Topics, Z

Z Particle > see electroweak theory.

Zassenhaus Formula > s.a. Baker-Campbell-Hausdorff Formula; path integrals.
* Idea: An expression for the product ea+b = ea eb Πn = 2 ecn for non-commuting a and b.
@ References: Sridhar & Jagannathan mp/02 [q-analog]; Scholz & Weyrauch JMP(06)mp [calculation of cn]; Casas et al CPC(12)-a1204 [efficient computation].

Zeeman Effect > see atomic physics.

Zeeman's Theorem
* Idea: Global causality implies the Lorentz group.
@ Simple proofs: Briginshaw IJTP(80); Kim a1311.

Zeeman Topology > see spacetime topology.

Zeno Effect (In quantum theory; including anti-Zeno effect)

Zeno's Paradox > see logic.

Zermelo's Axiom of Choice > see choice.

Zero
* History: Introduced by the Babylonians as a placeholder for a blank tablet entry; Explored by Indian and Muslim cultures, which did not have prejudices against representing "nothing"; Incorporated in Western thought using limits and the physics of empty space.
@ History: Rotman 93 [semiotics]; Kaplan 99, Seife 00 [I].

Zero Divisor > see ring.

Zero Modes > see operators.

Zero-Point Energy, Fluctuations > see vacuum; modified formulations of QED [without second quantization, zero-point radiation field].

Zeta Function > s.a. mathematical conjectures; regularization; series.
$ Def: The most common one is the Riemann zeta function,

ζ(x):= n = 1 nx ,   or   ζ(x) = Γ(x)–1 0 du ux–1/(eu–1) .

* History: 1974, H Montgomery found the first indication of a connection between the distribution of the zeros of the Riemann ζ-function and the distribution of the eigenvalues of random matrices; 1981, Numerical calculations by A Odlyzko of statistics of the zeros led to graphs illustrating the connection that Montgomery predicted.
* Properties: The summation in the definition is divergent for Re(x) ≤ 1, and ζ defined by analytic continuation; It has a simple pole at x = 1, and no other singularities; It encodes information about statistical propertis of the distribution of primes, and is the centerpiece of unsolved problems in number theory.
* Hilbert-Polya conjecture: The imaginary parts of the zeros of the Riemann zeta function are eigenvalues of a quantum Hamiltonian.
* And physics: There are several connections, including a surprising correspondence with freezing in disordered systems like glasses.
@ General references: Titchmarsh 30; Elizalde JPA(97)ht/96 [singularity structure]; Katz & Sarnak BAMS(99) [zeros and symmetry]; Bogomolny et al JPA(06) [spacing distribution of zeros]; Tyagi & Holm mp/07 [new integral representation for 0 < Re(x) < 1]; Kuznetsov PRS(07) [generalization of Riemann-Siegel asymptotic formula]; Kirsten et al JPA(08)-a0812 [meromorphic properties]; Kirsten a1005-in [introduction and use in the Casimir effect and Bose-Einstein condensation]; Coffey a1203 [series representation]; Milgram JoM(13)-a1208 [integral and series representations]; Arakawa et al 14 [and Bernoulli numbers].
@ Special properties: Berry PRS(95) [on the critical line x = \(1\over2\)+ i t]; Fujii & Suzuki IJMCS-a0805 [ζ(2n+1) in terms of {ζ(2k) | k ≥ 1}].
@ Zeros of the zeta function and quantum mechanics: Sierra NJP(08)-a0712; Tanaka JPSJ(11)-a1010 [correlation structure of 1D Fermi gas]; Srednicki PRL(11)-a1105; Bender et al a1608; Mueller a1704.
@ Other physics connections: Fyodorov et al PRL(12) [freezing transitions in glasses]; Elizalde IJMPA(12)-a1205 [operator zeta functions and physical applications].
@ Hurwitz zeta function: Espinosa & Moll TRJ(02)m.CA/00 [integrals]; Coffey a1106 [series representation]; > s.a. Stieltjes Constants; thermodynamical systems.
@ Other zeta functions: Cacciatori a0902 [polynomial]; Chaudhry et al a1004 [and extended Fermi-Dirac and Bose-Einstein functions]; > s.a. Dirichlet Eta Function.
> Online resources: see Wikipedia page.

