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Multidimensional phase space
thesisposted on 2023-05-27, 17:28 authored by Roberts, Martin
This thesis investigates the dimensional continuation of the standard 4-dimensional relativistic phase space to higher dimensions where all of the higher dimensions, beyond the usual four, are considered purely as flat spatial dimensions. Using standard approaches we are able to obtain simple closed expressions for the two- and three-body cases. We then take an alternative approach by recognizing an intimate connection between phase space and the 'sunset' diagram in co-ordinate space. This allows us to develop a kind of threshold expansion series for the phase space for an arbitrary number of massive particles in an arbitrary number of dimensions, in terms of the hypergeometric function. The beauty of this method is that the series terminates for odd dimensions, and easily describes the leading terms for even dimensions. We then develop a method for calculating phase space that is complementary to the Almgren recurrence relations. That is, it establishes relations between N‚ÄövÑvÆparticle phase space pN, in D‚ÄövÑvÆdimensions with N‚ÄövÑvÆparticle phase space in lower dimensions. This is done through a novel approach of considering phase space in higher dimensions where one or more of the (spatial) coordinates of space-time are confined to a torus or sphere of radius R and the limit as R ‚Äövúv¿ ‚Äöv†vª is taken. Summing over all the associated discrete modes leads to a mass integral, and hence the the resulting phase space is expressible as a set of mass integrals of pN in a dimension lower than D. Finally, we review some work done by Davydychev et al., which establishes an intimate connection between the integral associated with the one loop N‚ÄövÑvÆpoint Feynman diagram (the 'sun' diagram) and geometric aspects of its 'dual diagram' - an N‚ÄövÑvÆdimensional simplex (hyper-tetrahedron). This is of particular interest because the phase space integral is also closely associated with an N‚ÄövÑvÆdimensional simplex in Euclidean space. Therefore, by looking at the N‚ÄövÑvÆpoint single loop diagram and the N‚ÄövÑvÆloop two-point diagrams, we suggest that the geometry of the Feynman diagram itself contains valuable information that allows us to deduce a deeper structure to the field theory expressions. That is, inasmuch as the Feynman diagrams are a geometric interpretation of field theory interactions, this thesis intimates a geometric interpretation of the Feynman diagrams, themselves. This in turn, suggests that it may be possible to find a geometric interpretation, to a level deeper than ever before, of quantum field theory.
Rights statementCopyright 2003 the author - The University is continuing to endeavour to trace the copyright owner(s) and in the meantime this item has been reproduced here in good faith. We would be pleased to hear from the copyright owner(s). Thesis (Ph.D.)--University of Tasmania, 2003. Includes bibliographical references