# Dirac and Weyl Fermions - the Only Causal Systems

Domenico P. L. Castrigiano∗ ∗Technische Universität München, Fakultät für
Mathematik, München, Germany<EMAIL_ADDRESS>
###### Abstract.

Causal systems describe the localizability of relativistic quantum systems
complying with the principles of special relativity and elementary causality.
At their classification we restrict ourselves to real mass and finite spinor
systems. It follows that (up to certain not yet discarded unitarily related
systems) the only irreducible causal systems are the Dirac and the Weyl
fermions. Their wave-equations are established as a mere consequence of causal
localization. — The compact localized Dirac and Weyl wave-functions are
studied in detail. One finds that, at the speed of light, the carriers shrink
in the past and expand in the future. For every direction in space there is a
definite time at which the change from shrinking to expanding occurs. A late
changing time characterizes those states, which shrink to a $\delta$-strip if
boosted in the opposite direction. Using a density result for these late-
change states one shows that all Dirac and Weyl wave-functions are subjected
to Lorentz contraction. — We tackle the question whether a causal system
induces a representation of a causal logic and thus provides a localization in
proper space-time regions rather than on spacelike hyperplanes. The causal
logic generated by the spacelike relation is shown to do not admit
representations at all. But the logic generated by the non-timelike relation
in general does, and the necessary condition is derived that there is a
projection valued measure on every non-timelike non-spacelike hyperplane being
the high boost limit of the localization on the spacelike hyperplanes. Dirac
and Weyl systems are shown to satisfy this condition and thus to extend to all
non-timelike hyperplanes, which implies more profound properties of the causal
systems. The compact localized eigenstates of the projections to non-spacelike
flat strips are late-change states.

Keywords: Relativistic quantum systems, localization, causality, causal time
evolution, Lorentz contraction, representation of causal logic

###### Contents

  1. 1 Introduction
  2. 2 Newton-Wigner Position Operator and Localization by Wightman
  3. 3 Poincaré Covariance of Position
  4. 4 Positive Energy and Localized States
  5. 5 Three Contrasting Concepts
  6. 6 Positive Operator Valued Localization
  7. 7 Causal Time Evolution and Causal Localization
  8. 8 Point-Localized Sequences of States
  9. 9 Causal Systems
  10. 10 A Property of Causal Systems
  11. 11 The Lattice of Causally Complete Regions
  12. 12 Localization Operators for Causally Complete Regions
  13. 13 The Dirac System
  14. 14 Causal PO-Localization for the Electron
  15. 15 Negative Energy States
  16. 16 The Weyl Systems
  17. 17 No Causal Spin–$0$ Systems
  18. 18 Lattice Generated by the Non-Timelike Relation
  19. 19 A Rep of the Lattice of Borel Sets Generated by the Non-Timelike Relation
  20. 20 Extension of Causal Localizations to Non-Timelike Hyperplanes
  21. 21 Systems with Causal Time Evolution
  22. 22 Massive Causal Systems
  23. 23 Massless Causal Systems
  24. 24 Appendix

##  1\. Introduction

These investigations are concerned with causal systems $(W,E)$ (9), which
describe the localizability of relativistic quantum systems following Wightman
[1]. What is going beyond the physical ideas developed by Newton, Wigner [2]
and Wightman [1] is the inclusion of special relativity and causality. So the
representation $W$ of the Poincaré group $\tilde{\mathcal{P}}$ describing the
relativistic quantum system under consideration and the projection valued
measure (PM) $E$ describing the localization of the system on every spacelike
hyperplane of Minkowski space satisfy Poincaré covariance (i) and the
causality condition (ii), namely

  * (i)

$W(g)E(\Delta)W(g)^{-1}=E(g\cdot\Delta)$

  * (ii)

$E(\Delta)\leq E(\Delta_{\tau})$

Poincaré covariance (i) refers to every Poincaré transformation $g$ and every
spacelike flat region $\Delta$. As to the causality condition (ii), $\tau$ is
any spacelike hyperplane and $\Delta_{\tau}$ denotes the region of influence
of $\Delta$ in $\tau$ (7.2). Equivalent formulations of the causality
condition, as local orthogonality $E(\Delta)E(\Gamma)=0$ for spacelike
separated $\Delta$ and $\Gamma$, are given in (7.2).

$\tau$$\Delta_{\tau}$$\Delta$$t$$x$

Hence the concept of a causal system is the unique one satisfying a number of
minimal physical requirements. Moreover, the postulate (i) implies no
restriction for relativistic systems which are localized according to Wightman
[1], since as shown in [3] and (6.2) the localization automatically and
unambiguously extends to all spacelike hyperplanes in a Poincaré covariant
manner. What is really restrictive is the causality condition (ii). In
particular there is no causal system $(W,E)$ if the energy $H$ is semi-bounded
(4). Since bounded below energy is essential for the stability of the system
one could infer from this fact that localization in terms of position
operators is incompatible with causality in relativistic quantum physics [29].
We hasten to expound already at this point that this conclusion might be not
definitive. Indeed, in sec. 15 as in [4, sec. J], we argue that applying the
localization operator $E(\Delta)$ of a causal system to a particle (or
antiparticle) state does not create a pure state but a mixed state of particle
and antiparticle states, which of course have non-negative energy. We think
this feature of causal systems is fascinating as it puts first the fundamental
interplay in nature of matter and antimatter, as summed up at the creation of
matter from energy, where inevitably matter and antimatter occur in a
bookkeeping manner.
The localization properties of the resulting particle and antiparticle states
are described by a POL (positive operator valued localization), which is the
trace of the causal system on the particle and antiparticle subspace,
respectively. It is causal since it still satisfies Poincaré covariance (i)
and the causality condition (ii). POL are studied in [4, sec. G] and in sec.
6, 7, 8. The results are exploited for the causal electron POL $T^{e}$ in [4,
sec. I] and sec. 14 proving among other things that $T^{e}$ is separated which
implies that arbitrarily well localized electrons in any neighborhood of every
point exist. The analogous results are derived for the POL $T^{\chi\eta}$ of
the four Weyl particles (16).
The existence of a causal POL $T$ describing the localization of a positive-
energy system is not at all a trivial fact. As shown in [4, Lemma 4], the
localization operators $T(\Delta)\neq 0$ for bounded regions $\Delta$ are not
local observables of a local quantum theory, included the case of modular
localization. They might not be observables at all, although their expectation
values give the localization probabilities. We do not pursue these questions
here.

Proceeding in order, the first sections of this article are concerned with the
basic considerations on localizability, mainly in order to recapitulate and
elucidate the well-known difficulties one meets in relativistic quantum
mechanics. Let us mention that the outstanding problems maintain a vivid
interest in a theory of relativistic particle localization so that from the
beginning of relativistic quantum theory up to now a vast literature is
devoted to this subject. For a selection of about fifty articles from 1930 up
to 2000 see [5] 2005\. For further references see [6] 2017\. Also more
recently, in [4] 2015 a thorough investigation of causal time evolution for
massive relativistic quantum systems is presented, to which we will refer
recurrently. Some indicated parts of it are simply taken over for the sake of
completeness.

We start in sec. 2 with a brief presentation of the Wightman localization (WL)
and the particular Newton-Wigner localization (NWL). In sec. 3 Poincaré
covariance of any WL for a relativistic system (and the associated position
operator) is established, i.e., the ingredient (i) for a causal system is
ensured, and thus a widespread objection against WL, in particular NWL, is
removed. But despite of being Poincaré covariant, NWL does not localize frame-
independently. We do not renounce to give a rigorous proof of this property in
(3.3) (which certainly will exist in literature). This is one of the
inadequacies of NWL. Another shortcoming, still compatible with Poincaré
covariance, is the instantaneous spreading to infinity of the wave-functions.
On this it is worked intensively. In essence, causal time evolution and
localized state are contrasted under the premise of semi-bounded energy.
Particularly in the work of Hegerfeldt this is done in a concise manner. This
egged us on to use Occam’s razor for a further elaboration (similar in the
spirit to [28]). The achievement in sec. 4 is an unbiased view at the notion
of localized state being rid of localization operators.
Instantaneous spreading and frame-dependent localization both violate the
causality condition (ii). The former attracts up to now much more attention,
perhaps because it is perceived to be the quintessence of acausality. However,
it is not. Indeed, systems with causal time evolution (SCT) (21), i.e.,
systems satisfying (i) and the causality condition (ii) for time translations,
need not be causal. The massive respectively massless SCT in (22.2) and (23.4)
do not localize frame-independently and hence are not causal. Moreover, not
even causal time evolution and frame-independent localization together imply
causality as the example (23.4) shows.

So far this brief exposition already makes clear that the alternative to
pursue maintaining causality  is either to suppose a semi-bounded Hamiltonian
(stability) or to suppose states localized in bounded regions. However, as
already indicated, these assumptions seemingly opposed to each other are
simultaneously realized and intrinsically linked in causal systems.

One of our main results is the classification of the causal systems $(W,E)$.
We restrict ourselves to systems with non-negative mass squared operator and
finite spinor dimension. This is justified by the fact that representations of
$\tilde{\mathcal{P}}$ not satisfying these spectral conditions have not found
applications so far. (Only more recently it is suggested that the quantum
fields, which are covariant with respect to the massless infinite spin
representations, could be attributed to the galactic dark matter.) Within this
frame the result obtained in sec. 21, 22, 23 is that the Dirac system for
every positive mass and the two massless right- and left-handed Weyl systems
are the only irreducible causal systems.
As to this result, actually we did not yet succeed to rule out certain
obviously unlikely systems. See the details in 22.2, 23.4. At any rate the
Dirac equation and the Weyl equations are established as a mere consequence of
the principle of causality in relativistic quantum mechanics. In particular,
the requirement of a causal localization determines the right handedness of
particle and antiparticle constituting a Weyl system.
The classification of the causal systems proceeds in three steps. The first
one (a) consists in determining the causal time evolutions for WL. This is the
most involved part of the classification. Then (b) one establishes all
possible extensions of the representations in (a) of the little kinematical
group (i.e., the Euclidean transformations plus time translations) to
representations of $\tilde{\mathcal{P}}$. Moreover one applies (6.2) in order
to define in a Poincaré covariant manner the localizations operators for all
spacelike flat regions. Thus one obtains the SCT. They are more general than
causal systems. In the last step (c) one has to select the causal systems
among the SCT. In particular one discards all SCT which do not localize frame-
independently.
For massive systems the steps (a) and (b) are done in [4] yielding a complete
and explicit description of the SCT reported in sec. 22.1. The massless SCT
are also completely classified in an analogous manner (see sec. 23.1 for step
(a) and sec. 23.3 for step (b)), although the group theoretical part is more
involved. Step (c) (see sec. 22.2 and sec. 23.4) is tedious and, as mentioned,
not yet fully accomplished.

There are some causal systems for which the mass squared operator has a
spectrum containing $]-\infty,0]$. They are considered to be not physically
relevant for its imaginary mass spectrum. Nevertheless we present them in sec.
19, since they induce representations of the causal logic generated by the
non-timelike relation. As far as we know, these are the only examples of
representations of a causal logic. This topic, addressed below, is treated in
sec. 11, 12, and 18, 19, 20.

Much attention we pay to the fundamental causal systems, the Dirac system for
every positive mass in sec. 13, 14, 15 and the two Weyl systems in sec. 16,
obtaining important results. As the first act of course we verify the
causality condition (ii) for these systems (see (13.1), (16.1)).
We study in detail the free motion of the border of a compact localized (i.e.,
localized in a bounded region) wave function. One finds that at any time the
border moves at the speed of light. As expected, over the long term it moves
radially to infinity. In the short term however the movement of the wave
border is more complicated as parts of it may move in the opposite direction.
To be more explicit, for every compact localized wave function $\psi$ and any
direction in position space given by a unit vector $e\in\mathbb{R}^{3}$, let
$e(\psi)$ denote the maximal real number such that the half-space
$\\{x\in\mathbb{R}^{3}:xe\leq e(\psi)\\}$ and the carrier
$\\{x\in\mathbb{R}^{3}:\psi(x)\neq 0\\}$ of $\psi$ are disjoint up to a null
set. The plane $\\{x\in\mathbb{R}^{3}:xe=e(\psi)\\}$ is perpendicular to $e$
and tangential to the carrier.

carrier$xe\leq e(\psi)$$xe\geq e(\psi)$$e$

Now the result (13.2.1) for the evolution $\psi_{t}$ of $\psi$ is that there
is a unique time $t_{e}=t_{e}(\psi)$ with

$e(\psi_{t})=e(\psi)+|t_{e}|-|t-t_{e}|$

for all times $t\in\mathbb{R}$.

$(t,e(\psi_{t}))$$t_{e}$$t$$e(\psi_{t_{e}})$

As long as $t<t_{e}$ one has $e(\psi_{t})=e(\psi_{t_{e}})-t_{e}+t$. This means
that, at the speed of light, the carrier of $\psi_{t}$ retreats in the
direction $e\in\mathbb{R}^{3}$. At time $t_{e}$ there is an abrupt change of
motion. For $t>t_{e}$ the carrier advances in direction $-e$ with light
velocity as $e(\psi_{t})=e(\psi_{t_{e}})+t_{e}-t$.

carrier

Only then the wave function expands in direction $-e$ as expected. Hence,
despite of this simple motion, in the short term the picture is complicated
since the time of change $t_{e}$ depends in general on the direction $e$ (see
(13.2.1)). The carrier of the wave function performs the changes from
shrinking to expanding not isotropicly. But not later than the time
corresponding to the diameter of the carrier a simultaneous isotropic
expansion of the wave function at light speed takes place (13.2.1).
The changes of the carrier due to a boost are quite different. Let $\psi_{\rho
e}$ denote the wave function $\psi$ boosted along the direction $e$ with
rapidity $\rho$. Suppose that $\psi$ is compact localized in the half-space
$\\{x\in\mathbb{R}^{3}:xe\geq 0\\}$. Then by causality $e(\psi_{\rho
e})\geq\operatorname{e}^{-|\rho|}e(\psi)$ and $-\overline{e}(\psi_{\rho
e})\leq-\operatorname{e}^{|\rho|}\overline{e}(\psi)$ for $\overline{e}:=-e$.
This means that the expansion of the carrier in direction $e$ is limited by
the factor $\operatorname{e}^{|\rho|}$, whereas in direction $-e$ it is
limited by the factor $\operatorname{e}^{-|\rho|}$ thus not overcoming the
barrier $\\{x\in\mathbb{R}^{3}:xe=0\\}$. Actually one finds that $\psi_{\rho
e}$ for $\rho\in\mathbb{R}$ does not stay confined in any strip
$\\{x\in\mathbb{R}^{3}:0\leq xe\leq c\\}$. If
$|t_{\overline{e}}|<-\frac{1}{2}\overline{e}(\psi)$ then $\psi_{\rho e}$ is
confined neither for $\rho\to\infty$ nor for $\rho\to-\infty$. See (13.3).
Then there are the remarkable states satisfying
$|t_{\overline{e}}|\geq-\frac{1}{2}\overline{e}(\psi)$, which we call late-
change states or, more specifically, large - $t_{\overline{e}}$ \- states. If
$\psi$ is a large - $t_{\overline{e}}$ \- state with $\pm
t_{\overline{e}}\geq-\frac{1}{2}\overline{e}(\psi)$ then by (13.3) the carrier
is shrinking in direction $-e$ to a $\delta$-strip at the origin by the factor
$\operatorname{e}^{\mp\rho}$ for $\pm\rho\geq 0$, i.e.,
$-\overline{e}(\psi_{\rho
e})\leq-\operatorname{e}^{\mp\rho}\overline{e}(\psi)$, or equivalently, still
by (13.3), the time evolution $\psi_{\pm t}$ is localized in
$\\{x\in\mathbb{R}^{3}:0\leq xe\leq t\\}$ for some (and hence all)
$t\geq-\frac{1}{2}\overline{e}(\psi)$. What is more, every state localized in
$\\{x\in\mathbb{R}^{3}:xe\geq 0\\}$ can be approximated by large -
$t_{\overline{e}}$ \- states, see (13.3), (16.2). The proof of this density
result involves a property of causal systems shown in sec. 10 and an
asymptotic behavior of the evolution of the probability of localization. For
the Dirac systems it is the asymptotic causality (13.2.2), which is known for
massive systems endowed with the NWL. For the massless Weyl systems one has
(16.2)(c), which is harder to prove.
Eventually, as a consequence one obtains in (13.3), (16.6) the Lorentz
contraction

$\parallel\\!\\!E\big{(}\\{x\in\mathbb{R}^{3}:-\delta\leq
xe\leq\delta\\}\big{)}\,\psi_{\rho e}\\!\\!\parallel\,\to
1,\;\;|\rho|\to\infty$

for every wave-function $\psi$ and any $\delta>0$. One recalls that in
classical mechanics it is discussed whether the length contraction really
exists or not. Therefore we find it is worth attaching to (13.3) the rather
obvious considerations about the objectiveness of the Lorentz contraction in
quantum theory.

Causality is used to define various orthocomplemented lattices of subsets of
Minkowski spacetime (see e.g. [56] 2013 and the literature cited therein),
known under the general name of causal logics. The link to quantum theory is
thought to be provided attributing to the sets of the lattice projections on
the Hilbert space of states. By physical reasons as well mathematical
consequences this assignment in general cannot be required to be a lattice
homomorphism into the quantum logic as argued subsequent to (19). But it is
expected to be a normalized monotone locally orthogonal $\sigma$-orthoadditive
Poincaré covariant map, called a representation for short.
Most adequate from a physical point of view seems to be the lattice
$\mathcal{M}$ in sec. 11 of causally complete regions generated by the
spacelike relation. However for structural deficiencies $\mathcal{M}$ does not
allow for representations, see sec. 12. Mathematically more convenient is the
lattice $\mathcal{M}^{{}^{\prime}borel}$ in sec. 18.4 generated by the non-
timelike relation [68]. And, indeed there are representations of
$\mathcal{M}^{{}^{\prime}borel}$. Unfortunately the representations we know so
far (19) refer to unphysical systems with imaginary mass spectrum. Every
representation of $\mathcal{M}^{{}^{\prime}borel}$ determines in an obvious
way a causal system (19). By (20) this is uniquely extendable as a high boost
limit to a PM on every non-timelike non-spacelike hyperplane $\chi$, obeying
Poincaré covariance.

$\sigma$$\chi$$\sigma^{\prime\prime}$$\sigma^{\prime}$

Hence this extendability is a necessary condition for a causal system to
induce a representation of $\mathcal{M}^{{}^{\prime}borel}$. It implies more
profound properties of causal systems as e.g. the existence of dense sets of
late-change states. And, indeed, the Dirac and Weyl systems allow for such an
extension (20). The somewhat involved proof is based on the results on group
representations of the sec. 24.3 \- 24.5 in the appendix. In view of these
insights, $\mathcal{M}^{{}^{\prime}borel}$ and its representations apparently
have some physical relevance.

Acknowledgements: The author would like to thank Ulrich Mutze and Andreas D.
Leiseifer for many valuable discussions.

