Relative Normalizers of Automorphism Groups, Infravacua and the Problem of Velocity Superselection in QED

Abstract

We advance superselection theory of pure states on a $C^*$-algebra $\mathfrak {A}$ outside of the conventional (DHR) setting. First, we canonically define conjugate and second conjugate classes of such states with respect to a given reference state $\omega _{\mathrm {vac}}$ and background $a \in \mathrm {Aut}(\mathfrak {A})$. Next, for some subgroups $R \varsubsetneq S \varsubsetneq G\subset \mathrm {Aut}(\mathfrak {A})$ we study the family $\{\omega _{\mathrm {vac}}\circ s| s\in S \}$ of infrared singular states whose superselection sectors may be disjoint for different s. We show that their conjugate and second conjugate classes always coincide provided that R leaves the sector of $\omega _{\mathrm {vac}}$ invariant and a belongs to the relative normalizer$N_G(R,S):=\{g\in G| g\cdot S\cdot g^{-1}\subset R\}$. We study the basic properties of this apparently new group theoretic concept and show that the Kraus–Polley–Reents infravacuum automorphisms belong to the relative normalizers of the automorphism group of a suitable CCR algebra. Following up on this observation we show that the problem of velocity superselection in non-relativistic QED disappears at the level of conjugate and second conjugate classes, if they are computed with respect to an infravacuum background. We also demonstrate that for more regular backgrounds such merging effect does not occur.

Appendices

Some Auxiliary Lemmas About $C^*$-Algebras

For the reader’s convenience we recall some standard facts from the theory of $C^*$-algebras which we use in our paper. We refer to [KR, Chapter 10] for a more extensive discussion.

Lemma A.1

Let $(\pi ,\mathcal {H})$ be an irreducible representation of a $C^*$-algebra $\mathfrak {A}$ and let $\Psi \in \mathcal {H}$ be a unit vector. Then the GNS representation $(\tilde{\pi }, \tilde{\mathcal {H}}, \tilde{\Omega })$ induced by the state

$$\begin{aligned} \omega (A)=\langle \Psi , \pi (A)\Psi \rangle , \quad A\in \mathfrak {A}, \end{aligned}$$

(A.1)

is unitarily equivalent to $(\pi ,\mathcal {H})$.

Proof

The map $V: \mathcal {H}\rightarrow \tilde{\mathcal {H}}$, given by $V\pi (A)\Psi =\tilde{\pi }(A){\tilde{\Omega }}$, $A\in \mathfrak {A}$, is densely defined by irreducibility of $\pi$ and has a dense range by cyclicity of ${\tilde{\Omega }}$. By a straightforward computation one checks that V is an isometry and $V\pi (A)=\tilde{\pi }(A)V$ for all $A\in \mathfrak {A}$. $\quad \square$

In the following proposition we write $[\omega ]:=[\omega ]_{\mathrm {In}(\mathfrak {A})}$ for brevity.

Proposition A.2

Let $\omega$ be a pure state on $\mathfrak {A}$ and $(\pi , \mathcal {H}, \Omega )$ its GNS representation. Then $[\omega ]$ coincides with the set of all states whose GNS representations are unitarily equivalent to $\pi$. Furthermore, $[\omega \circ \gamma ]=[\omega ]$ iff $\gamma \in \mathrm {Aut}(\mathfrak {A})$ is unitarily implementable in $\pi$.

Proof

Let $\omega _1\in [\omega ]$ and denote by $(\pi _1, \mathcal {H}_1, \Omega _1)$ its GNS representation. Since $\omega _1=\omega \circ \mathrm {Ad}U$ for some unitary $U\in \mathfrak {A}$, we can write

$$\begin{aligned}&\omega (A)=\langle \Omega , \pi (A)\Omega \rangle , \, \end{aligned}$$

(A.2)

$$\begin{aligned}&\omega _1(A)=\langle \Omega _1, \pi _1(A)\Omega _1\rangle =\langle \Omega , \pi (UAU^*)\Omega \rangle =\langle \pi (U^*)\Omega , \pi (A) \pi (U^*)\Omega \rangle ,\qquad \qquad \end{aligned}$$

(A.3)

for $A\in \mathfrak {A}$. Now Lemma A.1 gives unitary equivalence of $\pi$ and $\pi _1$.

