qubits may designed differently depending full quantum computer 's computing model whether quantum logic gates quantum annealing adiabatic quantum computation currently number significant obstacles constructing useful quantum computers particularly difficult maintain qubits' quantum states suffer quantum decoherence state fidelity quantum computers therefore require error correction computational problem solved classical computer also solved quantum computer ref name nielsen p nielsen p ref conversely problem solved quantum computer also


solved classical computer least principle given enough time words quantum computers obey church turing thesis means quantum computers provide additional advantages classical computers terms computability quantum algorithms certain problems significantly lower time complexities corresponding known classical algorithms notably quantum computers believed able quickly solve certain problems classical computer could solve ''feasible'' amount time feat known quantum supremacy study computational complexity problems respect quantum computers known quantum

Formal science
Computer science
Quantum computing


complexity theory quantum circuit bloch sphere representation qubit fundamental building block quantum computers definition prevailing model quantum computation describes computation terms network quantum logic gates ref name nielsen chuang ref model thought abstract linear algebraic generalization classical circuit since circuit model obeys quantum mechanics quantum computer capable efficiently running circuits believed physically realizable memory consisting math display inline n math bits information math display inline n math possible states vector representing memory


states thus math display inline n math entries one state vector viewed ''probability vector'' represents fact memory found particular state classical view one entry would value ie probability state entries would zero quantum mechanics probability vectors generalized density operators quantum state vector formalism usually introduced first conceptually simpler used instead density matrix formalism ''for pure


states'' whole quantum system known begin considering simple memory consisting one bit memory may found one two states zero state one state may represent state memory using dirac notation math display block rangle begin pmatrix end pmatrix quad rangle begin pmatrix end pmatrix math quantum memory may found quantum superposition math display inline psi rangle math two classical states math display inline rangle math math display inline rangle math math display block psi rangle alpha rangle beta rangle begin pmatrix alpha


beta end pmatrix quad alpha beta math coefficients math display inline alpha math math display inline beta math complex numbers one qubit information said encoded quantum memory state math display inline psi rangle math probability vector connected probability vector via measurement operation quantum memory measured determine whether state math display inline rangle math math display inline rangle math known computational basis measurement zero state would observed probability math display inline alpha math one state probability math display inline beta math numbers math


display inline alpha math math display inline beta math called quantum amplitudes state one qubit quantum memory manipulated applying quantum logic gates analogous classical memory manipulated classical logic gates one important gate classical quantum computation gate represented matrix math display block x begin pmatrix end pmatrix math mathematically application logic gate quantum state vector modelled matrix multiplication thus math display inline x rangle rangle math math display inline x rangle rangle math mathematics single qubit


gates extended operate multi qubit quantum memories two important ways one way simply select qubit apply gate target qubit whilst leaving remainder memory unaffected another way apply gate target another part memory desired state two choices illustrated using another example possible states two qubit quantum memory math display block rangle begin pmatrix end pmatrix quad rangle begin pmatrix


words cnot applies gate math display inline x math second qubit first qubit state math display inline rangle math first qubit math display inline rangle math nothing done either qubit summary quantum computation described network quantum logic gates measurements however measurement deferred end quantum computation though deferment may come computational cost quantum circuits depict network consisting quantum logic gates measurements quantum computation


formalism unitary matrix math n math qubits represented network quantum logic gates fairly small family gates choice gate family enables construction known universal gate set since computer run circuits universal quantum computer one common set includes single qubit gates well cnot gate means quantum computation performed executing sequence single qubit gates together cnot gates though gate set infinite


replaced finite gate set appealing solovay kitaev theorem quantum algorithms progress finding quantum algorithms typically focuses quantum circuit model though exceptions like quantum adiabatic algorithm exist quantum algorithms roughly categorized type speedup achieved corresponding classical algorithms ref name zoo http mathnistgov quantum zoo quantum algorithm zoo stephen jordan 's homepage ref quantum algorithms offer polynomial speedup best known classical algorithm include shor 's algorithm factoring related quantum algorithms computing discrete logarithms solving pell 's equation generally solving hidden subgroup


problem abelian finite groups ref name zoo algorithms depend primitive quantum fourier transform mathematical proof found shows equally fast classical algorithm cannot discovered although considered unlikely certain oracle problems like simon 's problem bernstein vazirani problem give provable speedups though quantum query model restricted model lower bounds much easier prove n't necessarily translate speedups practical problems problems including simulation quantum physical processes chemistry solid state physics approximation


certain jones polynomials quantum algorithm linear systems equations quantum algorithms appearing give super polynomial speedups bqp complete problems bqp complete equally fast classical algorithm would imply ''no quantum algorithm'' gives super polynomial speedup believed unlikely ref name nielsen p nielsen p ref quantum algorithms like grover 's algorithm amplitude amplification give polynomial speedups corresponding classical algorithms ref name zoo though algorithms give comparably modest quadratic speedup widely applicable thus give speedups wide range problems many examples provable quantum


eg products two digit primes comparison quantum computer could efficiently solve problem using shor 's algorithm find factors ability would allow quantum computer break many cryptographic systems use today sense would polynomial time number digits integer algorithm solving problem particular popular public key ciphers based difficulty factoring integers discrete logarithm problem solved shor 's algorithm particular rsa diffie hellman


elliptic curve diffie hellman algorithms could broken used protect secure web pages encrypted email many types data breaking would significant ramifications electronic privacy security identifying cryptographic systems may secure quantum algorithms actively researched topic field ''post quantum cryptography'' ref name pqcrypto survey ref public key algorithms based problems integer factorization discrete logarithm problems shor 's algorithm applies like mceliece cryptosystem based problem coding theory ref name pqcrypto survey lattice based cryptosystems also known broken


maximal possible probability finding desired element number oracle lookups problems addressed grover 's algorithm following properties searchable structure collection possible answers number possible answers check number inputs algorithm exists boolean function evaluates input determines whether correct answer problems properties running time grover 's algorithm quantum computer scales square root number inputs


elements database opposed linear scaling classical algorithms general class problems grover 's algorithm applied boolean satisfiability problem ' 'database'' algorithm iterates possible answers example possible application password cracker attempts guess password symmetric ciphers triple des aes particularly vulnerable kind attack application quantum computing major interest government agencies simulation quantum systems since chemistry nanotechnology rely


understanding quantum systems systems impossible simulate efficient manner classically many believe quantum simulation one important applications quantum computing quantum simulation could also used simulate behavior atoms particles unusual conditions reactions inside collider quantum simulations might used predict future paths particles protons superposition double slit experiment annual global energy output used nitrogen fixation produce ammonia haber process agricultural fertilizer industry naturally


occurring organisms also produce ammonia quantum simulations might used understand process increasing production quantum annealing adiabatic optimization quantum annealing adiabatic quantum computation relies adiabatic theorem undertake calculations system placed ground state simple hamiltonian slowly evolved complicated hamiltonian whose ground state represents solution problem question adiabatic theorem states evolution slow enough system stay ground state times process machine learning since quantum


computers produce outputs classical computers cannot produce efficiently since quantum computation fundamentally linear algebraic express hope developing quantum algorithms speed machine learning tasks ref name preskill ref example quantum algorithm linear systems equations hhl algorithm named discoverers harrow hassidim lloyd believed provide speedup classical counterparts ref name quantum algorithm solving linear systems equations harrow et al ref ref name preskill research groups recently explored use quantum annealing hardware training boltzmann machines deep neural networks computational biology


field computational biology quantum computing played big role solving many biological problems one well known examples would computational genomics computing drastically reduced time sequence human genome given computational biology using generic data modeling storage applications computational biology expected arise well computer aided drug design generative chemistry deep generative chemistry models emerge powerful tools expedite drug discovery however immense size complexity structural space possible drug like molecules pose significant obstacles


could overcome future quantum computers quantum computers naturally good solving complex quantum many body problems thus may instrumental applications involving quantum chemistry therefore one expect quantum enhanced generative models including quantum gans may eventually developed ultimate generative chemistry algorithms hybrid architectures combining quantum computers deep classical networks quantum variational autoencoders already trained commercially available annealers used generate novel drug like molecular structures developing physical quantum computers challenges number technical challenges


building large scale quantum computer physicist david divincenzo listed requirements practical quantum computer physically scalable increase number qubits qubits initialized arbitrary values quantum gates faster decoherence time universal gate set qubits read easily sourcing parts quantum computers also difficult many quantum computers like constructed google ibm need helium nuclear research byproduct special superconducting cables made japanese company coax co control multi qubit systems requires generation coordination large


physical system used implement qubits decoherence irreversible effectively non unitary usually something highly controlled avoided decoherence times candidate systems particular transverse relaxation time ''t'' sub sub nmr mri technology also called ' 'dephasing time'' typically range nanoseconds seconds low temperature ref name divincenzo ref currently quantum computers require qubits cooled millikelvin usually using dilution refrigerator order prevent significant decoherence study argues ionizing radiation cosmic rays nevertheless cause


certain systems decohere within milliseconds result time consuming tasks may render quantum algorithms inoperable maintaining state qubits long enough duration eventually corrupt superpositions issues difficult optical approaches timescales orders magnitude shorter often cited approach overcoming optical pulse shaping error rates typically proportional ratio operating time decoherence time hence operation must completed much quickly decoherence time described quantum threshold theorem error rate small


