Meta tags:

Headings (most frequently used words):

conformal, field, theory, symmetry, vs, dimensions, and, critical, model, invariance, two, global, in, theories, algebra, contents, scale, higher, correlation, functions, bootstrap, features, examples, applications, see, also, references, further, reading, external, links, local, with, virasoro, definition, jacobian, group, issues, minkowski, space, behaviour, under, transformations, dependence, on, positions, operator, product, expansion, blocks, crossing, unitarity, compactness, extra, symmetries, mean, ising, potts, gauge, continuous, phase, transitions, string, ads, cft, correspondence, perturbation,

Text of the page (most frequently used words):
the (306), conformal (167), displaystyle (164), #theory (123), field (109), and (104), delta (56), fields (51), are (50), two (46), for (44), that (43), #functions (42), quantum (41), dimensions (41), point (39), symmetry (37), edit (37), critical (33), model (31), theories (29), this (28), primary (28), correlation (27), cft (27), space (26), can (26), algebra (25), dimensional (25), not (25), left (25), right (25), with (24), which (23), from (22), group (22), langle (22), rangle (22), transformations (21), operator (20), phi (20), physics (19), dimension (19), blocks (19), partial (18), doi (18), bibcode (18), frac (18), tensor (17), product (17), invariance (17), invariant (17), function (17), under (16), non (16), scalar (16), arxiv (16), three (16), scale (16), have (15), where (15), all (14), euclidean (14), representation (13), ising (13), bootstrap (13), unitary (13), but (13), four (13), mathbb (13), there (13), structure (13), string (12), analysis (12), higher (12), global (12), case (12), ope (12), statistical (11), numerical (11), also (11), mean (11), s2cid (11), ads (11), called (11), its (11), terms (10), differential (10), equations (10), issn (10), expansion (10), local (10), example (10), their (10), such (10), has (10), sum (10), vector (9), 1007 (9), particular (9), only (9), positions (9), more (9), minkowski (9), toggle (8), wikipedia (8), may (8), mathematical (8), mechanics (8), potts (8), state (8), main (8), article (8), transformation (8), cdots (8), determined (8), mathematics (7), applications (7), perturbation (7), gauge (7), number (7), links (7), models (7), phase (7), states (7), unitarity (7), fixed (7), techniques (7), one (7), correspondence (7), possible (7), rotation (7), conformally (7), any (7), compact (7), exactly (7), virasoro (7), descendant (7), constants (7), order (7), rho (7), ell (7), special (7), scaling (6), symmetries (6), computational (6), methods (6), applied (6), related (6), other (6), spacetime (6), renormalization (6), physical (6), software (6), finite (6), type (6), yang (6), mills (6), qft (6), rychkov (6), slava (6), 2018 (6), cfts (6), infinite (6), known (6), examples (6), central (6), corresponding (6), then (6), lorentz (6), prod (6), these (6), vert (6), 123 (6), omega (6), they (6), subsection (6), page (5), description (5), supersymmetry (5), stochastic (5), arbitrary (5), information (5), theorem (5), super (5), free (5), lattice (5), matter (5), principle (5), isbn (5), 2016 (5), 1103 (5), journal (5), high (5), energy (5), analytic (5), integer (5), zamolodchikov (5), charge (5), should (5), continuous (5), transitions (5), points (5), some (5), real (5), spectrum (5), into (5), than (5), implies (5), channel (5), depend (5), constant (5), over (5), move (5), contents (4), search (4), text (4), using (4), commons (4), articles (4), industrial (4), structures (4), supersymmetric (4), topological (4), linear (4), systems (4), geometry (4), constraint (4), see (4), history (4), minimal (4), condensed (4), relations (4), 1997 (4), 978 (4), 1016 (4), david (4), 2019 (4), 1029 (4), 8479 (4), pdf (4), 0550 (4), 3213 (4), computed (4), rotations (4), translations (4), form (4), many (4), solved (4), limit (4), infty (4), varepsilon (4), exists (4), defined (4), does (4), conserved (4), neq (4), whose (4), extra (4), depends (4), flat (4), crossing (4), same (4), while (4), because (4), coefficient (4), dots (4), traces (4), eta (4), dependence (4), behaviour (4), dilations (4), mathrm (4), hide (4), sidebar (4), view (3), statistics (3), additional (3), wikimedia (3), was (3), categories (3), wikidata (3), sciences (3), particle (3), bosonic (3), potential (3), classical (3), lagrangian (3), calculus (3), geometric (3), chaos (3), dynamical (3), discrete (3), gravity (3), supergravity (3), superconformal (3), wess (3), polyakov (3), logarithmic (3), liouville (3), boltzmann (3), 076 (3), sphere (3), data (3), wilson (3), fisher (3), physrevd (3), review (3), 2013 (3), 2020 (3), sense (3), original (3), without (3), boundary (3), resulting (3), equivalent (3), extended (3), constraints (3), spatial (3), described (3), lorentzian (3), necessary (3), condition (3), wick (3), often (3), however (3), will (3), results (3), constructed (3), includes (3), geq (3), stress (3), needed (3), built (3), out (3), generalized (3), appear (3), them (3), must (3), obey (3), powerful (3), axioms (3), features (3), coefficients (3), symmetric (3), written (3), independent (3), 12p (3), combination (3), used (3), given (3), characterized (3), been (3), cylinder (3), generate (3), jacobian (3), definition (3), copies (3), killing (3), sometimes (3), tools (3), languages (2), table (2), contact (2), about (2), privacy (2), policy (2), category (2), link (2), unsourced (2), statements (2), 2021 (2), clarification (2), 2023 (2), short (2), retrieved (2), title (2), topics (2), portal (2), community (2), society (2), natural (2), optimization (2), integral (2), algebraic (2), dynamics (2), variational (2), random (2), ordinary (2), exterior (2), harmonic (2), clifford (2), logic (2), thermodynamics (2), science (2), shell (2), loop (2), chern (2), simons (2), iib (2), zumino (2), witten (2), thirring (2), schwinger (2), chromodynamics (2), electrodynamics (2), sigma (2), heisenberg (2), thermal (2), system (2), entropy (2), size (2), transition (2), phenomena (2), equation (2), approaches (2), force (2), maxwell (2), thermodynamic (2), external (2), springer (2), 540 (2), 319 (2), further (2), reading (2), matrix (2), 1002 (2), 125025 (2), 0010 (2), simmons (2), duffin (2), 217 (2), 1809 (2), poland (2), 004 (2), 3616 (2), alexander (2), 105001 (2), 119 (2), jhep06 (2), 2018jhep (2), jhep (2), 117 (2), 180 (2), convergence (2), 105043 (2), francesco (2), new (2), york (2), recursion (2), lüscher (2), mack (2), lett (2), 2011 (2), belavin (2), 241 (2), nuclear (2), 2005 (2), 622 (2), hep (2), 303 (2), references (2), construct (2), perturbatively (2), play (2), role (2), dual (2), denotes (2), superstring (2), translation (2), happen (2), equal (2), describing (2), absence (2), exceptional (2), reflection (2), positive (2), either (2), abelian (2), charged (2), when (2), understood (2), scalars (2), spins (2), continuum (2), generalization (2), corresponds (2), orthogonal (2), hypercubic (2), just (2), laplacian (2), power (2), generic (2), similarly (2), trivial (2), name (2), contains (2), big (2), instance (2), deduced (2), addition (2), hand (2), identity (2), fact (2), conditions (2), unique (2), vacuum (2), finitely (2), compactness (2), inequalities (2), bound (2), osterwalder (2), schrader (2), dilation (2), hilbert (2), cross (2), ratios (2), lie (2), cases (2), decomposition (2), min (2), contributions (2), descendants (2), involve (2), several (2), compute (2), fundamental (2), expansions (2), efficient (2), reason (2), derivatives (2), therefore (2), primaries (2), runs (2), 12k (2), general (2), zero (2), traceless (2), rank (2), set (2), basis (2), vanishes (2), even (2), common (2), important (2), subgroup (2), induced (2), subalgebra (2), verma (2), module (2), here (2), factor (2), associated (2), transform (2), themselves (2), linearly (2), focus (2), minkowskian (2), approach (2), context (2), viewed (2), shown (2), poincaré (2), patch (2), causality (2), issues (2), metric (2), aligned (2), inversion (2), leave (2), circ (2), generators (2), centrally (2), moving (2), flow (2), witt (2), infinitesimal (2), ideas (2), total (2), derivative (2), psl (2), riemann (2), infinity (2), bar (2), development (2), assumptions (2), appearance (2), upload (2), file (2), changes (2), read (2), log (2), create (2), account (2), donate (2), menu (2), add, topic, mobile, cookie, statement, developers, code, conduct, legal, safety, contacts, disclaimers, available, apply, site, you, agree, registered, trademark, profit, organization, foundation, inc, use, creative, attribution, sharealike, license, rendered, parsoid, last, edited, june, 2026, utc, hidden, needing, april, different, https, org, index, php, conformal_field_theory, oldid, 1361182529, list, outline, european, international, council, société, mathématiques, appliquées, industrielles, japan, organizations, unreasonable, effectiveness, sociology, psychology, chemistry, biology, finance, economics, social, choice, operations, research, game, decision, superalgebra, poisson, feynman, effective, hamiltonian, analytical, path, processes, variables, distributions, probability, validated, numerics, multivariable, multilinear, fourier, functional, control, forms, approximation, graph, combinatorics, computer, solvers, proof, assistants, libraries, interactive, element, precision, arithmetic, cryptography, programming, satisfaction, coding, automated, proving, automata, design, algorithms, template, hydrodynamics, hadrodynamics, fluctuations, foam, off, cosmology, cosmic, casimir, effect, stueckelberg, standard, nmssm, nambu, jona, lasinio, mssm, fermi, chiral, 11d, iia, pure, abjm, qcd, seiberg, massless, toda, sine, gordon, gross, neveu, bullough, dodd, low, yukawa, higgs, soler, quartic, interaction, proca, ginzburg, landau, euler, born, infeld, regular, curved, noncommutative, axiomatic, machine, complex, superfluidity, elementary, von, neumann, rényi, tsallis, shannon, gibbs, length, exponents, process, bbgky, hierarchy, vlasov, particles, lennard, jones, depletion, percolation, ferromagnetism, partition, ensembles, ergodic, maximum, media, zoo, martin, schottenloher, verlag, berlin, heidelberg, 2nd, edition, 2008, 68625, 61753, introduction, springerbriefs, 119192484, 43625, 43626, 1601, 05000, epfl, lectures, läuchli, andreas, herviou, loïc, wilhelm, patrick, 2025, 21468, scipostphys, 2025scpp, 76l, 2504, 00842, scipost, exact, diagonalization, study, fuzzy, henriksson, johan, 0370, 1573, physrep, 2022, 002, 2023phr, 2201, 09520, reports, hogervorst, matthijs, van, rees, balt, violation, 55817425, 2470, 2016phrvd, 93l5025h, 1512, 00013, karateev, denis, kravchuk, petr, jhep10, 2019jhep, 217k, 05111, fitzpatrick, liam, kaplan, jared, superhorizon, locality, 45739036, jhep12, 2013jhep, 004f, 1212, yao, manashov, mixing, fermionic, 59401427, 2018phrvd, 98j5001j, 00021, d98, levy, 119441506, 119l, 1804, 02283, 1806, binder, damon, deligne, making, 208158396, jhep04, 2020jhep, 117b, 1911, 07895, isachenkov, mikhail, schomerus, volker, jhep07, 180i, 1711, 06609, integrability, part, calogero, sutherland, scattering, pappadopulo, duccio, espin, johnny, rattazzi, riccardo, 2012, 119155687, 2012phrvd, 86j5043p, 1208, 6449v3, vichi, alessandro, 15002, 119317682, 0034, 6861, revmodphys, 015002, 2019rvmp, 91a5002p, 1805, 04405, reviews, modern, philippe, 104, 4612, 