What is the meaning of "representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space"?

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    $\begingroup$ Is this from a reference? $\endgroup$ – Qmechanic Jan 14 at 16:03
  • $\begingroup$ @Qmechanic no, it's a question I've been given from my uni. $\endgroup$ – KatherinD Jan 14 at 16:35
  • $\begingroup$ It sounds like precious elliptical talk mathematicians often employ to refer to Moyal Brackets, but one can't be sure... $\endgroup$ – Cosmas Zachos Jan 14 at 16:42

The canonical commutation relations form a "well-defined" algebra of observables when one considers the unitary groups associated to the canonical quantum variables (Heisenberg group). In fact if one instead considers the Lie algebra induced by operators themselves, with the commutator as a Lie bracket, there are ambiguities: there are operators satisfying the Lie algebra of canonical commutation relations whose corresponding Lie group is not the Heisenberg group.

Therefore, in OP's question, I would interpret the phrase "in the form of Heisenberg" as referring to the Heisenberg group form of CCR, as opposed to the Lie algebra form.

Now, the Heisenberg group, or better the C*-algebra of Canonical Commutation Relations can be defined abstractly for any symplectic linear space (no actual need of a locally convex topology on it). In fact, given a symplectic space $X$ with symplectic form $\sigma$, the CCR C*-algebra is defined (uniquely up to $*$-isomorphisms) as $\mathbb{W}(X,\sigma)=\text{C*}\{W(x),x\in X\}$ (i.e. the smallest C*-algebra that contains such set of elements), where the elements $W(x)$ satisfy:

  • $W(x)\neq 0$ for all $x\in X$;
  • $W(-x)=W(x)^*$ for all $x\in X$;
  • $W(x)W(y)=e^{-i\sigma(x,y)}W(x+y)$ for all $x,y\in X$ (Weyl relations).

Finally, a representation of a C*-algebra is a map that identifies the elements of the algebra with an algebra of bounded operators on some given Hilbert space. Therefore, a representation of $\mathbb{W}(X,\sigma)$ is a couple $(\mathscr{H},\pi)$, where $\mathscr{H}$ is a Hilbert space, and $$\pi:\mathbb{W}(X,\sigma)\to \mathcal{L}(\mathscr{H})$$ is a map that preserves the algebraic operations ($*$-homomorphism).

Let me remark that if $X$ is finite dimensional, and $x\mapsto\pi(W(x))$ is strongly continuous from $X$ to $\mathcal{L}(\mathscr{H})$, then there is only one, up to unitary equivalence, irreducible representation of the CCR. This is the famous Stone-von Neumann Theorem. Irreducible means that only subspaces of $\mathscr{H}$ invariant under the action of $\pi\bigl(\mathbb{W}(X,\sigma)\bigr)$ are $\mathscr{H}$ and $\{0\}$. If $X$ is infinite dimensional instead, then there are usually uncountably many inequivalent irreducible representations of the CCR (and that plays a relevant role in rigorous Quantum Field Theory).

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    $\begingroup$ +1 ...as a minor remark: SvN theorem also requires that the representation is a strongly continuos map from the symplectic space to $B(H)$. $\endgroup$ – Valter Moretti Jan 14 at 18:17

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