enough thought possible use quantum error correction suppress errors decoherence allows total calculation time longer decoherence time error correction scheme correct errors faster decoherence introduces often cited figure required error rate gate fault tolerant computation sup sup assuming noise depolarizing meeting scalability condition possible wide range systems however use error correction brings cost greatly increased number required qubits number required


factor integers using shor 's algorithm still polynomial thought ''l'' ''l'' sup sup ''l'' number digits number factored error correction algorithms would inflate figure additional factor ''l'' bit number implies need sup sup bits without error correction error correction figure would rise sup sup bits computation time ''l'' sup sup sup sup steps nbsp mhz seconds different approach stability decoherence problem create topological quantum computer anyons


require computational time million years generate number samples quantum processor generate seconds ref name ref november quantum computing summit ibm presented qubit microprocessor named ibm eagle skepticism researchers expressed skepticism scalable quantum computers could ever built typically issue maintaining coherence large scales bill unruh doubted practicality quantum computers paper published back paul davies argued qubit computer would even come conflict cosmological information bound implied


physical system used realize qubits superconducting quantum computing ref name clarkewilhelm ref qubit implemented state small superconducting circuits nowiki josephson junctions trapped ion quantum computer qubit implemented internal state trapped ions neutral atoms optical lattices qubit implemented internal states neutral atoms trapped optical lattice quantum dot computer spin based eg loss divincenzo quantum computer qubit given spin states trapped electrons quantum dot computer spatial based qubit given electron position double quantum dot quantum computing using engineered quantum wells could principle enable construction quantum computers


operate room temperature coupled quantum wire qubit implemented pair quantum wires coupled quantum point contact nuclear magnetic resonance quantum computer nmrqc implemented nuclear magnetic resonance molecules solution qubits provided nuclear spins within dissolved molecule probed radio waves solid state nmr kane quantum computers qubit realized nuclear spin state phosphorus donors silicon electrons helium quantum computers qubit electron spin cavity quantum electrodynamics cqed qubit provided internal state trapped atoms coupled high finesse cavities molecular magnet qubit given


spin states fullerene based esr quantum computer qubit based electronic spin atoms molecules encased fullerenes nonlinear optical quantum computer qubits realized processing states different modes light linear nonlinear elements ref name qc ref ref name chuang ref linear optical quantum computer qubits realized processing states different modes light linear elements eg mirrors beam splitters phase shifters ref name klm ref diamond based quantum computer ref name nizovtsevetal ref ref name neumannetal ref qubit realized electronic nuclear spin nitrogen vacancy centers diamond bose einstein condensate based quantum computer transistor based quantum computer string quantum computers entrainment positive


holes using electrostatic trap rare earth metal ion doped inorganic crystal based quantum computers ref name ohlsson ref ref name longdell ref qubit realized internal electronic state dopants optical fibers metallic like carbon nanospheres based quantum computers ref name nafradi ref large number candidates demonstrates quantum computing despite rapid progress still infancy number quantum computing models distinguished basic elements computation decomposed practical implementations four relevant models computation quantum gate array computation decomposed sequence qubit quantum gates one way quantum computer computation decomposed sequence one qubit measurements


applied highly entangled initial state cluster state adiabatic quantum computer based quantum annealing computation decomposed slow continuous transformation initial hamiltonian final hamiltonian whose ground states contain solution ref name das ref topological quantum computer ref name nayaketal ref computation decomposed braiding anyons lattice quantum turing machine theoretically important physical implementation model feasible four models computation shown equivalent simulate polynomial overhead relation computability complexity


theory computability theory computational problem solvable classical computer also solvable quantum computer ref name nielsen p intuitively believed physical phenomena including operation classical computers described using quantum mechanics underlies operation quantum computers conversely problem solvable quantum computer also solvable classical computer possible simulate quantum classical computers manually paper pen given enough time formally quantum computer simulated turing machine


words quantum computers provide additional power classical computers terms computability means quantum computers cannot solve undecidable problems like halting problem existence quantum computers disprove church turing thesis quantum complexity theory quantum computers cannot solve problems classical computers cannot already solve suspected solve certain problems faster classical computers instance known quantum computers efficiently factor integers believed case classical computers class problems


powerful classical computers terms time complexity suspected relationship bqp several classical complexity classes ref name nielsen p exact relationship bqp p np pspace known however known math mathsf p subseteq bqp subseteq pspace math problems efficiently solved deterministic classical computer also efficiently solved quantum computer problems efficiently solved quantum computer also solved deterministic classical computer polynomial space resources suspected bqp


strict superset p meaning problems efficiently solvable quantum computers efficiently solvable deterministic classical computers instance integer factorization discrete logarithm problem known bqp suspected outside p relationship bqp np little known beyond fact np problems believed p also bqp integer factorization discrete logarithm problem np example suspected math mathsf np nsubseteq bqp math


believed efficiently checkable problems efficiently solvable quantum computer direct consequence belief also suspected bqp disjoint class np complete problems np complete problem bqp would follow np hardness problems np bqp ref name bernvazi ref relationship bqp basic classical complexity classes summarized follows math mathsf p subseteq bpp subseteq bqp subseteq pp subseteq pspace math also known bqp contained complexity class color blue mathsf p precisely


natural computing photonic computing post quantum cryptography quantum algorithm quantum annealing quantum bus quantum cognition quantum circuit quantum complexity theory quantum cryptography quantum logic gate quantum machine learning quantum supremacy quantum threshold theorem quantum volume rigetti computing supercomputer superposition theoretical computer science timeline quantum computing topological quantum computer valleytronics references reading textbooks academic papers table lists switching dephasing times various systems external links stanford encyclopedia philosophy http platostanfordedu entries qt quantcomp quantum computing amit hagar michael e cuffaro https quantumcountry qcvc quantum computing curious andy matuschak michael nielsen


theoretical physics '''quantum field theory''' '''qft''' theoretical framework combines classical field theory special relativity quantum mechanics ref name peskin ref qft used particle physics construct physical models subatomic particles condensed matter physics construct models quasiparticles qft treats particles excited states also called quanta underlying quantum fields fundamental particles interactions particles described interaction terms lagrangian involving corresponding quantum fields interaction visually represented feynman diagrams according perturbation theory quantum mechanics

Natural science
Physics
Mathematical physics
Quantum field theory


history quantum field theory emerged work generations theoretical physicists spanning much th century development began description interactions light electrons culminating first quantum field theory quantum electrodynamics major theoretical obstacle soon followed appearance persistence various infinities perturbative calculations problem resolved invention renormalization procedure second major barrier came qft 's apparent inability describe weak strong interactions point theorists called abandonment


field theoretic approach development gauge theory completion standard model led renaissance quantum field theory theoretical background magnetic field lines visualized using iron filings piece paper sprinkled iron filings placed bar magnet filings align according direction magnetic field forming arcs quantum field theory result combination classical field theory quantum mechanics special relativity ref name peskin brief overview theoretical precursors follows earliest successful classical field


theory one emerged newton 's law universal gravitation despite complete absence concept fields treatise ''philosophi naturalis principia mathematica'' force gravity described newton action distance effects faraway objects instantaneous matter distance exchange letters richard bentley however newton stated inconceivable inanimate brute matter without mediation something else material operate upon affect matter without mutual contact ref name hobson th century


mathematical physicists discovered convenient description gravity based fields numerical quantity vector assigned every point space indicating action gravity particle point however considered merely mathematical trick ref name weinberg ref fields began take existence development electromagnetism th century michael faraday coined english term field introduced fields properties space even devoid matter physical effects argued action distance proposed interactions objects occur


lines atomic spectra distribution blackbody radiation different wavelengths ref name weisskopf ref max planck 's study blackbody radiation marked beginning quantum mechanics treated atoms absorb emit electromagnetic radiation tiny oscillators crucial property energies take series discrete rather continuous values known quantum harmonic oscillators process restricting energies discrete values called quantization ref name heisenberg ref building idea albert einstein proposed explanation photoelectric effect light composed individual packets


energy called photons quanta light implied electromagnetic radiation waves classical electromagnetic field also exists form particles ref name weisskopf niels bohr introduced bohr model atomic structure wherein electrons within atoms take series discrete rather continuous energies another example quantization bohr model successfully explained discrete nature atomic spectral lines louis de broglie proposed hypothesis wave particle duality microscopic particles exhibit wave like particle like properties different circumstances ref name weisskopf


observers different velocities ie physical laws invariant lorentz transformations two difficulties remained observationally schr dinger equation underlying quantum mechanics could explain stimulated emission radiation atoms electron emits new photon action external electromagnetic field unable explain spontaneous emission electron spontaneously decreases energy emits photon even without action external electromagnetic field theoretically schr dinger equation could describe photons inconsistent principles special relativity treats time ordinary number


promoting spatial coordinates linear operators ref name weisskopf quantum electrodynamics quantum field theory naturally began study electromagnetic interactions electromagnetic field known classical field works born heisenberg pascual jordan quantum theory free electromagnetic field one interactions matter developed via canonical quantization treating electromagnetic field set quantum harmonic oscillators exclusion interactions however theory yet incapable making quantitative predictions real world


