho Brahe the works of these men alongside Galileo s can perhaps be considered to constitute the Scientific Revolution The great push toward the modern concept of explanation started with Galileo one of the few physicists who was both a consummate theoretician and a great experimentalist The analytic geometry and mechanics of Descartes were incorporated into the calculus and mechanics of Isaac Newton another theoretician experimentalist of the highest order writing Principia Mathematica In it contained a grand synthesis of the work of Copernicus Galileo and Kepler as well as Newton s theories of mechanics and gravitation which held sway as worldviews until the early th century Simultaneously progress was also made in optics in particular colour theory and the ancient science of geometrical optics courtesy of Newton Descartes and the Dutchmen Snell and Huygens In the th and th centuries Joseph Louis Lagrange Leonhard Euler and William Rowan Hamilton would extend the


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kristina-king kristina-klevits kristina-soderszk kristine-heller kristin-steen krisztina-ventura krystal-de-boor krystal-steal kylee-karr kylee-nash kylie-brooks kylie-channel kylie-haze kylie-wylde kym-wilde kyoto-sun lachelle-marie lacy-rose lady-amanda-wyldefyre lady-stephanie laetitia-bisset lana-burner lana-cox lana-wood lara-amour lara-roxx lara-stevens lataya-roxx latoya laura-clair laura-lazare laura-lion laura-may laura-orsolya laura-paouck laura-zanzibar lauren-black laurence-boutin lauren-montgomery laurien-dominique laurien-wilde laurie-smith lauryl-canyon lauryn-may leah-wilde lea-magic lea-martini leanna-foxxx lee-caroll leigh-livingston leilani lenora-bruce leslie-winston lesllie-bovee letizia-bruni lexi-lane lexi-matthews lezley-zen lia-fire liliane-gray liliane-lemieuvre lili-marlene lily-gilder lily-labeau lily-rodgers lily-valentine linda-shaw linda-vale linda-wong linnea-quigley lisa-bright lisa-de-leeuw lisa-k-loring lisa-lake lisa-melendez lisa-sue-corey lise-pinson little-oral-annie liza-dwyer liza-harper lizzy-borden logan-labrent lois-ayres lola-cait long-jean-silver loni-bunny loni-sanders loona-luxx lorelei-lee lorelei-rand lorena-sanchez lori-alexia lori-blue lorrie-lovett luci-diamond lucie-doll lucie-theodorova lucy-van-dam lydia-baum lynn-franciss lynn-lemay lynn-ray lynn-stevens lynx-canon lysa-thatcher madelina-ray madison-parker magdalena-lynn maggie-randall mai-lin mandi-wine mandy-bright mandy-malone mandy-may mandy-mistery mandy-starr marcia-minor maren margit-ojetz margitta-hofer margo-stevens margot-mahler mariah-cherry marianne-aubert maria-tortuga marie-anne marie-christine-chireix marie-christine-veroda marie-claude-moreau marie-dominique-cabannes marie-france-morel marie-luise-lusewitz marie-sharp marilyn-chambers marilyne-leroy marilyn-gee marilyn-jess marilyn-martyn marilyn-star marina-hedman marion-webb marita-ekberg marita-kemper marlena marlene-willoughby marry-queen martine-grimaud martine-schultz maryanne-fisher mary-hubay mary-ramunno mary-stuart mascha-mouton maud-kennedy mauvais-denoir maxine-tyler maya-black maya-france megan-leigh megan-martinez megan-reece mei-ling melanie-hotlips melanie-scott melba-cruz melinda-russell melissa-bonsardo melissa-del-prado melissa-golden melissa-martinez melissa-melendez melissa-monet mercedes-dragon mercedes-lynn merle-michaels mesha-lynn mia-beck mia-lina mia-smiles michele-raven michelle-aston michelle-ferrari michelle-greco michelle-maren michelle-maylene michelle-monroe micki-lynn mika-barthel mika-tan mikki-taylor mimi-morgan mindy-rae ming-toy miranda-stevens miss-bunny miss-meadow miss-pomodoro missy missy-graham missy-stone missy-vega misti-jane mistress-candice misty-anderson misty-dawn misty-rain misty-regan mona-lisa mona-page moni monica-baal monica-swinn monika-peta monika-sandmayr monika-unco monique-bruno monique-cardin monique-charell monique-demoan monique-gabrielle monique-la-belle morgan-fairlane morrigan-hel moxxie-maddron mulani-rivera mysti-may nadege-arnaud nadia-styles nadine-bronx nadine-proutnal nadine-roussial nadi-phuket nanci-suiter nancy-hoffman nancy-vee natacha-delyro natalia-wood natalli-diangelo natascha-throat natasha-skyler naudia-nyce nessa-devil nessy-grant nesty nicki-hunter nicky-reed nicole-berg nicole-bernard nicole-black nicole-grey nicole-london nicole-parks nicole-scott nicole-taylor nicolette-fauludi nicole-west nika-blond nika-mamic niki-cole nikita-love nikita-rush nikki-charm nikki-grand nikki-king nikki-knight nikki-randall nikki-rhodes nikki-santana nikki-steele nikki-wilde niko nina-cherry nina-deponca nina-hartley nina-preta oana-efria obaya-roberts olesja-derevko olga-cabaeva olga-conti olga-pechova olga-petrova olivia-alize olivia-del-rio olivia-flores olivia-la-roche olivia-outre ophelia-tozzi orchidea-keresztes orsolya-blonde paige-turner paisley-hunter pamela-bocchi pamela-jennings pamela-mann pamela-stanford pamela-stealt pandora paola-albini pascale-vital pat-manning pat-rhea patricia-dale patricia-diamond patricia-kennedy patricia-rhomberg patrizia-predan patti-cakes patti-petite paula-brasile paula-harlow paula-morton paula-price paula-winters pauline-teutscher penelope-pumpkins penelope-valentin petra-hermanova petra-lamas peyton-lafferty phaedra-grant pia-snow piper-fawn pipi-anderson porsche-lynn porsha-carrera precious-silver priscillia-lenn purple-passion queeny-love rachel-ashley rachel-love rachel-luv rachel-roxxx rachel-ryan rachel-ryder racquel-darrian rane-revere raven reagan-maddux rebecca-bardoux regan-anthony regine-bardot regula-mertens reina-leone reka-gabor renae-cruz renee-foxx renee-lovins renee-morgan renee-perez renee-summers renee-tiffany rhonda-jo-petty rikki-blake riley-ray rio-mariah rita-ricardo roberta-gemma roberta-pedon robin-byrd robin-cannes robin-everett robin-sane rochell-starr rosa-lee-kimball rosemarie roxanne-blaze roxanne-hall roxanne-rollan ruby-richards sabina-k sabre sabrina-chimaera sabrina-dawn sabrina-jade sabrina-johnson sabrina-love-cox sabrina-mastrolorenzi sabrina-rose sabrina-scott sabrina-summers sacha-davril sahara sahara-sands sai-tai-tiger samantha-fox samantha-ryan samantha-sterlyng samantha-strong samueline-de-la-rosa sandra-cardinale sandra-de-marco sandra-kalermen sandra-russo sandy-lee sandy-pinney sandy-reed sandy-samuel sandy-style sandy-summers sara-brandy-canyon sara-faye sarah-bernard sarah-cabrera sarah-hevyn sarah-mills sarah-shine sara-sloane sasha sasha-hollander sasha-ligaya sasha-rose satine-phoenix satin-summer savannah-stern savanna-jane scarlet-scarleau scarlet-windsor seka selena serena serena-south severine-amoux shana-evans shanna-mccullough shannon-kelly shannon-rush shantell-day sharon-da-vale sharon-kane sharon-mitchell shaun-michelle shawna-sexton shawnee-cates shay-hendrix shayne-ryder sheena-horne sheer-delight shelby-star shelby-stevens shelly-berlin shelly-lyons sheri-st-clair sheyla-cats shonna-lynn shyla-foxxx shy-love sierra-sinn sierra-skye sigrun-theil silver-starr silvia-bella silvia-saint silvie-de-lux silvy-taylor simone-west sindee-coxx sindy-lange sindy-shy siobhan-hunter skylar-knight skylar-price skyler-dupree smokie-flame smoking-mary-jane solange-shannon sonya-summers sophia-santi sophie-call sophie-duflot sophie-evans sophie-guers stacey-donovan stacy-lords stacy-moran stacy-nichols stacy-silver stacy-thorn starla-fox starr-wood stefania-bruni stella-virgin stephanie-duvalle stephanie-rage stephanie-renee stevie-taylor summer-knight summer-rose sunny-day sunset-thomas sunshine-seiber susan-hart susanne-brend susan-nero susi-hotkiss suzanne-mcbain suzan-nielsen suzie-bartlett suzie-carina suzi-sparks sweet-nice sweety-pie sybille-rossani sylvia-benedict sylvia-bourdon sylvia-brand sylvia-engelmann syreeta-taylor syren-de-mer syvette szabina-black szilvia-lauren tai-ellis taija-rae taisa-banx talia-james tamara-lee tamara-longley tamara-n-joy tamara-west tami-white tammy tammy-lee tammy-reynolds tania-lorenzo tantala-ray tanya-danielle tanya-fox tanya-foxx tanya-lawson tanya-valis tara-aire tasha-voux tatjana-belousova tatjana-skomorokhova tawnee-lee tawny-pearl tayla-rox taylor-wane teddi-austin teddi-barrett tera-bond tera-heart tera-joy teresa-may teresa-orlowski teri-diver teri-weigel terri-dolan terri-hall tess-ferre tess-newheart thais-vieira tia-cherry tianna tiara tiffany-blake tiffany-clark tiffany-duponte tiffany-rayne tiffany-rousso tiffany-storm tiffany-towers tiffany-tyler tiger-lily tigr timea-vagvoelgyi tina-blair tina-burner tina-evil tina-gabriel tina-loren tina-marie tina-russell tish-ambrose tommi-rose tonisha-mills topsy-curvey tori-secrets tori-sinclair tori-welles tracey-adams traci-lords traci-topps traci-winn tracy-duzit tracy-love tracy-williams tricia-devereaux tricia-yen trinity-loren trisha-rey trista-post trixie-tyler ultramax ursula-gaussmann ursula-moore uschi-karnat valentina valerie-leveau valery-hilton vanessa-chase vanessa-del-rio vanessa-michaels vanessa-ozdanic vanilla-deville velvet-summers veri-knotty veronica-dol veronica-hart veronica-hill veronica-rayne veronica-sage veronika-vanoza via-paxton vicky-lindsay vicky-vicci victoria-evans victoria-gold victoria-knight victoria-luna victoria-paris victoria-slick victoria-zdrok viper virginie-caprice vivian-valentine vivien-martines wendi-white wendy-divine whitney-banks whitney-fears whitney-wonders wonder-tracey wow-nikki xanthia-berstein yasmine-fitzgerald yelena-shieffer yvonne-green zara-whites zsanett-egerhazi zuzie-boobies

