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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 8th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexFrom Erhard Sholtz's [Weyl entering the 'new' quantum mechanics discourse](http://www2.math.uni-wuppertal.de/~scholz/preprints/HQ_1_ES.pdf) >Already earlier in the 1920s,Weyl had found two topics in modern physics, in which group representations became important. The first topic was in general relativity and differential geometry. The representation theory of the special linear group $S L_n \mathbb{R}$, showed that there is a _mathematical reason for the structural importance of tensors in differential geometry_. >Footnote: All irreducible representations of $S L_2 \mathbb{R}$, arise as subrepresentations of tensor products of the natural representation with certain symmetry properties. Thus infinitesimal structures of classical differential geometry have a good chance to be expressible in terms of vector and tensor fields. Perhaps that '2' should be $n$, but is something like this commonly understood? It seems very similar to the wish to find irreducible representation of powers of the natural representation of the symmetic group to yield logical operations, as reported at the Café [here](http://golem.ph.utexas.edu/category/2012/01/logic_as_invarianttheory.html) ([cached](http://webcache.googleusercontent.com/search?q=cache:LuzUAJB0MugJ:golem.ph.utexas.edu/category/2012/01/logic_as_invarianttheory.html) since out of action). Mautner used tensor notation for the logical operations.

   From Erhard Sholtz’s Weyl entering the ’new’ quantum mechanics discourse

   Already earlier in the 1920s,Weyl had found two topics in modern physics, in which group representations became important. The first topic was in general relativity and differential geometry. The representation theory of the special linear group $S L_n \mathbb{R}$, showed that there is a mathematical reason for the structural importance of tensors in differential geometry.

   Footnote: All irreducible representations of $S L_2 \mathbb{R}$, arise as subrepresentations of tensor products of the natural representation with certain symmetry properties. Thus infinitesimal structures of classical differential geometry have a good chance to be expressible in terms of vector and tensor fields.

   Perhaps that ’2’ should be $n$, but is something like this commonly understood?

   It seems very similar to the wish to find irreducible representation of powers of the natural representation of the symmetic group to yield logical operations, as reported at the Café here (cached since out of action). Mautner used tensor notation for the logical operations.