Topics, P

p-Adic Number / Structure > s.a. differential equations.
$ Def: A uniformity on Z defined by giving, as fundamental set of entourages,

Wn := {(x, y) | x = y mod pn} Z × Z ,   for all n (p is a prime) .

@ In cosmology and gravitation: Dragovich AIP-ht/06 [cosmology]; > s.a quantum cosmology, quantum spacetime.
@ Other physics: Khrennikov NCB(98)-a0906, Dubischar et al NCB(99)-a0906 [and correlations between quantum particles]; Dragovich et al pUAA-a0904 [rev]; Rodríguez-Vega & Zúñiga-Galindo PJM-a0907 [p-adic fields, pseudo-differential equations and Sobolev spaces]; > s.a. modified classical mechanics; modified uncertainty relations; path integrals.

Pachner Moves, Pachner Theorem > s.a. types of manifolds [PL, combinatorial].
@ References: Korepanov a0911 [4D, algebraic relations with anticommuting variables and topological field theory].

Packings > see sphere.

Padé Approximation

Painlevé 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 that vanishes on an arbitrary non-characteristic surface; (2) Verify the possibility of truncating the series at some finite power of .
* 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); Steib & Euler 89; Lakshmanan & Sahadevan PRP(93).
@ 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).
@ Related topics: Sakovich Sigma(06)n.SI/04-in [quadratic H that fails integrability test].

Painlevé-Gullstrand Metric > s.a. types of spacetimes; coordinates for schwarzschild spacetime; kerr metric; kerr-newman metric.
@ References: Lin & Soo PLB(09)-a0810 [generalized].

Pais-Uhlenbeck Model > s.a. quantum oscillators.
* Idea: A field theory with a higher-derivative field equation; 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.
@ References: Pais & Uhlenbeck PR(50); Bender & Mannheim JPA(08)-a0807, Nucci & Leach a0810 [no ghosts]; Di Criscienzo & Zerbini a0907 [euclidean path integral and propagator].

Palatini Action > see first-order actions for general relativity; dilaton; higher-dimensional and higher-order gravity; kaluza-klein theories.

Pancharatnam Phase > see geometric phase.

Paneitz Equation > see partial differential equations.

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.

Paracompact Topological Space

Paradoxes > see arrow of time; Fermi Paradox; Parrondo's Paradox; probability; Trouton-Noble Paradox; Zeno's Paradox.
> In gravitational 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.
> In special relativity: see clocks; Ehrenfest, Lock and Key, Submarine, Twin Paradox; special relativistic kinematics.
> In statistical physics: see Gibbs Paradox; probability in physics; quantum statistical mechanics; Recurrence Paradox; statistical mechanics.
@ References: Klein 96; Cucic a0812.

Parafermions > see generalized particle statistics.

Parallax, Cosmic > see cosmological observations.

Parallel Transport > s.a. Fermi Transport; connection; foliation [web].
* Idea: Defined on a manifold that has a connection; A tensor T is parallel transported along a curve with tanngent 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].
@ Specific spaces and metrics: Bini et al IJMPD(04)gq [circular orbits, stationary axisymmetric spacetime]; Chatterjee et al a0906 [over path spaces].

Parallel Universes > see multiverse.

Parallelizable Manifold > see types of manifolds.

Parallelotope > a special type of Polytope.

Paramagnetism > see magnetism.

Parametric Excitation / Resonance > see resonances.

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

Parametrized Post-Newtonian Formalism > see under PPN Formalism.

Parametrized Theories

Paraphotons
* 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].

