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from revision #62 to #63: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='gravity'>Gravity</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/gravity'>gravity</a>, <a class='existingWikiWord' href='/nlab/show/diff/supergravity'>supergravity</a> Formalism <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+Lie+algebra'>Poincaré Lie algebra</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/super+Poincar%C3%A9+Lie+algebra'>super Poincaré Lie algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/D%3D11+supergravity+Lie+3-algebra'>supergravity Lie 3-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/supergravity+Lie+6-algebra'>supergravity Lie 6-algebra</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a> </li> <li> (<a class='existingWikiWord' href='/nlab/show/diff/super-Cartan+geometry'>super</a>-)<a class='existingWikiWord' href='/nlab/show/diff/Cartan+geometry'>Cartan geometry</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+manifold'>pseudo</a>-<a class='existingWikiWord' href='/nlab/show/diff/Riemannian+manifold'>Riemannian manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a> </li> </ul> </li> </ul> Definition <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Einstein-Hilbert+action'>Einstein-Hilbert action</a>, <a class='existingWikiWord' href='/nlab/show/diff/Einstein+equation'>Einstein's equations</a>, <a class='existingWikiWord' href='/nlab/show/diff/general+relativity'>general relativity</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/first-order+formulation+of+gravity'>first-order formulation of gravity</a>, <a class='existingWikiWord' href='/nlab/show/diff/D%27Auria-Fr%C3%A9-Regge+formulation+of+supergravity'>D'Auria-Fre formulation of supergravity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/gravity+as+a+BF+theory'>gravity as a BF-theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Plebanski+formulation+of+gravity'>Plebanski formulation of gravity</a>, <a class='existingWikiWord' href='/nlab/show/diff/teleparallel+gravity'>teleparallel gravity</a> </li> </ul> </li> </ul> Spacetime configurations <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Minkowski+space'>Minkowski spacetime</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/black+hole'>black hole spacetime</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Schwarzschild+spacetime'>Schwarzschild spacetime</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kerr+spacetime'>Kerr spacetime</a> </li> </ul> </li> </ul> Properties <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kaluza-Klein+mechanism'>Kaluza-Klein mechanism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Bekenstein-Hawking+entropy'>Bekenstein-Hawking entropy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/generalized+second+law+of+thermodynamics'>generalized second law of thermodynamics</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cosmic+censorship+hypothesis'>cosmic censorship hypothesis</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/weak+gravity+conjecture'>weak gravity conjecture</a> </li> </ul> Spacetimes <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/standard+model+of+cosmology'>standard model of cosmology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Minkowski+space'>Minkowski spacetime</a> </li> </ul> <table><thead><tr><th><a class='existingWikiWord' href='/nlab/show/diff/black+hole'>black hole</a> <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetimes</a></th><th>vanishing <a class='existingWikiWord' href='/nlab/show/diff/angular+momentum'>angular momentum</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/positive+number'>positive</a> <a class='existingWikiWord' href='/nlab/show/diff/angular+momentum'>angular momentum</a></th></tr></thead><tbody><tr><td style='text-align: left;'>vanishing <a class='existingWikiWord' href='/nlab/show/diff/charge'>charge</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Schwarzschild+spacetime'>Schwarzschild spacetime</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Kerr+spacetime'>Kerr spacetime</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/positive+number'>positive</a> <a class='existingWikiWord' href='/nlab/show/diff/charge'>charge</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Reissner-Nordstr%C3%B6m+spacetime'>Reissner-Nordstrom spacetime</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Kerr-Newman+spacetime'>Kerr-Newman spacetime</a></td></tr> </tbody></table> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/asymptotically+flat+spacetime'>asymptotically flat spacetime</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/gravitational+wave'>gravitational wave</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/anti+de+Sitter+spacetime'>anti de Sitter spacetime</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/FRW+model'>FRW spacetime</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Taub-NUT+space'>Taub-NUT spacetime</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Einstein+manifold'>Einstein manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pp-wave+spacetime'>pp-wave spacetime</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kaluza-Klein+monopole'>KK-monopole</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Malament%E2%80%93Hogarth+spacetime'>Malament–Hogarth spacetime</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/super+spacetime'>super spacetime</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/super+Minkowski+spacetime'>super Minkowski spacetime</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/black+brane'>black brane</a> </li> </ul> Quantum theory <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/quantum+gravity'>quantum gravity</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/graviton'>graviton</a>, <a class='existingWikiWord' href='/nlab/show/diff/gravitino'>gravitino</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/string+theory'>string theory</a> </li> </ul> </li> </ul> </div> <h4 id='physics'>Physics</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/physics'>physics</a>, <a class='existingWikiWord' href='/nlab/show/diff/mathematical+physics'>mathematical physics</a>, <a class='existingWikiWord' href='/nlab/show/diff/philosophy+of+physics'>philosophy of physics</a> <h2 id='surveys_textbooks_and_lecture_notes'>Surveys, textbooks and lecture notes</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory+and+physics'>(higher) category theory and physics</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics'>geometry of physics</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/books+and+reviews+in+mathematical+physics'>books and reviews</a>, <a class='existingWikiWord' href='/nlab/show/diff/physics+resources'>physics resources</a> </li> </ul> <hr /> <a class='existingWikiWord' href='/nlab/show/diff/theory+%28physics%29'>theory (physics)</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+%28in+theoretical+physics%29'>model (physics)</a> <a class='existingWikiWord' href='/nlab/show/diff/experimental+observation'>experiment</a>, <a class='existingWikiWord' href='/nlab/show/diff/measurement'>measurement</a>, <a class='existingWikiWord' href='/nlab/show/diff/computable+physics'>computable physics</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/mechanics'>mechanics</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/mass'>mass</a>, <a class='existingWikiWord' href='/nlab/show/diff/charge'>charge</a>, <a class='existingWikiWord' href='/nlab/show/diff/momentum'>momentum</a>, <a class='existingWikiWord' href='/nlab/show/diff/angular+momentum'>angular momentum</a>, <a class='existingWikiWord' href='/nlab/show/diff/moment+of+inertia'>moment of inertia</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hamiltonian+dynamics+on+Lie+groups'>dynamics on Lie groups</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/rigid+body+dynamics'>rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/field+%28physics%29'>field (physics)</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lagrangian+density'>Lagrangian mechanics</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/configuration+space'>configuration space</a>, <a class='existingWikiWord' href='/nlab/show/diff/state'>state</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/action+functional'>action functional</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lagrangian+density'>Lagrangian</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/phase+space'>covariant phase space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+equation'>Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hamiltonian+mechanics'>Hamiltonian mechanics</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/phase+space'>phase space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symplectic+geometry'>symplectic geometry</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Poisson+manifold'>Poisson manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symplectic+manifold'>symplectic manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symplectic+groupoid'>symplectic groupoid</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/multisymplectic+geometry'>multisymplectic geometry</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/symplectic+Lie+n-algebroid'>n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+Lorentzian+space'>smooth Lorentzian manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/special+relativity'>special relativity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/general+relativity'>general relativity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/gravity'>gravity</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/supergravity'>supergravity</a>, <a class='existingWikiWord' href='/nlab/show/diff/dilaton'>dilaton gravity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/black+hole'>black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classical+field+theory'>Classical field theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/classical+physics'>classical physics</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/classical+mechanics'>classical mechanics</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/wave'>waves</a> and <a class='existingWikiWord' href='/nlab/show/diff/optics'>optics</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/thermodynamics'>thermodynamics</a></li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/quantum+mechanics'>Quantum Mechanics</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/quantum+information+theory+via+dagger-compact+categories'>in terms of ∞-compact categories</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/quantum+information'>quantum information</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hamiltonian'>Hamiltonian operator</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/density+matrix'>density matrix</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kochen-Specker+theorem'>Kochen-Specker theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Bell%27s+inequality'>Bell's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Gleason%27s+theorem'>Gleason's theorem</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/quantization'>Quantization</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometric+quantization'>geometric