theory of classical mechanics considerably They picked up the interactive intertwining of mathematics and physics begun two millennia earlier by Pythagoras Among the great conceptual achievements of the th and th centuries were the consolidation of the idea of energy as well as its global conservation by the inclusion of heat electricity and magnetism and then light The laws of thermodynamics and most importantly the introduction of the singular concept of entropy began to provide a macroscopic explanation for the properties of matter Statistical mechanics followed by statistical physics emerged as an offshoot of thermodynamics late in the th century Another important event in the th century was the discovery of electromagnetic theory unifying the previously separate phenomena of electricity magnetism and light The pillars of modern physics and perhaps the most revolutionary theories in the history of physics have been relativity theory and quantum mechanics Newtonian mechanics was subsumed under special relativity and Newton s gravity was given a kinematic explanation by general relativity Quantum mechanics led to an understanding of blackbody radiation which indeed was an original motivation for the theory and of anomalies in the specific heats of solids — and finally to an understanding of the internal structures of atoms and molecules Quantum mechanics soon gave way to the formulation of quantum field theory QFT begun in the late s In the aftermath of World War more progress brought much renewed interest in QFT which had since the early efforts stagnated The same period also saw fresh attacks on the problems of superconductivity and phase transitions as well as the first applications of QFT in the area of theoretical condensed matter The s and s saw the formulation of the Standard model of particle physics using QFT and progress in condensed matter physics theoretical foundation of superconductivity and critical phenomena among others in parallel to the applications of relativity to problems in astronomy and cosmology respectively All of these achievements depended on the theoretical physics as a moving force both to suggest experiments and to consolidate results — often by ingenious application of existing mathematics or as in the case of Descartes and Newton with Leibniz by inventing new mathematics Fourier s studies of heat conduction led to a new branch of mathematics infinite orthogonal series Modern theoretical physics attempts to unify theories and explain phenomena in further attempts to understand the Universe from the cosmological to the elementary particle scale Where experimentation cannot be done theoretical physics still tries to advance through the use of mathematical models Mainstream theoriesMainstream theories sometimes referred to as central theories are the body of knowledge of both factual and scientific views and possess a usual scientific quality of the tests of repeatability consistency with existing well established science and experimentation There do exist mainstream theories that are generally accepted theories based solely upon their effects explaining a wide variety of data although the detection explanation and possible composition are subjects of debate ExamplesBlack hole thermodynamics Classical mechanics Condensed matter physics including solid state physics and the electronic structure of materials Conservation of energy Dark Energy Dark matter Dynamics Electromagnetism Field theory Fluid dynamics General relativity Particle physics Physical cosmology Quantum chromodynamics Quantum computers Quantum electrochemistry Quantum electrodynamics Quantum field theory Quantum information theory Quantum mechanics Quantum Gravity Solid mechanics Special relativity Standard Model Statistical mechanics Thermodynamics Proposed theoriesThe proposed theories of physics are usually relatively new theories which deal with the study of physics which include scientific approaches means for determining the validity of models and new types of reasoning used to arrive at the theory However some proposed theories include theories that have been around for decades and have eluded methods of discovery and testing Proposed theories can include fringe theories in the process of becoming established and sometimes gaining wider acceptance Proposed theories usually have not been tested ExamplesCausal Sets Dark energy or Einstein s Cosmological Constant Einstein–Rosen Bridge The EPR paradox of is an influential thought experiment in quantum mechanics with which Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen EPR claimed to demonstrate that the wave function does not provide a complete description of physical reality and hence that the Copenhagen interpretation is unsatisfactory resolutions of the paradox have important implications for the interpretation of quantum mechanics The essence of the paradox is that particles can interact in such a way that it is possible to measure both their position and their momentum more accurately than Heisenberg s uncertainty principle allows unless measuring one particle instantaneously affects the other to prevent it which would involve information being transmitted faster than light as forbidden by the theory of relativity spooky action at a distance This consequence had not previously been noticed and seemed unreasonable at the time the phenomenon involved is now known as quantum entanglement While EPR felt that the paradox showed that quantum theory was incomplete and should be extended with hidden variables the usual modern resolution is to say that due to the common preparation of the two particles for example the creation of an electron positron pair from a photon the property we want to measure has a well defined meaning only when analyzed for the whole system while the same property for the parts individually remains undefined Therefore if similar measurements are being performed on the two entangled subsystems there will always be a correlation between the outcomes resulting in a well defined global outcome i e for both subsystems together However the outcomes for each subsystem separately at each repetition of the experiment will not be well defined or predictable This correlation does not imply any action of the measurement of one particle on the measurement of the other therefore it doesn t imply any form of action at a distance This modern resolution eliminates the need for hidden variables action at a distance or other structures introduced over time in order to explain the phenomenon A preference for the latter resolution is supported by experiments suggested by Bell s theorem of which exclude some classes of hidden variable theory According to quantum mechanics under some conditions a pair of quantum systems may be described by a single wave function which encodes the probabilities of the outcomes of experiments that may be performed on the two systems whether jointly or individually At the time the EPR article discussed below was written it was known from experiments that the outcome of an experiment sometimes cannot be uniquely predicted An example of such indeterminacy can be seen when a beam of light is incident on a half silvered mirror One half of the beam will reflect and the other will pass If the intensity of the beam is reduced until only one photon is in transit at any time whether that photon will reflect or transmit cannot be predicted quantum mechanically The routine explanation of this effect was at that time provided by Heisenberg s uncertainty principle Physical quantities come in pairs called conjugate quantities Examples of such conjugate pairs are position and momentum of a particle and components of spin measured around different axes When one quantity was measured and became determined the conjugated quantity became indeterminate Heisenberg explained this as a disturbance caused by measurement The EPR paper written in was intended to illustrate that this explanation is inadequate It considered two entangled particles referred to as A and B and pointed out that measuring a quantity of a particle A will cause the conjugated quantity of particle B to become undetermined even if there was no contact no classical disturbance The basic idea was that the quantum states of two particles in a system cannot always be decomposed from the joint state of the two An example in bra–ket notation is Heisenberg s principle was an attempt to provide a classical explanation of a quantum effect sometimes called non locality According to EPR there were two possible explanations Either there was some interaction between the particles even though they were separated or the information about the outcome of all possible measurements was already present in both particles The EPR authors preferred the second explanation according to which that information was encoded in some hidden parameters The first explanation of an effect propagating instantly across a distance is in conflict with the theory of relativity They then concluded that quantum mechanics was incomplete since its formalism does not permit hidden parameters Violations of the conclusions of Bell s theorem are generally understood to have demonstrated that the hypotheses of Bell s theorem also assumed by Einstein Podolsky and Rosen do not apply in our world Most physicists who have examined the issue concur that experiments such as those of Alain Aspect and his group have confirmed that physical probabilities as predicted by quantum theory do exhibit the phenomena of Bell inequality violations that are considered to invalidate EPR s preferred local hidden variables type of explanation for the correlations to which EPR first drew attention Contents History of EPR developments Quantum mechanics