##  2\. Newton-Wigner Position Operator and Localization by Wightman

Two main achievements in particle localization are the position operator
$X^{\textsc{nw}}$ of Newton, Wigner [2] from 1949 and Wightman’s formulation
of localization [1] in 1962. The latter furnishes also a rigorous mathematical
frame of the first using group representation theory.
The central notion in [1], which we like to call a Wightman localization (WL),
is a projection valued measure (PM) $E$ on the Borel sets of the Euclidean
space $\mathbb{R}^{3}$ and a representation $U$ of the covering group $ISU(2)$
of the group of Euclidean motions, both acting on the Hilbert space of states
$\mathcal{H}$, such that $(U,E)$ satisfies the covariance

$U(g)E(\Delta)U(g)^{-1}=E(g\cdot\Delta)$ (2.1)

for all $g\in ISU(2)$ and Borel sets $\Delta\subset\mathbb{R}^{3}$. Note that
here a representation of a topological group is always unitary continuous on a
separable complex Hilbert space.
With every WL a Euclidean covariant position operator $X=(X_{1},X_{2},X_{3})$
is associated by spectral integration

$X_{k}:=\int\pi_{k}\operatorname{d}E,\quad\pi_{k}(x):=x_{k}$ (2.2)

Clearly, $\pi_{k}(E)$ is the spectral measure of $X_{k}$ and $E$ is the joint
spectral measure of the three components of $X$. Hence $E$ is uniquely
determined by $X$.
For every irreducible relativistic system, which is massive or massless
helicity zero, a WL exist. Let this system be described by the representation
$W$ of the universal covering group $\tilde{\mathcal{P}}=ISL(2,\mathbb{C})$ of
the Poincaré group. Then the former means that there is a PM $E$ which is
covariant with respect to $U:=W|_{ISU(2)}$. Any other covariant PM
$E^{\prime}$ is related to $E$, i.e., there is a unitary transformation $S$
commuting with $U$ such that $E^{\prime}=SES^{-1}$. A particular
representative is the Newton-Wigner localization (NWL) $E^{\textsc{nw}}$,
which is singled out by the requirement of time inversion invariance and by a
mathematical smoothness condition. We prefer to distinguish NWL from all other
related WL by the Newton-Wigner-Pryce formula (see [10]), also called first
Bakamjian-Thomas-Foldy formula Eq. (2.3) below. The Newton-Wigner operator
$X^{\textsc{nw}}$ is the position operator associated with $E^{\textsc{nw}}$.
We add that $E^{\textsc{nw}}$ and hence $X^{\textsc{nw}}$ exist also for the
related antisystems with negative energy operator $H$, and, plainly, for every
system, which is an orthogonal integral of the former systems, antisystems
included. Such a system is massive if the mass-squared operator

$C:=H^{2}-P^{2}$

is positive ($C>0$). Here $P=(P_{1},P_{2},P_{3})$ is the linear momentum and
$P^{2}=P_{1}^{2}+P_{2}^{2}+P_{3}^{2}$.
For massive systems the remarkable Bakamjian-Thomas-Foldy formulae Eq. (2.3),
Eq. (2.4) hold on a common core. The first

$X^{\textsc{nw}}=\small{\frac{1}{2}}(H^{-1}N+NH^{-1})-C^{-1/2}(C^{1/2}+|H|)^{-1}\,P\times\big{(}J+H^{-1}(P\times
N)\big{)}$ (2.3)

may be regarded as the defining equation for the Newton-Wigner position
operator as it expresses $X^{\textsc{nw}}$ in terms of the ten generators of
the Poincaré group $\tilde{\mathcal{P}}=ISL(2,\mathbb{C})$, i.e., in addition
to the Hamiltonian $H$ and the linear momentum $P=(P_{1},P_{2},P_{3})$, the
angular momentum $J=(J_{1},J_{2},J_{3})$, which generates spatial rotations,
and the Lorentz booster $N=(N_{1},N_{2},N_{3})$. The second

$N=\small{\frac{1}{2}}(HX^{\textsc{nw}}+X^{\textsc{nw}}H)+(C^{1/2}+|H|)^{-1}\operatorname{sgn}(H)\,P\times(J-X^{\textsc{nw}}\times
P)$ (2.4)

shows that the generators of the little kinematical group $\mathbb{R}\otimes
ISU(2)$, representing time translations plus Euclidean motions, together with
$X^{\textsc{nw}}$ determine the booster and hence the whole representation of
$\tilde{\mathcal{P}}$. Note that Eq. (2.4) generalizes the original formula to
not semi-bounded $H$, as for instance in the case of the Dirac representation.
It is characteristic for the NWL that it commutes with the three Casimir
operators, i.e., the sign of the energy, the mass-squared operator, and the
Pauli-Lubanski scalar.

##  3\. Poincaré Covariance of Position

The principle of special relativity is automatically satisfied for a quantum
system if the Hilbert space of states of the system is the carrier space of a
representation $W$ of $\tilde{\mathcal{P}}$. This is thoroughly explained in
Currie, Jordan, Sudarshan 1962 [7] and, specifically for localization in space
regions, in Amrein 1969 [8, sec. VIII]. There is also clearly pointed out that
relativistic symmetry must be distinguished from manifest relativistic
invariance. This rather formal concept, moreover, has not found a generally
accepted formulation, above all not for the localization problem (see e.g.
[7], [8], [9], [15], [12]). $X^{\textsc{nw}}$ and $E^{\textsc{nw}}$ are not
manifest relativistic invariant, as argued e.g. in [3, sec. III] and in [10]
1997, respectively.

In the following for the group elements
$g=(\mathfrak{a},A)=(t,b,A)\in\tilde{\mathcal{P}}$ we often use abbreviations
as $\mathfrak{a}\equiv(\mathfrak{a},I_{2})$, $b\equiv(0,b,I_{2})$,
$t\equiv(t,0,I_{2})$, $A\equiv(0,A)$. The elements in $ISU(2)$ are denoted by
$(b,B)$.

###  3.1. Poincaré covariance of localization

By the foregoing remark it is clear that NWL does not violate the principle of
special relativity. By means of $W$ the localization on the particular
hyperplane $\mathbb{R}^{3}\equiv\\{0\\}\times\mathbb{R}^{3}$ is transported to
any other spacelike hyperplane.
Independently of this fact we will reason now following [4, sec. C] that for a
relativistic system, rather than a Euclidean covariant localization, physics
requires a fully Poincaré covariant localization. This generalization is more
than a merely manifest Poincaré covariant formulation of localization in the
above sense. On its necessity we are in accord with [9, sec. 7.3] (restricting
the permissive view in [3, sec. I]). It is a consequence of the physical ideas
on Euclidean covariance in [1] applied to general Poincaré transformations,
thus giving grounds for a unique extension of a WL to a Poincaré covariant
localization in regions, i.e., Borel subsets, of spacelike hyperplanes. In
Castrigiano, Mutze 1982 [3, Theorem] existence and uniqueness of this
extension is proven. The result is proven more generally for POL in (6.2).
First let us briefly report the physical idea behind Euclidean covariance of
the localization in [1]. Imagine an apparatus $\mathcal{A}$ suited for the
measurement of the position of the system in the region (i.e. Borel subset)
$\Delta$ and thus realizing the observable $E(\Delta)$. If this apparatus is
shifted by $b\in\mathbb{R}^{3}$ then relativistic symmetry means that it
realizes the observable $W(b)E(\Delta)W(b)^{-1}$. On the other hand, the
shifted apparatus obviously measures the position in the translated region
$\Delta+b$ and thus realizes the observable $E(\Delta+b)$. Consequently
translational covariance $W(b)E(\Delta)W(b)^{-1}=E(\Delta+b)$ must hold.
Analogously one deduces covariance under all Euclidean transformations.
If now the apparatus $\mathcal{A}$ is boosted by a Lorentz transformation $L$
given by $A\in SL(2,\mathbb{C})$ then, again by relativistic symmetry, the
boosted apparatus realizes the observable $W(A)E(\Delta)W(A)^{-1}$. Moreover,
by special relativity, the boosted apparatus is still suited for position
measurement, namely in the spatial region $L\,\Delta$ of the spacelike
hyperplane $L\,\mathbb{R}^{3}$ in Minkowski space $\mathbb{R}^{4}$. Here, for
convenience, $\mathbb{R}^{3}$ is identified with
$\\{0\\}\times\mathbb{R}^{3}$. Let $E(L\,\Delta)$ denote the yes-no observable
realized by the boosted apparatus. So $E(L\,\Delta)$ has to be equated with
$W(A)E(\Delta)W(A)^{-1}$. An analogous reasoning is valid for every
transformation $g=(\mathfrak{a},A)\in\tilde{\mathcal{P}}$.
There arises the following question of consistency. Let
$\Delta^{\prime}\subset\mathbb{R}^{3}$ be a region and let $A^{\prime}\in
SL(2,\mathbb{C})$ such that $L^{\prime}\,\Delta^{\prime}=L\,\Delta$. Further,
let the apparatus $\mathcal{A}^{\prime}$ realize the observable
$E(\Delta^{\prime})$. Then the apparatus $\mathcal{A}^{\prime}$ boosted by
$L^{\prime}$ realizes the observable
$W(A^{\prime})E(\Delta^{\prime})W(A^{\prime})^{-1}$, which by special
relativity equals the observable $E(L^{\prime}\,\Delta^{\prime})$ of position
in $L^{\prime}\,\Delta^{\prime}$. Hence physics imposes on $E$ the condition
$W(A)E(\Delta)W(A)^{-1}=W(A^{\prime})E(\Delta^{\prime})W(A^{\prime})^{-1}$.
However, as shown in [3, Theorem] and in (6.2), in the final analysis it is
due to Euclidean covariance that $E$ automatically satisfies this condition.
This allows the claimed unique extension of the localization to all spatial
regions $\Gamma$ of spacelike hyperplanes $\sigma$ such that the general
Poincaré covariance

$W(g)E(\Gamma)W(g)^{-1}=E(g\cdot\Gamma)$ (3.1)

holds. Here

$g\cdot\mathfrak{x}:=\mathfrak{a}+\Lambda(A)\mathfrak{x}\quad\text{ for
}g=(\mathfrak{a},A)\in\tilde{\mathcal{P}},\,\mathfrak{x}\in\mathbb{R}^{4}$
(3.2)

and $g\cdot\Gamma:=\\{g\cdot\mathfrak{x}:\mathfrak{x}\in\Gamma\\}$ with
$\Lambda:SL(2,\mathbb{C})\to O(1,3)_{0}$ the universal covering homomorphisms
onto the proper orthochronous Lorentz group. As a consequence of covariance
Eq. (3.1), one has a WL on every spacelike hyperplane $\sigma$ by mapping
$\Gamma\mapsto E(\Gamma)$. In particular, $E^{\textsc{nw}}$ is Poincaré
covariant.

###  3.2. Poincaré covariance of position operator

Poincaré covariance carries over from the WL to the position operator
associated with the WL. Indeed, let $\sigma$ be a spacelike hyperplane, choose
an origin $\mathfrak{o}\in\sigma$ and a direction
$\mathfrak{n}\in\sigma-\mathfrak{o}$. Then, by [3, (3.1)(3.2)], the (four-
component) position operator $X_{\sigma,\mathfrak{o}}$ for the three-
dimensional Euclidean space $\sigma-\mathfrak{o}$ is given by the integral of
$\operatorname{id}_{\sigma}-\,\mathfrak{o}$ with respect to the WL on
$\sigma$, whence the position operator $X_{\sigma,\mathfrak{o},\mathfrak{n}}$
in direction $\mathfrak{n}$ follows to be
$-\mathfrak{n}\cdot\,X_{\sigma,\mathfrak{o}}$ with $\cdot$ the Minkowski
product on Minkowski space.111 Representing Minkowski space by
$\mathbb{R}^{4}$ the Minkowski product is given by
$\mathfrak{a}\cdot\mathfrak{a}^{\prime}:=a_{0}a^{\prime}_{0}-a_{1}a^{\prime}_{1}-a_{2}a^{\prime}_{2}-a_{3}a^{\prime}_{3}$.
Often we use the notation $\mathfrak{a}^{\cdot
2}:=\mathfrak{a}\cdot\mathfrak{a}$. Then the Poincaré covariance

$W(g)\,X_{\sigma,\mathfrak{o},\mathfrak{n}}\,W(g)^{-1}=X_{g\cdot\sigma,g\cdot\mathfrak{o},g\cdot\mathfrak{n}}$
(3.3)

holds. This yields, in particular, Poincaré covariance of the Newton-Wigner
position operator $X^{\textsc{nw}}$. For this kind of covariance see Fleming
1965 [11, (2.6)]. One more derivation of the Poincaré covariance of
$X^{\textsc{nw}}$ can be found in [12] 2002.

###  3.3. Frame dependence of Newton-Wigner localization

Regarding NWL we are going to show that a state $\varphi$, $||\varphi||=1$,
localized in a bounded region $\Delta$, i.e.,
$E^{\textsc{nw}}(\Delta)\varphi=\varphi$, when boosted, is no longer localized
in any bounded region $\Delta^{\prime}$. We give a rigorous proof. Most
probably any WL related to NWL localizes frame-dependently as well, but we do
not pursue this question.
Occasionally we will say that a state is compact localized if it is localized
in a bounded region.

(1) Canonical cross section. In the sequel we repeatedly will need the
canonical cross section $Q(\mathfrak{k})$ for $\mathfrak{k}\in\mathbb{R}^{4}$
with $\mathfrak{k}\cdot\mathfrak{k}>0$. It is the unique positive $2\times
2$-matrix satisfying $Q(\mathfrak{k})\cdot(\eta m,0,0,0)=\mathfrak{k}$, where
$m=\sqrt{\mathfrak{k}\cdot\mathfrak{k}}$, $\eta=\operatorname{sgn}(k_{0})$.
Explicitly

$Q(\mathfrak{k})=\sqrt{\frac{m}{2(m+|k_{0}|)}}\left(I_{2}+\frac{\eta}{m}\sum_{j=0}^{3}k_{j}\sigma_{j}\right)$
(3.4)

Here $\sigma_{0}=I_{2}$ and $\sigma_{j}$ are the Pauli matrices.222
$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$,
$\sigma_{2}=\left(\begin{array}[]{cc}0&-\operatorname{i}\\\
\operatorname{i}&0\end{array}\right)$,
$\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$ Let $A\in
SL(2,\mathbb{C})$. Then
$R(\mathfrak{k},A):=Q(\mathfrak{k})^{-1}AQ(A^{-1}\cdot\mathfrak{k})$ is called
the Wigner rotation. It satisfies $R(\mathfrak{k},A)\in SU(2)$ and
$R(\mathfrak{k},B)=B$ for $B\in SU(2)$. For $p\in\mathbb{R}^{3}$,
$\epsilon(p):=\sqrt{|p|^{2}+m^{2}}$, $\eta=+,-$ and
$\mathfrak{p}^{\eta}:=(\eta\epsilon(p),p)$, one has
$Q(\mathfrak{p}^{+})^{-1}=Q(\mathfrak{p}^{-})$. Further useful formulae are
$Q(\mathfrak{k})^{2}=\frac{\eta}{m}\sum_{j=0}^{3}k_{j}\sigma_{j}$ and
$AQ(\mathfrak{k})^{2}A^{*}=Q(A\cdot\mathfrak{k})^{2}$,
$A^{*}Q(\mathfrak{k})^{-2}A=Q(A^{-1}\cdot\mathfrak{k})^{2}$. Furthermore,
$Q(\mathfrak{k})^{4\varsigma}=-I_{2}+\frac{2\epsilon(k)}{m}Q(\mathfrak{k})^{2\varsigma}$,
$\varsigma=\pm 1$.∎

Let $[m,j,\eta]$ denote any irreducible representation of
$\tilde{\mathcal{P}}$ for mass $m>0$, spin $j\in\mathbb{N}_{0}/2$, and sign of
energy $\eta\in\\{+,-\\}$. In the massless case $[0,s,\eta]$ spin $j$ is
replaced by helicity $s\in\mathbb{Z}/2$.

(2) Theorem. Let $W$ be an orthogonal sum of irreducible representations
$[m,j,\eta]$ and $[0,0,\eta]$ with $m>0$, $j\in\mathbb{N}_{0}/2$,
$\eta\in\\{+,-\\}$ provided with the NWL $E^{\textsc{nw}}$ (see _sec. 2)_.
Then $\varphi$ and $W(A)\varphi$ are not localized in a bounded region for
every state $\varphi$ and every $A\in SL(2,\mathbb{C})\setminus SU(2)$.

Proof. We may assume at once that $W$ is irreducible. Consider first the
massive case. Then $W$ acts in momentum space
$L^{2}(\mathbb{R}^{3},\mathbb{C}^{2j+1})$ by

$\big{(}W^{mom}(\mathfrak{a},A)\varphi\big{)}(p)=\sqrt{\epsilon(q^{\eta})/\epsilon(p)}\,\operatorname{e}^{i\mathfrak{p}^{\eta}\cdot\,\mathfrak{a}}\,D^{(j)}\big{(}R(\mathfrak{p}^{\eta},A)\big{)}\,\varphi(q^{\eta})$
(3.5)

where $D^{(j)}$ is the irreducible $2j+1$ dimensional unitary representation
of $SU(2)$, $\mathfrak{p}^{\eta}=(\eta\epsilon(p),p)$,
$\mathfrak{q}^{\eta}=(q_{0}^{\eta},q^{\eta}):=A^{-1}\cdot\mathfrak{p}^{\eta}$.
In position space representation NWL is the canonical PM given by
$E^{\textsc{nw}}(\Delta)\psi=1_{\Delta}\psi$.
One passes to the momentum space representation by the Fourier transformation
$\varphi(p)=\big{(}\mathcal{F}\psi\big{)}(p)=(2\pi)^{-\frac{3}{2}}\int\operatorname{e}^{-ixp}\psi(x)\operatorname{d}^{3}x$.
Assume now that $\varphi$ and $W^{mom}(A)\varphi$ are compact localized. Note
that $A=B^{\prime}A_{\rho}B$ for some $B,B^{\prime}\in SU(2)$ and $\rho\neq
0$.333 Frequently we use explicitly
$A_{\rho}:=\operatorname{e}^{\,\rho\,\sigma_{3}/2}$,
$\Lambda(A_{\rho})=\footnotesize{\left(\begin{array}[]{cccc}\cosh(\rho)&0&0&\sinh(\rho)\\\
0&1&0&0\\\ 0&0&1&0\\\ \sinh(\rho)&0&0&\cosh(\rho)\end{array}\right)}$ for
$\rho\in\mathbb{R}$. It follows that $\varphi^{\prime}:=W^{mom}(B)\varphi$ and
$W^{mom}(A_{\rho})\varphi^{\prime}$ are compact localized. By the Paley-Wiener
Theorem each component of these functions is the restriction of an entire
function on $\mathbb{C}^{3}$. Note further that $R(\mathfrak{p},A)\in SU(2)$
is diagonal and real for $A=A_{\rho}$ and $p_{1}=p_{2}=0$ and equal to $I_{2}$
for $A=I_{2}$. One infers $R(\mathfrak{p}^{\eta},A_{\rho})=I_{2}$ if
$p_{1}=p_{2}=0$ (see also Eq. (22.5)). Therefore it follows from Eq. (3.5)
that every component of $\varphi^{\prime}$ yields an entire function $f$ on
$\mathbb{C}$ such that
$z\mapsto\sqrt[4]{\frac{u(z)^{2}+m^{2}}{z^{2}+m^{2}}}f(u(z))$ with
$u(z):=z\cosh(\rho)-\eta\sqrt{z^{2}+m^{2}}\sinh(\rho)$ has an entire
extension. However, for $f\neq 0$, this is impossible for the singularities at
$z=\pm im\cosh(\rho)$.
We turn to the massless case $[0,0,\eta]$. Then, in momentum space
$L^{2}(\mathbb{R}^{3},\mathbb{C})$,

