**Topics, P**

**p****-Adic Number /
Structure** > s.a. differential equations;
knot invariants; Non-Archimedean Structures.

* __Idea__: For each
prime number *p*, the *p*-adic number system is an
extension of the rational numbers different from the real number system.

* __Motivation, use__:
Initially motivated by an attempt to use power-series methods in number
theory; Now *p*-adic analysis essentially provides an alternative form of calculus.

$ __Def__: A uniformity
on \(\mathbb Z\) defined by giving, as fundamental set of entourages,

*W*_{n} :=
{(*x*, *y*) | *x* = *y* mod *p*^{n}}
⊂ \(\mathbb Z \times \mathbb Z\) , 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-conf
[massless quantum field theory]; Dragovich a1205-proc [*p*-adic matter in the universe];
Abdesselam et al a1302;
Zelenov TMP(14) [*p*-adic dynamical systems];
Hu & Zong a1502
[*p*-adic quantum mechanics, symplectic group and Heisenberg group];
Dragovich et al pNUAA(17)-a1705 [rev];
> s.a. classical mechanics [generalizations];
modified uncertainty relations; path integrals.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Pachner Moves, Pachner Theorem **
> s.a. types of manifolds [PL, combinatorial].

@ __In 4D__: Korepanov a0911
[algebraic relations with anticommuting variables and topological field
theory]; Banburski et al PRD(15)-a1412 [in a Riemannian spin-foam model];
Kashaev a1504.

> __And physics__:
see regge calculus.

**Packings** > s.a. sphere.

@ __References__: Jaoshvili et al PRL(10) +
Frenkel Phy(10)
[random packings of tetrahedral dice].

**Padé Approximant / Approximation**

* __Idea__: The "best"
approximation of a function by a rational function of given order; It
often gives better approximation of the function than truncating its
Taylor series, and it may still work where the Taylor series does not
converge.

@ __References__: Wei et al JCAP(14)-a1312 [cosmological applications].

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Painlevé Equations / Analysis / Test**

* __Idea__: A criterion
of integrabilty for partial differential equations, which involves the
following steps, (1) Show that the general solution can be represented as
a (formal Laurent) series in powers of some function Φ that vanishes on an
arbitrary non-characteristic surface; (2) Verify the possibility of
truncating the series at some finite power of Φ.

* __Consequences__: If
satisfied, the equation is integrable, and we can get Bäcklund
transformations and a (weak) Lax pair; If not satisfied, we cannot
conclude the opposite.

@ __General references__: Weiss et al JMP(83);
Weiss JMP(83);
Ramani et al PRP(89);
Steeb & Euler 88; Lakshmanan &
Sahadevan PRP(93);
Guzzetti JPA(06)-a1010,
IMRN(12)-a1010
[Painlevé VI equation].

@ __Integrable equations without Painlevé property__:
Ramani et al JPA(00)-a0709;
Tamizhmani et al Sigma(07)-a0706.

@ __And general relativity__: see García-Díaz et al JMP(93);
> s.a. chaos in gravitation.

@ __Discrete versions__: Grammaticos et al PRL(91);
Ramani et al PRL(91);
Grammaticos & Ramani PS(14)
[rev]; Kajiwara et al JPA(17)-a1509 [geometrical aspects].

@ __Related topics__: Sakovich Sigma(06)n.SI/04-conf [quadratic *H* that fails the integrability test];
Aminov et al a1306
[multidimensional versions of the Painlevé VI equation];
Bermudez et al JPA(16)-a1512
[solutions to the Painlevé V equation using supersymmetric quantum mechanics].

> __Online resources__:
see The Painlevé Project site.

**Painlevé-Gullstrand Coordinates
/ Metric** > see spherically symmetric geometries.

@ __References__: Jaén & Molina GRG(16)-a1611 [natural extension].

> __Generalized to
rotating spacetimes__: see kerr metric;
kerr-newman metric.

**Pair Creation / Production**
> s.a. particle effects [Schwinger
effect]; quantum field theory effects in
curved spacetime.

@ __References__: Petrat & Tumulka JPA(14) [multi-time formulation].

**Pais-Uhlenbeck Model**
> s.a. quantum oscillators.

* __Idea__: A field
theory with a higher-derivative field equation.

* __The ghost issue__:
Applying the Ostrogradski approach to the Pais-Uhlenbeck oscillator yields
a Hamiltonian which is unbounded from below, which leads to a ghost
problem in quantum theory; It was believed for many years that the model
possesses ghost states attributable to the field equation having more than
two derivatives, and therefore that it is a physically unacceptable
quantum theory; In reality, the Pais-Uhlenbeck model does not possess
ghost states, when quantized according to the rules of PT quantum
mechanics, and is a perfectly acceptable quantum theory.

@ __General references__: Pais & Uhlenbeck PR(50);
Kaparulin & Lyakhovich a1506-proc
[energy and stability]; Kaparulin et al JPA(16)-a1510
[interactions]; Avendaño-Camacho et al a1703 [stability, perturbation-theory approach].

@ __Ghost-free formulations__: Bender & Mannheim JPA(08)-a0807;
Nucci & Leach PS(10)-a0810,
JMP(09);
Banerjee a1308.

@ __Hamiltonian formulation__: Mostafazadeh PLA(10)-a1008;
Andrzejewski NPB(14)-a1410
[and symmetries]; Masterov NPB(16)-a1505 [without ghost problem];
Sarkar et al a1507
[resolving the issue of the branched Hamiltonian];
Masterov a1603
[(2*n*+1)-order generalization].

@ __Quantum theory__: Mannheim & Davidson PRA(05)ht/04 [Dirac quantization];
Di Criscienzo & Zerbini JMP(09)-a0907 [euclidean path integral and propagator];
Mostafazadeh PRD(11)-a1107 [consistent quantization];
Bagarello IJTP(11),
Pramanik & Ghosh MPLA(13)-a1205 [coherent states];
Cumsille et al IJMPA(16)-a1503 [polymer quantization];
Berra-Montiel et al AP(15)-a1505 [deformation quantization];
Fernández a1605 [and its PT-variant].

@ __Applications__: Ketov et al a1110-ch
[as a toy-model for quantizing *f*(*R*) gravity theories].

**Palatini Action / Formulation of Gravity Theory**

* __Idea__: A
formulation in which the metric and connection are assumed to be
independent fields, as in metric-affine theories; Conceptually this amounts to considering the geodesic
structure and the causal structure of the spacetime as independent.

> __Theoretical
aspects__: see first-order actions
for general relativity; higher-dimensional
and higher-order gravity; kaluza-klein
theories; non-local gravity.

> __Phenomenology__:
see cosmology of
higher-order theories; dilaton.

**PAMELA (Payload for Antimatter/Matter
Exploration and Light-nuclei Astrophysics)** > s.a. cosmic rays.

* __Idea__: A space
mission onboard an Earth-orbiting spacecraft, that studies cosmic rays.

@ __References__: Adriani et al PRL(09),
PRL(13)
[results on positron excess].

> __Online resources__:
see PAMELA
website; Wikipedia page.

**Pancharatnam Phase** > see geometric phase.

**Paneitz Equation** > see partial differential equations.

**Paneitz Operator**

* __Idea__: A 4th-order
differential operator which occurs in the theory of conformal anomalies;
According to a conjecture, it gives 8π when acting upon the invariant
volume of the past light cone.

@ __References__: Park & Woodard GRG(10)-a0910 [and volume of the past light cone].

**Papapetrou Field** > see gravitomagnetism.

**Papapetrou Solution**
> s.a. kerr solutions [Papapetrou gauge].

@ __References__: Khugaev & Ahmedov IJMPD(04) [generalization].

**Papapetrou Theorem**

* __Idea__: A theorem
about the equivalence of two sets of circularity conditions for
(pseudo)stationary, asymptotically flat empty spacetimes; For stationary
axisymmetric sources, *g*_{ab}
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.

**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
relativity; kinematics.

> __In statistical
physics__: see Gibbs Paradox; probability
in physics; quantum statistical
mechanics; Recurrence Paradox; statistical
mechanics.

@ __General references__: Klein 96;
Chang 12
[in scientific inference].

@ __In thermodynamics__: Cucić a0812,
a0912 [and
statistical physics]; Yoder & Adkins AJP(11)aug
[ellipsoid paradox]; Sheehan et al FP(14)
[diatomic gas in a cavity].

**Parafermions** > see generalized particle statistics.

**Parallax** > s.a. cosmological observations [cosmic parallax].

* __Stellar parallax__:
The annual apparent displacement of the stars that occurs because of
Earth's orbit around the Sun.

@ __References__: Timberlake TPT(13)-a1208
[history, and aberration]; Räsänen JCAP(14)-a1312
[cosmic parallax, covariant treatment].

**Parallel Transport** > s.a. Fermi
Transport; connection; foliation [web];
Path.

* __Idea__: Defined on
a manifold that has a connection; A tensor *T* is parallel
transported along a curve with tangent vector *X* if ∇_{X}*T*
= 0.

@ __General references__: Anandan & Stodolsky PLA(00)qp/99
[classical and quantum physics]; Wagh & Rakhecha JPA(99)
[gauge-independent form]; Iliev IJGMP(05)m.DG
[and connections], IJGMP(08)
[axiomatic approach]; Iurato a1608 [history, Levi-Civita].

@ __Specific spaces and metrics__: Bini et al IJMPD(04)gq
[circular orbits, stationary axisymmetric spacetime]; Chatterjee et al RVMP(10)-a0906
[over path spaces].

@ __Generalizations__: Soncini & Zucchini JGP(15)-a1410
[higher parallel transport in higher gauge theory].