seminal paper ''the quantum theory emission absorption radiation'' dirac coined term quantum electrodynamics qed theory adds upon terms describing free electromagnetic field additional interaction term electric current density electromagnetic vector potential using first order perturbation theory successfully explained phenomenon spontaneous emission according uncertainty principle quantum mechanics quantum harmonic oscillators cannot remain stationary non zero minimum energy must always oscillating even lowest energy state ground state therefore even perfect vacuum


remains oscillating electromagnetic field zero point energy quantum fluctuation electromagnetic fields vacuum stimulates spontaneous emission radiation electrons atoms dirac 's theory hugely successful explaining emission absorption radiation atoms applying second order perturbation theory able account scattering photons resonance fluorescence well non relativistic compton scattering nonetheless application higher order perturbation theory plagued problematic infinities calculations ref name weisskopf dirac wrote wave equation described relativistic electrons dirac equation


following important consequences spin electron electron ''g'' factor led correct sommerfeld formula fine structure hydrogen atom could used derive klein nishina formula relativistic compton scattering although results fruitful theory also apparently implied existence negative energy states would cause atoms unstable since could always decay lower energy states emission radiation ref name weisskopf prevailing view time world composed two


different ingredients material particles electrons quantum fields photons material particles considered eternal physical state described probabilities finding particle given region space range velocities hand photons considered merely excited states underlying quantized electromagnetic field could freely created destroyed jordan eugene wigner heisenberg pauli enrico fermi discovered material particles could also seen excited states quantum fields photons


excited states quantized electromagnetic field type particle corresponding quantum field electron field proton field etc given enough energy would possible create material particles building idea fermi proposed explanation beta decay known fermi 's interaction atomic nuclei contain electrons ''per se'' process decay electron created surrounding electron field analogous photon created surrounding electromagnetic field radiative decay excited atom realized


dirac others negative energy states implied dirac equation could removed assuming existence particles mass electrons opposite electric charge ensured stability atoms also first proposal existence antimatter indeed evidence positrons discovered carl david anderson cosmic rays enough energy absorbing photon electron positron pair could created process called pair production reverse process annihilation could also occur emission


photon showed particle numbers need fixed interaction historically however positrons first thought holes infinite electron sea rather new kind particle theory referred dirac hole theory ref name weisskopf qft naturally incorporated antiparticles formalism infinities renormalization robert oppenheimer showed higher order perturbative calculations qed always resulted infinite quantities electron self energy vacuum zero point energy electron photon fields ref name weisskopf suggesting computational methods


time could properly deal interactions involving photons extremely high momenta years later systematic approach remove infinities developed series papers published ernst stueckelberg established relativistically invariant formulation qft stueckelberg also independently developed complete renormalization procedure unfortunately achievements understood recognized theoretical community ref name weisskopf faced infinities john archibald wheeler heisenberg proposed respectively supplant problematic qft called matrix


theory since specific details microscopic interactions inaccessible observations theory attempt describe relationships small number observables ''eg'' energy atom interaction rather concerned microscopic minutiae interaction richard feynman wheeler daringly suggested abandoning qft altogether proposed action distance mechanism particle interactions willis lamb robert retherford measured minute difference sup sup ''s'' sub sub sup sup ''p'' sub sub energy levels hydrogen atom also called lamb shift ignoring contribution photons


whose energy exceeds electron mass hans bethe successfully estimated numerical value lamb shift ref name weisskopf subsequently norman myles kroll lamb james bruce french victor weisskopf confirmed value using approach infinities cancelled infinities result finite quantities however method clumsy unreliable could generalized calculations ref name weisskopf breakthrough eventually came around robust method eliminating infinities developed julian schwinger richard feynman freeman dyson shinichiro tomonaga main idea replace calculated values


mass charge infinite though may finite measured values systematic computational procedure known renormalization applied arbitrary order perturbation theory ref name weisskopf tomonaga said nobel lecture blockquote since parts modified mass charge due field reactions become infinite impossible calculate theory however mass charge observed experiments original mass charge mass charge modified field reactions finite hand mass


charge appearing theory values modified field reactions since particularly since theory unable calculate modified mass charge may adopt procedure substituting experimental values phenomenologically procedure called renormalization mass charge long laborious calculations less skillful schwinger 's obtained result agreement americans' blockquote applying renormalization procedure calculations finally made explain electron 's anomalous magnetic moment deviation electron ''g'' factor vacuum polarisation


results agreed experimental measurements remarkable degree thus marking end war infinities ref name weisskopf time feynman introduced path integral formulation quantum mechanics feynman diagrams latter used visually intuitively organise help compute terms perturbative expansion diagram interpreted paths particles interaction vertex line corresponding mathematical expression product expressions gives scattering amplitude interaction represented diagram invention


fact devoted nobel speech describing work einstein talked relativity nobel speech photoelectric effect theory got award blockquote relativistic quantum theory fields born thirty five years ago paternal efforts dirac heisenberg pauli others somewhat retarded youngster however first reached adolescence seventeen years later event gathered celebrate subsequent development mature phase subject wish


discuss briefly today blockquote schwinger 's version qft fields described simple numbers described vectors infinite dimensional hilbert space field corresponding operator acts vectors hence schwinger 's name operator field theory use hilbert space leads concept field quanta blockquote two distinct classical concepts particles waves merged become transcended something classical counterpart quantized field new conception unity replaces classical duality blockquote quanta


sometimes called excitations field n't tell whole story quantum holistic unit field cannot subdivided blockquote electron quantized ripple electron quantum field acts particle travels holistically conserved quantities always sustained unit blockquote blockquote quantum nothing character 's entirely present entirely absent ca n't part photon nothing character entails must add remove entire quantum instantaneously even 's spread many kilometers ca n't alter part


quantum n't parts 's single thing blockquote despite success schwinger 's theory answering paradoxes mysteries quantum mechanics ref name brooks ref largely overlooked forgotten one reasons idea instantaneous collapse troubling many physicists including einstein called spooky action distance however experimental fact violate principle relativity information transmitted process removing field chance anything


changing spin property field changed anything changing something already happened ref name brooks another reason later work schwinger well understood physics community blockquote tragedy ensued schwinger 's need things way made develop language approaches techniques became isolated fewer people understood spoke newer languages created contributing isolation mutual loss schwinger community losers blockquote


possible small class theories called renormalizable theories qed example however theories including fermi theory weak interaction non renormalizable perturbative calculation theories beyond first order would result infinities could removed redefining finite number physical quantities second major problem stemmed limited validity feynman diagram method based series expansion perturbation theory order series converge low order calculations good approximation coupling constant


series expanded must sufficiently small number coupling constant qed fine structure constant small enough simplest lowest order feynman diagrams need considered realistic calculations contrast coupling constant strong interaction roughly order one making complicated higher order feynman diagrams important simple ones thus way deriving reliable quantitative predictions strong interaction using perturbative qft methods difficulties looming many theorists began turn away qft


focused symmetry principles conservation laws others picked old matrix theory wheeler heisenberg qft used heuristically guiding principles basis quantitative calculations schwinger however took different route decade students nearly exponents field theory found way around problem infinities new method called source theory developments pion physics new viewpoint successfully applied convinced great advantages


mathematical simplicity conceptual clarity use bestowed ref name perseus books ref source theory divergences renormalization may regarded calculational tool field theory general using source theory schwinger able calculate anomalous magnetic moment electron done time distracting remarks infinite quantities schwinger also applied source theory qft theory gravity able reproduce four einstein classic results gravitational red shift deflection slowing light


gravity perihelion precession mercury neglect source theory physics community major disappointment schwinger blockquote lack appreciation facts others depressing understandable j schwinger ref name perseus books blockquote standard model elementary particles standard model six types quarks six types leptons four types gauge bosons carry fundamental interactions well higgs boson endow elementary particles mass yang chen ning robert mills generalised local symmetry qed leading non abelian gauge theories also known yang mills


theories based complicated local symmetry groups ref name thooft ref qed electrically charged particles interact via exchange photons non abelian gauge theory particles carrying new type charge interact via exchange massless gauge bosons unlike photons gauge bosons carry charge sheldon glashow developed non abelian gauge theory unified electromagnetic weak interactions abdus salam john clive ward arrived theory different path theory nevertheless non renormalizable ref name coleman ref peter higgs robert brout fran ois englert gerald guralnik carl hagen


theories renormalizable electroweak theory weinberg salam extended leptons quarks glashow john iliopoulos luciano maiani marking completion ref name coleman harald fritzsch murray gell mann heinrich leutwyler discovered certain phenomena involving strong interaction could also explained non abelian gauge theory quantum chromodynamics qcd born david gross frank wilczek hugh david politzer showed non abelian gauge theories asymptotically free meaning renormalization coupling constant strong interaction decreases interaction energy increases similar discoveries made


numerous times previously largely ignored therefore least high energy interactions coupling constant qcd becomes sufficiently small warrant perturbative series expansion making quantitative predictions strong interaction possible theoretical breakthroughs brought renaissance qft full theory includes electroweak theory chromodynamics referred today standard model elementary particles standard model successfully describes fundamental interactions except gravity many predictions met remarkable experimental confirmation subsequent decades higgs boson central mechanism