2256, penedones, joão, trevisani, emilio, yamazaki, masahito, jhep09, 070, 2016jhep, 070p, 1509, 00428, 1975, 234, 120910626, bf01608988, 1975cmaph, 203l, 203, communications, 1986, 732, archived, 2015, 1986jetpl, 730z, 730, jetp, irreversibility, flux, rajabpour, 118397215, 2011jhep, 076r, 3625, mathieu, sénéchal, 387, 94785, 1984, 380, 90052, 1984nuphb, 333b, 333, elasticity, riva, cardy, counterexample, 342, 119175109, physletb, 010, 2005phlb, 339r, 0504197, 339, phys, polchinski, joseph, 1988, 236, 90179, 1988nuphb, 226p, 226, 1989, published, les, houches, elsevier, publishers, zinn, justin, brézin, ecole, eté, physique, théorique, champs, cordes, phénomènes, critiques, strings, 9108028, paul, ginsparg, consists, discretizing, studying, effects, perturbing, technique, prominent, gravitational, notable, prefix, degree, possessed, time, anti, sitter, world, sheet, involves, coupled, consistency, imposes, coordinates, lives, correspond, apart, exponent, quenched, disorder, always, sufficient, admit, sufficiently, banks, zaks, qed, large, means, well, hubbard, stratonovich, alternatively, becomes, question, vectors, interacting, fortuin, kasteleyn, reformulation, clusters, colors, range, depending, permutation, flipping, describes, magnetization, fluids, opalescence, universality, class, mathcal, quadratic, action, involving, raised, determines, citation, starting, spin, antisymmetric, decomposing, belong, modulo, integers, gamma, reads, believed, find, epsilon, insertion, modify, comes, satisfy, target, less, obeys, turn, certain, based, exploiting, bounded, below, lower, transforms, positivity, definite, self, adjoint, endows, thus, satisfying, collectively, referred, converges, analytically, continued, pairwise, coinciding, values, single, valued, except, circle, singly, transposed, 1324, converge, cyclic, equality, decompositions, 14p, p23, 13p, p24, block, contributes, integrable, p34, existence, axiom, generally, operators, rather, computing, practice, viewing, relation, between, shows, associative, furthermore, commutative, series, reduced, rewriting, contribution, equivalently, actually, absent, due, takes, radius, provided, close, enough, rewrites, chosen, technical, convenience, introduce, necessarily, allows, each, 2x_, choose, normalization, diagonalizable, diagonal, differ, severely, constrain, freedom, combinations, collection, representations, complicated, thereof, sums, behaves, highest, weight, generated, characterizes, appears, throughout, literature, maximally, larger, recently, appreciated, parabolic, cartan, would, quad, via, acts, commute, spacelike, separated, thanks, obtained, continuation, wightman, parameters, make, notations, writing, preserve, observables, restore, extending, region, violate, instead, outside, generates, begin, im_, end, commutation, exchanges, cup, lambda, locally, isomorphic, preserves, angles, algebras, contain, unless, cannot, exist, leaves, entire, unbroken, best, rest, spontaneously, broken, being, complexified, holomorphic, antiholomorphic, both, decreases, monotonically, tells, irreversible, extension, anomaly, studied, done, purpose, testing, tested, applying, comparing, follow, strictly, speaking, possessing, still, exhibiting, turns, biharmonic, property, requires, second, reduce, cauchy, modes, coordinate, yield, cdot, möbius, earlier, deeper, after, 1983, term, meaning, textbook, become, popular, late, 1990s, 2000s, makes, much, constraining, share, solvable, contrast, dominate, rare, interchangeably, completely, rule, hence, prove, stronger, needs, argue, nature, basic, idea, behind, plausibility, currents, include, trace, implying, indeed, classified, enjoying, encyclopedia, item, projects, printable, version, download, print, export, switch, legacy, parser, get, shortened, url, cite, permanent, what, actions, english, talk, українська, svenska, русский, português, polski, ਪੰਜਾਬੀ, nederlands, 한국어, 日本語, italiano, français, español, deutsch, català, العربية, top, personal, pages, recent, learn, help, contribute, current, events, navigation, jump, content,