$\big{(}W^{mom}(\mathfrak{a},A)\varphi\big{)}(p)=\sqrt{|q^{\eta}|/|p|}\,\operatorname{e}^{i\mathfrak{p}^{\eta}\cdot\,\mathfrak{a}}\,\varphi(q^{\eta})$
(3.6)

with $\mathfrak{p}^{\eta}=(\eta|p|,p)$. Proceeding as in the massive case, one
gets an entire function $f$ such that
$f_{\rho}(z):=\sqrt{\frac{|u(z)|}{|z|}}f(u(z))$ with $u(z):=\alpha z+\beta|z|$
for $\alpha:=\cosh(\rho)$, $\beta:=-\eta\sinh(\rho)$ has an entire extension.
Hence, $f_{\rho}(z)=\sqrt{\alpha+\beta}f\big{(}(\alpha+\beta)z\big{)}$ if
$z>0$, and $f_{\rho}(z)=\sqrt{\alpha-\beta}f\big{(}(\alpha-\beta)z\big{)}$ if
$z<0$, whence
$\sqrt{\alpha+\beta}f\big{(}(\alpha+\beta)z\big{)}=\sqrt{\alpha-\beta}f\big{(}(\alpha-\beta)z\big{)}$
$\forall$ $z\in\mathbb{C}$. However, for $f\neq 0$, this is impossible. $\Box$

The result in (3.3) is not inconsistent with Poincaré covariance Eq. (3.1), as
(3.3) concerns a single fixed Lorentz frame. By means of $W$, (3.3) can be
translated to any other Lorentz frame. Poincaré covariance states that if the
state $\varphi$ is localized in $\Delta$ (bounded or not) then the boosted
state $W(A)\varphi$ is localized in $E(A\cdot\Delta)$,
$\Delta\equiv\\{0\\}\times\Delta$. Indeed,
$E(A\cdot\Delta)W(A)\varphi=W(A)E(\Delta)W(A)^{-1}W(A)\varphi=W(A)\varphi$
because of $E(\Delta)\varphi=\varphi$.
The result in (3.3) is stated roughly also in the following way: Every state
$\varphi$ is localized in a bounded region with respect to one rest frame at
the most. Referring to the generalized eigenstates of the Newton-Wigner
position operator it is observed in Weidlich, Mitra 1963 [13, sec. 3] that “a
state strictly localized in one coordinate system $\sigma$ $\dots$ appears to
be smeared out if observed from a system $\sigma^{\prime}$ moving relative to
$\sigma$ ”. See also e.g. [12], [16]. This is the so-called frame dependence
of NWL. More precisely, let $E^{\textsc{nw}}(\Delta)\varphi=\varphi$ for a
bounded region $\Delta$ of some spacelike hyperplane $\sigma$, and let
$\sigma^{\prime}$ be a spacelike hyperplane different from $\sigma$ with the
same origin. Then $E^{\textsc{nw}}(\Gamma^{\prime})\varphi\neq\varphi$ for
every bounded region $\Gamma^{\prime}\subset\sigma^{\prime}$. Indeed, since
$\sigma^{\prime}=A\cdot\sigma$ for some $A\in SL(2,\mathbb{C})\setminus
SU(2)$,
$E^{\textsc{nw}}(A\cdot\Gamma)\varphi=W(A)E^{\textsc{nw}}(\Gamma)W(A^{-1})\varphi\neq
W(A)W(A^{-1})\varphi=\varphi$ holds for all bounded
$\Delta\subset\Gamma\subset\sigma$ by (3.3).
Although the frame dependence of NWL is in accord with relativistic symmetry,
it is still a worrying feature. Indeed, let $\varphi$ be localized in a
bounded region. Then $E^{\textsc{nw}}(\\{|x|\geq R\\})W(A)\varphi\neq 0$ for
all $R>0$ and every however small boost $\Lambda(A)$, despite of
$E^{\textsc{nw}}(\\{|x|\geq R_{0}\\})\varphi=0$ for some $R_{0}>0$. So imagine
an electron localized in a bounded region. Exert temporarily on it an electric
force. Then any moment later there is a non-vanishing probability to observe
the electron arbitrarily far away. This is acausal.

Let us state that Newton-Wigner localization would be a satisfactory solution
to localizability of irreducible relativistic systems, if there were not three
problems. First recall the fact that massless irreducible systems with non-
zero helicity, like the photon are not localizable according to Newton-Wigner.
Then there is the frame dependence of NWL as discussed above. The third
problem already mentioned 1955 in [14, sec. 2] and studied 1963 in [13, sec.
3] regards a similar phenomenon. It is the instantaneous spreading to infinity
of wave-functions which initially vanish outside a bounded region. In
Ruijsenaars 1981 [15] one finds a conscientious discussion of quantum
localization and the possibility to observe superluminal propagation with
respect to Newton-Wigner position. (For more recent engagements in this topic
see e.g. [16] 2011, [6] 2017.) The latter results in a violation of causality
and hence constitutes a serious objection against NWL. The following section 4
studies in detail the phenomenon of instantaneous spreading as a consequence
of positivity of energy mainly referring to the work of Hegerfeldt.
From sec. 9 on we will be concerned with relativistic quantum systems, which
implement Einstein causality. These systems are endowed with a WL for which
there is no transmission of signals faster than light. In particular
instantaneous spreading and frame dependence of localization are excluded on
principle.

##  4\. Positive Energy and Localized States

It is well-known by the work of Hegerfeldt (see [17] for a brief survey) that
the phenomenon of superluminal propagation is closely related to positivity of
energy, more precisely, to semi-boundedness of the Hamiltonian. For localized
states one has the alternative, either instantaneous spreading or confinement.
The latter means that once the quantum system is localized in a closed region
(different from the whole space) it stays there all the time. See Hegerfeldt
2001 [17, sec. 4] and [18, II. A]. Within the frame of WL this alternative is
shown in Schlieder 1971 [19, Kap. 3] (see also Jadczyk 1977 in [20, sec. 1],
Castrigiano 1984 [21, sec. I] ) using Borchers 1967 [22, III.1 Theorem].

Indeed, semi-boundedness of energy and little else already imply the above
alternative. We follow Hegerfeldt [17, sec. 4] reducing the assumptions. Let
time evolution be described by $V(t)=\operatorname{e}^{itH}$ with Hamiltonian
$H$. Let $\Delta_{0}\subset\mathbb{R}^{3}$ and $r_{0}>0$ be arbitrary but
fixed throughout this section 4. Set
$\Delta_{r_{0}}:=\\{y\in\mathbb{R}^{3}:\,\exists\,x\in\Delta_{0}\text{ with
}|y-x|\leq r_{0}\\}$. Then the assumptions are as follows:

  * ($L1$)

The set of all scalar multiples of the states localized in $\Delta_{r_{0}}$ is
a closed subspace.

  * ($L2$)

If the state $\varphi$ is localized in $\Delta_{0}$, then $V(t)\varphi$ is
localized in $\Delta_{r_{0}}$ for all $t\in[0,\tau]$ for some $\tau>0$
depending on $\varphi$.

The meaning of the first assumption is obvious. The second one is a minimal
requirement regarding non-instantaneous spreading. Note that the term
localized state is not and need not be specified here. In the examples
following (4) the states localized in a region $\Delta_{0}$ are the unit
eigenvectors with eigenvalue $1$ of the localization operator assigned to
$\Delta_{0}$, which is a positive operator with spectrum contained in $[0,1]$,
thus satisfying $(L1)$. Within the frame of POL with causal time evolution
(7.1) the assumption $(L2)$ is satisfied for all Borel $\Delta_{0}$ and all
$r_{0}>0$.

(3) Theorem. Let the Hamiltonian be semi-bounded and let the localized states
of the quantum system satisfy the assumptions _( $L\,1,2$)_. Then, if the
system is localized at some time $t_{0}$ in $\Delta_{0}$, it stays localized
in $\Delta_{r_{0}}$ for all times $t\in\,]-\infty,+\infty\,[$.

Proof. Without restriction assume $t_{0}=0$. Let $\varphi$ be a state
localized in $\Delta_{0}$, and denote by $\mathcal{L}_{r_{0}}$ the subspace in
($L1$). By ($L2$), $V(t)\varphi\in\mathcal{L}_{r_{0}}$ for $t\in[0,\tau]$. Let
$\chi\in\mathcal{L}_{r_{0}}^{\perp}$. Then $\langle\chi,V(t)\varphi\rangle=0$
holds for $t\in[0,\tau]$ and hence for all $t$, see [18, II. A] or [23, Proof
of Theorem]. The argument is as follows. Let without restriction $H\geq 0$.
Applying the spectral theorem to $H$, one has
$F(t):=\langle\chi,V(t)\varphi\rangle=\int\operatorname{e}^{it\lambda}\operatorname{d}m(\lambda)$
for a complex measure $m$ with $\operatorname{supp}(m)\subset[0,\infty[$.
Therefore $F$ has a continuous extension onto $\\{t\in\mathbb{C}:\Im(t)\geq
0\\}$, which is analytic on $\\{t\in\mathbb{C}:\Im(t)>0\\}$. Moreover, since
$F(t)=0$ for $t\in I:=]0,\tau[$, by Schwarz reflection principle, there is a
further analytic extension of $F$ onto $(\mathbb{C}\setminus\mathbb{R})\cup
I$. This is identical to zero by the principle of analytic continuation. By
continuity, $F=0$. — It follows $V(t)\varphi\in\mathcal{L}_{r_{0}}$ for all
$t$. $\Box$

Essentially Hegerfeldt’s result in [17, sec. 4] is equivalent to (4).
Obviously (4) finds application in case that localizability is described (i)
by a WL, or (ii) by a supra-additive, continuous from above, projection valued
map as used by Jauch, Piron 1967 [24] for weak localization, or (iii) by a
positive operator valued measure (POM), introduced by several authors (see
e.g. Toigo 2005 [25] for a survey and sec. 6). So, if the Hamiltonian is semi-
bounded and if $\Delta_{0}$ is closed and ($L\,1,2$) holds for some $r_{n}$ in
place of $r_{0}$ with $r_{n}\to 0$, then
$\Delta_{0}=\bigcap_{n}\Delta_{r_{n}}$ and in all three cases states localized
in $\Delta_{0}$ obey the alternative, either instantaneous spreading or
confinement in $\Delta_{0}$.
Moreover, in the cases (i), (ii), confinement occurs in $\Delta_{0}$ (if and)
only if the time translation operators commute with the localization operator
for $\Delta_{0}$. Indeed, by (4),
$V(t)\big{(}E(\Delta_{0})\mathcal{H}\big{)}\subset
E(\Delta_{r_{n}})\mathcal{H}$ holds for all $n$ and for all $t$. Since $E$ is
continuous from above, $V(t)\big{(}E(\Delta_{0})\mathcal{H}\big{)}\subset
E(\Delta_{0})\mathcal{H}$ follows by $n\to\infty$, and hence
$V(t)\big{(}E(\Delta_{0})\mathcal{H}\big{)}=E(\Delta_{0})\mathcal{H}$ for all
$t$. This implies $V(t)E(\Delta_{0})=E(\Delta_{0})V(t)$ $\forall$ $t$.

According to (4), a quantum system with semi-bounded Hamiltonian without
instantaneous spreading and without confinement does not have states, which
are localized in bounded regions. This result will now be refined exploiting
translational symmetry. It applies to massive and massless relativistic
systems and also to systems with a more general energy-momentum spectrum.
Partially we follow Thaller 1992 [26, Theorem 1.6] minimizing the
assumptions.444 and avoiding an error in [26, Theorem 1.6] regarding the
spectrum of $H=\lambda(P)$, as there the assumptions on $\lambda$ do not imply
that $\lambda^{-1}([a,b])$ is compact Lemma (4) is a version of Hegerfeldt,
Ruijsenaars 1980 [18, II]. Let $S$ be a representation of the group
$\mathbb{R}^{3}$ of spatial translations. To ($L\,1,2$) we add the assumption

  * ($L3$)

If the state $\varphi$ is localized in $\Delta_{r_{0}}$, then $S(b)\varphi$ is
localized in $\Delta_{2r_{0}}$ for all $|b|\leq\beta$ for some $\beta>0$
depending on $\varphi$.

Let $E^{\textsc{h}}$ denote the spectral measure of $H$ and let
$E^{\textsc{p}}$ denote the joint spectral measure on the Borel sets of
$\mathbb{R}^{3}$ of the generators $P=(P_{1},P_{2},P_{3})$ of the spatial
translations.

(4) Definition. The spectral condition holds if there are Borel sets
$A_{n}\subset\mathbb{R}$ with $\bigcup_{n\in\mathbb{N}}A_{n}=\mathbb{R}$ such
that for every $n$ and state $\varphi$ the vector valued measure
$E^{\textsc{p}}(\cdot)E^{\textsc{h}}(A_{n})\varphi$ on $\mathbb{R}^{3}$ has
compact support.

Obviously the spectral condition (4) holds if there are Borel sets
$A_{n}\subset\mathbb{R}$ with $\bigcup_{n\in\mathbb{N}}A_{n}=\mathbb{R}$ such
that $E^{\textsc{p}}(\mathbb{R}^{3}\setminus B_{n})E^{\textsc{h}}(A_{n})=0$
for some ball $B_{n}$, $n\in\mathbb{N}$. Hence the spectral condition is
satisfied, if $H=h(P)$ for some measurable semi-bounded function
$h:\mathbb{R}^{3}\to\mathbb{R}$ with $|h(p)|\to\infty$ if $|p|\to\infty$. Take
e.g. $A_{n}:=[-n,n]$. — A relativistic system with semi-bounded energy $H$
satisfies (4). Indeed, by the representation theory of $\tilde{\mathcal{P}}$
it follows $C\geq 0$ so that $H^{2}\geq H^{2}-C=P^{2}$, whence $H\geq|P|$.

(5) Lemma. Let the Hamiltonian be semi-bounded and let the spectral condition
hold. Let the localized states of the quantum system satisfy the assumptions
_( $L\,1,2,3$)_, and $(L\,1)$ also for $2r_{0}$. Let $\varphi$ be a state
localized in $\Delta_{0}$. Then $S(b)\varphi$ is localized in
$\Delta_{2r_{0}}$ for all $b\in\mathbb{R}^{3}$.

Proof. Let $\varphi$ be a state localized in $\Delta_{0}$, and denote by
$\mathcal{L}_{r_{0}}$ the subspace of states localized in $\Delta_{r_{0}}$
according to ($L1$). Let $\chi\in\mathcal{L}_{r_{0}}^{\perp}$. Then
$\int\operatorname{e}^{i\lambda
t}\operatorname{d}\langle\chi,E^{\textsc{h}}(\lambda)\varphi\rangle=\langle\chi,V(t)\varphi\rangle=0$
holds for all $t$ by (4). Fourier uniqueness theorem, valid also for complex
measures,555 Let $m$ be a complex measure on $\mathbb{R}$. By the Radon-
Nikodym Theorem $m=u|m|$ holds with a measurable $u:\mathbb{R}\to\mathbb{C}$,
$|u|=1$. Let $f\in L^{1}(\mathbb{R})$. Then, by Fubini’s theorem,
$\int\operatorname{e}^{-i\lambda t}\operatorname{d}m(\lambda)=0$ $\forall$ $t$
implies $\int f(t)\int\operatorname{e}^{-i\lambda
t}\operatorname{d}m(\lambda)\operatorname{d}t=\int\hat{f}(\lambda)u(\lambda)\operatorname{d}|m|(\lambda)=0$.
Hence $|m|=0$ since the set of Fourier transforms $\hat{f}$ is dense in
$(C_{0}(\mathbb{R}),||\cdot||_{\infty})$ and $C_{c}(\mathbb{R})\subset
C_{0}(\mathbb{R})$.  yields $\langle\chi,E^{\textsc{h}}(A)\varphi\rangle=0$
for all Borel sets $A\subset\mathbb{R}$. Hence $E^{\textsc{h}}(A)\varphi$ is
localized in $\Delta_{r_{0}}$. So, by assumption ($L\,3$),
$S(b)E^{\textsc{h}}(A)\varphi$ is localized in $\Delta_{2r_{0}}$ for all
$|b|\leq\beta$. Let $\chi\in\mathcal{L}_{2r_{0}}^{\perp}$. Then
$\int\operatorname{e}^{-ibp}\operatorname{d}\langle\chi,E^{\textsc{p}}(p)E^{\textsc{h}}(A)\varphi\rangle=\langle\chi,S(b)E^{\textsc{h}}(A)\varphi\rangle=0$
holds for all $|b|\leq\beta$. Obviously, due to the spectral condition, for
$A=A_{n}$ the left hand side is entire in $b\in\mathbb{C}^{3}$, whence
$\langle\chi,S(b)E^{\textsc{h}}(A_{n})\varphi\rangle=0$ holds for all
$b\in\mathbb{R}^{3}$ and every $n$. It follows
$\langle\chi,S(b)\varphi\rangle=0$. Hence $S(b)\varphi$ is localized in
$\Delta_{2r_{0}}$ for all $b\in\mathbb{R}^{3}$. $\Box$

(6) Corollary. Suppose that for every $b\in\mathbb{R}^{3}$ the state
$S(b)\varphi$ in _( 4)_ is localized in $\Delta_{0}+b$. Suppose further that a
state cannot be localized in each of two widely separated regions. Then
$\Delta_{0}$ is not bounded.

Regarding the result (4) see also Hegerfeldt 1974 [27]. Obviously by (4) there
is no causal system (see (9)) in case of semi-bounded energy $H$. Under the
premise of the relativistic spectral condition (i.e., the joint spectrum of
$(H,P)$ lies in the closed forward cone) this follows also from Perez, Wilde
1977 [28, Theorem 1] and it is shown in [29, Problems with Position operators,
Theorem] 2012 based on [28, Lemma].

In the following theorem we refer to POL with causal time evolution introduced
in sec. 6. See in particular Eq. (6.1) and Eq. (7.1). Actually, rotational
covariance is not needed for (4). Call a Borel subset
$\Delta\subset\mathbb{R}^{3}$ essentially dense if $\overline{\Delta\setminus
N}=\mathbb{R}^{3}$ for every Lebesgue null set $N$.

(7) Theorem. Let localizability be described by a POL $T$ with causal time
evolution. Let the spectral condition _( 4)_ hold. Suppose that there is a
state localized in the region $\Delta_{0}$. Then $\Delta_{0}$ is essentially
dense.
Proof. Let $\varphi$ be a state localized in $\Delta_{0}$, i.e.,
$T(\Delta_{0})\varphi=\varphi$. As $T$ is translational covariant it vanishes
at Lebesgue null sets $N$ (see (6.2)(a)). So one checks that the assumptions
in (4) are satisfied for $\Delta_{0}\setminus N$ in place of $\Delta_{0}$ and
for every $r_{0}>0$. Then from (4) we get immediately
$T(A)S(b)\varphi=S(b)\varphi$ $\forall$ $b\in\mathbb{R}^{3}$ with
$A:=\overline{\Delta_{0}\setminus N}$. For $\Gamma:=\mathbb{R}^{3}\setminus A$
this implies $T(\Gamma)S(b)\varphi=(I-T(A))S(b)\varphi=0$. By translation
covariance $T(\Gamma-b)\varphi=0$ $\forall$ $b\in\mathbb{R}^{3}$ follows. If
$\Gamma$ were not empty then, since $\Gamma$ is open, $\sigma$-subadditivity
of $T$ would imply the contradiction $\varphi=0$. $\Box$

See the consequence (14.1) of (4). The result (4) does not answer the question
if there is a state localized in an open dense set $\Delta$, for which
$\mathbb{R}^{3}\setminus\Delta$ has positive measure.

So (4) confirms that relativistic systems endowed with NWL show instantaneous
spreading of localized states rather than confinement. If a state $\varphi$ is
compact localized, then the state $V(t)\varphi$ evolved in time $t\neq 0$ is
not compact localized. This can directly be verified by Eq. (3.5), since in
momentum representation, each component of the state $V(t)\varphi$ evolved in
time is like $\operatorname{exp}\big{(}\pm it\sqrt{|p|^{2}+m^{2}}\big{)}f(p)$,
$p\in\mathbb{R}^{3}$, with $m\geq 0$ and $f$ the Fourier transform of an
$L^{2}$-function with compact support,666 An $L^{2}$-function $\varphi$ is
said to have compact support if there is a compact set $C$ and a null set $N$
such that $\\{\varphi\neq 0\\}\setminus N\subset C$. whence the result by the
Paley-Wiener Theorem.
This violation of causality holds also for the weak localization of the two-
component photon by Amrein [8], since by construction the localized states are
those of a NWL. Cf. also [30]. At last, localizability of the two-component
photon described by an Euclidean system of covariance in Castrigiano [31,
4.2.2] and Kraus [32] is not causal, too, since the localized states are the
same as those of the weak localization in [8].

