Topics, P

p-Adic Number / Structure > s.a. differential equations; knot invariants; Non-Archimedean Structures.
* Idea: For each prime number p, the p-adic number system is an extension of the rational numbers different from the real number system.
* Motivation, use: Initially motivated by an attempt to use power-series methods in number theory; Now p-adic analysis essentially provides an alternative form of calculus.
$ Def: A uniformity

Wn := {(x, y) | x = y mod pn} ⊂ \(\mathbb Z \times \mathbb Z\) ,    n (p is a prime) .

@ General references: Gouvêa 97.
@ In cosmology and gravitation: Dragovich AIP(06)ht [cosmology]; Huang et al a1901 [action for bosonic p-adic strings in curved spacetime, and the vacuum Einstein equation]; Frampton a2001-JPA [p-adic strings and Peter Freund's work]; > s.a quantum cosmology; quantum spacetime.
@ Quantum theory: Khrennikov NCB(98)-a0906, Dubischar et al NCB(99)-a0906 [and correlations between quantum particles]; Dragovich NPPS(01) [quantum mechanics and quantum field theory]; Abdesselam a1104-conf [massless quantum field theory]; Hu & Zong a1502 [p-adic quantum mechanics, symplectic group and Heisenberg group]; Palmer a1804/PRS [FTQP, finite theory of qubit physics]; > s.a. Invariant Set Theory; modified uncertainty relations; path integrals.
@ Other physics: Dragovich et al pUAA-a0904 [rev]; Rodríguez-Vega & Zúñiga-Galindo PJM-a0907 [p-adic fields, pseudo-differential equations and Sobolev spaces]; Dragovich a1205-proc [p-adic matter in the universe]; Abdesselam et al a1302; Zelenov TMP(14) [p-adic dynamical systems]; Dragovich et al pNUAA(17)-a1705 [rev]; Stoica a1809 [free quantum particle and Euclidean 2D Einstein gravity]; Gubser et al JHEP(19)-a1811 [mixed field theory]; Huang et al a2001 [particle in a box]; Frampton JPA(20)-a2001 [p-adic strings]; Terasawa & Nojiri a2105 [statistical system, phase transition]; > s.a. classical mechanics [generalizations].
> Online resources: see MathWorld page; Wikipedia page.

Pachner Moves, Pachner Theorem > s.a. types of manifolds [PL, combinatorial].
@ In 4D: Korepanov a0911 [algebraic relations with anticommuting variables and topological field theory]; Banburski et al PRD(15)-a1412 [in a Riemannian spin-foam model]; Kashaev a1504.
> And physics: see regge calculus.

Packings > s.a. sphere.
@ References: Jaoshvili et al PRL(10) + Frenkel Phy(10) [random packings of tetrahedral dice].

Padé Approximant / Approximation
* Idea: The "best" approximation of a function by a rational function of given order; It often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge.
@ References: Wei et al JCAP(14)-a1312 [cosmological applications].
> Online resources: see MathWorld page; Wikipedia page.

Painlevé Equations / Analysis / Test
* Idea: A criterion of integrabilty for partial differential equations, which involves the following steps, (1) Show that the general solution can be represented as a (formal Laurent) series in powers of some function Φ that vanishes on an arbitrary non-characteristic surface; (2) Verify the possibility of truncating the series at some finite power of Φ.
* Consequences: If satisfied, the equation is integrable, and we can get Bäcklund transformations and a (weak) Lax pair; If not satisfied, we cannot conclude the opposite.
@ General references: Weiss et al JMP(83); Weiss JMP(83); Ramani et al PRP(89); Steeb & Euler 88; Lakshmanan & Sahadevan PRP(93); Guzzetti JPA(06)-a1010, IMRN(12)-a1010 [Painlevé VI equation]; Schiff & Twiton a1810 [Painlevé IV equation, dynamical systems approach].
@ Integrable equations without Painlevé property: Ramani et al JPA(00)-a0709; Tamizhmani et al Sigma(07)-a0706.
@ And general relativity: see García-Díaz et al JMP(93); > s.a. chaos in gravitation.
@ Discrete versions: Grammaticos et al PRL(91); Ramani et al PRL(91); Grammaticos & Ramani PS(14) [rev]; Kajiwara et al JPA(17)-a1509 [geometrical aspects].
@ Related topics: Sakovich Sigma(06)n.SI/04-conf [quadratic H that fails the integrability test]; Aminov et al a1306 [multidimensional versions of the Painlevé VI equation]; Bermudez et al JPA(16)-a1512 [solutions to the Painlevé V equation using supersymmetric quantum mechanics].
> Online resources: see The Painlevé Project site.

Painlevé-Gullstrand Coordinates / Metric > see spherically symmetric geometries.
@ References: Jaén & Molina GRG(16)-a1611 [natural extension]; Faraoni & Vachon a2006 [breakdown].
> Generalized to rotating spacetimes: see kerr metric; kerr-newman metric.

Pair Creation / Production > s.a. particle effects [Schwinger effect]; quantum field theory effects in curved spacetime.
@ References: Petrat & Tumulka JPA(14) [multi-time formulation].

Pais-Uhlenbeck Model > s.a. quantum oscillators.
* Idea: A field theory with a higher-derivative field equation.
* The ghost issue: Applying the Ostrogradski approach to the Pais-Uhlenbeck oscillator yields a Hamiltonian which is unbounded from below, which leads to a ghost problem in quantum theory; It was believed for many years that the model possesses ghost states attributable to the field equation having more than two derivatives, and therefore that it is a physically unacceptable quantum theory; In reality, the Pais-Uhlenbeck model does not possess ghost states, when quantized according to the rules of PT quantum mechanics, and is a perfectly acceptable quantum theory.
@ General references: Pais & Uhlenbeck PR(50); Kaparulin & Lyakhovich a1506-proc [energy and stability]; Kaparulin et al JPA(16)-a1510 [interactions]; Avendaño-Camacho et al JMP(17)-a1703 [stability, perturbation-theory approach].
@ Ghost-free formulations: Bender & Mannheim JPA(08)-a0807; Nucci & Leach PS(10)-a0810, JMP(09); Banerjee a1308.
@ Hamiltonian formulation: Mostafazadeh PLA(10)-a1008; Andrzejewski NPB(14)-a1410 [and symmetries]; Masterov NPB(16)-a1505 [without ghost problem]; Sarkar et al a1507 [resolving the issue of the branched Hamiltonian]; Masterov a1603 [(2n+1)-order generalization].
@ Quantum theory: Mannheim & Davidson PRA(05)ht/04 [Dirac quantization]; Di Criscienzo & Zerbini JMP(09)-a0907 [euclidean path integral and propagator]; Mostafazadeh PRD(11)-a1107 [consistent quantization]; Bagarello IJTP(11), Pramanik & Ghosh MPLA(13)-a1205 [coherent states]; Cumsille et al IJMPA(16)-a1503 [polymer quantization]; Berra-Montiel et al AP(15)-a1505 [deformation quantization]; Fernández a1605 [and its PT-variant].
@ Applications: Ketov et al a1110-ch [as a toy-model for quantizing f(R) gravity theories].

Palatini Action / Formulation of Gravity Theory
* Idea: A formulation in which the metric and connection are assumed to be independent fields, as in metric-affine theories; Conceptually, this amounts to considering the geodesic structure and the causal structure of the spacetime as independent.
> Theoretical aspects: see first-order actions for general relativity; higher-dimensional and higher-order gravity; kaluza-klein theories; non-local gravity.
> Phenomenology: see cosmology of higher-order theories; dilaton.

PAMELA (Payload for Antimatter/Matter Exploration and Light-nuclei Astrophysics) > s.a. cosmic rays.
* Idea: A space mission onboard an Earth-orbiting spacecraft, that studies cosmic rays.
@ References: Adriani et al PRL(09), PRL(13) [results on positron excess].
> Online resources: see PAMELA website; Wikipedia page.

Pancharatnam Phase > see geometric phase; Parallel Transport.

Paneitz Equation > see partial differential equations.

Paneitz Operator
* Idea: A 4th-order differential operator which occurs in the theory of conformal anomalies; According to a conjecture, it gives 8π when acting upon the invariant volume of the past light cone.
@ References: Park & Woodard GRG(10)-a0910 [and volume of the past light cone].

Papapetrou Field > see gravitomagnetism.

Papapetrou Solution > s.a. kerr solutions [Papapetrou gauge].
@ References: Khugaev & Ahmedov IJMPD(04) [generalization].

Papapetrou Theorem
* Idea: A theorem about the equivalence of two sets of circularity conditions for (pseudo)stationary, asymptotically flat empty spacetimes; For stationary axisymmetric sources, gab shares these symmetries.

Papapetrou-Majumdar Metrics [> black-hole solutions].
* Idea: A family of electrovac solutions of Einstein's equation which are static because of balance between gravitational and electromagnetic forces, for special charge/mass ratios.
@ General references: Papapetrou PRIA(47); Majumdar PR(47); Hartle & Hawking CMP(72) [interpretation]; Heusler CQG(97)gq/96 [uniqueness].
@ Related topics: Gürses PRD(98)gq [dust generalization]; Varela GRG(03)gq/02 [charged dust sources].

Parabola > see conical sections.

Paraboloidal Coordinates > see coordinates.

Parabosons > see generalized particle statistics.

Paracompact Topological Space

Paradoxes > s.a. Fermi Paradox; Trouton-Noble Paradox.
> In mathematics: see Lindley Paradox; logic; Parrondo's Paradox; probability; Russell Paradox; Zeno's Paradox.
> In gravitation and cosmology: see black-hole information paradox; causality violations; expansion; Olbers' Paradox.
> In quantum theory: see EPR paradox; Klein Paradox; quantum correlations; quantum effects; quantum foundations; wave-function collapse.
> In special relativity: see arrow of time [causal paradoxes]; clocks; Ehrenfest, Lock and Key, Submarine, Twin Paradox; special relativity; kinematics.
> In statistical physics: see Gibbs Paradox; probability in physics; quantum statistical mechanics; Recurrence Paradox; statistical mechanics.
@ General references: Klein 96; Chang 12 [in scientific inference].
@ In thermodynamics: Cucić a0812, a0912 [and statistical physics]; Yoder & Adkins AJP(11)aug [ellipsoid paradox]; Sheehan et al FP(14) [diatomic gas in a cavity].

Parafermions, Paraparticles > see generalized particle statistics.

Parallax > s.a. cosmological observations [cosmic parallax].
* Stellar parallax: The annual apparent displacement of the stars that occurs because of Earth's orbit around the Sun.
@ References: Timberlake TPT(13)-a1208 [history, and aberration]; Räsänen JCAP(14)-a1312 [cosmic parallax, covariant treatment].

Parallel Transport > s.a. Fermi Transport; connection; foliation [web]; Path.
* Idea: Defined on a manifold that has a connection; A tensor T is parallel transported along a curve with tangent vector X if ∇XT = 0.
@ General references: Anandan & Stodolsky PLA(00)qp/99 [classical and quantum physics]; Wagh & Rakhecha JPA(99) [gauge-independent form]; Iliev IJGMP(05)m.DG [and connections], IJGMP(08) [axiomatic approach]; Iurato a1608 [history, Levi-Civita].
@ Specific spaces and metrics: Bini et al IJMPD(04)gq [circular orbits, stationary axisymmetric spacetime]; Chatterjee et al RVMP(10)-a0906 [over path spaces].
@ Generalizations: Soncini & Zucchini JGP(15)-a1410 [higher parallel transport in higher gauge theory]; Leone a1903 [in quantum mechanics, generalized Pancharatnam connection].

Parallel Universes > see multiverse.

Parallelizable Manifold > see types of manifolds.

Parallelotope > a special type of Polytope.

