**Topics, Z**

** Z Particle** > see electroweak
theory.

**Zassenhaus Formula** > s.a. Baker-Campbell-Hausdorff Formula; path
integrals.

* __Idea__: An expression
for the product e^{a+b} = e^{a} e* ^{b}* Π

@

**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}^{∞}* n*^{–x}
, or *ζ*(*x*)
= Γ(*x*)^{–1} ∫_{0}^{∞} d*u* *u ^{x}*

* __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*(2*n*+1)
in terms of {*ζ*(2*k*) | *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 PRL(17)-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 2*H*/\(\hbar\),
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(16)-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 3*π* is
suppressed with respect to decay into 2*K*.

@ __Proposals__: Zweig pr(64); Okubo PL(63);
Iizuka PTP(66),
PTP(66).

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send feedback and suggestions to bombelli at olemiss.edu – modified
17 dec 2017