advanced_tools:group_theory:desitter

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advanced_tools:group_theory:desitter [2018/03/24 11:09] jakobadmin [Student] |
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The deSitter group becomes the Poincare group in the contraction limit $R \rightarrow \infty$, where $R$ is the so-called deSitter radius. Oftentimes, people prefer to work with the [[open_problems:cosmological_constant|cosmological constant]] $ \Lambda \propto \frac{1}{R^2}$ instead. Analogously, the Poincare group becomes the Galilean group in the $c \rightarrow \infty$ limit. | The deSitter group becomes the Poincare group in the contraction limit $R \rightarrow \infty$, where $R$ is the so-called deSitter radius. Oftentimes, people prefer to work with the [[open_problems:cosmological_constant|cosmological constant]] $ \Lambda \propto \frac{1}{R^2}$ instead. Analogously, the Poincare group becomes the Galilean group in the $c \rightarrow \infty$ limit. | ||

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+ | The fact that the deSitter group contracts to the Poincare group in the $R\rightarrow \infty$ limit, means that the Poincare group is a good approximation as long as we consider systems with a length scale that is small compared to $R$. This is analogous to how the Galilean group is good enough as long as we are only dealing with velocities much smaller than the invariant velocity $c$. | ||

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+ | Expressed differently: the deSitter group is only important for cosmological systems, which have a length scale comparable to $R$. | ||

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+ | Alternatively, we can talk about the invariant energy scale $\Lambda$. The deSitter group contracts to the Poincare group in the $\Lambda \rightarrow 0$ limit. Thus the deSitter structure is not important, as long as we are dealing with energies much larger than $\Lambda$. In systems with energies much larger than $\Lambda$ such a small constant energy has no effect. The present day value for the cosmological constant is $\Lambda \approx 10^{-56} \mathrm{m^{-2}}$ and this means that present day effects of the deSitter group structure are tiny. This means, the Poincare group is a great approximate symmetry nowadays, because $\Lambda$ is almost zero. | ||

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+ | However, the deSitter group could be very important in the early universe, too. For example, because it seems plausible that there was [[https://en.wikipedia.org/wiki/Inflation_(cosmology)|a phase when the cosmological constant was much higher]]. | ||

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- | --> Example2:# | + | -->It ist confusing what a "five-dimensional" group like $SO(4,1)$ has to say about our four-dimensional world. Does this mean deSitter theories predict a fifth dimension?# |

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+ | No, recall that an explicit representation of the Poincare group is given by $(5 \times 5)$ matrices, too! This is, because the Lorentz group transformations are $(4 \times 4)$ matrices themselves and in oredr to describe translations, we need to make these matrices into $(5 \times 5)$ matrices. See, for example, chapter 4 "The Poincaré transformations" in [[http://www.springer.com/us/book/9783642154812|Symmetries and Group Theory in Particle Physics]] by Costa and Fogly. | ||

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advanced_tools/group_theory/desitter.1521886177.txt.gz · Last modified: 2018/03/24 10:09 (external edit)

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