Paramagnetism > see magnetism.

Parametric Excitation / Resonance > see resonances.

Parametrix > see approaches to canonical quantum gravity.

Parametrized Post-Friedmannian Formalism > see under Post-Friedmannian.

Parametrized Post-Newtonian Formalism > see under PPN Formalism.

Parametrized Theories

* Idea: Low-mass extra U(1) gauge bosons with gauge-kinetic mixing with the ordinary photon.
@ References: Jaeckel & Ringwald PLB(08)-a0707 [search, cavity experiment].

Parastatistics > see particle statistics.

Parisi-Sourlas Mechanism
@ References: Magpantay IJMPA(00)ht/99 [in Yang-Mills theory].


Parrondo's Paradox
* Idea: The proposition that two losing strategies can, by alternating randomly, produce a winner.
@ References: Martin & von Baeyer AJP(04)may.

Parseval's Integral > see bessel functions.

Parseval's Relation / Theorem > see fourier analysis.

Part > see Subsystem.

Partially Massless Fields > see spin-2 fields; types of field theories; types of yang-mills theories.

Partially Massless Gravity Theory > see massive gravity.

Partially Ordered Set > see poset.

Particle Descriptions and Types > see effects [including creation and decay], models, statistics, types; classical and quantum models; spinning particles.

Particle Horizon > see horizons.

Particle Physics > s.a. experimental particle physics.

Particle Physics Phenomenology > see lattice field theory; QCD, QED, and string phenomenology; Zweig Rule.

Particle Statistics > s.a. generalized particle statistics.

Partition, Partition of Unity, Partition Relation > see partition.

Partition Function > see states in statistical mechanics.

Parton Models > see hadrons.

Pascal > see programming languages.

Paschen-Back Effect > see Zeeman Effect.

Past > see spacetime subsets; photons and Trajectory in Quantum Mechanics [past of a quantum particle].

Past Hypothesis > see arrow of time.

Pataplectic Hamiltonian Formulation > see hamiltonian dynamics.

Path > s.a. loops; Parallel Transport.
* For a field: The path in a region Ω of spacetime is a cross-section of the bundle of internal degrees of freedom over Ω.
@ Path group: Mensky G&C(02)gq [gravity and paths in Minkowski spacetime], gq/02-conf [in gauge theory and general relativity]; > s.a. types of groups.
@ Path space: Cho & Hong a0706 [Morse theory]; Biswas & Chatterjee IJGMP(11) [geometric structures]; Chatterjee et al JGP(13) [bundles and connections over path spaces]; Chatterjee IJGMP(15)-a1401 [double category of geodesics on path space]; Gerstenhaber a1403 [path algebras and de Broglie waves]; > s.a. measure [Wiener measure].
@ Path-dependent functions: Reyes JMP(07)ht/06 [operators].
> Related topics: see formulations of general relativity [as a theory of paths]; Trajectory [in classical and quantum mechanics].

Path-Integral Quantization > s.a. for gauge theories and other theories.

Patterns > s.a. many-body quantum systems and Self-Organization for pattern formation.
@ Pattern theory: Grenander 76-81.

Pauli Equation > s.a. Scale Relativity.
@ References: Mancini et al JPA(01)qp/00 [for probability distributions]; Zhalij JMP(02)mp [separation of variables].

Pauli Exclusion Principle > see spin-statistics.

Pauli Matrices > see SU(2).

Pauli Theorem > see time in quantum theory.

Pauli-Fierz Lagrangian / Theory > s.a. spin-2 field theories; path-integral formulation of quantum field theory [spin-1/2].
* Idea: A theory of massive charged spin-2 fields hμν,

\(\cal L\) = |g|1/2 [R up to quadratic terms + m2 (hμνhμνh2)] ;

The theory arises also as an effective 4D theory in brane models; It does not reproduce linearized general relativity in the m → 0 limit, and has a ghost problem.
* van Dam-Veltman discontinuity: A discontinuity in the Pauli-Fierz formulation; The deflection angle in the background of a spherically symmetric gravitational field converges to 3/4 of the value predicted by the massless theory (linearized general relativity) as m → 0.
@ General references: Fierz & Pauli PRS(39); Groot Nibbelink & Peloso CQG(05)ht/04 [covariant]; Obukhov & Pereira PRD(03) [teleparallel origin]; Georgescu et al CMP(04) [massless, spectral theory]; Leclerc gq/06 [gauge and reduction]; Osipov & Rubakov CQG(08)-a0805 [superluminal graviton propagation]; Hasler & Herbst RVMP(08) [Hamiltonians]; González et al JHEP(08) [duality]; Loss et al LMP(09) [degeneracy of eigenvalues of Hamiltonian]; de Rham & Gabadadze PLB(10)-a1006 [non-linear completion without ghosts]; Park CQG(11)-a1009 [effect of quantum interactions]; Deser CJP(15)-a1407 [action, and manifestly positive energy].
@ Variations: Boulanger & Gualtieri CQG(01)ht/00 [PT non-invariant deformation]; de Rham & Gabadadze PRD(10)-a1007 [with generalized mass and interaction terms]; Park JHEP(11)-a1011 [non-Pauli-Fierz theory, unitarization]; Deffayet & Randjbar-Daemi PRD(11)-a1103 [non-linear, from torsion]; Alberte IJMPD(12)-a1202 [on an arbitrary curved background]; Gambuti & Maggiore EPJC(21)-a2102 [alternative without vDVZ discontinuity]; > s.a. massive gravity [including non-Pauli-Fierz theory].
> Online resources: see Wikipedia page on Markus Fierz.

Pauli-Jordan Function > s.a. green functions in quantum field theory.
* Idea: A type of Green function for a quantum field.
* For a scalar field: The two-point function G(x, x'):= −i \(\langle\)0| [φ(x), φ(x')] |0\(\rangle\).
* Properties: It satisfies the homogeneous field equation.

Pauli-Villars (Covariant) Regularization > see regularization.

$ Meaning: Partial Conservation of Axial Current.

Peano's Axioms > see mathematics.

Peano Curve > see fractals.

Peccei-Quinn Mechanism / Symmetry > s.a. axion; CP violation [strong CP problem]; neutron.
* Idea: A field theory mechanism by which a discrete symmetry arises from the spontaneous breaking of a continuous symmetry.
@ References: Mercuri PRL(09)-a0902 [gravitational, and Barbero-Immirzi parameter]; Takahashi & Yamada JCAP(15)-a1507 [breaking, in the early universe].

Peeling Property of Spacetime
* Idea: A property of the Weyl tensor in asymptotically flat spacetimes.
@ References: Geroch in(77); in Wald 84, p285; Bressange & Hogan PRD(99) [lightlike signals in Bondi-Sachs]; Klainerman & Nicolò CQG(03) [and initial data set falloff]; Pravdová et al CQG(05)gq [even higher dimensions]; Friedrich CQG(18)-a1709 [and isolated systems, asymptotic flatness and simplicity].

Peierls Argument > see ising models [spontaneous magnetization].

Peierls Brackets > s.a. canonical general relativity; types of symplectic structures.
* Idea: A bracket defined on the covariant phase space of a field theory, corresponding to the Poisson bracket on the canonical phase space.
@ General references: Peierls PRS(52); DeWitt in(64), in(99); Esposito et al ht/02 [intro]; Bimonte et al IJMPA(03)ht [field theory], ht/03 [dissipative systems]; DeWitt & DeWitt-Morette AP(04) [and path integrals]; Esposito & Stornaiolo IJGMP(07)ht/06 [for type-I gauge theories, and Moyal bracket].
@ Generalizations: Marolf AP(94)ht/93; Sharapov IJMPA(14)-a1408 [in non-Lagrangian field theory].
> Online resources: see nLab page; Wikipedia page.

Peirce Logic > see clifford algebra; dirac field theory.

Peltier Effect > see electricity [thermoelectricity].
@ References: Heremans & Boona Phy(14) [spin Peltier effect].

Pendulum > s.a. kinematics of special relativity, oscillator.
* Non-linear or physical : The Hamiltonian and equation of motion are given by

H = \(1\over2\)p2ω2 cos x ,       d2x / dt2 + ω2 sin x = 0 .

* Linearization: Gives the simple harmonic oscillator.
@ General references: Baker & Blackburn 05 [r PT(06)jul]; Gitterman 08 [noisy]; Baker 11; Brizard CNSNS-a1108 [action-angle coordinates].
@ History: Matthews 00 [and education, r pw(01)feb]; Dahmen a1409/EPJH [Denis Diderot's paper on pendulums and air resistance]; Boulanger & Buisseret a2004 [Foucault's pendulum]; Greenslade TPT(21)apr [physicists who have pendula named after them].
@ Beyond the small-angle approximation: Lima & Arun AJP(06)oct; Turkyilmazoglu EJP(10); Bel et al EJP(12) [periodic solutions by the homotopy analysis method].
@ Foucault's pendulum: Hart et al AJP(87)jan; Khein & Nelson AJP(93)feb [Hannay angle]; Pardy ap/06 [astronomical analogs]; von Bergmann & von Bergmann AJP(07)oct [and geometry]; news THE(10)jun [pendulum is irreparably damaged]; Jordan & Maps AJP(10)nov [in pictures]; Boulanger & Buisseret a2004 [and historical overview of formulations of mechanics]; Delphenich a2007 [as example of motion on a pseudo-surface]; Olson TPT(21)apr [Paris sites related to Foucault].
@ Other types: Butikov AJP(01)jul [inverted, stabilization]; Rafat et al AJP(09)mar [double, with square plates]; Bassan et al PLA(13) [torsion pendulum, Lagrangian model and small misalignments].
@ Quantum: Cushman & Śniatycki a1603 [spherical pendulum, geometric quantization]; > s.a. path integrals.

Penning Trap > s.a. lorentz-symmetry violation phenomenology; proton [mass measurement].
* Idea: An electron trap, made with a special configuration of electric and magnetic fields.
@ References: Brown & Gabrielse RMP(86); Blaum et al CP(10) [and experiments in fundamental physics].

Penrose Diagram > s.a. asymptotic flatness.
* Idea: A diagram of spacetime, as compactified by a suitable conformal transformation.
@ General references: Penrose in(64); Jadczyk RPMP(12)-a1107 [geometry of Penrose's 'light cone at infinity']; Schindler & Aguirre CQG(18)-a1802 [algorithm].
@ Specific types of spacetimes: Brown & Lindesay CQG(09)-a0811 [accreting black holes]; Lindesay & Sheldon CQG(10) [transient black holes].

Penrose Dodecahedron
* Idea: A set of 40 states of a spin-3/2 particle used by Zimba and Penrose to give a proof of Bell's non-locality theorem.
@ References: Zimba & Penrose SHPSA(93); Massad & Aravind AJP(99)jul.

Penrose Inequality / Conjecture
* Idea: A generalization of the positive mass theorem, which provides a lower bound on the mass of a spacetime,

GMADM / c2R/2 ;

More generally, the total mass of a spacetime which contains black holes with event horizons of total area A satisfies

GM / c2 ≥ (A/16π)1/2 .

