A034997 Number of generalized retarded functions in quantum field theory.

2, 6, 32, 370, 11292, 1066044, 347326352, 419172756930, 1955230985997140

Offset: 1

Author

Tim S. Evans

Keywords

• nonn
• more

Comments

a(d) is the number of parts into which d-dimensional space (x_1,...,x_d) is split by a set of (2^d - 1) hyperplanes c_1 x_1 + c_2 x_2 + ... + c_d x_d =0 where c_j are 0 or +1 and we exclude the case with all c=0.

Also, a(d) is the number of independent real-time Green functions of quantum field theory produced when analytically continuing from Euclidean time/energy (d+1 = number of energy/time variables). These are also known as "generalized retarded functions".

The numbers up to d=6 were first produced by T. S. Evans using a Pascal program, strictly as upper bounds only. M. van Eijck wrote a C program using a direct enumeration of hyperplanes which confirmed these and produced the value for d=7. Kamiya et al. showed how to find these numbers and some associated polynomials using more sophisticated methods, giving results up to d=7. T. S. Evans added a(8) on Aug 01 2011 using an updated version of van Eijck's program, which took 7 days on a standard desktop computer.

Examples

			a(1)=2 because the point x=0 splits the real line into two parts, the positive and negative reals.
a(2)=6 because we can split two-dimensional space into 6 parts using lines x=0, y=0 and x+y=0.
		

References

• Björner, Anders. "Positive Sum Systems", in Bruno Benedetti, Emanuele Delucchi, and Luca Moci, editors, Combinatorial Methods in Topology and Algebra. Springer International Publishing, 2015. 157-171.
• M. van Eijck, Thermal Field Theory and Finite-Temperature Renormalisation Group, PhD thesis, Univ. Amsterdam, 4th Dec. 1995.

Links

• Louis J. Billera, Sara C. Billey, and Vasu Tewari, Boolean product polynomials and Schur-positivity, arXiv:1806.02943 [math.CO], 2018.
• L. J. Billera, J. Tatch Moore, C. Dufort Moraites, Y. Wang, and K. Williams, Maximal unbalanced families, arXiv preprint arXiv:1209.2309 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 26 2012
• Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
• Antoine Deza, Mingfei Hao, and Lionel Pournin, Sizing the White Whale, arXiv:2205.13309 [math.CO], 2022.
• Antoine Deza, George Manoussakis, and Shmuel Onn, Primitive Zonotopes, Discrete & Computational Geometry, 2017, p. 1-13. (See p. 5.)
• Nick Early, Lukas Kühne, and Leonid Monin, When alcoved polytopes add, arXiv:2501.17249 [math.CO], 2025. See p. 7.
• Tim S. Evans, N-point finite temperature expectation values at real times, Nuclear Physics B 374 (1992) 340-370.
• Tim S. Evans, What is being calculated with Thermal Field Theory?, arXiv:hep-ph/9404262, 1994-2011 and in "Particle Physics and Cosmology: Proceedings of the Ninth Lake Louise Winter School", World Scientific, 1995 (ISBN 9810221002).
• Samuel C. Gutekunst, Karola Mészáros, and T. Kyle Petersen, Root Cones and the Resonance Arrangement, arXiv:1903.06595 [math.CO], 2019.
• Lukas Kühne, The Universality of the Resonance Arrangement and its Betti Numbers, arXiv:2008.10553 [math.CO], 2020.
• Hidehiko Kamiya, Akimichi Takemura, and Hiroaki Terao, Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47 (2011) 379-400.
• Lars Kastner and Marta Panizzut, Hyperplane arrangements in polymake, arXiv:2003.13548 [math.CO], 2020.
• Zhengwei Liu, William Norledge, and Adrian Ocneanu, The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations, arXiv:1901.03243 [math.CO], 2019.
• William Norledge and Adrian Ocneanu, Hopf monoids, permutohedral tangent cones, and generalized retarded functions, arXiv:1911.11736 [math.CO], 2019.

Extensions

a(9) from Zachary Chroman, Feb 19 2021