theorems:stone-von_neumann
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theorems:stone-von_neumann [2017/11/16 10:11] jakobadmin [Why is it interesting?] |
theorems:stone-von_neumann [2018/07/18 13:24] (current) jakobadmin [Intuitive] |
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| ====== Stone-von Neumann Theorem ====== | ====== Stone-von Neumann Theorem ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | The Stone-von Neumann theorem is important, for example, to understand why symmetry breaking is possible in quantum field theory. | ||
| - | <tabbox Layman> | ||
| - | <note tip> | + | <tabbox Intuitive> |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | <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> |
| - | </note> | + | |
| - | | + | <blockquote>In typical physics quantum mechanics textbooks, one often sees calculations |
| - | <tabbox Student> | + | 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 | ||
| + | to be justified by the remarkable fact that, for the Heisenberg group, once one | ||
| + | picks the constant with which Z acts, all irreducible representations are unitarily | ||
| + | equivalent. In a sense, the representation theory of the Heisenberg group is very simple: | ||
| + | there’s only one irreducible representation. This is very different from the | ||
| + | theory for even the simplest compact Lie groups (U(1) and SU(2)) which have | ||
| + | an infinity of inequivalent irreducibles labeled by weight or by spin. Representations | ||
| + | of a Heisenberg group will appear in different guises (we’ve seen two, | ||
| + | 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 | ||
| + | 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> | ||
| <blockquote> In the algebraic | <blockquote> In the algebraic | ||
| Line 22: | Line 42: | ||
| but in the infinite-dimensional case there are unitarily inequivalent representations | but in the infinite-dimensional case there are unitarily inequivalent representations | ||
| of the algebra.<cite>http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf</cite></blockquote> | of the algebra.<cite>http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf</cite></blockquote> | ||
| - | + | ||
| - | <tabbox Researcher> | + | |
| + | <blockquote>Theorem (Stone-von Neumann). Any irreducible representation π of the group H3 on a Hilbert space, satisfying $$π 0 (Z) = −i1$$ is unitarily equivalent to the Schrödinger representation (ΓS, L2 (R)).<cite>https://www.math.columbia.edu/~woit/QM/qmbook.pdf</cite></blockquote> | ||
| + | <tabbox Abstract> | ||
| <note tip> | <note tip> | ||
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| | | ||
| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| - | + | The Stone-von Neumann theorem is important, for example, to understand why symmetry breaking is possible in quantum field theory. | |
| - | --> Example1# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| - | + | ||
| - | --> Example2:# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| - | + | ||
| - | <tabbox FAQ> | + | |
| | | ||
| <tabbox History> | <tabbox History> | ||
| - | * A[[https://www.math.umd.edu/~jmr/StoneVNart.pdf| Selective History of the Stone-von Neumann Theorem]] by Jonathan Rosenberg | + | * [[https://www.math.umd.edu/~jmr/StoneVNart.pdf|A Selective History of the Stone-von Neumann Theorem]] by Jonathan Rosenberg |
| </tabbox> | </tabbox> | ||
theorems/stone-von_neumann.1510823514.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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