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--- theorems:stone-von_neumann [2018/03/28 15:24]
jakobadmin
+++ theorems:stone-von_neumann [2018/07/18 13:24] (current)
jakobadmin [Intuitive]
@@ Line 3: / Line 3: @@
 <tabbox Intuitive>
+<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 tip>
+<blockquote>In typical physics quantum mechanics textbooks, one often sees calculations
-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.
+made just using the [[formulas:canonical_commutation_relations|Heisenberg commutation relations]], without picking a specific
-</note>
+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>
@@ Line 21: / Line 42: @@
 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>
+<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>

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