Parity > s.a. canonical general relativity; hadrons [parity doubling]; matter phenomenology in quantum gravity.
* Idea: An operation defined on 3D space with a flat background, which consists of inverting all axes by mapping (x, y, z) to (–x, –y, –z) in the background.
* Remark: In higher-dimensional theories, we do not change the extra dimensions, which correspond to "internal charges".
* In field theory: One wants to have a representation of this on the space of fields; This may not always be possible (like for Dirac spinors in 5 dimensions, where one has to use a covering space, to get a faithful representation of the Clifford algebra).
* Status as symmetry: In classical physics laws are invariant under P reversal; 1957, Lee & Yang argued that P can be violated in nuclear -decay; 1957, Violation observed in -decay of polarized Co nuclei; 1957, L Landau argued that although P can be violated, CP should not be; 1982, First violation in atomic physics reported; > s.a. CP violation; CPT theorem.
* Violation, atomic physics: The nuclear decay results are explained in the standard model by assuming that the W +– bosons that govern the weak interaction only exist in a left-handed version; The different absorption of left- and right-circularly polarized light is explained by the Z 0-boson exchange in atoms between electrons and nuclei.
@ General references: Rosen AJP(73)apr [form electromagnetic quantities]; Bender et al qp/02 [in quantum mechanics], qp/04 [in PT-symmetric quantum theory], mp/04 [Lorentz transformation properties].
@ Violation, atomic physics: Bouchiat et al PLB(82), Wood et al Sci(97)mar [cesium]; Guéna et al MPLA(05) [atomic physics]; Tsigutkin et al PRL(09) + Jungmann Phy(09) [large violation observed in ytterbium].
@ Violation: Wu et al PR(57) [observation in nuclear decay]; Anthony et al PRL(04) + pw(04)may [observation in e collisions]; Alexander PLB(08)ht/06 [and WMAP anomalies]; Andrianov & Espriu PLB(08)-a0709 [in QCD, spontaneous at finite baryon density]; Contaldi et al PRL(08)-a0806 [in gravity, and cmb polarization]; Wu et QUaD PRL(09) [bounds from cmb polarization].

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.

Partially Ordered Set > see poset.

Particle Descriptions and Types > see effects, 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

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

Partition Function > see states in statistical mechanics.

Parton Models > see hadrons.

Paschen-Back Effect > see Zeeman Effect.

Pataplectic Hamiltonian Formulation > see hamiltonian dynamics.

Path > s.a. loops.
* For a field: The path in a region of spacetime is a cross-section of the bundle of internal degrees of freedom over .
@ Path group: Mensky G&C(02)gq [gravity and paths in Minkowski], gq/02-in [in gauge theory and general relativity].
@ Path space: Cho & Hong a0706 [Morse theory]; > s.a. measure [Wiener measure].
@ Path-dependent functions: Reyes JMP(07)ht/06 [operators].

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

Patterns > s.a. composite quantum systems.
@ 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 > see spin-2 field theories; path-integral formulation of quantum field theory [spin-1/2].

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 0| [(x), (x')] |0.
* Properties: It satisfies the homogeneous field equation.

Pauli-Villars (Covariant) Regularization > see regularization.

PCAC
$ Meaning: Partial Conservation of Axial Current.

Peano's Axioms > see mathematics.

Peano Curve > see fractals.

Peccei-Quinn Mechanism > s.a. axion; neutron.
* Idea: A field theory mechanism by which a discrete symmetry arises from the spontaneous breaking of a continuous symmetry.
@ References: Mercuri PRL-a0902 [gravitational, and Barbero-Immirzi parameter].

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].

Peierls Brackets > s.a. canonical general relativity.
@ References: Peierls PRS(52); DeWitt in(64); Marolf AP(94)ht/93 [generalization]; 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].

Peirce Logic > see clifford algebra; dirac field theory.

Peltier Effect > see electricity [thermoelectricity].

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

H = p2 – 2 cos x ,       d2x/dt2 + 2 sin x = 0 .

* Linearization: Gives the simple harmonic oscillator.
@ General references: Matthews 00 [history, education, r pw(01)feb]; Baker & Blackburn 05 [r PT(06)jul]; Gitterman 08 [noisy].
@ 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].
@ Other types: Butikov AJP(01)jul [inverted, stabilization]; Rafat et al AJP(09)mar [double, with square plates].
@ Related topics: Lima & Arun AJP(06)oct [period, beyond small-angle approximation].

Penning Trap > s.a. phenomenology of lorentz symmetry violations.
* Idea: An electron trap, made with electric and magnetic fields.
@ References: Brown & Gabrielse RMP(86).

Penrose Diagram > s.a. asymptotic flatness.
* Idea: A diagram of spacetime, as compactified by a suitable conformal transformation.
@ References: Penrose in(64); Brown & Lindesay CQG(09)-a0811 [for accreting 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
* Idea: For a spherically symmetric metric, on any apparent horizon

GMADM / c2 R/2 ;

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

GM / c2 (A/16)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 a0901 [violated for prolate black holes]; Mars CQG(09)-a0906 [rev].
@ Riemannian: & Huisken & Ilmanen (97) [proof, single black hole]; Bray JDG(01) [proof]; Bray & Chrusciel in(04)gq/03.
@ 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].

Penrose Limit
* Idea: Given a metric written in coordinates adapted to a null geodesic (can always be done), the procedure consists in replacing (u, v, yi) by (u, v, yi) in the line element, and then taking the limit as → 0 of ds2/2; One is then left with a metric of the form ds2 = 2 dudv + Cij dyidyj; 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) [curvature and matter]; Philip JGP(06) [of homogeneous spaces].