@ General references: Penrose NYAS(73); Ludvigsen & Vickers JPA(83) [partial proof]; Malec & Ó Murchadha PRD(94) [and refs]; Frauendiener PRL(01)gq [towards a proof]; Malec et al PRL(02)gq [general horizons]; Malec & Ó Murchadha CQG(04)gq [re use of Jang equation]; Karkowski & Malec APPB(05)gq/04 [numerical evidence]; Ben Dov PRD(04) [(counter)example]; Tippett PRD(09)-a0901 [violated for prolate black holes]; Mars CQG(09)-a0906 [rev]; Bengtsson & Jakobsson GRG(16)-a1608 [toy version with proof].
@ Charged black holes: Disconzi & Khuri CQG(12)-a1207 [charged black holes]; Khuri GRG(13)-a1308; Lopes de Lima et al a1401 [in higher dimensions]; Khuri et al CQG(15)-a1410 [extensions].
@ Riemannian: & Huisken & Ilmanen (97) [proof, single black hole]; Bray JDG(01) [proof]; Bray & Chruściel gq/03-in; Ohashi et al PRD(10)-a0906; Khuri et al CM-a1308 [with charge, for multiple black holes].
@ Other generalizations: Gibbons in(84); Karkowski et al CQG(94) [gravitational waves]; Herzlich CMP(97) [asymptotically flat, R ≥ 0]; Khuri CMP(09) [general initial data sets]; Carrasco & Mars CQG(10) [generalized-apparent-horizons version, counterexample]; Brendle & Wang CMP(14)-a1303 [2D spacelike surfaces in Schwarzschild spacetime]; Alexakis a1506 [perturbations of the Schwarzschild exterior]; Roesch a1609, Bray & Roesch a1708 [null Penrose conjecture]; Husain & Singh PRD(17)-a1709, Engelhardt & Horowitz PRD(19)-a1903 [in AdS space]; Bousso et al PRL(19)-a1908 [quantum version]; Kulczycki & Malec a2010 [with angular momentum].

Penrose Limit
* Idea: A procedure whereby the immediate neighborhood of an arbitrary null geodesic is "blown up" to yield a pp-wave as a limit; Given a metric written in coordinates adapted to the null geodesic (can always be done), the procedure consists in replacing (u, v, yi) by (u, λ2v, λyi) in the line element, and then taking the limit as λ → 0 of ds2/λ2; One is then left with a metric of the form ds2 = 2 dudv + Cij dyi dyj; Ricci-flat metrics and Einstein metrics both give Ricci-flat metrics as results.
@ References: Floratos & Kehagias JHEP(02)ht [orbifolds and orientifolds]; Siopsis PLB(02)ht, MPLA(04)ht/02 [AdS, and holography]; Hubeny et al JHEP(02)ht [non-local theories]; Kunze PRD(05)gq/04 [curvature and matter]; Philip JGP(06) [of homogeneous spaces].

Penrose Mechanism / Process > s.a. black-hole phenomenology; matter and radiation near black holes.
* Idea: A method for extracting energy from a rotating black hole; Send a mass into a trajectory inside the ergosphere, against the black hole's rotation; Separate the mass into two parts and let one fall inward; The outgoing one may have more energy than the initial one, obtained by slowing the black hole down; Results in an increase of the black hole's \(m_{\rm irr}\).
* Variations: The collisional Penrose, or super-Penrose process consists of particle collisions in the ergoregion.
@ General references: Penrose RNC(69), & Floyd NPS(71); Christodoulou & Ruffini PRD(71); Wald AJ(74); Wagh & Dadhich PRP(89); Fayos & Llanta GRG(91) [limitations]; Williams phy/04; Heller a0908; Schnittman PRL(14)-a1410 [upper limit to energy extraction]; Bravetti et al PRD(16)-a1511 [thermodynamic optimization].
@ Collisional Penrose process: Schnittman PRL(14)-a1410; Berti et al PRL(15)-a1410; Zaslavskii MPLA(15)-a1411; Zaslavskii IJMPD-a1510; Leiderschneider & Piran PRD(16)-a1510 [maximal efficiency]; Patil et al PRD(16)-a1510 [efficiency]; Zaslavskii GRG(16)-a1511, PRD(16)-a1511; Ogasawara et al PRD(16)-a1511 [heavy particle production]; Schnittman GRG(18).
@ Other variations: Lasota et al PRD(14)-a1310 [generalized].
@ Related topics: Williams ap/02/PRD [Compton scattering and \(e^+ e^-\) production]; Cen a1102-wd [astrophysical scenario].

Penrose Tiling > see tiling.

Pentaquark > see hadrons.

Percolation > s.a. ising models; in lattice field theory; Transport; voronoi tilings.
* Idea: The theory was initiated by Broadbent and Hammersley PPCS(57) as a mathematical framework for the study of random physical processes, such as flow through a disordered porous medium with randomly blocked channels in a gravitational field; It has proved to be a remarkably rich theory, with applications beyond natural phenomena to topics such as network modelling and the contact process for epidemic spreading.
* Phase transition: It turns out that the system undergoes a continuous phase transition with a non-trivial critical behavior, at which it becomes macroscopically permeable.
@ General references: Stauffer & Aharony 94 [intro]; Bollobás & Riordan 06; Duminil-Copin a1712-proc [rev, historical].
@ Theory: Cardy mp/01-ln [conformal field theory methods]; Smirnov & Werner MRL-m.PR/01 [triangular 2D lattice]; Bollobás & Riordan RSA(06)m.PR/04; Janssen & Täuber AP(05) [field theory approach, rev]; Gliozzi et al NPB(05) [random, as gauge theory]; Ziff et al JPA(11) [factorization of the three-point density correlation function]; Curien & Kortchemski PTRF-a1307 [on random triangulations].
@ Critical: Grassberger JPA(99); Cardy JPA(02)mp; Ridout NPB(09)-a0808 [and Watts' crossing probability].
@ Directed: Grassberger JSP(95); Janssen et al JPA(99) [equation of state]; Grimmett & Hiemer m.PR/01; Takeuchi et al PRL(07), PRE(09) + Hinrichsen Phy(09) [experimental realization]; Chen PhyA(11) [square lattice, asymptotic behavior].

Perfect Fluid > s.a. fluid; gas.

Perfect Group > see group types.

Perfect Number > see number theory.

Perfect Space > see types of topologies.

Periastron / Perihelion Precession > see Precession; black-hole binaries; orbits in newtonian gravity; test-body orbits; tests of general relativity.

Periodic Orbits > see classical systems [Bertrand's theorem; non-linear systems].

Perl > see programming languages.

Permanent of a Matrix > see matrix.

Permeability > see magnetism.

Permittivity > see electricity in matter.

Permutations > see finite groups; particle statistics [identical particles].
@ References: Huggett BJPS(99) [as a symmetry in quantum mechanics]; Olshanski a1104-ch [random permutations]; Cori et al EJC(12) [formulas for the number of factorizations of permutations]; Baker PhSc(13) [in quantum field theory, and theories with no particle interpretation].

Permutons > see phase transitions [in combinatorial systems].

Perpetual Motion Machine / Perpetuum Mobile > s.a. thermodynamics [violations of second law].
@ References: Chernodub a1203 [permanently rotating devices]; Jenkins AJP(13)-a1301 [early 18th century demonstrations by Orffyreus, con man].
> Related topics: see Maxwell's Demon; de sitter space [example].
> Online resources: see Continuous Frictioned Motion Machine page.

Perplex Numbers > see types of numbers.

Perron-Frobenius Operator > see under Frobenius-Perron.

Persistent Homology > see types of homology.

Perspectivalism > see causality; quantum foundations.

Perspectivism > see philosophy of science.

Perturbation Methods / Theory > s.a. fluids; gauge theories; quantum field theory techniques.
* In classical mechanics – Example: Delicate stuff – If initially stationary, Venus and Earth would collide in less than 370 yrs; If isolated in orbit around each other, never; So, what is the effect of Venus on Earth's trajectory?
* In quantum mechanics – Approaches: The usual time-dependent perturbation theory for solving the Schrödinger equation does not preserve unitarity; The Magnus expansion (also known as exponential perturbation theory) does provide unitary approximate solutions.
@ Texts: Giacaglia 72; Kevorkian & Cole 81; Gallavotti 83; Bender & Orszag 99; Holmes 13.
@ For differential equations: Odibat & Momani PLA(07) [homotopy perturbation method].
@ Hamiltonian systems: Lewis et al PLA(96) [time-dependent, invariants]; Laskar & Robutel ap/00 [symplectic integrators].
@ Related topics: Marmi m.DS/00-ln [small denominators, intro]; Amore mp/04-proc [anharmonic oscillator, classical and quantum], et al EJP(05)mp/04 [removal of secular terms]; Pound PRD(10)-a1003 [singular]; > s.a. classical systems; oscillator; series [convergence acceleration and divergent series].
@ In quantum mechanics: Sen IJMPA(99)cm/98 [singular potentials]; Fernández 01, JPA(06)qp/04; Franson & Donegan PRA(02)qp/01 [t-dependent]; Teufel 03 [adiabatic perturbation theory]; Ciftci et al PLA(05)mp [iterative]; Weinstein ht/05, NPPS(06)ht/05 [adaptive]; Albeverio et al RPMP(06) [singular, rigged Hilbert space approach]; Harlow a0905 [bound on the error]; Fernández a1004 [confined systems]; Blanes et al EJP(10) [Magnus expansion or exponential perturbation theory, pedagogical]; Hayata PTP(10)-a1010 [without weak-coupling assumption]; Faupin et al CMP(11) [for embedded eigenvalues, second-order]; Kerley a1306 [time-independent]; Rigolin & Ortiz PRA(14)-a1403 [degenerate adiabatic perturbation theory].
> Gravity-related areas: see black-hole perturbations; cosmological perturbations; metric perturbations in general relativity.

Peter-Weyl Theorem > see quantum mechanics representations [and Segal-Bargmann transform].

Petrov, Petrov-Pirani Classification

Pfaff Derivative of a Function
$ Def: ∂k f:= ek(f), with ek a basis for Tx X, such that df |X = ek(f) θk |x, with θk the dual basis.
* Idea: Just a generalization of the regular partial derivatives to the case in which ek is not necessarily the coordinate basis ∂/∂xk.

Pfaffian of a Matrix
* Idea: Given an antisymmetric 2m × 2m matrix, its Pfaffian is a polynomial in its entries, whose square gives the determinant of the matrix.

Phantom Divide
* Idea: The point in cosmological history at which w (the ratio of pressure to energy density for the effective fluid matter used to describe cosmological models) crossed the value −1, or the value −1 itself in the range of possible values for w.
@ References: Zhang a0909-ch [approaches].

Phantom Field > s.a. born-infeld theory; Quintom; wormholes.
* Idea: An exotic scalar field with a negative kinetic term (as a fluid, it has an equation of state with w < −1), that violates most of the classical energy conditions; 2005, Considered by some as a real possibility for dark energy, although it has serious problems like instability and lack of a well-posed initial-value formulation.
@ General references: Sami & Toporensky MPLA(04) [and fate of universe]; Majerotto et al ap/04/JCAP [and SN Ia data]; Santos & Alcaniz PLB(05)ap [Segre classification]; Giacomini & Lara GRG(06) [+ gravity + arbitrary potential, dynamics]; Pereira & Lima PLB(08)-a0806 [thermodynamics].
@ Black holes, isolated objects: Svetlichny ap/05 [possible production by black holes]; Berezin et al CQG(05)gq [shell around Schwarzschild]; Bronnikov & Fabris PRL(06) [regular asymptotically flat, de Sitter and AdS]; Rahaman et al NCB(06)gq; Gao et al PRD(08)-a0802 [mass increase]; Martins et al GRG(09)-a1006 [3D, phantom fluid]; Gyulchev & Stefanov PRD(13) [lensing]; > s.a. gravitational thermodynamics; models of topology change.
@ Cosmology: Dąbrowski et al PRD(03) [+ standard matter]; Chimento & Lazkoz MPLA(04) [big rip]; Curbelo et al CQG(06)ap/05 [avoidance of big rip]; Faraoni CQG(05)gq [general potential]; Capozziello et al PLB(06) [dark energy and dark matter]; Bouhmadi-López et al PLB(08)gq/06 [future singularity]; Dąbrowski gq/07-MGXI [dark energy]; Sanyal IJMPA(07) [inflation rather than big rip]; Hrycyna & Szydłowski PLB(07) [conformally coupled, acceleration]; Shatskiy JETP(07)-a0711; Chaves & Singleton SIGMA(08)-a0801 [and dark matter]; Chen et al JCAP(09)-a0812 [phase-space analysis]; Myung PLB(09) [thermodynamics]; Regoli PhD-a1104; Astashenok et al PLB(12)-a1201 [without big rip singularity]; Novosyadlyj et al PRD(12), Ludwick PRD(15)-a1507 [as dark energy]; Ludwick MPLA(17)-a1708 [rev]; > s.a. FLRW models; gravitational thermodynamics.
@ Loop quantum cosmology: Samart & Gumjudpai PRD(07)-a0704; Gumjudpai TJP-a0706-proc; Fu et al PRD(08)-a0808; Wu & Zhang JCAP(08)-a0805; > s.a. FLRW quantum cosmology.

