**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_{n
= 2}^{∞}
e^{cn}
for non-commuting *a* and *b*.

@ __References__:
Sridhar & Jagannathan mp/02 [*q*-analog];
Scholz & Weyrauch JMP(06)mp
[calculation of *c*_{n}];
Casas et al CPC(12)-a1204 [efficient computation];
Wang et al FMCh(19)-a1903 [multivariable form].

**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].

@ __Related topics__:
news sn(18)jun [bees get that zero is less than 1].

**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 properties
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;
Savvidy & Savvidy a1809 [interpretation].

@ __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.

**Zilch Tensor**

* __Idea__: A conserved Lorentz
covariant tensor *Z*_{abc}
for the electromagnetic field, representing a collection of conserved
currents parametrized by its first two indices; Its 000 component with
respect to an observer's 4-velocity is the optical chirality.

@ __References__:
Aghapour et al a1904.

**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 PRD(17)-a1610,
Zahiri Abyaneh & Farhoudi IJMPA(19)-a1903 [in non-commutative geometry].

@ __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, 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 27 oct 2019