##  5\. Three Contrasting Concepts

The discussion so far shows that there are three contrasting concepts
regarding relativistic quantum systems: causality, positivity of energy
(stability), and localized state. The least likely of the three to be
discarded in the literature seems to be Einstein causality. It means that any
signal, which could serve to synchronize clocks, cannot move faster than
light. It is hard to imagine a scenario where cause precede the effect, even
if the detection probability of acausal events as predicted by the Newton-
Wigner position operator is smaller than $10^{-10^{8}}$ according to
Ruijsenaars [15, sec. 3]. Another reason why to uphold causality is the fact
that in the Dirac theory it is the causal Dirac position operator which serves
for the interaction of the electron to the electromagnetic field through the
minimal coupling rather then the Newton-Wigner one. Also in case of the
massless Weyl fermions, for which the Newton-Wigner position operator does not
even exist, the causal position operator plays the analogous role.
There is a third rather weighty argument in favor of causality. We study
relativistic quantum systems endowed with a WL for which, by definition, there
is no transmission of signals faster than light, thus implementing Einstein
causality. Considering only physical systems, viz. systems with non-negative
mass-squared operator and finite spinor dimension, there are exactly three
such irreducible systems, the Dirac system and the two Weyl
systems.777Actually we did not succeed as yet to rule out certain irreducible
massive and massless SCT, see (22.2), (23.4).  This proves uniqueness and
fundamental importance of the Dirac system for massive quantum systems. In
view of the recently discovered Weyl fermions (see [33], [34] 2015) this may
equally hold true for the Weyl systems with regard to massless quantum
systems. One notes that the requirement of a causal localization also predicts
the opposite handedness of particle and antiparticle constituting a Weyl
system. First, however, the above classification shows the far-reaching impact
of the principle of causality.

Maintaining causality two possibilities are left for relativistic quantum
systems:

  * ($C1$)

causality plus semi-bounded Hamiltonian and localization without localized
states in bounded regions

  * ($C2$)

causality plus localized states in bounded regions and non-semi-bounded
Hamiltonian

Sec. 6, 7, 8 are concerned with ($C1$) in detail. Referring to ($C2$), in sec.
9 causal systems are introduced. Then in sec. 13, 14, 15 and sec. 16 the
concepts developed to meet ($C1$) and ($C2$) are illustrated by the most
important Dirac system and the Weyl systems. A further development of the
theory proceeds in sec. 20. It is an important issue of these investigations
that ($C1$) and ($C2$) are not opposed to each other but that they are
simultaneously present and intrinsically linked in causal systems.

##  6\. Positive Operator Valued Localization

Regarding ($C1$), in the literature there is the intensively studied concept
of Euclidean systems of covariance, henceforth called positive operator valued
localization (POL). See Neumann 1972 [35], Scutaru 1977 [36], Castrigiano,
Henrichs 1980 [37], and [25] for several other authors. POL were introduced
independently by Castrigiano 1976 [31], and Kraus 1977 [32] for the
localization of massless particles with non-zero helicity and the two-
component photon, respectively. This subject is also treated in [4, sections
F-H], from which some parts of this section are taken. Sec. 7 and 8 are still
concerned with POL. Particularly important is sec. 7.2, where causality for
localizable relativistic quantum systems is introduced. This is the central
notion, which will determine the investigations in the sequel.

POL are characterized by the fact that the localization observables assigned
to the space regions $\Delta$, which for Wightman localizations are projection
operators $E(\Delta)$ with spectrum in $\\{0,1\\}$, are permitted to be
positive operators $T(\Delta)$ with extended spectrum in $[0,1]$. Thus $T$ is
a positive operator valued measure (POM), which is Euclidean covariant

$U(g)T(\Delta)U(g)^{-1}=T(g\cdot\Delta)$ (6.1)

Here $U$ is the representation of $ISU(2)$ describing the Euclidean symmetry
of the system.

###  6.1. Interpretation of POL

POL is an unconventional concept and requires explanation. So [35], [38], [32]
refer to Ludwig’s reformulation of quantum theory (see Ludwig 1970, 1983
[39]), where a yes-no measurement, called an effect, is described by an
operator $A$ with $0\leq A\leq I$. Being the system in the state $\psi$, the
probability for the outcome yes is still given by $\langle\psi,A\psi\rangle$.
Following [38], [32] every single localization observable $T(\Delta)$ is an
effect with $\langle\psi,T(\Delta)\psi\rangle$ the probability for finding the
particle in $\Delta$. Let us mention that in this theory an observable is
described by a POM on the Borel sets of $\mathbb{R}$ and is called a
generalized observable. The benefits for quantum physics of POM describing
observables is fully explained in Busch, Grabowski, Lahti 1997 [40]. There
these observables are called unsharp observables. According to Heinonen,
Lahti, Ylinen 2004 [41], the POM obtained by smearing out PM represent so-
called fuzzy observables endowed with the interpretation of inaccurate
measurements. Fuzzy observables were launched by Ali, Emch 1974 [42], who use
the concept just for the construction of a POL for the two-component photon.
Within the frame of conventional quantum mechanics one possibility to
interpret POL is the assumption that there are described additional degrees of
freedom for an elementary particle. But about their nature the kinematic
considerations give no information. One could help out intuition imagining, as
in [31], a property of the system like a substance which is spread over space
so that $\langle\varphi,T(\Delta)\varphi\rangle\in[0,1]$ indicates the
expected fraction of the substance contained in $\Delta$, if the system is in
the state $\varphi$.
However, in case that the POL is the trace of the WL of a causal system, as
the POL $T^{e}$ of the Dirac electron or $T^{\chi\eta}$ of the Weyl fermions,
we propose in sec. 15 an interpretation of the PO-localization operators still
within conventional quantum mechanics, which attributes an important role to
the negative energy states. In this case,
$\langle\varphi,T(\Delta)\varphi\rangle$ results to be the probability of
localization or spatial probability in $\Delta$ in the state $\varphi$. This
interpretation will reconcile the three contrasting concepts discussed in sec.
5.
Finally, as expressed in [1], it might appear more natural from a physical
point of view to require the existence of localizations operators $T(\Delta)$
only for boxes (cuboids) and finite unions thereof and to weaken the
$\sigma$-additivity of $T$ to finite additivity. But it is shown in (24.6)
that this generalization is only seeming since, essentially due to
translational covariance, $T$ can anyway be extended uniquely to a POL. For
the special case of WL compare [1, Appendix I].

###  6.2. Poincaré covariance of POL

The discussion of Poincaré covariance for WL in sec. 3.1 applies as well to
POL. So there is the necessity to extend a POL for a relativistic system to
regions of spacelike hyperplanes in Minkowski space. Since this extension has
to satisfy Poincaré covariance, it is obvious how to define this extension.
But there is the question of consistency of the definition as explained in
sec. 3.1. Moreover, we will see that the formulation of causality in sec. 7
necessitates the further extension of POL to Lebesgue measurable sets.
Let the physical system be described by the representation $W$ of
$\tilde{\mathcal{P}}$ in the separable Hilbert space $\mathcal{H}$. Let $T$ be
a POL of the system, i.e., a POM on the Borel sets of the hyperplane
$\varepsilon:=\mathbb{R}^{3}\equiv\\{0\\}\times\mathbb{R}^{3}$, which is
Euclidean covariant, i.e., $W(g)T(\Delta)W(g)^{-1}=T(g\cdot\Delta)$ for all
$g\in ISU(2)$ (see Eq. (6.1)).
Let $\mathfrak{S}$ denote the set of all Lebesgue measurable sets of spacelike
hyperplanes of Minkowski space.

(8) Theorem. Let $T$ be a POL.

  * (a)

Let $\Delta\subset\mathbb{R}^{3}$ be a Borel set. Then $T(\Delta)=0$ if and
only if $\Delta$ is a Lebesgue null set. Hence the completion of $T$ is a POL
defined for the Lebesgue measurable sets and vanishes just at the Lebesgue
null sets.

Let $W$ be a representation of $\tilde{\mathcal{P}}$ and let $T$ be Euclidean
covariant with respect to $W|_{ISU(2)}$.

  * (b)

There is a unique positive-operator-valued extension of $T$ on $\mathfrak{S}$,
still called $T$, such that Poincaré covariance
$W(g)T(\Gamma)W(g)^{-1}=T(g\cdot\Gamma)$ $\forall$ $\Gamma\in\mathfrak{S}$,
$g\in\tilde{\mathcal{P}}$ holds.

Proof. (a) Choose a total set $\\{\varphi_{n}:n\in\mathbb{N}\\}$ of unit
vectors in $\mathcal{H}$. Then the probability measure $\mu$ on the Borel sets
of $\varepsilon$ defined by
$\mu(\Delta):=\sum_{n}2^{-n}\langle\varphi_{n},T(\Delta)\varphi_{n}\rangle$
has the same null sets as $T$, since generally
$\langle\varphi,T(\Delta)\varphi\rangle=||\sqrt{T(\Delta)}\varphi||^{2}=0$ if
and only if $T(\Delta)\varphi=0$. So $\mu$ is quasi-invariant under
translations $b\in\varepsilon$, since due to translational covariance of $T$
(see Eq. (6.1)) one has $\mu(b+\Delta)=\sum_{n}2^{-n}\langle
U(b)^{-1}\varphi_{n},T(\Delta)U(b)^{-1}\varphi_{n}\rangle=0$ if and only if
$T(\Delta)=0$. Hence $\mu$ is equivalent to the Lebesgue measure (see e.g.
[43, § 9.1]).
The well-known completion of an ordinary measure (see e.g. [44, sec. 1.1]) is
equally available for POM. Due to the foregoing result, the completion of $T$,
still denoted by $T$, is defined on the $\sigma$-algebra of Lebesgue
measurable subsets of $\mathbb{R}^{3}$, is zero just at Lebesgue null sets,
and obviously preserves Euclidean covariance.
(b) Let $\Gamma\in\mathfrak{S}$. There are $\Delta\subset\varepsilon$ and
$g\in\tilde{\mathcal{P}}$ with $\Gamma=g\cdot\Delta$. Note
$\Delta\in\mathfrak{S}$. Set $T(\Gamma):=W(g)T(\Delta)W(g)^{-1}$. This
definition of $T(\Gamma)$ obviously is forced from Poincaré covariance, and,
if this definition is unambiguous, then Poincaré covariance of $T$ holds.
So it remains to show that $T(\Gamma)$ is well-defined. To this end let
$\Delta^{\prime}\subset\varepsilon$ and $g^{\prime}\in\tilde{\mathcal{P}}$
also satisfy $\Gamma=g^{\prime}\cdot\Delta^{\prime}$. Introduce
$h:=g^{\prime-1}g$. Then $h\cdot\Delta=\Delta^{\prime}$. It suffices to show
$T(\Delta^{\prime})=W(h)T(\Delta)W(h)^{-1}$. If
$h\cdot\varepsilon=\varepsilon$ holds, then $h\in ISU(2)$. Hence $h$ yields an
Euclidean transformation and the assertion follows from Euclidean covariance
of $T$. Otherwise, the hyperplane $\sigma:=h\cdot\varepsilon$ is different
from $\varepsilon$. Then $\Delta^{\prime}\subset\varepsilon\cap\sigma$, which
is a plane in $\varepsilon$ and hence a null set with respect to the Lebesgue
measure on $\varepsilon$. Also $\Delta=h^{-1}\cdot\Delta^{\prime}$ is
contained in a plane and hence a Lebesgue null set. The proof is accomplished.
∎

In view of (6.2) let us fix that henceforth a POL is defined on the
$\sigma$-algebra of the Lebesgue measurable sets. Furthermore, a Lebesgue
measurable set will be called simply measurable or a region.

(9) Definition. Let $T$ be a positive operator valued map on $\mathfrak{S}$,
which is a POM on every spacelike hyperplane. Let $W$ be a representation of
$\tilde{\mathcal{P}}$. Then $T$ is called a Poincaré covariant POL for $W$ if
the covariance $W(g)T(\Gamma)W(g)^{-1}=T(g\cdot\Gamma)$ $\forall$
$\Gamma\in\mathfrak{S}$, $g\in\tilde{\mathcal{P}}$ holds.

##  7\. Causal Time Evolution and Causal Localization

###  7.1. POL with causal time evolution

For every region $\Delta\subset\mathbb{R}^{3}$ and time $t\in\mathbb{R}$ the
region of influence is

$\Delta_{t}:=\\{y\in\mathbb{R}^{3}:\,\exists\,x\in\Delta\text{ with
}|y-x|\leq|t|\\}$

It is the set of points, which are reached from $\Delta$ within time $|t|$
with velocity of light $c=1$. Every event $(t,x)\in\mathbb{R}^{4}$ determines
its region of influence $\\{x\\}_{t}\subset\mathbb{R}^{3}$, which is the
closed ball with center $x$ and radius $|t|$. So
$\Delta_{t}=\bigcup_{x\in\Delta}\\{x\\}_{t}$.
Causal time evolution requires that after time $t$, respectively before time
$t$, the probability of localization in $\Delta_{t}$ is not less than
originally in $\Delta$. Hence
$\langle\varphi,T(\Delta)\varphi\rangle\leq\langle
V(t)\varphi,T(\Delta_{t})V(t)\varphi\rangle$ holds for every state $\varphi$,
where $V(t)$ describes the time evolution by time $t$. In other words, for
causal time evolution one has

$V(t)T(\Delta)V(t)^{-1}\leq T(\Delta_{t})$ (7.1)

(10) Definition. One calls $(V,U,T)$ a POL with causal time evolution, if
$(V,U)$ is a representation of the little kinematical group, $(U,T)$ is a POL,
and Eq. (7.1) is satisfied.888 POL with causal time evolution are called
causal POL in [4], whereas causal POL in sec. 7.2 are not considered in [4].

In case of a Poincaré covariant POL (6.2) we check now that Eq. (7.1) does not
single out a particular frame. Indeed, by covariance one has a POL with causal
time evolution $(V_{\sigma},U_{\sigma},T_{\sigma})$ on every spacelike
hyperplane $\sigma$ by $T_{\sigma}(\Gamma):=T(\Gamma)$ for all regions
$\Gamma\subset\sigma$ with $U_{\sigma}(g):=W(g)$ for all $g\in
g_{0}\,ISU(2)g_{0}^{-1}$ and $V_{\sigma}(g):=W(g)$ for all
$g\in\\{g_{0}(t,0,I_{2})g_{0}^{-1}:t\in\mathbb{R}\\}$, where
$\sigma=g_{0}\cdot(\\{0\\}\times\mathbb{R}^{3})$ for some
$g_{0}\in\tilde{\mathcal{P}}$. Using
$g_{0}\cdot(g_{0}^{-1}\cdot\Gamma)_{t}=\Gamma_{g_{0}(t,0,I_{2})g_{0}^{-1}}$
one obtains the frame-independent formulation of Eq. (7.1)

$W(g)T(\Gamma)W(g)^{-1}\leq T(\Gamma_{g})$

for $\Gamma\subset\sigma=g_{0}\cdot(\\{0\\}\times\mathbb{R}^{3})$ and all time
translations $g=g_{0}(t,0,I_{2})g_{0}^{-1}$ with respect to $\sigma$. So the
concept of POL with causal time evolution obeys the principle of special
relativity.

###  7.2. Causal localization

Although a Poincaré covariant POL with causal time evolution does not violate
the principle of special relativity it is still a preliminary concept for the
description of causality.
Let $W$ be a representation of $\tilde{\mathcal{P}}$ describing a relativistic
system and let $T$ be a Poincaré covariant POL (6.2) for $W$. Then Einstein
causality requires more than the condition in Eq. (7.1). Indeed, let $\Gamma$
be a measurable subset of a spacelike hyperplane and let $\sigma$ be any
spacelike hyperplane. Then

$\Gamma_{\sigma}:=\bigcup_{\mathfrak{z}\in\Gamma}\\{\mathfrak{x}\in\sigma:(\mathfrak{x}-\mathfrak{z})\cdot(\mathfrak{x}-\mathfrak{z})\geq
0\\}$ (7.2)

is the region of influence of $\Gamma$ in $\sigma$. It is the set of all
points $\mathfrak{x}$ in $\sigma$, which can be reached from some point
$\mathfrak{z}$ in $\Gamma$ by a signal not moving faster than light.
Therefore, in particular, the probability of localization of the system in
$\Gamma_{\sigma}$ cannot be less than originally in $\Gamma$. Hence causality
requires

$T(\Gamma)\leq T(\Gamma_{\sigma})$ (7.3)

(11) Definition. A POL $T$ for $W$ is causal if it is Poincaré covariant and
satisfies Eq. (7.3).

Let $\Delta$ be a measurable set of a spacelike hyperplane $\sigma$, let
$g\in\tilde{\mathcal{P}}$. Then

$\Delta_{g}:=\bigcup_{\mathfrak{z}\in
g\cdot\Delta}\\{\mathfrak{x}\in\sigma:(\mathfrak{x}-\mathfrak{z})\cdot(\mathfrak{x}-\mathfrak{z})\geq
0\\}$ (7.4)

is the region of influence in $\sigma$ of $g\cdot\Delta$. One has

$\Delta_{g}=\bigcup_{\mathfrak{y}\in\Delta}\\{\mathfrak{y}\\}_{g},\quad\\{\mathfrak{y}\\}_{g}=\\{\mathfrak{x}\in\sigma:(\mathfrak{x}-g\cdot\mathfrak{y})^{\cdot
2}\geq 0\\}$ (7.5)

where $\\{\mathfrak{y}\\}_{g}$ is the region of influence in $\sigma$ of the
single event $g\cdot\mathfrak{y}$.

We like to mention that in Jancewicz 1977 [20] a Poincaré covariant
probability density current, which is a 4-vector operator valued distribution,
is related to what we call a causal POL and in particular causal WL.

(12) Example. Let $y\in\mathbb{R}^{3}\equiv\\{0\\}\times\mathbb{R}^{3}$,
$t\equiv(t,0,I_{2})$ for $t\in\mathbb{R}$, and $A_{\rho}\equiv(0,A_{\rho})$
for $A_{\rho}=\operatorname{e}^{\,\rho\,\sigma_{3}/2}$, $\rho\in\mathbb{R}$.
Then the regions of influence in $\mathbb{R}^{3}$ of $t\cdot y$ and
$A_{\rho}\cdot y$ are

_(a)_ $\\{y\\}_{t}$ the ball with center $y$ and radius $|t|$.

_(b)_ $\\{y\\}_{A_{\rho}}$ the ball with center
$(y_{1},y_{2},\cosh(\rho)\,y_{3})$ and radius $|\sinh(\rho)\,y_{3}|$.

Let $e\in\mathbb{R}^{3}$, $|e|=1$, $0\leq a<b$, and $A_{\rho
e}\equiv(0,A_{\rho e})$ for $A_{\rho
e}:=\operatorname{exp}(\frac{\rho}{2}\sum_{k=1}^{3}e_{k}\sigma_{k})$
representing the boost in direction $e$ with rapidity $\rho$. We consider a
strip.

_(c)_ $\\{x\in\mathbb{R}^{3}:a\leq xe\leq b\\}_{A_{\rho
e}}=\\{x\in\mathbb{R}^{3}:a\operatorname{e}^{-|\rho|}\leq xe\leq
b\operatorname{e}^{|\rho|}\\}$.

