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A034997
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Number of generalized retarded functions in quantum field theory.
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2
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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Antoine Deza, George Manoussakis, and Shmuel Onn, Primitive Zonotopes, Discrete & Computational Geometry, 2017, p. 1-13. (See p. 5.)
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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