Text of the page (random words):
and dilations i e scale transformations x μ λ x μ displaystyle x mu to lambda x mu this also includes special conformal transformations for any translation t a x x a displaystyle t_ a x x a there is a special conformal transformation s a i t a i displaystyle s_ a i circ t_ a circ i where i displaystyle i is the inversion such that i x μ x μ x 2 displaystyle i left x mu right frac x mu x 2 in the sphere s d r d displaystyle s d mathbb r d cup infty the inversion exchanges 0 displaystyle 0 with displaystyle infty translations leave displaystyle infty fixed while special conformal transformations leave 0 displaystyle 0 fixed conformal algebra edit the commutation relations of the corresponding lie algebra are p μ p ν 0 d k μ k μ d p μ p μ k μ k ν 0 k μ p ν η μ ν d i m μ ν displaystyle begin aligned p_ mu p_ nu 0 d k_ mu k_ mu d p_ mu p_ mu k_ mu k_ nu 0 k_ mu p_ nu eta _ mu nu d im_ mu nu end aligned where p displaystyle p generate translations d displaystyle d generates dilations k μ displaystyle k_ mu generate special conformal transformations and m μ ν displaystyle m_ mu nu generate rotations or lorentz transformations the tensor η μ ν displaystyle eta _ mu nu is the flat metric global issues in minkowski space edit in minkowski space the conformal group does not preserve causality observables such as correlation functions are invariant under the conformal algebra but not under the conformal group as shown by lüscher and mack it is possible to restore the invariance under the conformal group by extending the flat minkowski space into a lorentzian cylinder 8 the original minkowski space is conformally equivalent to a region of the cylinder called a poincaré patch in the cylinder global conformal transformations do not violate causality instead they can move points outside the poincaré patch correlation functions and conformal bootstrap edit in the conformal bootstrap approach a conformal field theory is a set of correlation functions that obey a number of axioms the n displaystyle n point correlation function o 1 x 1 o n x n displaystyle left langle o_ 1 x_ 1 cdots o_ n x_ n right rangle is a function of the positions x i displaystyle x_ i and other parameters of the fields o 1 o n displaystyle o_ 1 dots o_ n in the bootstrap approach the fields themselves make sense only in the context of correlation functions and may be viewed as efficient notations for writing axioms for correlation functions correlation functions depend linearly on fields in particular x 1 o 1 x 1 x 1 o 1 x 1 displaystyle partial _ x_ 1 left langle o_ 1 x_ 1 cdots right rangle left langle partial _ x_ 1 o_ 1 x_ 1 cdots right rangle we focus on cft on the euclidean space r d displaystyle mathbb r d in this case correlation functions are schwinger functions they are defined for x i x j displaystyle x_ i neq x_ j and do not depend on the order of the fields in minkowski space correlation functions are wightman functions they can depend on the order of the fields as fields commute only if they are spacelike separated a euclidean cft can be related to a minkowskian cft by wick rotation for example thanks to the osterwalder schrader theorem in such cases minkowskian correlation functions are obtained from euclidean correlation functions by an analytic continuation that depends on the order of the fields behaviour under conformal transformations edit any conformal transformation x f x displaystyle x to f x acts linearly on fields o x π f o x displaystyle o x to pi _ f o x such that f π f displaystyle