quantization</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/deformation+quantization'>deformation quantization</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/path+integral'>path integral quantization</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/semiclassical+approximation'>semiclassical approximation</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/quantum+field+theory'>Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/AQFT'>algebraic QFT</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Wightman+axioms'>Wightman axioms</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Haag-Kastler+axioms'>Haag-Kastler axioms</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/operator+algebra'>operator algebra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/causally+local+net+of+observables'>local net</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/conformal+net'>conformal net</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Reeh-Schlieder+theorem'>Reeh-Schlieder theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Osterwalder-Schrader+theorem'>Osterwalder-Schrader theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/PCT+theorem'>PCT theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Bisognano-Wichmann+theorem'>Bisognano-Wichmann theorem</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/modular+theory'>modular theory</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spin-statistics+theorem'>spin-statistics theorem</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/boson'>boson</a>, <a class='existingWikiWord' href='/nlab/show/diff/fermion'>fermion</a></li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/functorial+field+theory'>functorial QFT</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/cobordism'>cobordism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category+of+cobordisms'>(∞,n)-category of cobordisms</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cobordism+hypothesis'>cobordism hypothesis</a>-theorem </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/extended+functorial+field+theory'>extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/perturbative+quantum+field+theory'>perturbative quantum field theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/vacuum'>vacuum</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/effective+quantum+field+theory'>effective quantum field theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/renormalization'>renormalization</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/BV-BRST+formalism'>BV-BRST formalism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometric+infinity-function+theory'>geometric ∞-function theory</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/particle+physics'>particle physics</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/phenomenology+%28physics%29'>phenomenology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/model+%28in+theoretical+physics%29'>models</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/standard+model+of+particle+physics'>standard model of particle physics</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fields+and+quanta+-+table'>fields and quanta</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/GUT'>Grand Unified Theories</a>, <a class='existingWikiWord' href='/nlab/show/diff/MSSM'>MSSM</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/scattering+amplitude'>scattering amplitude</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/on-shell+recursion'>on-shell recursion</a>, <a class='existingWikiWord' href='/nlab/show/diff/KLT+relations'>KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/universality+class'>universality class</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/quantum+anomaly'>quantum anomaly</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Green-Schwarz+mechanism'>Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/instanton'>instanton</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spontaneously+broken+symmetry'>spontaneously broken symmetry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kaluza-Klein+mechanism'>Kaluza-Klein mechanism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/integrable+system'>integrable systems</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/holonomic+quantum+field'>holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+field+theory'>TQFT</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/D%3D2+TQFT'>2d TQFT</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dijkgraaf-Witten+theory'>Dijkgraaf-Witten theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/TCFT'>TCFT</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/A-model'>A-model</a>, <a class='existingWikiWord' href='/nlab/show/diff/B-model'>B-model</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mirror+symmetry'>homological mirror symmetry</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/QFT+with+defects'>QFT with defects</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/conformal+field+theory'>conformal field theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory'>(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%282%7C1%29-dimensional+Euclidean+field+theory'>(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/conformal+field+theory'>CFT</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Wess-Zumino-Witten+model'>WZW model</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/D%3D6+N%3D%282%2C0%29+SCFT'>6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/field+strength'>field strength</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/gauge+group'>gauge group</a>, <a class='existingWikiWord' href='/nlab/show/diff/gauge+transformation'>gauge