Topics, P

p-Adic Number / Structure > s.a. differential equations; 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 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) .

@ General references: Gouvêa 97.
@ In cosmology and gravitation: Dragovich AIP(06)ht [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 NPPS(01) [quantum mechanics and quantum field theory]; Dragovich et al pUAA-a0904 [rev]; Rodríguez-Vega & Zúñiga-Galindo PJM-a0907 [p-adic fields, pseudo-differential equations and Sobolev spaces]; Abdesselam a1104-in [massless quantum field theory]; > s.a. classical mechanics [generalizations]; modified uncertainty relations; path integrals.
> Online resources: see Wikipedia page; MathWorld page.

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

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 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 89; Lakshmanan & Sahadevan PRP(93); Guzzetti JPA(06)-a1010, a1010 [Painlevé VI equation].
@ 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].
> Online resources: see The Painlevé Project site.

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

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 PS(10)-a0810, JMP(09) [no ghosts]; Di Criscienzo & Zerbini JMP(09)-a0907 [euclidean path integral and propagator]; Mostafazadeh PLA(10)-a1008 [Hamiltonian formulation], PRD(11)-a1107 [consistent quantization]; Ketov et al a1110 [as a toy-model for quantizing f(R) gravity theories].

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.

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 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.

Paracompact Topological Space

Paradoxes > s.a. Fermi Paradox; Trouton-Noble Paradox.
> In mathematics: see logic; Parrondo's Paradox; probability; 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; wave-function collapse.
> In special relativity: see arrow of time [causal paradoxes]; 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.
@ General references: Klein 96; Chang 12 [in scientific inference].
@ In thermodynamics: Cucić a0812, a0912 [and statistical physics]; Yoder & Adkins AJP(11)aug [ellipsoid paradox].

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 RVMP(10)-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].
* 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 β decay; 1957, Violation observed in β 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.
* Violation, gravity: Can arise for example in Chern-Simons modified gravity, where the Einstein-Hilbert action is modified through the addition of the gravitational parity-violating Pontryagin density coupled to a field.
@ General references: Rosen AJP(73)apr [form electromagnetic quantities]; Bender et al JPA(03)qp/02 [in quantum mechanics], qp/04 [in PT-symmetric quantum theory], JMP(05)mp/04 [Lorentz transformation properties]; blog sa(11)aug, sa(11)sep [how to explain handedness to aliens].
@ Violation, atomic physics: Bouchiat et al PLB(82), Wood et al Sci(97)mar [cesium]; Guéna et al MPLA(05); Tsigutkin et al PRL(09) + Jungmann Phy(09) [large violation observed in ytterbium]; Darquié et al Chir(10)-a1007 [towards observation in chiral molecules].
@ Violation, gravity: Contaldi et al PRL(08)-a0806 [and cmb polarization]; Yunes et al PRD(10)-a0912 [and neutron-star moments of inertia]; Gluscevic & Kamionkowski PRD(10)-a1002 [TB/EB correlations in cmb]; Yunes et al PRD(10)-a1005 [gravitational waves and short GRBs]; > s.a. gravitational instantons; matter phenomenology in quantum gravity.
@ Violation, other: 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]; 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 > s.a. generalized 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 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-in [in gauge theory and general relativity].
@ Path space: Cho & Hong a0706 [Morse theory]; Biswas & Chatterjee IJGMP(11) [geometric structures]; > 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 / 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, e.g., the graviton; It arises also as an effective 4D theory in brane models.
* 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].
@ Variations: Boulanger & Gualtieri CQG(01)ht/00 [PT non-invariant deformation]; de Rham & Gabadadze PRD(10)-a1007 [with generalized mass and interaction terms]; Park a1011 [non-Pauli-Fierz theory, unitarization]; Deffayet & Randjbar-Daemi a1103 [non-linear, from torsion]; > s.a. Massive Gravity [including non-Pauli-Fierz theory].