Conversely, suppose that $\pi$ and $\pi _1$ are unitarily equivalent, i.e., $\pi _1(A)=V\pi (A)V^*$ for all $A\in \mathfrak {A}$ and some unitary $V: \mathcal {H}\rightarrow \mathcal {H}_1$. Thus we can write

$$\begin{aligned} \omega _1(A)=\langle \Omega _1, V\pi (A)V^* \Omega _1\rangle =\langle \Omega , \pi (U)\pi (A)\pi (U)^*\Omega \rangle =\omega (UAU^*), \end{aligned}$$

(A.4)

where we used the irreducibility of $\pi$ and the resulting existence of a unitary $U\in \mathfrak {A}$ such that ${\pi (U)^* }\Omega =V^* \Omega _1$. This follows from the Kadison transitivity theorem [KR, Theorem 10.2.1].

As for the last statement, suppose that $\gamma$ is unitarily implementable in $\pi$, that is,

$$\begin{aligned} \pi \circ \gamma =\mathrm {Ad}U_{\gamma }\circ \pi \end{aligned}$$

(A.5)

for some unitary $U_{\gamma }$ on $\mathcal {H}$. Thus we can write for any $A\in \mathfrak {A}$

$$\begin{aligned} \omega (\gamma (A))&=\langle \Omega , \pi (\gamma (A)) \Omega \rangle =\langle {U_{\gamma }^*}\Omega , \pi (A) {U_{\gamma }^* }\Omega \rangle \end{aligned}$$

(A.6)

$$\begin{aligned}&=\langle \pi (V_{\gamma }^*)\Omega , \pi (A) \pi (V_{\gamma }^*)\Omega \rangle =\omega (V_{\gamma }AV_{\gamma }^*), \end{aligned}$$

(A.7)

where we used again [KR, Theorem 10.2.1] to find a unitary $V_{\gamma }\in \mathfrak {A}$ such that $\pi (V_{\gamma }^*)\Omega ={U_{\gamma }^* }\,\Omega$.

Now suppose that $\omega$ and $\omega \circ \gamma$ are in the same sector, i.e., $\omega =\omega \circ \gamma \circ i$ for some $i\in \mathrm {In}(\mathfrak {A})$. Then $\gamma \circ i$ leaves $\omega$ invariant, hence it is unitarily implementable by the GNS theorem. As any $i\in \mathrm {In}(\mathfrak {A})$ is unitarily implementable, we conclude the proof. $\quad \square$

Vector Valued Spherical Harmonics

As we did not find a satisfactory reference, in this appendix we summarize the basic properties of the vector valued spherical harmonics from Sect. 5:

$$\begin{aligned} \pmb {Y}_{\ell m \pm } = \frac{1}{\sqrt{\ell (\ell +1)}}\pmb {a}_\pm Y_{\ell m} \quad \text {with} \quad \pmb {a}_+ = |\pmb {k}| \nabla _{\pmb {k}} \quad \text {and} \quad \pmb {a}_- = \hat{\pmb {k}} \times \pmb {a}_+, \end{aligned}$$

(B.1)

where $Y_{\ell m}$ are the usual spherical harmonics, orthonormal with respect to the measure $d\Omega (\theta , \phi )=\sin \, \theta \, d\theta d\phi$. For this purpose we recall that the total angular momentum of a photon is a self-adjoint operator on $L^2_{\mathrm {tr}}(\mathbb {R}^3;\mathbb {C}^3)$ given by

$$\begin{aligned} \pmb {J} =\pmb {L}+\pmb {S} , \quad \pmb {L}:= -i \pmb {k} \times \nabla _{\pmb {k}}, \quad \pmb {S} =(S_1, S_2, S_3), \quad (S_k{\pmb {\psi }} ):=i\pmb {e}_k\times {\pmb {\psi }}, \end{aligned}$$

(B.2)

where ${\pmb {\psi }}\in L^2_{\mathrm {tr}}(\mathbb {R}^3;\mathbb {C}^3)$ and $\{ \pmb {e}_k \}_{k=1,2,3}$ is the canonical basis in $\mathbb {R}^3$. The operators ${\{S_k\}_{k=1,2,3} }$ satisfy the standard angular momentum commutation relations and $\pmb {S} ^2=2$ holds true (cf. [LL, §58, Problem 2]).