**Parallel Universes** > see multiverse.

**Parallelizable Manifold**
> see types of manifolds.

**Parallelotope** > a special type of Polytope.

**Paramagnetism** > see magnetism.

**Parametric Excitation / Resonance** > see resonances.

**Parametrix** > see approaches
to canonical quantum gravity.

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

**Parametrized Post-Newtonian Formalism** > see under PPN
Formalism.

**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].

**Parrondo's Paradox**

* __Idea__: The
proposition that two losing strategies can, by alternating randomly,
produce a winner.

@ __References__: Martin & von Baeyer AJP(04)may.

**Parseval's Integral** > see bessel
functions.

**Parseval's Relation / Theorem** > see fourier
analysis.

**Part** > see Subsystem.

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

**Partially Massless Gravity Theory** > see massive
gravity.

**Partially Ordered Set** > see poset.

**Particle Descriptions and Types** > see effects,
models, statistics,
types; classical
and quantum models; spinning
particles.

**Particle Horizon** > see horizons.

**Particle Physics** >
s.a. experimental particle physics.

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

**Particle Statistics** >
s.a. generalized particle statistics.

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

**Partition Function** > see states
in statistical mechanics.

**Parton Models** > see hadrons.

**Pascal** > see programming
languages.

**Paschen-Back Effect** > see Zeeman
Effect.

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

**Pataplectic Hamiltonian Formulation** > see hamiltonian
dynamics.

**Path** > s.a. loops;
Parallel Transport; Trajectory
[in classical and quantum mechanics].

* __For a field__: The
path in a region Ω of spacetime is a cross-section of the bundle of
internal degrees of freedom over Ω.

@ __Path group__: Mensky G&C(02)gq
[gravity and paths in Minkowski spacetime], gq/02-conf
[in gauge theory and general relativity]; > s.a. types
of groups.

@ __Path space__: Cho & Hong a0706
[Morse theory]; Biswas & Chatterjee IJGMP(11)
[geometric structures]; Chatterjee et al JGP(13)
[bundles and connections over path spaces]; Chatterjee IJGMP(15)-a1401
[double category of geodesics on path space]; Gerstenhaber a1403
[path algebras and de Broglie waves]; > s.a. measure
[Wiener measure].

@ __Path-dependent functions__: Reyes JMP(07)ht/06
[operators].

**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 *h*_{μν},

\(\cal L\) = |*g*|^{1/2}
[*R* up to quadratic terms + *m*^{2}
(*h*_{μν}*h*^{μν}
−*h*^{2})] ;

The theory arises also as an effective 4D theory in brane models; It does
not reproduce linearized general relativity in the *m* → 0 limit,
and has a ghost problem.

*
__van Dam-Veltman discontinuity__: A discontinuity in the
Pauli-Fierz formulation; The deflection angle in the background of a
spherically symmetric gravitational field converges to 3/4 of the value predicted
by the massless theory (linearized general relativity) as *m* → 0.

@ __General references__: Fierz & Pauli PRS(39);
Groot Nibbelink & Peloso CQG(05)ht/04 [covariant];
Obukhov & Pereira PRD(03) [teleparallel origin];
Georgescu et al CMP(04)
[massless, spectral theory];
Leclerc gq/06 [gauge and reduction];
Osipov & Rubakov CQG(08)-a0805 [superluminal graviton propagation];
Hasler & Herbst RVMP(08) [Hamiltonians];
González et al JHEP(08) [duality];
Loss et al LMP(09) [degeneracy
of eigenvalues of Hamiltonian]; de Rham & Gabadadze PLB(10)-a1006 [non-linear completion without ghosts];
Park CQG(11)-a1009 [effect of quantum interactions];
Deser CJP(15)-a1407 [action, and manifestly positive energy].

@ __Variations__: Boulanger & Gualtieri CQG(01)ht/00 [PT non-invariant deformation];
de Rham & Gabadadze PRD(10)-a1007 [with generalized mass and interaction terms];
Park JHEP(11)-a1011 [non-Pauli-Fierz theory, unitarization];
Deffayet & Randjbar-Daemi PRD(11)-a1103[non-linear, from torsion];
Alberte IJMPD(12)-a1202 [on an arbitrary curved background];
> s.a. massive gravity [including non-Pauli-Fierz theory].

> __Online resources__:
see Wikipedia page on Markus Fierz.

**Pauli-Jordan Function** >
s.a. green functions in quantum field theory.

* __Idea__: A type of
Green function for a quantum field.

* __For a scalar field__:
The two-point function *G*(*x*,* x'*):= −i
\(\langle\)0| [*φ*(*x*), *φ*(*x'*)] |0\(\rangle\).

* __Properties__: It
satisfies the homogeneous field equation.

**Pauli-Villars (Covariant) Regularization** > see regularization.

**PCAC**

$ __Meaning__: Partial
Conservation of Axial Current.

**Peano's Axioms** > see mathematics.

**Peano Curve** > see fractals.

**Peccei-Quinn Mechanism / Symmetry**
> 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];
Takahashi & Yamada JCAP(15)-a1507 [breaking, in the early universe].

**Peeling Property of Spacetime**

* __Idea__: A property
of the Weyl tensor in asymptotically flat spacetimes.

@ __References__: Geroch in(77);
in Wald 84, p285; Bressange & Hogan PRD(99) [lightlike signals in Bondi-Sachs];
Klainerman & Nicolò CQG(03) [and initial data set falloff];
Pravdová et al CQG(05)gq [even higher dimensions];
Friedrich a1709 [and isolated systems,
asymptotic flatness and simplicity].

**Peierls Argument** > see ising models [spontaneous magnetization].

**Peierls Brackets** > s.a. canonical
general relativity; types of symplectic structures.

* __Idea__: A bracket
defined on the covariant phase space of a field theory, corresponding to
the Poisson bracket on the canonical phase space.

@ __General references__: Peierls PRS(52);
DeWitt in(64), in(99);
Esposito et al ht/02 [intro];
Bimonte et al IJMPA(03)ht [field theory],
ht/03
[dissipative systems]; DeWitt &
DeWitt-Morette AP(04)
[and path integrals]; Esposito & Stornaiolo IJGMP(07)ht/06
[for type-I gauge theories, and Moyal bracket].

@ __Generalizations__: Marolf AP(94)ht/93;
Sharapov IJMPA(14)-a1408
[in non-Lagrangian field theory].

> __Online resources__:
see nLab page;
Wikipedia page.

**Peirce Logic** > see clifford
algebra; dirac field theory.

**Peltier Effect** > see electricity
[thermoelectricity].

@ __References__: Heremans & Boona Phy(14)
[spin Peltier effect].

**Pendulum** > s.a. kinematics
of special relativity, oscillator.

* __Non-linear or
physical pendulum__: The Hamiltonian and equation of motion are given by

*H* = \(1\over2\)*p*^{2}
− *ω*^{2} cos *x ,
* d* ^{2}x/*d

* __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 CNSNS-a1108
[action-angle coordinates]; Dahmen a1409/EPJH
[historical, Denis Diderot's paper on pendulums and air resistance].

@ __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];
Bassan et al PLA(13) [torsion pendulum, Lagrangian model and small misalignments].

@ __Quantum__: Cushman & Śniatycki a1603 [spherical pendulum, geometric quantization].

**Penning Trap**
> s.a. lorentz-symmetry violation phenomenology;
proton [mass measurement].

* __Idea__: An electron
trap, made with a special configuration of electric and magnetic fields.

@ __References__: Brown & Gabrielse RMP(86);
Blaum et al CP(10)
[and experiments in fundamental physics].

**Penrose Diagram** > s.a. asymptotic flatness.

* __Idea__: A diagram
of spacetime, as compactified by a suitable conformal transformation.

@ __General references__: Penrose in(64);
Jadczyk RPMP(12)-a1107 [geometry of Penrose's 'light cone at infinity'];
Schindler & Aguirre a1802 [algorithm].

@ __Specific types of spacetimes__: Brown & Lindesay CQG(09)-a0811 [accreting black holes];
Lindesay & Sheldon CQG(10) [transient black holes].

**Penrose Dodecahedron**

* __Idea__: A set of 40
states of a spin-3/2 particle used by Zimba and Penrose to give a proof of
Bell's non-locality theorem.

@ __References__: Zimba & Penrose SHPSA(93);
Massad & Aravind AJP(99)jul.

**Penrose Inequality / Conjecture
***

*GM*_{ADM} / *c*^{2} ≥ *R*/2 ;

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

*GM* / *c*^{2}
≥ (*A*/16π)^{1/2} .

@ __General references__: Penrose NYAS(73);
Ludvigsen & Vickers JPA(83) [partial proof];
Malec & Ó Murchadha PRD(94)
[and refs]; Frauendiener PRL(01)gq [towards a proof];
Malec et al PRL(02)gq [general horizons];
Malec & Ó Murchadha CQG(04)gq [re use of Jang equation];
Karkowski & Malec APPB(05)gq/04 [numerical evidence];
Ben Dov PRD(04)
[(counter)example]; Tippett PRD(09)-a0901
[violated for prolate black holes]; Mars CQG(09)-a0906
[rev]; Bengtsson & Jakobsson GRG(16)-a1608 [toy version with proof].

@ __Charged black holes__: Disconzi & Khuri CQG(12)-a1207
[charged black holes]; Khuri GRG(13)-a1308;
Lopes de Lima et al a1401
[in higher dimensions]; Khuri et al CQG(15)-a1410
[extensions].

