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--- theorems:stone-von_neumann [2018/05/02 09:10]
jakobadmin [Intuitive]
+++ theorems:stone-von_neumann [2018/05/02 09:11]
jakobadmin [Intuitive]
@@ Line 18: / Line 18: @@
 freedom, which is the case of interest in quantum field theory, the Stone-von
 Neumann theorem no longer holds and one has an infinity of inequivalent irreducible
-representations, leading to quite different phenomena. <cite>https://www.math.columbia.edu/~woit/QM/qmbook.pdf</cite></blockquote>
+representations, leading to quite different phenomena. [...]It is also important to note that the Stone-von Neumann theorem is formulated
+for Heisenberg group representations, not for Heisenberg Lie algebra
+representations. For infinite dimensional representations in cases like this, there
+are representations of the Lie algebra that are “non-integrable”: they aren’t
+the derivatives of Lie group representations. For such non-integrable representations
+of the Heisenberg Lie algebra (i.e., operators satisfying the Heisenberg
+commutation relations) there are counter-examples to the analog of the Stone
+von-Neumann theorem. It is only for integrable representations that the theorem
+holds and one has a unique sort of irreducible representation.<cite>https://www.math.columbia.edu/~woit/QM/qmbook.pdf</cite></blockquote>
 <tabbox Concrete>