Finally we treat a cylinder. Let $c>0$ and let $c_{\rho}:=\cosh(\rho)$,
$s_{\rho}:=\sinh(|\rho|)$, $t_{\rho}:=\tanh(|\rho|)$. Then

  * (d)

$\\{x\in\mathbb{R}^{3}:x_{1}^{2}+x_{2}^{2}\leq c^{2},\,a\leq x_{3}\leq
b\\}_{A_{\rho}}$ is the solid of revolution obtained by rotating the curve
around the $x_{3}$-axis, which joins the points
$P_{1}:=(c,0,\operatorname{e}^{-|\rho|}a)$ and
$P_{2}:=(c+t_{\rho}a,0,a/c_{\rho})$ by the short segment of the circle around
$(c,0,c_{\rho}a)$ with radius $s_{\rho}a$, $P_{2}$ and
$P_{3}:=(c+t_{\rho}b,0,b/c_{\rho})$ by a straight line, and finally $P_{3}$
and $P_{4}:=(c,0,\operatorname{e}^{|\rho|}b)$ by the short segment of the
circle around $(c,0,c_{\rho}b)$ with radius $s_{\rho}b$.

Proof. (a) is obvious. As to (b), $A_{\rho}\cdot
y=\big{(}\sinh(\rho)\,y_{3},y_{1},y_{2},\cosh(\rho)\,y_{3}\big{)}$, whence the
assertion by Eq. (7.5). Turn to (c). Note that $A_{\rho
e}=B(e)\,A_{\rho}\,B(e)^{-1}$ due to Eq. (21.9) and that $B(e)\cdot\\{x:a\leq
x_{3}\leq b\\}=\\{x:a\leq xe\leq b\\}$. Thus the assertion reduces to
$\\{a\leq x_{3}\leq b\\}_{A_{\rho}}=\\{a\operatorname{e}^{-|\rho|}\leq
x_{3}\leq b\operatorname{e}^{|\rho|}\\}$, which holds by (b) and Eq. (7.5).
Also (d) follows by (b) and Eq. (7.5) exploiting the rotational symmetry
around the $x_{3}$-axis. ∎

(13) Proposition. Let $T$ be a Poincaré covariant POL for $W$. It is causal if
and only if one of the equivalent conditions $(a)$ – $(e)$ holds.

  * $(a)$

$T(\Gamma)\leq T(\Gamma_{\sigma})$ for all $\Gamma\in\mathfrak{S}$ and all
spacelike hyperplanes $\sigma$

  * $(b)$

$T(g\cdot\Delta)\,\leq\,T(\Delta_{g})$ for all $\Delta\in\mathfrak{S}$ and
$g\in\tilde{\mathcal{P}}$

  * $(c)$

$W(g)T(\Delta)W(g)^{-1}\leq\,T(\Delta_{g})$ for all $\Delta\in\mathfrak{S}$
and $g\in\tilde{\mathcal{P}}$

  * $(d)$

$\langle\varphi,T(\Delta)\varphi\rangle\leq\langle
W(g)\varphi,T(\Delta_{g})W(g)\varphi\rangle$ for all $\Delta\in\mathfrak{S}$,
$g\in\tilde{\mathcal{P}}$, and states $\varphi$

  * $(e)$

$T(\Gamma)+T(\Gamma^{\prime})\leq I$ for all spacelike separated
$\Gamma,\Gamma^{\prime}\in\mathfrak{S}$

If $T$ is a WL $E$, then

  * $(f)$

$E(\Gamma)\,E(\Gamma^{\prime})=0$ for all spacelike separated
$\Gamma,\Gamma^{\prime}\in\mathfrak{S}$

is a further equivalent condition.

Proof. $(a)$ is the definition of causal POL. — The equivalence of $(a)$ and
$(b)$ is obvious setting $g\cdot\Delta=\Gamma$ and $\Delta=g^{-1}\cdot\Gamma$,
respectively, as $\Gamma_{\sigma}=\Delta_{g}$ holds. — $(b)$ and $(c)$ are
equivalent by (6.2). — The equivalence of $(c)$ and $(d)$ is obvious. — Now we
show the equivalence of $(a)$ and $(e)$. Assume first $(a)$. Then
$T(\Gamma^{\prime})\leq T(\Gamma^{\prime}_{\sigma})$. Since $\Gamma$ and
$\Gamma^{\prime}$ are spacelike separated,
$\Gamma\cap\Gamma^{\prime}_{\sigma}=\emptyset$ and hence
$T(\Gamma)+T(\Gamma^{\prime}_{\sigma})=T(\Gamma\cup\Gamma^{\prime}_{\sigma})\leq
I$. This yields $(e)$. Conversely, for $\Gamma$ and $\sigma$ in $(a)$,
obviously $\Gamma$ lies spacelike to $\sigma\setminus\Gamma_{\sigma}$.
Therefore, by $(e)$, $I\geq
T(\Gamma)+T(\sigma\setminus\Gamma_{\sigma})=T(\Gamma)+I-T(\Gamma_{\sigma})$,
whence $(a)$. — Finally we turn to the equivalence of $(e)$ for the WL $E$ and
$(f)$. By $(e)$, $E(\Gamma)\leq I-E(\Gamma^{\prime})$, whence the projections
$E(\Gamma)$ and $I-E(\Gamma^{\prime})$ commute. Then $E(\Gamma)$ and
$E(\Gamma^{\prime})$ commute. So $(f)$ follows. The converse implication is
obvious.∎

Let us comment on (7.2). As to $(a)$ note that Poincaré covariance of $T$
(6.2) implies Poincaré covariance of causality $W(h)T(\Gamma)W(h)^{-1}\leq
T\big{(}(h\cdot\Gamma)_{h\cdot\sigma}\big{)}$ as
$h\cdot\Gamma_{\sigma}=(h\cdot\Gamma)_{h\cdot\sigma}$ for all
$h\in\tilde{\mathcal{P}}$. — Regarding covariance for $(b)$ use the analogous
formula $h\cdot\Delta_{g}=(h\cdot\Delta)_{hgh^{-1}}$. — $(c)$ extends Eq.
(7.1) for causal time evolution. Indeed, recall the frame-independent
formulation of causal time evolution given in sec. 7.1. — Property $(f)$ is
the local orthogonality, well-known from local quantum physics. — We add some
obvious inequalities for a causal POL $T$ following from (7.2), which may be
useful: $T(g^{\prime}g\cdot\Delta)$ is less or equal to each of the four
positive operators $T(\Delta_{g^{\prime}g})$, $T((\Delta_{g})_{g^{\prime}})$,
$T((g\cdot\Delta)_{g^{\prime}})$, $T(g^{\prime}\cdot\Delta_{g})$, where
$\Delta_{g^{\prime}g}\subset\sigma$, $(\Delta_{g})_{g^{\prime}}\subset\sigma$,
$(g\cdot\Delta)_{g^{\prime}}\subset g\cdot\sigma$, and
$g^{\prime}\cdot\Delta_{g}\subset g^{\prime}\cdot\sigma$.
According to $(d)$ in (7.2) the probability of localization of the system in
the region of influence $\Delta_{g}$ in the state $W(g)\varphi$ is not less
than that in the original region $\Delta$ and state $\varphi$. Note that
$\Delta_{g}$ is bounded if $\Delta$ is bounded. In case of a causal WL this
implies frame independence of localization. This qualitative property means
that, if the system is localized in a bounded region with respect to some
frame, this holds true with respect to all frames. Cf. the discussion after
(3.3). So we keep

(14) Corollary. A causal WL is frame-independent.

As mentioned, the property $(c)$ implies causal time evolution Eq. (7.1). By
(22.2) and (23.4) there are massive, respectively massless Poincaré covariant
WL with causal time evolution, which do not localize frame-independently.
Consequently, by (7.2) they are not causal. Finally, there are Poincaré
covariant frame independent WL with causal time evolution, which are not
causal, see (23.4). Hence the converse of (7.2) does not hold.

###  7.3. Measurability of the region of influence

For the causality condition Eq. (7.3) it is necessary that the region of
influence $\Gamma_{\sigma}$ (7.2) is measurable. This will be shown in (7.3).
Let us remark that Borel measurability has been abandoned in sec. 6.2 just
because $\Gamma_{\sigma}$ might not be a Borel set although $\Gamma$ itself is
Borel. Note that $\Gamma_{\sigma}$ can be obtained from $\Gamma$ by harmless
mappings as $g:\tau\times\sigma\to\tau\times\mathbb{R}$,
$g(\mathfrak{y},\mathfrak{x}):=\big{(}\mathfrak{y},(\mathfrak{x}-\mathfrak{y})^{\cdot
2}\big{)}$ and the projection $\pi_{\sigma}$. One has
$\Gamma_{\sigma}=\pi_{\sigma}\big{(}g^{-1}(\Gamma\times[0,\infty[\,)\big{)}$.
The problem is that the projection of a Borel set need not be Borel. —
Regarding (7.3) compare Leiseifer 2014 [45, chapter 2], which treats a special
case.

(15) Lemma. Let $\sigma,\tau$ be two different spacelike hyperplanes. Let
$\Gamma$ be any subset of $\tau$ and
$\Gamma_{\sigma}=\bigcup_{\mathfrak{y}\in\Gamma}\\{\mathfrak{x}\in\sigma:(\mathfrak{x}-\mathfrak{y})^{\cdot
2}\geq 0\\}$ the region of influence of $\Gamma$ in $\sigma$. Define
$M(\Gamma,\sigma):=\bigcup_{\mathfrak{y}\in\Gamma}\\{\mathfrak{x}\in\sigma:(\mathfrak{x}-\mathfrak{y})^{\cdot
2}>0\\}$,
$N(\Gamma,\sigma):=\\{\mathfrak{x}\in\sigma:\sup_{\mathfrak{y}\in\Gamma}(\mathfrak{x}-\mathfrak{y})^{\cdot
2}=0\\}$. Then

  * (a)

$M(\Gamma,\sigma)\subset\Gamma_{\sigma}$ and $M(\Gamma,\sigma)$ is open in
$\sigma$

  * (b)

$N(\Gamma,\sigma)$ is closed in $\sigma$ and a null set of $\sigma$

  * (c)

$\Gamma_{\sigma}\subset M(\Gamma,\sigma)\cup N(\Gamma,\sigma)$

  * (d)

$\Gamma_{\sigma}$ is measurable

Proof. (a) $M(\Gamma,\sigma)\subset\Gamma_{\sigma}$ is obvious.
$M(\Gamma,\sigma)$ is open in $\sigma$, since
$M(\Gamma,\sigma)=\sigma\cap\bigcup_{\mathfrak{y}\in\Gamma}(\mathfrak{y}+C)$
with $C:=\\{\mathfrak{z}:\mathfrak{z}^{\cdot 2}>0\\}$ the open double cone.
(c) Let $\mathfrak{x}\in\Gamma_{\sigma}\setminus M(\Gamma,\sigma)$. Then there
is $\mathfrak{y}\in\Gamma$ with $(\mathfrak{x}-\mathfrak{y})^{\cdot 2}\geq 0$.
Therefore
$\sup_{\mathfrak{y}^{\prime}\in\Gamma}(\mathfrak{x}-\mathfrak{y}^{\prime})^{\cdot
2}\geq 0$. But
$\sup_{\mathfrak{y}^{\prime}\in\Gamma}(\mathfrak{x}-\mathfrak{y}^{\prime})^{\cdot
2}>0$ is excluded as $\mathfrak{x}\not\in M(\Gamma,\sigma)$. Hence
$\mathfrak{x}\in N(\Gamma,\sigma)$.
(d) is an obvious consequence of (a) – (c).
(b) We show that $N(\Gamma,\sigma)$ is closed. Let $\mathfrak{x}_{n}\in
N(\Gamma,\sigma)$ with $\mathfrak{x}_{n}\to\mathfrak{x}$ for some
$\mathfrak{x}\in\mathbb{R}^{4}$. Then, obviously, $\mathfrak{x}\in\sigma$ and
$\sup_{\mathfrak{y}^{\prime}\in\Gamma}(\mathfrak{x}-\mathfrak{y}^{\prime})^{\cdot
2}\leq 0$. For every $n$ there is $\mathfrak{y}_{n}\in\Gamma$ such that
$-\frac{1}{n}\leq(\mathfrak{x}_{n}-\mathfrak{y}_{n})^{\cdot 2}\leq 0$. We use
now the representation
$\tau=\\{\mathfrak{y}:\mathfrak{y}\cdot\mathfrak{e}_{\tau}=\rho_{\tau}\\}$
with $\mathfrak{e}_{\tau}=(1,e_{\tau})$, $|e_{\tau}|<1$ and
$\rho_{\tau}\in\mathbb{R}$, see after (11.5). Hence
$\mathfrak{y}\in\tau\Leftrightarrow y_{0}=\rho_{\tau}+ye_{\tau}$. Then
$-\frac{1}{n}\leq(x_{n0}-\rho_{\tau}-y_{n}e_{\tau})^{2}-(x_{n}-y_{n})^{2}\leq
0$. As $|y_{n}e_{\tau}|\leq|y_{n}||e_{\tau}|$ with $|e_{\tau}|<1$ this implies
that $(y_{n})_{n}$ is bounded. So there is a subsequence of
$(\mathfrak{y}_{n})_{n}$, which converges to some
$\mathfrak{y}\in\overline{\Gamma}$. It satisfies
$(\mathfrak{x}-\mathfrak{y})^{\cdot 2}=0$. Hence
$\sup_{\mathfrak{y}^{\prime}\in\Gamma}(\mathfrak{x}-\mathfrak{y}^{\prime})^{\cdot
2}=0$, whence $\mathfrak{x}\in N(\Gamma,\sigma)$.
It remains to show that $N(\Gamma,\sigma)$ is a null set of $\sigma$.
Obviously $N(\Gamma,\sigma)=N(\overline{\Gamma},\sigma)$. Hence assume without
restriction that $\Gamma$ is closed. Then the foregoing considerations show
that for every $\mathfrak{x}\in N(\Gamma,\sigma)$ there is
$\mathfrak{y}\in\Gamma$ with $(\mathfrak{x}-\mathfrak{y})^{\cdot 2}=0$.
Moreover, due to the covariance $g\cdot
N(\Gamma,\sigma)=N(g\cdot\Gamma,g\cdot\sigma)$, without restriction
$\sigma=\\{\mathfrak{x}:x_{0}=0\\}$ and either (i)
$\tau=\\{\mathfrak{y}:y_{0}=sy_{3}\\}$ for some $s\in]0,1[$ or (ii)
$\tau=\\{\mathfrak{y}:y_{0}=s\\}$ for some $s\neq 0$.
We turn to the case (i). Let $Q:=\\{q\in\mathbb{Q}^{3}:q_{3}\neq 0\\}$. For
$q\in Q$ let $r_{q}:=\frac{s}{4+s}|q_{3}|$ and
$K_{q}:=\\{y\in\mathbb{R}^{3}:|y-q|\leq r_{q}\\}$. Check that
$\mathbb{R}^{3}\setminus\\{y:y_{3}=0\\}=\bigcup_{q\in Q}K_{q}$. Put
$\Gamma_{q}:=\Gamma\cap\\{\mathfrak{y}\in\tau:y\in K_{q}\\}$, and
$\Gamma_{0}:=\\{\mathfrak{y}\in\tau:y_{3}=0\\}$. We claim
$N(\Gamma,\sigma)\subset\bigcup_{q\in Q\cup\\{0\\}}N(\Gamma_{q},\sigma)$.
Indeed, let $\mathfrak{x}\in N(\Gamma,\sigma)$. Choose $\mathfrak{y}\in\Gamma$
with $(\mathfrak{x}-\mathfrak{y})^{\cdot 2}=0$. There is $q\in Q\cup\\{0\\}$
with $\mathfrak{y}\in\Gamma_{q}$. Plainly
$\sup_{\mathfrak{y}^{\prime}\in\Gamma_{q}}(\mathfrak{x}-\mathfrak{y}^{\prime})^{\cdot
2}\leq 0$. Hence
$\sup_{\mathfrak{y}^{\prime}\in\Gamma_{q}}(\mathfrak{x}-\mathfrak{y}^{\prime})^{\cdot
2}=0$, whence $\mathfrak{x}\in N(\Gamma_{q},\sigma)$.
Note that $N(\Gamma_{0},\sigma)\subset\\{\mathfrak{x}\in\sigma:x_{3}=0\\}$ is
a null set of $\sigma$. Hence for the following fix $q\in Q$. Obviously it
suffices to consider $\Gamma\subset\\{\mathfrak{y}\in\tau:|y-q|\leq r_{q}\\}$
and to show that $N(\Gamma,\sigma)$ is a null set of $\sigma$.
For $\mathfrak{y}\in\Gamma$ one has
$|q_{3}|-|y_{3}|\leq|q_{3}-y_{3}|\leq|y-q|\leq r_{q}$, whence
$|y_{3}|\geq|q_{3}|-r_{q}=\frac{4}{4+s}|q_{3}|$ and hence
$|y_{0}|=s|y_{3}|\geq 4r_{q}$. Further note that $N(\Gamma,\sigma)$ is bounded
and hence compact. Indeed, every $\mathfrak{x}\in N(\Gamma,\sigma)$ satisfies
$|sy_{3}|=|x-y|$ for some $\mathfrak{y}\in\Gamma$, whence $|x|<2(r_{q}+|q|)$.
For a further reductional step consider
$U:=\\{u\in\mathbb{R}^{3}:(u_{1}-q_{1})^{2}+(u_{2}-q_{2})^{2}<r_{q}^{2}\\}$.
Obviously $\\{\mathfrak{x}\in\sigma:x\in R_{q}\,U\\}$, where
$R_{q}\equiv(q-Rq,R)\in ISO(3)$ is any rotation around $q$, provides an open
cover of $\sigma$. So finitely many of these sets cover $N(\Gamma,\sigma)$.
Then there are also finitely many $R_{q}$ such that the sets
$\\{\mathfrak{x}\in\sigma:x\in R_{q}\,H\\}$ with
$H:=\\{u\in\mathbb{R}^{3}:(u_{1}-q_{1})^{2}+(u_{2}-q_{2})^{2}\leq
r_{q}^{2},\,u_{3}-q_{3}\geq 0\\}$ cover $N(\Gamma,\sigma)$. So it suffices to
show that for every $R_{q}$ the set $S_{0}:=\\{\mathfrak{x}\in\sigma:x\in
R_{q}\,H\\}\cap N(\Gamma,\sigma)$ is a null set of $\sigma$. Recall
$\Gamma\subset\\{\mathfrak{y}\in\tau:|y-q|\leq r_{q}\\}$.
So far the reductional steps. Now put $S_{\alpha}:=\mathfrak{a}+S_{0}$ with
$\mathfrak{a}:=\alpha(0,R_{q}e)$, $e:=(0,0,1)$, $\alpha\geq 0$. We show now
that the sets $S_{\alpha}$ are mutually disjoint. Let
$0\leq\alpha\leq\alpha^{\prime}$ and let $\mathfrak{z}\in S_{\alpha}\cap
S_{\alpha^{\prime}}$. There are $\mathfrak{x},\mathfrak{x}^{\prime}\in S_{0}$
such that
$\mathfrak{z}=\mathfrak{a}+\mathfrak{x}=\mathfrak{a}^{\prime}+\mathfrak{x}^{\prime}$.
As $\mathfrak{x},\mathfrak{x}^{\prime}\in N(\Gamma,\sigma)$, there are
$\mathfrak{y},\mathfrak{y}^{\prime}\in\Gamma$ with
$(\mathfrak{x}-\mathfrak{y})^{\cdot 2}=0$,
$(\mathfrak{x}^{\prime}-\mathfrak{y}^{\prime})^{\cdot 2}=0$. As
$\mathfrak{x}^{\prime}\in N(\Gamma,\sigma)$ and $\mathfrak{y}\in\Gamma$ one
has $(\mathfrak{x}^{\prime}-\mathfrak{y})^{\cdot 2}\leq 0$. Use
$\mathfrak{x}^{\prime}=\mathfrak{x}+\mathfrak{a}-\mathfrak{a}^{\prime}$. Then
$0\geq(\mathfrak{x}-\mathfrak{y}+\mathfrak{a}-\mathfrak{a}^{\prime})^{\cdot
2}=(\mathfrak{x}-\mathfrak{y})^{\cdot
2}+2(\mathfrak{x}-\mathfrak{y})\cdot(\mathfrak{a}-\mathfrak{a}^{\prime})+(\mathfrak{a}-\mathfrak{a}^{\prime})^{\cdot
2}=-2(\alpha-\alpha^{\prime})(x-y)R_{q}e-(\alpha-\alpha^{\prime})^{2}=(\alpha^{\prime}-\alpha)\big{(}2(x-y)R_{q}e+(x^{\prime}-x)R_{q}e\big{)}=(\alpha^{\prime}-\alpha)(x+x^{\prime}-2y)R_{q}e$.
Introduce $u:=R_{q}^{-1}x\in H$, similarly $u^{\prime}$, and $v:=R_{q}^{-1}y$.
Note $|v-q|\leq r_{q}$. Then we find
$0\geq(\alpha^{\prime}-\alpha)(u+u^{\prime}-2v)e=(\alpha^{\prime}-\alpha)(u_{3}+u^{\prime}_{3}-2v_{3})$.
— We exploit now $(\mathfrak{x}-\mathfrak{y})^{\cdot 2}=0$. Recall
$4r_{q}\leq|y_{0}|$. So $16r_{q}^{2}\leq
y_{0}^{2}=|x-y|^{2}=|(u-q)-(v-q)|^{2}\leq 2(|u-q|^{2}+|v-q|^{2})\leq
2r_{q}^{2}+2(u_{3}-q_{3})^{2}+2r_{q}^{2}$, whence $u_{3}-q_{3}>2r_{q}$.
Analogously $u^{\prime}_{3}-q_{3}>2r_{q}$ holds. Therefore,
$u_{3}+u^{\prime}_{3}-2v_{3}=u_{3}-q_{3}+u^{\prime}_{3}-q_{3}-2(v_{3}-q_{3})>2r_{q}+2r_{q}-2r_{q}=2r_{q}>0$.
This implies $\alpha^{\prime}=\alpha$.
The closed sets $S_{\alpha}$, $0\leq\alpha\leq 1$ lie in a compact set of
$\sigma$, which has finite measure. By the translational invariance of the
Lebesgue measure each $S_{\alpha}$ has the same measure. Hence this measure is
zero, thus accomplishing the proof for the case (i).
The proof for the case (ii) is simpler as $|y_{0}|=|s|>0$ holds independently
of $y$. Therefore it suffices to consider the cover of $\mathbb{R}^{3}$ by
$K_{q}:=\\{y:|y-q|\leq|s|/4\\}$ with $q\in\mathbb{Q}^{3}$.∎

