\\{\mathfrak{x}\in\sigma\cap\Gamma^{\perp^{\prime}}:\mathfrak{x}\not\perp\Gamma\\}\subset\\{\mathfrak{x}\in\sigma:(\mathfrak{x}-\mathfrak{y})^{\cdot
2}\leq 0\textnormal{ for all
}\mathfrak{y}\in\Gamma,(\mathfrak{x}-\mathfrak{y})^{\cdot 2}\geq 0\textnormal{
for some
}\mathfrak{y}\in\Gamma\\}\subset\\{\mathfrak{x}\in\sigma:\sup_{\mathfrak{y}\in\Gamma}(\mathfrak{x}-\mathfrak{y})^{\cdot
2}=0\\}$, which is the null set $N(\Gamma,\sigma)$ in (7.3).∎

Moreover, the analogue of (12) for $\perp^{\prime}$-diamonds is valid. It
shows in particular that $E$ assigns to all measurable spacelike flat bases of
a $\perp^{\prime}$-diamond the same operator.

(122) Lemma. Let $\sigma,\tau$ be spacelike hyperplanes. Consider a
$\perp^{\prime}$-diamond $M=\Delta^{\land^{\prime}}$ with measurable flat base
$\Delta\subset\sigma$. Let $\Gamma\subset M\cap\tau$ be a measurable set. Then
$E(\Gamma)\leq E(\Delta)$. If $\Gamma^{\land^{\prime}}=M$ then
$E(\Delta)=E(\Gamma)$.

$\Gamma$$\Delta$$\tau$$\sigma$$M$

Proof. By (7.3),
$M(\Gamma,\sigma)=\bigcup_{\mathfrak{y}\in\Gamma}\\{\mathfrak{x}\in\sigma:(\mathfrak{x}-\mathfrak{y})^{\cdot
2}>0\\}\subset\Gamma_{\sigma}$ equals $\Gamma_{\sigma}$ up to a null set of
$\sigma$. Recall $\Gamma\subset\Delta^{\land^{\prime}}$. Then, by (18.2)(v),
$M(\Gamma,\sigma)\subset\Delta$. So causality (7.2)(a) implies $E(\Gamma)\leq
E(\Gamma_{\sigma})=E(M(\Gamma,\sigma))\leq E(\Delta)$. ∎

We turn to the announced properties of a rep $F^{\prime}$ of
$\mathcal{M}^{{}^{\prime}borel}$. To this end we need the results in (20) and
(20).

(123) Lemma. Let $F^{\prime}$ be a rep of $\mathcal{M}^{{}^{\prime}borel}$.

  * (a)

Let $L,M\in\mathcal{M}^{{}^{\prime}borel}$ with $L\subset M$. Then
$F^{\prime}(M\cap L^{\perp^{\prime}})=F^{\prime}(M)-F^{\prime}(L)$. In
particular, $F^{\prime}$ is monotone.

  * (b)

Let $M_{n}\in\mathcal{M}^{{}^{\prime}borel}$, $n\in\mathbb{N}$ with
$M_{1}\supset M_{2}\supset M_{3}\supset\dots$. Then
$F^{\prime}(\bigcap_{n}M_{n})=\lim_{n}F^{\prime}(M_{n})$.

  * (c)

Let $M_{n}\in\mathcal{M}^{{}^{\prime}borel}$, $n\in\mathbb{N}$ with
$M_{1}\subset M_{2}\subset M_{3}\subset\dots$. Then
$F^{\prime}(\bigvee_{n}M_{n})=\lim_{n}F^{\prime}(M_{n})$.

Proof. (a) By orthomodularity on has the orthogonal decomposition
$M=L\lor(M\cap L^{\perp^{\prime}})$. Hence orthoadditivity of $F^{\prime}$
yields $F^{\prime}(M)=F^{\prime}(L)+F^{\prime}(M\cap L^{\perp^{\prime}})\geq
F^{\prime}(L)$. — (b) Put $M:=\bigcap_{n}M_{n}$. Apply (11.3) and
$\sigma$-orthoadditivity of $F^{\prime}$ to $(M_{n})$ as well to
$(M_{1},M_{2},\dots,M_{m},\emptyset,\emptyset,\dots)$ for $m\in\mathbb{N}$.
Then $F^{\prime}(M_{1}\cap M^{\perp^{\prime}})=\sum_{n}F^{\prime}(M_{n}\cap
M_{n+1}^{\perp^{\prime}})=\lim_{m}\sum_{n=1}^{m}F^{\prime}(M_{n}\cap
M_{n+1}^{\perp^{\prime}})=\lim_{m}F^{\prime}(M_{1}\cap
M_{m}^{\perp^{\prime}})$, whence
$F^{\prime}(M_{1})-F^{\prime}(M^{\perp^{\prime}})=F^{\prime}(M_{1})-\lim_{m}F^{\prime}(M_{m})$
proving (b). — (c) follows from (b) for $(M_{n}^{\perp^{\prime}})$ as
generally $F^{\prime}(M^{\perp^{\prime}})=I-F^{\prime}(M)$ for
$M\in\mathcal{M}^{{}^{\prime}borel}$.∎

(124) Lemma. Let $\chi$ be the non timelike non spacelike hyperplane
$\\{\mathfrak{x}:x_{0}=x_{3}\\}$ and
$\tilde{\mathcal{P}}_{\chi}:=\\{g\in\tilde{\mathcal{P}}:g\cdot\chi\subset\chi\\}$
the invariance subgroup. Then

$\tilde{\mathcal{P}}_{\chi}=\\{(\mathfrak{a},A)\in\tilde{\mathcal{P}}:\mathfrak{a}\in\chi,\;A=\left(\begin{array}[]{cc}z&w\\\
0&1/z\end{array}\right),z\in\mathbb{C}\setminus\\{0\\},w\in\mathbb{C}\\}$

Hence $\tilde{\mathcal{P}}_{\chi}=IST(2)$, i.e., the inhomogeneous group of
upper triangular $2\times 2$ matrices with determinant $1$. So
$g\in\tilde{\mathcal{P}}_{\chi}$ if and only if

$g=\big{(}\mathfrak{a},\,\operatorname{e}^{\zeta\sigma_{3}/2}\big{)}\big{(}0,\,\left(\begin{array}[]{cc}1&w\\\
0&1\end{array}\right)\big{)}$

with $\mathfrak{a}\in\chi$ and
$\zeta=\rho+\operatorname{i}\varphi,w=u+\operatorname{i}v\in\mathbb{C}$. The
first factor
$\big{(}\mathfrak{a},\,\operatorname{e}^{\zeta\sigma_{3}/2}\big{)}$ represents
the obvious part consisting in a translation $\mathfrak{a}$ along $\chi$, a
boost $\Lambda(\operatorname{e}^{\rho\,\sigma_{3}/2})$ in the $x_{3}$
direction, and a rotation
$\Lambda(\operatorname{e}^{\operatorname{i}\varphi\,\sigma_{3}/2})$ in the
$x_{1}x_{2}$ plane. Put $r:=\frac{1}{2}|w|^{2}$. The second factor acts by

$\Lambda\big{(}\left(\begin{array}[]{cc}1&w\\\
0&1\end{array}\right)\big{)}=\left(\begin{array}[]{cccc}1+r&u&-v&-r\\\
u&1&0&-u\\\ -v&0&1&v\\\ r&u&-v&1-r\end{array}\right)$

Proof. Write $\chi=\\{\mathfrak{x}:\mathfrak{x}\cdot\mathfrak{e}=0\\}$ with
$\mathfrak{e}:=(1,0,0,1)$. Then
$g=(\mathfrak{a},A)\in\tilde{\mathcal{P}}_{\chi}$ if and only if
$\chi=g\cdot\chi=\\{\mathfrak{x}:(g^{-1}\cdot\mathfrak{x})\cdot\mathfrak{e}=0\\}=\\{\mathfrak{x}:\mathfrak{x}\cdot(A\cdot\mathfrak{e})=\mathfrak{a}\cdot(A\cdot\mathfrak{e})\\}$,
whence equivalently $\mathfrak{a}\cdot(A\cdot\mathfrak{e})=0$ and
$A\cdot\mathfrak{e}=\lambda\mathfrak{e}$ for some $\lambda\neq 0$, see after
(18.1). Actually $\lambda>0$ as $A$ acts orthocronously. This means
$\mathfrak{a}\in\chi$ and $A=A_{\ln(\lambda)}A^{\prime}$ with $A^{\prime}\in
SL(2,\mathbb{C})_{\mathfrak{e}}=E(2)$. Recall $A\in E(2)\Leftrightarrow
A_{22}=\overline{A}_{11}$, $A_{21}=0$. The remainder follows by explicit
computation. ∎

Now let $W(\alpha)$ denote the time shift by $\alpha\in\mathbb{R}$ and recall
that $A_{\rho
e}=\operatorname{exp}(\frac{\rho}{2}\sum_{k=1}^{3}e_{k}\sigma_{k})$ acts as
the boost in direction $e\in\mathbb{R}^{3}$, $|e|=1$ with rapidity $\rho$. Let
$\varsigma\in\\{1,-1\\}$.
Let $\mathfrak{S}_{\infty}$ denote the set of all measurable subsets of non-
timelike non-spacelike hyperplanes.

(125) Theorem. Let $(W,F^{\prime})$ be a rep of
$\mathcal{M}^{{}^{\prime}borel}$. Set
$E(\Delta):=F^{\prime}(\Delta^{\land^{\prime}})$ for Borel
$\Delta\in\mathfrak{S}$ and $P(\Delta):=F^{\prime}(\Delta^{\land^{\prime}})$
for Borel $\Delta\in\mathfrak{S}_{\infty}$. Then $E$ and $P$ extend to all
sets of $\mathfrak{S}$ and $\mathfrak{S}_{\infty}$, respectively, and

  * (a)

$(W,E)$ is a causal system

  * (b)

$(W,P)$ is a Poincaré covariant localization. More precisely, on every non-
timelike non-spacelike hyperplane $P$ is a PM and
$W(g)P(\Delta)W(g)^{-1}=P(g\cdot\Delta)$ holds for all
$g\in\tilde{\mathcal{P}}$ and $\Delta\in\mathfrak{S}_{\infty}$

There is a close relationship between $E$ and $P$. Let $\alpha>0$ and
$e\in\mathbb{R}^{3}$, $|e|=1$. One has

  * (c)

$P(\\{0\leq\varsigma
x_{0}=xe\leq\alpha\\})=\lim_{\varsigma\rho\to\infty}E(A_{\rho
e}\cdot\\{x_{0}=0,0\leq xe\leq
2\alpha\,\operatorname{e}^{-\varsigma\rho}\\})$, where the limit exists since
the projections decrease as $\varsigma\rho$ increases

  * (d)

$P(\\{0\leq\varsigma
x_{0}=xe\leq\alpha\\})=E(\\{x_{0}=\varsigma\alpha,xe\leq\alpha\\})E(\\{x_{0}=0,xe\geq
0\\})$

  * (e)

$P(\\{0\leq\varsigma x_{0}=xe\\})=E(\\{\mathfrak{x}:x_{0}=0,xe\geq 0\\})$

Proof. (a) $(W,E)$ is a causal system by (19).

(b) Poincaré covariance of $P$ follows immediately from the Poincaré
covariance of $F^{\prime}$ since
$(g\cdot\Delta)^{\land^{\prime}}=g\cdot\Delta^{\land^{\prime}}$. — As noted
after (18.1) a non-timelike hyperplane is a maximal $\perp^{\prime}$-set and
hence by Eq. (18.2) a $\perp^{\prime}$-base. Now $\sigma$-additivity of $P$ on
every non-timelike non-spacelike hyperplane follows as the $\sigma$-additivity
of $E$ in (19). — Now extend $P$ to all of $\mathfrak{S}_{\infty}$, see (6.2).

(c) Obviously the sets
$M_{\rho}:=\\{-2\alpha\operatorname{e}^{-2\varsigma\rho}\leq\varsigma
x_{0}-xe\leq 0,0\leq\varsigma x_{0}+xe\leq 2\alpha\\}$, $\rho\in\mathbb{R}$
converge to $M:=\\{0\leq\varsigma x_{0}=xe\leq\alpha\\}$ for
$\varsigma\rho\to\infty$, as

$M_{\rho^{\prime}}\subset M_{\rho}\quad\textnormal{if\,
}\varsigma\rho<\varsigma\rho^{\prime}\quad\textnormal{and\;
}M=\cap_{\rho}\,M_{\rho}$

We check $M_{\rho}=\Gamma_{\rho}^{\land^{\prime}}$ for $\Gamma_{\rho}:=A_{\rho
e}\cdot\\{x_{0}=0,0\leq xe\leq 2\alpha\operatorname{e}^{-\varsigma\rho}\\}$
($=\\{0\leq-\sinh(\rho)\,x_{0}+\cosh(\rho)\,xe\leq
2\alpha\operatorname{e}^{-\varsigma\rho}\\}$). Indeed,
$\Gamma_{\rho}^{\land^{\prime}}=A_{\rho e}\cdot\\{x_{0}=0,0\leq xe\leq
2\alpha\operatorname{e}^{-\varsigma\rho}\\}^{\land^{\prime}}=A_{\rho
e}\cdot\\{-2\alpha\operatorname{e}^{-\varsigma\rho}\leq x_{0}-xe\leq 0,0\leq
x_{0}+xe\leq
2\alpha\operatorname{e}^{-\varsigma\rho}\\}=\\{-2\alpha\operatorname{e}^{-\varsigma\rho-\rho}\leq
x_{0}-xe\leq 0,0\leq x_{0}+xe\leq
2\alpha\operatorname{e}^{-\varsigma\rho+\rho}\\}=M_{\rho}$. — Hence $M_{\rho}$
is a $\perp^{\prime}$-diamond with the spacelike flat base $\Gamma_{\rho}$,
and $M=M^{\land^{\prime}}$, cf. also (18.3)(2). So using (20)(b) one infers
$P(M)=F^{\prime}(M)=\lim_{\varsigma\rho\to\infty}F^{\prime}(M_{\rho})=\lim_{\varsigma\rho\to\infty}E(\Gamma_{\rho})$
showing (c).

$\varsigma=+$$\rho>0$$\Gamma_{\rho}$$\Gamma_{0}$$2\alpha$$xe$$x_{0}$$M$

(d) By Eq. (10.3) the right hand side of (d) equals
$I-E(\\{x_{0}=\varsigma\alpha,xe>\alpha\\})-E(\\{x_{0}=0,xe<0\\})$ and hence
by definition
$=I-F^{\prime}(\\{x_{0}=\varsigma\alpha,xe>\alpha\\}^{\land^{\prime}})-F^{\prime}(\\{x_{0}=0,xe<0\\}^{\land^{\prime}})$,
and further $=I-F^{\prime}(\\{\varsigma
x_{0}=xe>\alpha\\}^{\land^{\prime}})-F^{\prime}(\\{\varsigma
x_{0}=xe<0\\}^{\land^{\prime}})$ due to (18.3), and finally
$=F^{\prime}(\\{0\leq\varsigma
x_{0}=xe\leq\alpha\\}^{\land^{\prime}})=P(\\{0\leq\varsigma
x_{0}=xe\leq\alpha\\})$ by orthoadditivity of $F^{\prime}$.

$xe$$x_{0}$$\alpha$$\alpha$$M$$\varsigma=+$

(e) By (18.3) one has $P(\\{0\leq\varsigma
x_{0}=xe\\})=F^{\prime}(\\{0\leq\varsigma
x_{0}=xe\\}^{\land^{\prime}})=F^{\prime}(\\{\mathfrak{x}:x_{0}=0,xe\geq
0\\}^{\land^{\prime}})=E(\\{\mathfrak{x}:x_{0}=0,xe\geq 0\\}$.∎

Let us comment on (20). The question is what does it mean that a state
$\varphi$ is localized in $\\{0\leq\varsigma x_{0}=xe\leq\alpha\\}$, i.e.
$P(\\{0\leq\varsigma x_{0}=xe\leq\alpha\\})\varphi=\varphi$. Clearly it is
senseless saying that the spatial probability of the system in
$\\{0\leq\varsigma x_{0}=xe\leq\alpha\\}$ in the state $\varphi$ is $1$, since
$\\{\varsigma x_{0}=xe\\}$ is not a reference frame. In the following one
recognizes $\varphi$ to be a late-change state in case of a Dirac or Weyl
state.
According to (c), the answer is that $\varphi$ is just a state for which the
boosted state $W(A_{\varsigma\rho e})\varphi$ is localized in
$\\{x_{0}=0,0\leq xe\leq 2\alpha\operatorname{e}^{-\rho}\\}$ for every
rapidity $\rho$. (Recall Poincaré covariance (6.2)(b).) This is particularly
interesting for $\rho\to\infty$ as the region $\\{x_{0}=0,0\leq xe\leq
2\alpha\operatorname{e}^{-\rho}\\}$ of localization shrinks to an arbitrarily
narrow strip at the origin. This is pure Lorentz contraction shown for Dirac
and Weyl wave functions in sec. 13.3 and (16.1), respectively.
Furthermore, according to (d), $\varphi$ is just a state which is localized in
$\\{x_{0}=0,xe\geq 0\\}$ and which at time $\varsigma\alpha$, i.e.,
$W(\varsigma\alpha)\varphi$, is localized in $\\{x_{0}=0,xe\leq\alpha\\}$.
(Note that the right hand side of (d) is the product of two projection which
commute due to causality, and recall Poincaré covariance (6.2)(b).) The
equality of the right hand sides of (c) and (d) is shown to be valid for Dirac
and Weyl states in (13.3) and (16.1), respectively. Moreover, by Eq. (10.3),
Eq. (10.4), and (10), the right hand side of (d) is a non-trivial projection
with infinite dimensional eigenspaces.
Since $P(\\{0\leq\varsigma x_{0}=xe\\})=\cup_{\alpha>0}P(\\{0\leq\varsigma
x_{0}=xe\leq\alpha\\})$ by (b), actually (e) is a completeness property. Every
state localized in $\\{x_{0}=0,x_{3}\geq 0\\}$ can be approximated by the
particular states described in (c), (d). This is shown to be true for Dirac
and Weyl states in (13.3) and (16.2), respectively.

Hence to every rep $(W,F^{\prime})$ of $\mathcal{M}^{{}^{\prime}borel}$ there
are related a causal system $(W,E)$ and a Poincaré covariant localization
$(W,P)$ on $\mathfrak{S}_{\infty}$ satisfying (20)(c), (d), (e). These
relations, which refer to more profound properties of causal systems, are
easily elucidated by its geometrical meaning (cf. above). Now we are going to
prove the existence of the localization $(W,P)$ for the Dirac and the Weyl
systems. It is obvious that much more efforts have to be made towards a
possible construction of a rep $(W,F^{\prime})$ of
$\mathcal{M}^{{}^{\prime}borel}$. But, at least, the Dirac and the Weyl
systems allow for a further Poincaré covariant extension of $E$ (recall (6.2)
for the first extension) to PM on all non-timelike hyperplanes. More precisely
on has (20).
Let $\chi$ denote the non-timelike non-spacelike hyperplane
$\\{\mathfrak{x}:x_{0}=x_{3}\\}$ and recall $\tilde{\mathcal{P}}_{\chi}$ in
(20).

(126) Theorem. Let $(W,E)$ be a causal system, which is the finite orthogonal
sum of Dirac and Weyl systems. Then there is a unique Poincaré covariant
localization $(W,P)$ on $\mathfrak{S}_{\infty}$ such that

$P(\\{\mathfrak{x}\in\chi:0\leq\varsigma
x_{3}\leq\alpha\\})=\lim_{\rho\to\infty}E\big{(}\operatorname{e}^{\,\rho\,\sigma_{3}/2}\cdot\\{x_{0}=0,0\leq\varsigma
x_{3}\leq 2\alpha\operatorname{e}^{-\rho}\\}\big{)}$ (1)

for all $\alpha>0$, $\varsigma\in\\{-1,1\\}$. The limit in _(1)_ exists since
the projections decrease as $\rho$ increases.
Proof. Obviously it is no restriction assuming that $(W,E)$ is the Dirac
system or a Weyl system. First, in (a) and (b), we prove the existence of $P$.
Put $\Delta_{\rho}:=\\{x_{0}=0,0\leq\varsigma x_{3}\leq
2\alpha\operatorname{e}^{-\rho}\\}$.

(a) Let $(W,E)$ be the Dirac system. Then
$E^{\textsc{d}\,mom}=\mathcal{F}E^{can}\mathcal{F}^{-1}$,
$\big{(}E^{\textsc{d}\,mom}(\Delta_{\rho})\varphi\big{)}(p)=\int
k_{\rho}(s-p_{3})\varphi(p_{1},p_{2},s)\operatorname{d}s$ for integrable
$\varphi\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$ with
$k_{\rho}(u):=\frac{\exp(\,\operatorname{i}2\alpha\operatorname{e}^{-\rho}\varsigma\,u\,)-1}{2\pi\operatorname{i}\varsigma\,u}$.
Hence, using Poincaré covariance one finds according to Eq. (13.1)

$\big{(}E^{\textsc{d}\,mom}(A_{\rho}\cdot\Delta_{\rho})\varphi\big{)}(p)=\sum_{\eta,\eta^{\prime}}\int
k_{\rho}(s-q^{\eta}_{3})\pi^{\eta}(p)s(A_{\rho})^{-1}\pi^{\eta^{\prime}}(p_{1},p_{2},s)s(A_{\rho})\varphi(p_{1},p_{2},s^{\prime})\operatorname{d}s$

with $q_{3}^{\eta}=\cosh(\rho)p_{3}-\eta\sinh(\rho)\epsilon(p)$ and
$s^{\prime}:=\eta^{\prime}\sinh(\rho)\epsilon(p_{1},p_{2},s)+\cosh(\rho)s$.
Note that
$s=\cosh(\rho)s^{\prime}-\eta^{\prime}\sinh(\rho)\epsilon(p_{1},p_{2},s^{\prime})$.
We apply Eq. (13.3) in order to move $s(A_{\rho})$ to the left and substitute
the variable of integration $s$ by $s^{\prime}$. One finds

$\big{(}E^{\textsc{d}\,mom}(A_{\rho}\cdot\Delta_{\rho})\varphi\big{)}(p)=\sum_{\eta,\eta^{\prime}}\int
k_{\rho}(s-q^{\eta}_{3})\pi^{\eta}(p)s(A_{\rho})^{-2}\pi^{\eta^{\prime}}(p_{1},p_{2},s^{\prime})\varphi(p_{1},p_{2},s^{\prime})\operatorname{d}s^{\prime}$
(2)

Now it is easy to perform in (1) the limit $\rho\to\infty$. Check
$|k_{\rho}(u)|\leq\frac{1}{\pi}\alpha\operatorname{e}^{-\rho}$ and
$s(A_{\rho})^{-2}=\operatorname{diag}(\operatorname{e}^{-\rho},\operatorname{e}^{\rho},\operatorname{e}^{\rho},\operatorname{e}^{-\rho})$,
whence
$\parallel\\!\\!k_{\rho}(s-q^{\eta}_{3})s(A_{\rho})^{-2}\\!\\!\parallel\leq\frac{\alpha}{\pi}$
for $\rho\geq 0$. Due to dominated convergence one gets for every
$p\in\mathbb{R}^{3}$

$\lim_{\rho\to\infty}\big{(}E^{\textsc{d}\,mom}(A_{\rho}\cdot\Delta_{\rho})\varphi\big{)}(p)=\sum_{\eta,\eta^{\prime}}\int
k(\mathfrak{p},\mathfrak{p}^{\prime})\pi^{\eta}(p)S\pi^{\eta^{\prime}}(p^{\prime})\varphi(p^{\prime})\operatorname{d}p_{3}^{\prime}$
(3)

where $\mathfrak{p}:=(\eta\epsilon(p),p)$ and
$\mathfrak{p}^{\prime}=(p_{0}^{\prime},p^{\prime})$ with
$p^{\prime}_{1}:=p_{1}$, $p^{\prime}_{2}:=p_{2}$,
$p^{\prime}_{0}:=\eta^{\prime}\epsilon(p^{\prime})$, and
$k(\mathfrak{p},\mathfrak{p}^{\prime}):=\frac{\exp(\operatorname{i}\alpha\varsigma(p_{0}-p_{3}-p^{\prime}_{0}+p^{\prime}_{3}))-1}{\operatorname{i}\pi\varsigma(p_{0}-p_{3}-p^{\prime}_{0}+p^{\prime}_{3})}$,
$S:=\operatorname{diag}(0,1,1,0)$.