Phases of Matter
* Idea: The phases that have been known for a long time are solid, liquid, gas and plasma, but experiments with matter cooled to within a few degrees of 0 K have turned up a number of exotic phases, such as superfluids, superconductors and topological phases; In these new types of phases one can see quantum mechanical effects at work in materials, unencumbered by the random motions of atoms.
* Topological phases: Thouless, Kosterlitz and Haldane won the 2016 Nobel Prize for their work on these phases; A variety of such phases are known.
@ General references: issue JPCM(98)#49 [matter under extreme conditions]; Pinheiro phy/07 [plasma, genesis of the word]; Kadanoff a1002; Baas IJGS-a1012 + news ns(11)jan [topology and generalization of Efimov states]; Bauer et al a1903 [mathematical formalism for classifying]; > s.a. magnetism [plasma physics or magnetohydrodynamics].
@ Topological phases: Read PT(12)jul; > s.a. matter [mathematical models].
> Type of phases: see condensed matter [gases, liquids]; crystals; fluid; gas; Plasma; bose-einstein condensate.

Phase of a Quantum State > s.a. arrow of time [phase squeezing]; geometric phase; pilot-wave interpretation [and quantum phase]; quantum states.
@ References: Barnett & Pegg JMO(89) [optical phase operator]; Lynch PRP(95); Koprinkov PLA(00)qp/06; Kastrup qp/01 [and modulus]; Lahti & Pellonpää PS(02) [formalisms]; Pellonpää JMP(02) [observables]; Heinonen et al JMP(03) [covariant phase difference]; de Gosson JPA(04) [general definition]; Gour et al PRA(04) [self-adjoint extensions]; Saxena a0803 [in terms of inverses of creation and annihilation operators]; Hall & Pegg PRA(12)-a1205.

Phase Curve > see phase space.

Phase Diagram
* Idea: A plot showing the boundaries between thermodynamically distinct phases in an equilibrium system.
> Gravity: see dynamical triangulations; phenomenology of gravity; quantum-gravity renormalization.
> Other field theories: see Gross-Neveu Model; QCD, QCD phenomenology; Wess-Zumino Model.
> Other physics: see Critical Points; matter [dense matter]; Potts Model; water.
> Online resources: see Wikipedia page.

Phase Space

Phase Transition > s.a. quantum phase transition.

Phase Velocity > see velocity.

Philosophy > s.a. philosophy of physics; philosophy of science.

Phoenix Universe > see cosmological models.

Phonon > s.a. Quasiparticle; specific heat [for a solid]; sound ["phonon tunneling"].
* Idea: Phonons are collective vibrational modes of solids; In a quantum theory, they are quanta of sound waves, a type of quasiparticle.
* Importance: They influence many physical properties of materials, such as free energy and phase stability, thermal expansion, and thermal conductivity, all of which are important to a wide range of technological applications.
* Applications: Theoretical applications include models for fundamental quantum field theory effects (such as the acoustic Casimir effect) and black-hole analogs; Practical ones include "phonon optics" (mirrors, filters, lenses, etc) used to look inside solids for point defects.
@ General references: Baym AP(61), re AP(00) [Green function, quantum field theory methods]; Kokkedee 63; Hu & Nori PRL(96) + pn(96)mar [squeezed].
@ Specific types of systems: Quilichini & Janssen RMP(97) [quasicrystals]; Gorishnyy et al pw(05)dec [phononic crystals]; Lukkarinen LNP(06)-a1509 [in weakly anharmonic particle chains, kinetic theory].
@ Related topics: Schwab et al Nat(00)apr [quantized thermal conductivity]; Johnson & Gutierrez AJP(02)mar [wave function visualization]; news tcd(15)mar [controlling phonons with magnetic fields]; Iachello et al PRB(15)-a1506 [algebraic theory, energy dispersion relation and density of states]; Viennot et al PRL(18) [counting phonons]; > s.a. Thermal Conductivity; phenomenology of magnetism.
> Online resources: see Wikipedia page.

Photoelectric Effect > s.a. photon phenomenology.
* Idea: The effect by which light (in particular, UV) incident on a metal causes electrons to be emitted by the metal surface; The quantitative explanation of observations related to this effect was one of the key arguments in favor of the idea that light is made of discrete photons.
@ General references: Einstein AdP(05); Zenk RVMP(08) [variant of standard approach with wider applicability].
@ Without quanta: Wentzel ZP(27); Franken in(69); Milonni AJP(97)jan.
@ Other topics: Bach et al ATMP(01)mp/02 [mathematical].
> Online resources: see Wikipedia page.

Photon > s.a. photon phenomenology.

Photon Sphere / Surface > see spacetime subsets.

Physical Constants

Physical Laws > see under Laws.

Physical Process > see Process [including astrophysical, mathematical, ... processes].

Physicalism > see philosophy of physics.

Physically Reasonable Model
* Idea: A model for a physical system that is considered as having values for the properties under study that reflect those that can occur in a real system.
* Rem: A stronger expression would be "physically realistic model".

Physically Significant Property
* Idea: A property of a model for a physical system is physically significant if, whenever the model has the property, the real system is expected to have it as well.
* Rem: Hawking has stated that "the only properties of spacetime that are physically significant are those that are stable in some appropriate topology".

Physics > s.a. history of physics; physical theories; physics teaching.

Pi, π

Picard-Lefschetz Theory > see quantum field theory techniques.

Piecewise Flat Metrics > see regge calculus and quantum regge calculus.

Pierre Auger Observatory
* Idea: A network of detectors in the pampa of Western Argentina for the study of high-energy cosmic rays.
@ References: Anchordoqui et al PRD(03)hp; Anchordoqui ap/04-proc; Kampert NPPS(06)ap/05; Van Elewyck ap/06-ln, MPLA(08); Nitz a0706-conf [north site]; Van Elewyck a0709-proc; Parizot et al a0709-conf; de Mello APPS-a0712-conf, Matthiae a0802-conf [status and results]; Abraham et PA a0906-conf [status and plans]; Etchegoyen et al a1004-conf; Roulet a1101-conf; Smida et al a1109-proc, Kampert a1207-proc [results]; Pierre Auger Collaboration NIMA(15)-a1502 [design and performance]; > s.a. ultra-high-energy cosmic rays.
> Online resources: see Pierre Auger website; Wikipedia page.

Pigeonhole Principle (A.k.a. Dirichlet box principle.)
* Theorem: If more than n pigeons are roosting in n pigeonholes, at least one hole contains more than one pigeon.
* Applications: There are at least two people in Los Angeles with the same net worth, to the nearest dollar; In mathematics research, it is used to prove the existence of things which are difficult to construct, for example in Ramsey theory.
* In quantum physics: There are instances when three quantum particles are put in two boxes, yet no two particles are in the same box.
@ General references: Olivastro ThSc(90)sep.
@ In quantum physics: Aharonov et al a1407 + sn(14)jul [it doesn't always hold]; Yu & Oh a1408 [and the quantum Cheshire cat]; Svensson a1412; Sun et al a1806 [it is not violated]; news sn(19)feb [three photons can occupy two states without any two being in the same state]; Aharonov et al PLA-a2103 [response to criticism].

Pilot-Wave Interpretation of Quantum Mechanics [including non-equilibrium theory] > s.a. phenomenology [systems and effects].

Pin Groups / Structures and Pinors > A generalization of spin.
* Idea: Double covers of the full Lorentz group, that can be used to describe the transformation behavior of fermions under parity and time reversal; Pin(1, 3) is to O(1, 3) what Spin(1, 3) is to SO(1, 3).
@ References: Dąbrowski & Percacci JMP(88) [2D]; DeWitt-Morette & DeWitt PRD(90); in Gibbons IJMPD(94); Cahen et al JGP(95); Alty & Chamblin JMP(96) [on Kleinian manifolds]; Trautman AIP(98)ht, APPB(95)ht/98; Berg et al RVMP(01)mp/00 [long]; Bonora et al BUMI-a0907 [and spinors and orientability]; Janssens PRD(20)-a1709 [and general relativity].

Pinch Technique > see green functions for differential operators and quantum field theories.

Pioneer Anomaly > see anomalous acceleration.

Pions, π > see hadrons.

PL Manifold / Space (Piecewise Linear) > see manifold types.

Plancherel Theorem > see Symmetric Space.

Planck Constant and Units > s.a. constants; Wikipedia page.
* Value: 1998, h = 6.62606891(58) × 10−34 J · s or × 10−27 erg · s; \(\hbar\) = 1.05457266(63) × 10−34 J · s, or × 10−27 erg · s; The best values are obtained from measurements of the flux quantum φ0 = h/2e using the Josephson effect, and the quantum of conductance G0 = 2e2/h from the quantum Hall effect; 2016, h = 6.62606983 × \(10^{-34}\) J · s, achieved with NIST's new watt balance; 2018, defined to be 6.62607015 × 10−34 kg · m2 · s−1, which then defines the kg.
* Length: lP = (G\(\hbar\)/c3)1/2 = 1.6 × 10−33 cm.
* Time: tP = lP / c = 5.4 × 10−44 s.
* Energy: EP = lP c4/G = 2 × 1016 erg = 1.3 × 1019 GeV.
* Mass and density: MP = 2.2 × 10−5 g, and ρP = 5.1 × 1096 kg/m3.
@ General references: Planck SBAW(1899); Fischbach et al PRL(91) [quantum mechanics with different \(\hbar\)]; Cooperstock & Faraoni MPLA(03)ht, IJMPD(03)gq [including e and s]; Wilczek PT(05)oct [absolute units].
@ Measurements: Williams et al PRL(98) + pn(98)sep + pw(98)sep; Steiner RPP(13); news pt(16)jul [precise determination in preparation for a new, refined SI in 2018]; news pt(16)sep.
@ Related topics: Zeilinger AJP(90)feb [Planck stroll]; Casher & Nussinov ht/97 [pP is unattainable]; Sivaram a0707 [Planck mass]; Ramanathan a1402 [Planck's constant as diffusion constant]; Calmet PTRS(15)-a1504 [effective enery-scale dependence, motivated by quantum gravity].

Planck Cube
* Idea: A cube with axes labeled by \(\hbar\), G and \(c^{-1}\), whose vertices correspond to various types of physical theories; Can be considered as illustrating the concept of deformation.

Planck Distribution / Formula / Law for Black Body > see thermal radiation.

Planck Mission / Satellite > see cosmic microwave background.

Planck Stars > see astronomical objects.

Plane Wave Solutions > see gravitational wave solutions; types of waves.

Planets > see extrasolar planets; solar planets [including "Planet X" and "Planet 9"].

Planetary Nebulae > see interstellar matter.

Plasma Physics > see phenomenology of magnetism.