@ __Riemannian__: & Huisken & Ilmanen (97) [proof, single
black hole]; Bray JDG(01)
[proof]; Bray & Chruściel in(04)gq/03;
Ohashi et al PRD(10)-a0906;
Khuri et al CM-a1308
[with charge, for multiple black holes].

@ __Other generalizations__: Gibbons in(84);
Karkowski et al CQG(94)
[gravitational waves]; Herzlich CMP(97)
[asymptotically
flat, *R* ≥ 0]; Khuri CMP(09)
[general
initial data sets]; Carrasco & Mars CQG(10)
[generalized-apparent-horizons version, counterexample]; Brendle &
Wang CMP(14)-a1303
[2D spacelike surfaces in Schwarzschild spacetime]; Alexakis a1506
[perturbations of the Schwarzschild exterior]; Roesch a1609, Bray & Roesch a1708 [null Penrose conjecture]; Husain & Singh PRD(17)-a1709 [in AdS space].

**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*, *y*^{i})
by (*u*, *λ*^{2}*v*,
*λy*^{i}) in the line
element, and then taking the limit as *λ* → 0 of d*s*^{2}/*λ*^{2};
One is then left with a metric of the form d*s*^{2}
= 2 d*u*d*v* + *C*_{ij}
d*y*^{i}d*y*^{j};
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 *m*_{irr}.

* __Variations__: The
collisional Penrose, or super-Penrose process consists of particle
collisions in the ergoregion.

@ __General references__: Penrose RNC(69),
& Floyd NPS(71); Christodoulou &
Ruffini PRD(71);
Wald AJ(74); Wagh & Dadhich PRP(89);
Fayos & Llanta GRG(91)
[limitations]; Williams phy/04;
Heller a0908;
Schnittman PRL(14)-a1410
[upper limit to energy extraction]; Bravetti et al PRD(16)-a1511
[thermodynamic optimization].

@ __Collisional Penrose process__: Schnittman PRL(14)-a1410;
Berti et al PRL(15)-a1410;
Zaslavskii MPLA(15)-a1411;
Zaslavskii IJMPD-a1510;
Leiderschneider & Piran PRD(16)-a1510
[maximal efficiency]; Patil et al PRD(16)-a1510
[efficiency]; Zaslavskii GRG(16)-a1511,
PRD(16)-a1511;
Ogasawara et al PRD(16)-a1511
[heavy particle production].

@ __Other variations__: Lasota et al PRD(14)-a1310 [generalized].

@ __Related topics__: Williams ap/02/PRD [Compton
scattering and *e*^{+}*e*^{−}
production]; Cen a1102-wd
[astrophysical scenario].

**Penrose Tiling** > see tiling.

**Pentaquark** > see hadrons.

**Percolation** > s.a. ising
models; in lattice field theory; Transport;
voronoi tilings.

* __Idea__: The theory
was initiated by Broadbent and Hammersley PPCS(57)
as a mathematical framework for the study of random physical processes,
such as flow through a disordered porous medium with randomly blocked
channels in a gravitational field; It has proved to be a remarkably rich
theory, with applications beyond natural phenomena to topics such as
network modelling and the contact process for epidemic spreading.

* __Phase transition__:
It turns out that the system undergoes a continuous phase transition with
a non-trivial critical behavior, at which it becomes macroscopically permeable.

@ __General references__: Stauffer & Aharony 94 [intro];
Bollobás & Riordan 06;
Duminil-Copin a1712-proc [rev, historical].

@ __Theory__: Cardy mp/01-ln
[conformal field theory methods]; Smirnov & Werner MRL-m.PR/01 [triangular 2D lattice];
Bollobás & Riordan RSA(06)m.PR/04;
Janssen & Täuber AP(05) [field theory approach, rev];
Gliozzi et al NPB(05) [random, as gauge theory];
Ziff et al JPA(11)
[factorization of the three-point density correlation function];
Curien & Kortchemski PTRF-a1307
[on random triangulations].

@ __Critical__: Grassberger JPA(99);
Cardy JPA(02)mp;
Ridout NPB(09)-a0808
[and Watts' crossing probability].

@ __Directed__: Grassberger JSP(95);
Janssen et al JPA(99)
[equation of state]; Grimmett & Hiemer m.PR/01;
Takeuchi et al PRL(07),
PRE(09) +
Hinrichsen Phy(09)
[experimental realization];
Chen PhyA(11) [square lattice, asymptotic behavior].

**Perfect Fluid** >
s.a. fluid; gas.

**Perfect Group** > see group types.

**Perfect Number** > see number theory.

**Perfect Space** > see types of topologies.

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

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

**Perl** > see programming languages.

**Permanent of a Matrix** > see matrix.

**Permeability** > see magnetism.

**Permittivity** > see electricity in matter.

**Permutations** > see finite
groups; particle statistics [identical particles].

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

**Permutons** > see phase
transitions [in combinatorial systems].

**Perpetual Motion Machine /
Perpetuum Mobile** > s.a. thermodynamics
[violations of second law].

@ __References__: Chernodub a1203
[permanently rotating devices]; Jenkins AJP(13)-a1301
[early 18th century demonstrations by Orffyreus, con man].

> __Related topics__:
see Maxwell's Demon; de
sitter space [example].

> __Online resources__:
see Continuous Frictioned Motion Machine page.

**Perplex Numbers** > see types
of numbers.

**Perron-Frobenius Operator** > see under Frobenius-Perron.

**Persistent Homology** > see types
.

**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 less than 370 yrs; If isolated in orbit around
each other, never; So, what is the effect of Venus on Earth's trajectory?

* __In quantum mechanics
– Approaches__: The usual time-dependent perturbation theory for
solving the Schrödinger equation does not preserve unitarity; The Magnus
expansion (also known as exponential perturbation theory) does provide
unitary approximate solutions.

@ __Texts__: Giacaglia 72;
Kevorkian & Cole 81; Gallavotti
83; Bender & Orszag 99;
Holmes 13.

@ __For differential equations__: Odibat & Momani PLA(07)
[homotopy perturbation method].

@ __Hamiltonian systems__: Lewis et al PLA(96) [time-dependent, invariants];
Laskar & Robutel ap/00 [symplectic integrators].

@ __Related topics__: Marmi m.DS/00-ln
[small denominators, intro]; Amore mp/04-proc [anharmonic oscillator, classical and quantum],
et al EJP(05)mp/04 [removal of secular terms];
Pound PRD(10)-a1003 [singular];
> s.a. classical systems;
oscillator;
series [convergence acceleration and divergent series].

@ __In quantum mechanics__: Sen IJMPA(99)cm/98 [singular potentials];
Fernández 01,
JPA(06)qp/04;
Franson & Donegan PRA(02)qp/01 [*t*-dependent];
Teufel 03 [adiabatic perturbation theory];
Ciftci et al PLA(05)mp [iterative];
Weinstein ht/05,
NPPS(06)ht/05 [adaptive];
Albeverio et al RPMP(06)
[singular, rigged Hilbert space approach];
Harlow a0905 [bound on the error];
Fernández a1004 [confined systems];
Blanes et al EJP(10)
[Magnus expansion or exponential perturbation theory, pedagogical];
Hayata PTP(10)-a1010 [without weak-coupling assumption];
Faupin et al CMP(11) [for embedded eigenvalues, second-order];
Kerley a1306 [time-independent];
Rigolin & Ortiz PRA(14)-a1403 [degenerate adiabatic perturbation theory].

> __Gravity-related
areas__: see black-hole perturbations; cosmological perturbations;
metric perturbations in general relativity.

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

**Petrov, Petrov-Pirani Classification**

**Pfaff Derivative of a Function
**$

*

**Pfaffian of a Matrix**

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

**Phantom Divide**

* __Idea__: The point
in cosmological history at which *w* (the ratio of pressure to energy density
for the effective fluid matter used to describe cosmological models) crossed the
value −1, or the value −1 itself in the range of possible values for *w*.

@ __References__: Zhang a0909-ch
[approaches].

**Phantom Field** > s.a. born-infeld
theory; Quintom; wormholes.

* __Idea__: An exotic
scalar field with a negative kinetic term (as a fluid, it has an equation
of state with *w* < −1), that violates most of the classical
energy conditions; 2005, Considered by some as a real possibility for dark
energy, although it has serious problems like instability and lack of a
well-posed initial-value formulation.

@ __General references__: Sami & Toporensky MPLA(04)
[and fate of universe]; Majerotto et al ap/04/JCAP
[and SN Ia data]; Santos & Alcaniz PLB(05)ap
[Segre classification]; Giacomini & Lara GRG(06)
[+ gravity + arbitrary potential, dynamics]; Pereira & Lima PLB(08)-a0806
[thermodynamics].

@ __Black holes, isolated objects__: Svetlichny ap/05
[possible production by black holes]; Berezin et al CQG(05)gq
[shell around Schwarzschild]; Bronnikov & Fabris PRL(06)
[regular
asymptotically flat, de Sitter and AdS]; Rahaman et al NCB(06)gq;
Gao et al PRD(08)-a0802
[mass increase]; Martins et al GRG(09)-a1006
[3D, phantom fluid]; Gyulchev & Stefanov PRD(13)
[lensing]; > s.a. gravitational
thermodynamics; models of
topology change.