The right hand side of (1) is a special case of the right hand side of
(20)(c). Hence, in particular, the limit
$\lim_{\rho\to\infty}E^{\textsc{d}\,mom}(A_{\rho}\cdot\Delta_{\rho})\varphi$
exists, whence for integrable $\varphi$ its value at every
$p\in\mathbb{R}^{3}$ is given by the right hand side of (3).
One notes that $k(\mathfrak{p},\mathfrak{p}^{\prime})$ is just the kernel for
$P^{shell}(M)$ in (24.3)(b). Actually the next step is to show that the shell
representation of
$P(M):=\lim_{\rho\to\infty}E^{\textsc{d}}(A_{\rho}\cdot\Delta_{\rho})$
coincides with $P^{shell}(M)$ in (24.3)(b). To this, according to (24.4), we
have to perform several equivalence transformations, namely
$P^{shell}(M)=(Y\xi Z)^{-1}P^{mom}(M)(Y\xi Z)$. We indicate some of the
intermediate steps.
Straightforward computations yield the unwieldy $4\times 4$ matrices $V(p)$
and $2\times 4$ matrices $U_{\eta}(p)$, which satisfy $(Y\xi
Z\phi)(p)=V(p)\big{(}\phi(\mathfrak{p}^{+}),(\phi(\mathfrak{p}^{-})\big{)}$
for $p\in\mathbb{R}^{3}$, $\mathfrak{p}^{\eta}=(\eta\epsilon(p),p)$ and
$((Y\xi Z)^{-1}\varphi)(\mathfrak{p})=U_{\eta}(p)\varphi(p)$ for
$\mathfrak{p}\in\mathcal{O}^{m}$, $\mathfrak{p}=(\eta\epsilon(p),p)$. However,
after repeated applications of the formulae regarding the canonical cross
section in (3.3) the relevant expressions became clear:

$\displaystyle S\pi^{\eta}(p)V(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2\epsilon(p)}\left(\begin{array}[]{cc}\Tiny{\big{(}\begin{array}[]{cc}0&0\\\
0&1\end{array}\big{)}}\delta_{+,\eta}&\Tiny{\big{(}\begin{array}[]{cc}0&0\\\
0&1\end{array}\big{)}}\delta_{-,\eta}\\\
\Tiny{\big{(}\begin{array}[]{cc}1&0\\\
0&0\end{array}\big{)}}\delta_{+,\eta}&-\Tiny{\big{(}\begin{array}[]{cc}1&0\\\
0&0\end{array}\big{)}}\delta_{-,\eta}\end{array}\right)\left(\begin{array}[]{cc}\sqrt{\epsilon(p)-p_{3}}I_{2}&0\\\
0&\sqrt{\epsilon(p)+p_{3}}I_{2}\end{array}\right)$ $\displaystyle
U_{\eta}(p)\pi^{\eta^{\prime}}(p)S$ $\displaystyle=$
$\displaystyle\delta_{\eta,\eta^{\prime}}\sqrt{\epsilon(p)-\eta
p_{3}}\left(\begin{array}[]{cc}\Tiny{\big{(}\begin{array}[]{cc}0&0\\\
0&1\end{array}\big{)}}&\Tiny{\big{(}\begin{array}[]{cc}1&0\\\
0&0\end{array}\big{)}}\end{array}\right)$

Now it is easy to verify that in the shell representation the right hand side
of (3) coincides for every $\mathfrak{p}\in\mathcal{O}^{m}$ with the right
hand side of (24.3)(b). This confirms the claim.
The considerations so far show by (24.3), (24.4) that
$(W^{\textsc{d}}|_{\tilde{\mathcal{P}}_{\chi}},P)$ is a covariant PM on $\chi$
satisfying (1). Now $P$ is extended to $\mathfrak{S}_{\infty}$ in a Poincaré
covariant manner. One achieves this unique extension arguing as in (6.2). This
concludes the proof of existence of $P$ in the Dirac case.

(b) Now we consider the Weyl system $(W^{\textsc{w}\chi},E^{\textsc{w}})$.
(Here $\chi=\pm$ denotes the handedness (cf. after Eq. (16.1)) not to be
confused with the hyperplane $\chi=\\{x_{0}=x_{3}\\}$.) One gets quite
analogously to the Dirac case

$\lim_{\rho\to\infty}\big{(}E^{\textsc{w}\,mom}(A_{\rho}\cdot\Delta_{\rho})\varphi\big{)}(p)=\sum_{\eta,\eta^{\prime}}\int
k(\mathfrak{p},\mathfrak{p}^{\prime})\pi^{\chi\eta}(p)S^{\chi}\pi^{\chi\eta^{\prime}}(p^{\prime})\varphi(p^{\prime})\operatorname{d}p_{3}^{\prime}$
(4)

with $S^{+}:=\operatorname{diag}(0,1)$, $S^{-}:=\operatorname{diag}(1,0)$.
It remains to check that the cone representation of
$P(M):=\lim_{\rho\to\infty}E^{\textsc{w}}(A_{\rho}\cdot\Delta_{\rho})$ in (4)
coincides with $P^{cone}(M)$ in (24.3). According to (24.5), the former is
$P^{cone}(M)=(Y^{\chi}\xi_{0}Z^{\chi})^{-1}P^{mom}(M)(Y^{\chi}\xi_{0}Z^{\chi})$.
The explicit computations just use repeatedly Eq. (21.8), Eq. (21.9).

This concludes the proof of the existence of $P$. We turn to its uniqueness.
Let $(W,E)$ be a causal system and let $P$, $P^{\prime}$ be two PM on $\chi$
which are covariant with respect to $W|_{\tilde{\mathcal{P}}_{\chi}}$ and
which satisfy (1). The assertion is $P=P^{\prime}$.

(c) Put $\overline{\mathbb{R}}:=\mathbb{R}\cup\\{-\infty,\infty\\}$. For
$a,b\in\overline{\mathbb{R}}^{3}$ with $a_{k}\leq b_{k}$, $k=1,2,3$ define the
box

$\langle a,b[_{\chi}:=\\{\mathfrak{x}\in\chi:a_{1}\leq
x_{3}-x_{1}<b_{1},a_{2}\leq x_{3}+x_{2}<b_{2},a_{3}\leq x_{3}<b_{3}\\}$

in $\chi$. Call a figure the union of finitely many boxes, and let
$\mathcal{Q}_{\chi}$ denote the set of all figures. The claim is that every
figure is the union of finitely many disjoint boxes and that
$\mathcal{Q}_{\chi}$ is an algebra.
Indeed, it is well-known that the unions of finitely many disjoint boxes
$\langle a,b[:=\\{x\in\mathbb{R}^{3}:a_{k}\leq x_{k}<b_{k},k=1,2,3\\}$,
$a,b\in\overline{\mathbb{R}}^{3}$ form an algebra $\mathcal{Q}^{3}$ in
$\mathbb{R}^{3}$.222222 $\mathcal{Q}^{3}$ is $\cap$-stable since the
intersection of two boxes is a box. Check that the complement of a box
$\mathbb{R}^{3}\setminus\langle a,b[$ is the union of disjoint boxes. Then
also the complement of a figure is the union of finitely many disjoint boxes.
Therefore $\mathcal{Q}^{3}$ is an algebra. Finally, let $M=B_{1}\cup\dots\cup
B_{n}$ be a figure with boxes $B_{i}$. Then $M=B_{1}\cup(B_{2}\setminus
B_{1})\cup\dots\cup\big{(}B_{n}\setminus(B_{1}\cup\dots\cup B_{n-1})\big{)}$
is the union of disjoint boxes. The bijective linear map
$l:\chi\to\mathbb{R}^{3}$, $l(\mathfrak{x}):=(x_{3}-x_{1},x_{2}+x_{3},x_{3})$
(with $l^{-1}(x)=(x_{3},x_{3}-x_{1},x_{2}-x_{3},x_{3})$) satisfies
$\mathcal{Q}_{\chi}=l^{-1}(\mathcal{Q}^{3})$, $\langle
a,b[_{\chi}\,=l^{-1}(\langle a,b[\,)$. This confirms the claim.

(d) The claim now is
$P|_{\mathcal{Q}_{\chi}}=P^{\prime}|_{\mathcal{Q}_{\chi}}$. Indeed, due to
(1), $P$ and $P^{\prime}$ coincide at the boxes $\\{0\leq x_{3}<\alpha\\}$ and
$\\{-\alpha\leq x_{3}<0\\}$, $\alpha>0$. (Note that $P$ and $P^{\prime}$
vanish at the Lebesgue null sets $\\{x_{3}=\gamma\\}$, $\gamma\in\mathbb{R}$.)
By continuity $P$, $P^{\prime}$ are equal also at the boxes $\\{0\leq
x_{3}<\infty\\}$, $\\{-\infty<x_{3}<0\\}$. Then one easily infers from
additivity that $P$, $P^{\prime}$ coincide at all boxes
$B(\alpha,\beta):=\\{\mathfrak{x}\in\chi:\alpha\leq x_{3}<\beta\\}$,
$\alpha,\beta\in\overline{\mathbb{R}}$, $\alpha\leq\beta$.
Next check $\\{\mathfrak{x}\in\chi:a_{1}\leq x_{3}-x_{1}<b_{1}\\}=\kappa\cdot
B(a_{1},b_{1})$ and $\\{\mathfrak{x}\in\chi:a_{2}\leq
x_{3}+x_{2}<b_{2}\\}=\kappa^{\prime}\cdot B(a_{2},b_{2})$ for
$\kappa:=I_{2}+\frac{1}{2}(\sigma_{1}+\operatorname{i}\sigma_{2})$,
$\kappa^{\prime}:=I_{2}+\frac{1}{2}(\operatorname{i}\sigma_{1}-\sigma_{2})$ in
$ST(2)$, see (20). Hence $\langle a,b[_{\chi}=\big{(}\kappa\cdot
B(a_{1},b_{1})\big{)}\cap\big{(}\kappa^{\prime}\cdot B(a_{2},b_{2})\big{)}\cap
B(a_{3},b_{3})$. Therefore $P(\langle a,b[_{\chi})=P\big{(}\kappa\cdot
B(a_{1},b_{1})\big{)}P\big{(}\kappa^{\prime}\cdot
B(a_{2},b_{2})\big{)}P\big{(}B(a_{3},b_{3})\big{)}$ and hence $P(\langle
a,b[_{\chi})=W(\kappa)P\big{(}B(a_{1},b_{1})\big{)}W(\kappa)^{-1}\,W(\kappa^{\prime})P\big{(}B(a_{2},b_{2})\big{)}W(\kappa^{\prime})^{-1}\,P\big{(}B(a_{3},b_{3})\big{)}$
by $ST(2)$-covariance. This equation holds equally for $P^{\prime}$ in place
of $P$. Since generally
$P\big{(}B(\alpha,\beta)\big{)}=P^{\prime}\big{(}B(\alpha,\beta)\big{)}$, it
follows $P(\langle a,b[_{\chi})=P^{\prime}(\langle a,b[_{\chi})$ for all
boxes, whence the claim.

(e) Now we are ready for the proof of $P=P^{\prime}$. For every vector
$\varphi$,
$G(M):=\langle\varphi,P(M)\varphi\rangle=\langle\varphi,P^{\prime}(M)\varphi\rangle$
defines a finite $\sigma$-additive content on $\mathcal{Q}_{\chi}$. It allows
a unique extension to a measure on the $\sigma$-algebra of Borel sets of
$\chi$ generated by $\mathcal{Q}_{\chi}$ (see e.g. [79, 5.7]). One infers that
$P$ and $P^{\prime}$ coincides at all Borel sets of $\chi$, whence
$P=P^{\prime}$ arguing as in (6.2).∎

(127) Corollary. $P$ from _( 20)_ satisfies _(c), (d), (e)_ in _( 20)_. One
obtains an alternative proof of _( 13.2.2)_ and _( 16.2)_.

Proof. Note $B(\varsigma e)\cdot\\{x_{0}=x_{3},0\leq\varsigma
x_{3}<\alpha\\}=\\{0\leq\varsigma x_{0}=xe<\alpha\\}$. Hence by covariance one
infers that (c) and (1) are equivalent. So (c) holds. Therefore, due to (13.3)
and Eq. (16.5) also (d) holds. Hence, by (10)(b), $P(\\{0\leq\varsigma
x_{0}=xe\\})\leq E(\\{\mathfrak{x}:x_{0}=0,xe\geq 0\\})$. Replacing
$(e,\varsigma)$ by $(-e,-\varsigma)$ one gets $P(\\{0\geq\varsigma
x_{0}=xe\\})\leq E(\\{\mathfrak{x}:x_{0}=0,xe\leq 0\\})$. The sum of the left
hand sides is $I$. Also the sum of the right hand sides equals $I$. This
proves that the inequalities actually are equalities. Hence (e) holds and
$I=\lim_{|\alpha|\to\infty}E(\\{x_{0}=\alpha,|xe|\leq|\alpha|\\})$ follows.∎

As an example consider the maximal $\perp^{\prime}$-set $\sigma_{M}\cup
M\cup\tau_{M}$ from (18.1). One has

$I=E(\sigma_{M})+P(M)+E(\tau_{M})$

Indeed, it suffices to verify this equation in the special case Eq. (18.1).
This follows easily from $P(\\{\mathfrak{x}:0\leq
x_{0}=xe\leq\alpha\\})=E(\\{\mathfrak{x}:x_{0}=\alpha,xe\leq\alpha\\})-E(\\{\mathfrak{x}:x_{0}=0,xe\leq
0\\})$, which holds by (20)(d) because of Eq. (10.4), Eq. (10.3). Furthermore,
because of (18.3) and (20), the spacelike hyperplanes $\sigma$ and $\tau$ may
even intersect as in the figure below.

$\sigma_{M}$$\tau_{M}$$M$

##  21\. Systems with Causal Time Evolution

After having studied in detail the massive and massless causal systems of
fundamental importance, i.e., in sec. 13, 14, 15 the Dirac system
$(W^{\textsc{d}},E^{\textsc{d}})$ for every positive mass and in sec. 16 the
two Weyl systems $(W^{\textsc{w}\chi},E^{\textsc{w}})$, $\chi\in\\{+,-\\}$,
now the goal is to determine all other causal systems. In so doing we will
consider only physical systems, i.e., systems with non-negative mass-squared
operator

$C=H^{2}-P_{1}^{2}-P_{2}^{2}-P_{3}^{2}\,\,\geq\,0$

and finite spinor dimension (sec. 21.2). Then it will turn out that the Dirac
and the Weyl systems are the only irreducible causal systems. Actually we will
not succeed as yet to rule out certain obviously unlikely irreducible massive
and massless SCT, see (22.2), (23.4). Recall that there exist causal systems
(19), which do not satisfy the properties $C\geq 0$ and finite spinor
dimension. We start studying a preliminary object, the SCT. For this recall
the definitions in sec. 6.2 and 7.1, in particular (7.1).

(128) Definition. A system with causal time evolution (SCT) consists of a
representation $W$ of $\tilde{\mathcal{P}}$ and, with respect to $W$, a
Poincaré covariant WL $E$ such that the time evolution is causal, i.e.,

$W(t)E(\Delta)W(t)^{-1}\leq E(\Delta_{t})$

for all measurable $\Delta\subset\mathbb{R}^{3}$ and all $t\in\mathbb{R}$. An
SCT is called massive if $C>0$, and massless if $C=0$. If $(W,E)$ is an SCT
then any SCT, which is unitarily equivalent to $(SWS^{-1},E)$, where $S$ is
some unitary operator commuting with the representation $W|_{\mathbb{R}\otimes
ISU(2)}$ of the little kinematical group, is said to be unitarily related to
$(W,E)$.

Clearly, causal systems are SCT. A complete explicit description of the SCT is
provided in the following sections. The massive ones are treated in [4] and
here we summarize the relevant results in (22.1). The massless SCT are derived
in sec. 23. The main result on massless SCT is (23.3).

###  21.1. Position representation

In analyzing the SCT $(W,E)$ the imprimitivity theorem by Mackey 1949, 1955
[74],[75] (see (24.2) is essential. By this theorem $W|_{ISU(2)}$ is an
induced representation of a representation $D$ of $SU(2)$ and $E$ is the
related system of imprimitivity. $D$ is uniquely determined up to unitary
equivalence. Let $D$ act on the Hilbert space $\mathcal{S}$. Then $(W,E)$ is
unitarily equivalent to the position representation $(W^{pos},E^{can})$ (cf.
[1],[76]), which is characterized by

$(W^{pos}(b,B)\psi)(x):=D(B)\,\psi(B^{-1}\cdot(x-b)),\quad
E^{can}(\Delta)\psi:=1_{\Delta}\psi$ (21.1)

on $L^{2}(\mathbb{R}^{3},\mathcal{S})$ with $(b,B)\in ISU(2)$ and
$\Delta\subset\mathbb{R}^{3}$ a measurable set. $E^{can}$ is called the
canonical PM. Note that $W$ is not determined by $W|_{ISU(2)}$.
The spin$-j$ components for $j\in\mathbb{N}_{0}/2$ of $W|_{ISU(2)}$ are the
eigenspaces to the eigenvalues $j(j+1)$ of the square of the spin vector
operator $S:=J-X\times P$. The orbital angular momentum $X\times P$ refers to
the causal position operator $X$, which in position representation is given by
$\big{(}X_{k}^{pos}\psi\big{)}(x)=x_{k}\psi(x)$.

###  21.2. Spinor basis

By Eq. (21.1), for $x\in\mathbb{R}^{3}$ the vector $\psi(x)$ is a spinor,
whence $\mathcal{S}$ is called the spinor space. Its dimension is called the
spinor dimension of $W$. Clearly it is an invariant and characteristic for the
SCT.
We shall write $D$ as the orthogonal sum $\oplus_{j}\nu_{j}D^{(j)}$ of the
spin components, where

$\nu_{j}\in\\{0,1,2,\dots\\}\cup\\{\infty\\}$ (21.2)

is the multiplicity of the irreducible representation $D^{(j)}$ for every spin
$j\in\mathbb{N}_{0}/2$. Then the spinor dimension is $\sum_{j}(2j+1)\nu_{j}$.
The standard matrix form of $D^{(j)}$ satisfies

$D^{(j)}(\operatorname{diag}(c,\overline{c}))=\operatorname{diag}(c^{2j},c^{2j-2},\dots,c^{-2j+2},c^{-2j})$
(21.3)

for $c\in\mathbb{C}$, $|c|=1$. Accordingly the standard basis for
$\mathbb{C}^{2j+1}$ is denoted by $(|j,s\rangle)$, where the label $s$ runs
through the helicity values $2j,2j-2,\dots,-2j+2,-2j$. In order to obtain a
basis of the spinor space $\mathcal{S}$ one has to take account of the
multiplicities $\nu_{j}$ by an additional index $\iota$ indicating the
$\iota$-th copy of $D^{(j)}$. One gets the orthogonal spinor basis

$(|j,\iota,s\rangle)\text{ with
}j\in|s|+\mathbb{N}_{0},\;\iota=1,2,\dots,\nu_{j},\,s=2j,2j-2,\dots,-2j+2,-2j$
(21.4)

From now on we will consider only SCT with finite spinor dimension. So all
$\nu_{j}$ are finite and only finitely many of them are non-zero. From a
physical point of view this is a reasonable restriction as infinite
dimensional spinor theories have not found application so far.

###  21.3. Momentum representation of time evolution

The Fourier transformation $\mathcal{F}$ on
$L^{2}(\mathbb{R}^{3},\mathcal{S})\simeq\mathcal{S}\otimes
L^{2}(\mathbb{R}^{3})$ acts as
$I_{\mathcal{S}}\otimes\mathcal{F}_{\mathbb{C}}$. Then the momentum
representation $W^{mom}:=\mathcal{F}\,W^{pos}\,\mathcal{F}^{-1}$ satisfies

$\big{(}W^{mom}(b,B)\varphi\big{)}(p)=\operatorname{e}^{-\operatorname{i}bp}D(B)\,\varphi(B^{-1}\cdot
p)$ (21.5)

Let $H$ be the generator of the time evolution
$W(t)=\operatorname{e}^{\operatorname{i}tH}$, $t\in\mathbb{R}$. Since $H$ is
translational invariant there are self-adjoint matrices $h(p)$ acting on
$\mathcal{S}$ such that

$(H^{mom}\varphi)(p)=h(p)\varphi(p)$ (21.6)

defined on its natural domain. $H$ is rotational invariant, too. So $h$ is
rotational covariant and $h$ can be chosen such that $h(\rho\,e_{3})$,
$\rho\geq 0$, commutes with $D(B)$ if $B$ is diagonal, and
$h(p)=D(B(p))h(|p|e_{3})D(B(p))^{-1}$, whence

$h(B\cdot p)=D(B)\,h(p)\,D(B)^{-1}$ (21.7)

for all $p\in\mathbb{R}^{3}$ without exception (see [4, Eq. (14)]).
Here $B(p)$ denotes the helicity cross section satisfying $B(p)\in SU(2)$ and
$|p|B(p)\cdot e_{3}=p$ for all $p\in\mathbb{R}^{3}$. Explicitly

$B(p)=\left(\begin{array}[]{cc}a_{+}&-\overline{b}\,a_{-}\\\
b\,a_{-}&a_{+}\end{array}\right),\quad a_{\pm}:=\sqrt{\frac{|p|\pm
p_{3}}{2|p|}},\quad b:=\frac{p_{1}+ip_{2}}{|p_{1}+ip_{2}|}\textrm{ for
}(p_{1},p_{2})\neq 0$ (21.8)

and $B(\alpha e_{3})$ equals $I_{2}$ if $\alpha\geq 0$ and $-i\sigma_{2}$ if
$\alpha<0$. Note $B(\lambda p)=B(p)$ for all $\lambda>0$ and for $|p|=1$ the
important relation

$B(p)\,\sigma_{3}\,B(p)^{-1}=\sum_{k=1}^{3}p_{k}\sigma_{k}$ (21.9)

Analogous formulae hold for $W(t)$, $t\in\mathbb{R}$ in place of $H$ such that

$\big{(}W^{mom}(t)\varphi\big{)}(p)=v_{t}(p)\varphi(p),\quad\,v_{t}(p)=\operatorname{e}^{\operatorname{i}th(p)}$
(21.10)

###  21.4. Causal time evolution

As proven in [4, Theorem 1] the causal time evolution of the SCT $(W,E)$
implies that $h$ is linear in momentum, i.e., for all $p\in\mathbb{R}^{3}$

$h(p)=A_{1}p_{1}+A_{2}p_{2}+A_{3}p_{3}+M$ (21.11)

where $A_{k}$, $M$ are constant self-adjoint matrices in spinor space with
$||\sum_{k}p_{k}A_{k}||=\gamma|p|$ for some $\gamma\in[0,1]$. By rotational
covariance Eq. (21.7) of $h$ one has

$D(B)\big{(}\sum_{k=1}^{3}p_{k}A_{k}\big{)}\,D(B)^{-1}=\sum_{k=1}^{3}(B\cdot
p)_{k}A_{k},\quad D(B)MD(B)^{-1}=M$ (21.12)

Further, since the mass-squared operator $C$ is supposed to be non-negative,
by [4, Lemma 16, Lemma 1] one has
$\big{(}C^{mom}\varphi\big{)}(p)=M^{2}\varphi(p)$. Hence $C$ commutes with
$(W,E)$ and the spectrum of $C$ is finite non-negative. This implies

(129) Lemma. Every SCT is the finite orthogonal sum of SCT with $C=m^{2}I$ for
pairwise different scalars $m\geq 0$. The matrices $A_{k}$, $M$ satisfy
$\\{A_{k},A_{j}\\}=2\delta_{kj}I$, $\\{A_{k},M\\}=0$.
In particular $\gamma=1$ follows. Further analysis will reveal the structure
of the matrices $A_{k}$, $M$.