Penrose Mechanism / Process > s.a. black-hole phenomenology.
* 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 mirr.
@ 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.
@ Related topics: Williams ap/02/PRD [Compton scattering and e+e– production].

Penrose Tiling > see tiling.

Pentaquark > see hadrons.

Percolation > s.a. ising models; in lattice field theory; Transport; voronoi tilings.
* Idea: The thory was initiated by Broadbent and Hammersley in 1957 as a mathematical framework for the study of random physical processes, such as flow through a disordered porous medium with randomly blocked channels; It has proved to be a remarkably rich theory, with applications beyond natural phenomena to topics such as network modelling.
@ Theory: Stauffer & Aharony 94; Cardy mp/01-in [conformal field theory methods]; Smirnov & Werner 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]; Bollobás & Riordan 06.
@ Critical: Grassberger JPA(99); Cardy JPA(02)mp; Ridout NPB(09)-a0808 [and Watts' crossing probability].
@ Directed: Janssen et al JPA(99) [equation of state]; Grimmett & Hiemer m.PR/01.

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; test-body orbits; tests of general relativity.

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

Permanent of a Matrix > see matrix.

Permeability > see magnetism.

Permittivity > see electricity.

Permutations > see finite groups; particle statistics.
@ References: Huggett BJPS(99) [as a symmetry in quantum mechanics].

Perpetuum Mobile > see de sitter space [example]; thermodynamics [violations of second law].

Perplex Numbers > see numbers.

Perron-Frobenius Operator > see under Frobenius-Perron.

Persistent Homology > see types of homology theories.

Perturbation Methods > s.a. black-hole perturbations; cosmological; fluids; quantum field theory techniques; spacetime metric perturbations.
* Example: Delicate stuff – If initially stationary, Venus and Earth would collide in < 370 yrs; If isolated in orbit around each other, never; So, what is the effect of Venus on Earth's trajectory?
@ Texts: Giacaglia 72; Kevorkian & Cole 81; Gallavotti 83; Bender & Orszag 99.
@ 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-in [anharmonic oscillator, classical and quantum], et al EJP(05)mp/04 [removal of secular terms]; > s.a. 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].

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) k |x, with k 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 [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]; > s.a. gravitational thermodynamics, models of topology change.
@ Cosmology: Dabrowski 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]; Dabrowski gq/07-in [dark energy]; Sanyal IJMPA(07) [inflation rather than big rip]; Hrycyna & Szydlowski 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]; > s.a. FRW models, gravitational thermodynamics.
@ Loop quantum cosmology: Samart & Gumjudpai PRD(07)-a0704; Fu et al PRD(08)-a0808; Wu & Zhang JCAP(08)-a0805; > s.a. FRW quantum cosmology.

Phase
@ In quantum theory: 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]; > s.a. arrow of time [phase squeezing], geometric phase, quantum states.

Phase Curve > see phase space.

Phase Space

Phase Transition > s.a. quantum phase transition.

Phase Velocity > see velocity.

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

Phonon > s.a. sound.
* Idea: A quantum of a sound wave, a type of quasiparticle.
* Applications: Theoretical applications include models for fundamental quantum field theory effects (such as the acoustic Casimir effect) and black-holke analogs; Practical ones include "phonon optics" (mirrors, filters, lenses, etc) used to look inside solids for point defects.
@ References: Baym AP(61), re AP(00) [Green function, quantum field theory methods]; Hu & Nori PRL(96) + pn(96)mar [squeezed]; Quilichini & Janssen RMP(97) [quasicrystals]; Schwab et al Nat(00)apr [quantized thermal conductivity]; Johnson & Gutierrez AJP(02)mar [wave function visualization]; Gorishnyy et al pw(05)dec [phononic crystals].
> Online resources: Wikipedia page.

Photoelectric Effect > see photon phenomenology.

Photon > s.a. photon phenomenology.

Photon Sphere / Surface > see spacetime subsets.

Physical Constants > see under Constants.

Physical Process
@ References: Spaans gq/05 [background independence].

Physics

Physics Teaching

Pi,

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.
@ References: Olivastro ThSc(90)sep.

Pilot-Wave Interpretation of Quantum Mechanics

Pin Groups / Structures and Pinors > A generalization of spin.
* Idea: Double covers of the full Lorentz group; Pin(1,3) is to O(1,3) what Spin(1,3) is to SO(1,3).
@ References: Dabrowski & 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 a0907 [and spinors and orientability].