##  8\. Point-Localized Sequences of States

Provided the spectral condition (4) holds, (4) shows that for a POL with
causal time evolution there cannot exist a state localized in a non-
essentially dense region $\Delta$, i.e., which is an eigenvector of
$T(\Delta)$ with eigenvalue $1$. This means that in any state the probability
of localization in a closed $\Delta\neq\mathbb{R}^{3}$ is less than $1$. So in
[46] is introduced a property, which is mathematically weaker than that of a
localized state in $\Delta$ but physically equivalent to it, i.e.,
$||T(\Delta)||=1$. Indeed, as

$||T(\Delta)||=\sup\left\\{\langle\varphi,T(\Delta)\varphi\rangle:\,||\varphi||=1\right\\}$

norm $1$ means that the system can be localized within that region $\Delta$ by
a suitable preparation, not strictly but as accurately as desired.
For obvious physical reasons one is interested in POM with $||T(B)||=1$ for
every however small open ball $B\neq\emptyset$. In [4, sec. G] they are called
separated POM. If the POM $T$ is separated, then obviously for every point
$b\in\mathbb{R}^{3}$ there is a sequence $(\varphi_{n})$ of states satisfying

$\big{\langle}\varphi_{n},T(B)\,\varphi_{n}\big{\rangle}\to 1,\quad
n\to\infty$ (8.1)

for every open ball $B$ around $b$. This means that by a suitable preparation
the system can be localized around $b$ as good as desired, thus distinguishing
$b$ from any other point. According to [4, sec. G], any $(\varphi_{n})$
satisfying Eq. (8.1) is called a sequence of states localized at $b$.

###  8.1. Properties of point-localized sequences

Euclidean covariance Eq. (6.1) and causal time evolution Eq. (7.1) of a POL
$T$ imply the following transformation properties of point-localized sequences
of states.

(16) Theorem. Let $(\varphi_{n})$ be localized at $b$. Then
$(U(g)\varphi_{n})$ is localized at $g\cdot b$ for $g\in ISU(2)$. If $V$ is a
causal time evolution for $T$ then $(V(t)\varphi_{n})$ for $t\in\mathbb{R}$
satisfies $\langle V(t)\varphi_{n},T(B)\,V(t)\varphi_{n}\rangle\to 1$ for
every ball $B$ around $b$ with radius greater than $|t|$.

Proof. If $B$ is a ball around $g\cdot b$, then $g^{-1}\cdot B$ is a ball
around $b$ and, $\langle
U(g)\varphi_{n},T(B)U(g)\varphi_{n}\rangle=\langle\varphi_{n},T(g^{-1}\cdot
B)\varphi_{n}\rangle\to 1$. — Let $V$ be causal time evolution and $B$ a ball
around $b$ with radius $r>|t|$. Then $(\mathbb{R}^{3}\setminus
B)_{t}=\mathbb{R}^{3}\setminus B^{\prime}$ with $B^{\prime}$ the ball around
$b$ with radius $r-|t|$. Therefore $0\leq 1-\langle
V(t)\varphi_{n},T(B)\,V(t)\varphi_{n}\rangle=\langle
V(t)\varphi_{n},T(\mathbb{R}^{3}\setminus
B)\,V(t)\varphi_{n}\rangle\leq\langle\varphi_{n},T((\mathbb{R}^{3}\setminus
B)_{t})\varphi_{n}\rangle=\langle\varphi_{n},T(\mathbb{R}^{3}\setminus
B^{\prime})\varphi_{n}\rangle=1-\langle\varphi_{n},T(B^{\prime})\varphi_{n}\rangle\to
0$. Hence $\langle V(t)\varphi_{n},T(B)\,V(t)\varphi_{n}\rangle\to 1$. $\Box$

In (8.1) one cannot expect $\langle
V(t)\varphi_{n},T(B)\,V(t)\varphi_{n}\rangle\to 1$ for a ball $B$ around $b$
with radius smaller than $|t|$, as
$\\{b\\}_{t}=\\{x\in\mathbb{R}^{3}:|x-b|\leq|t|\\}$ is the region of influence
of the event $(t,b)$. If $T$ is even causal then due to causality (7.2)$(c)$
one has more generally

(17) Theorem. Let $T$ be a causal POL for $W$. Let $(\varphi_{n})$ be
localized at $b$. Let $g\in\tilde{\mathcal{P}}$ and put
$(t,b^{\prime}):=g\cdot(0,b)$. Then

$\langle W(g)\varphi_{n},T(B)\,W(g)\varphi_{n}\rangle\to 1$

for every ball $B$ around $b^{\prime}$ with radius greater than $|t|$. In
particular, if $(\varphi_{n})$ is localized at $0$, then so is
$(W(A)\varphi_{n})$ for $A\in SL(2,\mathbb{C})$.

Proof. Note that
$g=(\mathfrak{a},A)=(t,0,I_{2})(0,b^{\prime},I_{2})(0,0,B^{\prime})(0,0,A_{\rho})(0,0,B)(0,-b,I_{2})$
as $(t,b^{\prime})=\mathfrak{a}+A\cdot(0,b)$ and $A=B^{\prime}A_{\rho}B$ some
$B,B^{\prime}\in SU(2)$ and
$A_{\rho}:=\operatorname{e}^{\frac{\rho}{2}\sigma_{3}}$, $\rho\in\mathbb{R}$.
Let $\varphi^{\prime}_{n}:=W(B)W(-b)\varphi_{n}$. By (8.1),
$(\varphi^{\prime}_{n})$ is localized at $0$. Therefore it suffices to show
that $(W(A_{\rho})\varphi_{n})$ is localized at $0$ if $(\varphi_{n})$ is so.
The result follows applying (8.1).
Let $B$ be a ball around $0$ with radius $r>0$. First we verify that
$\mathbb{R}^{3}\setminus(\mathbb{R}^{3}\setminus B)_{A_{\rho}^{-1}}$ contains
the ball $B^{\prime}$ around $0$ with radius $\frac{r}{c+|s|}$,
$c:=\cosh(\rho)$, $s:=\sinh(\rho)$. Indeed,
$\mathbb{R}^{3}\setminus(\mathbb{R}^{3}\setminus
B)_{A^{-1}_{\rho}}=\\{y\in\mathbb{R}^{3}:|y-z|>|z_{0}|\text{ for all
}(z_{0},z)\in A_{\rho}^{-1}\cdot(\mathbb{R}^{3}\setminus B)\\}$. Here
$A_{\rho}^{-1}\cdot(0,x)=(-x_{3}s,x_{1},x_{2},x_{3}c)$. Consider $y\in
B^{\prime}$. Then for all $x\in\mathbb{R}^{3}\setminus B$ one has
$|y|<\frac{|x|}{c+|s|}\leq\frac{|x|^{2}}{|z|+|z_{0}|}=|z|-|z_{0}|$, i.e.,
$|z_{0}|<|z|-|y|$, and hence $|z_{0}|<|z-y|$. — Now, by (7.2)$(c)$, $0\leq
1-\langle W(A_{\rho})\varphi_{n},T(B)\,W(A_{\rho})\varphi_{n}\rangle=\langle
W(A_{\rho})\varphi_{n},T(\mathbb{R}^{3}\setminus
B)\,W(A_{\rho})\varphi_{n}\rangle\leq\langle\varphi_{n},T\big{(}(\mathbb{R}^{3}\setminus
B)_{A_{\rho}^{-1}}\big{)}\varphi_{n}\rangle\leq\langle\varphi_{n},T(\mathbb{R}^{3}\setminus
B^{\prime})\varphi_{n}\rangle=1-\langle\varphi_{n},T(B^{\prime})\varphi_{n}\rangle\to
0$. $\Box$

Finally we add

(18) Lemma. A normalized linear combination of two sequences of states
localized at $b$ is localized at $b$. More generally, let
$(\alpha_{j})\in\ell^{1}$ and, for every $j\in\mathbb{N}$, let
$(\varphi_{jn})$ be a sequence of states localized at $b$. Suppose that
$\varphi_{n}:=c_{n}\sum_{j}\alpha_{j}\varphi_{jn}$ with
$\sup_{n}|c_{n}|<\infty$ is normalized for every $n$. Then $(\varphi_{n})$ is
localized at $b$.

Proof. Let $B$ be a ball around $b$. One has to show that
$||\sqrt{T(\mathbb{R}^{3}\setminus
B)}\varphi_{n}||^{2}=\langle\varphi_{n},T(\mathbb{R}^{3}\setminus
B)\varphi_{n}\rangle\to 0$. Now, $||\sqrt{T(\mathbb{R}^{3}\setminus
B)}\varphi_{n}||\leq\sum_{j}|\alpha_{j}|\,|c_{n}|\,||\sqrt{T(\mathbb{R}^{3}\setminus
B)}\varphi_{jn}||\leq\sum_{j}|\alpha_{j}|\,c<\infty$ with
$c:=\sup_{n}|c_{n}|$. By assumption
$c>|c_{n}|\,||\sqrt{T(\mathbb{R}^{3}\setminus B)}\varphi_{jn}||\to 0$,
$n\to\infty$ for every $j$. The result follows by dominated convergence.
$\Box$

###  8.2. Dilational covariance

Dilational covariance is a useful tool in order to show that a POM is
separated (cf. [46], [4]).

(19) Definition. A POM $T$ is said to admit dilational covariance if there is
a representation $D$ of the group $\mathbb{R}_{+}$ of dilations satisfying
$D_{\lambda}T(\Delta)D_{\lambda}^{-1}=T(\lambda\Delta)$ for all $\lambda>0$.
Plainly, since PM are special POM, (8.2) concerns also PM. — By [46, Lemma 3]
a POM is separated if it admits dilational covariance. Such POM seem to be
good candidates for the norm-$1$-property, i.e. $T(\Delta)\neq
0\;\Rightarrow\;||T(\Delta)||=1$ (cf. [41] for this property). The following
criterion (8.2) for separated POM generalizes [46, Lemma 3] and applies to
cases like the Dirac electron considered in sec. 13, and the Weyl particles
(16).

(20) Definition. A POM $T$ on $\mathcal{H}$ is said to be the trace of the PM
$E$ on a Hilbert space $\mathcal{K}$, if $\mathcal{H}$ is a subspace of
$\mathcal{K}$ and $T(\Delta)\varphi=PE(\Delta)\varphi$ $\forall$
$\varphi\in\mathcal{H}$, where $P$ is the projection in $\mathcal{K}$ on
$\mathcal{H}$.
One easily checks that the trace of a PM is a POM. It is well known that
actually every POM is the trace of some PM (see [47, Appendix]).

(21) Theorem [4, Theorem 7]. Let the POM $T$ be the trace of a PM $E$. Suppose
that $E$ admits dilation covariance. Suppose further that
$Q:=\text{s-}\lim_{\lambda\to\infty}D_{\lambda}PD_{\lambda}^{-1}$ exists with
$Q\neq 0$. Let $\phi\in\mathcal{K}$ with $Q\phi\neq 0$. Then
$||PD_{\lambda}^{-1}\phi||\to||Q\phi||\neq 0$ and
$\varphi_{\lambda}:=\frac{1}{||PD_{\lambda}^{-1}\phi||}PD_{\lambda}^{-1}\phi$
satisfies

$||T(B)\varphi_{\lambda}||\to 1,\quad\lambda\to\infty$

for every ball $B$ around $0$. In particular, $||T(B)||=1$ and
$(\varphi_{n})$, up to finitely many $n$, is a sequence of states localized at
$0$.

Proof. First,
$||PD_{\lambda}^{-1}\phi||=||D_{\lambda}PD_{\lambda}^{-1}\phi||\to||Q\phi||\neq
0$, whence $PD_{\lambda}^{-1}\phi\neq 0$ for all large $\lambda>0$. — Now, put
$P_{\lambda}:=D_{\lambda}PD_{\lambda}^{-1}$. Keep in mind that $Q^{2}=Q$. Then
$||T(B)PD_{\lambda}^{-1}\phi||=||P_{\lambda}E(\lambda B)P_{\lambda}\phi||$,
and $||P_{\lambda}E(\lambda
B)P_{\lambda}\phi-Q\phi||\leq||P_{\lambda}E(\lambda
B)\big{(}P_{\lambda}\phi-Q\phi\big{)}||+||P_{\lambda}\big{(}E(\lambda
B)-I\big{)}Q\phi||+||\big{(}P_{\lambda}-Q\big{)}Q\phi||\leq||P_{\lambda}\phi-Q\phi||+||\big{(}E(\lambda
B)-I\big{)}Q\phi||+||\big{(}P_{\lambda}-Q\big{)}Q\phi||\to 0$ for
$\lambda\to\infty$, since $E(\lambda B)\to I$ strongly. This proves
$||T(B)PD_{\lambda}^{-1}\phi||\to||Q\phi||$. $\Box$

For the proof of (8.2) the existence of the limit $Q$ is crucial. — In (8.2),
in case of a POL, $T$ is separated because of translational covariance.
Moreover, by translating $(\varphi_{n})$ in (8.2) one gets a sequence of
states localized at any given point. If $E$ in (8.2) is Euclidean covariant,
i.e., if $E$ is a WL, then $E$ automatically admits dilation covariance [46,
sec. 2]. This will be exploited for the Dirac system and the Weyl systems, see
sec. 14 and (16).

###  8.3. Decay of spatial probability

Acording to Hegerfeldt 1985 [48] the spatial probability in any state
$\varphi$ satisfies

$\langle\varphi,T(\\{x\in\mathbb{R}^{3}:\,|x|>r\\})\varphi\rangle\notin\mathcal{O}(\operatorname{e}^{-Kr}),\quad
r\to\infty$ (8.2)

for $K>2\frac{mc}{\hbar}$, if the causal time evolution for the POL $T$ is
relativistic with mass $m\geq 0$ and semi-bounded energy. So causal time
evolution not only forbids localized states in non-essentially dense regions
but requires also a limited exponential decay of the spatial probability. The
limit is determined by the Compton wavelength $\lambda_{C}=\frac{h}{mc}$. This
is an interesting behavior of free relativistic systems. But certainly it is
not a paradox as sometimes it is referred to in the literature, neither does
it mean that “arbitrarily good localization” [17, sec. 5] is impossible. We
consider in sec. 14 the POL $T^{e}$ for the Dirac electron and in (16) the POL
$T^{\chi\eta}$ for the Weyl particles, which are causal and separated.

There are myriads of inequivalent POL for massive as well massless systems.
The POL studied for the two-component photon in [31], [32] are not causal and
have states localized in bounded regions. POL for massless irreducible systems
with non-zero helicity never have states localized in bounded regions. This is
shown in [31, (2.5)] adapting the proof in Galindo 1968 [49, sec. 3]. For the
latter systems, a complete description of POL admitting dilation covariance
can be given, see Castrigiano 1981 [46, Theorem].
Up to here there are two related problems concerning POL for irreducible
relativistic systems, viz., their great variety and their unknown nature.
However, as we will see, imposing causality will lead to uniqueness and
furnish an interpretation of the POL.

##  9\. Causal Systems

We turn to ($C2$) from sec. 5. Within this frame there is no a-priori-
objection to returning to PM describing localizability. In the following
definition of a causal system we resume the considerations in sec. 3.1, 6.2
and (7.2) on Poincaré covariance and causality of localization.

(22) Definition. Let $W$ be a representation of $\tilde{\mathcal{P}}$
describing a relativistic quantum system. Let $E$ be a causal WL for $W$
according to (7.2). Then $(W,E)$ is called a causal system.999 Let us note
that Poincaré covariant WL with causal time evolution (sec. 7.1) are called
causal systems in [4], whereas causal systems according to (9) are not
considered in [4]. The present notations leave the name causal system to the
more fundamental concept.

Recall that $E$ in (9) is a map on the set $\mathfrak{S}$ of all measurable
subsets of spacelike hyperplanes, which is a PM on every spacelike hyperplane
and which is Poincaré covariant, viz. $W(g)E(\Gamma)W(g)^{-1}=E(g\cdot\Gamma)$
for all $\Gamma\in\mathfrak{S}$, $g\in\tilde{\mathcal{P}}$, and which is
causal, viz.

$E(\Gamma)\leq E(\Gamma_{\sigma})$

for all $\Gamma\in\mathfrak{S}$ and spacelike hyperplanes $\sigma$. Recall
that
$\Gamma_{\sigma}=\\{\mathfrak{x}\in\sigma:(\mathfrak{x}-\mathfrak{z})^{\cdot
2}\geq 0\textrm{ for some }\mathfrak{z}\in\Gamma\\}$ is the region of
influence in $\sigma$ of $\Gamma$, see Eq. (7.2). Recall also that by (7.2)(f)
$E$ is causal if and only if it is locally orthogonal, i.e.,

$E(\Gamma)\,E(\Gamma^{\prime})=0$

for all spacelike separated $\Gamma,\Gamma^{\prime}\in\mathfrak{S}$.

The trace (8.2) of a causal WL on a group-invariant subspace clearly yields a
causal POL. This is worth to be stated explicitly.