spontaneous symmetry breaking finally detected cern marking complete verification existence constituents standard model developments saw development non perturbative methods non abelian gauge theories 't hooft polyakov monopole discovered theoretically 't hooft alexander polyakov flux tubes holger bech nielsen poul olesen instantons polyakov coauthors objects inaccessible perturbation theory supersymmetry also appeared period first supersymmetric qft four dimensions built yuri golfand evgeny likhtman


theory arose study interactions elementary particles successfully applied physical systems particularly many body systems condensed matter physics historically higgs mechanism spontaneous symmetry breaking result yoichiro nambu 's application superconductor theory elementary particles concept renormalization came study second order phase transitions matter soon introduction photons einstein performed quantization procedure vibrations crystal leading first quasiparticle phonons lev landau claimed low energy excitations many condensed matter systems could described


terms interactions set quasiparticles feynman diagram method qft naturally well suited analysis various phenomena condensed matter systems ref name wilczek ref gauge theory used describe quantization magnetic flux superconductors resistivity quantum hall effect well relation frequency voltage ac josephson effect ref name wilczek principles simplicity natural units used following sections reduced planck constant speed light set one classical fields


classical field function spatial time coordinates ref name tong ref examples include gravitational field newtonian gravity electric field magnetic field classical electromagnetism classical field thought numerical quantity assigned every point space changes time hence infinitely many degrees freedom ref name tong many phenomena exhibiting quantum mechanical properties cannot explained classical fields alone phenomena photoelectric effect best explained discrete particles photons rather spatially continuous field goal quantum


field theory describe various quantum mechanical phenomena using modified concept fields canonical quantisation path integrals two common formulations qft ref name zee ref motivate fundamentals qft overview classical field theory order simplest classical field real scalar field real number every point space changes time denoted position vector time suppose lagrangian field math l math math l int x mathcal l int x left frac dot phi frac


nabla phi frac phi right math math mathcal l math lagrangian density math dot phi math time derivative field gradient operator real parameter mass field applying euler lagrange equation lagrangian math frac partial partial left frac partial mathcal l partial partial phi partial right sum frac partial partial x left frac partial mathcal l partial partial phi partial x right frac partial mathcal l partial phi math obtain equations motion field describe way varies time space math left frac partial partial nabla right phi math known klein gordon equation klein gordon equation wave equation solutions


expressed sum normal modes obtained via fourier transform follows math phi mathbf x int frac p pi frac sqrt omega mathbf p left mathbf p e omega mathbf p mathbf p cdot mathbf x mathbf p e omega mathbf p mathbf p cdot mathbf x right math complex number normalised convention denotes complex conjugation frequency normal mode math omega mathbf p sqrt mathbf p math thus normal mode corresponding single seen classical harmonic oscillator frequency canonical quantisation quantisation procedure classical field quantum operator field analogous promotion classical


e omega math complex numbers replaced annihilation operator math hat math creation operator math hat dagger math respectively denotes hermitian conjugation commutation relation two math left hat hat dagger right math vacuum state math rang math lowest energy state defined math hat rang math quantum state single harmonic oscillator obtained math rang math successively applying creation operator math hat dagger math math n rang left hat dagger right n rang math token aforementioned real scalar field corresponds single harmonic


p math quantum state field obtained math rang math successively applying creation operators math hat mathbf p dagger math eg math left hat mathbf p dagger right hat mathbf p dagger left hat mathbf p dagger right rang math although quantum field appearing lagrangian spatially continuous quantum states field discrete state space single quantum harmonic oscillator contains discrete energy states one oscillating particle state space quantum field contains discrete energy levels arbitrary number particles latter space known fock space account fact


particle numbers fixed relativistic quantum systems process quantising arbitrary number particles instead single particle often also called second quantisation foregoing procedure direct application non relativistic quantum mechanics used quantise complex scalar fields dirac fields vector fields ''eg'' electromagnetic field even strings however creation annihilation operators well defined simplest theories contain interactions called free theory case real scalar field existence operators consequence decomposition


solutions classical equations motion sum normal modes perform calculations realistic interacting theory perturbation theory would necessary lagrangian quantum field nature would contain interaction terms addition free theory terms example quartic interaction term could introduced lagrangian real scalar field math mathcal l frac partial mu phi left partial mu phi right frac phi frac lambda phi math spacetime index math partial partial partial partial partial partial x math etc summation index omitted following einstein


notation parameter sufficiently small interacting theory described lagrangian considered small perturbation free theory path integrals path integral formulation qft concerned direct computation scattering amplitude certain interaction process rather establishment operators state spaces calculate probability amplitude system evolve initial state math phi rang math time final state math phi f rang math ''t'' total time divided small intervals


overall amplitude product amplitude evolution within interval integrated intermediate states let hamiltonian ''ie'' generator time evolution math lang phi f e iht phi rang int phi int phi cdots int phi n lang phi f e iht n phi n rang cdots lang phi e iht n phi rang lang phi e iht n phi rang math taking limit product integrals becomes feynman path integral math lang phi f e iht phi rang int mathcal phi exp left int dt l right math lagrangian involving derivatives respect spatial time coordinates obtained hamiltonian via legendre transformation initial final conditions path integral respectively math phi phi quad phi


interacting theory different free ground state math rang math expression represents probability amplitude field propagate goes multiple names like two point propagator two point correlation function two point green 's function two point function short free two point function also known feynman propagator found real scalar field either canonical quantisation path integrals math lang phi x phi rang equiv f x lim epsilon int frac p pi frac p mu p mu epsilon e ip mu x mu mu math interacting theory


lagrangian hamiltonian contains terms math l math math h math describe interactions two point function difficult define however canonical quantisation formulation path integral formulation possible express infinite perturbation series ''free'' two point function canonical quantisation two point correlation function written math lang omega phi x phi omega rang lim infty epsilon frac left lang left left phi x phi exp left int dt h right right right right rang left lang left left exp left int dt h right right right right rang math infinitesimal number field operator free theory exponential understood power series expansion example math phi math theory


interacting term hamiltonian math display inline h int x frac lambda phi x math expansion two point correlator terms math lambda math becomes math display block lang omega phi x phi omega rang frac displaystyle sum n infty frac lambda n n n int z cdots int z n lang phi x phi phi z cdots phi z n rang displaystyle sum n infty frac lambda n n n int z cdots int z n lang phi z cdots phi z n rang math perturbation expansion expresses interacting two point function terms quantities math lang cdots rang math evaluated


''free'' theory path integral formulation two point correlation function written math lang omega phi x phi omega rang lim infty epsilon frac int mathcal phi phi x phi exp left int z mathcal l right int mathcal phi exp left int z mathcal l right math math mathcal l math lagrangian density previous paragraph exponential expanded series reducing interacting two point function quantities free theory wick 's theorem reduce point correlation function free theory sum products two point correlation functions example math begin align lang phi x phi x phi x phi x rang lang phi x phi x rang lang phi x phi x rang lang phi x phi x rang lang phi x phi x rang lang phi x phi x rang lang phi x phi x rang end align math since interacting correlation functions


expressed terms free correlation functions latter need evaluated order calculate physical quantities perturbative interacting theory makes feynman propagator one important quantities quantum field theory feynman diagram correlation functions interacting theory written perturbation series term series product feynman propagators free theory represented visually feynman diagram example term two point correlation function theory math frac lambda int z lang


phi x phi phi z phi z phi z phi z rang math applying wick 's theorem one terms math cdot frac lambda int z f x z f z f z z math term instead obtained ''feynman diagram'' px diagram consists ''external vertices'' connected one edge represented dots labelled math x math math math ''internal vertices'' connected four edges represented dots labelled math z math ''edges'' connecting vertices represented lines every vertex corresponds single math phi math field factor corresponding point spacetime edges correspond propagators spacetime points term perturbation series corresponding


diagram obtained writing expression follows '''feynman rules''' every internal vertex math z math write factor math display inline lambda int z math every edge connects two vertices math z math math z j math write factor math f z z j math divide symmetry factor diagram symmetry factor math math following rules yields exactly expression fourier transforming propagator feynman rules reformulated position space momentum space order compute point correlation function th order list valid


particle computed matrix found using feynman diagram method feynman diagrams devoid loops called tree level diagrams describe lowest order interaction processes containing loops referred loop diagrams describe higher order contributions radiative corrections interaction lines whose end points vertices thought propagation virtual particles renormalization feynman rules used directly evaluate tree level diagrams however na computation loop diagrams one shown result divergent momentum integrals


seems imply almost terms perturbative expansion infinite renormalisation procedure systematic process removing infinities parameters appearing lagrangian mass coupling constant physical meaning field strength experimentally measurable quantities referred bare mass bare coupling constant bare field respectively physical mass coupling constant measured interaction process generally different bare quantities computing physical quantities interaction


process one may limit domain divergent momentum integrals momentum cut obtain expressions physical quantities take limit example regularisation class methods treat divergences qft regulator approach illustrated called bare perturbation theory calculations involve bare quantities mass coupling constant different approach called renormalised perturbation theory use physically meaningful quantities beginning case theory field strength first redefined


