Schrödinger representation

For $q,p\in\mathbb R^n$ , the Schrödinger representation of the Weyl relations on $L^2(\mathbb R^n)$ (with Lebesgue measure) is the unitary operator-valued map

[V(q,p)f](x)=\exp\!\left(i\left(q\mathbin{\cdot}x+\tfrac12q\mathbin{\cdot}p\right)\right)f(x+p).

Weyl relation

With $B((q,p),(q',p'))=q\mathbin{\cdot}p'-p\mathbin{\cdot}q'$ , it satisfies

V(q,p)V(q',p')=e^{-iB((q,p),(q',p'))/2}V(q+q',p+p').

Why the phase has this sign

The formula is the exponential form $V(q,p)=e^{i(q\cdot Q+p\cdot P)}$ , where $Q_jf(x)=x_jf(x)$ and $P_jf(x)=-i\,\partial_jf(x)$ on their standard common smooth core. Direct multiplication gives

[V(q,p)V(q',p')f](x) =e^{i((q+q')\cdot x+(q\cdot p+q'\cdot p')/2+q'\cdot p)}f(x+p+p'),

whose extra phase relative to $V(q+q',p+p')$ is $e^{i(q'\cdot p-q\cdot p')/2}=e^{-iB((q,p),(q',p'))/2}$ .

Properties

Translations and modulations are unitary on $L^2(\mathbb R^n)$ , and their dependence on $(q,p)$ is strongly continuous. In finite dimension this is the standard irreducible model singled out, up to unitary equivalence, by the Stone–von Neumann theorem.

References

Gerald B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies 122, Princeton University Press, 1989, Chapter 1, §1.3, beginning on p. 21 (the Schrödinger representation). Electronic edition.