Exploring Fermions in Odd Space-Time Dimensions: A Review of Theoretical Frameworks
This compilation reviews significant papers that explore the theory of fermions in odd-dimensional spaces, highlighting the complexities of chiral symmetry and parity in these contexts. We discuss the findings of Bashir and de Jesus Anguiano, Burden and Burkitt, Bietenholz and Nishimura, as well as Kikukawa and Neuberger, focusing on the implications of using various representations of the Clifford algebra. The interactions of different spinor components and the resultant symmetries are analyzed, emphasizing the nuances in formulating fermionic fields within odd dimensions.
Exploring Fermions in Odd Space-Time Dimensions: A Review of Theoretical Frameworks
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Presentation Transcript
Some reference papers A. Bashir, Ma. de Jesus Anguiano Galicia Fermions in odd space-time dimensions: back to basics Few-Body Systems 0, 1–8 (2008),hep-ph/0502089v1 C. Burden, A. N. Burkitt Lattice Fermions in Odd Dimensions Europhys. Lett., 3 (5), pp. 545-552 (1987) W. Bietenholz, J, Nishimura Ginsparg-Wilson fermions in odd dimensions JHEP07(2001)015, hep-lat/0012020v2 Y. Kikukawa, H. Neuberger Overlap in odd dimensions Nucl. Phys. B, 513-3 (1998) 735, hep-lat/9707016v1 Fermions in odd dimensions
In odd dimensions the Clifford algebra does not admit a , so chiral symmetry cannot be defined Two inequivalent representation of the algebra exist Parity anomaly C. Burden, A. N. Burkitt Lattice Fermions in Odd Dimensions Europhys. Lett., 3 (5), pp. 545-552 (1987) Starting from staggered formulation, analogous to the 4d case, shows that its continuum equivalent are flavors of components spinors. Parity symmetry is recovered by using both representations A set of chiral symmetries can be defined.
A. Bashir, Ma. de Jesus Anguiano Galicia Fermions in odd space-time dimensions: back to basics Few-Body Systems 0, 1–8 (2008), hep-ph/0502089v1 They show that it is possible to write a formulation of odd dimensional fermions using the fundamental representation of the Clifford algebra. The massless action has more symmetry than the massive case. “We demonstrate that by taking into account both the inequivalent fundamental representations of the gamma matrices for odd number of space-time dimensions, the resulting Lagrangian is not only parity invariant but also that we can write out chiral transformations within the two-component description of the fermion spinors which mix the fields belonging to the different inequivalent representations.”