latter three terms referred counterterms lagrangian contains terms feynman diagrams include additional elements feynman rules procedure outlined follows first select regularisation scheme cut regularisation introduced dimensional regularization call regulator compute feynman diagrams divergent terms depend define feynman diagrams counterterms exactly cancel divergent terms normal feynman diagrams limit taken way meaningful finite


quantities obtained possible eliminate infinities obtain finite result renormalisable theories whereas non renormalisable theories infinities cannot removed redefinition small number parameters standard model elementary particles renormalisable qft quantum gravity non renormalisable renormalisation group renormalisation group developed kenneth wilson mathematical apparatus used study changes physical parameters coefficients lagrangian system viewed different scales way parameter changes scale described '' ''


function correlation functions underlie quantitative physical predictions change scale according callan symanzik equation example coupling constant qed namely elementary charge following '' '' function math beta e equiv frac lambda frac de lambda frac e pi mathord left e right math energy scale measurement performed differential equation implies observed elementary charge increases scale increases renormalized coupling constant changes energy scale also called running coupling constant coupling constant quantum chromodynamics non abelian gauge theory based


symmetry group following '' '' function math beta g equiv frac lambda frac dg lambda frac g pi left frac n f right mathord left g right math number quark flavours case standard model coupling constant decreases energy scale increases hence strong interaction strong low energies becomes weak high energy interactions phenomenon known asymptotic freedom conformal field theories cfts special qfts admit conformal symmetry insensitive changes scale coupling constants vanishing '' '' function


converse true however vanishing '' '' functions imply conformal symmetry theory examples include string theory ref name polchinski ref ''n'' supersymmetric yang mills theory according wilson 's picture every qft fundamentally accompanied energy cut ''ie'' theory longer valid energies higher degrees freedom scale omitted example cut could inverse atomic spacing condensed matter system elementary particle physics could associated


fundamental graininess spacetime caused quantum fluctuations gravity cut scale theories particle interactions lies far beyond current experiments even theory complicated scale long couplings sufficiently weak must described low energies renormalisable effective field theory difference renormalisable non renormalisable theories former insensitive details high energies whereas latter depend according view non renormalisable theories seen low energy effective theories fundamental theory failure


second terms lagrangian density correspond free dirac field free vector fields respectively last term describes interaction electron photon fields treated perturbation free theories px shown example tree level feynman diagram qed describes electron positron annihilating creating shell photon decaying new pair electron positron time runs left right arrows pointing forward time represent propagation positrons pointing backward time represent


propagation electrons wavy line represents propagation photon vertex qed feynman diagrams must incoming outgoing fermion positron electron leg well photon leg gauge symmetry following transformation fields performed every spacetime point local transformation qed lagrangian remains unchanged invariant math psi x e alpha x psi x quad mu x mu x ie e alpha x partial mu e alpha x math function spacetime coordinates theory 's lagrangian precisely action invariant certain local transformation transformation


built non abelian groups giving rise non abelian gauge theories also known yang mills theories quantum chromodynamics describes strong interaction non abelian gauge theory gauge symmetry contains three dirac fields representing quark fields well eight vector fields representing gluon fields gauge bosons qcd lagrangian density math mathcal l bar psi gamma mu mu ij psi j frac f mu nu af mu nu bar psi psi math gauge covariant derivative math mu partial mu iga mu math coupling constant eight


generators fundamental representation matrices math f mu nu partial mu nu partial nu mu gf abc mu b nu c math structure constants repeated indices implicitly summed following einstein notation lagrangian invariant transformation math psi x u ij x psi j x quad mu x u x left mu x ig partial mu right u dagger x math element every spacetime point math u x e alpha x math preceding discussion symmetries level lagrangian words classical symmetries quantisation theories longer exhibit classical


symmetries phenomenon called anomaly instance path integral formulation despite invariance lagrangian density math mathcal l phi partial mu phi math certain local transformation fields measure math display inline int mathcal phi math path integral may change theory describing nature consistent must contain anomaly gauge symmetry standard model elementary particles gauge theory based group anomalies exactly cancel theoretical foundation general relativity equivalence principle also understood form gauge symmetry making general relativity


gauge theory based lorentz group noether 's theorem states every continuous symmetry ''ie'' parameter symmetry transformation continuous rather discrete leads corresponding conservation law example symmetry qed implies charge conservation gauge transformations relate distinct quantum states rather relates two equivalent mathematical descriptions quantum state example photon field four vector four apparent degrees freedom actual state photon described two degrees freedom corresponding polarisation remaining


vacuum state classical theory uniform field satisfying math phi phi frac mu lambda math without loss generality let ground state th direction math phi left cdots frac mu sqrt lambda right math original fields rewritten math phi x left pi x cdots pi n x frac mu sqrt lambda sigma x right math original lagrangian density math mathcal l frac left partial mu pi k right left partial mu pi k right frac left partial mu sigma right left partial mu sigma right frac left mu right sigma sqrt lambda mu sigma sqrt lambda mu pi k pi k sigma frac lambda pi k pi k sigma frac lambda left pi k pi k right math ''n'' original global symmetry longer manifest leaving subgroup larger symmetry spontaneous symmetry


breaking said hidden spontaneously broken goldstone 's theorem states spontaneous symmetry breaking every broken continuous global symmetry leads massless field called goldstone boson example continuous symmetries dimension lie algebra number broken symmetries difference corresponds massless fields hand gauge opposed global symmetry spontaneously broken resulting goldstone boson eaten corresponding gauge boson becoming additional degree freedom


gauge boson goldstone boson equivalence theorem states high energy amplitude emission absorption longitudinally polarised massive gauge boson becomes equal amplitude emission absorption goldstone boson eaten gauge boson qft ferromagnetism spontaneous symmetry breaking explain alignment magnetic dipoles low temperatures standard model elementary particles w z bosons would otherwise massless result gauge symmetry acquire mass spontaneous symmetry breaking higgs boson process


called higgs mechanism supersymmetry experimentally known symmetries nature relate bosons bosons fermions fermions theorists hypothesised existence type symmetry called supersymmetry relates bosons fermions standard model obeys poincar symmetry whose generators spacetime translations lorentz transformations ref name weinbergqft ref addition generators supersymmetry dimensions includes additional generators called supercharges transform weyl fermions symmetry group generated generators known super poincar group general one set


supersymmetry generators ''n'' generate corresponding supersymmetry supersymmetry supersymmetry also constructed dimensions notably dimensions application superstring theory lagrangian supersymmetric theory must invariant action super poincar group examples theories include minimal supersymmetric standard model mssm ''n'' supersymmetric yang mills theory superstring theory supersymmetric theory every fermion bosonic superpartner vice versa supersymmetry promoted local symmetry resultant gauge theory


extension general relativity called supergravity ref name natharnowitt ref supersymmetry potential solution many current problems physics example hierarchy problem standard model mass higgs boson radiatively corrected renormalisation high scale grand unified scale planck scale resolved relating higgs field superpartner higgsino radiative corrections due higgs boson loops feynman diagrams cancelled corresponding higgsino loops supersymmetry also offers answers grand unification gauge coupling constants standard


model well nature dark matter nevertheless experiments yet provide evidence existence supersymmetric particles supersymmetry true symmetry nature must broken symmetry energy symmetry breaking must higher achievable present day experiments spacetimes theory qed qcd well whole standard model assume dimensional minkowski space spatial time dimensions background quantum fields defined however qft ''a priori'' imposes restriction


schwarzschild metric describing black hole used math mu mu g mu nu mu nu quad partial mu phi partial mu phi g mu nu partial mu phi partial nu phi math inverse real scalar field lagrangian density general spacetime background math mathcal l sqrt g left frac g mu nu nabla mu phi nabla nu phi frac phi right math det ''g sub sub '' denotes covariant derivative lagrangian qft hence calculational results physical predictions depends geometry spacetime background topological quantum field theory correlation functions physical predictions qft depend


spacetime metric special class qfts called topological quantum field theories tqfts correlation functions independent continuous changes spacetime metric qfts curved spacetime generally change according ''geometry'' local structure spacetime background tqfts invariant spacetime diffeomorphisms sensitive ''topology'' global structure spacetime means calculational results tqfts topological invariants underlying spacetime chern simons theory example tqft used construct models quantum gravity applications tqft include fractional quantum


hall effect topological quantum computers world line trajectory fractionalized particles known anyons form link configuration spacetime relates braiding statistics anyons physics link invariants mathematics topological quantum field theories tqfts applicable frontier research topological quantum matters include chern simons witten gauge theories spacetime dimensions new exotic tqfts spacetime dimensions beyond perturbative non perturbative methods using perturbation theory total effect small interaction term approximated order order series expansion


number virtual particles participating interaction every term expansion may understood one possible way physical particles interact via virtual particles expressed visually using feynman diagram electromagnetic force two electrons qed represented first order perturbation theory propagation virtual photon similar manner w z bosons carry weak interaction gluons carry strong interaction interpretation interaction sum intermediate states involving exchange various virtual particles makes sense


