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theorems:stone-von_neumann [2018/03/28 15:24] jakobadmin |
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 | |
- | <note tip> | + | made just using the [[formulas:canonical_commutation_relations|Heisenberg commutation relations]], without picking a specific |
- | 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. | + | representation of the operators that satisfy these relations. This turns out |
- | </note> | + | 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> | <tabbox Concrete> | ||
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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> | ||
- | + | ||
+ | |||
+ | <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> | <tabbox Abstract> | ||