transformation</a>, <a class='existingWikiWord' href='/nlab/show/diff/gauge+fixing'>gauge fixing</a> </li> <li> examples <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/electromagnetic+field'>electromagnetic field</a>, <a class='existingWikiWord' href='/nlab/show/diff/quantum+electrodynamics'>QED</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/electric+charge'>electric charge</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/magnetic+charge'>magnetic charge</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Yang-Mills+field'>Yang-Mills field</a>, <a class='existingWikiWord' href='/nlab/show/diff/QCD'>QCD</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Yang-Mills+theory'>Yang-Mills theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/The+Dirac+Electron'>spinors in Yang-Mills theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+Yang-Mills+theory'>topological Yang-Mills theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kalb-Ramond+field'>Kalb-Ramond field</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/supergravity+C-field'>supergravity C-field</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/RR+field'>RR field</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/first-order+formulation+of+gravity'>first-order formulation of gravity</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/general+covariance'>general covariance</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/supergravity'>supergravity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/D%27Auria-Fr%C3%A9-Regge+formulation+of+supergravity'>D'Auria-Fre formulation of supergravity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/gravity+as+a+BF+theory'>gravity as a BF-theory</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sigma-model'>sigma-model</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/particle'>particle</a>, <a class='existingWikiWord' href='/nlab/show/diff/relativistic+particle'>relativistic particle</a>, <a class='existingWikiWord' href='/nlab/show/diff/fundamental+particle'>fundamental particle</a>, <a class='existingWikiWord' href='/nlab/show/diff/spinning+particle'>spinning particle</a>, <a class='existingWikiWord' href='/nlab/show/diff/superparticle'>superparticle</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/string'>string</a>, <a class='existingWikiWord' href='/nlab/show/diff/spinning+string'>spinning string</a>, <a class='existingWikiWord' href='/nlab/show/diff/superstring'>superstring</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/membrane'>membrane</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/AKSZ+sigma-model'>AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/string+theory'>String Theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/string+theory+results+applied+elsewhere'>string theory results applied elsewhere</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/number+theory+and+physics'>number theory and physics</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Riemann+hypothesis+and+physics'>Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href='/nlab/edit/physicscontents'>Edit this sidebar</a> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#details'>Details</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#general'>General</a></li><li><a href='#CovariantPhaseSpaceReferences'>Covariant phase space</a></li><li><a href='#nonrenormalizability'>Non-renormalizability</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> A <a class='existingWikiWord' href='/nlab/show/diff/field+%28physics%29'>field</a> configuration of the <a class='existingWikiWord' href='/nlab/show/diff/theory+%28physics%29'>physical theory</a> of gravity on a <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a> <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is equivalently <ul> <li> a <a class='existingWikiWord' href='/nlab/show/diff/vielbein'>vielbein</a> field, hence a <a class='existingWikiWord' href='/nlab/show/diff/reduction+and+lift+of+structure+groups'>reduction of the structure group</a> of the <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a> along <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>O</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>→</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{B} O(n-1,1) \to \mathbf{B}GL(n)</annotation></semantics></math>, defining a <a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+metric'>pseudo-Riemannian metric</a>; </li> <li> a <a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+bundle'>connection</a> that is locally a <a class='existingWikiWord' href='/nlab/show/diff/groupoid+of+Lie-algebra+valued+forms'>Lie algebra-valued 1-form</a> with values in the <a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+Lie+algebra'>Poincare Lie algebra</a>. <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>E</mi><mo>,</mo><mi>Ω</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>T</mi><mi>X</mi><mo>→</mo><mi>𝔦𝔰𝔬</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (E, \Omega) : T X \to \mathfrak{iso}(d-1,1) </annotation></semantics></math></div> such that this is a <a class='existingWikiWord' href='/nlab/show/diff/Cartan+connection'>Cartan connection</a>. </li> </ul> (This parameterization of the gravitational field is called the <a class='existingWikiWord' href='/nlab/show/diff/first-order+formulation+of+gravity'>first-order formulation of gravity</a>.) The component <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> of the connection is the <a class='existingWikiWord' href='/nlab/show/diff/vielbein'>vielbein</a> that encodes a <a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+metric'>pseudo-Riemannian