$ Def: An ultrametric is a metric (a distance) such that d(x, y)
max[d(x, z), d(z, y)],
for all x, y, z.@ And physics: Rammal, Toulouse & Virasoro RMP(86); Parisi & Ricci-Tersenghi JPA(00) [origin].
Topics, U
U(1) Problem
* Types: The fact that
in QCD there is an apparent U(1) symmetry
that is not realized in the real world; The violation can
be attributed to an anomaly in the regularization of the theory, which
in instantons can be seen to give rise
to interactions that explicitly break the symmetry.
@ References: 't Hooft PRP(86)
[and instantons]; Fort & Gambini ht/97 [in
loop formulation].
Ultradistributions > see distributions.
Ultralocality > see field theory; types of quantum field theories.
Ultrametric / Ultrametricity > s.a. spin
models.
$ Def: An ultrametric
is a metric (a distance) such that d(x, y)
max[d(x, z), d(z, y)],
for all x, y, z.
@ And physics: Rammal, Toulouse & Virasoro RMP(86); Parisi & Ricci-Tersenghi
JPA(00)
[origin].
Ultrastatic Spacetime
* Idea: A spacetime is
ultrastatic if it is static, and gtt is a constant.
* Result: For any such
solution, 
tab =
0, atb
= 0, for all a, b, and
Rabcd =
0 with t in any position.
Umbral Calculus
* Idea: Can be viewed
as an abstract theory of the Heisenberg commutation relation [p, q]
= 1.
@ References: Dattoli et al JMP(08)
[and orthogonal polynomials].
Umbral Deformation > see discrete geometry.
Uncertainty, Uncertainty Relations > s.a. modified forms.
Undecidability > see Decidability.
Understanding > s.a. Explanation;
philosophy of science.
* Types: There are many
levels of understanding (of a physical phenomenon), from the 0-th level knowing
that the phenomenon is claimed to exist, to understanding that it is possible
(consistent with models),
to understanding (a model of) how it occurs.
@ References: Cat SHPMP(01)
[by illustration and metaphor]; de Regt PhSc(04)
[non-objectivist, pragmatic
conception]; De Regt & Dieks Syn(05);
Grimm BJPS(06)
[understanding as a species of knowledge]; Collins SHPSA(07)
[mathematical physics and expertise].
Unexpected Hanging
@ References: Weiss Mind(52)
[non-classical logic]; Quine Mind(53); Sharpe
Mind(65);
Kiefer & Ellison Mind(65).
Unified Theories > s.a. GUTS; particle physics.
Uniform Continuity > see uniformity.
Uniform Cover > see cover.
Uniform Equivalence
$ Def: A function f :
X → Y which is one-to-one, onto, and uniformly continuous,
together with its inverse.
Union > see set theory [including one-point union].
Unit > see ring.
Unitarity in Quantum Theory > s.a. causality
violation; CPT.
* Idea: Mathematical
expression for the conservation of probability in quantum theory.
* Formalism: Unitarity
says
that
exp(iHt/
)
must be a unitary operator, and means that the amplitudes 
q2, t2 | q1, t1
satisfy 
2 2|1'* 2|1 = 
(1',1);
Whether or not system is unitary can also be seen from the way it, if perturbed,
relaxes
back to equilibrium; A unitary theory in finite volume has correlation functions
for the perturbations which are quasi-periodic functions of time and in general
show Poincaré recurrences.
* Remark: One usually
assumes that the
Hamiltonian
operator must be Hermitian, H = H
,
where denotes the usual
Dirac Hermitian conjugation – transpose
and complex conjugate; However, the requirement of Hermiticity, which is often
stated as an axiom of quantum mechanics, may be replaced by the requirement of
spacetime
reflection symmetry (PT symmetry) without losing any of the essential physical
features of quantum mechanics; > s.a. modified quantum
mechanics.
@ General references: in Streater & Wightman 64; Hulpke et al FP(06)
[as preservation of entropy
and entanglement].
@ In quantum gravity: Ralph PRA(07)-a0708 [qubit
+ exotic spacetime]; Hsu & Reeb a0804 [and allowed quantum states].
Unitarity Triangle > see CP violation.
Unitary Group > see examples of lie groups.
