Anomalies in Quantum Theory| Anomalies in Quantum Theory |
In General > s.a. heat [heat kernel].
* Idea: The breakdown, upon
quantization of a theory, of conservation laws that hold classically; In
field theory, if \(j^a\) is a classically conserved current built from the
dynamical variables, \(\nabla_{\!a}\,j^a = 0\), we have an anomaly if the
corresponding quantum operator equation is not satisfied.
* In quantum mechanics: There
are at least two cases in quantum mechanics, the 2D δ-function interaction
and the 1/r 2 potential.
* And path integrals: In the
quantum path-integral formulation of a field theory, an anomaly arises when
the functional measure is not invariant under a symmetry transformation of
the Lagrangian.
* Consequences: They pose
consistency problems if they appear in quantized gauge theories.
References > s.a. early-universe
cosmology [baryogenesis]; regularization.
@ Intros, reviews: Jackiw in(88);
Bertlmann 96;
Zinn-Justin LNP(05)ht/02 [especially chiral];
Fujikawa & Suzuki 04 [and path integrals];
Adler ht/04-en;
Harvey ht/05-ln;
Bastianelli & van Nieuwenhuizen 06 [r CQG(07)];
Bilal a0802-ln.
@ General references: Aldaya et al ht/97-conf [algebraic vs topological, group quantization];
Fujikawa hl/00-proc,
IJMPA(01)ht/00 [and regularization];
Grigore a1011-conf [second-order anomalies and off-shell fields];
Monnier a1903-proc
[anomaly field theory and higher-dimensional field theory functor].
@ In quantum mechanics: Holstein AJP(93)feb;
Kirzhnits & Shpatakovskaya TMP(96)qp/99;
Coon & Holstein AJP(02)may-qp [1/r 2 potential];
Holstein AJP(14)jun [2D δ-function potential].
@ Consistency conditions:
Wess & Zumino PLB(71);
Becchi et al AP(76);
Grigore a1804-conf
[and absence of anomalies in SU(2) Yang-Mills theories].
@ Geometrical / topological view: Jackiw in(84);
Bardeen & White ed-85;
Catenacci & Pirola LMP(90);
Perrot CM(07)ht/06 [and non-commutative index theory];
Nikolov a0903,
a0907-conf [and cohomologies of configuration spaces];
Antoniadis & Savvidy EPJC(12)-a1205 [and topological invariants].
@ Hamiltonian view:
Nelson & Álvarez-Gaumé CMP(85);
Esteve PRD(02)ht;
Monnier CMP(15)-a1410 [anomalous field theories as relative field theories].
@ Related topics:
Dubois-Violette JGP(86);
Bowick & Rajeev NPB(88) [and complex geometry];
Kirzhnits & Shpatakovskaya TMP(96)qp/99 [singular potentials];
Balachandran & de Queiroz PRD(12)-a1108,
IJGMP(12)
[anomalous symmetries and mixed states with non-zero entropies];
Moss JPA(12)-a1201 [in the 'in-in', or closed-time path formulation of quantum field theory];
Duff a2003-in [hist];
> s.a. Nieh-Yan Form.
Gravitational Anomalies > s.a. 2D quantum gravity;
black-hole radiation; entanglement entropy.
@ General references:
Álvarez-Gaumé & Witten NPB(84);
Alvarez et al CMP(84) [and family's index theorem];
Witten CMP(85);
Hwang PRD(87);
Kim & Yoon PLB(88);
Brandt et al NPB(90);
Shimono PTP(90) [Kähler fermions and lattice gravity];
Estrada-Jiménez et al ht/04 [in non-commutative field theory];
Abe & Nakanishi PTP(06)ht/05 [criticism of Álvarez-Gaumé & Witten];
Salvio JPCS(09)-a0906 [role of Lorentz symmetry];
Landsteiner et al PRL(11)-a1103 [and transport phenomena].
@ Gravitational trace anomaly:
Pascual et al PRD(88);
Bilić et al PLA(07)-a0707 [and cosmology, effective cosmological constant].
@ Gravitational-Yang-Mills: Perrot JGP(01)mp/00 [topological interpretation];
Monnier CMP(14)-a1110 [self-dual field theory].
@ In 2D: Bertlmann & Kohlprath AP(01)ht/00 [Einstein & Weyl anomaly];
Habara et al a1206 [derivation of the Weyl anomaly from the Dirac sea];
Majhi GRG(13)-a1210 [and entropy].
@ In quantum gravity:
Rovelli PLB(87);
Surya & Vaidya NPB(98)ht/97.
Other Anomalies and Types of Theories > s.a. diffeomorphisms [in
canonical quantum gravity]; dualities [electromagnetic duality anomaly].
* In string theory:
Gauge and gravitational anomalies cancel in certain string theories.
@ In gauge theory: Grigore JPA(02),
a1804-conf [causal approach];
Golterman & Shamir PRD(10)-a1004 [supersymmetric gauge theories].
@ In string theory: Schwarz IJMPA(02)ht/01-conf [cancellation, review];
Bilal & Metzger NPB(03) [M-theory, cancellation].
@ Scale anomalies: Gomm et al PRD(86);
Visser PLB(95);
Lin & Ordóñez PRD(15)-a1508 [path-integral approach, finite temperature].
@ Axial anomalies: Ioffe IJMPA(06),
Jackiw IJMPA(10) [rev];
Kopper & Lévêque JMP(12)-a1112 [U(1) axial gauge anomaly with regularized path integrals];
Alfaro a2012 [in Very Special Relativity].
@ CPT anomalies: Klinkhamer NPB(00);
Klinkhamer & Schimmel NPB(02)ht.
@ Non-commutative gauge theory: Bonora et al PLB(00) [Yang-Mills];
Brandt et al JHEP(03)ht.
@ Related topics: Bär NPB(03) [higher SU(2) representations];
Kapustin & Thorngren PRL(14)-a1403 [3D discrete symmetries];
Dowker a1412 [functional determinant multiplicative anomaly];
Adler a1910 [in spin-3/2 theories].
Other types: see chiral and trace anomalies.
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