and its interpretation Einstein s opposition Description of the paradox EPR paper Measurements on an entangled state Locality in the EPR experiment Resolving the paradox Hidden variables Bell s inequality Einstein s hope for a purely algebraic theory Acceptable theories and the experiment Implications for quantum mechanics Mathematical formulation See also References Selected papers Notes Books External links History of EPR developments edit The article that first brought forth these matters Can Quantum Mechanical Description of Physical Reality Be Considered Complete was published in The paper prompted a response by Bohr which he published in the same journal in the same year using the same title There followed a debate between Bohr and Einstein about the fundamental nature of reality Einstein had been skeptical of the Heisenberg uncertainty principle and the role of chance in quantum theory But the crux of this debate was not about chance but something even deeper Is there one objective physical reality which every observer sees from his own vantage Einstein s view Or does the observer co create physical reality by the questions he poses with experiments Bohr s view Einstein struggled to the end of his life for a theory that could better comply with his idea of causality protesting against the view that there exists no objective physical reality other than that which is revealed through measurement interpreted in terms of quantum mechanical formalism However since Einstein s death experiments analogous to the one described in the EPR paper have been carried out starting in by French scientists Lamehi Rachti and Mittig at the Saclay Nuclear Research Centre These experiments appear to show that the local realism idea is false vindicating Bohr Quantum mechanics and its interpretation edit Main article Interpretations of quantum mechanics Since the early twentieth century quantum theory has proved to be successful in describing accurately the physical reality of the mesoscopic and microscopic world in multiple reproducible physics experiments Quantum mechanics was developed with the aim of describing atoms and explaining the observed spectral lines in a measurement apparatus Although disputed especially in the early twentieth century it has yet to be seriously challenged Philosophical interpretations of quantum phenomena however are another matter the question of how to interpret the mathematical formulation of quantum mechanics has given rise to a variety of different answers from people of different philosophical persuasions see Interpretations of quantum mechanics Quantum theory and quantum mechanics do not provide single measurement outcomes in a deterministic way According to the understanding of quantum mechanics known as the Copenhagen interpretation measurement causes an instantaneous collapse of the wave function describing the quantum system into an eigenstate of the observable that was measured Einstein characterized this imagined collapse in the Solvay Conference He presented a thought experiment in which electrons are introduced through a small hole in a sphere whose inner surface serves as a detection screen The electrons will contact the spherical detection screen in a widely dispersed manner Those electrons however are all individually described by wave fronts that expand in all directions from the point of entry A wave as it is understood in everyday life would paint a large area of the detection screen but the electrons would be found to impact the screen at single points and would eventually form a pattern in keeping with the probabilities described by their identical wave functions Einstein asks what makes each electron s wave front collapse at its respective location Why do the electrons appear as single bright scintillations rather than as dim washes of energy across the surface Why does any single electron appear at one point rather than some alternative point The behavior of the electrons gives the impression of some signal having been sent to all possible points of contact that would have nullified all but one of them or in other words would have preferentially selected a single point to the exclusion of all others Einstein s opposition edit Einstein was the most prominent opponent of the Copenhagen interpretation In his view quantum mechanics is incomplete Commenting on this other writers such as John von Neumann and David Bohm hypothesized that consequently there would have to be hidden variables responsible for random measurement results something which was not expressly claimed in the original paper The EPR paper condensed the philosophical discussion into a physical argument The authors claim that given a specific experiment in which the outcome of a measurement is known before the measurement takes place there must exist something in the real world an element of reality that determines the measurement outcome They postulate that these elements of reality are local in the sense that each belongs to a certain point in spacetime Each element may only be influenced by events which are located in the backward light cone of its point in spacetime i e the past These claims are founded on assumptions about nature that constitute what is now known as local realism Though the EPR paper has often been taken as an exact expression of Einstein s views it was primarily authored by Podolsky based on discussions at the Institute for Advanced Study with Einstein and Rosen Einstein later expressed to Erwin Schrödinger that it did not come out as well as I had originally wanted rather the essential thing was so to speak smothered by the formalism In Einstein presented an individual account of his local realist ideas Description of the paradox edit The original EPR paradox challenges the prediction of quantum mechanics that it is impossible to know both the position and the momentum of a quantum particle This challenge can be extended to other pairs of physical properties EPR paper edit The original paper purports to describe what must happen to two systems I and II which we permit to interact and after some time we suppose that there is no longer any interaction between the two parts As explained by Manjit Kumar the EPR description involves two particles A and B which interact briefly and then move off in opposite directions According to Heisenberg s uncertainty principle it is impossible to measure both the momentum and the position of particle B exactly However according to Kumar it is possible to measure the exact position of particle A By calculation therefore with the exact position of particle A known the exact position of particle B can be known Also the exact momentum of particle A can be measured so the exact momentum of particle B can be worked out Kumar writes EPR argued that they had proved that particle B can have simultaneously exact values of position and momentum Particle B has a position that is real and a momentum that is real EPR appeared to have contrived a means to establish the exact values of either the momentum or the position of B due to measurements made on particle A without the slightest possibility of particle B being physically disturbed EPR tried to set up a paradox to question the range of true application of Quantum Mechanics Quantum theory predicts that both values cannot be known for a particle and yet the EPR thought experiment purports to show that they must all have determinate values The EPR paper says We are thus forced to conclude that the quantum mechanical description of physical reality given by wave functions is not complete The EPR paper ends by saying While we have thus shown that the wave function does not provide a complete description of the physical reality we left open the question of whether or not such a description exists We believe however that such a theory is possible Measurements on an entangled state edit We have a source that emits electron–positron pairs with the electron sent to destination A where there is an observer named Alice and the positron sent to destination B where there is an observer named Bob According to quantum mechanics we can arrange our source so that each emitted pair occupies a quantum state called a spin singlet The particles are thus said to be entangled This can be viewed as a quantum superposition of two states which we call state I and state II In state I the electron has spin pointing upward along the z axis z and the positron has spin pointing downward along the z axis z In state II the electron has spin z and the positron has spin z Therefore it is impossible without measuring to know the definite state of spin of either particle in the spin singlet – The EPR thought experiment performed with electron–positron pairs A source center sends particles toward two observers electrons to Alice left and positrons to Bob right who can perform spin measurements Alice now measures the spin along the z axis She can obtain one of two possible outcomes z or z Suppose she gets z According to the Copenhagen interpretation of quantum mechanics the quantum state of the system collapses into state I The quantum state determines the probable outcomes of any measurement performed on the system In this case if Bob subsequently measures spin along the z axis there is probability that he will obtain z Similarly if Alice gets z Bob will get z There is of course nothing special about choosing the z axis according to quantum mechanics the spin singlet state may equally well be expressed as a superposition of spin states pointing in the x direction Suppose that Alice and Bob had decided to measure spin along the x axis