Universal Bundle
* Idea: The (n–k–1)-universal
bundle with (compact) fiber G, called 
(n–k–1, G),
is a bundle with total space the Stiefel manifold
O(n)/O(n–k)
and base space O(n)/(G × O(n–k)).
* Application: It is
used to classify principal fiber bundles with group G over
any base space B, with dimension B < n–k–1.
Universal Coefficient Theorem
* Idea: It essentially
says that if homology groups H(X,G) agree when the group G is Z, then they will
agree for all groups.
Universal Covering Group
* Idea: The universal
covering space of a topological group.
* Examples: For SO(3),
or S3 as a topological group, SU(2), the
unit quaternions; For SO(n),
the spinor
groups; For the proper Lorentz group
P3 × R3,
SL(2, C).
Universal Covering Space
$ Def1: Given a topological
space X, its universal covering space
is a covering space (E, 
:
E → X), with E simply connected (unique, up
to equivalence).
$ Def2: The set of equivalence
classes (x, c), x 
X
and c a path from x to
some fixed p X,
under the equivalence relation of homotopy.
* Examples: The universal
covering space of Tn is Rn;
For SO(3), it is SU(2).
* Conditions: A necessary
condition for the existence of the universal covering space is that X be
(connected, locally pathwise connected and) semi-locally simply connected.
* Result: Every connected manifold has a universal covering space.
* Result: If G is the group of covering transformations, 1(X)
G, and X E/G.
Universal Map
* Idea: Given a category A,
a map f : X → Y with Y A is
universal among maps into an object in A, if for any other map g: X → Z with Z A there
is a
unique k: Y →
Z with g = kf. [Extrapolated from
partial
definition by Rafael in spatial posets paper.]
Universality
@ References: Deift mp/06 [mathematical and physical systems].
Universe
$ Souriau: A space E
on which a recueil R acts transitively.
Unparticles > s.a. effective
quantum field theory.
* Idea: Confined states
of a scale-invariant theory with an infrared fixed point at high energy, which
have continuous masses because of scale invariance,
coupled with the standard
model matter via a higher-dimensional
operator suppressed by a high cutoff scale.
@ General references: Georgi PLB(07)
[peculiarities of propagator and interactions]; Goldberg & Nath PRL(08)-a0707 [ungravity
and modification of inverse square law]; Nakayama a0707 [with
supersymmetry]; Lee a0710 [holographic
duals]; Licht a0801,
a0801 [and gauge
fields]; Grinstein et al PLB(08)-a0801 [comments];
Gaete & Spallucci PLB(08)-a0801 [effective
actions]; Nikolic a0801 [as
arbitrary-mass particle]; Schroer a0804 [and
infraparticles]; McDonald a0805 [interpretation
and cosmology bounds]; Gaete & Spallucci a0807 [in
lower dimensions].
@ And particle physics phenomenology:
Luo & Zhu PLB(08)-a0704;
Kikuchi & Okada PLB(08)-a0707
[Higgs
phenomenology], a0711 [as
CDM candidate]; Lenz PRD(07)
[BS-BSbar
mixing]; Aliev et al PLB(07)
[lepton flavor volation]; Cheung et al PRL(07)
[collider signals]; Mureika PLB(08)-a0712
[enhanced black hole formation at LHC];
Choudhury
et al PLB(08)
[muon decay]; Cacciapaglia et al JHEP(08)
[colored unparticles]; Kikuchi et al a0801 [at
photon collider]; Feng et al a0801 [self-interactions].
@ And cosmology: Davoudiasl PRL(07)-a0705;
McDonald a0710;
Kikuchi & Okada PLB-a0711,
Gong & Chen a0803 [dark
matter].
@ And astrophysics: Hannestad et al PRD(07)
[bounds from SN1987A]; Das PRD(07) [supernova cooling].
Unruh Effect / Vacuum > see quantum field theory effects in curved spacetime.
Unstable State, System > see particles [decay]; quantum states; quantum systems; zeno effect.
Urysohn Lemma
$ Def: A topological space
X is normal iff for each pair of disjoint closed sets F, G 
X there
is a Urysohn function for F and G, i.e., a continuous function
whose value is 0 on F and 1 on G.
Utiyama Theorem
@ References: Janyska RPMP(06) [higher-order generalization].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 jul 2008