theorems:stone-von_neumann
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theorems:stone-von_neumann [2018/05/02 09:10] jakobadmin [Concrete] |
theorems:stone-von_neumann [2018/07/18 13:24] (current) jakobadmin [Intuitive] |
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| + | <blockquote>The Stone-von Neumann theorem roughly says that for any operators $A$ and $B$ satisfying the canonical commutation relation, we can get away with using the standard representation $A \rightarrow u,\ B \rightarrow -i \hbar \frac{d}{du}$ without loss of generality. (More precisely, it says that any representation of the *exponentiated* canonical commutation relation on a sufficiently smooth Hilbert space is unitarily equivalent to the standard representation, so any other representation basically just describes the same physics in a different coordinate system.)<cite>https://physics.stackexchange.com/a/264587/37286</cite></blockquote> | ||
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| <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 | ||
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| will see another in the discussion of the harmonic oscillator, and there are yet | will see another in the discussion of the harmonic oscillator, and there are yet | ||
| others that appear in the theory of theta-functions), but they are all unitarily | others that appear in the theory of theta-functions), but they are all unitarily | ||
| - | equivalent, a statement known as the Stone-von Neumann theorem.<cite>https://www.math.columbia.edu/~woit/QM/qmbook.pdf</cite></blockquote> | + | equivalent, a statement known as the Stone-von Neumann theorem. [...] In the case of an infinite number of degrees of |
| + | 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. [...]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.1525245001.txt.gz · Last modified: 2018/05/02 07:10 (external edit)
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