metric</a> <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>=</mo><mi>E</mi><mo>⋅</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>g = E \cdot E</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and makes <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+manifold'>pseudo-Riemannian manifold</a>. Its quanta are the <a class='existingWikiWord' href='/nlab/show/diff/graviton'>graviton</a>s. The “non-propagating field” <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ω</mi></mrow><annotation encoding='application/x-tex'>\Omega</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/spin+connection'>spin connection</a>. The <a class='existingWikiWord' href='/nlab/show/diff/action+functional'>action functional</a> on the space of such connection which defines the <a class='existingWikiWord' href='/nlab/show/diff/classical+field+theory'>classical field theory</a> of gravity is the <a class='existingWikiWord' href='/nlab/show/diff/Einstein-Hilbert+action'>Einstein-Hilbert action</a>. More generally, <a class='existingWikiWord' href='/nlab/show/diff/supergravity'>supergravity</a> is a <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a> over a <a class='existingWikiWord' href='/nlab/show/diff/supermanifold'>supermanifold</a> <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/super+Euclidean+group'>super Poincare group</a>. The field of supergravity is a Lie-algebra valued form with values in the <a class='existingWikiWord' href='/nlab/show/diff/super+Poincar%C3%A9+Lie+algebra'>super Poincare Lie algebra</a>. <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>E</mi><mo>,</mo><mi>Ω</mi><mo>,</mo><mi>Ψ</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>T</mi><mi>X</mi><mo>→</mo><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (E,\Omega, \Psi) : T X \to \mathfrak{siso}(d-1,1) </annotation></semantics></math></div> The additional <a class='existingWikiWord' href='/nlab/show/diff/fermion'>fermion</a>ic field <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding='application/x-tex'>\Psi</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/gravitino'>gravitino</a> field. So the <a class='existingWikiWord' href='/nlab/show/diff/configuration+space'>configuration space</a> of gravity on some <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is essentially the <a class='existingWikiWord' href='/nlab/show/diff/moduli+space+of+Riemannian+metrics'>moduli space of Riemannian metrics</a> on <math class='maruku-mathml' display='inline' id='mathml_4cb95db795078b3c6e5728ce87217dbf3467417d_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. <h2 id='details'>Details</h2> <blockquote> for the moment see <a class='existingWikiWord' href='/nlab/show/diff/D%27Auria-Fr%C3%A9-Regge+formulation+of+supergravity'>D'Auria-Fre formulation of supergravity</a> for further details </blockquote> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Einstein-Hilbert+action'>Einstein-Hilbert action</a>, <a class='existingWikiWord' href='/nlab/show/diff/Einstein+equation'>Einstein equation</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/elasticity'>rubber-sheet analogy of gravity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/general+covariance'>general covariance</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Penrose+singularity+theorem'>Penrose-Hawking theorem</a>, <a class='existingWikiWord' href='/nlab/show/diff/cosmic+censorship+hypothesis'>cosmic censorship hypothesis</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/weak+gravity+conjecture'>weak gravity conjecture</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/first-order+formulation+of+gravity'>first order formulation of gravity</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Plebanski+formulation+of+gravity'>Plebanski formulation of gravity</a>, <a class='existingWikiWord' href='/nlab/show/diff/gravity+as+a+BF+theory'>gravity as a BF-theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/teleparallel+gravity'>teleparallel gravity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+gravity'>Chern-Simons gravity</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Einstein-Maxwell+theory'>Einstein-Maxwell theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Einstein-Yang-Mills+theory'>Einstein-Yang-Mills theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+curvature+correction'>higher order curvature corrections</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/supergravity'>supergravity</a> </li> <ins class='diffins'><li> <a class='existingWikiWord' href='/nlab/show/diff/Ho%C5%99ava%E2%80%93Lifshitz+gravity'>Hořava–Lifshitz gravity</a> </li></ins><ins class='diffins'> </ins><li> <a class='existingWikiWord' href='/nlab/show/diff/D%3D2+gravity'>D=2 gravity</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Liouville+theory'>Liouville theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jackiw-Teitelboim+gravity'>Jackiw-Teitelboim gravity</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/D%3D5+gravity'>D=5 gravity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/black+hole'>black hole</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/gravitational+wave'>gravitational wave</a> </li> </ul> </li> <li> gravitational entropy <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Bekenstein-Hawking+entropy'>Bekenstein-Hawking entropy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/generalized+second+law+of+thermodynamics'>generalized second law of thermodynamics</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/energy+condition'>energy condition</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/positive+energy+theorem'>positive energy theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/asymptotic+safety'>asymptotic