(23) Lemma. Let $\mathcal{K}$ be a Hilbert space and $\mathcal{H}$ a Hilbert
subspace of $\mathcal{K}$. Let $(W,E)$ be a causal system in $\mathcal{K}$.
Suppose that $\mathcal{H}$ is invariant under $W$. Then the trace $T$ of $E$
on $\mathcal{H}$ is a causal POL for the subrepresentation $W^{\prime}$of $W$
on $\mathcal{H}$.
As mentioned after (8.2), every POM is the trace of a PM. Similarly, every POL
is the trace of a WL (see e.g. [35], [31], [36], [37]). We do not know whether
every causal POL is the trace of a causal WL.

##  10\. A Property of Causal Systems

Let $(W,E)$ be a causal system (9). By causality
$E(\Delta_{\tau})-E(\Delta)\geq 0$ holds for all $\Delta\in\mathfrak{S}$ and
spacelike hyperplanes $\tau$. Recall that
$\Delta_{\tau}:=\\{\mathfrak{x}\in\tau:(\mathfrak{x}-\mathfrak{z})\cdot(\mathfrak{x}-\mathfrak{z})\geq
0\textrm{ for some }\mathfrak{z}\in\Delta\\}$ is the region of influence in
$\tau$ of $\Delta$. We study the particular case where $\Delta$ is an open
spacelike half-hyperplane $\sigma^{>}$. We write
$\sigma^{>}_{\tau}:=(\sigma^{>})_{\tau}$.
The section heading refers to the results (10) – (10) concerning
$E(\sigma^{>}_{\tau})-E(\sigma^{>})$. The proposition (10) implies the no-go
result in (12)(c) about the representability of the lattice of causally
complete regions, and (10)(f) is needed for the proof of the Lorentz
contraction of the Dirac wave-functions in (13.3) and of the Weyl wave-
functions (16.6).

Consider a spacelike hyperplane $\sigma$ and let $\pi\subset\sigma$ be a
plane. Let $\sigma^{>}$ denote one of the two open half-hyperplanes
constituting $\sigma\setminus\pi$.101010 To be explicit write
$\sigma=\\{\mathfrak{x}:\mathfrak{x}\cdot\mathfrak{e}_{\sigma}=\rho_{\sigma}\\}$
for unique $\mathfrak{e}_{\sigma}=(1,e_{\sigma})$ with $|e_{\sigma}|<1$ and
$\rho_{\sigma}\in\mathbb{R}$ (cf. after (11.5)). Moreover, there are unique
$e\in\mathbb{R}^{3}$ with $|e|=1$ and $\rho\in\mathbb{R}$ such that
$\pi=\\{\mathfrak{x}\in\sigma:xe=\rho\\}$ and
$\sigma^{>}=\\{\mathfrak{x}\in\sigma:xe>\rho\\}$.  Let $\tau$ be a further
spacelike hyperplane.

(24) Lemma. If $\tau$ contains $\pi$ and if $\tau^{>}$ is the half-hyperplane
in $\tau\setminus\pi$, which is at the same side of $\pi$ as
$\sigma^{>}$,111111 $M\subset\mathbb{R}^{4}$ is said to be at the same side of
$\pi$ as $\sigma^{>}$ if
$M\subset\Pi^{\sigma,>}:=\\{\mathfrak{x}:\frac{e_{\sigma}e}{1-e^{2}_{\sigma}}(\mathfrak{x}\cdot\mathfrak{e}_{\sigma}-\rho_{\sigma})-xe>\rho\\}$,
which means that the spatial component of every $\mathfrak{y}\in M$ lies in
$\sigma^{>}$. Indeed, note that
$\hat{\mathfrak{e}}_{\sigma}:=(1-e^{2}_{\sigma})^{-1/2}\mathfrak{e}_{\sigma}$
is the direction of time referring to $\sigma$, as
$\hat{\mathfrak{e}}_{\sigma}^{\cdot 2}=1$ and
$\hat{\mathfrak{e}}_{\sigma}\cdot(\mathfrak{x}-\mathfrak{x}^{\prime})=0$ for
all $\mathfrak{x},\mathfrak{x}^{\prime}\in\sigma$, and that
$\mathfrak{y}-s\hat{\mathfrak{e}}_{\sigma}$ lies in $\sigma$ exactly for
$s=(1-e^{2}_{\sigma})^{-1/2}(\mathfrak{y}\cdot\mathfrak{e}_{\sigma}-\rho_{\sigma})$.
— Note that $\\{\mathfrak{y}\\}$ is at the same side of $\pi$ as $\sigma^{>}$
if the timelike line $l=\mathfrak{y}+\mathbb{R}\mathfrak{e}_{\sigma}$
intersects $\sigma^{>}$. then
$\tau^{>}_{\sigma}=\sigma^{>},\quad\sigma^{>}_{\tau}=\tau^{>}$ and
$E(\tau^{>})=E(\sigma^{>})$. In particular,
$E(\sigma^{>}_{\tau})-E(\sigma^{>})=0$.

Proof. Because of Poincaré symmetry it satisfies to check
$\tau^{>}_{\sigma}=\sigma^{>}$, $\sigma^{>}_{\tau}=\tau^{>}$ for the half-
hyperplanes of $\sigma:=\\{\mathfrak{x}:x_{0}=0\\}$ and
$\tau:=\\{\mathfrak{x}:x_{0}-\alpha x_{3}=0\\}$, $\alpha\in]0,1[$ determined
by $x_{3}>0$.121212 Explicitly one has
$\sigma=g\cdot\\{\mathfrak{x}:x_{0}=0\\}$ for $g\in\tilde{\mathcal{P}}$ if and
only if $g=\big{(}(\rho_{\sigma},0),Q(\mathfrak{e}_{\sigma})\big{)}(b,B)$ with
arbitrary $(b,B)\in ISU(2)$ and $Q$ the canonical cross section. Put
$\mathfrak{d}=(d_{0},d):=Q(\mathfrak{e}_{\sigma})^{-1}\cdot(0,e)$. Note
$\mathfrak{d}^{\cdot 2}=-1$, whence $|d|^{2}=1+d_{0}^{2}>0$. Then, in addition
to $\sigma=g\cdot\\{\mathfrak{x}:x_{0}=0\\}$, one has also
$\pi=g\cdot\\{\mathfrak{x}:x_{0}=x_{3}=0\\}$ if and only if $bd=\rho$,
$B^{-1}\cdot d=\pm|d|(0,0,1)$. If $(b,B)$ is specified in this way then one
checks $\Pi^{\sigma,>}=g\cdot\\{\mathfrak{x}:x_{3}>0\\}$ using
$d_{0}=(1-e^{2}_{\sigma})^{-1/2}e_{\sigma}e$, and one finds
$\tau=g\cdot\\{\mathfrak{x}:x_{0}-\alpha x_{3}=0\\}$ for some $|\alpha|<1$. If
necessary then replace $B$ by $B\operatorname{i}\sigma_{2}$ to ensure
$\alpha\geq 0$, as $\operatorname{i}\sigma_{2}$ acts on $\mathbb{R}^{4}$ by
$\operatorname{diag}(1,-1,1,-1)$. So $E(\tau^{>})=E(\sigma^{>})$ holds by
causality (7.2)(a) applied to $\sigma^{>}$ and $\tau^{>}$.∎

Analogous equalities hold true for the closed half-hyperplanes
$\sigma^{\geq}$, $\tau^{\geq}$. Since the closed and respective open half-
hyperplanes differ by a Lebesgue null set only, one has also
$E(\sigma^{>})=E(\sigma^{\geq})$, $E(\tau^{>})=E(\tau^{\geq})$. — Note that
$\tau=g\cdot\sigma$ holds for some $g\in\tilde{\mathcal{P}}$ satisfying
$g\cdot\pi=\pi$. Then $W(g)$ commutes with $E(\sigma^{>})$ by Eq. (3.1) as
$E(\sigma^{>})=E(g\cdot\sigma^{>})$.

Now we turn to the more interesting case that $\sigma$ and $\tau$ do not
intersect.131313 Explicitly,
$\sigma=\\{\mathfrak{x}:\mathfrak{x}\cdot\mathfrak{e}=\rho_{\sigma}\\}$,
$\tau=\\{\mathfrak{x}:\mathfrak{x}\cdot\mathfrak{e}=\rho_{\tau}\\}$ with
$\mathfrak{e}=(1,e)$, $|e|<1$. Then $\sigma=g\cdot\\{\mathfrak{x}:x_{0}=0\\}$
and $\tau=g\cdot\\{\mathfrak{x}:x_{0}=t\\}$ with
$t:=(1-e^{2}_{\sigma})^{-1/2}(\rho_{\tau}-\rho_{\sigma})$ for
$g=\big{(}(\rho_{\sigma},0),Q(\mathfrak{e})\big{)}(b,B)$ with arbitrary
$(b,B)\in ISU(2)$.

(25) Lemma. If $\tau$ does not intersect $\sigma$, and if
$g\in\tilde{\mathcal{P}}$ is any rotation satisfying $g\cdot\pi=\pi$ and
$g\cdot\sigma^{>}=\sigma\setminus\sigma^{\geq}$, then

$\big{(}E(\sigma^{>}_{\tau})-E(\sigma^{>})\big{)}+\big{(}E(g\cdot\sigma^{>}_{\tau})-E(g\cdot\sigma^{>})\big{)}=E(\sigma^{>}_{\tau}\cap(g\cdot\sigma^{>}_{\tau}))$

and the eigenspaces of the non-trivial projection
$E(\sigma^{>}_{\tau}\cap(g\cdot\sigma^{>}_{\tau}))$ are infinite dimensional.

Proof. Note
$E(g\cdot\sigma^{>})=E(\sigma\setminus\sigma^{\geq})=I-E(\sigma^{\geq})=I-E(\sigma^{>})$,
$I-E(\sigma^{>}_{\tau})=E(\tau\setminus\sigma^{>}_{\tau})$, and
$(g\cdot\sigma^{>}_{\tau})\supset(\tau\setminus\sigma^{>}_{\tau})$. As to the
latter, it suffices to check that
$\sigma^{>}_{\tau}=\\{\mathfrak{x}:x_{0}=\alpha,x_{3}>-|\alpha|\\}$ and
$g\cdot\sigma^{>}_{\tau}=\\{\mathfrak{x}:x_{0}=\alpha,x_{3}<|\alpha|\\}$ for
$\sigma^{>}=\\{\mathfrak{x}:x_{0}=0,x_{3}>0\\}$,
$\tau=\\{\mathfrak{x}:x_{0}=\alpha\\}$, $\alpha\in\mathbb{R}\setminus\\{0\\}$,
and $g=\operatorname{i}\sigma_{2}$, which acts on $\mathbb{R}^{4}$ by
$\operatorname{diag}(1,-1,1,-1)$. This shows in addition that neither
$(g\cdot\sigma^{>}_{\tau})\setminus(\tau\setminus\sigma^{>}_{\tau})=(g\cdot\sigma^{>}_{\tau})\cap\sigma^{>}_{\tau}$
nor its complement is a Lebesgue null set. Hence the eigenspaces of
$E(\sigma^{>}_{\tau}\cap(g\cdot\sigma^{>}_{\tau}))$ are infinite dimensional.
∎

Using obvious notations the formula in (10) reads more easily as

$\big{(}E(\sigma^{>}_{\tau})-E(\sigma^{>})\big{)}+\big{(}E(\sigma^{<}_{\tau})-E(\sigma^{<})\big{)}=E(\sigma^{>}_{\tau}\cap\sigma^{<}_{\tau})$
(10.1)

due to the relations $g\cdot\sigma^{>}=\sigma^{<}$ and
$g\cdot\sigma^{>}_{\tau}=\sigma^{<}_{\tau}$. Moreover, remember

$E(\sigma^{<}_{\tau})-E(\sigma^{<})=W(g)\big{(}E(\sigma^{>}_{\tau})-E(\sigma^{>})\big{)}W(g)^{-1}$
(10.2)

with $g$ from (10).

(26) Proposition. Let $\sigma$, $\tau$ be two non-intersecting spacelike
hyperplanes. Then the eigenspaces of the non-trivial projection
$E(\sigma^{>}_{\tau})-E(\sigma^{>})$ are infinite dimensional. The same holds
true for the projection $E(\sigma^{<}_{\tau})-E(\sigma^{<})$. Moreover,
$E(\sigma^{>}_{\tau})-E(\sigma^{>})$ and $E(\sigma^{<}_{\tau})-E(\sigma^{<})$
are orthogonal.

Proof. Since $E(\sigma^{>}_{\tau})\geq E(\sigma^{>})$ by causality,
$P^{<}:=E(\sigma^{>}_{\tau})-E(\sigma^{>})\geq 0$ is a projection. Analogously
define $P^{>}$. Let $g\in\tilde{\mathcal{P}}$ be as in (10). Due to Eq. (3.1),
$E(g\cdot\sigma^{>}_{\tau})-E(g\cdot\sigma^{>})=W(g)P^{<}W(g)^{-1}$ is a
projection, too. By (10) their sum is a projection with infinite dimensional
eigenspaces. Therefore $P^{<}$ and $W(g)P^{<}W(g)^{-1}$ are orthogonal and
none of the four eigenspaces is finite dimensional. It remains to note that
$P^{>}=W(g)P^{<}W(g)^{-1}$. ∎

$\sigma^{<}$$\tau\setminus\sigma^{>}_{\tau}$$\sigma^{>}$$\tau\setminus\sigma^{<}_{\tau}$$\tau$$\sigma$

Recall $E(\sigma^{>}_{\tau})\geq E(\sigma^{>})$. Hence

$E(\sigma^{>}_{\tau})-E(\sigma^{>})=E(\sigma^{>}_{\tau})E(\sigma^{<})$ (10.3)

as well the analogous formula, where $<$ and $>$ are interchanged.
We are going to characterize the ranges of these projections. Let
$\mathfrak{e}\in\mathbb{R}^{4}$ determine the time coordinate for the
spacelike hyperplane $\sigma$. It satisfies $\mathfrak{e}\cdot\mathfrak{e}=1$,
$\mathfrak{e}\cdot(\mathfrak{x}-\mathfrak{x}^{\prime})=0$ for all
$\mathfrak{x},\mathfrak{x}^{\prime}\in\sigma$, and $e_{0}>0$. $\mathfrak{e}$
is uniquely determined. Then $\tau=\alpha\mathfrak{e}+\sigma$ for some unique
$\alpha\in\mathbb{R}$. Let $\tau^{>}$ be determined with respect to the plane
$\alpha\mathfrak{e}+\pi$ in $\tau$ such that
$\tau^{>}=\alpha\mathfrak{e}+\sigma^{>}$. Then
$\sigma^{>}_{\tau}=\alpha\mathfrak{e}+\tau^{>}_{\sigma}$. By Poincaré symmetry
it suffices to check all the foregoing relations for
$\sigma=\mathfrak{a}+\\{\mathfrak{x}:x_{0}=0\\}$,
$\sigma^{>}=\mathfrak{a}+\\{\mathfrak{x}:x_{0}=0,x_{3}>0\\}$,
$\mathfrak{e}=(1,0,0,0)$, and
$\tau=\mathfrak{a}+\\{\mathfrak{x}:x_{0}=\alpha\\}$,
$\mathfrak{a}\in\mathbb{R}^{4}$. As a consequence one has

$E(\sigma^{>}_{\tau})=W(\alpha\mathfrak{e})E(\tau^{>}_{\sigma})W(\alpha\mathfrak{e})^{-1}$
(10.4)

as well the formula for $>$ replaced with $<$.

(27) Proposition. Let $\sigma$, $\tau$ be two non-intersecting spacelike
hyperplanes. Then a state $\varphi$ is in the range of
$E(\sigma^{>}_{\tau})-E(\sigma^{>})$ if and only if $\varphi$ is localized in
$\sigma^{<}$ and $W(-\alpha\mathfrak{e})\varphi$ is localized in
$\tau^{>}_{\sigma}$. The analogous statement with $>$ replaced with $<$ holds
true, too.

Proof. By Eqs. (10.3), (10.4) $\varphi$ is in the range of
$E(\sigma^{>}_{\tau})-E(\sigma^{>})$ if and only if $\varphi$ is in the range
of $E(\sigma^{<})$ and in the range of
$W(\alpha\mathfrak{e})E(\tau^{>}_{\sigma})W(\alpha\mathfrak{e})^{-1}$. The
proof is easily accomplished.∎

Recall $\tau=\alpha\mathfrak{e}+\sigma$ with $\alpha\in\mathbb{R}$. By
causality $E(\sigma^{>}_{\tau})\leq E(\sigma^{>}_{\tau^{\prime}})$ if
$0\leq\alpha\leq\alpha^{\prime}$ or $\alpha^{\prime}\leq\alpha\leq 0$. Hence
the strong limit
$P^{<}_{\sigma}:=\lim_{\alpha\to\infty}\big{(}E(\sigma^{>}_{\tau})-E(\sigma^{>})\big{)}=\lim_{\alpha\to\infty}E(\sigma^{>}_{\tau})E(\sigma^{<})$
exists. Analogously define $P^{>}_{\sigma}$. For the strong limits for
$\alpha\to-\infty$ see (10)(g).

(28) Proposition. Let $\sigma$ be a spacelike hyperplane. Let
$\tau:=\alpha\mathfrak{e}+\sigma$ with $\alpha\in\mathbb{R}$ and
$\mathfrak{e}$ the four-vector of time with respect to $\sigma$. Recall
$P^{<}_{\sigma}=\lim_{\alpha\to\infty}E(\sigma^{>}_{\tau})E(\sigma^{<})$ and
$P^{>}_{\sigma}=\lim_{\alpha\to\infty}E(\sigma^{<}_{\tau})E(\sigma^{>})$. Then

  * (a)

the eigenspaces of $P^{<}_{\sigma}$ and $P^{>}_{\sigma}$ are infinite
dimensional

  * (b)

$P^{<}_{\sigma}\leq E(\sigma^{<})$, $P^{>}_{\sigma}\leq E(\sigma^{>})$

  * (c)

$P^{<}_{\sigma}P^{>}_{\sigma}=0$

  * (d)

$P^{>}_{\sigma}=W(g)P^{<}_{\sigma}W(g)^{-1}$ with $g$ from _( 10)_.

  * (e)

$P^{<}_{\sigma}+P^{>}_{\sigma}=\lim_{\alpha\to\infty}E(\sigma^{>}_{\tau}\cap\sigma^{<}_{\tau})=\lim_{\alpha\to\infty}W(\alpha\mathfrak{e})E(\tau^{>}_{\sigma}\cap\tau^{<}_{\sigma})W(\alpha\mathfrak{e})^{-1}$

  * (f)

$P^{<}_{\sigma}=E(\sigma^{<})$ $\Leftrightarrow$
$P^{>}_{\sigma}=E(\sigma^{>})$ $\Leftrightarrow$
$P^{<}_{\sigma}+P^{>}_{\sigma}=I$ $\Leftrightarrow$
$\not\exists$ state $\varphi$ satisfying
$E(\tau^{>}_{\sigma}\cap\tau^{<}_{\sigma})W(\alpha\mathfrak{e})^{-1}\varphi=0$
$\forall$ $\alpha>0$ $\Leftrightarrow$
$\not\exists$ state $\varphi$ satisfying
$E(\tau^{>}_{\sigma})W(\alpha\mathfrak{e})^{-1}\varphi=0$ $\forall$ $\alpha>0$
$\Leftrightarrow$
$\not\exists$ state $\varphi$ satisfying
$E(\tau^{<}_{\sigma})W(\alpha\mathfrak{e})^{-1}\varphi=0$ $\forall$ $\alpha>0$

  * (g)

Analogously, the strong limits for $\alpha\to-\infty$ exist and they satisfy
_(a) – (f)_ for $\alpha<0$ and $\alpha\to-\infty$.