@ __Cosmology__:
Dąbrowski et al PRD(03) [+
standard matter]; Chimento & Lazkoz MPLA(04) [big rip];
Curbelo et al CQG(06)ap/05
[avoidance of big rip]; Faraoni CQG(05)gq
[general potential]; Capozziello et al PLB(06) [dark energy and dark matter];
Bouhmadi-López et al PLB(08)gq/06 [future singularity];
Dąbrowski gq/07-MGXI [dark energy];
Sanyal IJMPA(07) [inflation
rather than big rip]; Hrycyna & Szydłowski PLB(07) [conformally
coupled, acceleration]; Shatskiy JETP(07)-a0711;
Chaves & Singleton SIGMA(08)-a0801
[and dark matter]; Chen et al JCAP(09)-a0812
[phase-space analysis]; Myung PLB(09) [thermodynamics];
Regoli PhD-a1104;
Astashenok et al PLB(12)-a1201
[without big rip singularity]; Novosyadlyj et al PRD(12),
Ludwick PRD(15)-a1507
[as dark energy]; Ludwick MPLA(17)-a1708 [rev]; > s.a. FLRW models;
gravitational thermodynamics.

@ __Loop quantum cosmology__: Samart & Gumjudpai PRD(07)-a0704;
Gumjudpai TJP-a0706-proc;
Fu et al PRD(08)-a0808;
Wu & Zhang JCAP(08)-a0805;
> s.a. FLRW quantum cosmology.

**Phases of Matter**

* __Idea__: The phases that have been known for a long time are solid, liquid, gas and plasma, but experiments with matter cooled to within a few degrees of 0 K have turned up a number of exotic phases, such as superfluids, superconductors and topological phases; In these new types of phases one can see quantum mechanical effects at work in materials, unencumbered by the random motions of atoms.

* __Topological phases__: Thouless, Kosterlitz and Haldane won the 2016 Nobel Prize for their work on these phases; A variety of such phases are known.

@ __General references__: issue JPCM(98)#49
[matter under extreme conditions]; Pinheiro phy/07
[plasma, genesis of the word]; Kadanoff a1002;
Baas IJGS-a1012
+ news ns(11)jan
[topology and generalization of Efimov states]; > s.a. magnetism
[plasma physics or magnetohydrodynamics].

@ __Topological phases__: Read PT(12)jul; > s.a.
matter [mathematical models].

> __Type of phases__: see condensed
matter [gases, liquids]; crystals; fluid; gas; Plasma; bose-einstein
condensate.

**Phase of a Quantum State** > s.a. arrow
of time [phase squeezing]; geometric phase;
pilot-wave interpretation [and quantum phase]; quantum
states.

@ __References__: Barnett & Pegg JMO(89)
[optical phase operator]; Lynch PRP(95);
Koprinkov PLA(00)qp/06;
Kastrup qp/01 [and modulus];
Lahti & Pellonpää PS(02) [formalisms];
Pellonpää JMP(02) [observables];
Heinonen et al JMP(03) [covariant phase difference];
de Gosson JPA(04) [general definition];
Gour et al PRA(04) [self-adjoint extensions];
Saxena a0803 [in terms of inverses of creation and annihilation operators];
Hall & Pegg PRA(12)-a1205.

**Phase Curve** > see phase space.

**Phase Diagram**

* __Idea__: A plot showing
the boundaries between thermodynamically distinct phases in an equilibrium system.

> __Gravity__: see dynamical triangulations;
phenomenology of gravity; quantum-gravity renormalization.

> __Other field theories__:
see Gross-Neveu Model; QCD,
QCD phenomenology; Wess-Zumino
Model.

> __Other physics__:
see Critical Points; matter
[dense matter]; Potts Model; Water.

> __Online resources__:
see Wikipedia page.

**Phase 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.

@ __General references__: Baym AP(61),
re AP(00)
[Green function, quantum field theory methods]; Kokkedee 63;
Hu & Nori PRL(96)
+ pn(96)mar
[squeezed].

@ __Specific types of systems__: Quilichini & Janssen RMP(97)
[quasicrystals]; Gorishnyy et al pw(05)dec
[phononic
crystals];
Lukkarinen a1509
[in weakly anharmonic particle chains, kinetic theory].

@ __Related topics__: Schwab et al Nat(00)apr
[quantized thermal conductivity]; Johnson & Gutierrez AJP(02)mar
[wave
function visualization]; news tcd(15)mar
[controlling phonons with magnetic fields]; Iachello et al PRB(15)-a1506
[algebraic theory, energy dispersion relation and density of states].

> __Online resources__:
see Wikipedia page.

**Photoelectric Effect**
> s.a. photon phenomenology.

* __Idea__: The effect
by which light (in particular, UV) incident on a metal causes electrons to
be emitted by the metal surface; The quantitative explanation of
observations related to this effect was one of the key arguments in favor
of the idea that light is made of discrete photons.

@ __General references__: Einstein AdP(05);
Zenk RVMP(08)
[variant of standard approach with wider applicability].

@ __Without quanta__: Wentzel ZP(27);
Franken in(69); Milonni AJP(97)jan.

@ __Other topics__: Bach et al ATMP(01)mp/02
[mathematical].

> __Online resources__:
see Wikipedia page.

**Photon** > s.a. photon
phenomenology.

**Photon Sphere / Surface** > see spacetime
subsets.

**Physical Constants** > see under Constants.

**Physical Laws** > see under Laws.

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

**Physicalism** > see philosophy
of
physics.

**Physically Reasonable Model**

* __Idea__: A model for
a physical system that is considered as having values for the properties
under study that reflect those that can occur in a real system.

* __Rem__: A stronger
expression would be "physically realistic model".

**Physically Significant
Property**

* __Idea__: A property
of a model for a physical system is physically significant if, whenever
the model has the property, the real system is expected to have it as
well.

* __Rem__: Hawking has
stated that "the only properties of spacetime that are physically
significant are those that are stable in some appropriate topology".

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

**Picard-Lefschetz Theory** > see quantum
field theory techniques.

**Pierre Auger Observatory**

* __Idea__: A network
of detectors in the pampa of Western Argentina for the study of
high-energy cosmic rays.

@ __References__: Anchordoqui et al PRD(03)hp;
Anchordoqui ap/04-proc;
Kampert NPPS(06)ap/05;
Van Elewyck ap/06-ln,
MPLA(08);
Nitz a0706-conf
[north site]; Van Elewyck a0709-proc;
Parizot et al a0709-conf;
de Mello APPS-a0712-conf,
Matthiae a0802-conf
[status and results]; Abraham et PA a0906-conf [status and plans];
Etchegoyen et al a1004-conf;
Roulet a1101-conf;
Smida et al a1109-proc,
Kampert a1207-proc
[results]; Pierre Auger Collaboration NIMA(15)-a1502 [design and performance];
> s.a. ultra-high-energy cosmic rays.

> __Online resources__:
see Pierre Auger website;
Wikipedia page.

**Pigeonhole Principle**
(A.k.a. Dirichlet box principle.)

* __Theorem__: If more
than *n* pigeons are roosting in *n* pigeonholes, at least
one hole contains more than one pigeon.

* __Applications__:
There are at least two people in Los Angeles with the same net worth, to
the nearest dollar; In mathematics research, it is used to prove the
existence of things which are difficult to construct, for example in Ramsey theory.

* __In quantum physics__:
There are instances when three quantum particles are put in two boxes, yet
no two particles are in the same box.

@ __General references__: Olivastro ThSc(90)sep.

@ __In quantum physics__: Aharonov et al a1407
+ sn(14)jul
[it doesn't always hold]; Yu & Oh a1408
[and the quantum Cheshire cat]; Svensson a1412.

**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, that can be used to describe the transformation behavior of fermions
under parity and time reversal; Pin(1, 3) is to O(1, 3) what Spin(1, 3) is to SO(1, 3).

@ __References__: 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 BUMI-a0907
[and spinors and orientability]; Janssens a1709 [and general relativity].

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

**Pioneer Anomaly** > see anomalous acceleration.

**Pions, π** > see hadrons.

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

**Plancherel Theorem** > see Symmetric Space.

**Planck Constant and Units**
> s.a. constants;
Wikipedia page.

* __Value__: 1998, *h*
= 6.62606891(58) × 10^{−34} J · s or × 10^{−27}
erg · s; \(\hbar\) = 1.05457266(63) × 10^{−34}
J · s, or × 10^{−27} erg · s; The best
values are obtained from measurements of the flux quantum *φ*_{0}
= *h*/2*e* using the Josephson effect, and the quantum of
conductance *G*_{0} = 2*e*^{2}/*h* from the quantum Hall effect;
2016, *h* = 6.62606983 × 10^{−34} J · s,
achieved with NIST's new watt balance.

* __Length__: *l*_{P}
= (*G*\(\hbar\)/*c*^{3})^{1/2}
= 1.6 × 10^{−33} cm.

* __Time__: *t*_{P}
= *l*_{P} / *c*
= 5.4 × 10^{−44} s.

* __Energy__: *E*_{P}
= *l*_{P} *c*^{4}/*G*
= 2 × 10^{16} erg = 1.3 × 10^{19} GeV.

* __Mass and density__:
*M*_{P} = 2.2 × 10^{−5}
g, and *ρ*_{P}= 5.1 × 10^{96}
kg/m^{3}.

@ __General references__: Planck SBAW(1899);
Fischbach et al PRL(91)
[quantum mechanics with different \(\hbar\)];
Cooperstock & Faraoni MPLA(03)ht,
IJMPD(03)gq [including *e* and *s*];
Wilczek PT(05)oct [absolute units].

@ __Measurements__: Williams et al PRL(98) +
pn(98)sep +
pw(98)sep;
Steiner RPP(13);
news pt(16)jul
[precise determination in preparation for a new, refined SI in 2018];
news pt(16)sep.