safety</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cosmology'>cosmology</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/standard+model+of+cosmology'>standard model of cosmology</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/MOND'>MOND</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/experimental+mathematics'>computer experiment</a>: <a class='existingWikiWord' href='/nlab/show/diff/gevolution'>gevolution</a> </li> </ul> <h2 id='references'>References</h2> <h3 id='general'>General</h3> Historical texts: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Albert+Einstein'>Albert Einstein</a>, <a class='existingWikiWord' href='/nlab/show/diff/Marcel+Grossmann'>Marcel Grossmann</a>: Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation, Teubner (2013) [<a class='existingWikiWord' href='/nlab/files/EinsteinGrossmann-Entwurf.pdf' title='pdf'>pdf</a>] </li> <li id='Infeld1962'> <a class='existingWikiWord' href='/nlab/show/diff/Leopold+Infeld'>Leopold Infeld</a> (ed.), Relativistic Theories of Gravitation, Proceedings of a conference held in Warsaw and Jablonna 1962, Pergamon Press (1964) [[pdf](https://cds.cern.ch/record/2282975/files/warsaw-1962.pdf)] </li> </ul> On the early history of the idea: <ul> <li id='EarmanGlymor78'>John Earman, Clark Glymour: Lost in the tensors: Einstein’s struggles with covariance principles 1912–1916, Studies in History and Philosophy of Science Part A Volume 9, Issue 4, (1978) 251-278 [<a href='https://doi.org/10.1016/0039-3681(78)90008-0'>doi:10.1016/0039-3681(78)90008-0</a>]</li> </ul> Monographs: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Robert+Geroch'>Robert Geroch</a>: General Relativity – 1972 Lecture Notes, Minkowski Institute Press (2013) [[ISBN:78-0-9879871-7-4](https://www.minkowskiinstitute.com/one/#!/products/r-0x2e–robert-geroch-0x2c–general-relativity), <a href='http://strangebeautiful.com/other-texts/geroch-gr.pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Steven+Weinberg'>Steven Weinberg</a>: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley (1972, 2013) [[ISBN:978-0-471-92567-5](https://www.wiley.com/en-us/Gravitation+and+Cosmology%3A+Principles+and+Applications+of+the+General+Theory+of+Relativity-p-9780471925675), <a href='https://archive.org/details/WeinbergS.GravitationAndCosmology..PrinciplesAndApplicationsOfTheGeneralTheoryOf/page/n1/mode/2up'>ark:/13960/t13n7rw1f</a>, <a href='https://inspirehep.net/literature/1410180'>spire:1410180</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Charles+Misner'>Charles Misner</a>, <a class='existingWikiWord' href='/nlab/show/diff/Kip+Thorne'>Kip Thorne</a>, <a class='existingWikiWord' href='/nlab/show/diff/John+Wheeler'>John Wheeler</a>, <a class='existingWikiWord' href='/nlab/show/diff/Gravitation'>Gravitation</a> (1973) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Theodore+Frankel'>Theodore Frankel</a>: Gravitational Curvature, Freeman, San Francisco (1979) [[ark:13960/t58d7nn19](https://archive.org/details/gravitationalcur0000fran/)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Robert+Wald'>Robert Wald</a>, General Relativity, University of Chicago Press (1984) [[doi:10.7208/chicago/9780226870373.001.0001](https://doi.org/10.7208/chicago/9780226870373.001.0001), <a href='http://fma.if.usp.br/~mlima/teaching/PGF5292_2021/Wald_GR.pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Garth+Warner'>Garth Warner</a>: Mathematical Aspects of General Relativity, EPrint Collection, University Of Washington (2006) [[hdl:1773/2637](http://hdl.handle.net/1773/2637), <a href='https://sites.math.washington.edu//~warner/MAGR_Warner.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Warner-GeneralRelativity.pdf' title='pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Thanu+Padmanabhan'>Thanu Padmanabhan</a>, Gravitation – Foundations and Frontiers, Cambridge University Press (2012) [[doi:10.1017/CBO9780511807787](https://doi.org/10.1017/CBO9780511807787), <a href='https://inspirehep.net/literature/852758'>spire:852758</a>, toc: <a href='https://web.iucaa.in/~paddy/books/GRTOC.pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Pietro+Fr%C3%A9'>Pietro Fré</a>, Gravity, a Geometrical Course, Volume 1: Development of the Theory and Basic Physical Applications, Spinger (2013) [[doi:10.1007/978-94-007-5361-7](https://doi.org/10.1007/978-94-007-5361-7)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Pietro+Fr%C3%A9'>Pietro Fré</a>, Gravity, a Geometrical Course, Volume 2: Black Holes, Cosmology and Introduction to Supergravity, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)] <blockquote> on <a class='existingWikiWord' href='/nlab/show/diff/black+hole'>black holes</a>, <a class='existingWikiWord' href='/nlab/show/diff/cosmology'>cosmology</a> and the (<a class='existingWikiWord' href='/nlab/show/diff/D%27Auria-Fr%C3%A9-Regge+formulation+of+supergravity'>D'Auria-Fré formulation of</a>) <a class='existingWikiWord' href='/nlab/show/diff/supergravity'>supergravity</a> </blockquote> </li> </ul> Background on <a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+manifold'>pseudo-</a><a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Barrett+O%27Neill'>Barrett O'Neill</a>, Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics 103, Academic Press (1983) [[ISBN:9780125267403](https://shop.elsevier.com/books/semi-riemannian-geometry-with-applications-to-relativity/oneill/978-0-12-526740-3)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Shlomo+Sternberg'>Shlomo Sternberg</a>, Semi-Riemannian Geometry and General Relativity (2003) [[pdf](https://people.