Canonical
Commutation
Relations
\([Q,P] = i\)
\(\{q,p\}=1\)
\([Q,P] = i\)
\(Q\psi(x):=x\psi(x)\)
\(P\psi(x):=-i\nabla\psi(x)\)
\([Q,P] = i\)
Spatial Homogeneity
\([Q,P] = i\)
Spatial Location
Spatial Location
Spatial Location
Spatial Location
$\Delta\in B$ is a Borel set of \(\mathbb{R}\)
\(E_\Delta\) is a Hilbert space projection. The proposition,
'The system is in $\Delta$' is represented by eigenvalue 1.
\(\{E_{\Delta} \;|\; \Delta \text{ is Borel}\}\) is a projection-valued measure (PVM)
\(Q := \int_{-\infty}^{\infty} \lambda dE_\lambda\) is a self-adjoint operator, 'position'
Spatial Translation
Spatial Translation
\(\overset{a}{\longrightarrow}\)
\(\Delta' = \Delta-a \;\;\; (a\in\mathbb{R})\)
\(E_{\Delta'} = E_{\Delta-a} \;\;\; (a\in\mathbb{R})\)
Spatial Homogeneity
Local experiments display the same statistics everywhere in space
Spatial Homogeneity
Local experiments display the same statistics everywhere in space
Spatial Homogeneity
Local experiments display the same statistics everywhere in space
Spatial Homogeneity
Local experiments display the same statistics everywhere in space
i.e. \(E_{\Delta}\) and \(E_{\Delta'}\) are related by a unitary
\(\exists U_s\) such that \(U_a E_{\Delta}U_a^* = E_{\Delta'} = E_{\Delta + a} \)
Spatial Homogeneity
Local experiments display the same statistics everywhere in space
i.e. \(E_{\Delta}\) and \(E_{\Delta'}\) are related by a unitary
\(\exists U_a\) such that \(U_aE_{\Delta}U_a^* = E_{\Delta'} \)
Spatial Homogeneity
Local experiments display the same statistics everywhere in space
means
There exists a strongly continuous set of unitaries \(\{U_a \;|\; a\in\mathbb{R}\}\) such that for all \(a,a'\in\mathbb{R}\), we have \(U_aU_{a'} = U_{a+a'}\) and
\(U_aE_{\Delta}U^*_a = E_{\Delta - a}\)
i.e. 'spatial translation is unitary'
Spatial Homogeneity
\(U_aE_{\Delta}U^*_a = E_{\Delta - a}\)
\(Q := \int_{-\infty}^\infty \lambda dE_\lambda\)
\(\Rightarrow UQU^* =\) \(\int_{-\infty}^\infty\lambda dUE_\lambda U^*\) \(= \int_{-\infty}^\infty \lambda dE_{\lambda - a}\)
\(= \int_{-\infty}^\infty (\lambda + a)dE_\lambda\) \( = \int_{-\infty}^\infty \lambda dE_\lambda \;+\; a\int_{-\infty}^\infty dE_\lambda \)
\(= Q + aI\)
Spatial Homogeneity
Local experiments display the same statistics everywhere in space
means
There exists a strongly continuous set of unitaries \(\{U_a \;|\; a\in\mathbb{R}\}\) such that for all \(a,a'\in\mathbb{R}\), we have \(U_aU_{a'} = U_{a+a'}\) and
\(U_aE_{\Delta}U^*_a = E_{\Delta - a}\)
i.e. 'spatial translation is unitary'
Spatial Homogeneity
There exists a strongly continuous set of unitaries \(\{U_a \;|\; a\in\mathbb{R}\}\) such that for all \(a,a'\in\mathbb{R}\), we have \(U_aU_{a'} = U_{a+a'}\) and
by Stone's theorem
there is a unique self-adjoint \(P\) such that for all \(a\in\mathbb{R}\)
\(U_a = e^{-iaP}\)
Spatial Homogeneity
Position: \(U_aQU^*_a = Q+aI\)
Momentum: \(e^{-iaP}Qe^{iaP} = Q+aI\)
\(\Rightarrow \tfrac{1}{ia}(e^{-iaP} - I)Q = Q\tfrac{1}{ia}(e^{-iaP}-I) + ie^{-iaP}\)
Note \(P = \displaystyle\lim_{a\rightarrow 0}\tfrac{1}{ia}(e^{-iaP} - I)\) so...
Spatial Homogeneity
Position: \(U_aQU^*_a = Q+aI\)
Momentum: \(e^{-iaP}Qe^{iaP} = Q+aI\)
\(\Rightarrow \tfrac{1}{ia}(e^{-iaP} - I)Q = Q\tfrac{1}{ia}(e^{-iaP}-I) + ie^{-iaP}\)
Note \(P = \displaystyle\lim_{a\rightarrow 0}\tfrac{1}{ia}(e^{-iaP} - I)\) so...