Pauli-Jordan Function > s.a. green .
* Idea: A type of Green function for a quantum field.
* For a scalar field: The two-point function G(x, x'):= –i ⟨0| [φ(x), φ(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(09)-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 – ω2 cos x ,       d2x/dt2 + ω2 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]; Baker 11; Brizard a1108 [action-angle coordinates].
@ 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].
@ Other types: Butikov AJP(01)jul [inverted, stabilization]; Rafat et al AJP(09)mar [double, with square plates].

Penning Trap > s.a. phenomenology .
* Idea: An electron trap, made with 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 a1107 [geometry of Penrose's 'light cone at infinity'].
@ 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: 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/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].
@ Riemannian: & Huisken & Ilmanen (97) [proof, single black hole]; Bray JDG(01) [proof]; Bray & Chruściel in(04)gq/03; Ohashi et al PRD(10)-a0906.
@ 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].

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 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)gq/04 [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]; Cen a1102/ApJL [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.
@ 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; Ziff et al JPA(11) [factorization of the three-point density correlation function].
@ 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; 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 in matter.

Permutations > see finite groups; particle statistics.
@ References: Huggett BJPS(99) [as a symmetry in quantum mechanics]; Olshanski in-a1104 [random permutations].

Perpetual Motion Machine / Perpetuum Mobile > s.a. de sitter space [example]; thermodynamics [violations of second law].
@ References: Chernodub a1203 [permanently rotating devices].
> 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 theories.

Perturbation Methods / Theory > s.a. fluids; quantum field theory techniques.
* In classical mechanics – 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?
* 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.
@ 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]; Pound PRD(10)-a1003 [singular]; > 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]; 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].
> 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-in [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]; > 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; > s.a. FRW models; gravitational thermodynamics.
@ Loop quantum cosmology: Samart & Gumjudpai PRD(07)-a0704; Gumjudpai a0706-in; Fu et al PRD(08)-a0808; Wu & Zhang JCAP(08)-a0805; > s.a. [FRW quantum cosmology].

Phase of Matter > s.a. condensed matter [including liquids]; crystals; fluid; gas; matter; Plasma; bose-einstein condensate.
@ References: issue JPCM(98)#49 [matter under extreme conditions]; Pinheiro phy/07 [plasma, genesis of the word]; Kadanoff a1002; Baas a1012 + news ns(11)jan [topology and generalization of Efimov states]; > s.a. magnetism [plasma physics or magnetohydrodynamics].

Phase of a Quantum State
@ 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]; > 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.

Phoenix Universe > see cosmological models.

Phonon > s.a. specific heat [for a solid]; sound ["phonon tunneling"].
* 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-hole 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: see 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].

Physicalism > see philosophy of physics.

Physics

Physics Teaching

Pi, π

Pierre Auger Observatory
@ References: Anchordoqui et al PRD(03)hp; Anchordoqui ap/04-in; Kampert NPPS-ap/05; Van Elewyck ap/06-ln, MPLA(08); Nitz a0706-in [north site]; Van Elewyck a0709-in; Parizot et al a0709-in; de Mello APPS-a0712-in, Matthiae a0802-in [status and results]; Abraham et PA a0906-in [status and plans]; Etchegoyen et al a1004-in; Roulet a1101-in; Smida et al a1109-in [results]; > s.a. ultra-high-energy cosmic rays.

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 > s.a. phenomenology [systems and effects].

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].

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.
* Value: 1998, h = 6.62606891(58) × 10–34 J · s or × 10–27 erg · s; = 1.05457266(63) × 10–34 J · s, or × 10–27 erg · s; The best values are obtained frm measurement of the flux quantum φ0 = h/2e using the Josephson effect, and the quantum of conductance G0 = 2e2/h from the quantum Hall effect.
* Length: lP = (G/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 = 2.2 × 10–5 g.
@ General references: Planck SBAW(1899); Fischbach et al PRL(91) [quantum mechanics with different ]; Cooperstock & Faraoni MPLA(03)ht, IJMPD(03)gq [including e and s]; Wilczek PT(05)oct [absolute units].
@ Related topics: Zeilinger AJP(90)feb [Planck stroll]; Casher & Nussinov ht/97 [pP is unattainable]; Williams et al PRL(98) + pn(98)sep + pw(98)sep [measurement]; Sivaram a0707 [Planck mass].