@ __Related topics__: Zeilinger AJP(90)feb [Planck stroll];
Casher & Nussinov ht/97
[*p*_{P} is unattainable]; Sivaram a0707
[Planck mass]; Ramanathan a1402
[Planck's constant as diffusion constant]; Calmet PTRS(15)-a1504
[effective enery-scale dependence, motivated by quantum gravity].

**Planck Cube**

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

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

**Planck Mission / Satellite** > see cosmic
microwave background.

**Planck Stars** > see astronomical objects.

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

**Planets** > see extrasolar
; solar planets
[including "planet X"].

**Planetary Nebulae** > see interstellar
matter.

**Plasma Physics** > see phenomenology of magnetism.

**Plasticity** > s.a. Elasticity.

* __Idea__: The
phenomenon by which many materials maintain their deformed shape after
forces are applied to them; It is often irreversible; In some materials
the plastic deformation occurs when the applied forces exceed a certain
threshold, below which the materials are elastic.

* __Microscopically__:
Plasticity is a result of the propensity of solids to "flow", usually
because of the motion of dislocations within them; It relies therefore on
the presence of many dislocations that can easily move through the
crystal, and on the bonds that hold the crystal together not being too
localized, making it brittle.

* __Examples__:
Materials with delocalized bonds are metals (in which they are due to
conduction electrons) and quantum crystals (in which they are due to the
atoms or molecules in the lattice, which are light, making their quantum
properties important).

@ __References__: Castaing Phy(13)
[giant, anisotropic plastic deformation that is also reversible in the
quantum solid Helium-4].

> __Online resources__:
see Wikipedia page.

**Plateau Problem** > see extrinsic
geometry [minimal surface].

**Platonic Solids** > see euclidean geometry.

**Plausibility Measures**

* __Idea__: Structures
for reasoning in the face of uncertainty that generalize probabilities,
unifying them with weaker structures like possibility measures and
comparative probability relations.

@ __References__: Fritz & Leifer a1505/QPL [on test spaces].

**Plebański Action for Gravity** > s.a. first-order
actions; BF theories; unified theories.

@ __References__: Bennett et al IJMPA(13)-a1206
[several theories of four-dimensional gravity in the Plebański formulation].

**Plebański-Demiański Solutions** > see types of geodesics.

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

**PMNS (Pontecorvo-Maki-Nakagawa-Sakata) Matrix**

* __Idea__: The lepton
flavor mixing matrix in the Standard Model of particle physics.

> __Online resources__:
see Wikipedia page.

**PN Formalism** > see under Post-Newtonian Expansion.

**Podolsky Theory** > see modified theories of electrodynamics.

**Pohlmeyer Invariants** > see bosonic
strings and superstrings.

**Pohlmeyer's Theorem**

* __Idea__: A result
proving that any critical fixed point for a field theory (in integer
dimension) with vanishing anomalous dimension must be the Gaussian one.

@ __References__: Rosten JPA(10)-a1005 [extension to non-integer dimension].

**Poincaré Conjecture** > see conjectures.

**Poincaré Duality** > see
cohomology.

**Poincaré 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 *N*_{dof}
− *N*_{com} ≤ 2, otherwise regular
behavior can be misinterpreted as chaos.

@ __Examples__: in Murray & Dermott 99 [solar system].

@ __Generalization__: Gaeta JNMP(03)mp/02 [Poincaré-Nekhoroshev].

**Poincaré Recurrence**
> see Recurrence; Unitarity.

**Poincaré-Hopf Theorem**

* __Idea__: A
relationship between the Euler characteristic of a manifold *M*
and the indices of a vector field on *M* over its zeroes; A
special case is the "hairy ball theorem", which states that there is no
smooth vector field on a sphere having no sources or sinks.

@ __References__: Cima et al Top(98)
[non-compact manifolds]; Szczęsny et al IJGMP(09)-a0810
[new elementary proof].

> __Online resources__:
see Wikipedia page.

**Point** > see spacetime.

**Point-Present Theories** > see time.

**Point Process** > see statistical geometry.

**Point Transformation** > see symplectic structure.

**Point-Splitting Regularization** > see regularization.

**Pointed Topological Spaces** > see types of topological spaces.

**Poisson Algebra / Bracket / Structure**

**Poisson Distribution**
> s.a. probability.

$ __Def__: The
distribution on \(\mathbb N\) given by *P*(*n*) = e^{−a}
*a*^{n}/*n*!.

* __Properties__: It
has mean *a*, and standard deviation *a*^{1/2}.

@ __General references__: de Groot 75, ch5.

@ __Applications__: Elizalde & Gaztañaga PLA(88) [of galaxies].

@ __Modified__: Laskin JMP(09) [fractional].

**Poisson Equation** > s.a.
partial differential equations.

* __Idea__: The
elliptic partial differential equation ∇^{2}*u*
= −*f*, where ∇^{2} is the
Laplacian operator for a Riemannian metric, often flat.

@ __General references__: Ma et al a1208/JCP
[efficient numerical solution for arbitrary 2D shapes].

__Generalizations__: Sebastian & Gorenflo a1307
[fractional].

> __Online resources__:
see MathWorld page.

**Poisson Formula**

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

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

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

**Poisson Integral** > see integration.

**Poisson Process** > see statistical
geometry.

**Poisson Ratio** > see Strain
Tensor.

**Poisson σ-Model** > see sigma
model.

**Poisson-Boltzmann Equation** > see partial
differential
equations.

**Poisson-Lie Group**

* __Applications__:
Useful for quantum deformations of a group.

@ __References__: Drinfeld SMD(83);
Lu & Weinstein JDG(90).

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

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

**Polariton** > see Quasiparticles.

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

**Polarization of Waves** > see polarization.

**Polarization in Symplectic Geometry**

* __Idea__: A
polarization is an *n*-dimensional completely degenerate subspace
of a symplectic vector space, or integrable distribution on a 2*n*-dimensional
symplectic manifold (it thus forms Lagrangian submanifolds).

* __Example__: Given a
symplectic vector space (*V*, Ω) and a map *P*: *V*
→ *V* such that *P*^{2} =
\(\mathbb 1\) and *P* Ω = − Ω *P*, we can construct a
polarization defined by the eigenvectors of *P*_{+}:=
\(\frac12\)(\(\mathbb 1\) + *P*) (so *P*_{+}
Ω *P*_{+} = 0), with eigenvalue 1.

**Polaron**

* __Idea__: A
quasiparticle used in condensed matter physics to understand interactions
between electrons and atoms in a solid.

__References__: Emin 13.

> __Online resources__:
see Wikipedia page.

**Polish Space** > see types
of distances.

**Polygamma Function**

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

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

**Polygroup Theory** > see group
theory.

**Polyhomogeneous Spacetimes** > see types
of
spacetimes.

**Polymer** > s.a. condensed
matter [soft matter]; molecular physics.

@ __Statistical mechanics__: Brereton JPA(01);
Ioffe & Velenik BJPStat(10)-a0908
[stretched by an external force]; Sabbagh & Eu PhyA(10)
[van der Waals equation of state, self-diffusion coefficient];
De Roeck & Kupiainen CMP(11)-a1005
[polymer expansion]; Rodrigues & Oliveira JPA(14)
[Monte Carlo simulations].

@ __Related topics__: Jitomirskaya et al CMP(03)mp/04
[random, and delocalization]; Imbrie JPA(04)
[branched directed, dimensional reduction]; > s.a. solitons
[in polyacetylene].

**Polymer Quantization**
> s.a. representations of quantum mechanics.

* __Idea__: The name
given to one of four related non-regular representations of the Heisenberg
algebra, in which the spectrum of the configuration or the momentum
variable is not continuous, and the corresponding infinitesimal generator
is not defined; This approach to quantization is related to and inspired
by, but distinct from that used in loop quantum gravity.

@ __General references__: Fredenhagen & Reszewski CQG(06)gq;
Corichi et al CQG(07)gq/06,
PRD(07)-a0704;
Chiou CQG(07)gq/06
[and the Galileo group]; Hossain et al CQG(10)-a1003
[and the uncertainty principle]; Campiglia a1111
[and geometric quantization]; Date & Kajuri CQG(13)-a1211
[and symmetries]; Chacón-Acosta et al Sigma(12)
[statistical thermodynamics]; Barbero et al PRD(14)-a1403 [separable Hilbert space];
Gorji et al CQG(15)-a1506 [versus the Snyder non-commutative space];
Morales-Técotl et al PRD(15)-a1507 [and the saddle point approximation of partition functions];
Morales-Técotl et al PRD(17)-a1608 [particles, path-integral propagator];
Amelino-Camelia et al PLB(17)-a1707
[and deformed symmetries, non-commutative geometry].

@ __Simple systems__: Husain et al PRD(07)-a0707
[Coulomb potential]; Kunstatter et al PRA(09)-a0811
[1/*r*^{2} potential]; Kunstatter
& Louko JPA(12)-a1201 [on the half line];
Majumder & Sen PLB(12)-a1207 [and GUP];
Flores-González et al AP(13)-a1302 [particle propagators];
Barbero et al CQG(13)
[band structure]; Gorji et al PRD(14)-a1408 [ideal gas, partition function];
Martín-Ruiz et al PRD(15)-a1506 [bouncing particle];
Berra-Montiel & Molgado a1610
[and zeros of the Riemann zeta function]; > s.a. gas.