Proposition B.1

The vector valued spherical harmonics $\pmb {Y}_{\ell m \pm }$, $\ell \in \mathbb {N}$, $-\ell \le m \le \ell$, given by equation (B.1)

• (1) satisfy $\pmb {J}^2 \pmb {Y}_{\ell m \pm }= \ell (\ell +1)\pmb {Y}_{\ell m \pm }$ and $\pmb {J}_3 \pmb {Y}_{\ell m \pm }= m \pmb {Y}_{\ell m \pm }$;

• (2) form (a) an orthonormal and (b) complete basis of $L^2_{\mathrm {tr}}(S^2; {\mathbb {C}}^3)$.

Proof

• (1) Following [BLP], we compute on $C^2$ functions from $L^2_{\mathrm {tr}}(S^2; {\mathbb {C}}^3)$

  $$\begin{aligned}{}[L_i, a_{\pm p}] = i \epsilon _{ipq} a_{\pm q}, \end{aligned}$$

  (B.3)

  where $L_i$ and $a_p$ are the Cartesian components of $\pmb {L},\pmb {a}$, respectively, and $\epsilon _{ip q}$ is the Levi-Civita symbol. Since $(S_i \pmb {\psi })_p = -i\epsilon _{ipq}\psi _q$, equation (B.3) applied to $Y_{\ell m}$ (which are smooth functions [Ho65, Chapter IV]) can be written as

  $$\begin{aligned} L_i a_{\pm p}Y_{\ell m} - a_{\pm p}L_i Y_{\ell m}= -(S_i \pmb {a}_\pm Y_{\ell m})_p \end{aligned}$$

  (B.4)

  which implies

  $$\begin{aligned} (J_i \pmb {a}_\pm Y_{\ell m})_p = L_i a_{\pm p}Y_{\ell m} + (S_i \pmb {a}_\pm Y_{\ell m})_p = a_{\pm p} L_i Y_{\ell m}. \end{aligned}$$

  (B.5)

  Consequently, for $i= 3$ the above equation yields

  $$\begin{aligned} J_3 (\pmb {a}_\pm Y_{\ell m}) = \pmb {a}_\pm L_3 Y_{\ell m}. \end{aligned}$$

  (B.6)

  Instead, by applying $J_i$ twice and summing over the index $i=1,2,3$, we find from (B.5),

  $$\begin{aligned} \pmb {J}^2(\pmb {a}_\pm Y_{\ell m}) = \pmb {a}_\pm \pmb {L}^2 Y_{\ell m}. \end{aligned}$$

  (B.7)

  Since the scalar spherical harmonics are eigenfunctions of the operators $\pmb {L}^2$ and $L_3$ with eigenvalues $\ell (\ell +1)$ and m, respectively, we arrive at (1).

• (2a) We denote by $\nabla _t = \pmb {a}_+$ the gradient on the sphere. Then the $\pmb {Y}_{\ell m +}$ are orthogonal to the $\pmb {Y}_{\ell ' m' -}$, which follows by Green’s identity on $S^2$. Similarly for $\pmb {Y}_{\ell m +}$,

  $$\begin{aligned}&\frac{1}{\ell (\ell +1)}\int _{S^2} \nabla _t \overline{Y_{\ell m}(\hat{\pmb {k}})} \cdot \nabla _t Y_{\ell ' m'}( \hat{\pmb {k}} ) \; d\Omega \nonumber \\&\quad = -\frac{1}{\ell (\ell +1)} \int _{S^2} \Delta _t \overline{Y_{\ell m}( \hat{\pmb {k}} )} Y_{\ell ' m'}( \hat{\pmb {k}} ) \; d\Omega = \delta _{\ell \ell '}\delta _{mm'}, \end{aligned}$$

