Chiral Anomaly in Contorted Spacetimes

Abstract

The Dirac equation in Riemann-Cartan spacetimes with torsion is reconsidered. As is well-known, only the axial covector torsion $A$, a one-form, couples to massive Dirac fields. Using diagrammatic techniques, we show that besides the familiar Riemannian term only the Pontrjagin type four-form $dA\wedge dA$ does arise additionally in the chiral anomaly, but not the Nieh-Yan term $d ^* A$, as has been claimed recently. Implications for cosmic strings in Einstein-Cartan theory as well as for Ashtekar's canonical approach to quantum gravity are discussed.Comment: 17 pages, REVTE

Comment on “Topological invariants, instantons, and the chiral anomaly on spaces with torsion”

“…We have shown in detail in Ref. [17] that the NY term (2.2) vanishes identically for this example of a spinning cosmic string exhibiting a torsion line defect. The instantons of Ref.…”

“…which establishes a link to the NY four form [18], but only for massive fields [17]. However, if we restore chiral invariance for the Dirac fields in the limit m → 0, we find within the dynamical framework of ECD theory that the NY four-form tends to zero "on shell", i.e.…”

Conservation laws in the teleparallel theory of gravity

“…Among various successful applications of the teleparallel approach, one should mention a pure tensorial proof of the positivity of energy in GR [20], a transparent introduction of Ashtekar's complex variables [21], and a formulation of a five-dimensional teleparallel equivalent of the Kaluza-Klein theory [22]. Although quantum properties of PGT are in general not so attractive [23,24], the related behavior in the specific case of the teleparallel theory might be better [25,26], and should be further explored.…”

The topological structure of Nieh-Yan form and the chiral anomaly in spaces with torsion

“…Though it has been under investigation for more than two thirds of a century, there is no general agreement on its mathematical formulation nor on its physical significance. Vast funds of work on torsion had been done by many physicists (see [1][2][3][4][5][6][7][8][9][10][11] for examples) since it was proposed by E. Cartan [12] in 1920s.…”

“…This component of the torsion tensor is the one that couples to the spin 1/2 fields [7]. Recently, there have been some discussions on the question of further contributions to the Chiral anomaly in the presence of space-time with torsion [5][6][7][8][9][10][11]. It is shown the Nieh-Yan form does contribute to the chiral anomaly for massive field.…”