@ __Phenomenology__: Martín-Ruiz PRD(14)-a1406 [beam of particles, and diffraction in time];
Chacón & Hernández IJMPD(15)-a1408 [semiclassical Hamiltonian and compact stars];
Martín-Ruiz et al a1408,
Demir & Sargın PLA(14)-a1409 [tunneling, Zeno effect];
Kajuri CQG(16)-a1508 [radiation in inertial frames];
Ali & Seahra PRD(17)-a1709 [natural inflation];
Kajuri & Sardar PLB(18)-a1711 [Lorentz violation, at low energies];
Khodadi et al sRep-a1801 [optomechanical setup];
> s.a. phenomenology of cosmological perturbations;
unruh effect.

> __Related topics__:
see Bohr Compactification; entropy in quantum theory;
fock space; holography;
renormalization; tunneling.

> __Gravity /
cosmology__: see black-hole quantization;
loop quantum gravity; minisuperspace; models in canonical quantum gravity; 2D quantum gravity.

> __Other field
theories__: see bose-einstein condensates; Pais-Uhlenbeck
Model [with higher-order time derivatives]; quantum
field theories [scalar].

**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 \(\mathbb R\)^{d} 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];
Enciso a1408
[volumes of polytopes in any dimension without triangulations].

@ __Regular polytopes__: Cantwell JCTA(07) [all regular polytopes are Ramsey];
Boya & Rivera RPMP(13)-a1210.

@ __Delaunay polytopes__: Dutour EJC(04);
Erdahl et al m.NT/04-(proc);
Sikiric & Grishukhin EJC(07)
[computing the rank].

@ __In 3D spaces of constant curvature__: Abrosimov & Mednykh a1302 [volume formulas].

@ __Other special types__: Neiman GD(14)-a1212
[null-faced 4-polytopes in Minkowski spacetime].

> __Related topics__:
see Schlegel Diagram; statistical
geometry [from random point set].

**Pomeransky-Senkov Black Hole** > see causality
conditions.

**Pomeron**

@ __General references__: Levin hp/98-conf;
cern(99);
Brower et al JHEP(07)ht/06
[and gauge/string duality]; Swain a1110-fs
[and the nature of particles].

@ __And QCD__: Donnachie et al 02;
Nachtmann hp/03-conf.

> __Online resources__:
see Wikipedia page.

**Pontrjagin / Pontryagin Classes,
Numbers**

**Ponzano-Regge Model** > s.a.
spin-foam models / 3D
gravity; SU(2).

* __Idea__: 3D spin
coupling theory, giving a non-perturbative definition of the path integral
for (Euclidean) 3D gravity.

@ __General references__: Ponzano & Regge in(68);
Lewis
PLB(83)
[renormalizability];
Iwasaki gq/94,
JMP(95)gq
[in terms of surfaces]; O'Loughlin ATMP(02)gq/00
[boundary actions]; Barrett & Naish-Guzman CQG(09)-a0803;
Wieland PRD(14)-a1402
[action from a 1D spinor action].

@ __Variations__: Carfora et al PLB(93)
[4D,
and 12*j* symbols]; Carbone et al CMP(00);
Freidel
NPPS(00)gq/01
[Lorentzian]; Livine & Oeckl ATMP(03)ht/03
[supersymmetric]; Li CMP(14)-a1110
[*κ*-deformation]; Vargas a1307
[on a manifold with torsion].

@ __Related topics__: Barrett & Foxon CQG(94)gq/93
[semiclassical limit]; Petryk & Schleich PRD(03)gq/01
[geometric quantities]; Arcioni et al NPB(01)ht
[and holography]; Freidel & Louapre CQG(04)ht
[gauge fixing], gq/04
[and Chern-Simons theory]; Freidel & Livine CQG(06)ht/05
[effective field theory for particles]; Hackett & Speziale CQG(07)gq/06
[geometry and clasping rules]; Barrett & Naish-Guzman gq/06-MGXI
[and
Reidemeister torsion]; Livine & Ryan CQG(09)-a0808
[B-observables]; Caravelli & Modesto a0905
[spectral dimension of spacetime].

**Popper's Thought Experiment**

* __Idea__: A thought
experiment proposed by Karl Popper designed to check for possible
violations of the uncertainty principle.

@ __General references__: Qureshi IJQI(04)qp/03,
AJP(05)jun-qp/04;
Richardson & Dowling IJQI(12)-a1102
[no violation of the uncertainty principle, fundamental flaw]; Qureshi Quanta(12)-a1206
[modern perspective]; Cardoso a1504
[non-linear quantum theory and uncertainty principle violation].

@ __With photons__: Kim & Shih FP(99)
[entangled photon pairs]; Peng et al EPL(15)
+ news pw(15)jan
[photon number fluctuation correlations in a thermal state]; Reintjes
& Bashkansky a1501.

> __Online resources__:
see Net Advance of Physics page;
Wikipedia page.

**Porosity of a Measure** > see measure.

**Pöschl-Teller Potential** >
s.a. types of coherent states.

@ __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].

@ __Tests of local position invariance__:
Peil et al PRA(13) [using continuously-running atomic clocks];
Shao & Wex CQG(13) [bounds].

**Positioning Systems**
> s.a. coordinates;
minkowski spacetime [secure positioning].

@ __Relativistic positioning systems__:
Coll et al a0906-rp [status];
Tartaglia a1212-conf [principles and strategies];
Coll a1302-conf [rev];
Puchades & Sáez ASS(14)-a1404 [errors due to uncertainties in the satellite world lines].

@ __GPS__: Parkinson & Spilker ed-96;
in Hartle 03;
Puchades & Sáez ASS(12)-a1112.

**Positive Action Conjecture** > see action for general relativity.

**Positive Frequency Function** > see functions.

**Positive Map** > see Maps.

**Positivism** > see philosophy of science.

**Positron** > see electron.

**Possibilism** > see time.

**Possibility** > see many-worlds interpretation.

**Post-Friedmannian
Formalism** > see cosmological models.

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

**Potential for a Field**

* __Idea__: Originally,
a potential was a scalar function whose gradient gives a force on a test
particle (per unit charge); It was extended to a vector field whose curl
gives a (magnetic) field, and then to the general mathematical notion of a
function (or a higher-rank tensor field) which gives, by differentiation,
a field of interest, possibly a dynamical tensor field.

> __Vector potential__:
see aharonov-bohm effect; connection;
electromagnetism.

**Potential in Physics** >
s.a. scattering; thermodynamics
[thermodynamic potentials].

* __Retarded potential__:
It has to be used for systems with large velocities (corrections are of
order *v*^{2}/*c*^{2}),
or pairs of systems with large separations compared to the internal
motions (even if slow).

@ __General references__: Kellogg 29;
Grant & Rosner AJP(94)apr
[orbits in power law *V*].

@ __Retarded potential__: Spruch & Kelsey PRA(78) [elementary
derivation]; > s.a. arrow of time.

> __In classical
mechanics__: see Bertrand's Theorem; classical
systems [including central potentials]; Coulomb
Potential.

> __In classical field
theory__: see electromagnetism; newtonian
gravitation.

> __In quantum
mechanics__: see schrödinger equation;
special potentials; pilot-wave
interpretation [quantum potential].

> __In quantum field
theory__: see effective field theories
[effective potential]; quantum field theory.

**Potts Model** > s.a. lattice
field theory; Yang-Baxter Equation.

* __Idea__: A 2D
generalization of the Ising model of interacting spins on a lattice; The
chiral Potts model is a challenging one, it is "exactly solvable'' in the
sense that it satisfies the Yang-Baxter relation, but actually obtaining
the solution is not easy; Its free energy was calculated in 1988, the
order parameter was conjectured in full generality in 1989 and derived in
2005.

@ __General references__: Baxter 82;
Wu RMP(82);
Sokal
MPRF(01)cm/00-in
[unsolved problems]; Baxter JPCS(06)cm/05
[rev]; Beaudin et al DM(10)
[introduction from a graph theory perspective].

@ __Phase transitions__: Baxter JSP(05)cm,
PRL(05)cm
[chiral, order parameter]; Georgii et al JSM(05)mp
[continuum, order-disorder transition]; Ahmed & Gehring JPA(05)
[anisotropic,
phase diagram]; Jacobsen & Saleur NPB(06) [antiferromagnetic transition];
Fernandes et al PhyA(06) [alternative order parameter];
Gobron & Merola JSP(07) [first-order];
Johansson PLA(08)
[2D with open boundary conditions, Monte Carlo]; Aluffi & Marcolli JGP(13)-a1102 [motivic approach].

@ __Coupled to gravity__: Ambjørn et al NPB(09)-a0806,
Cerda Hernández a1603
[causal dynamical triangulations].

@ __Related topics and variations__: Richard & Jacobsen NPB(07) [on a torus];
Barré & Gonçalves PhyA(07) [on a random graph, canonical and microcanonical ensembles];
Ganikhodjaev PLA(08)
[next-nearest-neighbor interactions, on the Bethe lattice];
De Masi et al JSP(09) [continuum version, phases];
Contucci et al CMP(13)-a1106 [on a random graph];
Dasu & Marcolli JGP(15)-a1412 [in an external magnetic field, sheaf-theoretic interpretation];
> s.a. Confinement [model for];
renormalization.

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

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

**Powder** > see metamaterials.

**Power of a Graph** > see graph theory.

**Power Spectrum of Perturbations in Field Theory**

* __Idea__: Usually
defined as the Fourier transform of the two-point correlation function of
the field in a quantum state.

**Power-Law Distributions**
> s.a. critical phenomena; states
in statistical mechanics.

@ __References__: Simkin & Roychowdhury PRP(11)
[mechanism for producing them].

