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A034997 Number of Generalized Retarded Functions in Quantum Field Theory. 1
2, 6, 32, 370, 11292, 1066044, 347326352, 419172756930 (list; graph; refs; listen; history; text; internal format)



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 the last number on Aug 01 2011 using an updated version of van Eijck's program, which took 7 days on a standard desktop computer.


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.

T. S. Evans, N-point finite temperature expectation values at real times, Nuclear Physics B 374 (1992) 340-370.

H. Kamiya, A. Takemura and H. Terao, Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47 (2011) 379 - 400.

M. van Eijck, Thermal Field Theory and Finite-Temperature Renormalisation Group, PhD thesis, Univ. Amsterdam, 4th Dec. 1995.


Table of n, a(n) for n=1..8.

L. J. Billera, J. T. Moore, C. D. Moraites, Y. Wang and K. Williams, Maximal unbalanced families, arXiv preprint arXiv:1209.2309, 2012. - From N. J. A. Sloane, Dec 26 2012

T. S. Evans, What is being calculated with Thermal Field Theory?, arXiv:hep-ph/9404262 and in "Particle Physics and Cosmology: Proceedings of the Ninth Lake Louise Winter School", World Scientific, 1995 (ISBN 9810221002)


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.


Sequence in context: A056642 A001199 A232469 * A067735 A118077 A013976

Adjacent sequences:  A034994 A034995 A034996 * A034998 A034999 A035000




Tim S. Evans



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Last modified September 26 14:59 EDT 2016. Contains 276546 sequences.