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf), <a href='https://archive.org/details/Shlomo_Sternberg__SemiRiemann_Geometry_and_General_Relativity/'>ark:/13960/t5m927d2v</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Shlomo+Sternberg'>Shlomo Sternberg</a>, Curvature in Mathematical Physics, Dover (2012) [[ISBN:9780486478555](https://store.doverpublications.com/products/9780486478555)] </li> </ul> Lecture notes: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Matthias+Blau'>Matthias Blau</a>, Lecture notes on general relativity (<a href='http://www.blau.itp.unibe.ch/GRLecturenotes.html'>web</a>) </li> <li> Emil T. Akhmedov, Lectures on General Theory of Relativity (<a href='https://arxiv.org/abs/1601.04996'>arXiv:1601.04996</a>) </li> <li> Pietro Menotti, Lectures on gravitation (<a href='https://arxiv.org/abs/1703.05155'>arXiv:1703.05155</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Daniel+Baumann'>Daniel Baumann</a>, <a href='http://cosmology.amsterdam/general-relativity/'>General Relativity</a> (2021-24) [[pdf](http://cosmology.amsterdam/wp-content/uploads/2021/10/GR-Oct19.pdf), <a href='https://cdn.prod.website-files.com/65c089cfdfce11a0392e5c42/67469a196f855821380fffa4_GR-2024.pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Edward+Witten'>Edward Witten</a>, Light Rays, Singularities, and All That, Rev. Mod. Phys. 92 45004 (2020) [[arXiv:1901.03928](https://arxiv.org/abs/1901.03928), <a href='https://doi.org/10.1103/RevModPhys.92.045004'>doi:10.1103/RevModPhys.92.045004</a>, lecture: <a href='https://www.youtube.com/watch?v=-vS_WIa0vYo'>video 1</a>, <a href='https://www.youtube.com/watch?v=4kEiNotDhck'>2</a>, <a href='https://www.youtube.com/watch?v=Qje1ak6XF20'>3</a>, <a href='https://www.youtube.com/watch?v=tzt25nkGg3Q'>4</a>, <a href='https://www.youtube.com/watch?v=yC5BHqPw38Q'>5</a>, <a href='https://www.youtube.com/watch?v=GHPL2_aomo8'>6</a>, <a href='https://www.youtube.com/watch?v=tDeQ9ght3aI'>7</a>, <a href='https://www.youtube.com/watch?v=fsmcGpKFbFU'>8</a>, <a href='https://www.youtube.com/watch?v=Z_crNwSKpW0'>9</a>] <blockquote> (with focus on <a class='existingWikiWord' href='/nlab/show/diff/causal+structure'>causal structure</a>, the <a class='existingWikiWord' href='/nlab/show/diff/Penrose+singularity+theorem'>Penrose singularity theorem</a> and related aspects) </blockquote> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kirill+Krasnov'>Kirill Krasnov</a>, Formulations of General Relativity, PIRSA lecture series (Feb 2019) [part 1:doi:10.48660/19020079, 2:doi:10.48660/19020080, 3:doi:10.48660/19020081, 4:doi:10.48660/19020082] </li> <li> Peter Hayman: A Lean and Mean Introduction to Modern General Relativity [[arXiv:2412.08026](https://arxiv.org/abs/2412.08026)] </li> </ul> See also <ul> <li id='Coley18'><a class='existingWikiWord' href='/nlab/show/diff/Alan+Coley'>Alan Coley</a>, Mathematical General Relativity (<a href='https://arxiv.org/abs/1807.08628'>arXiv:1807.08628</a>)</li> </ul> With focus on methods of <a class='existingWikiWord' href='/nlab/show/diff/conformal+geometry'>conformal geometry</a> (<a class='existingWikiWord' href='/nlab/show/diff/conformal+compactification'>conformal boundaries</a>, <a class='existingWikiWord' href='/nlab/show/diff/conformal+compactification'>conformal compactification</a>): <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Juan+A.+Valiente+Kroon'>Juan A. Valiente Kroon</a>: Conformal Methods in General Relativity, Cambridge University Press (2017, 2022) [[doi:10.1017/9781009291309](https://doi.org/10.1017/9781009291309), <a href='https://library.oapen.org/handle/20.500.12657/59217'>oapen:20.500.12657/59217</a>, <a href='https://library.oapen.org/bitstream/id/548cfabe-a25b-4562-b95e-aa2894525f32/9781009291309.pdf'>pdf</a>]</li> </ul> On gravity in relation to <a class='existingWikiWord' href='/nlab/show/diff/thermodynamics'>thermodynamics</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Thanu+Padmanabhan'>Thanu Padmanabhan</a>, Gravity and/is Thermodynamics, Current Science, 109 (2015) 2236-2242 [[doi:10.18520/v109/i12/2236-2242](https://doi.org/10.18520/v109/i12/2236-2242)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Thanu+Padmanabhan'>Thanu Padmanabhan</a>, Exploring the Nature of Gravity, talk notes [[arXiv:1602.01474](https://arxiv.org/abs/1602.01474)] </li> </ul> Discussion of classical gravity via its <a class='existingWikiWord' href='/nlab/show/diff/perturbative+quantum+field+theory'>perturbative quantum field theory</a>: <ul> <li id='Scharf01'> <a class='existingWikiWord' href='/nlab/show/diff/G%C3%BCnter+Scharf'>Günter Scharf</a>, <a class='existingWikiWord' href='/nlab/show/diff/Quantum+Gauge+Theories+--+A+True+Ghost+Story'>Quantum Gauge Theories -- A True Ghost Story</a>, Wiley 2001 </li> <li> Gustav Uhre Jakobsen, General Relativity from Quantum Field Theory (<a href='https://arxiv.org/abs/2010.08839'>arXiv:2010.08839</a>) </li> </ul> This way the theory of gravity based on the standard <a class='existingWikiWord' href='/nlab/show/diff/Einstein-Hilbert+action'>Einstein-Hilbert action</a> may be regarded as just an <a class='existingWikiWord' href='/nlab/show/diff/effective+quantum+field+theory'>effective quantum field theory</a>, which makes some of its notorious problems be non-problems: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/John+Donoghue'>John Donoghue</a>, Introduction to the Effective Field Theory Description of Gravity (<a href='http://arxiv.org/abs/gr-qc/9512024'>arXiv:gr-qc/9512024</a>)</li> </ul> Relation of the <a class='existingWikiWord' href='/nlab/show/diff/first-order+formulation+of+gravity'>first-order formulation of gravity</a> to <a class='existingWikiWord' href='/nlab/show/diff/BF-theory'>BF-theory</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Alberto+Cattaneo'>Alberto