\(QP - PQ = i\)
Spatial Homogeneity
Position: \(U_aQU^*_a = Q+aI\)
Momentum: \(e^{-iaP}Qe^{iaP} = Q+aI\)
Write \(U(a):=e^{-iaP}\) and \(U(b):=e^{-ibQ}\)
Spatial Homogeneity
Position: \(U_aQU^*_a = Q+aI\)
Momentum: \(e^{-iaP}Qe^{iaP} = Q+aI\)
Write \(U(a):=e^{-iaP}\) and \(U(b):=e^{-ibQ}\)
Taylor \(\Downarrow\) Expand
\(U(a)U(b) = e^{iab}U(b)U(a)\)
'Weyl Form' of the Canonical Commutation Relations (CCRs)
\(U(a)U(b) = e^{iab}U(b)U(a)\)
- Weyl algebra: algebraic closure of \(U(a),U(b)\).
Uniqueness
\(U(a)U(b) = e^{iab}U(b)U(a)\)
The Stone-von Neumann Theorem
SCHRÖDINGER: \(Q\psi(x)=x\psi(x), P\phi(x)=i\nabla\phi(x)\)
for \(\psi(x),\phi(x)\in L^2(\mathbb{R})\) with \(\psi(x)\in D_Q\) and \(\phi(x)\in D_P\)
HEISENBERG: \(QP - PQ = i\)
for some non-commutative `quantum variables' \(Q,P\)
SCHRÖDINGER: \(Q\psi(x)=x\psi(x), P\phi(x)=i\nabla\phi(x)\)
for \(\psi(x),\phi(x)\in L^2(\mathbb{R})\) with \(\psi(x)\in D_Q\) and \(\phi(x)\in D_P\)
HEISENBERG: \(QP - PQ = i\)
for some non-commutative `quantum variables' \(Q,P\)
Do these make the same statistical predictions?
Unitary intertwining. Two Hilbert space representations \((\pi(\mathcal{A}),\mathcal{H})\) and \((\pi'(\mathcal{A}),\mathcal{H}')\) of an algebra \(\mathcal{A}\) have the same statistics iff there exists a unitary \(U:\mathcal{H}\rightarrow\mathcal{H}'\) (called a unitary intertwiner) such that for each \(A\in\mathcal{A}\),
\[ U\pi(A)U^* = \pi'(A).\]Then the two representations are called unitarily equivalent.
Stone-von Neumann Question. Is an arbitrary Heisenberg representation
\[QP-PQ=i\]unitarily equivalent to the Schrödinger representation
\[Q\psi(x)=x\psi(x), \;\; P\psi(x)=i\nabla\psi(x))\]on \(L^2(\mathbb{R})\)?
Answer: Only when the CCRs hold in Weyl form too:
\[ U(a)V(b)=e^{ia\cdot b}V(b)U(a). \]Equivalently: only if \(\Pi = Q^2 + P^2\) has a dense domain.
Stone: States the question, says it's true, but weirdly remarks it follows from something to do with the Cayley transform.
von Neumann: Proves it, using an unrelated technique.
von Neumann's (1931) first paragraph:
Sketch of Proof
1. Main idea. Restrict attention to representations on \(L^2(\mathbb{R}^{2n})\). It turns out that the following is a projection:
\[P_\varphi = \int\int U(x)V(y)\varphi(x,y)dxdy\]for \(\varphi\in S(\mathbb{R}^{2n})\) in the Schwarz space of functions satisfying,
\[ \varphi(-x,-y) = e^{-ix\cdot y}\overline{\varphi}(x,y) \]and
\[ \int\int\varphi(x,y)\varphi(u-x,v-y)e^{i(x-u)\cdot y}dxdy = \varphi(u,v) \]2. von Neumann shows: This projection \(P_\varphi\) vanishes if \(\varphi\equiv 0\). (This uses Fourier analysis, and the fact that Fourier is an automorphism of the Schwarz space.) And if
\[ \varphi(x,y) = \frac{1}{(2\pi)^n}e^{-ix\cdot y/2}e^{-(|x|^2+|y|^2)/4} \]then \(P_\varphi\) is indeed a projection.
3. Furthermore, one finds with this choice that \(P_\varphi U(x)P\varphi\) and \(P_\varphi V(y)P_\varphi\) agree with \(P_\varphi\) up to scalar factors depending on \(x\) and \(y\), respectively.
Thus if the representation of the \(U(x)\)'s and \(V(y)\)'s on \(\mathcal{H}\) is irreducible, \(P_\varphi\) must be a rank-one projection.
4. The theorem follows, since given two irreducible representations of the Weyl CCRs on \(\mathcal{H}\) and \(\mathcal{H}'\), the map sending a unit vector in the range of \(P_\varphi\) on \(\mathcal{H}\) to a unit vector in the range of the corresponding operator \(P'_\varphi\) on \(\mathcal{H}'\) will extend uniquely to a unitary intertwining operator.