f to pi _ f is a representation of the conformal group and correlation functions are invariant π f o 1 x 1 π f o n x n o 1 x 1 o n x n displaystyle left langle pi _ f o_ 1 x_ 1 cdots pi _ f o_ n x_ n right rangle left langle o_ 1 x_ 1 cdots o_ n x_ n right rangle primary fields are fields that transform into themselves via π f displaystyle pi _ f the behaviour of a primary field is characterized by a number δ displaystyle delta called its conformal dimension and a representation ρ displaystyle rho of the rotation or lorentz group for a primary field we then have π f o x ω x δ ρ r x o x where x f 1 x displaystyle pi _ f o x omega x delta rho r x o x quad text where x f 1 x here ω x displaystyle omega x and r x displaystyle r x are the scale factor and rotation that are associated to the conformal transformation f displaystyle f the representation ρ displaystyle rho is trivial in the case of scalar fields which transform as π f o x ω x δ o x displaystyle pi _ f o x omega x delta o x for vector fields the representation ρ displaystyle rho is the fundamental representation and we would have π f o μ x ω x δ r μ ν x o ν x displaystyle pi _ f o_ mu x omega x delta r_ mu nu x o_ nu x a primary field that is characterized by the conformal dimension δ displaystyle delta and representation ρ displaystyle rho behaves as a highest weight vector in an induced representation of the conformal group from the subgroup generated by dilations and rotations in particular the conformal dimension δ displaystyle delta characterizes a representation of the subgroup of dilations in two dimensions the fact that this induced representation is a verma module appears throughout the literature for higher dimensional cfts in which the maximally compact subalgebra is larger than the cartan subalgebra it has recently been appreciated that this representation is a parabolic or generalized verma module 9 derivatives of any order of primary fields are called descendant fields their behaviour under conformal transformations is more complicated for example if o displaystyle o is a primary field then π f μ o x μ π f o x displaystyle pi _ f partial _ mu o x partial _ mu left pi _ f o x right is a linear combination of μ o displaystyle partial _ mu o and o displaystyle o correlation functions of descendant fields can be deduced from correlation functions of primary fields however even in the common case where all fields are either primaries or descendants thereof descendant fields play an important role because conformal blocks and operator product expansions involve sums over all descendant fields the collection of all primary fields o p displaystyle o_ p characterized by their scaling dimensions δ p displaystyle delta _ p and the representations ρ p displaystyle rho _ p is called the spectrum of the theory dependence on field positions edit the invariance of correlation functions under conformal transformations severely constrain their dependence on field positions in the case of two and three point functions that dependence is determined up to finitely many constant coefficients higher point functions have more freedom and are only determined up to functions of conformally invariant combinations of the positions the two point function of two primary fields vanishes if their conformal dimensions differ δ 1 δ 2 o 1 x 1 o 2 x 2 0 displaystyle delta _ 1 neq delta _ 2 implies left langle o_ 1 x_ 1 o_ 2 x_ 2 right rangle 0 if the dilation operator is diagonalizable i e if the theory is not logarithmic there exists a basis of primary fields such that two point functions are diagonal i e i j o i o j 0 displaystyle i neq j implies left langle o_ i o_ j right rangle 0 in this case the two point function of a scalar primary field is 10 o x 1 o x 2 1 x 1 x 2 2 δ displaystyle left langle o x_ 1 o x_ 2 