We ll call these states Ia and IIa In state Ia Alice s electron has spin x and Bob s positron has spin x In state IIa Alice s electron has spin x and Bob s positron has spin x Therefore if Alice measures x the system collapses into state Ia and Bob will get x If Alice measures x the system collapses into state IIa and Bob will get x Whatever axis their spins are measured along they are always found to be opposite This can only be explained if the particles are linked in some way Either they were created with a definite opposite spin about every axis—a hidden variable argument—or they are linked so that one electron feels which axis the other is having its spin measured along and becomes its opposite about that one axis—an entanglement argument Moreover if the two particles have their spins measured about different axes once the electron s spin has been measured about the x axis and the positron s spin about the x axis deduced the positron s spin about the z axis will no longer be certain as if a it knows that the measurement has taken place or b it has a definite spin already about a second axis—a hidden variable However it turns out that the predictions of Quantum Mechanics which have been confirmed by experiment cannot be explained by any local hidden variable theory This is demonstrated in Bell s theorem In quantum mechanics the x spin and z spin are incompatible observables meaning the Heisenberg uncertainty principle applies to alternating measurements of them a quantum state cannot possess a definite value for both of these variables Suppose Alice measures the z spin and obtains z so that the quantum state collapses into state I Now instead of measuring the z spin as well Bob measures the x spin According to quantum mechanics when the system is in state I Bob s x spin measurement will have a probability of producing x and a probability of x It is impossible to predict which outcome will appear until Bob actually performs the measurement Here is the crux of the matter You might imagine that when Bob measures the x spin of his positron he would get an answer with absolute certainty since prior to this he hasn t disturbed his particle at all Bob s positron has a probability of producing x and a probability of x—so the outcome is not certain Bob s positron knows that Alice s electron has been measured and its z spin detected and hence B s z spin has been calculated but the x spin of Bob s positron remains uncertain Put another way how does Bob s positron know which way to point if Alice decides based on information unavailable to Bob to measure x i e to be the opposite of Alice s electron s spin about the x axis and also how to point if Alice measures z since it is only supposed to know one thing at a time The Copenhagen interpretation rules that say the wave function collapses at the time of measurement so there must be action at a distance entanglement or the positron must know more than it s supposed to know hidden variables Here is the paradox summed up It is one thing to say that physical measurement of the first particle s momentum affects uncertainty in its own position but to say that measuring the first particle s momentum affects the uncertainty in the position of the other is another thing altogether Einstein Podolsky and Rosen asked how can the second particle know to have precisely defined momentum but uncertain position Since this implies that one particle is communicating with the other instantaneously across space i e faster than light this is the paradox Incidentally Bell used spin as his example but many types of physical quantities—referred to as observables in quantum mechanics—can be used The EPR paper used momentum for the observable Experimental realisations of the EPR scenario often use photon polarization because polarized photons are easy to prepare and measure Locality in the EPR experiment edit The principle of locality states that physical processes occurring at one place should have no immediate effect on the elements of reality at another location At first sight this appears to be a reasonable assumption to make as it seems to be a consequence of special relativity which states that information can never be transmitted faster than the speed of light without violating causality It is generally believed that any theory which violates causality would also be internally inconsistent and thus useless – It turns out that the usual rules for combining quantum mechanical and classical descriptions violate the principle of locality without violating causality – Causality is preserved because there is no way for Alice to transmit messages i e information to Bob by manipulating her measurement axis Whichever axis she uses she has a probability of obtaining and probability of obtaining completely at random according to quantum mechanics it is fundamentally impossible for her to influence what result she gets Furthermore Bob is only able to perform his measurement once there is a fundamental property of quantum mechanics known as the no cloning theorem which makes it impossible for him to make a million copies of the electron he receives perform a spin measurement on each and look at the statistical distribution of the results Therefore in the one measurement he is allowed to make there is a probability of getting and of getting regardless of whether or not his axis is aligned with Alice s However the principle of locality appeals powerfully to physical intuition and Einstein Podolsky and Rosen were unwilling to abandon it Einstein derided the quantum mechanical predictions as spooky action at a distance The conclusion they drew was that quantum mechanics is not a complete theory In recent years however doubt has been cast on EPR s conclusion due to developments in understanding locality and especially quantum decoherence The word locality has several different meanings in physics For example in quantum field theory locality means that quantum fields at different points of space do not interact with one another However quantum field theories that are local in this sense appear to violate the principle of locality as defined by EPR but they nevertheless do not violate locality in a more general sense Wavefunction collapse can be viewed as an epiphenomenon of quantum decoherence which in turn is nothing more than an effect of the underlying local time evolution of the wavefunction of a system and all of its environment Since the underlying behaviour doesn t violate local causality it follows that neither does the additional effect of wavefunction collapse whether real or apparent Therefore as outlined in the example above neither the EPR experiment nor any quantum experiment demonstrates that faster than light signaling is possible Resolving the paradox edit Hidden variables edit There are several ways to resolve the EPR paradox The one suggested by EPR is that quantum mechanics despite its success in a wide variety of experimental scenarios is actually an incomplete theory In other words there is some yet undiscovered theory of nature to which quantum mechanics acts as a kind of statistical approximation albeit an exceedingly successful one Unlike quantum mechanics the more complete theory contains variables corresponding to all the elements of reality There must be some unknown mechanism acting on these variables to give rise to the observed effects of non commuting quantum observables i e the Heisenberg uncertainty principle Such a theory is called a hidden variable theory – To illustrate this idea we can formulate a very simple hidden variable theory for the above thought experiment One supposes that the quantum spin singlet states emitted by the source are actually approximate descriptions for true physical states possessing definite values for the z spin and x spin In these true states the positron going to Bob always has spin values opposite to the electron going to Alice but the values are otherwise completely random For example the first pair emitted by the source might be z x to Alice and z x to Bob the next pair z x to Alice and z x to Bob and so forth Therefore if Bob s measurement axis is aligned with Alice s he will necessarily get the opposite of whatever Alice gets otherwise he will get and with equal probability – Assuming we restrict our measurements to the z and x axes such a hidden variable theory is experimentally indistinguishable from quantum mechanics In reality there may be an infinite number of axes along which Alice and Bob can perform their measurements so there would have to be an infinite number of independent hidden variables However this is not a serious problem we have formulated a very simplistic hidden variable theory and a more sophisticated theory might be able to patch it up It turns out that there is a much more serious challenge to the idea of hidden variables Bell s inequality edit Main article Bell s theorem In John Bell showed that the predictions of quantum mechanics in the EPR thought experiment are significantly different from the predictions of a particular class of hidden variable theories the local hidden variable theories Roughly speaking quantum mechanics has a much stronger statistical correlation with measurement results performed on different axes than do these hidden variable theories These differences expressed using inequality relations known as Bell s inequalities are in principle experimentally detectable Later work by Eberhard showed that the key properties of local hidden variable theories which lead to Bell s inequalities are locality and counter factual definiteness Any theory in which these principles apply