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--- advanced_tools:group_theory:de_sitter_group [2017/09/23 13:05]
jakobadmin [Researcher]
+++ advanced_tools:group_theory:de_sitter_group [2017/11/05 16:03] (current)
jakobadmin ↷ Page moved from theories:general_relativity:cosmology:de_sitter_group to advanced_tools:group_theory:de_sitter_group
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 unique maximally symmetric curved spacetimes Weinberg, 1972).   <cite>Dirac Wave Equation in the de Sitter Universe E. A. Notte Cuello </cite></blockquote>
 <tabbox Researcher>
+<blockquote>As shown in [[https://projecteuclid.org/download/pdf_1/euclid.cmp/1103858248|Contractions of Representations of de Sitter Groups]] J. MICKELSSON* and J. NIEDERLE* (see also their references to Wigner and Inonu), it is fortunately the case that at least all the massive representations of the Poincaré group can be obtained by contracting representations of the deSitter group, meaning the approximation is also consistent on this level. What exactly the mass parameter itself means in deSitter space is a somewhat open question.
+A longer overview over the Wigner classification for O(1,4)O(1,4), i.e. 4D deSitter space, and the physical meaning of the different possible representations is given in "Group theory and de Sitter QFT" by Boers.
+<cite>https://physics.stackexchange.com/questions/299621/the-desitter-group-vs-the-poincare-group-for-a-non-zero-cosmological-constant</cite></blockquote>
+<blockquote>
+Recall that in the flat spacetime case the particle states are labeled by the Poincar´e label (p, λ), whose values are closely related to the UIR label (m, s); m is used to define the mass-shell condition and s determines the range of λ. We identified m with the mass of the field quanta, and s with the spin. In a group theoretical sense, we associate these parameters with the eigenvalues of the Casimir operators (see section 3.5). Since the notion of mass is not as clear in de Sitter space as in Minkowski space in a field theoretical sense, we may ask ourselves if the Casimir operators of SO(4, 1) might bring resolvement to the issue. There is a very straightforward link between the Casimir operators and the wave equation, which we shall investigate by using the ambient space formalism in the next section. [...]
+The classification of SO(4, 1) UIRs reveals that there are three types, belonging either
+to the so-called principal, complementary or discrete series. In the zero curvature limit the
+principal series contracts to the massive Poincar´e UIRs, one particular complementary series
+UIR contracts to the massless scalar Poincar´e UIR, and a certain class of the discrete series
+contracts to the massless nonzero-spin Poincar´e UIRs (see section 4.3.2 for details). These UIRs
+contracting to the massless Poincar´e UIRs are uniquely extendable to UIRs of the conformal
+group SO(4, 2). All other SO(4, 1) UIRs have no (or no physically interesting) link to the
+Poincar´e group in the zero curvature limit. This last observation excludes a range of de Sitter
+fields from having a Minkowskian interpretation of their mass parameter in this sense.
+<cite>http://thep.housing.rug.nl/sites/default/files/theses/Master%20thesis_Marco%20Boers.pdf</cite>
+</blockquote>
+For the representations of the De Sitter group, see http://thep.housing.rug.nl/sites/default/files/theses/Master%20thesis_Marco%20Boers.pdf and Gürsey, F.: Introduction to the De Sitter Group. In: Gürsey, F. (ed.) Group Theoretical Concepts and Methods in Elementary Particle Physics. Gordon and Breach, New York (1965
 A great, very complete discussion can be found in http://onlinelibrary.wiley.com/doi/10.1002/prop.19770250112/epdf. Unfortunately, the paper is only available in German.
-For the representations of the De Sitter group, see 249. Gürsey, F.: Introduction to the De Sitter Group. In: Gürsey, F. (ed.) Group Theoretical Concepts and Methods in Elementary Particle Physics. Gordon and Breach, New York (1965
 and "Positive Cosmological Constant and Quantum Theory" by Lev, Felix M.

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