right rangle frac 1 x_ 1 x_ 2 2 delta where we choose the normalization of the field such that the constant coefficient which is not determined by conformal symmetry is one similarly two point functions of non scalar primary fields are determined up to a coefficient which can be set to one in the case of a symmetric traceless tensor of rank ℓ displaystyle ell the two point function is o μ 1 μ ℓ x 1 o ν 1 ν ℓ x 2 i 1 ℓ i μ i ν i x 1 x 2 traces x 1 x 2 2 δ displaystyle left langle o_ mu _ 1 dots mu _ ell x_ 1 o_ nu _ 1 dots nu _ ell x_ 2 right rangle frac prod _ i 1 ell i_ mu _ i nu _ i x_ 1 x_ 2 text traces x_ 1 x_ 2 2 delta where the tensor i μ ν x displaystyle i_ mu nu x is defined as i μ ν x η μ ν 2 x μ x ν x 2 displaystyle i_ mu nu x eta _ mu nu frac 2x_ mu x_ nu x 2 the three point function of three scalar primary fields is o 1 x 1 o 2 x 2 o 3 x 3 c 123 x 12 δ 1 δ 2 δ 3 x 13 δ 1 δ 3 δ 2 x 23 δ 2 δ 3 δ 1 displaystyle left langle o_ 1 x_ 1 o_ 2 x_ 2 o_ 3 x_ 3 right rangle frac c_ 123 x_ 12 delta _ 1 delta _ 2 delta _ 3 x_ 13 delta _ 1 delta _ 3 delta _ 2 x_ 23 delta _ 2 delta _ 3 delta _ 1 where x i j x i x j displaystyle x_ ij x_ i x_ j and c 123 displaystyle c_ 123 is a three point structure constant with primary fields that are not necessarily scalars conformal symmetry allows a finite number of tensor structures and there is a structure constant for each tensor structure in the case of two scalar fields and a symmetric traceless tensor of rank ℓ displaystyle ell there is only one tensor structure and the three point function is o 1 x 1 o 2 x 2 o μ 1 μ ℓ x 3 c 123 i 1 ℓ v μ i traces x 12 δ 1 δ 2 δ 3 x 13 δ 1 δ 3 δ 2 x 23 δ 2 δ 3 δ 1 displaystyle left langle o_ 1 x_ 1 o_ 2 x_ 2 o_ mu _ 1 dots mu _ ell x_ 3 right rangle frac c_ 123 left prod _ i 1 ell v_ mu _ i text traces right x_ 12 delta _ 1 delta _ 2 delta _ 3 x_ 13 delta _ 1 delta _ 3 delta _ 2 x_ 23 delta _ 2 delta _ 3 delta _ 1 where we introduce the vector v μ x 13 μ x 23 2 x 23 μ x 13 2 x 12 x 13 x 23 displaystyle v_ mu frac x_ 13 mu x_ 23 2 x_ 23 mu x_ 13 2 x_ 12 x_ 13 x_ 23 four point functions of scalar primary fields are determined up to arbitrary functions g u v displaystyle g u v of the two cross ratios u x 12 2 x 34 2 x 13 2 x 24 2 v x 14 2 x 23 2 x 13 2 x 24 2 displaystyle u frac x_ 12 2 x_ 34 2 x_ 13 2 x_ 24 2 v frac x_ 14 2 x_ 23 2 x_ 13 2 x_ 24 2 the four point function is then 11 i 1 4 o i x i x 24 x 14 δ 1 δ 2 x 14 x 13 δ 3 δ 4 x 12 δ 1 δ 2 x 34 δ 3 δ 4 g u v displaystyle left langle prod _ i 1 4 o_ i x_ i right rangle frac left frac x_ 24 x_ 14 right delta _ 1 delta _ 2 left frac x_ 14 x_ 13 right delta _ 3 delta _ 4 x_ 12 delta _ 1 delta _ 2 x_ 34 delta _ 3 delta _ 4 g u v operator product expansion edit the operator product expansion ope is more powerful in conformal field theory than in more general quantum field theories this is because in conformal field theory the operator product expansion s radius of convergence is finite i e it is not zero 12 provided the positions x 1 x 2 displaystyle x_ 1 x_ 2 of two fields are close enough the operator product expansion rewrites the product of these two fields as a linear combination of fields at a given point which can be chosen as x 2 displaystyle x_ 2 for technical convenience the operator product expansion of two fields takes the form o 1 x 1 o 2 x 2 k c 12 k x 1 x 2 o k x 2 displaystyle o_ 1 x_ 1 o_ 2 x_ 2 sum _ k c_ 12k x_ 1 x_ 2 o_ k x_ 2 where c 12 k x displaystyle c_ 12k x is some coefficient function and the sum in principle runs over all fields in the theory equivalently by the state field correspondence the sum runs over all states in the space of states some fields may actually be absent in particular due to constraints from symmetry conformal symmetry or extra symmetries if all fields are primary or descendant the sum over