Zeta Matrix > see types of posets.

Zipf's Law
* Idea: Given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table: The most frequent word will occur twice as often as the second most frequent word, three times as often as the third most frequent word, etc; The scaling applies to all languages, and has been interpreted in terms of a least-effort principle–minimization of the efforts of both hearer and speaker in a conversation leads to a Zipf-like distribution law.
@ References: Bernhardsson et al NJP(09) [word-frequency distribution, etc]; Corominas-Murtra et al PRE(11)-a1008 [emergence in the evolution of communication]; Baek et al NJP(11); Zhang & Sornette PhyA(11) [empirical test and mechanism]; Visser NJP(13)-a1212 [and maximum entropy].
> Online resources: see MathWorld page; Wikipedia page.

Zitterbewegung > s.a. dirac equation; dirac quantum field theory / geometric phase; spinning particles.
* Idea: An oscillatory or "trembling" motion with frequency 2H/\(\hbar\), superimposed on the average translational motion, obtained for a Dirac particle when using the usual position operator x (as opposed to the Foldy-Wouthuysen representation); The term was coined by Schrödinger.
@ General references: Schrödinger SBAW(30); Lock AJP(84)mar; Hestenes FP(90) [and interpretation of quantum mechanics], FP(93) [modeling]; Ghose et al PLA(03)qp [not found for bosons]; Bolte & Glaser JPA(04)qp [and semiclassical observables]; Krekora et al PRL(04) [no effect for electrons]; Brovetto et al qp/05 [electron size and mass]; Sidharth IJTP(09); Singh & Mobed CQG(09)-a0903 [effect of spacetime curvature]; Dávid & Cserti PRB(10)-a0909 [general theory]; O'Connell MPLA(11)-a1103 [not observable]; Knuth AIP(15)-a1411 [statistical considerations and the relativistic addition of velocities]; Eckstein et al a1610 [non-commutative geometry model].
@ Special situations: Rusin & Zawadzki a1003, PRD(10)-a1008 [in a magnetic field, simulation by trapped ions]; Zawadzki & Rusin PLA(10) [in crystalline solids]; Zawadzki & Rusin JPCM(11)-a1101 [in semiconductors, rev]; Wang et al a1105 [significance for Hawking radiation]; Tarakanov JTP-a1201 [as a classical phenomenon, for particles with internal degrees of freedom]; Tenev & Vitanov PRA(13)-a1210 [neutral relativistic particles in static longitudinal fields]; Qu et al PRA(13)-a1301, LeBlanc NJP(13) [in a Bose-Einstein condensate, observation]; Weberszpil & Helayël-Neto JAP-a1406 [in a coarse-grained medium]; Kobakhidze et al PLB-a1508 [in non-inertial frames and curved spacetimes].
@ And spacetime algebra: Dreisigmeyer et al FPL(03)qp/01; Hestenes FP(10)-a0802 [self-contained dynamical model of the electron].
@ For photons: Kobe PLA(99); Wang et al PRA(09)-a0905, ChPB(12)-a1105 [and gravitational vacuum fluctuations].
@ Simulations: Rusin & Zawadzki PRA(12)-a1205 [spin-zero particles, and simulation by classical fields]; Ahrens et al NJP(15)-a1505 [in metamaterials].
> Online resources: see Wikipedia page.

Zoll, Zollfrei Metric > see types of metrics.

Zoo Hypothesis > see civilizations.

Zorn's Lemma > see axiom of choice.

Zweig Rule
* Idea: The phenomenological rule according to which strong processes in which the final states can only be reached through quark-antiquark annihilation are suppressed.
* Example: The φ (~ \(s\bar s\)) decay into 3π is suppressed with respect to decay into 2K.
@ Proposals: Zweig pr(64); Okubo PL(63); Iizuka PTP(66), PTP(66).


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