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group_theory:notions:unitary_representations [2016/12/22 09:35]
jakobadmin
group_theory:notions:unitary_representations [2017/12/06 09:33] (current)
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 To make sense of the probabilistic interpretation of quantum theory, we need to use unitary representations (see e.g. page 69 in [[http://​isites.harvard.edu/​fs/​docs/​icb.topic521209.files/​QFT-Schwartz.pdf|Schwartz'​s book]]). Expressed differently,​ we are interested in representations of given groups on the Hilbert space in a quantum field theory. To make sense of the probabilistic interpretation of quantum theory, we need to use unitary representations (see e.g. page 69 in [[http://​isites.harvard.edu/​fs/​docs/​icb.topic521209.files/​QFT-Schwartz.pdf|Schwartz'​s book]]). Expressed differently,​ we are interested in representations of given groups on the Hilbert space in a quantum field theory.
  
 +Many important groups are non-compact (e.g. the Poincare group and the conformal group) and there is a theorem that tells us that all unitary representations of a non-compact group are infinite-dimensional.
  
 +"//The infinite-dimensional representations are
 +considered unphysical because we never see particle states in nature labelled by extra continuous
 +parameters.//"​ http://​homepages.uc.edu/​~argyrepc/​cu661-gr-SUSY/​susy2001.pdf
 +
 +Thus it is often a non-trivial problem to find make sense of the unitary representations of such groups.
  
 ===== Examples ===== ===== Examples =====
group_theory/notions/unitary_representations.txt ยท Last modified: 2017/12/06 09:33 (external edit)