

A034997


Number of generalized retarded functions in quantum field theory.


2




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 a(8) 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

Antoine Deza, George Manoussakis, and Shmuel Onn, Primitive Zonotopes, Discrete & Computational Geometry, 2017, p. 113. (See p. 5.)


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 twodimensional space into 6 parts using lines x=0, y=0 and x+y=0.


CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS



STATUS

approved