##  22\. Massive Causal Systems

In [4] massive SCT are treated in detail.232323 In [4] a massive SCT is called
a massive causal system. We summarize the main results from [4] providing a
complete description of the massive SCT. Then the aim is to single out the
massive causal systems. To this, in view of (7.2), we study the dependence of
their localizations on the frame of reference.

###  22.1. Construction of the massive SCT

One starts from the fundamental Dirac localization
$(V^{\textsc{d}},U^{\textsc{d}},E^{\textsc{d}}):=(W^{\textsc{d}}|_{\mathbb{R}\otimes
ISU(2)},E^{\textsc{d}})$ (see sec. 13). By a simple construction, which
enlarges the spinor space, one gets for every positive mass $m$ and every
$n\in\mathbb{N}$ the $n$-th Dirac tensor-localization

$(V_{n},U_{n},E_{n}):=(V^{\textsc{d}},U^{\textsc{d}},E^{\textsc{d}})\otimes(I_{n},U^{(\frac{n}{2}-\frac{1}{2})},I_{n})$
(22.1)

with $I_{n}$ the identity on $\mathbb{C}^{n}$ and
$U^{(\frac{n}{2}-\frac{1}{2})}(b,B):=D^{(\frac{n}{2}-\frac{1}{2})}(B)$. In
particular, $h_{n}(p)=h^{\textsc{d}}(p)\otimes I_{n}$ with
$h^{\textsc{d}}(p)=\sum_{k=1}^{3}\alpha_{k}p_{k}+\beta m$, whence the Dirac
tensor-matrices are $\alpha_{k}\otimes I_{n}$, $\beta\otimes I_{n}$. In the
position representation $E_{n}$ is the canonical PM. The time evolutions
$V_{n}$ are causal. The Dirac tensor-localizations are irreducible and
mutually unitarily inequivalent.
Every representation $(V_{n},U_{n})$ for mass $m>0$ of the little kinematical
group can be extended to a representation $W_{n}$ of $\tilde{\mathcal{P}}$ on
$L^{2}(\mathbb{R}^{3},\mathbb{C}^{4}\times\mathbb{C}^{n})$. For
$(t,b,A)\in\tilde{\mathcal{P}}$ the latter is given by

$\big{(}W^{mom}_{n}(t,b,A)\varphi\big{)}(p):=\operatorname{e}^{-\operatorname{i}bp}\sum_{\eta=\pm
1}\left(\operatorname{e}^{\operatorname{i}t\eta\epsilon(p)}\pi^{\eta}(p)s(A)^{*-1}\otimes
D^{(\frac{n}{2}-\frac{1}{2})}\big{(}R(\mathfrak{p}^{\eta},A)\big{)}\right)\varphi(q^{\eta})$
(22.2)

As to the notations recall Eq. (13.1) and following. Furthermore recall the
Wigner rotation $R(\mathfrak{p}^{\eta},A)$ with respect to the canonical cross
section $Q$ with the important property $R(\mathfrak{p}^{\eta},B)=B$ for $B\in
SU(2)$ (see (3.3)). Note that $W_{1}$ is just the Dirac representation
$W^{\textsc{d}}$. Hence for every mass $m>0$ a sequence $(W_{n},E_{n})$ of
irreducible and mutually unitarily inequivalent massive SCT is constructed.
$(W_{1},E_{1})$ is the Dirac system $(W^{\textsc{d}},E^{\textsc{d}})$ studied
in sec. 13.
From [4, sec. B,D,E] we extract

(130) Theorem. _(i)_ Every massive SCT $(W,E)$ is a unique orthogonal sum of
finitely many massive SCT for different masses $m>0$.
_(ii)_ If $(W,E)$ is a massive SCT for mass $m>0$, then it is unitarily
related to a finite orthogonal sum of $(W_{n},E_{n})$ for mass $m$ with
uniquely determined multiplicities.
_(iii)_ Finally, a massive SCT $(W,E)$ is unitarily equivalent to an
orthogonal sum of finitely many $(W_{n},E_{n})$ with possibly various masses
if and only if the booster of $W$ satisfies

$N=\frac{1}{2}\\{X,H\\}+\frac{\operatorname{sgn}(H)}{C^{1/2}+|H|}P\times\left(J-X\times
P-\frac{\operatorname{i}}{4}[X,H]\times[X,H]\right)$ (22.3)

where $X=(X_{1},X_{2},X_{3})$ denotes the causal position operator associated
with $E$.

###  22.2. Determination of the massive causal systems

By (13.1) the Dirac localization is causal and hence, by (7.2), frame-
independent. In contrast, we recall, there is no Dirac state localized by NWL
in a bounded region with respect to two frames moving relative to each other
(see sec. 3.3). This may be not surprising as, with respect to NWL, the Dirac
time evolution is not causal. But, surprisingly, as shown in (22.2), an SCT
need not localize frame-independently.

Let $(W,E)$ be an irreducible massive SCT. By (22.1)(i),(ii) it is unitarily
related to a finite orthogonal sum of $(W_{n},E_{n})$ for some mass $m>0$. So,
in the position representation, $E$ is the canonical PM and, in the momentum
representation,

$\big{(}W^{mom}(t,b,A)\varphi\big{)}(p)=\operatorname{e}^{-\operatorname{i}bp}s(p)^{-1}\sum_{\eta=\pm
1}\operatorname{e}^{\operatorname{i}t\eta\epsilon(p)}w^{\eta}(A,p)s(q^{\eta})\varphi(q^{\eta})$
(22.4)

for almost all $p\in\mathbb{R}^{3}$. Here,
$w^{\eta}(A,p):=\pi^{\eta}(p)s(A)^{*-1}\otimes
D^{\prime}\big{(}R(\mathfrak{p}^{\eta},A)\big{)}$ with some finite dimensional
representation $D^{\prime}$ of $ISU(2)$. Further, according to (21), $S$ is
some unitary operator commuting with $W|_{\mathbb{R}\times ISU(2)}$ so that
$(S^{mom}\varphi)(p)=s(p)\varphi(p)$ with unitary matrices $s(p)$ acting on
spinor space, satisfying rotational covariance $s(B\cdot p)=D(B)s(p)D(B)^{-1}$
for $D:=(D^{\frac{1}{2}}\oplus D^{\frac{1}{2}})\otimes D^{\prime}$, and
commuting with the projections on the energy eigenspaces
$s(p)(\pi^{\eta}(p)\otimes I^{\prime})=(\pi^{\eta}(p)\otimes I^{\prime})s(p)$
with $I^{\prime}:=D^{\prime}(I_{2})$. Therefore
$s(p)=D(B(p))s(|p|e_{3})D(B(p))^{-1}$, and $s(re_{3})$, $r\geq 0$ commutes
with $D(B)$ for $B$ diagonal and with $(r\alpha_{3}+m\beta)\otimes I^{\prime}$
(see Eq. (21.12) with $h(p)=h^{\textsc{d}}(p)\otimes I^{\prime}$ in Eq.
(21.11)).

(131) Theorem. Let $(W,E)$ be an irreducible massive SCT, which localizes
frame-independently. Suppose that $p\mapsto s(p)$ (see _Eq. ( 22.4)_) has an
entire extension. Then $(W,E)$ is unitarily related to a finite multiple of
$(W_{1},E_{1})$.
Proof. Let $\varphi$ be localized in a bounded region. Then, by assumption, so
is $\varphi_{\rho}:=W^{mom}(A_{\rho})\varphi$, where
$A_{\rho}:=\operatorname{e}^{\frac{\rho}{2}\sigma_{3}}$, $\rho\neq 0$. Hence,
by the Paley-Wiener Theorem, these functions are the restriction of entire
functions on $\mathbb{C}^{3}$, still denoted by $\varphi$ and
$\varphi_{\rho}$. As $q^{\eta}=(p_{1},p_{2},-\eta\beta\epsilon+\alpha p_{3})$
with $\alpha:=\cosh\rho$, $\beta:=\sinh\rho$, and
$\epsilon=\sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+m^{2}}$, there are entire
functions $\phi_{1}$, $\phi_{2}$ such that $(s(A_{\rho})^{*-1}\otimes
I_{n})s(q^{\eta})\varphi(q^{\eta})=\phi_{1}(p)-\eta\epsilon\phi_{2}(p)$.
Furthermore, explicit computation yields

$R(\mathfrak{p}^{\eta},A_{\rho})=d(\eta
p)^{-1/2}\left(\begin{array}[]{cc}\gamma(m+\epsilon)-\delta\eta
p_{3}&\delta\eta(p_{1}-\operatorname{i}p_{2})\\\
-\delta\eta(p_{1}+\operatorname{i}p_{2})&\gamma(m+\epsilon)-\delta\eta
p_{3}\end{array}\right)$ (22.5)

with $d(p):=(m+\epsilon)(m+\alpha\,\epsilon-\beta p_{3})$ and constants
$\gamma:=\cosh\frac{\rho}{2}$, $\delta:=\sinh\frac{\rho}{2}$.
Note that each component $D^{(j)}_{kl}(M)$ is a homogeneous polynomial of
degree $2j$ of the entries of $M$. Hence there are matrix-valued functions
$A^{(j)}$, $B^{(j)}$ with polynomial entries such that
$D^{(j)}\big{(}R(\mathfrak{p}^{\eta},A_{\rho})\big{)}=\varDelta^{(j)}(\eta
p)\big{(}A^{(j)}(\eta p)+\epsilon B^{(j)}(\eta p)\big{)}$ with
$\varDelta^{(j)}(\eta p):=d(\eta p)^{-j}I_{2j+1}$. Recall that $D^{\prime}$ is
the orthogonal sum of some $D^{(j)}$. Accordingly we write
$D^{\prime}\big{(}R(\mathfrak{p}^{\eta},A_{\rho})\big{)}=\varDelta^{\prime}(\eta
p)\big{(}A^{\prime}(\eta p)+\epsilon B^{\prime}(\eta p)\big{)}$. Hence, by Eq.
(22.4), one has

$2s(p)\varphi_{\rho}(p)=\sum_{\eta}\left(\big{(}\text{{\it\small{I}}}_{4}+\frac{\eta}{\epsilon}h^{\textsc{d}}(p)\big{)}\otimes\varDelta^{\prime}(\eta
p)\big{(}A^{\prime}(\eta p)+\epsilon B^{\prime}(\eta
p)\big{)}\right)\big{(}\phi_{1}(p)-\eta\epsilon\phi_{2}(p)\big{)}$

Now we examine the point $p_{*}:=(\operatorname{i}m,0,0)$. There $\epsilon$
vanishes. Hence one infers $\sum_{\eta}\big{(}\eta
I_{4}\otimes\varDelta^{\prime}(\eta p_{*})A^{\prime}(\eta
p_{*})\big{)}\,\xi_{*}=0$ for $\xi_{*}:=(h^{\textsc{d}}(p_{*})\otimes
I^{\prime})\phi_{1}(p_{*})=\big{(}h^{\textsc{d}}(p_{*})s(A_{\rho}^{*-1})\otimes
I^{\prime}\big{)}s(p_{*})\varphi(p_{*})$. Here, plainly, $s(p_{*})$ denotes
the analytic continuation of $s$ at $p_{*}$. As $\varDelta^{\prime}(\eta
p_{*})A^{\prime}(\eta
p_{*})=D^{\prime}\left(\begin{array}[]{cc}\gamma&\operatorname{i}\eta\delta\\\
-\operatorname{i}\eta\delta&\gamma\end{array}\right)$, one gets equivalently

$\xi_{*}=\left(I_{4}\otimes
D^{\prime}\left(\begin{array}[]{cc}\alpha&\operatorname{i}\beta\\\
-\operatorname{i}\beta&\alpha\end{array}\right)\right)\,\xi_{*}$ ($\star$)

This condition implies that $D^{\prime}$ is a multiple of the unit
representation. Indeed, assume the contrary. Recall that
$\varphi(p_{*})=\int\operatorname{e}^{mx_{1}}\psi(x)\operatorname{d}^{3}x$ for
some $\psi\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{4}\times\mathbb{C}^{n})$ with
bounded support. So one may alter $\psi$ a bit within its support to guarantee
$\varphi(p_{*})\neq 0$. Further, note that $s(p_{*})$ is invertible, as the
equation $I=s(z)s(\overline{z})^{*}$ holds true by analytic continuation for
all $z\in\mathbb{C}^{3}$. Hence one may multiply $\psi$ by a suitable constant
spinor matrix in order that $\xi_{*}$ does not satisfy Eq. $(\star)$. This
accomplishes the proof. ∎

(132) Corollary. The SCT $(W_{n},E_{n})$ for $n>1$ do not localize frame-
independently and hence are not causal.

Proof. The result in (22.2) applies with $s(p)=I_{4n}$ for all $p$.∎

We are sure that the assumption of analyticity of s in (22.2) can be dropped.
Compare also the analogous result (23.4) for massless SCT going without such
an assumption. Thus, due to (7.2), the higher Dirac tensor-systems
$(W_{n},E_{n})$, $n>1$ are revealed to be mathematical artefacts, interim
results in determining the massive causal systems.
In the end we are sure that the only irreducible massive SCT, which are
causal, are unitarily equivalent to the Dirac systems
$(W^{\textsc{d}},E^{\textsc{d}})$ for masses $m>0$ studied in sec. 13.

##  23\. Massless Causal Systems

The analysis of the massless causal systems proceeds analogously to that of
the massive ones. However there is an important difference arising from the
matrices $A_{1},A_{2},A_{3},M$ (21.4). In the massive case one is dealing with
the Clifford algebra with four basis elements, which has only one irreducible
representation, the $4$-dimensional Dirac representation (see [4, Lemma 2]).
The massless case $M=0$, however, is determined by the Clifford algebra
generated by three basis elements, which has just two inequivalent irreducible
representations, the $2$-dimensional Weyl representations (see (23)).

(133) Lemma. Let $A_{1},A_{2},A_{3}$ be any (respectively, self-adjoint)
square matrices. Then, they satisfy $\\{A_{k},A_{j}\\}=2\delta_{kj}I$ if and
only if, up to (respectively, unitary) equivalence,

$A_{k}=(\sigma_{k}\otimes I^{\prime})\oplus(-\sigma_{k}\otimes
I^{\prime\prime})$

with $\sigma_{k}$, $k=1,2,3$ the Pauli matrices and
$I^{\prime},I^{\prime\prime}$ unit matrices, and $\oplus$ the direct
(respectively, orthogonal) sum.

Proof. The group $G$ generated by the Clifford algebra
$\\{\alpha_{k},\alpha_{j}\\}=2\delta_{kj}1$, $k,j=1,2,3$ has order $16$. Its
elements are $1,-1$ and all words in $\alpha_{k}$ which result from
$\pm\,\alpha_{1}\alpha_{2}\alpha_{3}$ by canceling at most $2$ of them. There
are $10$ conjugacy classes, namely $\\{1\\}$, $\\{-1\\}$,
$\\{\alpha_{1}\alpha_{2}\alpha_{3}\\},$
$\\{-\alpha_{1}\alpha_{2}\alpha_{3}\\}$, and $\\{g,-g\\}$ for all other group
elements $g$. Every representation of $G$, which maps $-1$ to $I$, actually is
a representation of the abelian group $G/\\{1,-1\\}$. Hence there are $8$
inequivalent one-dimensional representations of $G$. So, up to equivalence,
there are just two inequivalent irreducible representation $\rho_{1},\rho_{2}$
left. Let $d_{1}$ and $d_{2}$ be their dimensions. They satisfy $16=8\cdot
1^{2}+d_{1}^{2}+d_{2}^{2}$, whence $d_{1}=d_{2}=2$. One easily checks that
$\rho_{1}(\pm 1):=\pm I_{2}$ and $\rho_{1}(\alpha_{k}):=\sigma_{k}$, $k=1,2,3$
defines an irreducible representation of $G$. Similarly does $\rho_{2}(\pm
1):=\pm I_{2}$ and $\rho_{2}(\alpha_{k}):=-\sigma_{k}$, $k=1,2,3$. Since
$\operatorname{tr}(\rho_{1}(\alpha_{1}\alpha_{2}\alpha_{3}))=2\operatorname{i}\neq\operatorname{tr}(\rho_{2}(\alpha_{1}\alpha_{2}\alpha_{3}))=-2\operatorname{i}$,
$\rho_{1}$ and $\rho_{2}$ are inequivalent. By the same reason, $\rho_{1}$,
$\rho_{2}$ are inequivalent even as representations of the Clifford algebra,
known as Weyl representations. Finally, if one limits oneself to self-adjoint
representations of the algebra, which means to unitary representations of the
group, equivalence becomes unitary equivalence. Moreover, the direct sum must
be replaced by an orthogonal sum. ∎

(134) Lemma. Let $D$ be a finite dimensional representation of $SU(2)$ and let
$A_{1},A_{2},A_{3}$ be self-adjoint matrices. They satisfy rotational
covariance
$D(B)\big{(}\sum_{k=1}^{3}p_{k}A_{k}\big{)}\,D(B)^{-1}=\sum_{k=1}^{3}(B\cdot
p)_{k}A_{k}$ (cf.  _Eq. ( 21.12)_) and $\\{A_{k},A_{j}\\}=2\delta_{kj}I$ (cf.
_( 21.4)_) if and only if, up to unitary equivalence,
$A_{k}=(\sigma_{k}\otimes I^{\prime})\oplus(-\sigma_{k}\otimes
I^{\prime\prime})$ and

$D=(D^{(\frac{1}{2})}\otimes D^{\prime})\oplus(D^{(\frac{1}{2})}\otimes
D^{\prime\prime})$

for some representations $D^{\prime}$, $D^{\prime\prime}$ of $SU(2)$.

Proof. By Lemma (23) we may assume at once $A_{k}=(\sigma_{k}\otimes
I^{\prime})\oplus(-\sigma_{k}\otimes I^{\prime\prime})$. Recall the relation
$B(\sum_{k}p_{k}\sigma_{k})B^{*}=\sum_{k}(B\cdot p)_{k}\sigma_{k}$. So the
representation $L:=(D^{(\frac{1}{2})}\otimes
I^{\prime})\oplus(D^{(\frac{1}{2})}\otimes I^{\prime\prime})$ of $SU(2)$
satisfies
$L(B)\big{(}\sum_{k=1}^{3}p_{k}A_{k}\big{)}\,L(B)^{-1}=\sum_{k=1}^{3}(B\cdot
p)_{k}A_{k}$. This implies the “if”part of the assertion. As to the “only
if”part, note that $L(B)^{-1}D(B)$ commutes with $A_{k}$, $k=1,2,3$. Note that
$A_{k}=A^{\prime}_{k}\oplus A^{\prime\prime}_{k}$ with
$A^{\prime}_{k}:=\sigma_{k}\otimes I^{\prime}$ and
$A^{\prime\prime}_{k}:=-\sigma_{k}\otimes I^{\prime\prime}$ determines the
primary decomposition of the representation of $G$ (see the proof of (23))
defined by $A_{k}$, $k=1,2,3$. Therefore $L(B)^{-1}D(B)=R^{\prime}(B)\oplus
R^{\prime\prime}(B)$ for some $R^{\prime}(B)$ and $R^{\prime\prime}(B)$, which
commute with $A^{\prime}_{k}$ and $A^{\prime\prime}_{k}$, respectively. Hence
$R^{\prime}(B)=I_{2}\otimes D^{\prime}(B)$ and
$R^{\prime\prime}(B)=I_{2}\otimes D^{\prime\prime}(B)$ with uniquely
determined matrices $D^{\prime}(B)$, $D^{\prime\prime}(B)$ for every $B\in
SU(2)$. Then $D(B)=L(B)\big{(}R^{\prime}(B)\oplus R^{\prime\prime}(B)\big{)}$
yields the formula on $D$. The latter implies that $D^{\prime}$,
$D^{\prime\prime}$ are representations of $SU(2)$.∎

###  23.1. The massless WL with causal time evolution

The following results (23.1), (23.1), and (23.1) are an immediate consequence
of the foregoing considerations (21.4), (23), (23). Recall that
$\big{(}U^{pos}(b,B)\psi\big{)}(x)=D(B)\,\psi(B^{-1}\cdot(x-b))$ and
$E^{pos}=E^{can}$, i.e., $E^{pos}(\Delta)\psi=1_{\Delta}\psi$. A WL with
causal time evolution (7.1) is called massless if $C=0$.

(135) Theorem. Let $(V,U,E)$ be a massless WL with causal time evolution of
minimal spinor dimension. Then the spinor dimension is $2$ and
$D=D^{(\frac{1}{2})}$. Moreover, there is $\chi\in\\{+,-\\}$ such that, up to
unitary equivalence,

$H^{\chi
pos}=\chi\left(\sigma_{1}\frac{1}{\operatorname{i}}\frac{\partial}{\partial
x_{1}}+\sigma_{2}\frac{1}{\operatorname{i}}\frac{\partial}{\partial
x_{2}}+\sigma_{3}\frac{1}{\operatorname{i}}\frac{\partial}{\partial
x_{3}}\right)$ (23.1)

The result in (23.1) is a derivation of the Weyl equations as a mere
consequence of the principle of causality in relativistic quantum mechanics.

(136) Definition. The massless WL with causal time evolution, for which in
position representation $H$ is given by Eq. (23.1) and $D=D^{(\frac{1}{2})}$,
is called the right-handed, respectively left-handed Weyl localization. It is
denoted by $(V^{\textsc{w}\chi},U^{\textsc{w}},E^{\textsc{w}})$ with $\chi=+$
and $\chi=-$, respectively.

(137) Theorem. Let $(V,U,E)$ be an irreducible massless WL with causal time
evolution. Then there exists and $n\in\mathbb{N}$ and $\chi\in\\{+,-\\}$ such
that $(V,U,E)$ is unitarily equivalent to the massless WL with causal time
evolution $(V_{n}^{\chi},U_{n},E_{n})$ given by

$\displaystyle V_{n}^{\chi}(t)$ $\displaystyle=V^{\textsc{w}\chi}(t)\otimes
I_{n}$ time translations $\displaystyle U_{n}(b)$
$\displaystyle=U^{\textsc{w}}(b)\otimes I_{n}$ space translations
$\displaystyle U_{n}(B)$ $\displaystyle=U^{\textsc{w}}(B)\otimes
D^{(\frac{n}{2}-\frac{1}{2})}(B)$ rotations $\displaystyle E_{n}(\Delta)$
$\displaystyle=E^{\textsc{w}}(\Delta)\otimes I_{n}$ localization operators

where $I_{n}$ is the identity on $\mathbb{C}^{n}$.

(138) Definition. Let the massless WL with causal time evolution
$(V_{n}^{\chi},U_{n},E_{n})$, $n\in\mathbb{N}$, $\chi\in\\{+,-\\}$ be called
Weyl tensor-localizations.