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.

Planck Mission / Satellite > see cosmic microwave background.

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.

Plebański 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.

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 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]; Szczęsny et al IJGMP(09)-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 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,C)].

Polariton > see bose-einstein condensation.

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: V → V such that P2 = I and P Ω = – Ω P, we can construct a polarization defined by the eigenvectors of P+:= (I + P) (so P+ Ω 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]; Sabbagh & Eu PhyA(10) [van der Waals equation of state, self-diffusion coefficient]; De Roeck & Kupiainen a1005 [polymer expansion].
@ 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.
@ General references: Fredenhagen & Reszewski CQG(06)gq; Corichi et al CQG(07)gq/06, PRD(07)-a0704; Chiou CQG(07)gq/06 [and Galileo group]; Hossain et al a1003 [and the uncertainty principle]; Campiglia a1111 [and geometric quantization].
@ Simple systems: Husain et al PRD(07)-a0707 [Coulomb potential]; Kunstatter et al PRA(09)-a0811 [1/r2 potential]; Kunstatter & Louko a1201 [on the half line]; > s.a. black-hole quantization; gas.
> Related topics: see Bohr Compactification; fock space; renormalization.
> Field theories: see klein-gordon quantum field theory; loop quantum gravity; types of quantum field theories; 2D quantum gravity.

Polynomials > see functions.

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

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 Rd is the union of all simplices σ ∈ 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].

Pomeransky-Senkov Black Hole > see causality conditions.

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

Pontrjagin 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.
@ 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 a1110 [κ-deformation].
@ 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-in [and Reidemeister torsion]; Livine & Ryan CQG(09)-a0808 [B-observables]; Caravelli & Modesto a0905 [spectral dimension of spacetime].

Popper's Thought Experiment
@ References: Qureshi IJQI(04)qp/03, AJP(05)jun-qp/04; Richardson & Dowling a1102 [no violation of the uncertainty principle, fundamental flaw].

Porosity of a Measure > see measure.

Pöschl-Teller Potential
@ Modified: Aldaya & Guerrero qp/04 [group quantization].
> Online resources: see 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; > s.a. localization in quantum mechanics.
@ 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.

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 quantum potential, see pilot-wave interpretation.

Potts Model > s.a. lattice field theory; Yang-Baxter Equation.
* 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 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 a1102 [motivic approach].
@ 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]; Contucci et al a1106 [on a random graph]; > s.a. Confinement [model for].

Pound-Rebka Experiment > see tests of general relativity with light [gravitational redshift].

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 .

@ 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]; Bini & Geralico CQG(10) [extended to spinning particles in Schwarzschild spacetime]; Bini et al CQG(11).

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 [perturbations and curved spaces].
> In various spacetimes: see reissner-nordström solutions; schwarzschild-de sitter spacetime; test bodies.

Precompactness > see compactness.

Prediction and Predictability > s.a. causality; Determinism; Explanation; 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 PRD(10)-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.

Preorder
@ References: Cameron et al DM(10) [random preorders and alignments]; Minguzzi a1108 [representation by continuous utilities].

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 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).

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.

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].

Prisoner's Dilemma > see games.

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

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]; 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; Castineiras et al PRD(11)-a1108 [in the Rindler wedge].
@ 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 = 2∞(1 – 1/k2) = 1/2   [prove by splitting into (1 – 1/k) (1 + 1/k) and using factorials] .

@ References: Roy 11 [series and products from the XV to the XXI century].

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.

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.

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.

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].

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.

Pseudosymmetric Spacetime > see 3D geometry.

Pseudotensor > see stress-energy pseudotensor.

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

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

PT Symmetry > s.a. modified quantum mechanics [PT-symmetric]; Unitarity.
@ General references: Mostafazadeh PS(10)-a1008 [rev].
@ Spontaneous breaking: Bender & Darg JMP(07) [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 > s.a. mixed states.
* 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.

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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