

A034997


Number of Generalized Retarded Functions in Quantum Field Theory.


1




OFFSET

1,1


COMMENTS

a(d) is the number of parts into which ddimensional 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 realtime 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.


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. 157171.
M. van Eijck, Thermal Field Theory and FiniteTemperature Renormalisation Group, PhD thesis, Univ. Amsterdam, 4th Dec. 1995.


LINKS

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 [math.CO], 2012.  From N. J. A. Sloane, Dec 26 2012
Antoine Deza, George Manoussakis, Shmuel Onn, Primitive Zonotopes, Discrete & Computational Geometry, 2017, p. 113. (See p. 5.)
T. S. Evans, Npoint finite temperature expectation values at real times, Nuclear Physics B 374 (1992) 340370.
T. S. Evans, What is being calculated with Thermal Field Theory?, arXiv:hepph/9404262 and in "Particle Physics and Cosmology: Proceedings of the Ninth Lake Louise Winter School", World Scientific, 1995 (ISBN 9810221002).
H. Kamiya, A. Takemura and H. Terao, Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47 (2011) 379  400.


EXAMPLE

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.


CROSSREFS

Sequence in context: A056642 A001199 A232469 * A067735 A118077 A013976
Adjacent sequences: A034994 A034995 A034996 * A034998 A034999 A035000


KEYWORD

nonn,more,changed


AUTHOR

Tim S. Evans


STATUS

approved



