Z ×
Z , for all n (p is a prime) .Topics, P
p-Adic Number / Structure
$ 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 ht/06-in
[cosmology]; > s.a quantum
cosmology, quantum
spacetime.
> Other physics: see modified
classical mechanics; modified uncertainty
relations; path
integrals.
> Applications: see differential equations.
Pachner Moves > see types of manifolds [PL].
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 n.SI/04-in
[quadratic H that fails integrability
test].
Painlevé-Gullstrand Metric > see types of spacetimes; also coordinates for schwarzschild and kerr metric.
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 [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.
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.
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].
@ Specific metrics: Bini et al IJMPD(04)gq [circular
orbits, stationary axisymmetric
spacetime].
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 Post-Newtonian.
Paraphotons
* Idea: Low-mass extra U(1) gauge bosons with gauge-kinetic mixing with the ordinary
photon.
@ References: Jaeckel & Ringwald 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; CP
violation; CPT; 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, L Landau argued that although P can be violated, CP should not be.
@ General references: Rosen AJP(73)
[form electromagnetic quantities]; Bender et al qp/02 [in
quantum mechanics], qp/04 [in
PT-symmetric quantum theory], mp/04 [Lorentz
transformation properties].
@ Violation: Guéna et al MPLA(05)
[atomic physics]; Anthony et al PRL(04)
+ pw(04)may
[observation
in e collisions]; Alexander ht/06 [and
WMAP anomalies]; Andrianov & Espriu a0709 [in
QCD, spontaneous at finite baryon
density]; Contaldi et al a0806 [in gravity, and 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).
Parseval's Integral > see bessel functions.
Parseval's Relation / Theorem > see fourier analysis.
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.
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].
@ 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.]
* For a scalar field: 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.
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].
@ Foucault's pendulum: Hart et al AJP(87);
Khein & Nelson AJP(93)
[Hannay angle]; Pardy ap/06 [astronomical
analogs]; von Bergmann & von Bergmann AJP(07) [and geometry].
@ Related topics: Butikov AJP(01)
[inverted, stabilization]; Lima & Arun AJP(06)
[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).
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).
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].
@ Riemannian: & Huisken & Ilmanen (97) [proof, single black
hole]; Bray
JDG(01)
[proof]; Bray & Chrusciel gq/03-in.
@ Generalizations: Gibbons in(84); Karkowski et al CQG(94)
[gravitational waves];
Herzlich CMP(97)
[asymptotically flat, R 0].
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.
@ 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]; Grassberger JPA(99),
Cardy JPA(02)mp [critical];
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.
@ 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].
Perplex Numbers > see numbers.
Perron-Frobenius Operator > see under Frobenius-Perron.
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 qp/01/PRA
[t-dependent];
Teufel 03 [adiabatic perturbation theory]; Ciftci et al PLA(05)mp [iterative];
Weinstein ht/05,
ht/05-in
[adaptive]; Albeverio et al RPMP(06)
[singular, rigged Hilbert space approach].
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 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 ap/04/JCAP
[and SN Ia data]; Santos & Alcaniz PLB(05)ap [Segre
classification]; Giacomini & Lara GRG(06)
[+ gravity + arbitrary potential, dynamics]; Pereira & Lima a0806 [thermodynamics].
@ Black holes: Svetlichny ap/05 [possible
production by black holes]; Berezin et al CQG(05)gq [shell
around Schwarzschild]; Bronnikov & Fabris PRL(06)
[regular asymptotically flat, dS and AdS]; Rahaman et al NCB(06)gq;
Gao
et al a0802 [mass
increase]; > s.a. gravitational
thermodynamics.
@ 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]; Samart & Gumjudpai PRD(07)-a0704 [in
lqc]; Hrycyna & Szydlowski PLB(07)
[conformally coupled, acceleration]; Shatskiy JETP(07)-a0711;
Chaves & Singleton SIGMA(08)-a0801 [and
dark matter]; > s.a. FRW
models and 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 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: "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)
[wave function visualization]; Gorishnyy et al pw(05)dec
[phononic
crystals].
Photoelectric Effect > see photon.
Photon Sphere / Surface > see spacetime subsets.
Physical Constants > see under Constants.
Physical Process
@ References: Spaans gq/05 [background
independence].
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].
Pioneer Anomaly > see anomalous acceleration.
Pions, > see
hadrons.
PL 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.
Pohlmeyer Invariants > see bosonic strings and superstrings.
Poincaré Conjecture > see conjectures.
Poincaré Duality > see cohomology.
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 >
s.a. Unitarity.
@ References: Buric et al JPA(03)
[area-preserving maps].
Poincaré-Hopf Theorem
@ References: Cima et al Top(98)
[non-compact manifolds].
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].
Polyhomogeneous Spacetimes > see types of spacetimes.
Polymer > s.a. molecular
physics.
@ References: Brereton JPA(01) [statistical mechanics]; Jitomirskaya et al CMP(03)mp/04 [random,
and delocalization]; Imbrie JPA(04) [dimensional reduction for directed branched
polymers].
Polymer Quantization > see 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.
@ 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 ht/06 [and
gauge/string duality].
Ponzano-Regge Model > see spin foam.
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 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) Formalism > see [gravitation]; modified newtonian gravity.
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].
@ 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 a0806 [coupled
to quantum gravity]; > s.a. Confinement [model
for].
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)
[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].
pp-Waves > see gravitational wave solutions.
PPN Formalism > see under PN formalism.
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);
Sigismondi ap/05-in;
Harper PhSc(07).
@ General references: Magli phy/04 [in
ancient
astronomy]; Stewart AJP(05)
[Mercury, due to other planets]; 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].
@ 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; fluid [including
gravity]; gravitating matter; radiation; thermodynamics.
@ References: Durand AJP(04) [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 Numbers > see number theory.
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].
@ Einstein-Proca: Dereli et al CQG(96)
[torsion and non-metricity]; Vollick gq/06;
> s.a. einstein-cartan.
@ Quantization: Zamani & Mostafazadeh a0805-AP.
@ Related topics: Comay NCB(98); Kim et al MPLA(98)ht [symmetries];
Vytheeswaran IJMPA(98)
[as gauge theory]; Zecca GRG(06)
[in FRW]; 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)
[with variation of atmospheric
pressure]; Price & Romano
AJP(98) [optimal
launch angles];
Warburton & Wang AJP(04);
Linthorne pw(06)jun [and soccer].
Projection Mapping > see bundles.
Projective Geometry, Structure, Limit, System > see projective.
Projective Relativity and Field Theory
@ References: in Schmutzer ed-83 [projective relativity]; Schmutzer AN(05)ap [projective
unified field theory and 2-body system].
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.
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)
[calculation]; Fulling & Güntürk AJP(03)
[1D particle in a box]; Kosut
et al qp/06 [distance
between propagators]; Moshinsky et al SIGMA(07)-a0711 [from
Green function].
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.
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.
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 gq/04-in
[Lorentzian version]; Crease pw(06)jan
[history and significance].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
27 jul 2008