produces the inequalities Arthur Fine subsequently showed that any theory satisfying the inequalities can be modeled by a local hidden variable theory After the publication of Bell s paper a variety of experiments to test Bell s inequalities were devised These generally relied on measurement of photon polarization All experiments conducted to date have found behavior in line with the predictions of standard quantum mechanics theory However Bell s theorem does not apply to all possible philosophically realist theories It is a common misconception that quantum mechanics is inconsistent with all notions of philosophical realism Realist interpretations of quantum mechanics are possible although as discussed above such interpretations must reject either locality or counter factual definiteness Mainstream physics prefers to keep locality while striving also to maintain a notion of realism that nevertheless rejects counter factual definiteness Examples of such mainstream realist interpretations are the consistent histories interpretation and the transactional interpretation first proposed by John G Cramer in Fine s work showed that taking locality as a given there exist scenarios in which two statistical variables are correlated in a manner inconsistent with counter factual definiteness and that such scenarios are no more mysterious than any other despite the fact that the inconsistency with counter factual definiteness may seem counter intuitive Violation of locality is difficult to reconcile with special relativity and is thought to be incompatible with the principle of causality However the Bohm interpretation of quantum mechanics keeps counter factual definiteness while introducing a conjectured non local mechanism in the form of the quantum potential that is defined as one of the terms of the Schrödinger equation Some workers in the field have also attempted to formulate hidden variable theories that exploit loopholes in actual experiments such as the assumptions made in interpreting experimental data although no theory has been proposed that can reproduce all the results of quantum mechanics Alternatives are still possible A recent review article based on the Wheeler Feynman time symmetric theory rewrites the entire theory in terms of retared Liénard Wiechert potentials only which becomes manifestly causal and establishes a conservation law for total generalized momenta held instantaneously for any closed system The outcome results in correlation between particles from a handshake principle based on a variational principle applied to a system as a whole an idea with a slightly non local feature but the theory is nonetheless in agreement with the essential results of quantum electrodynamics and relativistic quantum chemistry There are also individual EPR like experiments that have no local hidden variables explanation Examples have been suggested by David Bohm and by Lucien Hardy Einstein s hope for a purely algebraic theory edit The Bohm interpretation of quantum mechanics hypothesizes that the state of the universe evolves smoothly through time with no collapsing of quantum wavefunctions One problem for the Copenhagen interpretation is to precisely define wavefunction collapse Einstein maintained that quantum mechanics is physically incomplete and logically unsatisfactory In The Meaning of Relativity Einstein wrote One can give good reasons why reality cannot at all be represented by a continuous field From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers quantum numbers This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory for the representation of reality But nobody knows how to find the basis for such a theory If time space and energy are secondary features derived from a substrate below the Planck scale then Einstein s hypothetical algebraic system might resolve the EPR paradox although Bell s theorem would still be valid If physical reality is totally finite then the Copenhagen interpretation might be an approximation to an information processing system below the Planck scale Acceptable theories and the experiment edit According to the present view of the situation quantum mechanics flatly contradicts Einstein s philosophical postulate that any acceptable physical theory must fulfill local realism In the EPR paper the authors realised that quantum mechanics was inconsistent with their assumptions but Einstein nevertheless thought that quantum mechanics might simply be augmented by hidden variables i e variables which were at that point still obscure to him without any other change to achieve an acceptable theory He pursued these ideas for over twenty years until the end of his life in In contrast John Bell in his paper showed that quantum mechanics and the class of hidden variable theories Einstein favored would lead to different experimental results different by a factor of for certain correlations So the issue of acceptability up to that time mainly concerning theory finally became experimentally decidable There are many Bell test experiments e g those of Alain Aspect and others They support the predictions of quantum mechanics rather than the class of hidden variable theories supported by Einstein Implications for quantum mechanics edit Most physicists today believe that quantum mechanics is correct and that the EPR paradox is a paradox only because classical intuitions do not correspond to physical reality How EPR is interpreted regarding locality depends on the interpretation of quantum mechanics one uses In the Copenhagen interpretation it is usually understood that instantaneous wave function collapse does occur However the view that there is no causal instantaneous effect has also been proposed within the Copenhagen interpretation in this alternate view measurement affects our ability to define and measure quantities in the physical system not the system itself In the many worlds interpretation locality is strictly preserved since the effects of operations such as measurement affect only the state of the particle that is measured However the results of the measurement are not unique—every possible result is obtained The EPR paradox has deepened our understanding of quantum mechanics by exposing the fundamentally non classical characteristics of the measurement process Before the publication of the EPR paper a measurement was often visualized as a physical disturbance inflicted directly upon the measured system For instance when measuring the position of an electron one imagines shining a light on it thus disturbing the electron and producing the quantum mechanical uncertainties in its position Such explanations which are still encountered in popular expositions of quantum mechanics are debunked by the EPR paradox which shows that a measurement can be performed on a particle without disturbing it directly by performing a measurement on a distant entangled particle In fact Yakir Aharonov and his collaborators have developed a whole theory of so called Weak measurement – Technologies relying on quantum entanglement are now being developed In quantum cryptography entangled particles are used to transmit signals that cannot be eavesdropped upon without leaving a trace In quantum computation entangled quantum states are used to perform computations in parallel which may allow certain calculations to be performed much more quickly than they ever could be with classical computers – Mathematical formulation edit The above discussion can be expressed mathematically using the quantum mechanical formulation of spin The spin degree of freedom for an electron is associated with a two dimensional complex vector space V with each quantum state corresponding to a vector in that space The operators corresponding to the spin along the x y and z direction denoted Sx Sy and Sz respectively can be represented using the Pauli matrices where is the reduced Planck constant or the Planck constant divided by p The eigenstates of Sz are represented as and the eigenstates of Sx are represented as The vector space of the electron positron pair is the tensor product of the electron s and positron s vector spaces The spin singlet state is where the two terms on the right hand side are what we have referred to as state I and state II above From the above equations it can be shown that the spin singlet can also be written as where the terms on the right hand side are what we have referred to as state Ia and state IIa To illustrate how this leads to the violation of local realism we need to show that after Alice s measurement of Sz or Sx Bob s value of Sz or Sx is uniquely determined and therefore corresponds to an element of physical reality This follows from the principles of measurement in quantum mechanics When Sz is measured the system state collapses into an eigenvector of Sz If the measurement result is z this means that immediately after measurement the system state undergoes an orthogonal projection of onto the space of states of the form For the spin singlet the new state is Similarly if Alice s measurement result is z the system undergoes an orthogonal projection onto which means that the new state is This implies that the measurement for Sz for Bob s positron is now determined It will be z in the first case or z in the second case It remains only to show that Sx and Sz cannot simultaneously possess definite values in quantum mechanics One may show in a straightforward manner that no possible vector can be an eigenvector of both matrices More generally one may use the fact that the operators do not commute along with the Heisenberg uncertainty relation See also edit Bell test experiments Bell