Plasticity > s.a. Elasticity.
* Idea: The phenomenon by which many materials maintain their deformed shape after forces are applied to them; It is often irreversible; In some materials the plastic deformation occurs when the applied forces exceed a certain threshold, below which the materials are elastic.
* Microscopically: Plasticity is a result of the propensity of solids to "flow", usually because of the motion of dislocations within them; It relies therefore on the presence of many dislocations that can easily move through the crystal, and on the bonds that hold the crystal together not being too localized, making it brittle.
* Examples: Materials with delocalized bonds are metals (in which they are due to conduction electrons) and quantum crystals (in which they are due to the atoms or molecules in the lattice, which are light, making their quantum properties important).
@ References: Castaing Phy(13) [giant, anisotropic plastic deformation that is also reversible in the quantum solid Helium-4].
> Online resources: see Wikipedia page.

Plateau Problem > see extrinsic geometry [minimal surface].

Platonic Solids > see euclidean geometry.

Plausibility Measures
* Idea: Structures for reasoning in the face of uncertainty that generalize probabilities, unifying them with weaker structures like possibility measures and comparative probability relations.
@ References: Fritz & Leifer a1505/QPL [on test spaces].

Plebański Action for Gravity > s.a. first-order actions; BF theories; unified theories.
@ References: Bennett et al IJMPA(13)-a1206 [several theories of four-dimensional gravity in the Plebański formulation].

Plebański-Demiański Solutions > see types of geodesics.

Plenitude Principle > see the Totalitarian Principle.

Plurality of Worlds > see extrasolar astronomy; history of cosmology.

PMNS (Pontecorvo-Maki-Nakagawa-Sakata) Matrix
* Idea: The lepton flavor mixing matrix in the Standard Model of particle physics.
> Online resources: see Wikipedia page.

PN Formalism > see under Post-Newtonian Expansion.

Podolsky Theory > see modified theories of electrodynamics.

Pohlmeyer Invariants > see bosonic strings and superstrings.

Pohlmeyer's Theorem
* Idea: A result proving that any critical fixed point for a field theory (in integer dimension) with vanishing anomalous dimension must be the Gaussian one.
@ References: Rosten JPA(10)-a1005 [extension to non-integer dimension].

Poincaré Conjecture > see conjectures.

Poincaré Duality > see cohomology.

Poincaré Group

Poincaré Lemma > see differential forms.

Poincaré Map / Section / Surface
* Idea: A 2D scatter plot representing the position in phase space of a system at discrete values of independent variables; Useful indicator of chaos when NdofNcom ≤ 2, otherwise regular behavior can be misinterpreted as chaos.
@ Examples: in Murray & Dermott 99 [solar system].
@ Generalization: Gaeta JNMP(03)mp/02 [Poincaré-Nekhoroshev].

Poincaré Recurrence > see Recurrence; Unitarity.

Poincaré-Hopf Theorem
* Idea: A relationship between the Euler characteristic of a manifold M and the indices of a vector field on M over its zeroes; A special case is the "hairy ball theorem", which states that there is no smooth vector field on a sphere having no sources or sinks.
@ References: Cima et al Top(98) [non-compact manifolds]; Szczęsny et al IJGMP(09)-a0810 [new elementary proof].
> Online resources: see Wikipedia page.

Point > see spacetime.

Point-Present Theories > see time.

Point Process > see statistical geometry.

Point Transformation > see symplectic structure.

Point-Splitting Regularization > see regularization.

Pointed Topological Spaces > see types of topological spaces.

Poisson Algebra / Bracket / Structure

Poisson Distribution > s.a. probability.
$ Def: The distribution on \(\mathbb N\) given by P(n) = ea an/n!.
* Properties: It has mean a, and standard deviation a1/2.
@ General references: de Groot 75, ch5.
@ Applications: Elizalde & Gaztañaga PLA(88) [of galaxies].
@ Modified: Laskin JMP(09) [fractional].

Poisson Equation > s.a. partial differential equations.
* Idea: The elliptic partial differential equation ∇2u = −f, where ∇2 is the Laplacian operator for a Riemannian metric, often flat.
@ General references: Ma et al a1208/JCP [efficient numerical solution for arbitrary 2D shapes].
@ Generalizations: Sebastian & Gorenflo a1307 [fractional].
> Online resources: see MathWorld page.

Poisson Formula
* Idea: The name given to a set of summation formulas, the original one being

\[\sum_{k=-\infty}^{\infty}\exp\{{\rm i}kx\} = 2\pi\sum_{m=-\infty}^{\infty}\delta(x-2\pi mx)\;.\]

@ References: news PhysOrg(16)mar [new formulas].

Poisson Integral > see integration.

Poisson Process > see statistical geometry.

Poisson Ratio > see Strain Tensor.

Poisson σ-Model > see sigma model.

Poisson-Boltzmann Equation > see partial differential equations.

Poisson-Lie Group
* Applications: Useful for quantum deformations of a group.
@ References: Drinfeld SMD(83); Lu & Weinstein JDG(90).

Poisson-Vlasov Equations > see under Vlasov-Poisson Equations.

Polar Decomposition Theorem > see examples of lie groups [SL(2, \(\mathbb C\))].

Polariton > see Quasiparticles.

Polarization in Electricity and Field Theory > see electricity; quantum field theory states; vacuum.

Polarization of Waves > see polarization.

Polarization in Symplectic Geometry
* Idea: A polarization is an n-dimensional completely degenerate subspace of a symplectic vector space, or integrable distribution on a 2n-dimensional symplectic manifold (it thus forms Lagrangian submanifolds).
* Example: Given a symplectic vector space (V, Ω) and a map P: VV such that P2 = \(\mathbb 1\) and P Ω = − Ω P, we can construct a polarization defined by the eigenvectors of P+:= \(\frac12\)(\(\mathbb 1\) + P) (so P+ Ω P+ = 0), with eigenvalue 1.

* Idea: A quasiparticle used in condensed matter physics to understand interactions between electrons and atoms in a solid.
@ References: Emin 13.
> Online resources: see Wikipedia page.

Polish Space > see types of distances.

Polygamma Function
$ Def: The polygamma function of order m is the (m + 1)th derivative of the logarithm of the gamma function; > s.a. Wikipedia page.

Polygon, Polyhedron > see euclidean geometry; For quantum polyhedra, see quantum geometry.

Polygroup Theory > see group theory.

Polyhomogeneous Spacetimes > see types of spacetimes.

Polymer > s.a. condensed matter [soft matter]; molecular physics.
@ Statistical mechanics: Brereton JPA(01); Ioffe & Velenik BJPStat(10)-a0908 [stretched by an external force]; Sabbagh & Eu PhyA(10) [van der Waals equation of state, self-diffusion coefficient]; De Roeck & Kupiainen CMP(11)-a1005 [polymer expansion]; Rodrigues & Oliveira JPA(14) [Monte Carlo simulations].
@ Related topics: Jitomirskaya et al CMP(03)mp/04 [random, and delocalization]; Imbrie JPA(04) [branched directed, dimensional reduction]; > s.a. solitons [in polyacetylene].

Polymer Quantization > s.a. representations of quantum mechanics.
* Idea: The name given to one of four related non-regular representations of the Heisenberg algebra, in which the spectrum of the configuration or the momentum variable is not continuous, and the corresponding infinitesimal generator is not defined; This approach to quantization is related to and inspired by, but distinct from that used in loop quantum gravity.
@ General references: Fredenhagen & Reszewski CQG(06)gq; Corichi et al CQG(07)gq/06, PRD(07)-a0704; Chiou CQG(07)gq/06 [and the Galileo group]; Hossain et al CQG(10)-a1003 [and the uncertainty principle]; Campiglia a1111 [and geometric quantization]; Date & Kajuri CQG(13)-a1211 [and symmetries]; Chacón-Acosta et al Sigma(12) [statistical thermodynamics]; Barbero et al PRD(14)-a1403 [separable Hilbert space]; Gorji et al CQG(15)-a1506 [versus the Snyder non-commutative space]; Morales-Técotl et al PRD(15)-a1507 [and the saddle point approximation of partition functions]; Morales-Técotl et al PRD(17)-a1608 [particles, path-integral propagator]; Amelino-Camelia et al PLB(17)-a1707 [and deformed symmetries, non-commutative geometry]; Berra-Montiel & Molgado CQG(19)-a1805 [as deformation quantization]; Garcia-Chung PRD(20)-a2003 [representation of the symplectic group].
@ Simple systems: Husain et al PRD(07)-a0707 [Coulomb potential]; Kunstatter et al PRA(09)-a0811 [1/r2 potential]; Kunstatter & Louko JPA(12)-a1201 [on the half line]; Majumder & Sen PLB(12)-a1207 [and GUP]; Flores-González et al AP(13)-a1302 [particle propagators]; Barbero et al CQG(13) [band structure]; Gorji et al PRD(14)-a1408 [ideal gas, partition function]; Martín-Ruiz et al PRD(15)-a1506 [bouncing particle]; Berra-Montiel & Molgado IJGMP(18)-a1610 [and zeros of the Riemann zeta function]; > s.a. gas.
@ Phenomenology: Martín-Ruiz PRD(14)-a1406 [beam of particles, and diffraction in time]; Chacón & Hernández IJMPD(15)-a1408 [semiclassical Hamiltonian and compact stars]; Martín-Ruiz et al a1408, Demir & Sargın PLA(14)-a1409 [tunneling, Zeno effect]; Kajuri CQG(16)-a1508 [radiation in inertial frames]; Ali & Seahra PRD(17)-a1709 [natural inflation]; Kajuri & Sardar PLB(18)-a1711 [Lorentz violation, at low energies]; Khodadi et al sRep(18)-a1801 [optomechanical setup]; Amirfakhrian & Vakili IJGMP(19)-a1812 [particle tunneling from Schwarzschild black hole]; Barca et al PRD(19)-a1902 [flat isotropic cosmology]; Barman & Sardar PRD(19)-a1906 [entanglement entropy and area law]; Berra-Montiel et al a2007 [N-body systems, superfluids]; > s.a. phenomenology of cosmological perturbations; unruh effect.
> Related topics: see Bohr Compactification; entropy in quantum theory; fock space; holography; renormalization; tunneling.
> Gravity / cosmology: see loop quantum gravity; minisuperspace; models in canonical quantum gravity; quantum black holes; 2D quantum gravity; Taub Model.
> Other field theories: see bose-einstein condensates; Pais-Uhlenbeck Model [higher-order time derivatives]; QED phenomenology; quantum field theories [scalar].

Polynomials > see functions.

Polyomino > s.a. voronoi tilings.
* Idea: A finite and connected union of tiles.

Polysymplectic Formalism > see types of symplectic structures.

Polytope > s.a. Complex / simplex.
* Idea: An n-dimensional generalization of a polyhedron; The word was coined by Alicia Boole (daughter of George Boole).
$ Def: A polytope in an affine space is the convex hull of a finite set of points.
* Result: (Balinski's theorem) The graph of a d-polytope is d-connected.
* Simple polytope: One in which each vertex is on the boundary of d facets.
* Polytope of a collection of simplices: The polytope |K| of the collection K in \(\mathbb R\)d is the union of all simplices σK, adequately structured as a topological space [?]; If K is a simplicial complex, then its polytope is a polyhedron.
* Delaunay polytope: A polytope P such that the set of its vertices is SL, with S being an empty sphere of a given lattice L.
* Parallelotope: A polytope whose translation copies fill space without gaps and intersections by interior points; Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, i.e., a Voronoi polytope.
@ Books: Grünbaum 67, 03; Thomas 06 [geometric combinatorics].
@ General references: Kalai JCTA(88) [and graphs]; Walton in(04)mp [and Lie characters]; Deza & Grishukhin EJC(04) [parallelotopes]; Enciso a1408 [volumes of polytopes in any dimension without triangulations].
@ Regular polytopes: Cantwell JCTA(07) [all regular polytopes are Ramsey]; Boya & Rivera RPMP(13)-a1210.
@ Delaunay polytopes: Dutour EJC(04); Erdahl et al m.NT/04-(proc); Sikiric & Grishukhin EJC(07) [computing the rank].
@ In 3D spaces of constant curvature: Abrosimov & Mednykh a1302 [volume formulas].
@ Other special types: Neiman GD(14)-a1212 [null-faced 4-polytopes in Minkowski spacetime].
> Related topics: see Schlegel Diagram; statistical geometry [from random point set].

