We analyze the phenomenon of fermion pairing into an effective boson associated with anomalies and the anomalous commutators of currents, bilinear in the fermion fields. In two spacetime dimensions the chiral bosonization of the Schwinger model is determined by the chiral current anomaly of massless Dirac fermions. A similar bosonized description applies to the 2D conformal trace anomaly of the fermion stress-energy tensor. For both the chiral and conformal anomalies, correlation functions involving anomalous currents, j μ5 or T μν of massless fermions exhibit a massless boson 1/k 2 pole, and the associated spectral functions obey a UV finite sum rule, becoming δ-functions in the massless limit. In both cases the corresponding effective action of the anomaly is non-local, but may be expressed in a local form by the introduction of a new bosonic field, which becomes a bona fide propagating quantum field in its own right. In both cases this is expressed in Fock space by the anomalous Schwinger commutators of currents becoming the canonical commutation relations of the corresponding boson. The boson has a Fock space operator realization as a coherent superposition of massless fermion pairs, which saturates the intermediate state sums in quantum correlation functions of fermion currents. The Casimir energy of fermions on a finite spatial interval [0, L] can also be described as a coherent scalar condensation of pairs, and the one-loop correlation function of any number n of fermion stress-energy tensors 〈TT . . . T 〉 may be expressed as a combinatoric sum of n!/2 linear tree diagrams of the scalar boson.
A.J. Leggett, Quantum liquids: Bose condensation and Cooper pairing in condensed-matter systems, Oxford graduate texts in mathematics, Oxford University Press, Oxford U.K. (2006).
P.C.W. Davies, S.A. Fulling and W.G. Unruh, Energy momentum tensor near an evaporating black hole, Phys. Rev. D 13 (1976) 2720 [INSPIRE].
L.S. Brown, Stress tensor trace anomaly in a gravitational metric: scalar fields, Phys. Rev. D 15 (1977) 1469 [INSPIRE].
N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge Monogr. Math. Phys., Cambridge Univ. Press, Cambridge U.K. (1982).
J.S. Schwinger, Gauge invariance and mass, Phys. Rev. 125 (1962) 397 [INSPIRE].
J.S. Schwinger, Gauge invariance and mass. 2, Phys. Rev. 128 (1962) 2425 [INSPIRE].
L.S. Brown, Gauge invariance and mass in a two-dimensional model, Nuovo Cim. 29 (1963) 617.
J.H. Lowenstein and J.A. Swieca, Quantum electrodynamics in two-dimensions, Annals Phys. 68 (1971) 172 [INSPIRE].
A. Casher, J.B. Kogut and L. Susskind, Vacuum polarization and the absence of free quarks, Phys. Rev. D 10 (1974) 732 [INSPIRE].
M.B. Halpern, Equivalent-boson method and free currents in two-dimensional gauge theories, Phys. Rev. D 13 (1976) 337 [INSPIRE].
N.S. Manton, The Schwinger model and its axial anomaly, Annals Phys. 159 (1985) 220 [INSPIRE].
D. Wolf and J. Zittartz, Physics of the Schwinger model, Z. Phys. B 59 (1985) 117.
J.E. Hetrick and Y. Hosotani, QED on a circle, Phys. Rev. D 38 (1988) 2621 [INSPIRE].
R. Link, Eigenstates of the Schwinger model Hamiltonian, Phys. Rev. D 42 (1990) 2103 [INSPIRE].
A.V. Smilga, On the fermion condensate in Schwinger model, Phys. Lett. B 278 (1992) 371 [INSPIRE].
A.D. Dolgov and V.I. Zakharov, On conservation of the axial current in massless electrodynamics, Nucl. Phys. B 27 (1971) 525 [INSPIRE].
J. Horejsi, Ultraviolet and infrared aspects of the axial anomaly II, Czech. J. Phys. 42 (1992) 345 [INSPIRE].
J.S. Schwinger, Field theory commutators, Phys. Rev. Lett. 3 (1959) 296 [INSPIRE].
