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].

* 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 gtt is a constant.
* Result: For any such solution, Γ tab = 0, Γ atb = 0, for all a, b, and Rabcd = 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 : XY which is one-to-one, onto, and uniformly continuous, together with its inverse.

Uniformity / Uniform Space

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(iHt/\(\hbar\)) must be a unitary operator, and means that the amplitudes \(\langle\)q2, t2 | q1, t1\(\rangle\) 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.

Units of Measurement

Universal Bundle
* Idea: The (nk–1)-universal bundle with (compact) fiber G, called ξ(nk–1, G), is a bundle with total space the Stiefel manifold O(n)/O(nk) and base space O(n)/(G × O(nk)).
* Application: It is used to classify principal fiber bundles with group G over any base space B, with dimension B < nk–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 S3 as a topological group, SU(2), the unit quaternions; For SO(n), the spinor groups; For the proper Lorentz group P3 × \(\mathbb R\)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: EX), with E simply connected (unique, up to equivalence).
$ Def2: The set of equivalence classes (x, c), xX and c a path from x to some fixed pX, under the equivalence relation of homotopy.
* Examples: The universal covering space of Tn 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, π1(X) ≅ G, and XE/G.
> Online resources: see Wikipedia page.

Universal Horizon > see horizons.

Universal Map
* Idea: Given a category A, a map f : XY with YA is universal among maps into an object in A, if for any other map g: XZ with ZA there is a unique k: YZ with g = kf. [Extrapolated from partial definition by Rafael in the spatial posets paper.]

Universal Metrics > see general relativity solutions.

* 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) [BS-BSbar 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 Effect

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, GX 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.

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