state Bell s theorem Bra–ket notation CHSH Bell test Coherence physics Correlation does not imply causation Counter factual definiteness Ghirardi–Rimini–Weber theory GHZ experiment Interpretations of quantum mechanics Local hidden variable theory Many worlds interpretation Measurement in quantum mechanics The Bell states are a concept in quantum information science and represent the most simple examples of entanglement They are named after John S Bell because they are the subject of his famous Bell inequality An EPR pair is a pair of qubits or quantum bits which are in a Bell state together that is entangled with each other Unlike classical phenomena such as the nuclear electromagnetic and gravitational fields entanglement is invariant under distance of separation dubious – discuss and is not subject to relativistic limitations such as the speed of light though the no communication theorem prevents this behaviour being used to transmit information faster than light which would violate causality Contents The Bell states Bell state measurement Bell state correlations See also References Notes The Bell states edit The Bell states are four specific maximally entangled quantum states of two qubits The degree to which a state in a quantum system consisting of two particles is entangled is measured by the Von Neumann entropy of either of the two reduced density operators of the state The Von Neumann entropy of a pure state is zero also for the bell states which are specific pure states But the Von Neumann entropy of the reduced density operator of the Bell states is maximal The qubits are usually thought to be spatially separated Nevertheless they exhibit perfect correlation which cannot be explained without quantum mechanics In order to explain this it is important to first look at the Bell state This expression means the following The qubit held by Alice subscript A can be as well as If Alice measured her qubit in the standard basis the outcome would be perfectly random either possibility having probability But if Bob then measured his qubit the outcome would be the same as the one Alice got So if Bob measured he would also get a random outcome on first sight but if Alice and Bob communicated they would find out that although the outcomes seemed random they are correlated So far this is nothing special maybe the two particles agreed in advance when the pair was created before the qubits were separated which outcome they would show in case of a measurement Hence followed Einstein Podolsky and Rosen in in their famous EPR paper there is something missing in the description of the qubit pair given above—namely this agreement called more formally a hidden variable But quantum mechanics allows qubits to be in quantum superposition—i e in and simultaneously—that is a linear combination of the two classical states—for example the states or If Alice and Bob chose to measure in this basis i e check whether their qubit were or they would find the same correlations as above That is because the Bell state can be formally rewritten as follows Note that this is still the same state In his famous paper of John S Bell showed by simple probability theory arguments that these correlations the one for the basis and the one for the basis cannot both be made perfect by the use of any pre agreement stored in

some hidden variables—but that quantum mechanics predicts perfect correlations In a more formal and refined formulation known as the Bell CHSH inequality it is shown that a certain correlation measure cannot exceed the value if one assumes that physics respects the constraints of local hidden variable theory a sort of common sense formulation of how information is conveyed but certain systems permitted in quantum mechanics can attain values as high as Four specific two qubit states with the maximal value of are designated as Bell states They are known as the four maximally entangled two qubit Bell states and they form a convenient basis of the two qubit Hilbert space Bell test experiments or Bell s inequality experiments are designed to demonstrate the real world existence of certain theoretical consequences of the phenomenon of entanglement in quantum mechanics which could not possibly occur according to a classical picture of the world characterised by the notion of local realism Under local realism correlations between outcomes of different measurements performed on separated physical systems have to satisfy certain constraints called Bell inequalities John Bell derived the first inequality of this kind in his paper On the Einstein Podolsky Rosen Paradox Bell s Theorem states that the predictions of quantum mechanics cannot be reproduced by any local hidden variable theory The term Bell inequality can mean any one of a number of inequalities satisfied by local hidden variables theories in practice in present day experiments most often the CHSH earlier the CH inequality All these inequalities like the original inequality of Bell by assuming local realism place restrictions on the statistical results of experiments on sets of particles that have taken part in an interaction and then separated A Bell test experiment is one designed to test whether or not the real world satisfies local realism Contents Conduct of optical Bell test experiments A typical CHSH two channel experiment A typical CH single channel experiment Experimental assumptions Notable experiments Freedman and Clauser Aspect Tittel and the Geneva group Weihs experiment under strict Einstein locality conditions Pan et al s experiment on the GHZ state Rowe et al are the first to close the detection loophole Gröblacher et al test of Leggett type non local realist theories Salart et al Separation in a Bell Test Ansmann et al Overcoming the detection loophole in solid state Giustina et al Larsson et al Overcoming the detection loophole for photons Christensen et al Overcoming the detection loophole for photons Hensen et al A loophole free Bell test Loopholes See also References Further reading Conduct of optical Bell test experiments edit In practice most actual experiments have used light assumed to be emitted in the form of particle like photons produced by atomic cascade or spontaneous parametric down conversion rather than the atoms that Bell originally had in mind The property of interest is in the best known experiments the polarisation direction though other properties can be used Such experiments fall into two classes depending on whether the analysers used have one or two output channels A typical CHSH two channel experiment edit Scheme of a two channel Bell test The source S produces pairs of photons sent in opposite directions Each photon encounters a two channel polariser whose orientation can be set by the experimenter Emerging signals from each channel are detected and coincidences counted by the coincidence monitor CM The diagram shows a typical optical experiment of the two channel kind for which Alain Aspect set a precedent in Coincidences simultaneous detections are recorded the results being categorised as or and corresponding counts accumulated Four separate subexperiments are conducted corresponding to the four terms E a b in the test statistic S equation shown below The settings a a b and b are generally in practice chosen to be ° ° and ° respectively — the Bell test angles — these being the ones for which the quantum mechanical formula gives the greatest violation of the inequality For each selected value of a and b the numbers of coincidences in each category N N N and N are recorded The experimental estimate for E a b is then calculated as E N N N N N N N N Once all four E s have been estimated an experimental estimate of the test statistic S E a b E a b E a b E a b can be found If S is numerically greater than it has infringed the CHSH inequality The experiment is declared to have supported the QM prediction and ruled out all local hidden variable theories A strong assumption has had to be made however to justify use of expression It has been assumed that the sample of detected pairs is representative of the pairs emitted by the source That this assumption may not be true comprises the fair sampling loophole The derivation of the inequality is given in the CHSH Bell test page A typical CH single channel experiment edit Setup for a single channel Bell test The source S produces pairs of photons sent in opposite directions Each photon encounters a single channel e g pile of plates polariser whose orientation can be set by the experimenter Emerging signals are detected and coincidences counted by the coincidence monitor CM Prior to all actual Bell tests used single channel polarisers and variations on an inequality designed for this setup The latter is described in Clauser Horne Shimony and Holt s much cited article as being the one suitable for practical use As with the CHSH test there are four subexperiments in which each polariser takes one of two possible settings but in addition there are other subexperiments in which one or other polariser or both are absent Counts are taken as before and used to estimate the test statistic S N a b N a b N a b N a b N a N b N where the symbol indicates absence of a polariser If S exceeds then the experiment is declared to have infringed Bell s inequality and hence to have refuted local realism In order to derive CHSH in their paper had to make an extra assumption the so called fair sampling assumption This means that the probability of detection of a given photon once it has passed the polarizer is independent of the polarizer setting including the absence setting If this assumption were