login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
OFFSET

1,1

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 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

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

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

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.

LINKS

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

T. S. Evans, What is being calculated with Thermal Field Theory?

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

AUTHOR

Tim S. Evans

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 27 15:49 EST 2014. Contains 250225 sequences.