Pomeransky-Senkov Black Hole > see causality conditions.

@ General references: Levin hp/98-conf; cern(99); Brower et al JHEP(07)ht/06 [and gauge/string duality]; Swain a1110-fs [and the nature of particles].
@ And QCD: Donnachie et al 02; Nachtmann hp/03-conf.
> Online resources: see Wikipedia page.

Pontrjagin / Pontryagin Classes, Numbers

Ponzano-Regge Model > s.a. spin-foam models / 3D gravity; SU(2).
* Idea: 3D spin coupling theory, giving a non-perturbative definition of the path integral for (Euclidean) 3D gravity.
@ General references: Ponzano & Regge in(68); Lewis PLB(83) [renormalizability]; Iwasaki gq/94, JMP(95)gq [in terms of surfaces]; O'Loughlin ATMP(02)gq/00 [boundary actions]; Barrett & Naish-Guzman CQG(09)-a0803; Wieland PRD(14)-a1402 [action from a 1D spinor action].
@ Variations: Carfora et al PLB(93) [4D, and 12j symbols]; Carbone et al CMP(00); Freidel NPPS(00)gq/01 [Lorentzian]; Livine & Oeckl ATMP(03)ht/03 [supersymmetric]; Li CMP(14)-a1110 [κ-deformation]; Vargas a1307 [on a manifold with torsion].
@ Related topics: Barrett & Foxon CQG(94)gq/93 [semiclassical limit]; Petryk & Schleich PRD(03)gq/01 [geometric quantities]; Arcioni et al NPB(01)ht [and holography]; Freidel & Louapre CQG(04)ht [gauge fixing], gq/04 [and Chern-Simons theory]; Freidel & Livine CQG(06)ht/05 [effective field theory for particles]; Hackett & Speziale CQG(07)gq/06 [geometry and clasping rules]; Barrett & Naish-Guzman gq/06-MGXI [and Reidemeister torsion]; Livine & Ryan CQG(09)-a0808 [B-observables]; Caravelli & Modesto a0905 [spectral dimension of spacetime].

Popper's Thought Experiment
* Idea: A thought experiment proposed by Karl Popper designed to check for possible violations of the uncertainty principle.
@ General references: Qureshi IJQI(04)qp/03, AJP(05)jun-qp/04; Richardson & Dowling IJQI(12)-a1102 [no violation of the uncertainty principle, fundamental flaw]; Qureshi Quanta(12)-a1206 [modern perspective]; Cardoso a1504 [non-linear quantum theory and uncertainty principle violation].
@ With photons: Kim & Shih FP(99) [entangled photon pairs]; Peng et al EPL(15) + news pw(15)jan [photon number fluctuation correlations in a thermal state]; Reintjes & Bashkansky a1501.
> Online resources: see Net Advance of Physics page; Wikipedia page.

Porosity of a Measure > see measure.

Pöschl-Teller Potential > s.a. types of coherent states.
@ References: Aldaya & Guerrero qp/04 [modified, group quantization]; Rudoy & Oladimeji a1906 [detailed treatment].
> Online resources: see MathWorld page on Pöschl-Teller differential equations.

Poset > s.a. set of posets and types of posets.

* In quantum mechanics: Teller (1979) argued that a particle cannot have a sharp position; Others disagree; > s.a. localization in quantum mechanics.
@ In quantum mechanics: Chew SP(63); Halvorson JPL(01)qp/00 [sharp]; Kosiński & Maślanka a1806 [operator, for massless spinning particles].
@ Tests of local position invariance: Peil et al PRA(13) [using continuously-running atomic clocks]; Shao & Wex CQG(13) [bounds].

Positioning Systems > s.a. coordinates; minkowski spacetime [secure positioning].
@ Relativistic positioning systems: Coll et al a0906-rp [status]; Tartaglia a1212-conf [principles and strategies]; Coll a1302-conf [rev]; Puchades & Sáez ApSS(14)-a1404 [errors due to uncertainties in the satellite world lines].
@ GPS: Parkinson & Spilker ed-96; in Hartle 03; Puchades & Sáez ApSS(12)-a1112.

Positive Action Conjecture > see action for general relativity.

Positive-Energy Theorem

Positive Frequency Function > see functions.

Positive Map > see Maps.

Positivism > see philosophy of science.

Positron > see electron; elements [Positronium].

Possibilism > see time.

Possibility > see many-worlds interpretation.

Post-Friedmannian Formalism > see cosmological models.

Post-Newtonian (PN) Expansion > see gravitational phenomenology; gravitomagnetism; matter dynamics in gravitation.

Potential for a Field
* Idea: Originally, a potential was a scalar function whose gradient gives a force on a test particle (per unit charge); It was extended to a vector field whose curl gives a (magnetic) field, and then to the general mathematical notion of a function (or a higher-rank tensor field) which gives, by differentiation, a field of interest, possibly a dynamical tensor field.
> Vector potential: see aharonov-bohm effect; connection; electromagnetism.

Potential in Physics > s.a. scattering; thermodynamics [thermodynamic potentials].
* Retarded potential: A potential generated by sources in the past of the field point; It has to be used for systems with large velocities (corrections are of order v2/c2), or pairs of systems with large separations compared to the internal motions (even if slow).
@ General references: Kellogg 29; Grant & Rosner AJP(94)apr [orbits in power law V].
@ Retarded potential: Spruch & Kelsey PRA(78) [elementary derivation]; > s.a. arrow of time.
> In classical mechanics: see Bertrand's Theorem; classical systems [including central potentials]; Coulomb Potential.
> In classical field theory: see electromagnetism; newtonian gravitation.
> In quantum mechanics: see schrödinger equation; special potentials; pilot-wave interpretation [quantum potential].
> In quantum field theory: see effective field theories [effective potential]; quantum field theory.

Potts Model > s.a. lattice field theory; Yang-Baxter Equation.
* Idea: A 2D generalization of the Ising model of interacting spins on a lattice; The chiral Potts model is a challenging one, it is "exactly solvable'' in the sense that it satisfies the Yang-Baxter relation, but actually obtaining the solution is not easy; Its free energy was calculated in 1988, the order parameter was conjectured in full generality in 1989 and derived in 2005.
@ General references: Baxter 82; Wu RMP(82); Sokal MPRF(01)cm/00-in [unsolved problems]; Baxter JPCS(06)cm/05 [rev]; Beaudin et al DM(10) [introduction from a graph theory perspective].
@ Phase transitions: Baxter JSP(05)cm, PRL(05)cm [chiral, order parameter]; Georgii et al JSM(05)mp [continuum, order-disorder transition]; Ahmed & Gehring JPA(05) [anisotropic, phase diagram]; Jacobsen & Saleur NPB(06) [antiferromagnetic transition]; Fernandes et al PhyA(06) [alternative order parameter]; Gobron & Merola JSP(07) [first-order]; Johansson PLA(08) [2D with open boundary conditions, Monte Carlo]; Aluffi & Marcolli JGP(13)-a1102 [motivic approach].
@ Coupled to gravity: Ambjørn et al NPB(09)-a0806, Cerda Hernández JMP(17)-a1603 [causal dynamical triangulations].
@ Related topics and variations: Richard & Jacobsen NPB(07) [on a torus]; Barré & Gonçalves PhyA(07) [on a random graph, canonical and microcanonical ensembles]; Ganikhodjaev PLA(08) [next-nearest-neighbor interactions, on the Bethe lattice]; De Masi et al JSP(09) [continuum version, phases]; Contucci et al CMP(13)-a1106 [on a random graph]; Dasu & Marcolli JGP(15)-a1412 [in an external magnetic field, sheaf-theoretic interpretation]; > s.a. Confinement [model for]; renormalization.

Pound-Rebka Experiment > see tests of gravitational redshift.

POVM > Positive Operator-Valued Measure, see measure theory.

Powder > see metamaterials.

Power of a Graph > see graph theory.

Power Spectrum of Perturbations in Field Theory
* Idea: Usually defined as the Fourier transform of the two-point correlation function of the field in a quantum state.

Power-Law Distributions > s.a. critical phenomena; states in statistical mechanics.
@ References: Simkin & Roychowdhury PRP(11) [mechanism for producing them].

Poynting Vector > s.a. energy-momentum tensor.
* Idea: The vector S = E × B/μ0, giving the direction of propagation of energy-momentum in an electromagnetic field, and the power flux across a unit normal surface.
* As a 4-vector: Without sources (Poincaré pointed out a difficulty with sources), the vector Pa = (U, P), where

U:= (1/8π) (E2 + B2) dv = T00 dv ,   P:= (1/4πc) E × B dv = T0i dv .

* Gravitational: In general relativity an analog of the electromagnetic Poynting vector is the super-Poynting vector of the Weyl tensor.
@ General references: in Jackson; in Rohrlich; McDonald AJP(96)jan [meaning].
@ Gravitational: de Menezes gq/98; Manko et al CQG(06) [axistationary electrovac spacetimes]; Wylleman et al a2007 [and principal observers].

Poynting-Robertson Effect
* Idea: An effect that produces changes in the orbital plane of a particle; It has been applied to meteoroids and to high-energy astrophysics, where a particle orbiting a compact object feels the gravitational pull, radiation pressure, and the radiation drag force opposite to the orbital motion caused by the process of absorption and emission of radiation by the test particle, which efficiently removes energy and angular momentum from the particle and results in an effective dissipative force.
@ References: Klacka ap/00, ap/01, ap/02, ap/02; in Harwit 06; Klacka a0807 [paradox in astrophysical application]; Klacka et al a0904 [explanations].
@ In curved spacetime: Bini & Geralico CQG(10) [extended to spinning particles in Schwarzschild spacetime]; Bini et al CQG(11); De Falco et al PRD(18)-a1804 [relativistic, Lagrangian formulation]; De Falco & Battista PRD(20)-a2003 [analytic form of the Rayleigh potential]; De Falco et al PRD-a2006; De Falco a2006.

pp-Waves > see gravitational wave solutions.

PPN (Parametrized Post-Newtonian) Formalism > s.a. gravitation / higher-order gravity; modified newtonian gravity.
* Rem: It is not the same as PN (Post-Newtonian) expansion of general-relativistic results around the weak-field / slow-motion limit.

Prasad-Sommerfield Solution > see monopoles.

Pre-Recueil > see Recueil.

Pre-Acceleration > see self-force [Lorentz-Dirac equation].

Precanonical Quantization > see approaches to quantum field theory; approaches to quantum gravity; quantization of gauge theories.