  (B.8)

  since $Y_{\ell m}$ are orthonormal eigenfunctions of $\Delta _t$ with eigenvalues $-\ell (\ell +1)$. As for $\pmb {Y}_{\ell m -}$, we have

  $$\begin{aligned}&\int _{S^2} \big ( \hat{\pmb {k}} \times \overline{\pmb {Y}_{\ell m +}(\hat{\pmb {k}})} \big ) \cdot \big ( \hat{\pmb {k}} \times \pmb {Y}_{\ell ' m' +}(\pmb {\hat{k}}) \big ) \; d\Omega \nonumber \\&\quad =\int _{S^2} (\hat{\pmb {k}} \cdot \hat{\pmb {k}})\big ( \overline{\pmb {Y}_{\ell m +}(\hat{\pmb {k}}) } \cdot \pmb {Y}_{ \ell ' m' +}(\hat{\pmb {k}}) \big ) - \big ( \hat{\pmb {k}}\cdot \pmb {Y}_{\ell ' m' +}(\hat{\pmb {k}}) \big ) \big ( \overline{\pmb {Y}_{\ell m+}(\hat{\pmb {k}})}\cdot \hat{\pmb {k}}\big )\; d\Omega = \delta _{\ell \ell '}\delta _{m m'},\quad \qquad \end{aligned}$$

  (B.9)

  using the above result for $\pmb {Y}_{\ell m +}$ and the fact that these are orthogonal to $\hat{\pmb {k}}$.

• (2b) Completeness is a consequence of [Wi57, Theorem 3.4], which states that given a field of tangent vectors $\pmb {\alpha }$ in the class $C^3$ on the unit sphere $S^2$, there exist unique functions F and G of class $C^2$ on $S^2$ such that

  $$\begin{aligned} \int _{S^2} F \, d\Omega = \int _{S^2} G\, d\Omega =0 \end{aligned}$$

  (B.10)

  and

  $$\begin{aligned} \quad \pmb {\alpha } = \nabla _t F + \hat{\pmb {k}} \times \nabla _t G. \end{aligned}$$

  (B.11)

  We consider an arbitrary smooth field of tangent vectors $\pmb {\alpha }$ on $S^2$. As this is more restrictive than the hypothesis in Wilcox’s theorem above, we can apply this theorem to $\pmb {\alpha }$, yielding functions F and G of class $C^2$ on $S^2$ with the properties (B.10) and (B.11) above.

  Let us now decompose F, G into sums which converge in $L^2(S^2)$:

  $$\begin{aligned} F=\sum _{\ell ,m}c_{\ell m}Y_{\ell m}, \quad G=\sum _{\ell ,m}d_{\ell m}Y_{\ell m}, \end{aligned}$$

  (B.12)

  and substitute them to equation (B.11). In order to exchange the sums with the action of $\nabla _t$, $({\hat{\varvec{k}}}\times \nabla _t)$, we do the following computation for any $\pmb {\alpha }_1$ in the domain of the adjoint maps $\nabla _t^{*}, \, ({\hat{\varvec{k}}}\times \nabla _t)^*$. (For example, we can choose $\pmb {\alpha }_1$ smooth. As we indicate below, such vector fields form a dense subspace in $L^2_{\mathrm {tr}}(S^2;\mathbb {C}^3)$).

