theorems:stone-von_neumann
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theorems:stone-von_neumann [2018/05/02 09:10] jakobadmin [Intuitive] |
theorems:stone-von_neumann [2018/05/13 09:18] jakobadmin ↷ Links adapted because of a move operation |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| <blockquote>In typical physics quantum mechanics textbooks, one often sees calculations | <blockquote>In typical physics quantum mechanics textbooks, one often sees calculations | ||
| - | made just using the [[equations:canonical_commutation_relations|Heisenberg commutation relations]], without picking a specific | + | made just using the [[formulas:canonical_commutation_relations|Heisenberg commutation relations]], without picking a specific |
| representation of the operators that satisfy these relations. This turns out | representation of the operators that satisfy these relations. This turns out | ||
| to be justified by the remarkable fact that, for the Heisenberg group, once one | to be justified by the remarkable fact that, for the Heisenberg group, once one | ||
| Line 18: | Line 18: | ||
| freedom, which is the case of interest in quantum field theory, the Stone-von | 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 | 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> | <tabbox Concrete> | ||
theorems/stone-von_neumann.txt · Last modified: 2018/07/18 13:24 by jakobadmin
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