Precession > s.a. gravitating bodies; Gyroscope; Runge-Lenz Vector; test bodies; Thomas Precession.
* In general relativity: There are several types, perihelion (Einstein), geodetic (de Sitter), orbital plane (Lense-Thirring, gravitomagnetic), and spin-spin (Pugh-Schiff); > s.a. tests of general relativity with orbits.
@ General references: Magli phy/04 [in ancient astronomy]; Jonsson CQG(06)-a0708 [spin precession, covariant formalism]; Casotto & Bardella MNRAS(13)-a1210-conf [equations of motion of a secularly precessing elliptical orbit]; Lo et al AJP(13)sep, D'Eliseo AJP(15)apr [unified frameworks for perihelion advance, different causes].
@ In general relativity: Holstein AJP(01)dec; Sigismondi ap/05-MGX; Harper PhSc(07)dec; He & Zhao IJTP(09) [analytical solution]; Boyle et al PRD(11) [compact binaries, geometric approach]; D'Eliseo ApSS(12)-a1206 [precession of orbits, quick method]; Mashhoon & Obukhov PRD(13)-a1307 [in gravitational fields]; Hu et al AHEP(14)-a1312 [general spherically symmetric spacetimes]; Heydari-Fard et al GRG(19)-a1906 [using higher-order geodesic deviation approach]; > s.a. gravitational self-force [spin precession].
@ In modified gravity theories: Behera & Naik ap/03 [vector gravity]; Schmidt PRD(08) [modified Newtonian potential]; Fokas et al a1509 [relativistic gravitational law]; Friedman & Steiner EPL(16)-a1603 [in relativistic Newtonian dynamics].
@ Specific cases: Stewart AJP(05)aug [Mercury, due to other planets]; Iorio AJ(09)-a0811 [Saturn, anomalous]; Moniruzzaman & Faruque PS(13) [periastron precession due to gravitational spin-orbit coupling].
> In various theories: see Cogravity; gravity theories; newtonian gravity [perturbations and curved spaces].
> In various spacetimes: see reissner-nordström solutions; schwarzschild-de sitter spacetime [with a cosmological constant]; test bodies.

Precision > s.a. Accuracy.
* Idea: The size of the error bar in a series of measurements.

Precompactness > see compactness.

Prediction and Predictability > s.a. causality; paradigms in physics; time.
* Idea: Predictability is an epistemic property of a model for a physical system, related to what we are able to compute and predict with it; Prediction may refer to a theory predicting either effects, phenomena, values of quantities, or more specifically the evolution of a system and results of future measurements.
* Question: Does a physical law have to be predictive?
* Remark: Usually, for several practical and theoretical reasons, predictions in physics are statistical.
@ General references: Brush Sci(89)dec [light bending]; Hole IJTP(94) [and determinism]; Holt & Holt BJPS(93) [in classical mechanics]; Caves & Schack Compl(97)cd [types]; Coles 06 [I]; Manchak FP(08) [in general relativity]; Werndl BJPS(09) [and chaos]; Srednicki & Hartle PRD(10)-a0906 [in a very large universe]; Stuart et al PRL(12) + news physorg(12)jul, physorg(12)jul [experimental bound on the maximum predictive power]; Cecconi et al AJP(12)nov [intrinsic limitations]; Hosni & Vulpiani P&T(17)-a1705 [forecasting and big data].
@ Of effects: Hitchcock & Sober BJPS(04) [vs accommodation, and overfitting].
> Related topics: see chaos; determinism; electron [magnetic moment]; Explanation; wave phenomena [superluminal].

Prefixes > see units.

Pregeometry > see Matroid [mathematics]; quantum spacetime [physics].

Preons > see composite models.

Preorder > s.a. poset; Quasiorder [non-reflexive generalization].
$ Def: A reflexive and transitive binary relation; The concept generalizes that of (reflexive) partial orders and equivalence relations.
* Remark: One can always define an Alexandrov topology on a preorder by using the upper sets as open sets.
@ References: Cameron et al DM(10) [random preorders and alignments]; Minguzzi AGT(12)-a1108 [representation by continuous utilities].
> Online resources: see Wikipedia page.

Prequantization > s.a. geometric quantization.
@ References: Schreiber a1601-in [higher prequantum geometry].

Presentation of a Group
$ Def: A pair (S, D) of a set of generators S and a set of relations between the generators D = {Γi}; Each relation Γi is of the form wi = 1, where wi is a word; The group elements are equivalence classes of words.
* Example: One generator, S = {a}; If D = Ø, the group is \(\mathbb Z\), the infinite cyclic group generated by a, but if D = {aa = 1}, we get the group of order 2.
* Remark: Two presentations of the same group may look quite different, and it may be difficult or impossible to tell whether two groups are isomorphic by looking at their presentations; > s.a. group theory [isomorphism problem]; Word [word problem].

Presentation of a Topological Space
* Idea: An appropriate set of vertices, edges, faces, etc.
* Result: A finitely presented space has a finitely presented fundamental group (> s.a. Calculating Theorem).

Presentism > s.a. special relativity; time.
* Idea: The view that only the present is real (as opposed to possibilism, eternalism or the block-universe view, and their variants).
@ References: Wüthrich a1207 [fate in modern physics]; Romero & Pérez EJPS(14)-a1403 [and black holes]; Rovelli a1910 [reality has a more complex temporal structure].
> Online resources: see Wikipedia page.

Pressure > s.a. energy-momentum tensor; fluid; gravitating matter; Momentum; radiation; thermodynamics; turbulence.
@ General references: Durand AJP(04)aug [quantum, Bose and Fermi statistics]; Frontali PhysEd(13) [history of the concept].
@ Coupling to gravity: Ehlers et al PRD(05)gq; Narimani et al JCAP(14)-a1406 [and observational cosmology].

Presymplectic Structure > see symplectic geometry.

Prevalence [> s.a. measure theory.]
* Idea: The analogue of the finite-dimensional notions of 'Lebesgue almost every' and 'Lebesgue measure zero' in the infinite-dimensional setting
@ References: Ott & Yorke BAMS(05).

Price's Law > see perturbations of schwarzschild spacetime.

Primakoff Effect > s.a. axions.
* Idea: The production of an axion from the interaction of a photon with a classical electromagnetic field [Henry Primakoff 1951].

Prime Graphs > see types of graphs.

Prime Numbers
> see number theory.

Primordial Black Holes > see types of black holes.

Primordial Gravitational Waves > see gravitational-wave background.

Primordial Magnetic Fields > see magnetic fields in cosmology.

Primordial Perturbations > see phenomenology of cosmological perturbations.

Principal Fiber Bundle

Principal Ideal, Principal Ideal Domain, Principal Ideal Ring > see rings.

Principal Part / Value > see distribution.

Principal Principle > s.a. quantum measurements.
* Idea: A principle relating objective probabilities and subjective chance.
@ References: Meacham BJPS(10) [misconceptions].

Principle of Equivalence > see under Equivalence Principle.

Principle of the Excluded Middle > see Law of the Excluded Middle.

Principle of Mediocrity > see civilizations.

Principles in Mathematics, Physics, and Related Areas > s.a. Physical Laws.
> In gravitation and cosmology: see anthropic principle; Copernican Principle; cosmological principle; equivalence principle; mach's principle; Principle of Mediocrity; Relativity Principle.
> In quantum theory: see Correspondence Principle; (Pauli) Exclusion Principle; Landauer's Erasure Principle; Maximal Variety; (heisenberg's) uncertainty principle.
> In other physics: see Action-Reaction Principle; Boltzmann Principle; Causal Entropic Principle; Fermat's Principle; Hamilton's Principle; huygens' principle; Maupertuis Principle; Maximum Entropy Principle; Non-Demolition Principle; Superposition Principle; Symmetric Criticality; variational principles.
> In mathematics: see (Cauchy's) Argument Principle; Enumeration Principle; Pigeonhole Principle; Principal Principle; Well-Ordering Principle.
> In logic: see Common Cause Principle; Excluded Middle; Leibniz Principle; Principle of Sufficient Reason.

Prisoner's Dilemma > see games.

Probability > s.a. probability in physics and in quantum physics.

Probability Current > s.a. path integrals.
* In quantum mechanics: It can be constructed from the wave function by j:= # Im(ψ* ∇ψ); The integral lines for this current are analogous to trajectories.
@ References: Schumacher et al a1607 [generalization to finite-dimensional Hilbert spaces, open quantum systems].

Problems > see Coloring; matrix; orbits in newtonian gravity [Kepler], of gravitating objects; Three-Body Problem; Two-Body Problem.
* 2012.03: Lightning strikes produce free neutrons, and we're not sure how [@ news at(12)mar].

Proca Theory > s.a. modified electromagnetism / spin-1 field theories [and spin-3/2]; lagrangian systems [Proca Lagrangian].
* Idea: A "massive gauge theory", a gauge theory with a non-gauge-invariant mass term m2 A2 added to the Lagrangian,

L = − \(1\over4\)Fab Fab + \(1\over2\)m2 Aa Aa + Aa j a .

@ General references: Proca CRAS(36); in Wentzel 49; Goldhaber & Nieto RMP(71) [and photon mass limits]; in Gsponer & Hurni in(98)phy/05 [history]; Dvoeglazov CzJP(00)ht/97; Fabbri AFLB(11)-a0908 [most general consistent theory]; Errasti et al PRD(20)-a1905 [interacting Maxwell and Proca fields].
@ Einstein-Proca: Dereli et al CQG(96) [torsion and non-metricity]; Vollick gq/06; > s.a. black-hole hair; black-hole perturbations; einstein-cartan theory.
@ Other variations, generalizations: Kruglov IJMPA(06) [sqrt version, including spin-1/2]; Escalante et al a1402 [5D, canonical analysis]; Heisenberg JCAP(14)-a1402; Allys et al JCAP(16)-a1511; De Felice et al PRD(16)-a1602, JCAP(16)-a1603 [fifth-force screening and cosmology]; Heisenberg et al PLB(16)-a1605 [with higher-order derivative interactions]; Allys et al PRD(16)-a1609 [SU(2) Proca theory, or non-Abelian vector galileon]; Heisenberg a1705-proc [rev]; de Rham & Pozsgay PRD(20)-a2003 [new class of interactions].
@ Quantization: Aldaya et al IJMPA(97)ht/96; van Hees ht/03 [renormalizability]; Helesfai CQG(07)gq/06 [in lqg]; Zamani & Mostafazadeh JMP(09)-a0805; Castineiras et al PRD(11)-a1108 [in a Rindler wedge].
@ Quantum theory, in curved spacetime: Furlani JMP(99) [on a globally hyperbolic Lorentzian manifold, canonical]; Toms a1509 [with non-minimal terms, Faddeev-Jackiw approach to quantization]; Schambach a1709-MSc; Schambach & Sanders a1709 [and the zero mass limit].
@ Phenomenology: Brito et al PLB(16)-a1508 [self-gravitating BECs of Proca particles]; De Felice et al PRD(17)-a1703 [observational constraints].
@ Related topics: Comay NCB(98); Kim et al MPLA(98)ht [symmetries]; Vytheeswaran IJMPA(98) [as gauge theory]; Zecca GRG(06) [in FLRW spacetime].

Process > s.a. Ontology [process ontology].
* Quantum process: The operation performed by a quantum processor that transforms a quantum system's state into a different one.
@ Physical process: Spaans gq/05 [background independence]; Needham BJPS(13) [processes as autonomous entities, thermodynamic perspective].
@ Quantum process: Poyatos et al PRL(97) [characterization]; D'Ariano & Lo Presti PRL(01) [and quantum tomography]; Bendersky et al PRL(16)-a1407, Parke a1409 [implications of computer science principles]; Lee & Hoban PRS(16)-a1510 [tradeoff between quantum computation and communication complexity]; Yadin et al PRX(16) [operations which do not use coherence]; > s.a. creation operator; quantum effects.
> Specific physical processes: see diffusion; Drell-Yan Process; Joule-Thomson Process; Penrose Process; Transport.
> Processes theory in physics: see approaches to quantum field theory [process algebra approach]; causal structures.
> Types of mathematical processes: see markov processes; random processes; statistical geometry [point processes]; stochastic processes.
> Specific mathematical processes: see Airy Process; Lévy Process; Wiener Process.
> Astrophysical processes: see Accretion Process.

* Special infinite products:

k = 2(1 − 1/k2) = \(1\over2\)   [prove by splitting into (1 − 1/k) (1 + 1/k) and using factorials] .

