**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
the loop formulation].

**Übergravity**

* __Idea__: An approach to gravity in which one takes an ensemble average over all consistent models.

@ __References__: Khosravi PRD(16)-a1606 [proposal], a1703 [and the cosmological constant].

**Uhlmann's Geometric Phase** > see geometric phase.

**Ultradistributions** > see distributions.

**Ultralocality** > see field
theory; types of quantum field theories.

**Ultrametric Space / 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].

> __Online resources__:
see Wikipedia page.

**Ultrasound** > see acoustics.

**Ultrastatic Spacetime**

* __Idea__: A spacetime is
ultrastatic if it is static, and *g*_{tt} is a constant.

* __Result__: For any such
solution, Γ^{ t}_{ab} =
0, Γ^{ a}_{tb}
= 0, for all *a*, *b*, and
*R*^{a}_{bcd} =
0 with *t *in any position.

@ __References__: Sonego JMP(10)-a1004.

**Ultraviolet Completion of a Theory** > see under UV Completion.

**Umbral Calculus**

* __Idea__: A tool used to discretize continuum equations and systematically find solutions of difference equations; In quantum mechanics it can be viewed
as an abstract theory of the Heisenberg commutation relation [*p*, *q*] = 1.

@ __References__: Roman 84; Gessel math/01 [applications]; Dattoli et al JMP(08)
[and orthogonal polynomials]; López-Sendino et al a0805-proc [and quantum mechanics]; Curtright & Zachos FiP(13) [framework, and solitons].

> __Online resources__: see Mathworld page; Wikipedia page.

**Umbral Deformation** > see discrete geometry.

**Umbral Moonshine** > see finite groups.

**Uncertainty (Measurement, Statistical)** > see statistics and error analysis in physics; fluctuations [classical uncertainty relations].

**Uncertainty Principle / Relations in Quantum Theory** > s.a. deformed and modified uncertainty relations; phenomenology.

**Undecidability** > see Decidability.

**Understanding** > s.a. Explanation; Knowledge; philosophy of science; physics teaching [conceptual understanding].

* __Types__: There are many
levels of understanding (of a physical phenomenon), from the 0-th level knowledge
that the phenomenon is claimed to exist, to understanding that it is possible
(consistent with models), to understanding (a model of) how it occurs
[so is understanding something actually just understanding what it is like?];
It has both a psychological dimension and a truth connection.

@ __References__: Cat SHPMP(01)
[by illustration and metaphor]; de Regt PhSc(04)jan
[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]; Khalifa BJPS(13) [role of explanation]; Khalifa & Gadomski SHPMP(13).

**Unexpected Hanging Paradox**

@ __References__: Weiss Mind(52)
[non-classical logic]; Quine Mind(53); Sharpe
Mind(65);
Kiefer & Ellison Mind(65).

> __Online resources__: see MathWorld page; Wikipedia page.

**Unified Theories** > s.a. gravity and electromagnetism;
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.

**Unimodular Gravity / Relativity**

**Union** > see set theory [including one-point union].

**Unit** > see ring.

**Unitarity in Quantum Theory** > s.a. causality
violation; CPT symmetry; quantum
field theory in curved spacetime.

* __Idea__: A mathematical
expression for the conservation of probability in quantum theory.

* __Formalism__: Unitarity says that
exp(i*Ht*/\(\hbar\)) must be a unitary operator, and means that the amplitudes
\(\langle\)*q*_{2},
*t*_{2} | *q*_{1},
*t*_{1}\(\rangle\) satisfy

∑_{2}
\(\langle\) 2 | 1' \(\rangle\)\(\langle\) 2 | 1 \(\rangle\) = δ(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]; Mannheim PTRS-a0912 [PT
symmetry as necessary and sufficient condition]; Albrecht PRD(11)-a1012 [and weighted power-counting renormalizability].

@ __In quantum
field theory__: Anselmi PRD(16)-a1606 [perturbative unitarity]; > s.a. in curved spacetime.

@ __In quantum gravity__: Ralph PRA(07)-a0708 [qubit
+ exotic spacetime]; Hsu & Reeb CQG(08)-a0804 [and
allowed quantum states].

@ __Related topics__: van Wezel a1502-proc [possible fundamental violation].

**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 *ξ*(*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.

> __Online resources__: see Wikipedia page.

**Universal Coefficient Theorem** > s.a. geometric quantization.

* __Idea__: It essentially
says that if homology groups *H*(*X*,*G*) agree when the group *G* is \(\mathbb Z\), then they will
agree for all groups.

> __Online resources__:
see Wikipedia page.

**Universal Covering Group**

* __Idea__: The universal
covering space of a topological group.

* __Examples__: For SO(3),
or S^{3} as a topological group, SU(2), the
unit quaternions; For SO(*n*), the spinor groups; For the proper Lorentz group
P^{3} × \(\mathbb R\)^{3}, SL(2, \(\mathbb C\)).

> __Online resources__:
see Wikipedia page.

**Universal Covering Space**

$ __Def1__: Given a topological space *X*, its universal covering space
is a covering space (*E*, *π*: *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 T^{n} is \(\mathbb R\)^{n};
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, π_{1}(*X*)
≅ *G*, and *X* ≅ *E*/*G*.