**Poynting Vector** > s.a. energy-momentum tensor.

* __Idea__: The vector
**S** = ** E** × **B**/*μ*_{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 *P*_{a} = (*U*,
**P**), where

*U*:= (1/8π) ∫ (*E*^{2}
+ *B*^{2}) d*v* = ∫
*T*_{00} d*v*
, **P**:= (1/4π*c*) **∫
E** × **B** d*v* = ∫
*T*_{0i} d*v* .

@ __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__: Klacka ap/00,
ap/01,
ap/02,
ap/02;
in Harwit 06;
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**
> s.a. gravitation / higher-order
gravity; modified newtonian gravity.

* __Rem__: It is not
the same as PN (Post-Newtonian) expansion of general-relativistic results
around the weak-field / slow-motion limit.

**Prasad-Sommerfield Solution**
> see monopoles.

**Pre-Recueil** > see Recueil.

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

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

**Precession** > s.a. gravitating
bodies; Gyroscope; Runge-Lenz
Vector; test bodies; Thomas
Precession.

* __In general relativity__:
There are several types, perihelion (Einstein), geodetic (de Sitter),
orbital plane (Lense-Thirring, gravitomagnetic), and spin-spin
(Pugh-Schiff); > s.a. tests
of general relativity with orbits.

@ __General references__: Magli phy/04
[in ancient astronomy]; Jonsson CQG(06)-a0708
[spin precession, covariant formalism]; Casotto & Bardella MNRAS(13)-a1210-conf
[equations of motion of a secularly precessing elliptical orbit]; Lo et al
AJP(13)sep,
D'Eliseo
AJP(15)apr
[unified frameworks for perihelion advance, different causes].

@ __In general relativity__: Holstein AJP(01)dec;
Sigismondi ap/05-MGX;
Harper PhSc(07)dec;
He & Zhao IJTP(09)
[analytical solution]; Boyle et al PRD(11)
[compact binaries, geometric approach]; D'Eliseo ASS(12)-a1206
[precession of orbits, quick method]; Mashhoon & Obukhov PRD(13)-a1307
[in gravitational fields]; Hu et al AHEP(14)-a1312
[general spherically symmetric spacetimes]; > s.a. gravitational self-force [spin precession].

@ __In modified gravity theories__: Behera & Naik ap/03
[vector gravity]; Schmidt PRD(08)
[modified Newtonian potential]; Fokas et al a1509
[relativistic gravitational law]; Friedman & Steiner EPL(16)-a1603
[in relativistic Newtonian dynamics].

@ __Specific cases__: Stewart AJP(05)aug
[Mercury, due to other planets]; Iorio AJ(09)-a0811 [Saturn, anomalous];
Moniruzzaman & Faruque PS(13)
[periastron precession due to gravitational spin-orbit coupling].

> __In various
theories__: see Cogravity; gravity
theories; newtonian gravity [perturbations and curved spaces].

> __In various
spacetimes__: see reissner-nordström
solutions; schwarzschild-de sitter
spacetime [with a cosmological constant]; test bodies.

**Precision** > s.a. Accuracy.

* __Idea__: The size of
the error bar in a series of measurements.

**Precompactness** > see compactness.

**Prediction and Predictability**
> s.a. causality; paradigms
in physics; time.

* __Idea__:
Predictability is an epistemic property of a model for a physical system,
related to what we are able to compute and predict with it; Prediction may
refer to a theory predicting either effects, phenomena, values of
quantities, or more specifically the evolution of a system and results of
future measurements.

* __Question__: Does a physical law have to be predictive?

* __Remark__: Usually, for
several practical and theoretical reasons, predictions in physics are statistical.

@ __General references__: Brush Sci(89)dec [light bending];
Hole IJTP(94) [and determinism];
Holt & Holt BJPS(93) [in
classical mechanics]; Caves & Schack Compl(97)cd [types];
Coles 06 [I]; Manchak FP(08) [in general relativity];
Werndl BJPS(09) [and chaos];
Srednicki & Hartle PRD(10)-a0906 [in a very large universe];
Stuart et al PRL(12)
+ news physorg(12)jul,
physorg(12)jul
[experimental bound on the maximum predictive power]; Cecconi et al AJP(12)nov [intrinsic limitations];
Hosni & Vulpiani P&T(17)-a1705 [forecasting and big data].

@ __Of effects__: Hitchcock & Sober BJPS(04)
[vs accommodation, and overfitting].

> __Related topics__:
see chaos; Determinism;
Explanation; wave phenomena [superluminal].

**Prefixes** > see units.

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

**Preons** > see composite models.

**Preorder** > s.a. poset;
Quasiorder [non-reflexive generalization].

$ __Def__: A reflexive
and transitive binary relation; The concept generalizes that of
(reflexive) partial orders and equivalence relations.

* __Remark__: One can
always define an Alexandrov topology on a preorder by using the upper sets
as open sets.

@ __References__: Cameron et al DM(10) [random preorders and alignments];
Minguzzi AGT(12)-a1108 [representation by continuous utilities].

> __Online resources__:
see Wikipedia page.

**Prequantization** > s.a.
geometric quantization.

@ __References__: Schreiber a1601-in [higher prequantum geometry].

**Presentation of a Group**

$ __Def__: A pair (*S*,*
D*) of a set of generators *S* and a set of relations
between the generators *D* = {Γ* _{i}*};
Each relation Γ

*

*

**Presentation of a Topological Space**

* __Idea__: An
appropriate set of vertices, edges, faces, etc.

* __Result__: A
finitely presented space has a finitely presented fundamental group (>
s.a. Calculating Theorem).

**Presentism** > s.a. special
relativity; time.

* __Idea__: The view
that only the present is real (as opposed to possibilism, eternalism or
the block-universe view, and their variants).

@ __References__: Wüthrich a1207
[fate in modern physics]; Romero & Pérez EJPS(14)-a1403 [and black holes].

> __Online resources__:
see Wikipedia page.

**Pressure** > s.a. energy-momentum
tensor; fluid; gravitating
matter; Momentum; radiation;
thermodynamics; turbulence.

@ __General references__: Durand AJP(04)aug
[quantum,
Bose and Fermi statistics]; Frontali PhysEd(13)
[history of the concept].

@ __Coupling to gravity__: Ehlers et al PRD(05)gq;
Narimani et al JCAP(14)-a1406
[and observational cosmology].

**Presymplectic Structure** > see symplectic
geometry.

**Prevalence** [> s.a. measure
theory.]

* __Idea__: The
analogue of the finite-dimensional notions of 'Lebesgue almost every' and
'Lebesgue measure zero' in the infinite-dimensional setting

@ __References__: Ott & Yorke BAMS(05).

**Price's Law** > see perturbations
of schwarzschild spacetime.

**Primakoff Effect** > s.a. axions.

* __Idea__: The
production of an axion from the interaction of a photon with a classical
electromagnetic field [Henry Primakoff 1951].

**Prime Graphs** > see types
of
graphs.

**Prime Numbers** > see number
theory.

**Primordial Black Holes** > see types
of black holes.

**Primordial Gravitational Waves** > see gravitational-wave
background.

**Primordial Magnetic Fields** > see magnetic
fields in cosmology.

**Primordial Perturbations** > see phenomenology
of cosmological perturbations.

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

**Principal Part / Value **> see distribution.

**Principal Principle**
> s.a. quantum measurements.

* __Idea__: A principle
relating objective probabilities and subjective chance.

@ __References__: Meacham BJPS(10)
[misconceptions].

**Principle of Equivalence** > see under Equivalence
Principle.

**Principle of Mediocrity** >
see civilizations.

**Principles in Mathematics, Physics, and Related Areas**

> __In gravitation and
cosmology__: see anthropic principle;
Copernican Principle; cosmological
principle; equivalence principle; mach's
principle; Principle of Mediocrity; Relativity
Principle.

> __In quantum theory__:
see Correspondence Principle; (Pauli)
Exclusion Principle; Landauer's Erasure
Principle; Maximal Variety; (heisenberg's)
uncertainty principle.

> __In other physics__:
see Action-Reaction Principle;
Boltzmann Principle; Causal
Entropic Principle; Fermat's
Principle; Hamilton's Principle;
huygens' principle; Maupertuis
Principle; Maximum Entropy
Principle; Non-Demolition
Principle; Superposition Principle;
Symmetric Criticality; variational
principles.

> __In mathematics__:
see (Cauchy's) Argument Principle;
Enumeration Principle; Pigeonhole
Principle; Principal Principle; Well-Ordering
Principle.

> __In logic__: see
Common Cause Principle; Leibniz
Principle; Principle of
Sufficient Reason.

**Prisoner's Dilemma** > see games.

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

**Probability Current**
> s.a. path integrals.

* __In quantum mechanics__:
It can be constructed from the wave function by ** j**:= #
Im(*ψ** ∇*ψ*); The integral lines for this current are
analogous to trajectories.

@ __References__: Schumacher et al a1607
[generalization to finite-dimensional Hilbert spaces, open quantum
systems].

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

* __2012.03__:
Lightning strikes produce free neutrons, and we're not sure how [@ news at(12)mar].

**Proca Theory** > s.a. modified
electromagnetism / 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 *m*^{2}*
A*^{2} added to the Lagrangian,

*L* = − \(1\over4\)*F*_{ab}*
F*^{ab} + \(1\over2\)*m*^{2}
*A*_{a} *A*^{a}
+ *A*_{a} *j*^{
a} .

@ __General references__: Proca CRAS(36);
in Wentzel 49; Goldhaber & Nieto RMP(71)
[and photon mass limits]; in Gsponer & Hurni in(98)phy/05
[history]; Dvoeglazov CzJP(00)ht/97;
Fabbri AFLB(11)-a0908
[most general consistent theory].