framework perturbation theory contrast non perturbative methods qft treat interacting lagrangian whole without series expansion instead particles carry interactions methods spawned concepts 't hooft polyakov monopole domain wall flux tube instanton ref name shifman ref examples qfts completely solvable non perturbatively include minimal models conformal field theory thirring model mathematical rigour spite overwhelming success particle physics condensed matter physics qft lacks formal mathematical foundation example according haag 's theorem exist


well defined interaction picture qft implies perturbation theory qft underlies entire feynman diagram method fundamentally ill defined however ''perturbative'' quantum field theory requires quantities computable formal power series without convergence requirements given rigorous mathematical treatment particular kevin costello 's monograph ''renormalization effective field theory'' ref name costello kevin costello ''renormalization effective field theory'' mathematical surveys monographs volume american mathematical society ref provides rigorous formulation perturbative renormalization combines effective field theory approaches kadanoff wilson polchinski together


line study called constructive quantum field theory subfield mathematical physics ref name summers ref led results cpt theorem spin statistics theorem goldstone 's theorem ref name buchholz also mathematically rigorous constructions many interacting qfts two three spacetime dimensions eg two dimensional scalar field theories arbitrary polynomial interactions ref name simon ref three dimensional scalar field theories quartic interaction etc ref name glimm ref compared ordinary qft topological quantum field theory conformal field theory better supported mathematically classified framework representations cobordisms algebraic quantum field theory


another approach axiomatisation qft fundamental objects local operators algebraic relations axiomatic systems following approach include wightman axioms haag kastler axioms one way construct theories satisfying wightman axioms use osterwalder schrader axioms give necessary sufficient conditions real time theory obtained imaginary time theory analytic continuation wick rotation yang mills existence mass gap one millennium prize problems concerns well defined existence yang mills theories set axioms full problem statement


follows non trivial quantum yang mills theory exists math mathbb r math mass gap existence includes establishing axiomatic properties least strong cited see also abraham lorentz force ads cft correspondence axiomatic quantum field theory introduction quantum mechanics common integrals quantum field theory conformal field theory constructive quantum field theory form factor quantum field theory green kubo relations green 's function many body theory group field theory lattice field theory list quantum field theories local quantum field theory noncommutative quantum field theory quantization field quantum


electrodynamics quantum field theory curved spacetime quantum chromodynamics quantum flavordynamics quantum hadrodynamics quantum hydrodynamics quantum triviality relation schr dinger 's equation path integral formulation quantum mechanics relationship string theory quantum field theory schwinger dyson equation static forces virtual particle exchange symmetry quantum mechanics theoretical experimental justification schr dinger equation topological quantum field theory ward takahashi identity wheeler feynman absorber theory wigner 's classification wigner 's theorem references reading general readers introductory texts advanced texts external links ''stanford encyclopedia philosophy'' http platostanfordedu entries quantum field theory quantum field theory meinard kuhlmann


siegel warren '' http instiphysicssunysbedu esiegel erratahtml fields '' http wwwnatvunl mulders qft pdf quantum field theory p j mulders quantum mechanics mathematical physics


classical physics however description incomplete describing gravitational field black hole general theory relativity leads physical quantities spacetime curvature diverge center black hole signals breakdown general theory relativity need theory goes beyond general relativity quantum distances close center black hole closer planck length quantum fluctuations spacetime expected play important role describe quantum effects theory quantum gravity

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Physics
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Quantum gravity


needed theory allow description extended closer center might even allow understanding physics center black hole formal grounds one argue classical system cannot consistently coupled quantum one ref name flog ref field quantum gravity actively developing theorists exploring variety approaches problem quantum gravity popular theory loop quantum gravity approaches aim describe quantum behavior gravitational field


suggested testing tool theories overview diagram showing place quantum gravity hierarchy physics theories much difficulty meshing theories energy scales comes different assumptions theories make universe works general relativity models gravity curvature spacetime slogan john archibald wheeler spacetime tells matter move matter tells spacetime curve hand quantum field theory typically formulated ''flat'' spacetime used special relativity theory yet proven


successful describing general situation dynamics matter modeled quantum mechanics affect curvature spacetime one attempts treat gravity simply another quantum field resulting theory renormalizable ref name ref even simpler case curvature spacetime fixed ''a priori'' developing quantum field theory becomes mathematically challenging many ideas physicists use quantum field theory flat spacetime longer applicable ref name wald ref widely hoped theory quantum gravity would allow us understand problems high energy


small dimensions space behavior black holes origin universe ref name scholarpedia quantum mechanics general relativity gravity probe b gp b measured spacetime curvature near earth test related models application einstein 's general theory relativity graviton observation fundamental forces except gravity one known messenger particles leads researchers believe least one must exist gravity hypothetical particle known ''graviton'' particles act force particle similar photon electromagnetic interaction


mild assumptions structure general relativity requires follow quantum mechanical description interacting theoretical spin massless particles ref name kraichnan ref ref name gupta ref ref name gupta ref ref name gupta ref ref name deser ref many accepted notions unified theory physics since assume degree depend upon existence graviton weinberg witten theorem places constraints theories graviton composite particle gravitons important theoretical step quantum mechanical description gravity generally believed undetectable interact weakly nonrenormalizability gravity general


relativity like electromagnetism classical field theory one might expect electromagnetism gravitational force also corresponding quantum field theory however gravity perturbatively nonrenormalizable ref name flog quantum field theory well defined according understanding subject must asymptotically free asymptotically safe theory must characterized choice ''finitely many'' parameters could principle set experiment example quantum electrodynamics parameters charge mass electron measured particular energy scale


hand quantizing gravity perturbation theory ''infinitely many independent parameters'' counterterm coefficients needed define theory given choice parameters one could make sense theory since impossible conduct infinite experiments fix values every parameter argued one perturbation theory meaningful physical theory low energies logic renormalization group tells us despite unknown choices infinitely many parameters quantum gravity reduce usual einstein theory general relativity


hand could probe high energies quantum effects take ''every one'' infinitely many unknown parameters would begin matter could make predictions conceivable correct theory quantum gravity infinitely many unknown parameters reduce finite number measured one possibility normal perturbation theory reliable guide renormalizability theory really ''is'' uv fixed point gravity since question non perturbative


quantum field theory finding reliable answer difficult pursued asymptotic safety program another possibility new undiscovered symmetry principles constrain parameters reduce finite set route taken string theory excitations string essentially manifest new symmetries quantum gravity effective field theory effective field theory first infinite set parameters nonrenormalizable theory suppressed huge energy scales hence neglected


computing low energy effects thus least low energy regime model predictive quantum field theory ref name ref furthermore many theorists argue standard model regarded effective field theory nonrenormalizable interactions suppressed large energy scales whose effects consequently observed experimentally treating general relativity effective field theory one actually make legitimate predictions quantum gravity least low energy phenomena example well known calculation tiny first order quantum mechanical correction classical newtonian gravitational potential two masses ref name


spacetime background dependence fundamental lesson general relativity fixed spacetime background found newtonian mechanics special relativity spacetime geometry dynamic simple grasp principle complex idea understand general relativity consequences profound fully explored even classical level certain extent general relativity seen relational theory physically relevant information relationship different events space time hand quantum mechanics depended since


inception fixed background non dynamic structure case quantum mechanics time given dynamic newtonian classical mechanics relativistic quantum field theory classical field theory minkowski spacetime fixed background theory string theory interaction subatomic world world lines point like particles standard model world sheet swept closed strings string theory string theory seen generalization quantum field theory instead point particles string like objects propagate


fixed spacetime background although interactions among closed strings give rise space time dynamical way although string theory origins study quark confinement quantum gravity soon discovered string spectrum contains graviton condensation certain vibration modes strings equivalent modification original background sense string perturbation theory exhibits exactly features one would expect perturbation theory may exhibit strong dependence asymptotics seen example ads cft correspondence weak


form background dependence background independent theories loop quantum gravity fruit effort formulate background independent quantum theory topological quantum field theory provided example background independent quantum theory local degrees freedom finitely many degrees freedom globally inadequate describe gravity dimensions local degrees freedom according general relativity dimensions however gravity topological field theory successfully quantized several different ways including spin networks semi classical quantum gravity quantum field theory


variability time observed macroscopically removes possibility employing fixed notion time similar conception time quantum theory macroscopic level candidate theories number proposed quantum gravity theories currently still complete consistent quantum theory gravity candidate models still need overcome major formal conceptual problems also face common problem yet way put quantum gravity predictions experimental tests although hope change future


data cosmological observations particle physics experiments becomes available string theory projection calabi yau manifold one ways compactifying extra dimensions posited string theory central idea string theory replace classical concept point particle quantum field theory quantum theory one dimensional extended objects string theory energies reached current experiments strings indistinguishable point like particles crucially different modes oscillation one type fundamental string appear particles different electric charges


would constitute uniquely defined consistent theory quantum gravity presently understood however string theory admits large number sup sup estimates consistent vacua comprising called string landscape sorting large family solutions remains major challenge loop quantum gravity simple spin network type used loop quantum gravity loop quantum gravity seriously considers general relativity 's insight spacetime dynamical field therefore quantum object second idea quantum discreteness determines particle like behavior field theories