Note that the Weyl tensor-localizations for $n=1$ are just the Weyl
localizations. The spinor space of the Weyl tensor-localizations for $n$ is
$\mathbb{C}^{2}\otimes\mathbb{C}^{n}\simeq\mathbb{C}^{2n}$. Hence the spinor
dimension is $2n$. As $D_{n}\simeq D^{(\frac{1}{2})}\otimes
D^{(\frac{n}{2}-\frac{1}{2})}\simeq D^{(\frac{n}{2}-1)}\oplus
D^{(\frac{n}{2})}$, the helicity spectrum is
$\\{-\frac{n}{2},-\frac{n}{2}+1,\dots,\frac{n}{2}\\}$ with multiplicity $1$
for helicity $\frac{n}{2},-\frac{n}{2}$ and $2$ else.

(139) Theorem. Every massless WL with causal time evolution is unitarily
equivalent to a finite orthogonal sum of Weyl tensor-localizations with
uniquely determined multiplicities.

We will show that each representation $(V_{n}^{\chi},U_{n})$ of the little
kinematical group can be extended to an, up to unitary equivalence, unique
representation $W_{n}^{\chi}$ of $\tilde{\mathcal{P}}$. The following
insertion provides some facts used for this objective.

###  23.2. On irreducible massless representations of $\tilde{\mathcal{P}}$

Let $\mathcal{O}^{0,\eta}$ for $\eta=\pm$ denote the light half-cone
$\\{\mathfrak{p}\in\mathbb{R}^{4}\setminus\\{0\\}:p_{0}=\eta\,|p|\\}$ endowed
with the Lorentz invariant measure and let
$\mathcal{O}^{0}:=\mathcal{O}^{0,+}\cup\mathcal{O}^{0,-}=\\{\mathfrak{p}\in\mathbb{R}^{4}\setminus\\{0\\}:|p_{0}|=|p|\\}$
denote the light cone.

The $1$-dimensional representations
$(\mathfrak{a},A)\mapsto\operatorname{e}^{\operatorname{i}(\eta
a_{0}-a_{3})}(A_{11})^{2s}$ for $s\in\mathbb{Z}/2$ of the inhomogeneous little
group $IE(2)\sqsubset\tilde{\mathcal{P}}$ (recall $A\in E(2)$ if and only if
$A\in SL(2,\mathbb{C})$ with $A_{21}=0$, $A_{22}=\overline{A}_{11}$) induce
the irreducible finite spinor dimensional massless representations of
$\tilde{\mathcal{P}}$. They act on $L^{2}(\mathcal{O}^{0,\eta})$ by

$\big{(}U^{0,s,\eta}(\mathfrak{a},A)\phi\big{)}(\mathfrak{p}):=\operatorname{e}^{\operatorname{i}\mathfrak{a}\cdot\mathfrak{p}}k(\mathfrak{p},A)^{2s}\,\phi(A^{-1}\cdot\mathfrak{p})$
(23.2)

see e.g. [77, Eq. (1.1.15)]. Here

$k(\mathfrak{p},A):=H(\mathfrak{p},A)_{11}=\frac{(|p|+\eta
p_{3})\overline{A_{22}}-\eta(p_{1}-\operatorname{i}p_{2})\overline{A_{12}}}{|(|p|+\eta
p_{3})\overline{A_{22}}-\eta(p_{1}-\operatorname{i}p_{2})\overline{A_{12}}|}$
(23.3)

(if the denominator $\neq 0$) with the Wigner rotation
$H(\mathfrak{p},A):=H(\mathfrak{p})^{-1}A\,H(A^{-1}\cdot\mathfrak{p})$ with
respect to the helicity cross section
$H(\mathfrak{p}):=B\big{(}\eta\,p\big{)}A_{\ln|p|}$, cf. Eq. (21.8) and recall
$A_{\rho}=\operatorname{e}^{\frac{\rho}{2}\sigma_{3}}$. (The canonical cross
section is not defined for $\mathfrak{p}\in\mathcal{O}^{0,\eta}$). Since
$H(\mathfrak{p})\cdot\big{(}\eta(e_{0}+e_{3})\big{)}=\mathfrak{p}$, one has
$H(\mathfrak{p},A)\in E(2)$.

As expected, $\eta$ denotes the sign of the energy. Indeed, the representation
of $|H|^{-1}H$ by $U^{0,s,\eta}$ equals $\eta I$. We are going to clarify the
significance of $s$. By explicit computation one finds that for the
representation $U^{0,s,\eta}$ the helicity operator $|P|^{-1}JP$ equals $\eta
sI$.

(140) Lemma. $U^{0,s,-}$ is antiunitarily equivalent to $U^{0,-s,+}$. More
precisely, $U^{0,-s,-}=\mathcal{C}U^{0,s,+}\mathcal{C}^{-1}$ for the
antiunitary transformation $\mathcal{C}:L^{2}(\mathcal{O}^{0,+})\to
L^{2}(\mathcal{O}^{0,-})$,
$(\mathcal{C}\phi)(\mathfrak{p}):=\overline{\phi(-\mathfrak{p})}$.
$\mathcal{C}$ is uniquely determined up to a constant phase.

Proof. According to Eq. (23.3), $k(-\mathfrak{p},A)=k(\mathfrak{p},A)$ holds
for all $\mathfrak{p}\in\mathcal{O}^{0}$. Further recall that $U^{0,s,\eta}$
is irreducible. Then the assertion is easily verified.∎

We will denote by $[0,s,\eta]$ any representation of $\tilde{\mathcal{P}}$,
which is unitarily equivalent to $U^{0,s,\eta}$. Hence in view of (23.2) one
has the following consistent

(141) Interpretation. $[0,s,+]$ represents a massless particle with positive
energy and helicity $s\in\mathbb{Z}/2$, and $[0,-s,-]$ represents its
antiparticle, which has positive energy, zero mass, and helicity $s$, too.

For later use we cast $U^{0,s,\eta}$ into a more convenient form. By the
isomorphism $X_{\eta}:L^{2}(\mathcal{O}^{0,\eta})\to L^{2}(\mathbb{R}^{3})$,
$(X_{\eta}\phi)(p):=(2|p|)^{-\frac{1}{2}}\phi(\eta|p|,p)$ one passes to the
representation $X_{\eta}U^{0,s,\eta}X_{\eta}^{-1}$ on $L^{2}(\mathbb{R}^{3})$.
In the case of positive energy $\eta=+$ it has already the desired form

$\big{(}W^{0,s,+}(t,b,A)\varphi\big{)}(p):=(|q^{+}|/|p|)^{\frac{1}{2}}\operatorname{e}^{\operatorname{i}t|p|}\operatorname{e}^{-\operatorname{i}bp}k(\mathfrak{p}^{+},A)^{2s}\,\varphi(q^{+})$
(23.4)

In the case of negative energy $\eta=-$ one carries out a further unitary
transformation by $X_{s}:L^{2}(\mathbb{R}^{3})\to L^{2}(\mathbb{R}^{3})$,
$(X_{s}\varphi)(p):=b(p)^{2s}\varphi(p)$. Then
$X_{s}X_{-}U^{0,s,-}X_{-}^{-1}X_{s}^{-1}$ equals

$\big{(}W^{0,s,-}(t,b,A)\varphi\big{)}(p):=(|q^{-}|/|p|)^{\frac{1}{2}}\operatorname{e}^{-\operatorname{i}t|p|}\operatorname{e}^{-\operatorname{i}bp}\big{(}b(p)k(\mathfrak{p}^{-},A)\overline{b(q^{-})}\,\big{)}^{2s}\,\varphi(q^{-})$
(23.5)

Recall
$b(p)=\frac{p_{1}+\operatorname{i}p_{2}}{|p_{1}+\operatorname{i}p_{2}|}$ for
$(p_{1},p_{2})\neq 0$, and $\mathfrak{p}^{\eta}=(\eta|p|,p)$ and
$\mathfrak{q}^{\eta}=(\eta|q^{\eta}|,q^{\eta})=A^{-1}\cdot\mathfrak{p}^{\eta}$.

(142) Lemma. Let $s\in\mathbb{Z}/2$. The representations

$\big{(}K^{s,+}(t,b,B)\varphi\big{)}(p):=\operatorname{e}^{\operatorname{i}t|p|}\operatorname{e}^{-\operatorname{i}bp}\kappa(p,B)^{2s}\,\varphi(B^{-1}\cdot
p)$ (23.6)

and

$\big{(}K^{s,-}(t,b,B)\varphi\big{)}(p):=\operatorname{e}^{-\operatorname{i}t|p|}\operatorname{e}^{-\operatorname{i}bp}\kappa(p,B)^{-2s}\,\varphi(B^{-1}\cdot
p)$ (23.7)

on $L^{2}(\mathbb{R}^{3})$ of the little kinematical group are extended by
_Eq. ( 23.4)_ and _Eq. ( 23.5)_, respectively, to representations of
$\tilde{\mathcal{P}}$. Here $\kappa(p,B):=\big{(}B(p)^{-1}B\,B(B^{-1}\cdot
p)\big{)}_{11}$ for $p\neq 0$.
Proof. Note that for $p\neq 0$ the Wigner rotation $B(p)^{-1}B\,B(B^{-1}\cdot
p)\in SU(2)$ is diagonal as it leaves $(0,0,1)$ fixed. Since $A_{\ln|p|}$ is
diagonal, too, $H(\mathfrak{p}^{+},B)=B(p)^{-1}B\,B(B^{-1}\cdot
p)=\operatorname{diag}\big{(}\kappa(p,B),\overline{\kappa(p,B)}\,\big{)}$ with
$\kappa(p,B)=k(\mathfrak{p}^{+},B)$. This already shows
$W^{0,s,+}(t,b,B)=K^{s,+}(t,b,B)$.
For the negative energy case note first that
$H(-\mathfrak{p})=H(\mathfrak{p})$ for all $\mathfrak{p}\in\mathcal{O}^{0}$.
Hence $k(\mathfrak{p}^{-},B)=\kappa(-p,B)$. It remains to check

$\kappa(-p,B)=\overline{b(p)}\,\overline{\kappa(p,B)}\,b(q),\;q:=B^{-1}\cdot
p$ (1)

Indeed,
$\operatorname{diag}\big{(}\kappa(-p,B),\overline{\kappa(-p,B)}\,\big{)}=B(-p)^{-1}B\,B(-q)=C(p)B(p)^{-1}B\,B(q)C(q)^{-1}=C(p)\operatorname{diag}\big{(}\kappa(p,B),\overline{\kappa(p,B)}\,\big{)}C(q)^{-1}=\operatorname{diag}\big{(}\overline{b(p)}\,\overline{\kappa(p,B)}\,b(q),b(p)\kappa(p,B)\,\overline{b(q)}\big{)}$
where

$C(p):=B(-p)^{-1}B(p)=\left(\begin{array}[]{cc}0&-\overline{b(p)}\\\
b(p)&0\end{array}\right),\;b(p)=\frac{p_{1}+ip_{2}}{|p_{1}+ip_{2}|}$ (2)

∎

We will need the integral decomposition of the representations $K^{s,\eta}$ of
the little kinematical group.

(143) Lemma. There is the following set of mutually inequivalent irreducible
representations of the little kinematical group:
$U^{r,s,\eta}_{\mathbb{R}\otimes ISU(2)}$ on $L^{2}(S_{r}^{2})$ with
$S_{r}^{2}:=\\{p\in\mathbb{R}^{3}:|p|=r\\}$ for $r>0$, $s\in\mathbb{Z}/2$, and
$\eta=\pm$, given by

$\big{(}U^{r,s,\eta}_{\mathbb{R}\otimes
ISU(2)}(t,b,B)\varphi\big{)}(p)=\operatorname{e}^{\operatorname{i}\eta
tr}\operatorname{e}^{-\operatorname{i}bp}\kappa(p,B)^{2\eta
s}\varphi(B^{-1}\cdot p)$

Then the integral decomposition
$K^{s,\eta}=\int_{\mathbb{R}^{+}}U^{r,s,\eta}_{\mathbb{R}\otimes ISU(2)}4\pi
r^{2}\operatorname{d}r$ holds.
Proof. The first part of the assertion follows from general representation
theory for semi-direct products of locally compact groups. The second part is
easily verified.∎

Let us add a result on covariance of the helicity cross section which we need
in (23.3).

(144) Lemma. For all $p\in\mathbb{R}^{3}\setminus\\{0\\}$ and
$C=\operatorname{diag}(\gamma,\overline{\gamma})$ with $\gamma\in\mathbb{C}$,
$|\gamma|=1$ one has $B(C\cdot p)=CB(p)\,C^{-1}$. Also,
$H(\mathfrak{p},C)=B(p)^{-1}CB(C^{-1}\cdot p)=C$ for all
$\mathfrak{p}\in\mathcal{O}^{0}$, implying
$k(\mathfrak{p},C)=\kappa(p,C)=\gamma$.

Proof. Explicit calculation yields $(C\cdot
p)_{1}=\frac{1}{2}(\gamma^{2}+\overline{\gamma}^{2})p_{1}+\frac{1}{2\operatorname{i}}(\gamma^{2}-\overline{\gamma}^{2})p_{2},\quad(C\cdot
p)_{2}=-\frac{1}{2\operatorname{i}}(\gamma^{2}-\overline{\gamma}^{2})p_{1}+\frac{1}{2}(\gamma^{2}+\overline{\gamma}^{2})p_{2},\quad(C\cdot
p)_{3}=p_{3}$. Recall Eq. (21.8). One finds $a_{\pm}(C\cdot p)=a_{\pm}(p)$ and
$b(C\cdot p)=\overline{\gamma}^{2}b(p)$, whence the first part of assertion. —
Therefore $B(p)^{-1}CB(C^{-1}\cdot p)=C$ for all $p\neq 0$. Finally recall
that $A_{\ln|p|}$ is diagonal and hence commutes with $C$. ∎

###  23.3. Derivation of the massless SCT

As to the notations see sec. 23.2. Moreover let

$h^{\chi}(p):=\chi(\sigma_{1}p_{1}+\sigma_{2}p_{2}+\sigma_{3}p_{3}),\quad\pi^{\chi\eta}(p):=\frac{1}{2}\big{(}I_{2}+\frac{\eta}{|p|}h^{\chi}(p)\big{)}$

for $\chi\in\\{+,-\\}$. Particular attention requires the rotation
$R_{0}(\mathfrak{p},A)$ in Eq. (23.8), which replaces the Wigner rotation
$R(\mathfrak{p},A)$ in Eq. (22.2) in the massive case.

(145) Lemma. Let $\mathfrak{p}\in\mathcal{O}^{0}$ and $A\in SL(2,\mathbb{C})$.
Set $\mathfrak{q}:=A^{-1}\cdot\mathfrak{p}$. Choose $B,B^{\prime}\in SU(2)$
and $\rho\in\mathbb{R}$ such that $A=B^{\prime}A_{\rho}B$. Set

$R_{0}(\mathfrak{p},A):=B^{\prime}B(B^{\prime-1}\cdot p)B(B\cdot q)^{-1}B$
(23.8)

Then $R_{0}$ is well-defined, i.e. the right hand side does not depend on the
particular choice of $B^{\prime},B,\rho$. Furthermore,

  * $(\alpha)$

$R_{0}(\mathfrak{p},A)\in SU(2)$

  * $(\beta)$

$R_{0}(\mathfrak{p},B)=B$

  * $(\gamma)$

$R_{0}(\lambda\mathfrak{p},A)=R_{0}(\mathfrak{p},A)$ for all $\lambda>0$

  * $(\delta)$

$R_{0}(\mathfrak{p},A)=\lim_{m\to 0}R(\mathfrak{p}^{m},A)$ with
$\mathfrak{p}^{m}:=(\operatorname{sgn}(p_{0})\sqrt{|p|^{2}+m^{2}},p)$, $m>0$

  * $(\epsilon)$

$R_{0}(\mathfrak{p},A)R_{0}(\mathfrak{q},A^{\prime})=R_{0}(\mathfrak{p},AA^{\prime})$

for all $\mathfrak{p}\in\mathcal{O}^{0}$, $A,A^{\prime}\in SL(2,\mathbb{C})$,
and $B\in SU(2)$.
Proof. $(\alpha)$ is obvious. — If $\rho=0$, i.e., $A=B^{\prime}B\in SU(2)$,
then $B(B\cdot q)^{-1}=B(B^{\prime-1}\cdot p)^{-1}$, whence
$R_{0}(\mathfrak{p},A)=B^{\prime}B=A$. So $(\beta)$ holds.
We examine the ambiguity of the representation $A=B^{\prime}A_{\rho}B$ for
$A\in SL(2,\mathbb{C})\setminus SU(2)$. Note that
$A=(B^{\prime}B)(B^{-1}A_{\rho}B)$ is the polar decomposition of $A$. This is
unique. Therefore, the eigenvalues $\operatorname{e}^{\rho/2}$ and
$\operatorname{e}^{-\rho/2}$ of $B^{-1}A_{\rho}B$ determine $\rho\neq 0$.
Moreover, as the eigenvalues are not equal, $B$ is unique up to a diagonal
matrix $C\in SU(2)$ multiplied from the left. But when $B$ is replaced with
$CB$ then $B^{\prime}$ has to be replaced by $B^{\prime}C^{-1}$ in order to
leave unchanged $B^{\prime}B$. Conversely, $B^{\prime}C^{-1}A_{\rho}CB$ still
equals $A$ as $A_{\rho}$ and $C$ commute.
Now we show that $R_{0}$ is well-defined. In view of $(\beta)$ assume at once
$A\not\in SU(2)$. Consider the representation $A=B^{\prime}C^{-1}A_{\rho}CB$.
Then the right hand side of Eq. (23.8) reads $B^{\prime}C^{-1}B(C\cdot
B^{\prime-1}\cdot p)B(C\cdot B\cdot q)^{-1}CB$, which actually does not depend
on $C$ by (23.2).
$(\gamma)$ holds as $B(\lambda p)=B(p)$ for $\lambda>0$,
$p\in\mathbb{R}^{3}\setminus\\{0\\}$.
$(\delta)$ By definition $R_{0}(\mathfrak{p},A_{\rho})=B(p)B(q)^{-1}$ for
$A=A_{\rho}$. Recall $\alpha=\cosh\rho$, $\beta=\sinh\rho$ and
$\gamma=\cosh\frac{\rho}{2}$, $\delta=\sinh\frac{\rho}{2}$. Explicit
computation using Eq. (21.8) yields

$R_{0}(\mathfrak{p},A_{\rho})=\big{(}|p|(\alpha|p|-\beta\eta
p_{3})\big{)}^{-1/2}\left(\begin{array}[]{cc}\gamma|p|-\delta\eta
p_{3}&\delta\eta(p_{1}-\operatorname{i}p_{2})\\\
-\delta\eta(p_{1}+\operatorname{i}p_{2})&\gamma|p|-\delta\eta
p_{3}\end{array}\right)$ (23.9)

which shows $(\delta)$ for $A=A_{\rho}$ by Eq. (22.5). As to the general case
$A=B^{\prime}A_{\rho}B$ note that
$R(\mathfrak{p}^{m},A)=B^{\prime}R(B^{\prime-1}\cdot\mathfrak{p}^{m},A_{\rho})B$
by the analogous properties $(\beta)$ and $(\epsilon)$ for $R$, which are easy
to verify for $R$ (in place of $R_{0}$). Hence $R(\mathfrak{p}^{m},A)\to
B^{\prime}R_{0}(B^{\prime-1}\cdot\mathfrak{p}^{m},A_{\rho})B=B^{\prime}B(B^{\prime-1}\cdot
p)B(B\cdot q)B=R_{0}(\mathfrak{p},A)$ by definition.
$(\epsilon)$ follows immediately from $(\delta)$, as $R$ satisfies the
analogous property. ∎

It is interesting that the limit in (23.3)$(\delta)$ exists whereas the limit
of $Q(\mathfrak{p}^{m})$ for $m\to 0$ does not if $p\neq 0$. Moreover one
notes $R_{0}(\mathfrak{p},A)\cdot q=\frac{|q|}{|p|}p$ as an easy consequence
of $|p|B(p)\cdot e_{3}=p$ for all $p\in\mathbb{R}^{3}\setminus\\{0\\}$.

(146) Theorem. For every massless WL with causal time evolution there exists a
representation of $\tilde{\mathcal{P}}$ extending the representation of the
little kinematical group.
Let $\chi\in\\{+,-\\}$. For $(t,b,A)\in\tilde{\mathcal{P}}$ and $\varphi\in
L^{2}(\mathbb{R}^{3},\mathbb{C}^{2n})$ put

$\big{(}W_{n}^{\chi\,mom}(t,b,A)\varphi\big{)}(p):=\operatorname{e}^{-\operatorname{i}bp}\sum_{\eta=\pm
1}\left(\operatorname{e}^{\operatorname{i}t\eta|p|}\pi^{\chi\eta}(p)s^{\chi}(A)^{*-1}\otimes
D^{(\frac{n}{2}-\frac{1}{2})}\big{(}R_{0}(\mathfrak{p}^{\eta},A)\big{)}\right)\varphi(q^{\eta})$

with $s^{+}(A):=A$, $s^{-}(A):=A^{*-1}$. Then $W_{n}^{\chi}$ is a
representation of $\tilde{\mathcal{P}}$, which extends $(V_{n}^{\chi},U_{n})$.
_$(W_{1}^{\chi},E_{1})$ , $\chi\in\\{+,-\\}$ is called the $\chi$-handed Weyl
system._

Proof. Due to (23.1) we may turn at once to the Weyl tensor-localization
$(V_{n}^{\chi},U_{n},E_{n})$. Obviously $W_{n}^{\chi\,mom}(B)=U_{n}^{mom}(B)$
for $B\in SU(2)$, and
$\operatorname{e}^{\operatorname{i}th^{\chi}(p)}\pi^{\chi\eta}(p)=\operatorname{e}^{\operatorname{i}t\eta|p|}\pi^{\chi\eta}(p)$
holds. Hence $W_{n}^{\chi}$ is an extension of $(V_{n}^{\chi},U_{n})$. Then,
using the relation

$\pi^{\chi\eta}(p)s^{\chi}(A)^{*-1}=(|q^{\eta}|/|p|)\,s^{\chi}(A)\pi^{\chi\eta}(q^{\eta})$
(23.10)

which is mainly a consequence of the fundamental relation
$A(\sum_{\mu=0}^{3}\mathfrak{p}_{\mu}\sigma_{\mu})A^{*}=\sum_{\mu=0}^{3}(A\cdot\mathfrak{p})_{\mu}\sigma_{\mu}$
with $\sigma_{0}:=I_{2}$ for $A\in SL(2,\mathbb{C})$,
$\mathfrak{p}\in\mathbb{R}^{4}$, and by means of the property
(23.3)$(\epsilon)$, one verifies the group multiplication law
$W_{n}^{\chi\,mom}(\mathfrak{a}+A\cdot\mathfrak{a}^{\prime},AA^{\prime})=W_{n}^{\chi\,mom}(\mathfrak{a},A)W_{n}^{\chi\,mom}(\mathfrak{a}^{\prime},A^{\prime})$.
Finally one checks unitarity of $W_{n}^{\chi\,mom}(A)$.∎

The representation $W_{n}^{\chi}$ of $\tilde{\mathcal{P}}$ in (23.3) can be
constructed as follows giving an insight into its composition. By the unitary
transformation $(Y_{0}\varphi)(p):=D(B(p)^{-1})\,\varphi(p)$ with
$D:=D^{(\frac{1}{2})}\otimes D^{(\frac{n}{2}-\frac{1}{2})}$ we go to the
energy representation $Y_{0}(V_{n}^{\chi mom},U_{n}^{mom})Y_{0}^{-1}$ of
$(V_{n}^{\chi},U_{n})$. It reads

$\big{(}V_{n}^{\chi
erg}(t)U_{n}^{erg}(b,B)\varphi\big{)}(p)=\big{(}\operatorname{e}^{\operatorname{i}t\chi|p|\sigma_{3}}\otimes
I_{n}\big{)}\operatorname{e}^{-\operatorname{i}bp}D\left(\begin{array}[]{cc}\kappa(p,B)&0\\\
0&\overline{\kappa(p,B)}\end{array}\right)\varphi(B^{-1}\cdot p)$ (23.11)

Hence $(V_{n}^{\chi
erg},U_{n}^{erg})\simeq\big{(}\bigoplus_{i=0}^{n-1}K^{\chi(\frac{n}{2}-i),\chi}\big{)}\oplus\big{(}\bigoplus_{i=1}^{n}K^{-\chi(\frac{n}{2}-i),-\chi}\big{)}$.
Each $K^{s,\eta}$ is extended to $\tilde{\mathcal{P}}$ by $W^{0,s,\eta}$
according to (23.2). Thus one gets an extension to $\tilde{\mathcal{P}}$ of
$(V_{n}^{\chi erg},U_{n}^{erg})$ by

$W_{n}^{\chi
erg}\simeq\big{(}\bigoplus_{i=0}^{n-1}W^{0,\chi(\frac{n}{2}-i),\chi}\big{)}\oplus\big{(}\bigoplus_{i=1}^{n}W^{0,-\chi(\frac{n}{2}-i),-\chi}\big{)}$
(23.12)

It satisfies $\big{(}W_{n}^{\chi
erg}(A_{\rho})\varphi\big{)}(p)=\big{(}\sqrt{|q^{\chi}|/|p|}\,\varphi_{1}(q^{\chi}),\sqrt{|q^{-\chi}|/|p|}\,\varphi_{2}(q^{-\chi})\big{)}$
as $k(\mathfrak{p},A_{\rho})=1$ for all $\mathfrak{p}\in\mathcal{O}^{0}$ by
Eq. (23.3) and $b(p)=b(q^{-})$ for $A=A_{\rho}$. By some computations one goes
back to the momentum representation obtaining $W_{n}^{\chi\,mom}(A_{\rho})$.
For general $A=B^{\prime}A_{\rho}B$ with $B^{\prime},B\in SU(2)$ one obtains
$W_{n}^{\chi\,mom}(A)$ in (23.3) from
$W_{n}^{\chi\,mom}(A)=U_{n}^{mom}(B^{\prime})W_{n}^{\chi\,mom}(A_{\rho})U_{n}^{mom}(B)$.