@ __Einstein-Proca__: Dereli et al CQG(96)
[torsion and non-metricity]; Vollick gq/06;
> s.a. black-hole hair; black-hole
perturbations; einstein-cartan theory.

@ __Other variations, generalizations__: Kruglov IJMPA(06)
[sqrt version, including spin-1/2]; Escalante et al a1402
[5D, canonical analysis]; Heisenberg JCAP(14)-a1402;
Allys et al JCAP(16)-a1511;
De Felice et al a1602,
JCAP(16)-a1603
[fifth-force screening and cosmology]; Heisenberg et al PLB(16)-a1605
[with higher-order derivative interactions]; Allys et al a1609 [SU(2) Proca theory, or non-Abelian vector galileon]; Heisenberg a1705-proc [rev].

@ __Quantization__: Aldaya et al IJMPA(97)ht/96;
van Hees ht/03
[renormalizability]; Helesfai CQG(07)gq/06
[in lqg]; Zamani & Mostafazadeh JMP(09)-a0805;
Castineiras et al PRD(11)-a1108
[in a Rindler wedge].

@ __Quantum theory, in curved spacetime__: Furlani JMP(99)
[on a globally hyperbolic Lorentzian manifold, canonical]; Toms a1509
[with non-minimal terms, Faddeev-Jackiw approach to quantization]; Schambach a1709-MSc; Schambach & Sanders a1709 [and the zero mass limit].

@ __Phenomenology__: Brito et al PLB(16)-a1508
[self-gravitating BECs of Proca particles]; De Felice et al a1703 [observational constraints].

@ __Related topics__: Comay NCB(98); Kim
et al MPLA(98)ht
[symmetries]; Vytheeswaran IJMPA(98)
[as
gauge theory]; Zecca GRG(06)
[in
FLRW spacetime].

**Process** > s.a. Ontology
[process ontology].

* __Quantum process__:
The operation performed by a quantum processor that transforms a quantum
system's state into a different one.

@ __Physical process__: Spaans gq/05
[background independence]; Needham BJPS(13)
[processes as autonomous entities, thermodynamic perspective].

@ __Quantum process__: Poyatos et al PRL(97) [characterization];
D'Ariano & Lo Presti PRL(01)
[and quantum tomography]; Bendersky et al a1407,
Parke a1409 [implications of computer science principles];
Lee & Hoban PRS(16)-a1510 [tradeoff between quantum computation
and communication complexity]; Yadin et al PRX(16)
[operations which do not use coherence]; > s.a. creation
operator; quantum effects.

> __Specific physical
processes__: see diffusion; Drell-Yan
Process; Joule-Thomson Process;
Penrose Process; Transport.

> __Processes theory in physics__: see approaches
to quantum field theory [process algebra approach]; causal structures.

> __Types of
mathematical processes__: see markov
processes; random processes; statistical
geometry [point processes]; stochastic processes.

> __Specific
mathematical processes__: see Airy Process;
Lévy Process; Wiener Process.

> __Astrophysical
processes__: see Accretion Process.

**Products**

* __Special infinite
products__:

∏_{k = 2}^{∞}(1
− 1/*k*^{2}) =
\(1\over2\) [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];
Albert & Kiessling JSP(17)-a1610
[infinite trigonometric products and random walks on the real line].

@ __Generalized products__: van de Wetering a1803 [sequential product]; > s.a. Star Product; vectors [scalar, vector product].

**Programming** > see computation;
computer languages.

**Progressing Waves** > see types
of waves.

**Projectable Vector Field**

$ __Def__: A
differentiable vector field *v* is projectable by the map *f*
if *f* '(*v*) is differentiable.

**Projectile Motion** > s.a. kinematics
of special relativity.

@ __General references__: Klevgard a1501
[and XX century changes in physics].

@ __With air resistance__: Mohazzabi & Shea AJP(96)oct [with variation of atmospheric pressure];
Price & Romano AJP(98)feb [optimal launch angles];
Warburton & Wang AJP(04)nov;
Linthorne pw(06)jun [and soccer];
Goff & Carré AJP(09)nov
[soccer balls].

**Projection Mapping** > see bundles.

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

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

**Projective Relativity and Field Theory**

* __Projective relativity__:
Initially proposed by Fantappiè and subsequently developed by Arcidiacono.

@ __General references__: in Schmutzer ed-83
[projective relativity]; Schmutzer AN(05)ap
[projective unified field theory and 2-body system].

@ __And cosmology__: Licata & Chiatti IJTP(09)-a0808;
Benedetto IJTP(09)
[and varying speed of light].

**Projector, or Projection Operator**

$ __Def__: An operator
*P* on an inner product space which is self-adjoint and idempotent.

* __Projective methods__:
Used for systems of linear and non-linear algebraic equations and convex optimization.

@ __References__: Galántai 03;
Halliwell PLA(13)-a1207
[localized in a region of phase space].

**Prop** > see examples of categories.

**Propagator** > s.a. green
function [for differential operators]; feynman
propagator and green function [in quantum field theory].

* __In quantum mechanics__:
Can be calculated directly using the path-integral technique, or as
inverse Laplace transform of the Green function.

@ __In quantum mechanics__: Nardone AJP(93)mar [calculation];
Fulling & Güntürk AJP(03)jan [1D
particle in a box]; Kosut et al qp/06
[distance between propagators]; Moshinsky et al Sigma(07)-a0711
[from Green function]; Zanelli et al RPMP(08) [integral representations].

**Propensity** > see probability in physics.

**Proper Discontinuous Action of a Group** > see group action.

**Proper Time** > s.a. special-relativistic kinematics.

* __Idea__: The proper
time at a point along a timelike line in spacetime is the length of the
line from a reference initial point.

@ __References__: Wesson a1011
[adjustments from the possible existence of higher dimensions].

**Property** > s.a. Generic
Property; Physically Significant
Property; Stability.

$ __In mathematics__: A
property *P* defined for elements *x* of a set *X*
is an attribute that those elements may have or not have, i.e., a map *P*
: *X* → {0,1}.

$ __In physics__: A
property *P* is often an attribute that a physical system *s*
or theoretical model may have to varying degrees, i.e., a map *P*
: *S* → \(\mathbb R\) (sometimes \(\mathbb C\)); Important
examples are the values of observables, or the truth values of
propositions about the system.

* __Rem__: For the
purpose of discussing different types of properties, it is often
convenient to specify a topological space structure on *X* and
distinguish cases in which *P* behaves differently when
considering its values for elements in a neighborhood of a given *x*.

* __Terminology__: An
element *x* in *X* (or a subset *A* of *X*)
are said to have the property if *P*(*x*) = 1 (resp., *P*(*A*)
= {1}).

@ __References__: Hofmann et al a1605-proc
[of a quantum system, and observable effects].

> __Related topics__:
see measurement in quantum theory.

**Propositional Logic** > see logic.

**Proton** > see hadrons.

**Prout's Law** > see atomic
physics.

**Proximity-Force Approximation**
> s.a. casimir-effect examples.

* __Idea__: An
approximation method for the electrostatic interaction between two
perfectly conducting surfaces, used when the distance between them is much
smaller than the characteristic lengths associated to their shapes; The
electrostatic force is evaluated by first dividing each surface into a set
of small flat patches, and then adding up the forces due two opposite
pairs, approximated as pairs of parallel planes; It has been successfully
applied to contexts such as nuclear physics and Casimir-effect
calculations.

@ __References__: Fosco et al AP(12)-a1201
[improved approximation].

**Proximity Graphs** > see graph
types.

**Pseudoclassical Dynamical Systems**

* __Idea__: Models that
have classically anticommuting variables.

@ __References__: Allen et al a1509
[quantization].

**Pseudodifferential Operator** > see operator
theory.

**Pseudogroup** > s.a. differentiable
maps [local pseudogroup of transformations].

@ __In physics__: Woon ht/98
[intro and applications].

**Pseudomanifold** > see types
of
manifolds.

**Pseudometric Space** > see distance.

**Pseudorandomness** > see random
processes.

**Pseudosphere** > s.a. sphere.

* __Idea and history__:
A 2D surface with constant and negative Gaussian curvature; Discussed in
1868 by Eugenio Beltrami in terms of a disk on the plane, which is
isomorphic to the two-sheet hyperboloid in \(\mathbb R\)^{3}.

@ __References__: Bertotti et al gq/05-proc
[review,
geometry and physics].

**Pseudostationary Spacetime** > see types
of
spacetimes.

**Pseudosymmetric Spacetime** > see 3D
geometry.

**Pseudotensor** > see stress-energy
pseudotensor.

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

** ψ-Epistemic Quantum
Theory** > s.a. interpretations
of quantum theory [statistical interpretation]; quantum
foundations; types of
interpretations [type-II].

*

@

@

@

@

>

** ψ-Ontic
Quantum Theory** ("wave functions are real") > s.a. interpretations
of quantum theory [including PBR theorem]; types
of interpretations [type-I].

*

*

*

*

*

*

@

@

@

@

@

>

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

**PT Symmetry** > s.a. modified
quantum mechanics, statistical mechanical
systems [PT-symmetric]; Unitarity.

@ __General references__: Mostafazadeh PS(10)-a1008 [rev].

@ __Breaking__: Bender & Darg JMP(07) [in classical mechanics];
Ambichl et al PRX(13) [in scattering systems].

**Pullback Bundle** > see fiber bundle.

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

**Pure Sequence** > see exact sequence.

**Purity** > s.a. mixed
states; polarization.

* __Idea__: The
quantity *ζ* = tr *ρ*^{2},
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-fs [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 23 mar 2018