instance photons electromagnetic field also affects structure space main result loop quantum gravity derivation granular structure space planck length derived following considerations case electromagnetism quantum operator representing energy frequency field discrete spectrum thus energy frequency quantized quanta photons case gravity operators representing area volume surface space region likewise discrete spectra thus area


volume portion space also quantized quanta elementary quanta space follows spacetime elementary quantum granular structure planck scale cuts ultraviolet infinities quantum field theory quantum state spacetime described theory means mathematical structure called spin networks spin networks initially introduced roger penrose abstract form later shown carlo rovelli lee smolin derive naturally non perturbative quantization general relativity spin networks represent quantum


studies possibility experimental tests obtained increased attention widely pursued possibilities quantum gravity phenomenology include violations lorentz invariance imprints quantum gravitational effects cosmic microwave background particular polarization decoherence induced fluctuations space time foam esa 's integral satellite measured polarization photons different wavelengths able place limit granularity space less sup sup orders magnitude planck scale bicep experiment detected initially thought primordial b mode polarization caused gravitational waves


early universe signal fact primordial origin could indication quantum gravitational effects soon transpired polarization due interstellar dust interference ref name nature ref see also de sitter relativity dilaton doubly special relativity gravitational decoherence gravitomagnetism hawking radiation list quantum gravity researchers orders magnitude length penrose interpretation planck epoch planck units swampland physics virtual black hole weak gravity conjecture notes references reading external links http abyssuoregonedu js cosmo lectures lec html planck era planck time sup sup


seconds birth universe university oregon http wwwbbccouk programmes p c quantum gravity bbc radio discussion john gribbin lee smolin janna levin ''in time'' feb general relativity physics beyond standard model theories gravitation


painting depicting lecture knight academy painted pieter isaacsz reinhold timm rosenborg castle part series seven paintings depicting seven independent arts painting illustrates rhetoric '''rhetoric''' ''rh torik s'' oratorical ''rh r'' public speaker related ''rh ma'' said spoken word saying ultimately derived verb ''er '' say speak art persuasion along grammar logic dialectic nbsp see martianus capella one three ancient arts discourse rhetoric aims study techniques

Humanities
Languages and literature
Linguistics
Rhetoric


writers speakers utilize inform persuade motivate particular audiences specific situationstaking place athens early fifth century demos people created strategy effectively talking people juries forums senate aristotle defines rhetoric faculty observing given case available means persuasion since mastery art necessary victory case law passage proposals assembly fame speaker civic ceremonies calls combination science logic


ethical branch politics rhetoric typically provides heuristics understanding discovering developing arguments particular situations aristotle 's three persuasive audience appeals logos pathos ethos five canons rhetoric phases developing persuasive speech first codified classical rome invention arrangement style memory delivery ancient greece late th century rhetoric played central role western education training orators lawyers counsellors historians statesmen poets used form communication person person ref uses scope ezra calls rebuilding


temple woodcut julius schnorr von karolsfeld scholars debated scope rhetoric since ancient times although limited rhetoric specific realm political discourse many modern scholars liberate encompass every aspect culture contemporary studies rhetoric address much diverse range domains case ancient times classical rhetoric trained speakers effective persuaders public forums institutions courtrooms assemblies contemporary rhetoric investigates human discourse writ large rhetoricians studied discourses wide variety


communicating expertise politics ''encomium helen'' gorgias even applied rhetoric fiction seeking pleasure prove blamelessness mythical helen troy starting trojan war looking another key rhetorical theorist plato defined scope rhetoric according negative opinions art criticized sophists using rhetoric means deceit instead discovering truth gorgias one socratic dialogues plato defines rhetoric persuasion ignorant masses within courts assemblies rhetoric plato 's opinion


merely form flattery functions similarly cookery masks undesirability unhealthy food making taste good thus plato considered speech lengthy prose aimed flattery within scope rhetoric scholars however contest idea plato despised rhetoric instead view dialogues dramatization complex rhetorical principles aristotle redeemed rhetoric teacher narrowed focus defining three genres rhetoric deliberative forensic judicial epideictic yet even provided order existing rhetorical theories aristotle extended definition rhetoric


yet aristotle also outlined generic constraints focused rhetorical art squarely within domain public political practice restricted rhetoric domain contingent probable matters admit multiple legitimate opinions arguments contemporary neo aristotelian neo sophistic positions rhetoric mirror division sophists aristotle neo aristotelians generally study rhetoric political discourse neo sophistic view contends rhetoric cannot limited rhetorical scholar michael leff characterizes conflict positions viewing rhetoric thing contained versus container neo aristotelian view threatens study


rhetoric restraining limited field ignoring many critical applications rhetorical theory criticism practice simultaneously neo sophists threaten expand rhetoric beyond point coherent theoretical value past century people studying rhetoric tended enlarge object domain beyond speech texts kenneth burke asserted humans use rhetoric resolve conflicts identifying shared characteristics interests symbols nature humans engage identification either assign oneself another group definition rhetoric identification broadened scope strategic overt political persuasion


always trying construct new world persuading readers share world within text people engage rhetorical process anytime speak produce meaning even field science practices viewed merely objective testing reporting knowledge scientists must persuade audience accept findings sufficiently demonstrating study experiment conducted reliably resulted sufficient evidence support conclusions vast scope rhetoric difficult define however political discourse remains many


ways paradigmatic example studying theorizing specific techniques conceptions persuasion considered many synonym rhetoric civic art throughout european history rhetoric concerned persuasion public political settings assemblies courts associations democratic institutions rhetoric commonly said flourish open democratic societies rights free speech free assembly political enfranchisement portion population classify rhetoric civic art believe rhetoric power shape communities form character citizens


greatly affect civic life rhetoric viewed civic art several ancient philosophers aristotle isocrates two first see rhetoric light work antidosis isocrates states come together founded cities made laws invented arts generally speaking institution devised man power speech helped us establish statement argues rhetoric fundamental part civic life every society necessary foundation


aspects society argues piece sophists rhetoric although cannot taught anyone capable shaping character man writes think study political discourse help thing stimulate form qualities character aristotle writing several years isocrates supported many arguments continued make arguments rhetoric civic art words aristotle ''rhetoric'' rhetoric nbsp faculty observing given case available means


persuasion according aristotle art persuasion could used public settings three different ways writes book nbsp chapter nbsp iii member assembly decides future events juryman past events merely decide orator 's skill observers follows three divisions oratory nbsp political forensic ceremonial oratory display eugene garver critique aristotle 's rhetoric confirms aristotle viewed rhetoric civic art garver writes rhetoric articulates civic art rhetoric combining almost


incompatible properties techne appropriateness citizens aristotle 's divisions plays role civic life used different way affect cities rhetoric public art capable shaping opinion ancients including plato found fault claimed could used improve civic life could used equally easily deceive manipulate negative effects city masses incapable analyzing deciding anything would therefore swayed persuasive speeches thus


civic life could controlled one could deliver best speech plato explores problematic moral status rhetoric twice ''gorgias'' dialogue named famed sophist ''the phaedrus'' dialogue best known commentary love concern still maintained nowadays trusting power rhetoric support republic roman orator cicero argued art required something eloquence good orator needed also good man person enlightened variety civic topics describes proper training


orator major text rhetoric ''de oratore'' modeled plato 's dialogues modern day works continue support claims ancients rhetoric art capable influencing civic life work ''political style'' robert hariman claims furthermore questions freedom equality justice often raised addressed performances ranging debates demonstrations without loss moral content james boyd white argues rhetoric capable addressing issues political interest influence culture whole book ''when words lose


meaning'' argues words persuasion identification define community civic life states words produce methods culture maintained criticized transformed white hariman agree words rhetoric power shape culture civic life modern times rhetoric consistently remained relevant civic art speeches well non verbal forms rhetoric continues used tool influence communities local national levels course study rhetoric course study evolved significantly since ancient


beginnings ages study teaching rhetoric adapted particular exigencies time venue study rhetoric conformed multitude different applications ranging architecture literature although curriculum transformed number ways generally emphasized study principles rules composition means moving audiences generally speaking study rhetoric trains students speak write effectively well critically understand analyze discourse rhetoric began civic art ancient greece students


trained develop tactics oratorical persuasion especially legal disputes rhetoric originated school pre socratic philosophers known sophists circa bc demosthenes lysias emerged major orators period isocrates gorgias prominent teachers rhetorical education focused five particular canons ''inventio'' invention ' 'dispositio'' arrangement ''elocutio'' style ''memoria'' memory ''actio'' delivery modern teachings continue reference rhetorical leaders work discussions classical rhetoric persuasion rhetoric later taught universities middle ages one three original liberal arts trivium


along logic grammar medieval period political rhetoric declined republican oratory died emperors rome garnered increasing authority rise european monarchs following centuries rhetoric shifted courtly religious applications augustine exerted strong influence christian rhetoric middle ages advocating use rhetoric lead audiences truth understanding especially church study liberal arts believed contributed rhetorical study case keen ardent nature fine words come readily reading hearing