(147) Lemma. Let $(V,U)$ be a representation of the little kinematical group
with $C=0$ and finite helicity spectrum. Then all representations $W$ of
$\tilde{\mathcal{P}}$ extending $(V,U)$ are unitarily equivalent.
Proof. This is an immediate consequence of the representation theory of
$\tilde{\mathcal{P}}$ as the mass-squared operator $C$, the sign of the energy
operator $\operatorname{sgn}(H)$, and the helicity operator $|P|^{-1}JP$ are
already determined by $(V,U)$.∎

(148) Theorem. Let $(V,U,E)$ be a massless WL with causal time evolution. Let
$X$ be the associated causal position operator (recall _Eq. ( 2.2)_). Then
there is a unique representation $W_{E}$ of $\tilde{\mathcal{P}}$ which
extends $(V,U)$ such that its booster is given by

$N=\frac{1}{2}\\{X,H\\}+\frac{\operatorname{sgn}(H)}{|H|}P\times\left(J-X\times
P-\frac{\operatorname{i}}{4}[X,H]\times[X,H]\right)$ (23.13)

The representation $W_{n}^{\chi}$ from _( 23.3)_ satisfies this relation. Any
extension $W$ of $(V,U)$ to $\tilde{\mathcal{P}}$ is unitarily equivalent to
$W_{E}$.

Proof. Uniqueness of $W_{E}$ is obvious as the representation of all
generators of the Poincaré group is set. Due to (23.1) it suffices to prove
the existence of $W_{E}$ for $(V_{n}^{\chi},U_{n})$ of the Weyl tensor-
localization. It is not hard to show that the booster of $W_{n}^{\chi}$ from
(23.3) satisfies the formula for $N$. In particular check
$\frac{d}{dt}R_{0}(p^{\eta},\operatorname{e}^{\frac{t}{2}\sigma_{k}})|_{t=0}=\frac{\operatorname{i}\eta}{|p|}\frac{1}{2}(p_{l}\sigma_{m}-p_{m}\sigma_{l})$
with $(k,l,m)$ a cyclic permutation of $(1,2,3)$ and verify that
$I_{2}\otimes\frac{1}{2}\mathfrak{d}^{(\frac{n}{2}-\frac{1}{2})}(\sigma_{1},\sigma_{2},\sigma_{3})$
is given by $J-X\times P-\frac{\operatorname{i}}{4}[X,H]\times[X,H]$. The last
part of the assertion holds true by (23.3). ∎

The next result, which follows from (23.3), shows how a general massless SCT
is built of the systems $(W_{n}^{\chi},E_{n})$. Recall (21).

(149) Theorem. Every massless SCT $(W,E)$ is unitarily related to $(W_{E},E)$.
The latter is unitarily equivalent to a finite orthogonal sum of the systems
$(W_{n}^{\chi},E_{n})$, $\chi\in\\{+,-\\}$, $n\in\mathbb{N}$ with uniquely
determined multiplicities.

As to (23.3) note that a massless SCT $(W,E)$ in general is not unitarily
equivalent to $(W_{E},E)$. The example (23.3) shows that there are massless
SCT $(W,E)$ which are irreducible although $(W_{E},E)$ is not. In view of the
result (23.4) we study the following special case.

(150) Lemma. Let $\chi\in\\{+,-\\}$ and let $(W,E)$ be a massless SCT
unitarily related to a finite multiple of $(W^{\chi}_{1},E_{1})$. Let $Q$ be
an orthogonal projection commuting with $(W,E)$. Then there is a unitary
transformation $S$ such that $W=SW_{E}S^{-1}$ and which commutes with $Q$.
Proof. By assumption there are unitary $S$ and $U$ such that $W=SW_{E}S^{-1}$
and $(W_{E},E)=U\,\big{(}I_{\nu}\otimes(W^{\chi}_{1},E_{1})\big{)}\,U^{-1}$
with multiplicity $\nu\in\mathbb{N}$ and $I_{\nu}$ the identity on
$\mathbb{C}^{\nu}$. Therefore it is no restriction assuming
$W=S\,(I_{\nu}\otimes W_{1}^{\chi})\,S^{-1}$ and $E=I_{\nu}\otimes E_{1}$.
Recall that $S$ commutes with $W|_{\mathcal{K}}=(I_{\nu}\otimes
W_{1})|_{\mathcal{K}}$, where $\mathcal{K}$ denotes the little kinematical
group $\mathbb{R}\otimes ISU(2)$.
As $Q$ commutes with $(W,E)$, it commutes with its restriction to the
Euclidean group $ISU(2)$, which is $I_{\nu}\otimes(U_{1},E_{1})$. Now,
$(U_{1},E_{1})$ is the system of imprimitivities induced by
$D=D^{\frac{1}{2}}$ on $SU(2)$ and therefore irreducible. Hence
$Q=Q_{\nu}\otimes I$ with $Q_{\nu}$ an orthogonal projection matrix on
$\mathbb{C}^{\nu}$ and $I$ the identity on
$L^{2}(\mathbb{R}^{3},\mathbb{C}^{2})$.
Recall that $Q$ commutes with $W$. Hence $S^{-1}QS$ commutes with
$I_{\nu}\otimes W^{\chi}_{1}$. By Eq. (23.12), $I_{\nu}\otimes W_{1}^{\chi
erg}=(I_{\nu}\otimes W^{0,\chi/2,\chi})\oplus(I_{\nu}\otimes
W^{0,\chi/2,-\chi})$. Recall that $W^{0,s,\eta}$ is $[0,s,\eta]$ and, hence,
irreducible and mutually inequivalent. Consequently one has
$S^{erg-1}QS^{erg}=(Q_{\nu}^{\chi}\otimes
I^{\prime})\oplus(Q_{\nu}^{-\chi}\otimes I^{\prime})=(Q_{\nu}^{\chi}\oplus
Q_{\nu}^{-\chi})\otimes I^{\prime}$ with orthogonal projection matrices
$Q_{\nu}^{\chi}$, $Q_{\nu}^{-\chi}$ on $\mathbb{C}^{\nu}$ and $I^{\prime}$ the
identity on $L^{2}(\mathbb{R}^{3},\mathbb{C})$.
Similarly, as $S^{erg}$ commutes with $I_{\nu}\otimes W_{1}^{\chi
erg}|_{\mathcal{K}}=(I_{\nu}\otimes K^{\chi/2,\chi})\oplus(I_{\nu}\otimes
K^{\chi/2,-\chi})$, one infers from (23.2) that
$(S^{erg}\varphi)(p)=s(|p|)\varphi(p)$ with unitary matrices
$s(r)=s^{\chi}(r)\oplus s^{-\chi}(r)$ on
$\mathbb{C}^{\nu}\oplus\mathbb{C}^{\nu}$ for $r\geq 0$.
Therefore $s^{\chi}(r)^{-1}Q_{\nu}\,s^{\chi}(r)=Q^{\chi}_{\nu}$ for almost all
$r\geq 0$, and also for $-\chi$ in place of $\chi$. Hence there is $r_{0}\geq
0$ such that $s(r)s(r_{0})^{-1}$ commutes with $Q_{\nu}^{\chi}\oplus
Q_{\nu}^{-\chi}$ for almost all $r\geq 0$. Then
$(S^{erg}_{Q}\varphi)(p):=s(|p|)s(r_{0})^{-1}\varphi(p)$,
$(S^{erg}_{0}\varphi)(p):=s(r_{0})\varphi(p)$ define unitary transformations
$S_{Q}$, $S_{0}$ such that $S=S_{Q}S_{0}$. Moreover, $S_{0}$ commutes with
$I_{\nu}\otimes W_{1}^{\chi}$, whence $W=S(I_{\nu}\otimes
W_{1}^{\chi})S^{-1}=S_{Q}(I_{\nu}\otimes W_{1}^{\chi})S_{Q}^{-1}$. Finally,
$S_{Q}$ commutes with $Q$, thus concluding the proof.∎

(151) Example for an irreducible massless SCT $(W,E)$ unitarily related to
$2\,(W_{1}^{+},E_{1})$. For its construction we refer to the proof of (23.3).
Let $s^{\pm}(r):=I_{2}\cos r\pm\operatorname{i}\sigma_{2}\sin r$ for $r\geq
0$, which depends continuously on $r$. Choose $r_{0}=0$. Then $0,I_{2}$ are
the only projections invariant under $\\{s^{\pm}(r)s^{\pm}(0)^{-1}:r\geq
0\\}$. Hence $(W,E)$ is irreducible by (23.3).
For the momentum representation $S^{mom}=Y_{h}^{-1}S^{erg}Y_{h}$ of $S$ one
gets $(S^{mom}\varphi)(p)=s(p)\varphi(p)$ with
$s(p):=\exp(\operatorname{i}\sigma_{2}\otimes\sum_{k=1}^{3}\sigma_{k}p_{k})$.

###  23.4. Determination of the massless causal systems

It does not suffice to examine the $(W^{\chi}_{n},E_{n})$ for
$n\in\mathbb{N}$, $\chi\in\\{+,-\\}$, but all irreducible massless SCT have to
be considered which are unitarily related to finite orthogonal sums of the
former.

(152) Theorem. Let $(W,E)$ be an irreducible massless SCT, which localizes
frame-independently. Then $(W,E)$ is unitarily related to a finite multiple of
$(W_{1}^{\chi},E_{1})$ for some $\chi\in\\{+,-\\}$. In particular,
$(W_{n}^{\chi},E_{n})$, $\chi\in\\{+,-\\}$, $n>1$ do not localize frame-
independently.
Proof. Recall $A_{\rho}=\operatorname{e}^{\frac{\rho}{2}\sigma_{3}}$. By
(23.3) and (23.3), up to unitary equivalence, the momentum representation of
$W(A_{\rho})$ reads

$\big{(}W^{mom}(A_{\rho})\varphi\big{)}(p)=s(p)^{-1}\sum_{\eta=\pm
1}w^{\eta}(p)s(q^{\eta})\varphi(q^{\eta})$ ($1$)

for almost all $p\in\mathbb{R}^{3}$. Here,
$w^{\eta}(p):=\bigoplus_{\chi}\pi^{\chi\eta}(p)s^{\chi}(A_{\rho})^{*-1}\otimes
D^{\chi}\big{(}R_{0}(\mathfrak{p}^{\eta},A_{\rho})\big{)}$ with some finite
dimensional representation $D^{\chi}$ of $ISU(2)$, and, according to (21),
there is some unitary operator $S$ commuting with $W|_{\mathbb{R}\times
ISU(2)}$ so that $(S^{mom}\varphi)(p)=s(p)\varphi(p)$ with unitary matrices
$s(p)$ acting on spinor space.
Now let $\varphi$ be localized in a bounded region. By assumption so is
$\varphi_{\rho}:=W^{mom}(A_{\rho})\varphi$. Hence, by the Paley-Wiener
Theorem, these functions are the restriction of entire functions on
$\mathbb{C}^{3}$, still denoted by $\varphi$ and $\varphi_{\rho}$. We need to
exploit only their continuity, in particular at $p=0$.
Since $S$ is rotation-invariant, one has
$s(p)=D(B(p))s(|p|e_{3})D(B(p))^{-1}$, where $D:=(D^{(\frac{1}{2})}\otimes
D^{+})\oplus(D^{(\frac{1}{2})}\otimes D^{-})$. Let
$I^{\chi}:=D^{\chi}(I_{2})$. As $S$ is also time translation-invariant,
$s(|p|e_{3})$ commutes with
$(\operatorname{e}^{\operatorname{i}t|p|\sigma_{3}}\otimes
I^{+})\oplus(\operatorname{e}^{-\operatorname{i}t|p|\sigma_{3}}\otimes I^{-})$
for all $t$ (see the energy representation Eq. (23.11)). Hence, equivalently,
$s(re_{3})$, $r>0$ commutes with $P^{\eta}$, where
$P^{+}:=(\big{(}\tiny{\begin{array}[]{cc}1&0\\\ 0&0\end{array}}\big{)}\otimes
I^{+})\oplus(\big{(}\tiny{\begin{array}[]{cc}0&0\\\
0&1\end{array}}\big{)}\otimes I^{-})$ and $P^{-}:=I-P^{+}$.
One has
$D(B(p))^{-1}w^{\eta}(p)D(B(q^{\eta}))=\sqrt{|q^{\eta}|/|p|}\,P^{\eta}$ (see
the energy representation of $W^{\chi}_{n}(A_{\rho})$ after Eq. (23.12)). Let
$t^{\eta}(p):=s(|p|e_{3})^{-1}s(|q^{\eta}|e_{3})\sqrt{|q^{\eta}|/|p|}\,P^{\eta}$
for $p\neq 0$. Clearly $t^{\eta}$ commutes with $P^{\pm}$ and
$t^{\eta}(p)=t^{\eta}(p^{\prime})$ if $|p|=|p^{\prime}|$ and
$|q^{\eta}|=|q^{\prime\eta}|$. By $(1)$,

$\varphi_{\rho}(p)=\sum_{\eta=\pm
1}D(B(p))t^{\eta}(p)D(B(q^{\eta}))^{-1}\varphi(q^{\eta})$ ($2$)

holds for almost all $p\in\mathbb{R}^{3}$.

(a) First we will redefine $t^{\eta}$ on some null set such that it becomes
continuous on $\mathbb{R}^{3}\setminus Z$ with
$Z:=\\{0\\}\times\\{0\\}\times\mathbb{R}$.
Fix $p_{*}\in\mathbb{R}^{3}$. One has
$\varphi(p)=(2\pi)^{-3/2}\int\operatorname{e}^{-\operatorname{i}px}\psi(x)\operatorname{d}^{3}x$
for $\psi:=\mathcal{F}^{-1}\varphi$ with compact support. Hence, if necessary
it suffices to alter $\psi$ a bit on its support in order to achieve
$\varphi(q^{\eta}_{*})\neq 0$ for $\eta=\pm$. Choose matrices $u_{i}$ acting
on the spinor space such that the squared matrix
$\phi(p):=(u_{1}\varphi(p),u_{2}\varphi(p),\dots,u_{n}\varphi(p))$, where $n$
is the dimension of the spinor space, is invertible at $p=q_{*}^{+}$ and
$p=q_{*}^{-}$. By continuity there is a neighborhood $N(p_{*})$ of $p_{*}$
such that $\phi(q^{\eta})$ is still invertible for all $p\in N(p_{*})$,
$\eta=\pm$.
Clearly, the functions $u_{i}\psi(x)$ have compact support with
$u_{i}\varphi(p)$ their Fourier transforms. Define the matrix $\phi_{\rho}(p)$
replacing $\varphi$ with $\varphi_{\rho}$ in $\phi(p)$. Then $(2)$ holds for
$\phi_{\rho}(p)$ and $\phi(q^{\eta})$ in place of $\varphi_{\rho}(p)$ and
$\varphi(q^{\eta})$, respectively. Hence, as $t^{\eta}(p)$ commutes with
$P^{+}$ and $P^{-}$, one finds

$t^{\eta}(p)=P^{\eta}D(B(p))^{-1}\,\phi_{\rho}(p)\phi(q^{\eta})^{-1}D(B(q^{\eta}))$
($3$)

for almost all $p\in N(p_{*})$.
Assume $p_{*}\not\in Z$. Then the right hand side of $(3)$ is continuous on
$N(p_{*})\setminus Z$ because of Eq. (21.8) and since $q^{\eta}_{1}=p_{1}$,
$q^{\eta}_{2}=p_{2}$. Hence, redefining $t^{\eta}$ by $(3)$, one gets
$t^{\eta}$ continuous on $\mathbb{R}^{3}\setminus Z$ and
$t^{\eta}(p)=s(|p|e_{3})^{-1}s(|q^{\eta}|e_{3})\sqrt{|q^{\eta}|/|p|}\,P^{\eta}$
for all $p\in\mathbb{R}^{3}\setminus L$ for some Lebesgue null set
$L\subset\mathbb{R}^{3}$.

(b) Let $p,p^{\prime}\in\mathbb{R}^{3}\setminus Z$ and $\eta=\pm$. Clearly,
$t^{\eta}(p)$ still commutes with $P^{+}$ and $P^{-}$. We show that
$t^{\eta}(p)=t^{\eta}(p^{\prime})$ still holds if $|p|=|p^{\prime}|$ and
$|q^{\eta}|=|q^{\prime\eta}|$ or, equivalently, if $|p|=|p^{\prime}|$ and
$p_{3}=p^{\prime}_{3}$, since $q^{\eta}=(p_{1},p_{2},\alpha
p_{3}-\beta\eta|p|)$ and hence $|q^{\eta}|=\alpha|p|-\beta\eta p_{3}$.
Indeed, assume the contrary. Then there are
$p_{*},p_{*}^{\prime}\in\mathbb{R}^{3}\setminus Z$ with
$|p_{*}|=|p_{*}^{\prime}|$, $p_{*3}=p^{\prime}_{*3}$, and $t^{\eta}(p_{*})\neq
t^{\eta}(p_{*}^{\prime})$. By continuity $t^{\eta}(p)\neq
t^{\eta}(p^{\prime})$ holds true for all $p\in N$ and $p^{\prime}\in
N^{\prime}$, where $N$ and $N^{\prime}$ are some neighborhoods in
$\mathbb{R}^{3}\setminus Z$ of $p_{*}$ and $p^{\prime}_{*}$, respectively.
Using cylindrical coordinates $z,r,\varphi$ for $p\in\mathbb{R}^{3}$, there is
an open disc $Q\subset\mathbb{R}\times]0,\infty[$ around $(z_{*},r_{*})$ and
open intervals $I$ and $I^{\prime}$ around $\varphi_{*}$ and
$\varphi^{\prime}_{*}$, respectively, such that $Q\times I\subset N$ and
$Q\times I^{\prime}\subset N^{\prime}$. By Tonelli’s theorem there is
$\varphi\in I$ and a Lebesgue null set $M\subset Q$ with $(Q\setminus
M)\times\\{\varphi\\}\subset N\setminus L$. $\varphi^{\prime}\in I^{\prime}$
and $M^{\prime}\subset Q$ are determined analogously. Hence there is $q\in
Q\setminus(M\cup M^{\prime})$. Then $p$ and $p^{\prime}$ given by
$(q,\varphi)$ and $(q,\varphi^{\prime})$, respectively, yield the
contradiction $t^{\eta}(p)=t^{\eta}(p^{\prime})$.

(c) Consider $p\in\mathbb{R}^{3}\setminus Z$. Then $\lambda p\in N(0)\setminus
Z$ for all small $\lambda>0$. So, by $(3)$, $t^{\eta}(\lambda
p)=P^{\eta}D(B(\lambda p))^{-1}\phi_{\rho}(\lambda p)\phi(\lambda
q^{\eta})\,D(B(\lambda q^{\eta}))\to T^{\eta}(p)$ for $\lambda\to 0$ with

$T^{\eta}(p):=P^{\eta}D(B(p))^{-1}X\,D(B(q^{\eta}))$ ($4$)

and $X:=\phi_{\rho}(0)\phi(0)^{-1}$ independent of $p$. It follows from (b)
that $T^{\eta}(p)$ commutes with $P^{+}$ and $P^{-}$, and that
$T^{\eta}(p^{\prime})=T^{\eta}(p)$ if $|p^{\prime}|=|p|$,
$p^{\prime}_{3}=p_{3}$. We will infer from (4) and these properties that
$D^{+}$ and $D^{-}$ are multiples of $D^{(0)}$.

(d) Write $X$ as a $2\times 2$-block matrix according to the orthogonal sum of
$D$ with entries $X_{\chi\chi^{\prime}}$, $\chi,\chi^{\prime}=\pm$. Then
$T^{+}(p)_{++}=\left(\big{(}\tiny{\begin{array}[]{cc}1&0\\\
0&0\end{array}}\big{)}B(p)^{-1}\otimes
D^{+}(B(p))^{-1}\right)\,X_{++}\,\big{(}B(q^{+})\otimes
D^{+}(B(q^{+}))\big{)}$. According to the tensor structure of the factors
write $X_{++}$ as a $2\times 2$-block matrix with entries
$U_{\varsigma\varsigma^{\prime}}$, $\varsigma,\varsigma^{\prime}=\pm$. (The
indices refer to the helicity values $s=\varsigma\frac{1}{2}$.) So,
$T^{+}(p)=T^{+}(p)P^{+}$ implies
$T^{+}(p)_{++,+-}=D^{+}(B(p))^{-1}\big{(}(a_{+}(p)U_{++}+\overline{b}a_{-}(p)U_{-+})(-\overline{b})a_{-}(q^{+})+(a_{+}(p)U_{+-}+\overline{b}a_{-}(p)U_{--})a_{+}(q^{+})\big{)}D(B(q^{+}))=0$
using Eq. (21.8). Equivalently,
$(|p|^{2}-p^{2}_{3})C=(-p_{1}+\operatorname{i}p_{2})(|p|-p_{3})A+(p_{1}+\operatorname{i}p_{2})(|p|+p_{3})B$
with $A:=\operatorname{e}^{\rho/2}U_{-+}$,
$B:=\operatorname{e}^{-\rho/2}U_{+-}$, and
$C:=\operatorname{e}^{\rho/2}U_{++}-\operatorname{e}^{-\rho/2}U_{--}$. As
$A,B,C$ are independent of $p$, and since the equation holds for all
$p\in\mathbb{R}^{3}\setminus Z$, A=B=C=0 follows. Hence $U_{+-}=U_{-+}=0$ and
$U_{--}=\operatorname{e}^{\rho}U_{++}$. Let $U$ denote the constant matrix
$\operatorname{e}^{\rho/2}U_{++}$. Then a short computation yields

$T^{+}(p)_{++}=\sqrt{|q^{+}|/|p|}\left(\begin{array}[]{cc}D^{+}(B(p))^{-1}UD^{+}(B(q^{+}))&0\\\
0&0\end{array}\right)$ ($5$)

Similarly one computes $T^{+}(p)_{+-}=0$, $T^{+}(p)_{-+}=0$ and

$T^{+}(p)_{--}=\sqrt{|q^{+}|/|p|}\left(\begin{array}[]{cc}0&0\\\
0&D^{-}(B(p))^{-1}VD^{-}(B(q^{+}))\end{array}\right)$ ($6$)

for some constant matrix $V$. From the definition of $t^{+}(p)$ it follows
that $U$ and $V$ are unitary. Analogous formulae hold for $T^{-}(p)$.