violated then in principle an local hidden variable LHV model could violate the CHSH inequality In a later article Clauser and Horne replaced this assumption by a much weaker no enhancement assumption deriving a modified inequality see the page on Clauser and Horne s Bell test Experimental assumptions edit In addition to the theoretical assumptions made there are practical ones There may for example be a number of accidental coincidences in addition to those of interest It is assumed that no bias is introduced by subtracting their estimated number before calculating S but that this is true is not considered by some to be obvious There may be synchronisation problems — ambiguity in recognising pairs because in practice they will not be detected at exactly the same time Nevertheless despite all these deficiencies of the actual experiments one striking fact emerges the results are to a very good approximation what quantum mechanics predicts If imperfect experiments give us such excellent overlap with quantum predictions most working quantum physicists would agree with John Bell in expecting that when a perfect Bell test is done the Bell inequalities will still be violated This attitude has led to the emergence of a new sub field of physics which is now known as quantum information theory One of the main achievements of this new branch of physics is showing that violation of Bell s inequalities leads to the possibility of a secure information transfer which utilizes the so called quantum cryptography involving entangled states of pairs of particles Notable experiments edit Over the past thirty or so years a great number of Bell test experiments have now been conducted These experiments are subject to assumptions in particular the no enhancement hypothesis of Clauser and Horne above The experiments are commonly interpreted to rule out local hidden variable theories though so far no experiment has been performed which is not subject to either the locality loophole or the detection loophole An experiment free of the locality loophole is one where for each separate measurement and in each wing of the experiment a new setting is chosen and the measurement completed before signals could communicate the settings from one wing of the experiment to the other An experiment free of the detection loophole is one where close to of the successful measurement outcomes in one wing of the experiment are paired with a successful measurement in the other wing This percentage is called the efficiency of the experiment Advancements in technology have led to significant improvement in efficiencies as well as a greater variety of methods to test the Bell Theorem The challenge is to combine high efficiency with rapid generation of measurement settings and completion of measurements Some of the best known and recent experiments include Freedman and Clauser edit This was the first actual Bell test using Freedman s inequality a variant on the CH inequality Aspect edit Alain Aspect and his team at Orsay Paris conducted three Bell tests using calcium cascade sources The first and last used the CH inequality The second was the first application of the CHSH inequality The third and most famous was arranged such that the choice between the two settings on each side was made during the flight of the photons as originally suggested by John Bell Tittel and the Geneva group edit The Geneva Bell test experiments showed that distance did not destroy the entanglement Light was sent in fibre optic cables over distances of several kilometers before it was analysed As with almost all Bell tests since about a parametric down conversion PDC source was used Weihs experiment under strict Einstein locality conditions edit In Gregor Weihs and a team at Innsbruck led by Anton Zeilinger conducted an ingenious experiment that closed the locality loophole improving on Aspect s of The choice of detector was made using a quantum process to ensure that it was random This test violated the CHSH inequality by over standard deviations the coincidence curves agreeing with those predicted by quantum theory Pan et al s In Bell test experiments there may be problems of experimental design or set up that affect the validity of the experimental findings These problems are often referred to as loopholes See the article on Bell s theorem for the theoretical background to these experimental efforts see also J S Bell The purpose of the experiment is to test whether nature is best described using a local hidden variable theory or by the quantum entanglement theory of quantum mechanics The detection efficiency or fair sampling problem is the most prevalent loophole in optical experiments Another loophole that has more often been addressed is that of communication i e locality There is also the disjoint measurement loophole which entails multiple samples used to obtain correlations as compared to joint measurement where a single sample is used to obtain all correlations used in an inequality To date no test has simultaneously closed all loopholes Ronald Hanson of Delft University of Technology claims the first Bell experiment that closes both the detection and the communication loopholes This was not an optical experiment in the sense discussed below the entangled degrees of freedom were electron spins rather than photon polarization Nevertheless correlations of classical optical fields also violate Bell s inequality In some experiments there may be additional defects that make local realist explanations of Bell test violations possible these are briefly described below Many modern experiments are directed at detecting quantum entanglement rather than ruling out local hidden variable theories and these tasks are different since the former accepts quantum mechanics at the outset no entanglement without quantum mechanics This is regularly done using Bell s theorem but in this situation the theorem is used as an entanglement witness a dividing line between entangled quantum states and separable quantum states and is as such not as sensitive to the problems described here In October scientists from the Kavli Institute of Nanoscience reported that the quantum entanglement phenomenon is strongly supported based on a loophole free Bell test study Contents Loopholes Detection efficiency or fair sampling Fair sampling assumption Double detections Communication or locality Disjoint sampling Failure of rotational invariance Coincidence loophole Memory loophole Sources of error in optical Bell test experiments Example of typical experiment Sources of error in the light source Sources of error in the optical polarizer Sources of error in the detector or detector settings Free choice of detector orientations References Notes Sources Loopholes edit Detection efficiency or fair sampling edit In Bell test experiments one problem is that detection efficiency may be less than and this is always the case in optical experiments This problem was noted first by Pearle in and Clauser and Horne devised another result intended to take care of this Some results were also obtained in the s but the subject has undergone significant research in recent years The many experiments affected by this problem deal with it without exception by using the fair sampling assumption see below This loophole changes the inequalities to be used for example the CHSH inequality is changed When data from an experiment is used in the inequality one needs to condition on that a coincidence occurred that a detection occurred in both wings of the experiment This will change the inequality into In this formula the denotes the efficiency of the experiment formally the minimum probability of a coincidence given a detection on one side In Quantum mechanics the left hand side reaches which is greater than two but for a non efficiency the latter formula has a larger right hand side And at low efficiency below ˜ the inequality is no longer violated All optical experiments are affected by this problem having typical efficiencies around Several non optical systems such as trapped ions superconducting qubits and NV centers have been able to bypass the detection loophole Unfortunately they are all still vulnerable to the communication loophole There are tests that are not sensitive to this problem such as the Clauser Horne test but these have the same performance as the latter of the two inequalities above they cannot be violated unless the efficiency exceeds a certain bound For example in the Clauser Horne test the bound is ˜ Eberhard X Larsson Fair sampling assumption edit Usually the fair sampling assumption alternatively the no enhancement assumption is used in regard to this loophole It states that the sample of detected pairs is representative of the pairs emitted in which case the right hand side in the equation above is reduced to irrespective of the efficiency This comprises a third postulate necessary for violation in low efficiency experiments in addition to the two postulates of local realism There is no way to test experimentally whether a given experiment does fair sampling as the number of emitted but undetected pairs is by definition unknown citation needed Double detections edit In many experiments the electronics are such that simultaneous and – counts from both outputs of a polariser can never occur only one or the other being recorded Under quantum mechanics they will not occur anyway but under a wave theory the suppression of these counts will cause even the basic realist prediction to yield unfair sampling However the effect is negligible if the detection efficiencies are low citation needed Communication or