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.
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.
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.
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 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].
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 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].
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 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:=
(1/4
c)
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 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.
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.
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].
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified
9 nov 2009