(e) Next we exploit that $T^{\eta}(p)=T^{\eta}(p^{\prime})$ if
$p,p^{\prime}\in\mathbb{R}^{3}\setminus Z$ with $|p|=|p^{\prime}|$,
$p_{3}=p^{\prime}_{3}$. It implies by $(5)$ that the matrices
$D^{+}\big{(}B(q^{\prime+})B(q^{+})^{-1}\big{)}$ and
$D^{+}\big{(}B(p^{\prime})B(p)^{-1}\big{)}$ are unitarily equivalent by means
of $U$. Hence they have the same spectrum.
Choose $p,p^{\prime}$ such that $b(p)=b(q^{+})=1$ (i.e., $p_{2}=0$) and
$b(p^{\prime})=b(q^{\prime+})=\operatorname{e}^{\operatorname{i}\varphi}$ for
$\varphi\in\mathbb{R}\setminus\\{0\\}$. Then the trace of
$B(p^{\prime})B(p)^{-1}$ is $2-2a_{-}(p)^{2}(1-\cos\varphi)$. Hence its
eigenvalues are $\lambda(p),\overline{\lambda(p)}$ with $|\lambda(p)|=1$ and
$\Re(\lambda(p))=1-a_{-}(p)^{2}(1-\cos\varphi)$. Therefore
$B(p^{\prime})B(p)^{-1}$ is unitarily equivalent to
$\operatorname{diag}(\lambda(p),\overline{\lambda(p)})$ by some $B\in SU(2)$.
Consequently, $D^{+}\big{(}B(p^{\prime})B(p)^{-1}\big{)}$ is unitarily
equivalent to
$D^{+}\big{(}\operatorname{diag}(\lambda(p),\overline{\lambda(p)})\big{)}$ by
$D^{+}(B)$. As
$D^{(j)}\big{(}\operatorname{diag}(\lambda(p),\overline{\lambda(p)})\big{)}=\operatorname{diag}(\lambda(p)^{2j},\lambda(p)^{2j-2},\dots,\lambda(p)^{-2j})$
for $j\in\mathbb{N}_{0}/2$, the spectrum $\Sigma(p)$ of
$D^{+}\big{(}B(p^{\prime})B(p)^{-1}\big{)}$ equals
$\\{\lambda(p)^{2m}:m=-j_{f},-j_{f}+1,\dots,j_{f}\\}$ for some half-integer
$j_{f}$ or equals $\\{\lambda(p)^{2m}:m=-j_{b},-j_{b}+1,\dots,j_{b}\\}$ for
some integer $j_{b}$ or equals the union of these sets. Replacing $p$ by
$q^{+}$ one obtains the spectrum $\Sigma(q^{+})$ of
$D^{+}\big{(}B(q^{\prime+})B(q^{+})^{-1}\big{)}$.

(f) Now we claim that $D^{+}$ is a multiple of $D^{(0)}$. Check that
$\lambda(p)\neq 1$, $\lambda(q^{+})\neq 1$ and
$\Re(\lambda(p))\neq\Re(\lambda(q^{+}))$ as $\varphi\neq 0$. Hence
$\lambda(p)\neq\lambda(q^{+})$ and $\lambda(p)\neq\overline{\lambda(q^{+})}$.
Choose $\varphi\in\mathbb{R}$ such that $\lambda(p)$ is not a root of unity.
This is possible as $a_{-}(p)\neq 0$.
For the proof assume the contrary. As shown above, $\Sigma(p)$ and
$\Sigma(q^{+})$ coincide. Hence, if $\lambda(p)\in\Sigma(p)$, then there are
$a,b\in\mathbb{Z}\setminus\\{0,-1\\}$ such that
$\lambda(p)^{2a+1}=\lambda(q^{+})$ and $\lambda(q^{+})^{2b+1}=\lambda(p)$, and
the contradiction $\lambda(p)^{(2a+1)(2b+1)-1}=1$ follows. If
$\lambda(p)^{2}\in\Sigma(p)$, then there are
$a,b\in\mathbb{Z}\setminus\\{-1,1\\}$ such that
$\lambda(p)^{2a}=\lambda(q^{+})^{2}$ and $\lambda(q^{+})^{2b}=\lambda(p)^{2}$,
and the contradiction $\lambda(p)^{2ab-2}=1$ follows. This proves the claim.
Analogously one shows that $D^{-}$ is a multiple of $D^{(0)}$.

(g) By the result achieved so far, $W=\nu^{+}W^{+}_{1}\oplus\nu^{-}W^{-}_{1}$
with multiplicities $\nu^{\chi}\in\mathbb{N}_{0}$, $\chi=\pm$. It remains to
show that $S$ decomposes accordingly. Let $\mathcal{K}$ denote the little
kinematical group $\mathbb{R}\otimes ISU(2)$. As $S$ commutes with
$W|_{\mathcal{K}}$, it suffices to show that $W^{+}_{1}|_{\mathcal{K}}$ does
not contain an irreducible subrepresentation which is equivalent to a
subrepresentation of $W^{-}_{1}|_{\mathcal{K}}$. Hence we need the
decomposition of $W^{\chi}_{1}|_{\mathcal{K}}$. By Eqs. (23.12), (23.11),
$W^{\chi}_{1}|_{\mathcal{K}}\simeq K^{\chi/2,\chi}\oplus K^{\chi/2,-\chi}$
holds, whence the assertion by (23.2).∎

This result (23.4), due to (7.2), definitely proves that the higher Weyl
tensor-systems $(W^{\chi}_{n},E_{n})$, $\chi\in\\{+,-\\}$, $n>1$ are
mathematical artefacts, interim results in determining the massless causal
systems.

In sec. 16.1 it is shown that the systems $(W_{1}^{\chi},E_{1})$,
$\chi\in\\{+,-\\}$ are causal. The following example concerns a massless SCT
unitarily related to $(W_{1}^{\chi},E_{1})$, which localizes frame-
independently without being causal.

(153) Example. Let $W:=SW^{\chi}_{1}S^{-1}$ with $S:=W_{1}^{\chi}(r)$ for
$r\equiv(r,0,I_{2})$ with $r\in\mathbb{R}\setminus\\{0\\}$, and $E:=E_{1}$.
Then obviously $(W,E)$ is a massless SCT unitarily related to
$(W_{1}^{\chi},E_{1})$. For all $h\in\tilde{\mathcal{P}}$ and
$\Gamma\in\mathcal{S}$ one has
$W(h)E(\Gamma)W(h)^{-1}=W_{1}^{\chi}(rhr^{-1})E(\Gamma)W_{1}^{\chi}(rhr^{-1})^{-1}$,
whence

$W(h)E(\Gamma)W(h)^{-1}\leq E(\Gamma_{rhr^{-1}})$ (1)

by causality of $(W_{1}^{\chi},E_{1})$. Then
(a) $(W,E)$ localizes frame-independently.
(b) $(W,E)$ is not causal.
Proof. (a) is obvious by (1). — (b) In (1) specify $h=A_{\rho}$ for $\rho\neq
0$ and, for the moment, $\Gamma=\\{0\\}$ as subset of the spacelike hyperplane
$\\{0\\}\times\mathbb{R}^{3}\equiv\mathbb{R}^{3}$. Then $A_{\rho}\cdot 0=0$,
whence $\Gamma_{h}=\\{0\\}$. Furthermore,
$rhr^{-1}=(r(1-\cosh\rho),0,0,-r\sinh\rho,A_{\rho})$ and
$rhr^{-1}\cdot\Gamma=\\{(r(1-\cosh\rho),0,0,-r\sinh\rho)\\}$, whence
$x\in\Gamma_{rhr^{-1}}$ $\Leftrightarrow$
$r^{2}(\cosh\rho-1)^{2}-x_{1}^{2}-x_{2}^{2}-(x_{3}+r\sinh\rho)^{2}\geq 0$. For
$x\in\Gamma_{rhr^{-1}}$, therefore, $x_{3}$ lies in the closed interval with
endpoints $r(\operatorname{e}^{-\rho}-1)$ and $r(1-\operatorname{e}^{\rho})$,
which are either both positive or both negative, thus excluding $x_{3}=0$.
Hence $\Gamma_{h}\cap\Gamma_{rhr^{-1}}=\emptyset$. By continuity this still
holds true if $\Gamma=\\{0\\}$ is replaced by
$\Gamma=\\{y\in\mathbb{R}^{3}:|y|<\epsilon\\}$ for $\epsilon>0$ small enough.
This implies $W(h)E(\Gamma)W(h)^{-1}\not\leq E(\Gamma_{h})$ confirming the
claim.∎

(154) Lemma. Let the massless causal system $(W,E)$ be unitarily related to
$\nu\,(W_{1}^{\chi},E_{1})$ for some $\chi\in\\{+,-\\}$ with multiplicity
$\nu\in\mathbb{N}$ by the unitary transformation $S$. Suppose $SE(B)S^{-1}\leq
E(B^{\prime})$, where $B$, $B^{\prime}$ are two balls with equal radius. Then
$(W,E)$ is unitarily equivalent to $\nu\,(W_{1}^{\chi},E_{1})$.

Proof. Without restriction $W=S\,\nu W_{1}^{\chi mom}\,S^{-1}$ and
$E=\nu\,E_{1}^{mom}=\mathcal{F}E^{can}\mathcal{F}^{-1}$. Recall that then
$(S\varphi)(p)=s(p)\varphi(p)$, where
$s:\mathbb{R}^{3}\to\mathbb{C}^{2\nu\times 2\nu}$ is measurable and $s(p)$
unitary.

(a) First we show that
$s(p)=\operatorname{e}^{\operatorname{i}(a-a^{\prime})p}s_{0}$, where $a$ and
$a^{\prime}$ are the midpoints of $B$ and $B^{\prime}$, respectively, and
$s_{0}$ is some constant. Indeed, let $R$ denote the radius of $B$ and
$B^{\prime}$, let $B_{R}$ be the open ball around the origin with radius $R$
and set $S^{\prime}:=W(a^{\prime})^{-1}SW(a)$. Then $a+B_{R}=B$,
$B_{R}=B^{\prime}-a^{\prime}$, whence
$S^{\prime}E(B_{R})S^{\prime-1}=W(a^{\prime})^{-1}SW(a)E(B_{R})W(a)^{-1}S^{-1}W(a^{\prime})=W(a^{\prime})^{-1}SE(B)S^{-1}W(a^{\prime})\leq
W(a^{\prime})^{-1}E(B^{\prime})W(a^{\prime})=E(B_{R})$. Moreover check by
(23.3) for $n=1$ that
$(S^{\prime}\varphi)(p)=\operatorname{e}^{-\operatorname{i}(a-a^{\prime})p}s(p)\varphi(p)$.
Now, following the proof of [4, Theorem 10] for $\delta=0$ one infers that
$(S^{\prime}\varphi)(p)=s_{0}\varphi(p)$ for some constant matrix $s_{0}$.
This shows the assertion.

(b) Now one shows that $S$ commutes with $\nu W_{1}^{\chi mom}\equiv
W_{1}^{\chi mom}\otimes I_{\nu}$ thus concluding the proof. Indeed, as $S$
commutes with $\nu W_{1}^{\chi mom}|_{SU(2)}$ one has $s(B\cdot p)=(B\otimes
I_{\nu})s(p)(B^{-1}\otimes I_{\nu})$, whence
$\operatorname{e}^{\operatorname{i}bp}s_{0}=(B\otimes
I_{\nu})s_{0}(B^{-1}\otimes I_{\nu})$ with
$b:=\big{(}(B^{-1}-I_{2})\cdot(a-a^{\prime})\big{)}p$, for almost all
$p\in\mathbb{R}^{3}$ and all $B\in SU(2)$. This implies
$(B^{-1}-I_{2})\cdot(a-a^{\prime})=0$ and hence $a=a^{\prime}$. So $s_{0}$
commutes with $B\otimes I_{\nu}$ for all $B\in SU(2)$. Therefore there is
$s_{\nu}\in\mathbb{C}^{\nu\times\nu}$ such that $s_{0}=I_{2}\otimes s_{\nu}$,
whence the assertion. ∎

It remains to find out those irreducible SCT unitarily related to multiples of
$(W_{1}^{\chi},E_{1})$ for $\chi\in\\{+,-\\}$ that are causal. The foregoing
considerations clearly suggest that up to unitary equivalence the only ones
are the two Weyl systems $(W_{1}^{\chi},E_{1})$, $\chi\in\\{+,-\\}$ studied in
sec. 16.

##  24\. Appendix

###  24.1. A total set in $L^{p}_{\mu}(\mathbb{R}^{d},\mathbb{C}^{m})$ of
radially symmetric functions

The Lebesgue measure on $\mathbb{R}^{d}$ is a Radon measure. The following
result is valid for any Radon measure on $\mathbb{R}^{d}$. As to the
definition of a Radon measure on a topological Hausdorff space see e.g. [44,
2.1.1 (9)], cf. also [78, 1.5 Definition]. A subset $L$ of a topological
vector space $X$ is said to be total if the linear hull $\langle L\rangle$ of
$L$ is dense in $X$.

(155) Theorem. Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Let
$m,d\in\mathbb{N}$ and $1\leq p<\infty$. Let $e_{1},\dots,e_{m}$ denote the
elements of the standard basis of $\mathbb{C}^{m}$. Then
$\\{1_{B}e_{i}:B\subset\mathbb{R}^{d}\textnormal{ closed
ball},\;i=1,\dots,m\\}$ is total in
$L^{p}_{\mu}(\mathbb{R}^{d},\mathbb{C}^{m})$.
Proof. (a) Let $L$ be total in the topological vector space $X$. Let $M\subset
X$ with $L\subset\overline{\langle M\rangle}$. Then $M$ is total in $X$. This
is obvious as $X=\overline{\langle L\rangle}\subset\overline{\langle
M\rangle}$.
(b) Let also $L^{\prime}$ be total in the topological vector space
$X^{\prime}$. Then $M:=(L\times\\{0\\})\cup(\\{0\\}\times L^{\prime})$ is
total in $X\times X^{\prime}$. This is obvious as $\langle M\rangle=\langle
L\times L^{\prime}\rangle=\langle L\rangle\times\langle L^{\prime}\rangle$,
which is dense in $X\times X^{\prime}$.
Due to (b) the assertion is readily reduced to the claim that
$M:=\\{1_{B}:B\subset\mathbb{R}^{d}\textnormal{ closed ball}\\}$ is total in
$L^{p}:=L^{p}_{\mu}(\mathbb{R}^{d},\mathbb{C})$.
It follows from Lebesgue’s ladder that $\\{1_{A}:A\textnormal{
measurable},\;\mu(A)<\infty\\}$ is total in $L^{p}$. As $\mu$ is finite at
compact sets, for $\varepsilon>0$ and $A$ measurable with $\mu(A)<\infty$
there is $R>0$ great enough such $\mu(A\setminus B_{R})<\varepsilon$.
Therefore even $\\{1_{A}:A\textnormal{ measurable bounded}\\}$ is total in
$L^{p}$. Furthermore, due to the outer regularity of $\mu$, for
$\varepsilon>0$ and $A$ measurable bounded there is an open bounded $U\supset
A$ with $\mu(U\setminus A)<\varepsilon$. Consequently,
$L:=\\{1_{U}:U\textnormal{ open bounded}\\}$ is total in $L^{p}$.
Thus due to (a) it remains to show that $L\subset\overline{\langle M\rangle}$.
So for $U$ open bounded consider the set $\mathcal{B}:=\\{B:B\subset
U,B\textnormal{ closed ball}\\}$. Then according to Vitali’s covering theorem
[78, 2.8 Theorem] there are countably many mutually disjoint
$B_{i}\in\mathcal{B}$ with $\mu(U\setminus\bigcup_{i}B_{i})=0$. Hence, for
$\varepsilon>0$ there is $n\in\mathbb{N}$ with $\mu(U\setminus
V)<\varepsilon^{p}$ for $V:=\bigcup_{i=1}^{n}B_{i}$. Consequently,
$\parallel\\!\\!1_{U}-\sum_{i=1}^{n}1_{B_{i}}\\!\\!\parallel_{p}=\parallel\\!\\!1_{U}-1_{V}\\!\\!\parallel_{p}=\parallel\\!\\!1_{U\setminus
V}\\!\\!\parallel_{p}=\mu(U\setminus V)^{1/p}<\varepsilon$, accomplishing the
proof.∎

###  24.2. Imprimitivity Theorem

Let $G$ be a locally compact group with countable base acting on a locally
compact space $X$. Let $H$ be a separable Hilbert space. Then a system of
imprimitivity $(V,E)$ for $(G,X)$ on $H$ is a representation $V$ of $G$ on $H$
and a PM $E$ on $X$ in $H$ such that $V(g)E(M)V(g)^{-1}=V(g^{-1}M)$ for all
$g\in G$ and Borel sets $M\subset X$.
Let $G$ act continuously and transitively on $X$ and let $x_{0}\in X$ be any
point. Then every system of imprimitivity $(V,E)$ for $(G,X)$ on $H$ is
unitarily equivalent to a canonical system $(U^{\textsc{g}},E^{can})$ on
$L^{2}_{m}(X,H_{\textsc{u}})$. Here

$E^{can}(M)\psi:=1_{M}\psi$ (1)

for Borel set $M\subset X$ is the canonical PM on $X$, and $U$ is a
representation of the stabilizer subgroup $G_{x_{0}}$ in $x_{0}$ on the
Hilbert space $H_{\textsc{u}}$ and $U^{\textsc{g}}$ the representation of $G$
induced by $U$, i.e.,

$(U^{\textsc{g}}(g)\psi)(x):=\sqrt{(\operatorname{d}m^{g}/\operatorname{d}m)(x)}\,U\big{(}R(x,g)\big{)}\psi(g^{-1}x)$
(2)

where $m$ is the up equivalence unique non trivial regular Borel measure on
$X$, which is quasi-invariant under $G$, $m^{g}(M):=m(g^{-1}M)$ for Borel set
$M\subset X$, $\operatorname{d}m^{g}/\operatorname{d}m$ the Radon-Nikodym
derivative, $q:X\to G$ a Borel section with respect to the surjection $G\to
X$, $g\mapsto gx_{0}$, and the Wigner rotation
$R(x,g):=q(x)^{-1}g\,q(g^{-1}x)$.
Two canonical systems of imprimitivity for $(G,X)$ are equivalent,
respectively irreducible, if and only if the corresponding representations of
the stabilizer subgroup are equivalent, respectively irreducible.

###  24.3. Systems of imprimitivity for $(\tilde{\mathcal{P}}_{\chi},\chi)$

The aim is to extend a causal WL to non-timelike non-spacelike hyperplanes. As
studied in sec. 20, a rep of the lattice of the $\perp^{\prime}$-complete
regions (i.e., of the causal logic) would require such an extension.
By Poincaré covariance it suffices to consider the hyperplane
$\chi=\\{\mathfrak{x}:x_{0}=x_{1}\\}$. Then by (20) the invariance subgroup
$\tilde{\mathcal{P}}_{\chi}\subset\tilde{\mathcal{P}}$ equals

$IST(2)=\\{(\mathfrak{a},A):\mathfrak{a}\in\chi,A\in
SL(2,\mathbb{C}),A_{21}=0\\}$

One gets the canonical systems of imprimitivity for
$(\tilde{\mathcal{P}}_{\chi},\chi)$ by (24.2). First note that the Lebesgue
measure $\lambda_{\chi}$ on $\chi$ is quasi-invariant under
$\tilde{\mathcal{P}}_{\chi}$. Indeed, let $j:\mathbb{R}^{3}\to\chi$,
$j(x):=(x_{3},x)$. Then $\lambda_{\chi}=j(\lambda)$ with $\lambda$ the
Lebesgue measure on $\mathbb{R}^{3}$. Hence
$\lambda_{\chi}^{(\mathfrak{a},A)}(M)=\lambda\big{(}-j^{-1}(A^{-1}\cdot\mathfrak{a})+j^{-1}(A^{-1}\cdot
M)\big{)}=\det\big{(}j^{-1}\Lambda(A^{-1})j\big{)}\,\lambda_{\chi}(M)=|A_{11}|^{-2}\lambda_{\chi}(M)$,
where by (20) the last equation holds since
$j^{-1}\Lambda(\operatorname{e}^{-\rho\sigma_{3}/2})j=\operatorname{diag}(1,1,\operatorname{e}^{-\rho})$
and

$j^{-1}\Lambda\big{(}\left(\begin{array}[]{cc}1&w\\\
0&1\end{array}\right)\big{)}j=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
u&-v&1\end{array}\right)$

In particular,
$\operatorname{d}\lambda_{\chi}^{(\mathfrak{a},A)}/\operatorname{d}\lambda_{\chi}=|A_{11}|^{-2}$
is constant. — Next we choose $0\in\chi$ to be the fixed point. Then
$\tilde{\mathcal{P}}_{\chi}\to\chi$,
$(\mathfrak{a},A)\mapsto(\mathfrak{a},A)\cdot 0=\mathfrak{a}$ is the natural
surjection and obviously $q:\chi\to\tilde{\mathcal{P}}_{\chi}$,
$q(\mathfrak{a}):=(\mathfrak{a},I_{2})$ is a Borel section of it. Moreover,
$ST(2)$ is the stabilizer subgroup in $0$, and the Wigner rotation is
$R\big{(}\mathfrak{x},(\mathfrak{a},A)\big{)}=A$. Indeed,
$q(\mathfrak{x})^{-1}(\mathfrak{a},A)\,q\big{(}(\mathfrak{a},A)^{-1}\cdot\mathfrak{x}\big{)}=(-\mathfrak{x},I_{2})(\mathfrak{a},A)\,q\big{(}-A^{-1}\cdot\mathfrak{a},A^{-1})\cdot\big{)}=(-\mathfrak{x}+\mathfrak{a},A)\,q(-A^{-1}\cdot\mathfrak{a}+A^{-1}\cdot\mathfrak{x})=(-\mathfrak{x}+\mathfrak{a},A)(-A^{-1}\cdot\mathfrak{a}+A^{-1}\cdot\mathfrak{x},I_{2})=\big{(}-\mathfrak{x}+\mathfrak{a}+A\cdot(-A^{-1}\cdot\mathfrak{a}+A^{-1}\cdot\mathfrak{x}),A\big{)}=(0,A)\equiv
A$.
Now let $U$ be any representation of $ST(2)$ on the Hilbert space
$H_{\textsc{u}}$. Then the representation of $\tilde{\mathcal{P}}_{\chi}$
induced by $U$ acting on $L^{2}(\chi,H_{\textsc{u}})$, here called $V^{can}$,
reads

$\big{(}V^{can}(\mathfrak{a},A)\psi\big{)}(\mathfrak{x})=|A_{11}|^{-1}U(A)\psi\big{(}A^{-1}\cdot(\mathfrak{x}-\mathfrak{a})\big{)}$

and $(V^{can},P^{can})$, where $P^{can}$ denotes the canonical PM, is a system
of imprimitivity.