> __Online resources__:
see Wikipedia page.

**Universal Horizon** > see horizons.

**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 the spatial posets paper.]

**Universal Metrics** > see general relativity solutions.

**Universality**

* __Idea__: A characteristic of properties that are common to systems of different types arising in different contexts or even disciplines; Examples are the ability of being described by the laws of thermodynamics, the fractal nature of shapes, the scaling behavior of many distributions, the values of critical exponents in phase transitions.

@ __General references__: Deift mp/06 [recent history of universality ideas in mathematical and physical systems]; Sfondrini PoS-a1210-ln [and renormalization group techniques].

@ __In statistical mechanics__: Delfino AP(15)-a1502-ln [in two dimensions].

@ __Examples__: Lieberman & Melott Pal(13)-a1206 [declining volatility]; > s.a. topological defects.

> __Online resources__: see Wikipedia page on universality in dynamical systems.

**Universe (cosmology)**

@ __References__: news smith(14)sep [what is the universe?].

**Universe (mathematics)**

$ __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.

* __Phenomenology__: Because the unparticles' mass varies, they can mediate a hypothetical fifth force that can be thought of as a version of magnetism that does not weaken as quickly with distance; They may be involved in long range spin-spin interactions in the Earth's crust, and in conduction in cuprate superconductors.

@ __General references__: Georgi PRL(07) [introduction]; Georgi PLB(07)
[peculiarities of propagator and interactions]; Goldberg & Nath PRL(08)-a0707 [ungravity
and modification of inverse square law]; Nakayama PRD(07)-a0707 [with
supersymmetry]; Lee a0710 [holographic
duals]; Grinstein et al PLB(08)-a0801 [comments];
Gaete & Spallucci PLB(08)-a0801 [effective
actions]; Nikolić MPLA(08)-a0801 [as
arbitrary-mass particle]; Schroer a0804 [and
infraparticles]; McDonald a0805 [interpretation
and cosmology bounds]; Gaete & Spallucci PLB(08)-a0807 [in
lower dimensions]; Georgi & Kats PRL(08)
[2D example]; Georgi IJMPA(10)-proc; Rahaman MPLA(14) [in 1+1 dimensions].

@ __Searches__:
news ns(11)may [possible signal at Fermilab and matter-antimatter asymmetry]; Hunter et al Sci(13)feb + news pw(13)feb, ns(13)feb [using Earth's crust].

@ __Interactions, gauge theories__: Licht a0801, a0801;
Ilderton PRD(09)-a0810;
Georgi & Kats JHEP(10) [self-interactions].

@ __And particle physics phenomenology__:
Luo & Zhu PLB(08)-a0704;
Kikuchi & Okada PLB(08)-a0707
[Higgs
particles]; Lenz PRD(07)
[*B*_{S}-*B*_{S}bar
mixing]; Aliev et al PLB(07)
[lepton flavor volation]; Cheung et al PRL(07),
IJMPA(09)
[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 PRD(08)-a0801 [at
photon collider]; Feng et al PRD(08)-a0801 [self-interactions];
Alan IJMPA(09) [two-gluon jet production]; Frassino et al a1311 [un-Casimir effect].

@ __And cosmology__: Davoudiasl PRL(07)-a0705;
McDonald JCAP(09)-a0710;
Kikuchi & Okada PLB(08)-a0711,
Gong & Chen EPJC(08)-a0803 [as
dark
matter]; Wei EPJC(09)-a0812 [relaxing
the cosmological constraints]; Dai et al PRD(09)-a0909 [as
quintessence]; Grzadkowski & Wudka PRD(09).

@ __And astrophysics__: Hannestad et al PRD(07)
[bounds from SN1987A]; Das PRD(07)
[supernova cooling]; Bertolami et al PRD(09)-a0905 [constraints
on ungravity].

@ __Other phenomenology__: Wondrak et al PLB(16)-a1603 [contribution to the H atom ground-state energy].

> __Related topics__: see black-hole
radiation; schwarzschild solution [un-gravity
corrections].

> __Online resources__:
see Wikipedia page.

**Unruh-DeWitt Detectors** > see Detectors.

**Unstable State, System** > see particle effects
and quantum-mechanical effects [decay]; quantum
systems; types of quantum
states; zeno effect.

**Ur-Element** > see Wikipedia page.

**Ur-Object** > s.a. origin of quantum
theory.

* __Idea__: A system described
by an element of a two-dimensional complex Hilbert space.

* __Question__:
Is there any difference between this and a qubit?

**Urbantke Metric**

* __Idea__: A metric on the space of two-forms over a (4D) vector space.

**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__: Janyška RPMP(06) [higher-order generalization].

**UV Completion of a Field Theory** > s.a. quantum field theory formalism [and classicalization].

@ __References__: Ho & Lin EPJC(11)-a1010 [of scalar electrodynamics]; Crowther & Linnemann a1705 [and the search for quantum gravity].

> __Specific theories__: see modified QED; theories
of
gravity.

main page – abbreviations – journals – comments – other
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send feedback and suggestions to bombelli at olemiss.edu – modified
21 may 2017