  $$\begin{aligned} \langle \pmb {\alpha }_1, \pmb {\alpha }\rangle&= \langle \pmb {\alpha }_1, \nabla _t F + \hat{\pmb {k}} \times \nabla _t G\rangle \nonumber \\&= \langle (\nabla _t)^* \pmb {\alpha }_1, F\rangle + \langle ( \hat{\pmb {k}} \times \nabla _t)^*\pmb {\alpha }_1, G\rangle \nonumber \\&=\sum _{\ell ,m} c_{\ell ,m} \langle (\nabla _t)^* \pmb {\alpha }_1, Y_{\ell m}\rangle + \sum _{\ell ,m} d_{\ell ,m} \langle ( \hat{\pmb {k}} \times \nabla _t)^*\pmb {\alpha }_1, Y_{\ell m}\rangle \nonumber \\&=\sum _{\ell ,m} c_{\ell ,m} \langle \pmb {\alpha }_1, \nabla _tY_{\ell m}\rangle + \sum _{\ell ,m} d_{\ell ,m} \langle \pmb {\alpha }_1, ( \hat{\pmb {k}} \times \nabla _t)Y_{\ell m}\rangle \nonumber \\&=\sum _{\ell ,m} c_{\ell ,m}\sqrt{\ell (\ell +1)} \langle \pmb {\alpha }_1, \pmb {Y}_{\ell m+}\rangle + \sum _{\ell ,m} d_{\ell ,m}\sqrt{\ell (\ell +1)} \langle \pmb {\alpha }_1, \pmb {Y}_{\ell m-}\rangle . \end{aligned}$$

  (B.13)

  To proceed, we need to show that

  $$\begin{aligned} \sum _{\ell ,m} |c_{\ell ,m}|^2 \ell (\ell +1)<\infty , \quad \sum _{\ell ,m} |d_{\ell ,m}|^2 \ell (\ell +1)<\infty . \end{aligned}$$

  (B.14)

  To this end, we compute using formula (B.11)

  $$\begin{aligned} \langle \pmb {Y}_{\ell m +}, \pmb {\alpha } \rangle&= \frac{1}{\sqrt{\ell (\ell +1)} } \langle \nabla _t Y_{\ell m}, \pmb {\alpha }\rangle = \frac{1}{\sqrt{\ell (\ell +1)} } \langle \nabla _t Y_{\ell m}, \nabla _t F\rangle \nonumber \\&=-\frac{1}{\sqrt{\ell (\ell +1)} }\langle \Delta _t Y_{\ell m}, F\rangle =-\sqrt{\ell (\ell +1)} \langle Y_{\ell m}, F\rangle . \end{aligned}$$

  (B.15)

  Since $c_{\ell ,m}=\langle Y_{\ell m}, F\rangle$ and $\pmb {Y}_{\ell m +}$ form an orthonormal system, the first bound in (B.14) follows. The second bound is proven analogously.

  Given (B.14), computation (B.13) gives

  $$\begin{aligned} \langle \pmb {\alpha }_1, \pmb {\alpha }- \sum _{\ell m \pm } \tilde{c}_{\ell m \pm } \pmb {Y}_{\ell m \pm }\rangle =0 \end{aligned}$$

  (B.16)

  for some square-summable coefficients $\tilde{c}_{\ell m \pm }$. By taking supremum over all $\pmb {\alpha }_1$ in the dense domain specified above, subject to $\Vert \pmb {\alpha }_1\Vert \le 1$, we get

  $$\begin{aligned} \Vert \pmb {\alpha }- \sum _{\ell m \pm } \tilde{c}_{\ell m \pm } \pmb {Y}_{\ell m \pm }\Vert =0 \end{aligned}$$

  (B.17)

  which implies that any smooth $\pmb {\alpha }$ is in the closed subspace spanned by $\pmb {Y}_{\ell m \pm }$.

  Hence it only remains to be shown that the smooth vector fields on the unit sphere are $L^2$-dense in the space of all $L^2$ vector fields. For this, we consider a generic $L^2$ vector field $\pmb {\beta }$ and we split it into a sum $\pmb {\beta }= \pmb {\beta }_n + \pmb {\beta }_s$, where $\pmb {\beta }_{n,s}$ have support in the north and south hemisphere, respectively. Now stereographic projections map each hemisphere to a circle in ${\mathbb {R}}^2$, and the transformed $\pmb {\beta }_{n,s}$ can be approximated by smooth vector fields on a slightly larger circle. Applying the inverse transformation yields the result. $\quad \square$