Shell representation. Recall that $\mathcal{O}^{m,\eta}$ for $m>0$, $\eta=\pm$
denotes the mass shell
$\\{\mathfrak{p}\in\mathbb{R}^{4}:p_{0}=\eta\epsilon(p)\\}$ and let
$\mathcal{O}^{m}:=\mathcal{O}^{m,+}\cup\mathcal{O}^{m,-}$ be the double shell
$\\{\mathfrak{p}\cdot\mathfrak{p}=m^{2}\\}$ endowed with the Lorentz invariant
measure. We shall use the shell representation $(V^{shell},P^{shell})$, which
is equivalent to $(V^{can},P^{can})$. The former acts on
$L^{2}(\mathcal{O}^{m},H_{\textsc{u}})$ by

  * (a)

$\big{(}V^{shell}(\mathfrak{a},A)\phi\big{)}(\mathfrak{p})=\operatorname{e}^{\operatorname{i}\mathfrak{a}\cdot\mathfrak{p}}U(A)\phi(A^{-1}\cdot\mathfrak{p})$

The form of $P^{shell}(M)$, $M\subset\chi$ is complicated. For
$M:=\\{\mathfrak{x}\in\chi:0\leq\varsigma x_{3}\leq\alpha\\}$, $\alpha>0$,
$\varsigma=\pm 1$ and $\phi\in L^{1}\cap L^{2}$ we get explicitly

  * (b)

$\big{(}P^{shell}(M)\phi\big{)}(\mathfrak{p})=\textrm{\footnotesize{$\sqrt{|p_{0}-p_{3}|}$}}\sum_{\eta^{\prime}=+,-}\int
k(\mathfrak{p},\mathfrak{p}^{\prime})\frac{\sqrt{|p^{\prime}_{0}-p^{\prime}_{3}|}}{2|p^{\prime}_{0}|}\phi(\mathfrak{p}^{\prime})\operatorname{d}p^{\prime}_{3}$

where $p^{\prime}_{1}:=p_{1}$, $p^{\prime}_{2}:=p_{2}$,
$p^{\prime}_{0}:=\eta^{\prime}\epsilon(p^{\prime})$, and
$k(\mathfrak{p},\mathfrak{p}^{\prime}):=\frac{\exp(\operatorname{i}\alpha\varsigma(p_{0}-p_{3}-p^{\prime}_{0}+p^{\prime}_{3}))-1}{\operatorname{i}\pi\varsigma(p_{0}-p_{3}-p^{\prime}_{0}+p^{\prime}_{3})}$.
A similar formula regarding $P^{shell}(M)$ is available if $M$ is a Borel set
with finite Lebesgue measure.

Let us sketch how to get the shell representation. Let
$\mathfrak{e}:=(1,0,0,1)$, consider the bijection
$l:\mathbb{R}^{3}\to\mathbb{R}^{4}/\mathbb{R}\mathfrak{e}$,
$l(p):=(0,p)+\mathbb{R}\mathfrak{e}$ and endow
$\mathbb{R}^{4}/\mathbb{R}\mathfrak{e}$ with the measure $l(\lambda)$, where
$\lambda$ is the Lebesgue measure on $\mathbb{R}^{3}$. Then
$\psi\mapsto\varphi:=\mathcal{F}(\psi\circ j)\circ l^{-1}$ defines an
isomorphism $\iota:L^{2}(\chi,H_{\textsc{u}})\to
L^{2}(\mathbb{R}^{4}/\mathbb{R}\mathfrak{e},H_{\textsc{u}})$. For integrable
$\psi$ one has
$\varphi(\mathfrak{p}+\mathbb{R}\mathfrak{e})=\big{(}\frac{1}{2\pi}\big{)}^{3/2}\int\operatorname{e}^{\operatorname{i}\mathfrak{y}\cdot\mathfrak{p}}\psi(\mathfrak{y})\operatorname{d}^{3}y$
with $\mathfrak{y}:=j(y)$. Let $\hat{V}:=\iota V^{can}\iota^{-1}$. It is easy
to verify
$\big{(}\hat{V}(\mathfrak{a},A)\varphi\big{)}(\mathfrak{p}+\mathbb{R}\mathfrak{e})=|A_{11}|\operatorname{e}^{\operatorname{i}\mathfrak{a}\cdot\mathfrak{p}}U(A)\varphi(A^{-1}\cdot\mathfrak{p}+\mathbb{R}\mathfrak{e})$.
Put
$\mathcal{K}^{\eta}:=\\{\mathfrak{p}+\mathbb{R}\mathfrak{e}:\operatorname{sgn}(p_{0}-p_{3})=\eta\\}$
and $\mathcal{K}:=\mathcal{K}^{+}\cup\mathcal{K}^{-}$. Then
$\mathcal{K}=(\mathbb{R}^{4}/\mathbb{R}\mathfrak{e})\setminus(\chi/\mathbb{R}\mathfrak{e})$,
where $\chi/\mathbb{R}\mathfrak{e}$ is a $l(\lambda)$-null set.
Now we pass to the double shell by the bijection
$k:\mathcal{O}^{m}\to\mathcal{K}$,
$k(\mathfrak{p}):=\mathfrak{p}+\mathbb{R}\mathfrak{e}$. Indeed,
$k|_{\mathcal{O}^{m,\eta}}\to\mathcal{K}^{\eta}$ is a bijection since
$\operatorname{sgn}(p_{0})=\operatorname{sgn}(p_{0}-p_{3})$ for every
$\mathfrak{p}\in\mathcal{O}^{m}$ and since
$\mathcal{O}^{m,\eta}\cap(\mathfrak{p}+\mathbb{R}\mathfrak{e})=\\{\mathfrak{p}+\frac{m^{2}-\,\mathfrak{p}\cdot\mathfrak{p}}{2(p_{0}-p_{3})}\mathfrak{e}\\}$
is a one-point set (and not empty) just for every
$\mathfrak{p}\in\mathbb{R}^{4}$ with $\operatorname{sgn}(p_{0}-p_{3})=\eta$.
Moreover,
$k^{-1}(\mathfrak{p}+\mathbb{R}\mathfrak{e})=\mathfrak{p}+\frac{m^{2}-\,\mathfrak{p}\cdot\mathfrak{p}}{2(p_{0}-p_{3})}\mathfrak{e}$
for every $\mathfrak{p}\not\in\chi$.
One claims that
$\kappa:L^{2}(\mathbb{R}^{4}/\mathbb{R}\mathfrak{e},H_{\textsc{u}})\to
L^{2}(\mathcal{O}^{m},H_{\textsc{u}})$,
$\kappa(\varphi):=\sqrt{2|p_{0}-p_{3}|}\varphi\circ k$ is an isomorphism.
Indeed, note first $\kappa^{-1}(\phi)=\frac{1}{\sqrt{2|p_{0}-p_{3}|}}\phi\circ
k^{-1}$. Then, on the one hand
$\parallel\\!\\!\varphi\\!\\!\parallel^{2}=\int_{\mathcal{K}}\parallel\\!\\!\varphi(\cdot)\\!\\!\parallel^{2}\operatorname{d}l(\lambda)=\int_{\mathbb{R}^{3}\setminus\\{p_{3}=0\\}}\parallel\\!\\!\varphi\circ
l(\cdot)\\!\\!\parallel^{2}\operatorname{d}\lambda$. On the other hand
$\parallel\\!\\!\kappa(\varphi)\\!\\!\parallel^{2}=\sum_{\eta}\int_{\mathbb{R}^{3}}2(\epsilon(p)-\eta
p_{3})$
$\parallel\\!\\!\varphi\big{(}(\eta\epsilon(p),p)+\mathbb{R}\mathfrak{e}\big{)}\\!\\!\parallel^{2}\frac{\operatorname{d}^{3}p}{2\epsilon(p)}=\sum_{\eta}\int_{\mathbb{R}^{3}}\frac{\epsilon(p)-\eta
p_{3}}{\epsilon(p)}\parallel\\!\\!\varphi\circ
l(T^{\eta}(p)\\!\\!\parallel^{2}\operatorname{d}^{3}p$ with the coordinate
transformation $T^{\eta}(p):=p-\eta\epsilon(p)e_{3}$. Note
$T^{+}(\mathbb{R}^{3})=\mathbb{R}\times\mathbb{R}\times]-\infty,0[$,
$T^{-}(\mathbb{R}^{3})=\mathbb{R}\times\mathbb{R}\times]0,\infty[$, and the
Jacobian determinant equals $1-\eta\frac{p_{3}}{\epsilon(p)}$. Hence by the
substitution rule one has
$\parallel\\!\\!\kappa(\varphi)\\!\\!\parallel^{2}=\int_{\mathbb{R}\times\mathbb{R}\times]-\infty,0[}\parallel\\!\\!\varphi\circ
l(p^{\prime})\\!\\!\parallel^{2}\operatorname{d}^{3}p^{\prime}+\int_{\mathbb{R}\times\mathbb{R}\times]-0,\infty[}\parallel\\!\\!\varphi\circ
l(p^{\prime})\\!\\!\parallel^{2}\operatorname{d}^{3}p^{\prime}$, whence the
claim.
Finally, one easily verifies $V^{shell}=\kappa\hat{V}\kappa^{-1}$ as indicated
in (a). Also the formula regarding
$P^{shell}(M)=\kappa\iota\,P^{can}(M)(\kappa\iota)^{-1}$ in (b) follows by a
similar computation.

Cone representation. We will need also the cone representation
$(V^{cone},P^{cone})$ acting on $L^{2}(\mathcal{O}^{0},H_{\textsc{u}})$
equivalent to $(V^{can},P^{can})$ . Recall the light half-cone
$\mathcal{O}^{0,\eta}$ for $\eta=\pm$ and the light cone $\mathcal{O}^{0}$
endowed with the Lorentz invariant measure. The above formulae (a) and (b) for
the shell representation hold literally for the equivalent cone representation
simply putting $m=0$. In particular, $\epsilon(p)$ becomes $|p|$.
One passes to the cone replacing $k$ by
$k_{0}:\mathcal{O}^{0}\setminus\mathbb{R}\mathfrak{e}\to\mathcal{K}$,
$k_{0}(\mathfrak{p}):=\mathfrak{p}+\mathbb{R}\mathfrak{e}$. Note that
$k_{0}|_{\mathcal{O}^{0,\eta}}\to\mathcal{K}^{\eta}$ is a bijection since
$\operatorname{sgn}(p_{0})=\operatorname{sgn}(p_{0}-p_{3})$ for every
$\mathfrak{p}\in\mathcal{O}^{0}\setminus\mathbb{R}\mathfrak{e}$ and since
$(\mathcal{O}^{0,\eta}\setminus\mathbb{R}\mathfrak{e})\cap(\mathfrak{p}+\mathbb{R}\mathfrak{e})=\\{\mathfrak{p}-\frac{\mathfrak{p}\cdot\mathfrak{p}}{2(p_{0}-p_{3})}\mathfrak{e}\\}$
is a one-point set (and not empty) just for every
$\mathfrak{p}\in\mathbb{R}^{4}$ with $\operatorname{sgn}(p_{0}-p_{3})=\eta$.
Moreover,
$k_{0}^{-1}(\mathfrak{p}+\mathbb{R}\mathfrak{e})=\mathfrak{p}-\frac{\mathfrak{p}\cdot\mathfrak{p}}{2(p_{0}-p_{3})}\mathfrak{e}$
for every $\mathfrak{p}\not\in\chi$, and note that $\mathbb{R}\mathfrak{e}$ in
a null set of $\mathcal{O}^{0}$. Then like in the shell case one shows that
$\kappa_{0}:L^{2}(\mathbb{R}^{4}/\mathbb{R}\mathfrak{e},H_{\textsc{u}})\to
L^{2}(\mathcal{O}^{m},H_{\textsc{u}})$,
$\kappa_{0}(\varphi):=\sqrt{2|p_{0}-p_{3}|}\varphi\circ k_{0}$ is an
isomorphism with
$\kappa_{0}^{-1}(\phi)=\frac{1}{\sqrt{2|p_{0}-p_{3}|}}\phi\circ k_{0}^{-1}$.
Finally, one verifies $V^{cone}=\kappa_{0}\hat{V}\kappa_{0}^{-1}$ and
$P^{shell}(M)=\kappa_{0}\iota\,P^{can}(M)(\kappa_{0}\iota)^{-1}$ as claimed.

###  24.4. Energy and shell representation of the Dirac system and of
$W^{D}|_{\tilde{\mathcal{P}}_{\chi}}$

The energy representation $(W^{\textsc{d}\,erg},E^{\textsc{d}\,erg})$ of the
Dirac system is related to the momentum representation Eq. (13.1) by the
unitary self-adjoint transformation $Y(=Y^{-1})$ on
$L^{2}(\mathbb{R}^{2},\mathbb{C}^{4})$ given by

$(Y\varphi)(p)=Y(p)\varphi(p),\;Y(p):=\left(\frac{m}{2\epsilon(p)}\right)^{1/2}\left(\begin{array}[]{cc}Q(\mathfrak{p}^{+})&Q(\mathfrak{p}^{+})^{-1}\\\
Q(\mathfrak{p}^{+})^{-1}&-Q(\mathfrak{p}^{+})\end{array}\right)$

where $\mathfrak{p}^{+}=(\epsilon(p),p)$ and $Q(\mathfrak{p}^{+})$ denotes the
canonical cross section given in (3.3). One has
$(W^{\textsc{d}\,erg},E^{\textsc{d}\,erg}):=Y^{-1}(W^{\textsc{d}\,mom},E^{\textsc{d}\,mom})Y$
and

$W^{\textsc{d}\,erg}=\oplus_{\eta}W^{\textsc{d}\,\eta},\;\big{(}W^{\textsc{d}\,\eta}(t,b,A)\varphi_{\eta}\big{)}(p)=\sqrt{\epsilon(q^{\eta})/\epsilon(p)}\,\operatorname{e}^{\operatorname{i}(\eta
t\epsilon(p)-bp)}R(\mathfrak{p}^{\eta},A)\varphi_{\eta}(q^{\eta})$

where $W^{\textsc{d}\,\eta}$ acts on $L^{2}(\mathbb{R}^{3},\mathbb{C}^{2})$,
cf. Eq. (3.5) for $j=1/2$. The computations use the formulae regarding the
canonical cross section in (3.3) and the identity

$\pi^{\eta}(p)=\frac{1}{2}\big{(}I_{2}+\frac{\eta}{\epsilon(p)}h^{\textsc{d}}(p)\big{)}=\frac{m}{2\epsilon(p)}\left(\begin{array}[]{cc}Q(\mathfrak{p}^{+})^{2\eta}&\eta
I_{2}\\\ \eta I_{2}&Q(\mathfrak{p}^{+})^{-2\eta}\end{array}\right)$

As to the shell representation consider the isomorphism
$\xi:L^{2}(\mathcal{O}^{m},\mathbb{C}^{2})\to
L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$ with
$(\xi\phi)(p):=\frac{1}{\sqrt{2\epsilon(p)}}\big{(}\phi(\mathfrak{p}^{+}),\phi(\mathfrak{p}^{-})\big{)}$
and $(\xi^{-1}\varphi)(\mathfrak{p})=\sqrt{2\epsilon(p)}\varphi_{\eta}(p)$
with $\varphi=(\varphi_{+},\varphi_{-})$ and $\varphi_{\eta}\in
L^{2}(\mathbb{R}^{3},\mathbb{C}^{2})$. Then
$(W^{\textsc{d}\,shell},E^{\textsc{d}\,shell}):=\xi^{-1}(W^{\textsc{d}\,erg},E^{\textsc{d}\,erg})\xi$
acts on $L^{2}(\mathcal{O}^{m},\mathbb{C}^{2})$ and

$\big{(}W^{\textsc{d}\,shell}(\mathfrak{a},A)\phi\big{)}(\mathfrak{p})=\operatorname{e}^{\operatorname{i}\mathfrak{a}\cdot\mathfrak{p}}R(\mathfrak{p},A)\phi(A^{-1}\cdot\mathfrak{p})$

A last unitary transformation is performed in order to cast
$W^{\textsc{d}}|_{\tilde{\mathcal{P}}_{\chi}}$ into the form $V^{shell}$ in
sec. 24.3, namely
$(W^{{}^{\prime}\textsc{d}\,shell},E^{{}^{\prime}\textsc{d}\,shell}):=Z^{-1}(W^{\textsc{d}\,shell},E^{\textsc{d}\,shell})Z$
with $Z:L^{2}(\mathcal{O}^{m},\mathbb{C}^{2})\to
L^{2}(\mathcal{O}^{m},\mathbb{C}^{2})$

$(Z\phi)(\mathfrak{p})=Z(\mathfrak{p})\phi(\mathfrak{p}),\;Z(\mathfrak{p}):=(m/|p_{0}-p_{3}|)^{1/2}\left(Q(\mathfrak{p})\Tiny{\big{(}\begin{array}[]{cc}0&0\\\
0&1\end{array}\big{)}}+\eta\,Q(\mathfrak{p})^{-1}\Tiny{\big{(}\begin{array}[]{cc}1&0\\\
0&0\end{array}\big{)}}\right)$

Check $I_{2}=Z(\mathfrak{p})Z(\mathfrak{p})^{*}$ using
$(|p_{0}-p_{3}|/m)\,I_{2}=Q(\mathfrak{p})^{2}\Tiny{\big{(}\begin{array}[]{cc}0&0\\\
0&1\end{array}\big{)}}+\,\Tiny{\big{(}\begin{array}[]{cc}1&0\\\
0&0\end{array}\big{)}}Q(\mathfrak{p})^{-2}$. Finally one finds

$(W^{{}^{\prime}\textsc{d}\,shell}(\mathfrak{a},A)\phi)(\mathfrak{p})=\operatorname{e}^{\operatorname{i}\mathfrak{a}\cdot\mathfrak{p}}U(A)\phi(A^{-1}\cdot\mathfrak{p})$

for $(\mathfrak{a},A)\in\tilde{\mathcal{P}}_{\chi}$ and
$U(A):=\operatorname{diag}\big{(}\frac{A_{11}}{|A_{11}|},\frac{\overline{A}_{11}}{|A_{11}|}\big{)}$,
hence
$W^{{}^{\prime}\textsc{d}\,shell}|_{\tilde{\mathcal{P}}_{\chi}}=V^{shell}$,
see sec. 24.3 (a).

###  24.5. Energy and cone representation of the Weyl systems and of
$W^{\textsc{w}\chi}|_{\tilde{\mathcal{P}}_{\chi}}$

The energy representation $(W^{\textsc{w}\chi\,erg},E^{\textsc{d}\,erg})$ of
the Weyl system is defined as

$(W^{\textsc{w}\chi\,erg},E^{\textsc{w}\,erg}):=(Y^{\chi})^{-1}(W^{\textsc{w}\chi\,mom},E^{\textsc{w}\,mom})Y^{\chi}$

where the momentum representation is given in Eq. (16.1) and the unitary
transformation $Y^{\chi}$ on $L^{2}(\mathbb{R}^{2},\mathbb{C}^{2})$ reads

$(Y^{+}\varphi)(p)=B(p)\Tiny{\big{(}\begin{array}[]{cc}1&0\\\
0&b(p)\end{array}\big{)}}\varphi(p),\,(Y^{-}\varphi)(p)=B(p)\Tiny{\big{(}\begin{array}[]{cc}0&1\\\
1&0\end{array}\big{)}}\Tiny{\big{(}\begin{array}[]{cc}1&0\\\
0&\overline{b(p)}\end{array}\big{)}}\varphi(p)$

with $B(p)$ from Eq. (21.8). Cf. (23.3) and (23.12). Note that $Y^{\chi}$
relatively to $Y_{0}^{-1}$ has an additional unitary transformation yielding
$W^{\textsc{w}\chi\,erg}$ particularly clear. Explicitly one finds for
$\varphi=(\varphi_{+},\varphi_{-})\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{2})$

$\displaystyle\big{(}W^{\textsc{w}\chi\,erg}(t,b,A)\Tiny{\big{(}\begin{array}[]{c}\varphi_{+}\\\
\varphi_{-}\end{array}\big{)}}\big{)}(p)$ $\displaystyle=$
$\displaystyle\operatorname{e}^{-\operatorname{i}bp}\left(\begin{array}[]{c}(|q^{+}|/|p|)^{1/2}\;\operatorname{e}^{\operatorname{i}t|p|}\;k(\mathfrak{p}^{+},A)^{\chi}\;\varphi_{+}(q^{+})\\\
(|q^{-}|/|p|)^{1/2}\operatorname{e}^{-\operatorname{i}t|p|}k(\mathfrak{p}^{-},A)^{\chi}\;\varphi_{-}(q^{-})\end{array}\right)$

For the computations use Eq. (21.9), the fact that $B(\eta p)^{-1}AB(\eta
q)=A_{\ln|p|}H(\mathfrak{p},A)A_{\ln|q|}^{-1}\in E(2)$ by definition of
$H(\mathfrak{p},A)$, and $B(-p)=C(p)^{-1}B(p)$,
$C(p)\operatorname{diag}(\alpha,\beta)=\operatorname{diag}(\beta,\alpha)C(p)$
for $C(p)=-C(p)^{-1}$ in Eq. (23.2)(2).
One passes to the cone representation by the isomorphism
$\xi_{0}:L^{2}(\mathcal{O}^{0})\to L^{2}(\mathbb{R}^{3},\mathbb{C}^{2})$,
$(\xi_{0}\phi)(p):=\frac{1}{\sqrt{2|p|}}\big{(}\phi(\mathfrak{p^{+}}),\phi(\mathfrak{p^{-}})\big{)}$,
$(\xi_{0}^{-1}\varphi)(\mathfrak{p})=\sqrt{2|p|}\,\varphi_{\eta}(p)$. For
$(W^{\textsc{w}\chi\,cone},E^{\textsc{w}\,cone}):=\xi^{-1}_{0}(W^{\textsc{w}\chi\,erg},E^{\textsc{w}\,erg})\xi_{0}$
one finds

$\big{(}W^{\textsc{w}\chi\,cone}(\mathfrak{a},A)\phi\big{)}(\mathfrak{p})=\operatorname{e}^{\operatorname{i}\mathfrak{p}\cdot\mathfrak{a}}k(\mathfrak{p},A)^{\chi}\phi(A^{-1}\cdot\mathfrak{p})$

(cf. Eq. (23.2)).
A last unitary transformation is performed in order to cast
$W^{\textsc{w}\chi}|_{\tilde{\mathcal{P}}_{\chi}}$ into the form $V^{cone}$ in
sec. 24.3, namely
$(W^{{}^{\prime}\textsc{w}\chi\,cone},E^{{}^{\prime}\textsc{w}\,cone}):=(Z^{\chi})^{-1}(W^{\textsc{w}\chi\,cone},E^{\textsc{w}\,cone})Z^{\chi}$
with $Z^{\chi}:L^{2}(\mathcal{O}^{0})\to L^{2}(\mathcal{O}^{0})$,
$(Z^{\chi}\phi)(\mathfrak{p}):=\overline{b(\mathfrak{p})}^{\chi}\phi(\mathfrak{p})$
with $b(\mathfrak{p}):=b(p)$. Then
$\big{(}W^{{}^{\prime}\textsc{w}\chi\,cone}(\mathfrak{a},A)\phi\big{)}(\mathfrak{p})=\operatorname{e}^{\operatorname{i}\mathfrak{p}\cdot\mathfrak{a}}k^{\prime}(\mathfrak{p},A)^{\chi}\phi(A^{-1}\cdot\mathfrak{p})$
with
$k^{\prime}(\mathfrak{p},A):=b(\mathfrak{p})k(\mathfrak{p},A)\overline{b(A^{-1}\cdot\mathfrak{p})}$.
Now, by the remarkable property
$k^{\prime}(\mathfrak{p},A)=\frac{\overline{A}_{11}}{A_{11}}$ for all
$\mathfrak{p}\in\mathcal{O}^{0}$, $A\in ST(2)$ shown below one has

$W^{{}^{\prime}\textsc{w}\chi\,cone}|_{\tilde{\mathcal{P}}_{\chi}}=V^{cone}$

for
$U(A):=\operatorname{diag}\big{(}\frac{\overline{A}_{11}}{|A_{11}|},\frac{A_{11}}{|A_{11}|}\big{)}^{\chi}$.
It remains to show $k^{\prime}(\mathfrak{p},A)=\overline{A}_{11}/A_{11}$. Let
$A=CA_{\rho}A^{\prime}$ with
$C=\operatorname{diag}(\gamma,\overline{\gamma})$, $|\gamma|=1$ and
$A^{\prime}_{11}=1$ according to (20). Then
$k(\mathfrak{p},A)=k(\mathfrak{p},CA_{\rho})k(\mathfrak{q},A^{\prime})$ for
$\mathfrak{q}=(CA_{\rho})^{-1}\cdot\mathfrak{p}$. By Eq. (23.3),
$k(\mathfrak{p},CA_{\rho})=\gamma$. Explicit computation using the formula for
$\Lambda(A^{\prime})$ in (20) shows
$k(\mathfrak{q},A^{\prime})=b(q^{\prime})\overline{b(q)}$ for
$\mathfrak{q}^{\prime}:=A^{{}^{\prime}-1}\cdot\mathfrak{q}$. Now the assertion
follows, since $\mathfrak{q}^{\prime}:=A^{-1}\cdot\mathfrak{p}$ and since
$q=C^{-1}\cdot p$, whence $b(q)=\gamma^{2}b(p)$ by a short computation.