@ References: Albert & Kiessling JSP(17)-a1610 [infinite trigonometric products and random walks on the real line]; > s.a. series [history].
@ Generalized products: van de Wetering JMP(18)-a1803 [sequential product]; > s.a. Star Product; vectors [scalar, vector product].

Programming > see computation; computer languages.

Progressing Waves > see types of waves.

Projectable Vector Field
$ Def: A differentiable vector field v is projectable by the map f if f '(v) is differentiable.

Projectile Motion > s.a. kinematics of special relativity.
@ General references: Klevgard a1501 [and XX century changes in physics]; Walley a1804 [history, from Aristotle to Galileo and Newton].
@ With air resistance: Mohazzabi & Shea AJP(96)oct [with variation of atmospheric pressure]; Price & Romano AJP(98)feb [optimal launch angles]; Warburton & Wang AJP(04)nov; Linthorne pw(06)jun [and soccer]; Goff & Carré AJP(09)nov [soccer balls].

Projection Mapping > see bundles.

Projection Postulate in Quantum Theory > see axioms for quantum theory; wave function collapse.

Projective Geometry, Structure, Limit, System > see projective.

Projective Relativity and Field Theory
* Projective relativity: Initially proposed by Fantappiè and subsequently developed by Arcidiacono.
@ General references: in Schmutzer ed-83 [projective relativity]; Schmutzer AN(05)ap [projective unified field theory and 2-body system].
@ And cosmology: Licata & Chiatti IJTP(09)-a0808; Benedetto IJTP(09) [and varying speed of light].

Projector, or Projection Operator
$ Def: An operator P on an inner product space which is self-adjoint and idempotent.
* Projective methods: Used for systems of linear and non-linear algebraic equations and convex optimization.
@ References: Galántai 03; Halliwell PLA(13)-a1207 [localized in a region of phase space].

Proof Theory

Prop > see examples of categories.

Propagator > s.a. green function [for differential operators, wave equations]; feynman propagator and green function [in quantum field theory].
* In quantum mechanics: Can be calculated directly using the path-integral technique, or as inverse Laplace transform of the Green function.
@ In quantum mechanics: Nardone AJP(93)mar [calculation]; Fulling & Güntürk AJP(03)jan [1D particle in a box]; Kosut et al qp/06 [distance between propagators]; Moshinsky et al Sigma(07)-a0711 [from Green function]; Zanelli et al RPMP(08) [integral representations].

Propensity > see probability in physics.

Proper Discontinuous Action of a Group > see group action.

Proper Time > s.a. special-relativistic kinematics.
* Idea: The proper time at a point along a timelike line in spacetime is the length of the line from a reference initial point.
@ References: Wesson a1011 [adjustments from the possible existence of higher dimensions].

Property > s.a. Generic Property; Physically Significant Property; Stability.
$ In mathematics: A property P defined for elements x of a set X is an attribute that those elements may have or not have, i.e., a map P : X → {0,1}.
$ In physics: A property P is often an attribute that a physical system s or theoretical model may have to varying degrees, i.e., a map P : S → \(\mathbb R\) (sometimes \(\mathbb C\)); Important examples are the values of observables, or the truth values of propositions about the system.
* Rem: For the purpose of discussing different types of properties, it is often convenient to specify a topological space structure on X and distinguish cases in which P behaves differently when considering its values for elements in a neighborhood of a given x.
* Terminology: An element x in X (or a subset A of X) are said to have the property if P(x) = 1 (resp., P(A) = {1}).
@ General references: Szabó FP(19)-a1901 [the intrinsic-extrinsic distinction is flawed].
@ Of a quantum system: Hofmann et al a1605-proc [and observable effects]; Pandey et al a1812 [unmeasured properties]; > s.a. Beable.
> Related topics: see measurement in quantum theory.

Propositional Logic > see logic.


Prout's Law > see atomic physics.

Proximity-Force Approximation > s.a. casimir-effect examples.
* Idea: An approximation method for the electrostatic interaction between two perfectly conducting surfaces, used when the distance between them is much smaller than the characteristic lengths associated to their shapes; The electrostatic force is evaluated by first dividing each surface into a set of small flat patches, and then adding up the forces due two opposite pairs, approximated as pairs of parallel planes; It has been successfully applied to contexts such as nuclear physics and Casimir-effect calculations.
@ References: Fosco et al AP(12)-a1201 [improved approximation].

Proximity Graphs > see graph types.

Proximity Structure

Pseudoclassical Dynamical Systems
* Idea: Models that have classically anticommuting variables.
@ References: Allen et al a1509 [quantization].

Pseudodifferential Operator > see operator theory.

Pseudogroup > s.a. differentiable maps [local pseudogroup of transformations].
@ In physics: Woon ht/98 [intro and applications].

Pseudomanifold > see types of manifolds.

Pseudometric Space > see distance.

Pseudorandomness > see random processes.

Pseudosphere > s.a. sphere.
* Idea and history: A 2D surface with constant and negative Gaussian curvature; Discussed in 1868 by Eugenio Beltrami in terms of a disk on the plane, which is isomorphic to the two-sheet hyperboloid in \(\mathbb R^3\).
@ References: Bertotti et al gq/05-proc [review, geometry and physics].

Pseudostationary Spacetime > see types of spacetimes.

Pseudosymmetric Spacetime > see 3D geometry.

> see stress-energy pseudotensor.

Pseudovector (a.k.a. axial vector) > see vector.

ψ-Epistemic Quantum Theory > s.a. interpretations of quantum theory [statistical interpretation]; quantum foundations; types of interpretations [type-II].
* Idea: The view that quantum states are not descriptions of quantum systems but rather reflect the assigning agents' epistemic relations to the systems; Theories that try to reproduce the predictions of quantum mechanics, while viewing quantum states as ordinary probability distributions over underlying objects called "ontic states".
@ General references: Harrigan & Spekkens FP(10)-a0706; Friedrich SHPMP(11)-a1101; Aaronson et al PRA(13)-a1303 [conditions, no-go results, and the role of symmetry]; Patra et al PRA(13) [experiment]; Ballentine a1402 ["functionally ψ-epistemic" theories]; Wharton Info(14)-a1403 [quantum states as ordinary information]; Miller & Farr a1405 [quantum states apply only to ensembles, there are no ontic states]; Rovelli FP(16)-a1508, refutation Zeh a1508 [argument against the realist interpretation]; Boge a1603 [Einsteinian view, new developments]; Ruebeck et al Quant(20)-a1812 [measurement problem]; Shanahan IJQF-a1901; Sokolovski EPL-a2005 [minimalist view]; Hance et al a2101 [states can be both ontic and epistemic].
@ And distinguishability of quantum states: Barrett et al PRL(14)-a1310, Leifer PRL(14)-a1401, Branciard PRL(14)-a1407, news nat(15)may [no-go results].
@ Gravity-related theories: Evans et al SHPMP(16)-a1606 [quantum cosmology].
@ Other theories and applications: Kak a1607-conf [quantum communication]; Sen FP(19)-a1803 [retrocasual hidden-variable model].
> Related topics: see Epistemology; hidden-variable theories; quantum probabilities; realism [epistemological realism]; sub-quantum theories.

ψ-Ontic Quantum Theory ("wave functions are real") > s.a. interpretations of quantum theory [including PBR theorem]; types of interpretations [type-I].
* Idea: The view that quantum states are ontic, i.e., states of reality; A particle's wave function is a physical object that can be studied in wave mechanics.
* Schrödinger's original interpretation: The wave function is actual density of stuff, and can be identified with a particle's cherge density, for example.
* Problems with wave function = particle: Wave packets spread, particles don't; What about systems with N > 1 particles?
* Other possibilities: The wave function may be real but not to be identified with a physical object.
* The PBR theorem: (Pusey-Barrett-Rudolph) Models in which quantum states just represent information about underlying physical states contradict quantum mechanics.
* Experiments: 2015, Results obtained for photon systems indicate that no knowledge interpretation of quantum theory can fully explain the distinguishability of non-orthogonal quantum states; The results are not yet conclusive, because most of the photons were not detected, and other groups are working on experiments with ions; The Barrett-Cavalcanti-Lal-Maroney (BCLM) argument can be turned into an effective experimental test.
@ Schrödinger's interpretation: Barut AdP(88), FP(88), FPL(88) [revival].
@ General references: Liu BJPS(94); Jabs PE(96)qp; Lewis BJPS(04) [less problematic interpretation]; Colbeck & Renner PRL(12)-a1111 [and completeness of quantum theory]; Hardy IJMPB(13)-a1205; Mansfield a1306 [ontic and epistemic interpretations]; Shenoy & Srikanth a1311 [the wave function is real but non-physical]; Leifer Quanta(14)-a1409 [rev]; Cabello et al PRA(16) [thermodynamic constraints]; Durham a1807 [for field theories, based on the Wheeler-DeWitt equation]; Myrvold a1911-in [status], a1911-in [and subjective probabilities]; Rousselle a2008-PhD [wave mechanics]; Hermens a2101 [how real are states?].
@ The PBR theorem: Pusey et al nPhys(12)may-a1111 + news nat(12)may [the theorem]; Nigg et al NJP(16)-a1211 [experimental test using trapped ions]; Patra et al PRL(13)-a1211 [argument based on a continuity assumption]; Colbeck & Renner NJP(17)-a1312 [condition under which Ψ is uniquely determined by a complete description of the system's physical state]; Moseley a1401 [simpler proof]; Mansfield PRA(16)-a1412 [using a weaker, physically motivated notion of independence]; Mansfield EPTCS(14)-a1412; Ducuara et al JPA(17)-a1608 [under noisy channels]; Charrakh a1706, Schmelzer a1906 [criticisms of the argument]; Das Bhowmik et al a2005 [theorem with different assumption]; Hermens a2008 [critical note on Colbeck & Renner].
@ Other support of ψ-ontology: Allen QS:MF(15)-a1501 [quantum superpositions cannot be epistemic]; Gao SHPMP(15)-a1508 [in terms of protective measurements]; Bhaumik Quanta(15)-a1511; Kiefer a1908-conf; Gao BJPS(20)-a2001 [and protective measurements]; > s.a. Tidal Force; Tractor Beam [pulling force from a quantum-mechanical matter wave].
@ Experiments: Ringbauer et al nPhys(15)feb-a1412 + news NYT(15)feb [with single photons]; Knee NJP(17)-a1609 [towards optimal experimental tests].
> Related topics: see Beable; Ontology; realism [including ontic structural realism].

PSSC (Physical Sciences Study Committee) > see physics teaching.

PT Symmetry > s.a. modified quantum mechanics [including field theory], statistical mechanical systems [PT-symmetric]; Unitarity.
@ General references: Mostafazadeh PS(10)-a1008 [rev].
@ Breaking: Bender & Darg JMP(07) [in classical mechanics]; Ambichl et al PRX(13) [in scattering systems].

Pullback Bundle > see fiber bundle.

Pullback of a Function / Form under a Mapping > see differentiable maps.


Pure Sequence > see exact sequence.

Purity > s.a. mixed states; polarization.
* Idea: The quantity ζ = tr ρ2, a measure of how pure a quantum state is; Its value is one for pure states and 1/d for maximally mixed states of dimension d.
* Applications: It can be used for example to quantify entropy increase in decoherence.
> Online resources: see Quantiki page; Wikipedia page.

Push-Forward > see tangent structures.

Puzzles > see logic.

* Idea: One of the 4D particles corresponding to the non-zero modes of the harmonic expansions in mass eigenstates of the 5D fields in Kaluza-Klein theory.

Pythagorean Theorem
@ General references: Crease pw(06)jan, Overduin & Henry a2005 [history and significance].
@ Lorentzian version: Ungar FP(98); Brill & Jacobson GRG(06)gq/04-fs.

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