nLab QFT on non-commutative spacetime

Idea

The idea of generalizing quantum field theory from ordinary spacetimes to spacetimes in noncommutative geometry.

References

General

Survey:

• Richard Szabo, Quantum Field Theory on Noncommutative Spaces (arXiv:hep-th/0109162)

Quantum Hall effect via non-commutative geometry

Discussion of the integer quantum Hall effect via a Brillouin torus with noncommutative geometry and using the Connes-Chern character:

• J. Bellissard, A. van Elst, H. Schulz-Baldes: The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35 5373 (1994) [doi:10.1063/1.530758, cond-mat/9411052]

Generalization of BvESB94 to the fractional quantum Hall effect:

• Matilde Marcolli, Varghese Mathai: Towards the fractional quantum Hall effect: a noncommutative geometry perspective, in Noncommutative Geometry and Number Theory, Vieweg (2006) [doi:10.1007/978-3-8348-0352-8_12, arXiv:cond-mat/0502356]

See also exposition in:

• Vincent Pasquier: Quantum Hall Effect and Non-commutative Geometry, in: Quantum Spaces, Progress in Mathematical Physics 53, Birkhäuser (2007) [doi:10.1007/978-3-7643-8522-4_1, pdf]

Discussion of the fractional quantum Hall effect via abelian but noncommutative (matrix model-)Chern-Simons theory:

• Leonard Susskind: The Quantum Hall Fluid and Non-Commutative Chern Simons Theory [arXiv:hep-th/0101029]

• Simeon Hellerman, Leonard Susskind: Realizing the Quantum Hall System in String Theory [arXiv:hep-th/0107200]

  (relating this to M5-branes via the BFSS matrix model)

• Alexios P. Polychronakos: Quantum Hall states as matrix Chern-Simons theory, JHEP 0104:011 (2001) [doi:10.1088/1126-6708/2001/04/011, arXiv:hep-th/0103013]

• Simeon Hellerman, Mark Van Raamsdonk: Quantum Hall Physics = Noncommutative Field Theory, JHEP 0110:039 (2001) [doi:10.1088/1126-6708/2001/10/039, arXiv:hep-th/0103179]

• Eduardo Fradkin, Vishnu Jejjala, Robert G. Leigh: Non-commutative Chern-Simons for the Quantum Hall System and Duality, Nucl. Phys. B 642 (2002) 483-500 [doi:10.1016/S0550-3213(02)00616-8, arXiv:cond-mat/0205653]

• Andrea Cappelli, Ivan D. Rodriguez: Matrix Effective Theories of the Fractional Quantum Hall effect, J. Phys. A 42 (2009) 304006 [doi:10.1088/1751-8113/42/30/304006, arXiv:0902.0765]

• Zhihuan Dong, T. Senthil: Non-commutative field theory and composite Fermi Liquids in some quantum Hall systems, Phys. Rev. B 102 (2020) 205126 [doi:10.1103/PhysRevB.102.205126, arXiv:2006.01282]