###  24.6. Extension of a positive-operator-valued content to a POM

The main result is (24.6). By physical reasons we are interested in the
corollary (24.6).

(156) Theorem. Let $\mathcal{Q}$ be an algebra of subsets of a set $X$,
$\hat{\mathcal{Q}}$ the $\sigma$-algebra generated by $\mathcal{Q}$, and $G$ a
positive-operator-valued content on $\mathcal{Q}$. Let $m$ be a normalized
content on $\mathcal{Q}$ controlling $G$, i.e.,

$m(M_{i})\to 0\quad\Leftrightarrow\quad G(M_{i})\to 0$

as strong limit for every sequence $(M_{i})_{i}$ in $\mathcal{Q}$. Then there
is a POM $\hat{G}$ on $\hat{\mathcal{Q}}$ extending $G$ if and only if $m$ is
$\sigma$-additive. The extension $\hat{G}$ is unique. Finally, if $G$ is
projection-valued then $\hat{G}$ is a PM.
Proof. Let $\hat{G}$ be a POM extending $G$. By $\sigma$-additivity of
$\hat{G}$ one has $\lim_{i}G(M_{i})=\lim_{i}\hat{G}(M_{i})=0$ for $(M_{i})$ in
$\mathcal{Q}$ with $M_{i}\downarrow\emptyset$. Hence $\lim_{i}m(M_{i})=0$.
Therefore $m$ is $\sigma$-additive (see e.g. [79, 3.2]).
Conversely, let $m$ be $\sigma$-additive. There exists a unique extension
$\hat{m}$ of $m$ on $\hat{\mathcal{Q}}$ satisfying
$\hat{m}(\Delta)=\inf\big{\\{}\sum_{j}m(K_{j}):K_{j}\in\mathcal{Q},\Delta\subset\bigcup_{j}K_{j}\big{\\}}$
$\forall\;\Delta\in\hat{\mathcal{Q}}$ (see e.g. [79, 5.2, 5.7]). Put

$\mathcal{C}(\Delta):=\big{\\{}(M_{ij}):M_{ij}\in\mathcal{Q},\;M_{ij}\downarrow_{i},\;M_{ij}\uparrow_{j}\cup_{j}M_{ij}\supset\Delta,\;\hat{m}(\Delta)=\hat{m}(\cap_{i}\cup_{j}M_{ij})\big{\\}}$
(1)

It follows $\mathcal{C}(\Delta)\neq\emptyset$. Indeed, note first that there
are $K_{ij}\in\mathcal{Q}$ with $\Delta\subset\cup_{j}K_{ij}$ and
$\hat{m}(\Delta)=\inf_{i}\big{(}\sum_{j}m(K_{ij})\big{)}$. Then
$\hat{m}(\Delta)\leq\hat{m}(\cup_{j}K_{ij})\leq\sum_{j}m(K_{ij})$, whence
$\hat{m}(\Delta)=\inf_{i}\hat{m}(\cup_{j}K_{ij})$. Put
$L_{ij}:=\cup_{l=1}^{j}K_{il}$ and finally let $M_{ij}:=\cap_{l=1}^{i}L_{lj}$.
Now verify
$\cup_{j}M_{ij}=\cap_{l=1}^{i}\cup_{j}L_{lj}=\cap_{l=1}^{i}\cup_{j}K_{lj}$.
Then it is easy to show $(M_{ij})\in\mathcal{C}(\Delta)$. — From (1) it
follows immediately

$\forall\,(M_{ij})\in\mathcal{C}(\Delta),\;i,n\in\mathbb{N}\;\;\exists\;i_{n},\,j_{n,i}\in\mathbb{N}:\;\;\;\hat{m}(\Delta\vartriangle
M_{ij})\leq\textnormal{\tiny{$\frac{1}{n}$}}\;\;\;\forall\;i\geq i_{n},\,j\geq
j_{n,i}$ (2)

Put

$\hat{G}(\Delta):=\lim_{i}\lim_{j}G(M_{ij})\quad\forall\Delta\in\hat{\mathcal{Q}},\;(M_{ij})\in\mathcal{C}(\Delta)$
(3)

According to [80, 4.28 (b)] the strong limits exist and yield bounded
operators. Since $0\leq G(M_{ij})\leq I$ also $0\leq\hat{G}(\Delta)\leq I$
holds. If $G(M_{ij})$ are projections then $\hat{G}(\Delta)$ is a projection
by [80, 4.32].
(a) First one shows that $\hat{G}(\Delta)$ in (3) does not depend on the
choice of $(M_{ij})\in\mathcal{C}(\Delta)$. Let
$(M^{l}_{ij})\in\mathcal{C}(\Delta)$, $l=1,2$. Then
$(M^{0}_{ij})\in\mathcal{C}(\Delta)$ for $M^{0}_{ij}:=M^{1}_{ij}\cap
M^{2}_{ij}$. Let $\hat{G}^{l}(\Delta):=\lim_{i}\lim_{j}G(M^{l}_{ij})$,
$l=0,1,2$. Obviously $\hat{G}^{0}(\Delta)\leq\hat{G}^{1}(\Delta)$. Assume
$\hat{G}^{0}(\Delta)<\hat{G}^{1}(\Delta)$. Then there exists a vector
$\varphi$ with
$\langle\varphi,\hat{G}^{1}(\Delta)\varphi\rangle-\langle\varphi,\hat{G}^{0}(\Delta)\varphi\rangle=:2c>0$.
From (3) for $\hat{G}^{l}(\Delta)$, $l=0,1$ one infers that there exists
$i_{0}\in\mathbb{N}$ and for every $i\in\mathbb{N}$ a $j_{i}\in\mathbb{N}$
such that $\langle\varphi,G(M^{1}_{ij}\setminus M^{0}_{ij})\varphi\rangle\geq
c$ for all $i\geq i_{0}$ and $j\geq j_{i}$. This and (2) applied to
$(M^{l}_{ij})$, $l=0,1$ imply that for every $n\in\mathbb{N}$ there are
$i_{n},j_{n}\in\mathbb{N}$ such that $L^{l}_{n}:=M^{l}_{i_{n}j_{n}}$ satisfies
$\langle\varphi,G(L^{1}_{n}\setminus L^{0}_{n})\varphi\rangle\geq c$ and
$m(L^{1}_{n}\setminus L^{0}_{n})=\hat{m}(\Delta\vartriangle
L^{1}_{n}\vartriangle\Delta\vartriangle
L^{0}_{n})\leq\hat{m}(\Delta\vartriangle L^{1}_{n})+\hat{m}(\Delta\vartriangle
L^{0}_{n})\leq\frac{2}{n}$242424 The symmetric difference $\vartriangle$ is a
commutative and associative set-theoretic operation. contradicting the
assumption that $m$ controls $G$. Similarly one shows
$\hat{G}^{0}(\Delta)=\hat{G}^{2}(\Delta)$.
(b) Obviously (a) implies $\hat{G}|_{\mathcal{Q}}=G$. Hence, in particular,
$\hat{G}(\emptyset)=0$, $\hat{G}(X)=I$.
(c) Now additivity of $\hat{G}$ is shown. Let $\Delta^{l}\in\hat{\mathcal{Q}}$
and $(M^{l}_{ij})\in\mathcal{C}(\Delta^{l})$ for $l=1,2$. Put
$M^{l}:=\cap_{i}\cup_{j}M^{l}_{ij}$. Further put $L_{ij}:=M^{1}_{ij}\cup
M^{2}_{ij}$ and $K_{ij}:=M^{1}_{ij}\cap M^{2}_{ij}$. Keep in mind
$M_{ij}\downarrow_{i}$, $M_{ij}\uparrow_{j}$ and equally for $(L_{ij})$ and
$(K_{ij})$.
Note $\Delta^{1}\cup\Delta^{2}\subset\cap_{i}\cup_{j}L_{ij}=M^{1}\cup M^{2}$.
As to the less trivial inclusion $\subset$ of the equality let $x\not\in
M^{1}\cup M^{2}$. Then for $l=1,2$ there exists $i_{l}$ with
$x\not\in\cup_{j}M^{l}_{i_{l}j}$. Hence
$x\not\in(\cup_{j}M^{l}_{ij})\cup(\cup_{j}M^{l}_{ij})=\cup_{j}L_{ij}$ for
$i\geq i_{1},i_{2}$, whence $x\not\in\cap_{i}\cup_{j}L_{ij}$.
Note $\Delta^{1}\cap\Delta^{2}\subset\cap_{i}\cup_{j}K_{ij}=M^{1}\cap M^{2}$.
Indeed, clearly $\Delta^{1}\cap\Delta^{2}\subset M^{1}\cap M^{2}$. If $x\in
M^{1}\cap M^{2}$, then, for every $i$,
$x\in(\cup_{j}M^{1}_{ij})\cap(\cup_{j}M^{2}_{ij})$, whence $x\in
M^{1}_{ij_{1}}\cap M^{2}_{ij_{2}}$ for some $j_{1},j_{2}$ and hence $x\in
K_{ij}$ for $j\geq j_{1},j_{2}$. Therefore $x\in\cap_{i}\cup_{j}K_{ij}$. —
Conversely, if $x\not\in M^{1}\cap M^{2}$, then $x\not\in M^{l}$ for some $l$.
Hence $x\not\in\cap_{i}\cup_{j}K_{ij}$.
Now, by these inclusions,
$\hat{m}(\Delta^{1})+\hat{m}(\Delta^{2})=\hat{m}(\Delta^{1}\cup\Delta^{2})+\hat{m}(\Delta^{1}\cap\Delta^{2})\leq\hat{m}(M^{1}\cup
M^{2})+\hat{m}(M^{1}\cap
M^{2})=\hat{m}(M^{1})+\hat{m}(M^{2})=\hat{m}(\Delta^{1})+\hat{m}(\Delta^{2})$.
It follows $\hat{m}(\Delta^{1}\cup\Delta^{2})=\hat{m}(M^{1}\cup M^{2})$ and
$\hat{m}(\Delta^{1}\cap\Delta^{2})=\hat{m}(M^{1}\cap M^{2})$. This shows that
$(L_{ij})\in\mathcal{C}(\Delta^{1}\cup\Delta^{2})$ and
$(K_{ij})\in\mathcal{C}(\Delta^{1}\cap\Delta^{2})$.
By the last result one may perform the limits in (3) for each summand of
$G(L_{ij})+G(K_{ij})=G(M^{1}_{ij})+G(M^{2}_{ij})$ thus obtaining the
additivity of $\hat{G}$.
(d) The claim is $\lim_{l\to\infty}\hat{G}(\Delta^{l})=0$ for $(\Delta^{l})$
in $\hat{\mathcal{Q}}$ with $\Delta^{l}\downarrow_{l}\emptyset$, which
together with additivity is equivalent to $\sigma$-additivity of $\hat{G}$.
Assume $\lim_{l\to\infty}\hat{G}(\Delta^{l})>0$. Then there exists a vector
$\varphi$ with $\langle\varphi,\hat{G}(\Delta^{l})\varphi\rangle=:2c>0$ for
all $l\in\mathbb{N}$. Let $(M^{l}_{ij})\in\mathcal{C}(\Delta^{l})$. Then,
according to (3), for $l$ there exists $i_{l}$ and for every $i$ a $j_{l,i}$
such that $\langle\varphi,G(M^{l}_{ij})\varphi\rangle\geq c$ for all $i\geq
i_{l}$ and $j\geq j_{l,i}$. This and (2) applied to $(M^{l}_{ij})$ imply that
for every $n\in\mathbb{N}$ there are $l_{n},i_{n},j_{n}\in\mathbb{N}$ such
that $\hat{m}(\Delta^{l_{n}})\leq\frac{1}{n}$ and such that
$L_{n}:=M^{l_{n}}_{i_{n}j_{n}}$ satisfies
$\langle\varphi,G(L_{n})\varphi\rangle\geq c$ and
$m(L_{n})=\hat{m}(\Delta^{l_{n}}\vartriangle
L_{n}\vartriangle\Delta^{l_{n}})\leq\hat{m}(\Delta^{l_{n}}\vartriangle
L_{n})+\hat{m}(\Delta^{l_{n}})\leq\frac{2}{n}$. This contradicts the
assumption that $m$ controls $G$.
(e) It remains to show uniqueness of the extension $\hat{G}$. For every vector
$\varphi$, $M\mapsto\langle\varphi,G(M)\varphi\rangle$ is a finite
$\sigma$-additive content on $\mathcal{Q}$. It allows a unique extension to a
measure on $\hat{\mathcal{Q}}$ (see e.g. [79, 5.7]). Therefore this extension
coincides with $\Delta\mapsto\langle\varphi,\hat{G}(\Delta)\varphi\rangle$.
This ends the proof.∎

A normalized content controlling $G$ is easily available, namely

(157) Lemma. Let the Hilbert space be separable and
$\\{\varphi_{n}:n\in\mathbb{N}\\}$ a total set of unit vectors. Then
$m(M):=\sum_{n}2^{-n}\langle\varphi_{n},G(M)\varphi_{n}\rangle$ defines a
normalized content $m$ controlling $G$ in _( 24.6)_.
Proof. $m(M_{i})\to 0\quad\Leftrightarrow\quad G(M_{i})\varphi_{n}\to 0$ for
every $n$ by dominated convergence, since generally
$\langle\varphi,G(M)\varphi\rangle=\parallel\\!\\!\sqrt{G(M)}\varphi\\!\\!\parallel^{2}$
and
$\parallel\\!\\!G(M)\varphi\\!\\!\parallel\leq\parallel\\!\\!\sqrt{G(M)}\varphi\\!\\!\parallel$
using $\parallel\\!\\!G(M)\\!\\!\parallel\leq 1$. Hence $G(M_{i})\varphi\to 0$
for a dense set of vectors $\varphi$, whence $G(M_{i})\to 0$.∎

Recall that the unions of finitely many disjoint boxes $\langle
a,b[:=\\{x\in\mathbb{R}^{d}:a_{k}\leq x_{k}<b_{k},k=1,\dots,d\\}$,
$a,b\in\overline{\mathbb{R}}^{d}$ form an algebra $\mathcal{Q}^{d}$ in
$\mathbb{R}^{d}$, which generates the Borel $\sigma$-algebra of
$\mathbb{R}^{d}$ (cf. the proof of (20)).

(158) Lemma. Let $m$ be a finite content on
$(\mathbb{R}^{d},\mathcal{Q}^{d})$, which is translation quasi-invariant,
i.e., for every $b\in\mathbb{R}^{d}$ and every sequence $(M_{i})_{i}$ in
$\mathcal{Q}^{d}$

$m(M_{i})\to 0\quad\Leftrightarrow\quad m(b+M_{i})\to 0$

Moreover, for every box $B$ and every $\epsilon>0$ there is a bounded box
$B^{\prime}\subset B$ such that $m(B\setminus B^{\prime})<\epsilon$. Then $m$
is $\sigma$-additive.
Proof. (a) Let $B:=\langle a,b[\,\in\mathcal{Q}^{d}$ be a non-empty box with
$a,b\in\mathbb{R}^{d}$. Let $\epsilon>0$. The claim is that there is
$b^{\prime}\in\mathbb{R}^{d}$ with $a_{k}<b^{\prime}_{k}<b_{k}$, $k=1,\dots,d$
such that for $B^{\prime}:=\langle a,b^{\prime}[$

$m(B)\leq m(B^{\prime})+\epsilon$

Indeed, let $c\in\mathbb{R}^{d}$ with $c_{k}<b_{k}$, $k=1,\dots,d$ and put
$\Delta(c,b):=\langle a,b[\,\setminus\langle a,c[$. Obviously, the result
follows if $\inf_{c}m(\Delta(c,b))=0$. Assume the contrary and a fortiori
$a=(-\infty,\dots,-\infty)$. Then by translation quasi-invariance
$\inf_{c}m(d+\Delta(c,b))>0$ for every $d\in\mathbb{R}^{d}$. Choose $d:=tf$,
$t\in\mathbb{R}$, $f:=(1,1,\dots,1)$. Then $d+\Delta(c,b)=\Delta(c+tf,b+tf)$.
Since $]0,1[$ is uncountable, there are $\delta>0$ and a strictly monotonic
sequence $(t_{n})$ in $]0,1[$ such that
$\inf_{c}m(\Delta(c+t_{n}f,b+t_{n}f))\geq\delta$ for all $n$. If $(t_{n})$ is
increasing then
$\Delta\big{(}b+\frac{1}{2}(t_{n+1}+t_{n})f,b+t_{n+1}f\big{)}$,
$n\in\mathbb{N}$ are mutually disjoint and contained in $\langle a,b+f[$. Each
set has content $\geq\delta$. Hence additivity of $m$ implies the
contradiction $m(\langle a,b+f[\,)=\infty$. If $(t_{n})$ is decreasing then
consider $\big{(}\Delta(b+\frac{1}{2}(t_{n+1}+t_{n})f,b+t_{n}f)\big{)}_{n}$.
(b) Due to (a) and by the assumption on $m$ it follows that generally for
every non-empty box $B=\langle a,b\,[$ and every $\epsilon>0$ there are
$a^{\prime},b^{\prime}\in\mathbb{R}^{d}$ with $a_{k}\leq
a^{\prime}_{k}<b^{\prime}_{k}<b_{k}$ $\forall$ $k$ and $m(B)\leq
m(B^{\prime})+\epsilon$. Hence the closure of $B^{\prime}$ is compact and
contained in $B$. Now, by a well-known compactness argument we show for $m$
the continuity from above which is equivalent to $\sigma$-additivity.
(c) Let $(M_{n})_{n}$ be a decreasing sequence in $\mathcal{Q}^{d}$ such that
$\delta:=\lim_{n}m(M_{n})>0$. One has to show $\bigcap_{n}M_{n}\neq\emptyset$.
Recall that the figure $M_{n}$ is the union of finitely many disjoint boxes
$B$. Replace every $B$ by some $B^{\prime}$ as above forming a new figure
$M^{\prime}_{n}\subset M_{n}$ such that

$m(M_{n})-m(M_{n}^{\prime})\leq 2^{-n}\delta$

Let $L_{n}:=M_{1}^{\prime}\cap\dots\cap M_{n}^{\prime}$. Then $(L_{n})_{n}$ is
a decreasing sequence in $\mathcal{Q}^{d}$ with
$\overline{L}_{n}\subset\overline{M}_{n}^{\prime}\subset M_{n}$. In
particular, $\overline{L}_{n}$ is compact. It suffices to show
$\overline{L}_{n}\neq\emptyset$ for every $n$, since then by the finite
intersection property
$\emptyset\neq\bigcap_{n}\overline{L}_{n}\subset\bigcap_{n}M_{n}$.
Assume $\overline{L}_{n}=\emptyset$ and hence $L_{n}=\emptyset$ for some $n$.
Then $\delta\leq m(M_{n})=m(M_{n}\setminus
L_{n})=m\big{(}\bigcup_{k=1}^{n}(M_{n}\setminus
M_{k}^{\prime})\big{)}\leq\sum_{k=1}^{n}m(M_{n}\setminus
M_{k}^{\prime})\leq\sum_{k=1}^{n}m(M_{k}\setminus
M_{k}^{\prime})\leq\sum_{k=1}^{n}2^{-k}\delta$, which is a contradiction.
Hence the result follows.∎

(159) Corollary. Let the Hilbert space be separable. Let $G$ be a positive-
operator-valued content on $(\mathbb{R}^{d},\mathcal{Q}^{d})$, Let $U(b)$ be
unitary with $U(b)G(M)U(b)^{-1}=G(b+M)$ for all $M\in\mathcal{Q}^{d}$,
$b\in\mathbb{R}^{d}$. Moreover, let for every box $B$, every vector $\varphi$,
and every $\epsilon>0$ exist a bounded box $B^{\prime}\subset B$ with
$\langle\varphi,G(B\setminus B^{\prime})\varphi\rangle\leq\epsilon$.
Then there is a unique POM $T$ on the Borel sets of $\mathbb{R}^{d}$ with
$G=T|_{\mathcal{Q}^{d}}$. If $G$ is projection-valued, then $T$ is a PM. If
$S$ is a Borel-algebra homomorphism with
$S(\mathcal{Q}^{d})\subset\mathcal{Q}^{d}$ and $U(S)$ unitary with
$U(S)G(M)U(S)^{-1}=G\big{(}S(M)\big{)}$ for all $M\in\mathcal{Q}^{d}$, then
one has $U(S)T(\Delta)U(S)^{-1}=T\big{(}S(\Delta)\big{)}$ for all Borel sets
$\Delta\subset\mathbb{R}^{d}$.
Proof. Let $m$ be the normalized content on $\mathcal{Q}^{d}$ controlling $G$
from (24.6). Then $m$ is translation quasi-invariant. Indeed, let
$(M_{i})_{i}$ be a sequence in $\mathcal{Q}^{d}$, $b\in\mathbb{R}^{d}$, then
$\lim_{i}m(M_{i})=0\Rightarrow\lim_{i}G(M_{i})=0\Rightarrow\lim_{i}G(b+M_{i})=\lim_{i}U(b)G(M_{i})U(b)^{-1}=0\Rightarrow\lim_{i}m(b+M_{i})=0$.
We like to apply (24.6). Let $\epsilon>0$. Let $n^{\prime}\in\mathbb{N}$ with
$\sum_{n>n^{\prime}}2^{-n}<\frac{\epsilon}{2}$. Let $B$ be a box. By
assumption there is a bounded box $B_{1}\subset B$ with
$\langle\varphi_{1},G(B\setminus
B_{1})\varphi_{1}\rangle\leq\frac{\epsilon}{2}$.Then, by monotony of $G$,
there is a bounded box $B_{1}\subset B_{2}\subset B$ with
$\langle\varphi_{2},G(B\setminus
B_{2})\varphi_{2}\rangle\leq\frac{\epsilon}{2}$, and so on. Hence there is a