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].

blue bullet Other types: see chiral and trace anomalies.

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 1 jan 2021