Pioneer Anomaly > see anomalous acceleration.

Pions, > see hadrons.

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

Plancherel Theorem > see Symmetric Space.

Planck Constant and Units > see constants.

Planck Cube
* Idea: A cube with axes labeled by , G and c–1, whose vertices correspond to various types of physical theories; Can be considered as illustrating the concept of deformation.

Planck Formula for Black Body > see thermal radiation.

Plane Wave Solutions > see gravitational wave solutions; wave equations.

Planets > see extrasolar systems; solar planets

Planetary Nebulae > see interstellar matter.

Plasma Physics > see magnetism.

Platonic Solids > see euclidean geometry.

Plebanski Action for Gravity > see first-order actions; BF theories; unified theories.

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

PN Formalism > see under Post-Newtonian Expansion.

Pohlmeyer Invariants > see bosonic strings and superstrings.

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 Ndof – Ncom 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
@ References: Cima et al Top(98) [non-compact manifolds]; Szczesny et al a0810 [new elementary proof].

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 > see probability.

Poisson Equation > see partial differential equations.

Poisson Integral > see integration.

Poisson Process > see statistical geometry.

Poisson Sigma-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).

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

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, ) and a map P: V → V such that P2 = I and P = – P, we can construct a polarization defined by the eigenvectors of P+:= (I + P) (so P+ P+ = 0), with eigenvalue 1.

Polish Space > see types of distances.

Polygon, Polyhedron > s.a. euclidean geometry.
@ In Minkowski space: Foth JGP(08) [3D Minkowski].
@ Related topics: Charles a0806 [quantization of polygon spaces]; > s.a. markov processes [polygonal Markov fields].

Polyhomogeneous Spacetimes > see types of spacetimes.

Polymer > s.a. molecular physics.
@ Statistical mechanics: Brereton JPA(01); Ioffe Velenik a0908 [stretched by an external force].
@ Related topics: Jitomirskaya et al CMP(03)mp/04 [random, and delocalization]; Imbrie JPA(04) [branched directed, dimensional reduction].

Polymer Quantization > see black-hole quantization; Bohr Compactification; loop quantum gravity; representations of quantum mechanics; types of quantum field theories; 2D quantum gravity.

Polynomials > see functions.

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) 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 Rd is the union of all simplices K, 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 S L, 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]; Cantwell JCTA(07) [all regular polytopes are Ramsey].
@ Delaunay polytopes: Dutour EJC(04); Erdahl et al m.NT/04-in; Sikiric & Grishukhin EJC(07) [computing the rank].
> Related topics: see statistical geometry [from random point set].

Pomeron
@ References: Brower et al JHEP(07)ht/06 [and gauge/string duality].

Pontrjagin Classes, Numbers

Ponzano-Regge Model > see spin foam models.

Porosity of a Measure > see measure.

Pöschl-Teller Potential
@ Modified: Aldaya & Guerrero qp/04 [group quantization].
> Online resources: MathWorld page on Pöschl-Teller differential equations.

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

Position
* In quantum mechanics: Teller (1979) argued that a particle cannot have a sharp position; Others disagree.
@ Concept: Chew SP(63); Halvorson JPL(01)qp/00 [sharp, in quantum mechanics].

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 types of particles.

Post-Friedmannian Formalism > see cosmological models.

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

Potential > for quantum potential, see pilot-wave interpretation.

Potts Model > s.a. lattice field theory; Yang-Baxter.
* Idea: A 2D generalization of the Ising model; 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 cm/00-in [unsolved problems]; Baxter JPCS(06)cm/05 [rev].
@ 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].
@ Related topics and variations: Richard & Jacobsen NPB(07) [on a torus]; Barré & Gonçalves PhyA(07) [on a random graph, canonical and microcanonical ensembles]; Ambjørn et al NPB(09)-a0806 [coupled to quantum gravity]; Ganikhodjaev PLA(08) [next-nearest-neighbor interactions, on the Bethe lattice]; De Masi et al JSP(09) [continuum version, phases]; > s.a. Confinement [model for].

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.

Poynting Vector
* Idea: The vector S = E × B/0, 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:= (1/8) (E2 + B2) dv = T00 dv ,   P:= (1/4c) E × B dv = T0i dv .

@ General references: in Jackson; in Rohrlich; McDonald AJP(96)jan [meaning].
@ Gravitational: de Menezes gq/98; Manko et al CQG(06) [axistationary electrovac spacetimes].