A.M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].
B. Klaiber, The Thirring model in Boulder 1967, Lectures in theoretical physics vol. Xa — Quantum Theory and Statistical Physics, New York U.S.A. (1968), pg. 141 [INSPIRE].
S.R. Coleman, The quantum sine-Gordon equation as the massive Thirring model, Phys. Rev. D 11 (1975) 2088 [INSPIRE].
S.R. Coleman, R. Jackiw and L. Susskind, Charge shielding and quark confinement in the massive Schwinger model, Annals Phys. 93 (1975) 267 [INSPIRE].
E. Witten, Instantons, the quark model and the 1/n expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].
G. Veneziano, U(1) without instantons, Nucl. Phys. B 159 (1979) 213 [INSPIRE].
K. Johnson, γ5 invariance, Phys. Lett. 5 (1963) 253 [INSPIRE].
M. Nakahara, Geometry, topology and physics, second edition, Graduate student series in physics, Institute of Physics Publishing, Bristol U.K. (2003) [INSPIRE].
R. Jackiw, Topological investigations of quantized gauge theories, in Relativity, groups and topology, vol. II, B. DeWitt and R. Stora eds., North-Holland, Amsterdam The Netherlands (1983) [INSPIRE].
K. Fujikawa, Path integral measure for gauge invariant fermion theories, Phys. Rev. Lett. 42 (1979) 1195 [INSPIRE].
K. Fujikawa, Path integral for gauge theories with fermions, Phys. Rev. D 21 (1980) 2848 [Erratum ibid. D 22 (1980) 1499] [INSPIRE].
R.A. Bertlmann, Anomalies in quantum field theory, International series of monographs on physics 91, Clarendon, Oxford U.K. (1996) [INSPIRE].
S.R. Coleman, More about the massive Schwinger model, Annals Phys. 101 (1976) 239 [INSPIRE].
E.C.G. Stueckelberg, Interaction forces in electrodynamics and in the field theory of nuclear forces, Helv. Phys. Acta 11 (1938) 299 [INSPIRE].
A. Aurilia, Y. Takahashi and P.K. Townsend, The U(1) problem and the Higgs mechanism in two-dimensions and four-dimensions, Phys. Lett. B 95 (1980) 265 [INSPIRE].
C. Adam, R.A. Bertlmann and P. Hofer, Overview on the anomaly and Schwinger term in two-dimensional QED, Riv. Nuovo Cim. 16N8 (1993) 1 [INSPIRE].
D. Wolf and J. Zittartz, Bosons and fermions in one space dimension, Z. Phys. B 51 (1983) 65.
S.R. Coleman, D.J. Gross and R. Jackiw, Fermion avatars of the Sugawara model, Phys. Rev. 180 (1969) 1359 [INSPIRE].
M. Tomiya, The Schwinger terms and the gravitational anomaly, Phys. Lett. B 167 (1986) 411 [INSPIRE].
M. Ebner, R. Heid and G. Lopes Cardoso, Gravitational anomalies and Schwinger terms, Z. Phys. C 37 (1987) 85 [INSPIRE].
P. Goddard and D.I. Olive, Kac-Moody and Virasoro algebras in relation to quantum physics, Int. J. Mod. Phys. A 1 (1986) 303 [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer, New York U.S.A. (1997).
C. Corianò, L. Delle Rose, E. Mottola and M. Serino, Graviton vertices and the mapping of anomalous correlators to momentum space for a general conformal field theory, JHEP 08 (2012) 147 [arXiv:1203.1339] [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
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ArXiv ePrint: 1407.8523
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Blaschke, D.N., Carballo-Rubio, R. & Mottola, E. Fermion pairing and the scalar boson of the 2D conformal anomaly. J. High Energ. Phys. 2014, 153 (2014). https://doi.org/10.1007/JHEP12(2014)153
- 2D Gravity
- Anomalies in Field and String Theories
- Field Theories in Lower Dimensions