locality edit The Bell inequality is motivated by the absence of communication between the two measurement sites In experiments this is usually ensured simply by prohibiting any light speed communication by separating the two sites and then ensuring that the measurement duration is shorter than the time it would take for any light speed signal from one site to the other or indeed to the source In one of Alain Aspect s experiments inter detector communication at light speed during the time between pair emission and detection was possible but such communication between the time of fixing the detectors settings and the time of detection was not An experimental set up without any such provision effectively becomes entirely local and therefore cannot rule out local realism Additionally the experiment design will ideally be such that the settings for each measurement are not determined by any earlier event at both measurement stations John Bell supported Aspect s investigation of it p and had some active involvement with the work being on the examining board for Aspect s PhD Aspect improved the separation of the sites and did the first attempt on really having independent random detector orientations Weihs et al improved on this with a distance on the order of a few hundred meters in their experiment in addition to using random settings retrieved from a quantum system Scheidl et al improved on this further by conducting an experiment between locations separated by a distance of km Disjoint sampling edit John Bell assumed observations are obtained with a common hidden variable lambda However particle experiments violate that assumption To estimate the correlation when the two measurement devices have parameters a and b a sample of observations is taken To estimate the correlation when the devices have parameters a and c a second sample is taken To estimate the correlation when the devices have parameters b and c a third sample is taken Those three correlations are then used in Bell s original inequality and found to violate that inequality But the statistics of sequential disjoint samples are different from the statistics of a single joint sample where all of the parameters a b and c are set once and not changed That condition can only be met if there are particles not Bell s parameter inequality holds without ambiguity for particles measured jointly particle joint correlations inserted into Bell s inequality will not violate the inequality Using disjoint correlations in joint inequalities is claimed to be the cause of inequality violation citation needed Failure of rotational invariance edit The source is said to be rotationally invariant if all possible hidden variable values describing the states of the emitted pairs are equally likely The general form of a Bell test does not assume rotational invariance but a number of experiments have been analysed using a simplified formula that depends upon it It is possible that there has not always been adequate testing to justify this Even where as is usually the case the actual test applied is general if the hidden variables are not rotationally invariant this can result in misleading descriptions of the results Graphs may be presented for example of coincidence rate against the difference between the settings a and b but if a more comprehensive set of experiments had been done it might have become clear that the rate depended on a and b separately Cases in point may be Weihs experiment Weihs presented as having closed the locality loophole and Kwiat s demonstration of entanglement using an ultrabright photon source Kwiat Coincidence loophole edit In many experiments especially those based on photon polarization pairs of events in the two wings of the experiment are only identified as belonging to a single pair after the experiment is performed by judging whether or not their detection times are close enough to one another This generates a new possibility for a local hidden variables theory to fake quantum correlations delay the detection time of each of the two particles by a larger or smaller amount depending on some relationship between hidden variables carried by the particles and the detector settings encountered at the measurement station This loophole was noted by A Fine in and by S Pascazio in and by J Larsson and R D Gill in It turns out to be more serious that the detection loophole in that it gives more room for local hidden variables to reproduce quantum correlations for the same effective experimental efficiency the chance that particle is accepted coincidence loophole or measured detection loophole given that particle is detected The coincidence loophole can be ruled out entirely simply by working with a pre fixed lattice of detection windows which are short enough that most pairs of events occurring in the same window do originate with the same emission and long enough that a true pair is not separated by a window boundary Memory loophole edit In most experiments measurements are repeatedly made at the same two locations Under local realism there could be effects of memory leading to statistical dependence between subsequent pairs of measurements Moreover physical parameters might be varying in time It has been shown that provided each new pair of measurements is done with a new random pair of measurement settings that neither memory nor time inhomogeneity cannot have a serious effect on the experiment Sources of error in optical Bell test experiments edit This section does not cite any references sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed August In the case of Bell test experiments if there are sources of error that are not accounted for by the experimentalists that might be of enough importance to explain why a particular experiment gives results in favor of quantum entanglement as opposed to local realism they are called loopholes Here some examples of existing and hypothetical experimental errors are explained There are of course sources of error in all physical experiments Whether or not any of those presented here have been found important enough to be called loopholes in general or because of possible mistakes by the performers of some known experiment found in literature is discussed in the subsequent sections There are also non optical Bell test experiments which are not discussed here citation needed Example of typical experiment edit For the Unix command see chsh In physics the CHSH inequality can be used in the proof of Bell s theorem which states that certain consequences of entanglement in quantum mechanics cannot be reproduced by local hidden variable theories Experimental verification of violation of the inequalities is seen as experimental confirmation that nature cannot be described by local hidden variables theories CHSH stands for John Clauser Michael Horne Abner Shimony and Richard Holt who described it in a much cited paper published in Clauser et al They derived the CHSH inequality which as with John Bell s original inequality Bell is a constraint on the statistics of coincidences in a Bell test experiment which is necessarily true if there exist underlying local hidden variables local realism This constraint can on the other hand be infringed by quantum mechanics Contents Statement of the inequality A typical CHSH experiment Derivation of the CHSH inequality Bell s derivation Derivation from Clauser and Horne s inequality Experiments using the CHSH test See also References Statement of the inequality edit The usual form of the CHSH inequality is where a and a are detector settings on side A b and b on side B the four combinations being tested in separate subexperiments The terms E a b etc are the quantum correlations of the particle pairs where the quantum correlation is defined to be the expectation value of the product of the outcomes of the experiment i e the statistical average of A a B b where A and B are the separate outcomes using the coding for the channel and for the channel Clauser et al s derivation was oriented towards the use of two channel detectors and indeed it is for these that it is generally used but under their method the only possible outcomes were and In order to adapt it to real situations which at the time meant the use of polarised light and single channel polarisers they had to interpret as meaning non detection in the channel i e either or nothing They did not in the original article discuss how the two channel inequality could be applied in real experiments with real imperfect detectors though it was later proved Bell that the inequality itself was equally valid The occurrence of zero outcomes though means it is no longer so obvious how the values of E are to be estimated from the experimental data The mathematical formalism of quantum mechanics predicts a maximum value for S of v which is greater than and CHSH violations are therefore predicted by the theory of quantum mechanics A typical CHSH experiment edit Schematic of a two channel Bell test The source S produces pairs of photons sent in opposite directions Each photon encounters a two channel polariser whose orientation can be set by the experimenter Emerging signals from each channel are detected and coincidences counted by the coincidence monitor CM In practice most actual experiments have used light rather than the electrons that Bell originally had in mind The property of interest is in the best known experiments Aspect the polarisation direction though other properties can be used The diagram shows a typical optical experiment Coincidences simultaneous detections are recorded the results being categorised as or and