eloquent pursuing rules rhetoric poetry letter writing instance became central component rhetorical study middle ages fall republic rome poetry became tool rhetorical training since fewer opportunities political speech letter writing primary form business conducted state church became important aspect rhetorical education ref name bedfordstmartinscom brief history rhetoric composition ''the bedford bibliography teachers writing'' bedford st martin 's nd web april http wwwbedfordstmartinscom bb historyhtml bedfordstmartinscom ref rhetorical education


became restrained style substance separated th century france peter ramus attention turned scientific method influential scholars like ramus argued processes invention arrangement elevated domain philosophy rhetorical instruction chiefly concerned use figures forms ornamentation language scholars francis bacon developed study scientific rhetoric concentration rejected elaborate style characteristic classical oration plain language carried john locke 's teaching emphasized concrete knowledge steered


away ornamentation speech alienating rhetorical instruction identified wholly ornamentation pursuit knowledge th century rhetoric assumed social role initiating creation new education systems elocution schools arose predominantly england females analyzed classic literature notably works william shakespeare discussed pronunciation tactics study rhetoric underwent revival rise democratic institutions late th early th centuries scotland 's author theorist hugh blair served key leader movement late


th century famous work lectures rhetoric belles lettres advocates rhetorical study common citizens resource social success many american colleges secondary schools used blair 's text throughout th century train students rhetoric political rhetoric also underwent renewal wake us french revolutions rhetorical studies ancient greece rome resurrected studies era speakers teachers looked cicero others inspire defense new republic leading rhetorical theorists included john quincy adams harvard


social institution featuring group discussions guest lecturers programs cultivated democratic values promoted active participation political analysis throughout th century rhetoric developed concentrated field study establishment rhetorical courses high schools universities courses public speaking speech analysis apply fundamental greek theories modes persuasion ethos pathos logos well trace rhetorical development throughout course history rhetoric earned esteemed reputation field study emergence communication studies departments well


rhetoric composition programs within english departments universities conjunction linguistic turn rhetorical study broadened scope especially utilized fields marketing politics literature rhetoric area study concerned humans use symbols especially language reach agreement permits coordinated effort sort harvard university first university united states based european model taught basic curriculum including rhetoric rhetoric sense properly give speeches played important role training rhetoric soon taught


relevant crux matter selection true otherwise trivial statements hence rhetoric also closely related knowledge history rhetoric origins mesopotamia ref name ref earliest examples rhetoric found akkadian writings princess priestess enheduanna c bc ref name ref first named author history ref name ref name enheduanna 's writing exhibits numerous rhetorical features would later become canon ancient greece enheduanna 's exaltation inanna includes exordium argument peroration ref name well elements


ethos pathos logos ref name repetition metonymy also known describing process invention exaltation inanna moving first third person address relate composing process collaboration goddess inanna ref name reflecting mystical enthymeme drawing upon cosmic audience ref name later examples early rhetoric found neo assyrian empire time sennacherib bc ancient egypt rhetoric existed since least middle kingdom period c bc egyptians held eloquent speaking high esteem


skill high value society egyptian rules rhetoric also clearly specified knowing speak essential respected rhetorical knowledge approach rhetoric thus balance eloquence wise silence rules speech also strongly emphasized adherence social behaviors support conservative status quo held skilled speech support question society ancient china rhetoric dates back chinese philosopher confucius bc continued later followers tradition confucianism emphasized use


judicial decisions made philosophical ideas developed disseminated modern students today difficult remember wide use availability written texts phenomenon coming vogue classical greece classical times many great thinkers political leaders performed works audience usually context competition contest fame political influence cultural capital fact many known texts students followers detractors wrote already


noted ''rhetor'' greek term ''orator '' ''rhetor'' citizen regularly addressed juries political assemblies thus understood gained knowledge public speaking process though general facility language often referred ''log n techne'' skill arguments verbal artistry rhetoric thus evolved important art one provided orator forms means strategies persuading audience correctness orator 's arguments today term ''rhetoric'' used times refer form


argumentation often pejorative connotation rhetoric means obscuring truth classical philosophers believed quite contrary skilled use rhetoric essential discovery truths provided means ordering clarifying arguments giving everyone opportunity voice opinions sophists europe organized thought public speaking began ancient greece possibly first study power language may attributed philosopher empedocles c bc whose theories human knowledge would provide newfound basis many future rhetoricians first


written manual attributed corax pupil tisias work well many early rhetoricians grew courts law tisias example believed written judicial speeches others delivered courts teaching oratory popularized th century bc itinerant teachers known sophists best known protagoras c bc gorgias c bc isocrates bc aspasia miletus believed one first women engage private public rhetoric activities


sophist sophists disparate group travelled city city teaching public places attract students offer education central focus logos might broadly refer discourse functions powers defined parts speech analyzed poetry parsed close synonyms invented argumentation strategies debated nature reality claimed make students better words teach virtue thus claimed human excellence accident fate prerogative noble birth art


''techne'' could taught learned thus among first humanists several sophists also questioned received wisdom gods greek culture believed taken granted greeks time making among first agnostics example argued cultural practices function convention ''nomos'' rather blood birth ''phusis'' argued even morality immorality action could judged outside cultural context within occurred well known phrase man measure things


arises belief one famous infamous doctrines probability counter arguments taught every argument could countered opposing argument argument 's effectiveness derived likely appeared audience probability seeming true probability argument could countered inverted probability argument thus seemed likely strong poor man guilty robbing rich weak man strong poor man could argue contrary likelihood would suspect


makes unlikely committed crime since would likely apprehended crime also taught known ability make weaker worse argument stronger better aristophanes famously parodies clever inversions sophists known play ''the clouds'' word sophistry developed strong negative connotations ancient greece continue today ancient greece sophists nevertheless popular well paid professionals widely respected abilities also widely criticized excesses isocrates isocrates bc like sophists taught public


speaking means human improvement worked distinguish sophists saw claiming far could deliver suggested art virtue excellence exist one piece least process self improvement relied much heavily native talent desire constant practice imitation good models isocrates believed practice speaking publicly noble themes important questions would function improve character speaker audience also offering best service


city fact isocrates outspoken champion rhetoric mode civic engagement thus wrote speeches models students imitate way poets might imitate homer hesiod seeking inspire desire attain fame civic leadership first permanent school athens likely plato 's academy aristotle 's lyceum founded part response isocrates though left handbooks speeches '' antidosis '' '' sophists '' relevant students rhetoric


became models oratory one canonical ten attic orators keys entire educational program marked influence cicero quintilian entire educational system west plato plato bc famously outlined differences true false rhetoric number dialogues particularly ''gorgias'' ''phaedrus'' dialogues wherein plato disputes sophistic notion art persuasion sophists' art calls rhetoric exist independent art dialectic plato claims since sophists appeal seems


probable advancing students audiences simply flattering want hear plato 's condemnation rhetoric clear ''gorgias'' ''phaedrus'' suggests possibility true art wherein rhetoric based upon knowledge produced dialectic relies dialectically informed rhetoric appeal main character phaedrus take philosophy thus plato 's rhetoric actually dialectic philosophy turned toward yet philosophers thus unready pursue dialectic directly plato 's animosity rhetoric sophists


derives inflated claims teach virtue reliance appearances fact teacher socrates sentenced death sophists' efforts scholars however see plato opponent rhetoric rather nuanced rhetorical theorist dramatized rhetorical practice dialogues imagined rhetoric oratory aristotle marble bust aristotle aristotle bc student plato famously set forth extended treatise rhetoric still repays careful study today first sentence


dialectic ''nietzsche humanist'' claude pavur explains greek prefix 'anti' merely designate opposition also mean 'in place of' aristotle characterizes rhetoric antistrophe dialectic doubt means rhetoric used place dialectic discussing civic issues court law legislative assembly domain rhetoric civic affairs practical decision making civic affairs theoretical considerations operational definitions terms clarification thought domain dialectic


aristotle 's treatise rhetoric systematically describes civic rhetoric human art skill techne objective theory interpretive theory rhetorical tradition aristotle 's art rhetoric emphasizes persuasion purpose rhetoric definition rhetoric faculty observing given case available means persuasion essentially mode discovery limits art inventional process aristotle heavily emphasizes logical aspect process account rhetoric art discovering available means persuasion speaker supports


probability message logical ethical emotional proofs form logos ethos pathos present every possible public presentation exists treatise fact also discusses elements style briefly delivery also emotional appeals pathos characterological appeals ethos aristotle identifies three steps offices rhetoric invention arrangement style three different types rhetorical proof ethos aristotle 's theory character character credibility speaker influence audience consider believable three qualities contribute credible


ethos perceived intelligence virtuous character goodwill pathos use emotional appeals alter audience 's judgment metaphor amplification storytelling presenting topic way evokes strong emotions audience logos use reasoning either inductive deductive construct argument aristotle emphasized ''enthymematic reasoning'' central process rhetorical invention though later rhetorical theorists placed much less emphasis enthymeme would follow today 's form syllogism however would exclude either major minor premise enthymeme persuasive audience providing


missing premise audience able provide missing premise likely persuaded message aristotle identified three different types genres civic rhetoric ''forensic'' also known judicial concerned determining truth falseness events took place past issues guilt example forensic rhetoric would courtroom ''deliberative'' also known political concerned determining whether particular actions taken future making laws would example