fields can be reduced to a sum over primaries by rewriting the contributions of any descendant in terms of the contribution of the corresponding primary o 1 x 1 o 2 x 2 p c 12 p p p x 1 x 2 x 2 o p x 2 displaystyle o_ 1 x_ 1 o_ 2 x_ 2 sum _ p c_ 12p p_ p x_ 1 x_ 2 partial _ x_ 2 o_ p x_ 2 where the fields o p displaystyle o_ p are all primary and c 12 p displaystyle c_ 12p is the three point structure constant which for this reason is also called ope coefficient the differential operator p p x 1 x 2 x 2 displaystyle p_ p x_ 1 x_ 2 partial _ x_ 2 is an infinite series in derivatives which is determined by conformal symmetry and therefore in principle known viewing the ope as a relation between correlation functions shows that the ope must be associative furthermore if the space is euclidean the ope must be commutative because correlation functions do not depend on the order of the fields i e o 1 x 1 o 2 x 2 o 2 x 2 o 1 x 1 displaystyle o_ 1 x_ 1 o_ 2 x_ 2 o_ 2 x_ 2 o_ 1 x_ 1 the existence of the operator product expansion is a fundamental axiom of the conformal bootstrap however it is generally not necessary to compute operator product expansions and in particular the differential operators p p x 1 x 2 x 2 displaystyle p_ p x_ 1 x_ 2 partial _ x_ 2 rather it is the decomposition of correlation functions into structure constants and conformal blocks that is needed the ope can in principle be used for computing conformal blocks but in practice there are more efficient methods conformal blocks and crossing symmetry edit using the ope o 1 x 1 o 2 x 2 displaystyle o_ 1 x_ 1 o_ 2 x_ 2 a four point function can be written as a combination of three point structure constants and s channel conformal blocks i 1 4 o i x i p c 12 p c p 34 g p s x i displaystyle left langle prod _ i 1 4 o_ i x_ i right rangle sum _ p c_ 12p c_ p34 g_ p s x_ i the conformal block g p s x i displaystyle g_ p s x_ i is the sum of the contributions of the primary field o p displaystyle o_ p and its descendants it depends on the fields o i displaystyle o_ i and their positions if the three point functions o 1 o 2 o p displaystyle left langle o_ 1 o_ 2 o_ p right rangle or o 3 o 4 o p displaystyle left langle o_ 3 o_ 4 o_ p right rangle involve several independent tensor structures the structure constants and conformal blocks depend on these tensor structures and the primary field o p displaystyle o_ p contributes several independent blocks conformal blocks are determined by conformal symmetry and known in principle to compute them there are recursion relations 9 and integrable techniques 13 using the ope o 1 x 1 o 4 x 4 displaystyle o_ 1 x_ 1 o_ 4 x_ 4 or o 1 x 1 o 3 x 3 displaystyle o_ 1 x_ 1 o_ 3 x_ 3 the same four point function is written in terms of t channel conformal blocks or u channel conformal blocks i 1 4 o i x i p c 14 p c p 23 g p t x i p c 13 p c p 24 g p u x i displaystyle left langle prod _ i 1 4 o_ i x_ i right rangle sum _ p c_ 14p c_ p23 g_ p t x_ i sum _ p c_ 13p c_ p24 g_ p u x_ i the equality of the s t and u channel decompositions is called crossing symmetry a constraint on the spectrum of primary fields and on the three point structure constants conformal blocks obey the same conformal symmetry constraints as four point functions in particular s channel conformal blocks can be written in terms of functions g p s u v displaystyle g_ p s u v of the cross ratios while the ope o 1 x 1 o 2 x 2 displaystyle o_ 1 x_ 1 o_ 2 x_ 2 only converges if x 12 min x 23 x 24 displaystyle vert x_ 12 vert min vert x_ 23 vert vert x_ 24 vert conformal blocks can be analytically continued to all non pairwise coinciding values of the positions in euclidean space conformal blocks are single valued real analytic functions of the positions except when the four points x i displaystyle x_ i l...