Poynting-Robertson Effect
* Idea: An effect that produces changes in the orbital plane of a particle; Has been applied to meteoroids.
@ References: in Harwit 88; Klacka ap/00, ap/01, ap/02, ap/02; Klacka a0807 [paradox in astrophysical application]; Klacka et al a0904 [explanations].

pp-Waves > see gravitational wave solutions.

PPN (Parametrized Post-Newtonian) Formalism > see [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].

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.
@ In general relativity: Holstein AJP(01)dec; Sigismondi ap/05-in; Harper PhSc(07)dec; He & Zhao IJTP(09) [analytical solution].
@ In modified gravity theories: Behera & Naik ap/03 [vector gravity]; Schmidt PRD(08) [modified Newtonian potential].
@ Specific cases: Stewart AJP(05)aug [Mercury, due to other planets]; Iorio AJ(09)-a0811 [Saturn, anomalous].
@ General references: Magli phy/04 [in ancient astronomy]; Jonsson CQG(06)-a0708 [spin precession, covariant formalism].
> In various theories: see Cogravity; gravity theories; newtonian gravity.
> In various spacetimes: see reissner-nordström, schwarzschild-de sitter, test bodies.

Precompactness > see compactness.

Prediction and Predictability > s.a. causality; Determinism; paradigms in physics; time.
* Meaning: 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 a0906 [in a very large universe].
@ Of effects: Hitchcock & Sober BJPS(04) [vs accommodation, and overfitting].

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

Preons > see composite models.

Presentation of a Group
$ Def: A pair (S,D) of a set of generators S and a set of relations between the generators D = {i}; Each relation i 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 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; > see the 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.
* Idea: The view that only the present is real.

Pressure > s.a. energy-momentum tensor; fluid [including gravity]; gravitating matter; Momentum; radiation; thermodynamics; turbulence.
@ References: Durand AJP(04)aug [quantum, Bose and Fermi statistics].

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).

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.

Principal Fiber Bundle

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

Principal Part > see distribution.

Principal Principle > s.a. quantum measurements.
* Idea: A principle relating objective probabilities and subjective chance.

Prisoner's Dilemma > see games.

Probability > s.a. probability in physics.

Problems > see Coloring; matrix; orbits in newtonian gravity [Kepler], of gravitating objects; Three-Body; Two-Body.

Proca Theory > s.a. [modified electromagnetism]; black-hole hair; field theories [spin-1, 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 = – Fab Fab + m2 Aa Aa + Aa j a .

@ General references: Proca CRAS(36); in Wentzel 49; Goldhaber & Nieto RMP(71) [and photon mass limits]; Aldaya et al IJMPA(97)ht/96 [quantization]; in Gsponer & Hurni in(98)phy/05 [history]; Dvoeglazov CzJP(00)ht/97; Kruglov IJMPA(06) [sqrt version, including spin-1/2]; Fabbri a0908 [most general consistent theory].
@ Einstein-Proca: Dereli et al CQG(96) [torsion and non-metricity]; Vollick gq/06; > s.a. einstein-cartan theory.
@ Quantization: Zamani & Mostafazadeh JMP(09)-a0805.
@ Related topics: Comay NCB(98); Kim et al MPLA(98)ht [symmetries]; Vytheeswaran IJMPA(98) [as gauge theory]; Zecca GRG(06) [in FRW spacetime]; Helesfai CQG(07)gq/06 [in lqg].

Process > see Ontology; Physical Process.

Products
* Special infinite products:

k = 2infty(1 – 1/k2) = 1/2   [prove by splitting into (1 – 1/k) (1 + 1/k) and using factorials] .

Programming > see computation; computer languages.

Progressing Waves > see wave equation.

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.
@ 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.

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.

Proof Theory

Propagator > s.a. 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.

Propositional Logic > see logic.

Proton > see hadrons.

Prout's Law > see atomic physics.

Proximity Graphs > see graph types.

Proximity Structure

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.

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 R3.
@ References: Bertotti et al gq/05-in [review, geometry and physics].

Pseudostationary Spacetime > see types of spacetimes.

Pseudotensor > see stress-energy pseudotensor.

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

PT Symmetry > s.a. modified quantum mechanics [PT-symmetric].
@ References: Bender & Darg JMP(07) [spontaneous breaking, in classical mechanics].

Pullback Bundle > see fiber bundle.

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

Pulsars > see neutron stars.

Pure Sequence > see exact sequence.

Purity > see mixed state.

Push-Forward > see tangent structures.

Pyrgon
* 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
@ References: Ungar FP(98), Brill & Jacobson GRG(06)gq/04-in [Lorentzian version]; Crease pw(06)jan [history and significance].

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