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theorems:stone-von_neumann [2018/05/02 09:10] jakobadmin [Concrete] |
theorems:stone-von_neumann [2018/05/02 09:11] jakobadmin [Intuitive] |
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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> | ||