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];
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∞ 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 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(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 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 Zabc
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 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];
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];
Davis a2006
[and internal structure of the electron];
Silenko a2008 [massless particles].
@ For photons:
Kobe PLA(99);
Wang et al PRA(09)-a0905,
ChPB(12)-a1105 [and gravitational vacuum fluctuations].
@ For bosons: Ghose et al PLA(03)qp [not found];
Silenko a1912 [not observable].
@ 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].
@ 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 3π is suppressed with respect to decay
into 2K.
@ Proposals: Zweig pr(64);
Okubo PL(63);
Iizuka PTP(66),
PTP(66).
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 15 aug 2020