S. Cattaneo</a>, <a class='existingWikiWord' href='/nlab/show/diff/Leon+Menger'>Leon Menger</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michele+Schiavina'>Michele Schiavina</a>, Gravity with torsion as deformed BF theory [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]</li> </ul> See also the references at <a class='existingWikiWord' href='/nlab/show/diff/general+relativity'>general relativity</a>. <h3 id='CovariantPhaseSpaceReferences'>Covariant phase space</h3> The (reduced) <a class='existingWikiWord' href='/nlab/show/diff/phase+space'>covariant phase space</a> of gravity (presented for instance by its <a class='existingWikiWord' href='/nlab/show/diff/BV-BRST+formalism'>BV-BRST complex</a>, see there fore more details) is discussed for instance in <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Romeo+Brunetti'>Romeo Brunetti</a>, <a class='existingWikiWord' href='/nlab/show/diff/Klaus+Fredenhagen'>Klaus Fredenhagen</a>, Towards a Background Independent Formulation of Perturbative Quantum Gravity (<a href='http://arxiv.org/abs/gr-qc/0603079'>arXiv:gr-qc/0603079</a>)</li> </ul> which is surveyed in <ul> <li>Katarzyna Rejzner, The BV formalism applied to classical gravity (<a href='http://rejzner.com/talks/Karlsruhe2011.pdf'>pdf</a>)</li> </ul> Careful discussion of <a class='existingWikiWord' href='/nlab/show/diff/observable'>observables</a> in gravity is in <ul> <li id='Khavkine14'><a class='existingWikiWord' href='/nlab/show/diff/Igor+Khavkine'>Igor Khavkine</a>, Local and gauge invariant observables in gravity, talk at <a href='http://www.science.unitn.it/~moretti/convegno/convegno.html'>Operator and Geometric Analysis on Quantum Theory</a>, preprint <a href='https://arxiv.org/abs/1503.03754'>arXiv:1503.03754</a></li> </ul> Further discussion of the phase space of gravity in <a class='existingWikiWord' href='/nlab/show/diff/first-order+formulation+of+gravity'>first-order formulation</a> via <a class='existingWikiWord' href='/nlab/show/diff/BV-BRST+formalism'>BV-BFV formalism</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Michele+Schiavina'>Michele Schiavina</a>, BV-BFV approach to general relativity (2015) [[pdf](http://user.math.uzh.ch/cattaneo/schiavina.pdf), <a class='existingWikiWord' href='/nlab/files/Schiavina-Relativity.pdf' title='pdf'>pdf</a>] </li> <li id='CattaneoSchiavina15'> <a class='existingWikiWord' href='/nlab/show/diff/Alberto+Cattaneo'>Alberto Cattaneo</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michele+Schiavina'>Michele Schiavina</a>, BV-BFV approach to General Relativity, Einstein-Hilbert action, J. Math. Phys. 57 023515 (2016) [[arXiv:1509.05762](https://arxiv.org/abs/1509.05762), <a href='https://doi.org/10.1063/1.4941410'>doi:10.1063/1.4941410</a>] </li> <li id='CattaneoSchiavina17a'> <a class='existingWikiWord' href='/nlab/show/diff/Alberto+Cattaneo'>Alberto Cattaneo</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michele+Schiavina'>Michele Schiavina</a>, The reduced phase space of Palatini-Cartan-Holst theory, Ann. Henri Poincaré 20 (2019) 445 [[arXiv:1707.05351](https://arxiv.org/abs/1707.05351), <a href='https://doi.org/10.1007/s00023-018-0733-z'>doi:10.1007/s00023-018-0733-z</a>] </li> <li id='CattaneoSchiavina17b'> <a class='existingWikiWord' href='/nlab/show/diff/Alberto+Cattaneo'>Alberto Cattaneo</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michele+Schiavina'>Michele Schiavina</a>, BV-BFV approach to General Relativity: Palatini-Cartan-Holst action, Adv. Theor. Math. Phys. 23 (2019) 1801-1835 [[arXiv:1707.06328](https://arxiv.org/abs/1707.06328), <a href='https://doi.org/10.4310/ATMP.2019.v23.n8.a3'>doi:10.4310/ATMP.2019.v23.n8.a3</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Alberto+Cattaneo'>Alberto S. Cattaneo</a>, Phase space for gravity with boundaries, in <a class='existingWikiWord' href='/nlab/show/diff/Encyclopedia+of+Mathematical+Physics+2nd+ed'>Encyclopedia of Mathematical Physics 2nd ed</a> (2023) [[arXiv:2307.04666](https://arxiv.org/abs/2307.04666)] <blockquote> (following <a href='phase+space#KijowskiTulczyjew05'>Kijowski & Tulczyjew (2005)</a>) </blockquote> </li> </ul> Discussion of flux-observables: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Alberto+Cattaneo'>Alberto S. Cattaneo</a>, Alejandro Perez, A note on the Poisson bracket of 2d smeared fluxes in loop quantum gravity, Class. Quant. Grav. 34 (2017) 107001 [[arXiv:1611.08394](https://arxiv.org/abs/1611.08394), <a href='https://doi.org/10.1088/1361-6382/aa69b4'>doi:10.1088/1361-6382/aa69b4</a>]</li> </ul> <h3 id='nonrenormalizability'>Non-renormalizability</h3> The result that gravity is not <a class='existingWikiWord' href='/nlab/show/diff/renormalizable+interaction'>renormalizable</a> is due to: <ul> <li id='tHooftVelrman74'><a class='existingWikiWord' href='/nlab/show/diff/Gerard+%E2%80%99t+Hooft'>Gerard 't Hooft</a>, <a class='existingWikiWord' href='/nlab/show/diff/Martinus+Veltman'>Martinus Veltman</a>, One loop divergencies in the theory of gravitation, Ann. Inst. Poincar é 20 (1974) 69 (<a href='AIHPA_1974__20_1_69_0'>numdam:AIHPA_1974__20_1_69_0</a>)</li> </ul> Review: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Zvi+Bern'>Zvi Bern</a>, Perturbative quantum gravity and its relation to gauge theory, Living reviews in relativity, <a href='www.livingreviews.org/Articles/Volume5/2002-5bern'>www.livingreviews.org/Articles/Volume5/2002-5bern</a> .</li> <li>Assaf Shomer, A pedagogical explanation for the non-renormalizability of gravity (<a href='https://arxiv.org/abs/0709.3555'>arXiv:0709.3555</a>)</li> </ul> </div> <div class="revisedby"> Last revised on February 27, 2025 at 06:35:57. 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