Proof. Keep the definition of $P^{\lessgtr}_{\sigma}$ in mind. (g) is obvious.
Now (a) follows from (10), Eq. (10.3). (b) is obvious. (c) holds by (b). One
has (d) due to Eq. (10.2). (e) follows from Eqs. (10.1), (10.3), (10.4).
We turn to (f). The first two $\Leftrightarrow$ hold by (b), (d). As
$W(g)E(\tau^{>}_{\sigma})W(\alpha\mathfrak{e})^{-1}W(g)^{-1}\varphi=W(g)E(\tau^{>}_{\sigma})W(g)^{-1}W(\alpha\mathfrak{e})^{-1}\varphi=E(\tau^{<}_{\sigma})W(\alpha\mathfrak{e})^{-1}\varphi$,
also the last $\Leftrightarrow$ holds true. — As to the forth
$\Leftrightarrow$ , the direction $\Rightarrow$ is obvious. For the proof of
the reverse direction assume the contrary and define
$\varphi^{<}:=E(\sigma^{<})\varphi$, $\varphi^{>}:=E(\sigma^{>})\varphi$. Then
$\varphi=\varphi^{<}+\varphi^{>}$, whence $\varphi^{<}\neq 0$ or
$\varphi^{>}\neq 0$. Let
$Q:=W(\alpha\mathfrak{e})^{-1}E(\sigma^{<})W(\alpha\mathfrak{e})$. By
causality $I-E(\tau^{>}_{\sigma})\leq Q\leq E(\tau^{<}_{\sigma})$. Therefore
these three projections commute and
$E(\tau^{>}_{\sigma})W(\alpha\mathfrak{e})^{-1}\varphi^{<}=E(\tau^{>}_{\sigma})QW(\alpha\mathfrak{e})^{-1}\varphi=QE(\tau^{>}_{\sigma})W(\alpha\mathfrak{e})^{-1}\varphi=E(\tau^{<}_{\sigma})QE(\tau^{>}_{\sigma})W(\alpha\mathfrak{e})^{-1}\varphi=QE(\tau^{<}_{\sigma}\cap\tau^{>}_{\sigma})W(\alpha\mathfrak{e})^{-1}\varphi=0.$
Similarly $E(\tau^{<}_{\sigma})W(\alpha\mathfrak{e})^{-1}\varphi^{>}=0$ holds.
Hence the assertion follows, using the last $\Leftrightarrow$ in case that
$\varphi^{<}=0$.
It remains to prove the third $\Leftrightarrow$ of (f). The implication
$\Rightarrow$ is obvious by (e). As to $\Leftarrow$, suppose
$P^{<}_{\sigma}+P^{>}_{\sigma}<I$. Let
$V:=\\{\varphi:\big{(}I-P(\alpha)\big{)}\varphi_{-\alpha}\to
0,\alpha\to\infty\\}$, where
$P(\alpha):=E(\tau^{>}_{\sigma}\cap\tau^{<}_{\sigma})$,
$\varphi_{-\alpha}:=W(\alpha\mathfrak{e})^{-1}\varphi$. Check
$V=\operatorname{ran}(P^{<}_{\sigma}+P^{>}_{\sigma})$. Then
$V^{\perp}\neq\\{0\\}$ by (e). Let $\varphi^{*}\in V^{\perp}\setminus\\{0\\}$.
Let $\beta>0$ and $\chi\in\operatorname{ran}P(\beta)$. We claim
$\chi_{\beta}\in V$. Indeed, by causality
$\chi_{\beta-\alpha}\in\operatorname{ran}P(\beta+|\alpha-\beta|)$ for
$\alpha\geq 0$, whence $\chi_{\beta-\alpha}\in\operatorname{ran}P(\alpha)$ for
$\alpha\geq\beta$ and hence $\chi_{\beta}\in V$. So one infers
$0=\langle\varphi^{*},\chi_{\beta}\rangle=\langle\varphi^{*}_{-\beta},\chi\rangle$
for all $\chi\in\operatorname{ran}P(\beta)$, $\beta>0$. This means
$\varphi^{*}_{-\beta}\in\operatorname{ran}\big{(}I-P(\beta)\big{)}$ and hence
$P(\beta)\varphi^{*}_{-\beta}=0$ for all $\beta>0$. ∎

To comment on (10) consider $\sigma=\\{\mathfrak{x}:x_{0}=0\\}$,
$\sigma^{>}=\\{\mathfrak{x}:x_{0}=0,x_{3}>0\\}$, $\mathfrak{e}=(1,0,0,0)$,
$\tau=\\{\mathfrak{x}:x_{0}=\alpha\\}$, and
$\tau^{<}_{\sigma}=\\{x_{0}=0,x_{3}<|\alpha|\\}$. The question raised by the
last item of (f) is whether there exists a state $\varphi$ such that the
states $W(-t)\varphi$ evolved in time are localized in $\\{x_{0}=0,x_{3}>t\\}$
for $t>0$. In case of the Dirac system, (13.2.2) shows that such a state does
not exist, thus proving $I=P^{<}_{\sigma}+P^{>}_{\sigma}$ by (10)(f). In case
of the Weyl systems, $I=P^{<}_{\sigma}+P^{>}_{\sigma}$ holds as well. However
its proof in (16.2) (see the comments to (16.2) and (16.2)) requires a
different method. There is an interesting alternative proof of
$I=P^{<}_{\sigma}+P^{>}_{\sigma}$ for Dirac and Weyl systems. It arises from
group representation theoretical considerations leading to farther-reaching
results, see (20).

##  11\. The Lattice of Causally Complete Regions

The lattice $\mathcal{M}$ of causally complete regions of Minkowski space,
generated and orthocomplemented by the spacelike relation

$\mathfrak{x}\perp\mathfrak{y}\quad\Leftrightarrow\quad(\mathfrak{x}-\mathfrak{y})^{\cdot
2}<0$

is well-known from local quantum physics. In view of Poincaré covariance and
local orthogonality (equivalent to causality by (7.2)) one may ask whether a
causal system $(W,E)$ gives rise to a representation (rep) $(W,F)$ of
$\mathcal{M}$ by orthogonal projections on the space of states. The properties
to impose on $F$ are Poincaré covariance with respect to $W$, normalization
$F(\emptyset)=0$, $F(\mathbb{R}^{4})=I$, monotony, local orthogonality, and
$\sigma$-orthoadditivity. It is important to point out that $F$ is not
required to be a lattice homomorphism of $\mathcal{M}$ into the lattice of the
projections on the Hilbert space of states.
One is tempted to think such a rep $(W,F)$ to be possible just because of the
fact that the energy need not be semi-bounded. But actually is does not exist,
see (12) and (12)(c).
In local quantum theory to every finite open contractible region $M$ a von
Neumann ring $\mathcal{R}(M)$ is assigned. The causality principle relates the
orthocomplement $M^{\perp}$ to the commutant $\mathcal{R}(M)^{\prime}$. So in
Haag 1992 [51, III.4.2.1] it is postulated tentatively that for the vacuum
sector the assignment $\mathcal{R}$ actually extends to a homomorphism from
$\mathcal{M}$ into the lattice of von Neumann rings on a Hilbert space.
Obviously, the tentative to construct a rep $F$ describing localization within
the frame of local quantum theory may be inspired by these considerations. But
a localization operator $E(\Delta)$ in general cannot be regarded to be an
element of $\mathcal{R}(\hat{\Delta})$ by [4, Lemma 4] just because of the
semi-boundedness of energy.
Before showing the mentioned no-go results we like to study $\mathcal{M}$ in
some detail. Many examples at the level of exercises serve to become more
familiar with $\mathcal{M}$ and can be skipped. Only (11.5) is needed in the
sequel. In particular it shows that there are maximal spacelike sets, which
are not bases. In (11.3) it will be proven that causally complete sets are
automatically measurable, i.e., Lebesgue measurable subsets of
$\mathbb{R}^{4}$. Regarding the definition of $\mathcal{M}$ we follow [51,
III.4.1 The Lattice of Causally Complete Regions].

###  11.1. Causal completion

Let $M$ be any subset of Minkowski space and let $M^{\perp}$ denote the causal
complement of $M$ consisting of all points, which lie spacelike to all points
of $M$, i.e.,

$M^{\perp}:=\\{\mathfrak{x}:(\mathfrak{x}-\mathfrak{y})^{\cdot
2}<0\;\forall\,\mathfrak{y}\in M\\}$ (11.1)

Clearly, $M\cap M^{\perp}=\emptyset$, and $M^{\perp}\subset L^{\perp}$ if
$L\subset M$. The set $\hat{M}:=(M^{\perp})^{\perp}$ is called the causal
completion of $M$. If $M=\hat{M}$ then $M$ is said to be causally complete.

Clearly, $\emptyset$ and $\mathbb{R}^{4}$ are causally complete. Any causal
complement $M^{\perp}$ is causally complete and, equivalently,
$M^{\perp}=\hat{M}^{\perp}$ holds. (Indeed, $M\subset\hat{M}$ implies
$M^{\perp}\supset\hat{M}^{\perp}=M^{\perp\perp\perp}=(M^{\perp})^{\land}\supset
M^{\perp}$.) Therefore $(\hat{M})^{\land}=\hat{M}$. For any family
$(M_{\iota})_{\iota}$ of sets one has
$\big{(}\bigcup_{\iota}M_{\iota}\big{)}^{\perp}=\bigcap_{\iota}M_{\iota}^{\perp}$,
as both sides describe the set of points lying spacelike to all $M_{\iota}$.
This implies $\big{(}\bigcup
M_{\iota}^{\perp}\big{)}^{\perp}=\bigcap_{\iota}\hat{M}_{\iota}$, which proves
that the intersection of any family of causally complete sets is causally
complete. Moreover,
$\big{(}\bigcup_{\iota}M_{\iota}\big{)}^{\land}=\big{(}\bigcap_{\iota}M_{\iota}^{\perp}\big{)}^{\perp}$
and
$\big{(}\bigcup_{\iota}M_{\iota}^{\perp}\big{)}^{\land}=\big{(}\bigcap_{\iota}\hat{M}_{\iota}\big{)}^{\perp}$
follow. Finally one has
$\big{(}\bigcup_{\iota}M_{\iota}\big{)}^{\land}=\big{(}\bigcup_{\iota}\hat{M}_{\iota}\big{)}^{\land}$.
(Indeed, the inclusion $\subset$ holds by monotony of causal completion.
Conversely, still by this monotony,
$\big{(}\bigcup_{\iota}M_{\iota}\big{)}^{\land}\supset\hat{M}_{\kappa}$ for
every $\kappa$. Hence
$\big{(}\bigcup_{\iota}M_{\iota}\big{)}^{\land}\supset\bigcup_{\kappa}\hat{M}_{\kappa}$,
where $\supset$ still holds for the causal completion of the right hand side.)
Note, however, that in general $(M_{1}\cap
M_{2})^{\perp}\neq(M_{1}^{\perp}\cup M_{2}^{\perp})^{\land}$. (Indeed, let
$M_{1}:=\\{\mathfrak{x}:x_{0}-x_{3}=0\\},M_{2}:=\\{\mathfrak{x}:x_{0}+x_{3}=0\\}$.
Then $M_{1}^{\perp}=M_{2}^{\perp}=\emptyset$ and $M_{1}\cap
M_{2}=\\{x_{0}=x_{3}=0\\}$, whence $(M_{1}\cap
M_{2})^{\perp}=\\{|x_{0}|<|x_{3}|\\}$ and $(M_{1}^{\perp}\cup
M_{2}^{\perp})^{\land}=\emptyset$.)
Moreover, in general, $M_{1}\cup M_{2}$ is not causally complete, even if
$M_{1}$, $M_{2}$ are causally complete spacelike separated. (Indeed, let
$M_{1}:=\\{\mathfrak{x}:|x_{0}|\leq-x_{3}\\}$,
$M_{2}:=\\{\mathfrak{x}:|x_{0}|<x_{3}\\}$. Then, by (11.1)(b), $M_{1}=h(\geq
0)\cap k(\leq 0)$, $M_{2}=h(<0)\cap k(>0)$, whence $M_{1}=M_{2}^{\perp}$ are
causally complete spacelike separated. As $M_{1}\cup M_{2}$ contains the
spacelike hyperplane $\\{\mathfrak{x}:x_{0}=0\\}$, $(M_{1}\cup
M_{2})^{\land}=\mathbb{R}^{4}$ follows.)

(29) Lemma. The one-point sets $\\{\mathfrak{x}\\}$ are causally complete.

Proof. It suffices to consider $\mathfrak{x}=0$. Since
$\\{0\\}^{\perp}=\\{\mathfrak{y}:|y_{0}|<|y|\\}$, one has
$\mathfrak{x}\in\\{0\\}^{\land}$ if $|x_{0}-y_{0}|<|x-y|$ for all
$|y_{0}|<|y|$. Then $x=0$ as otherwise $\mathfrak{y}:=(0,x)$ yields the
contradiction $|x_{0}|<0$. Now $x_{0}=0$ follows similarly.∎

(30) Example for the causal completion of a two-point set. Let
$s\in\mathbb{R}$. Then

$\\{0,(s,0,0,0)\\}^{\land}=\\{\mathfrak{x}:|x_{0}-\textnormal{\tiny{$\frac{s}{2}$}}|+|x|\leq|\textnormal{\tiny{$\frac{s}{2}$}}|\\}=\\{\mathfrak{x}:x_{0}=\textnormal{\tiny{$\frac{s}{2}$}},|x|\leq|\textnormal{\tiny{$\frac{s}{2}$}}|\\}^{\land}$

Proof.
$\\{0,(s,0,0,0)\\}^{\land}=(\\{0\\}^{\perp}\cap\\{(s,0,0,0)\\}^{\perp})^{\perp}=(\\{|y_{0}|<|y|\\}\cap\\{|y_{0}-s|<|y|\\})^{\perp}=\\{|y_{0}-\frac{s}{2}|+|\frac{s}{2}|<|y|\\}^{\perp}=\\{\mathfrak{x}:|x_{0}-y_{0}|<|x-y|\textrm{
for all }\mathfrak{y}\textrm{ with
}|y_{0}-\frac{s}{2}|+|\frac{s}{2}|<|y|\\}=\\{\mathfrak{x}:|x_{0}-\frac{s}{2}|+|y_{0}-\frac{s}{2}|<\big{|}|y|-|x|\big{|}\textrm{
for all }\mathfrak{y}\textrm{ with }|y_{0}-\frac{s}{2}|+|\frac{s}{2}|<|y|\\}$,
whence $|x|\leq|\frac{s}{2}|$ and hence the first equality $=$. As to the
second $=$, the inclusion $\supset$ holds, since
$\\{\mathfrak{x}:x_{0}=\frac{s}{2},|x|\leq|\frac{s}{2}|\\}$ is contained in
the left hand side. For the inclusion $\subset$ it suffices to show that
$\\{0,(s,0,0,0)\\}$ is contained in the right hand side. Anticipating (11.2),
consider $\mathfrak{z}=(1,z)$ with $|z|\leq 1$ and note that
$\mathfrak{x}:=\frac{s}{2}\mathfrak{z}$ satisfies
$x_{0}=\frac{s}{2},|x|\leq|\frac{s}{2}|$. Hence the assertion holds true for
$0$. Similarly the assertion is verified for $(s,0,0,0)$.∎

(31) Example for some causally complete sets. Let $\gamma\in\mathbb{R}$.
Consider the half-spaces
$h(\leq\gamma):=\\{\mathfrak{x}:x_{0}-x_{3}\leq\gamma\\}$,
$k(\leq\gamma):=\\{\mathfrak{x}:x_{0}+x_{3}\leq\gamma\\}$ and, generally,
$h(\sim\gamma)$, $k(\sim\gamma)$ for $\sim\,\in\\{<,\,\leq,\,>,\,\geq\\}$.

Obviously, their causal complements $h^{\perp}$, $k^{\perp}$ are empty and,
hence, their causal completions $\hat{h}$, $\hat{k}$ equal $\mathbb{R}^{4}$.
For the causal complement of the intersection of two half-spaces of the type
$h$ and $k$ one finds

  * (a)

$\big{(}h(\sim\gamma)\cap k(\backsim\delta)\big{)}^{\perp}=\emptyset$

if $\sim$ and $\backsim$ are both in $\\{<,\,\leq\\}$ or both in
$\\{>,\,\geq\\}$, and

  * (b)

$\big{(}h(\sim\gamma)\cap
k(\backsim\delta)\big{)}^{\perp}=h(\not\sim\gamma)\cap k(\not\backsim\delta)$

otherwise. It follows that the intersections $h\cap k$ in _(b)_ are causally
complete. Note $h(\not\sim\gamma)=\mathbb{R}^{4}\setminus h(\sim\gamma)$,
$k(\not\backsim\,\delta)=\mathbb{R}^{4}\setminus k(\backsim\delta)$.

Proof. As to (a), for every $\mathfrak{y}\in\mathbb{R}^{4}$ there is
$s\in\mathbb{R}$ such that $(y_{0}+s)-y_{3}<\gamma$ and
$(y_{0}+s)+y_{3}<\delta$. Hence $\mathfrak{x}:=(y_{0}+s,y)\in h(<\gamma)\cap
k(<\delta)$ and $(\mathfrak{x}-\mathfrak{y})^{\cdot 2}=(s,0)^{\cdot
2}=s^{2}\geq 0$. Therefore $\mathfrak{y}\not\in\big{(}h(<\gamma)\cap
k(<\delta)\big{)}^{\perp}$. This proves the cases
$\sim,\backsim\,\in\\{<,\leq\\}$. The cases $\sim,\backsim\,\in\\{>,\geq\\}$
follow analogously. — We turn to (b). Consider the case
$(\sim,\backsim)=(\leq,>)$. Let $\mathfrak{y}\in\big{(}h(\leq\gamma)\cap
k(>\delta)\big{)}^{\perp}$. This means that for every $\mathfrak{x}$
satisfying $x_{0}-x_{3}\leq\gamma$, $x_{0}+x_{3}>\delta$ one has
$|y_{0}-x_{0}|<|y_{3}-x_{3}|$, i.e., either [ $y_{0}-y_{3}<x_{0}-x_{3}$ and
$y_{0}+y_{3}>x_{0}+x_{3}$ ] or [ $y_{0}-y_{3}>x_{0}-x_{3}$ and
$y_{0}+y_{3}<x_{0}+x_{3}$ ]. Obviously this holds true if and only if
$y_{0}-y_{3}>\gamma$ and $y_{0}+y_{3}\leq\delta$. — The other cases follow
analogously.∎

###  11.2. Set of determinacy

For any subset $M$ of Minkowski space its set of determinacy

$\tilde{M}:=\\{\mathfrak{x}:\forall\,\mathfrak{z}\neq 0,\,\mathfrak{z}^{\cdot
2}\geq 0\;\exists\,s\in\mathbb{R}\textrm{ with }\mathfrak{x}+s\mathfrak{z}\in
M\\}$ (11.2)

consists of all points $\mathfrak{x}$ such that every timelike or lightlike
line through $\mathfrak{x}$ meets $M$.

Obviously, $M\subset\tilde{M}$. Further, $\tilde{\tilde{M}}=\tilde{M}$ holds.
(Indeed, let $\mathfrak{x}\in\tilde{\tilde{M}}$. For $\mathfrak{z}\neq
0,\,\mathfrak{z}^{\cdot 2}\geq 0$ there is $s\in\mathbb{R}$ with
$\mathfrak{x}+s\mathfrak{z}\in\tilde{M}$. Hence, there is
$s^{\prime}\in\mathbb{R}$ with
$(\mathfrak{x}+s\mathfrak{z})+s^{\prime}\mathfrak{z}\in M$, whence
$\mathfrak{x}\in\tilde{M}$.) Clearly $\tilde{M}_{1}\subset\tilde{M}_{2}$ if
$M_{1}\subset M_{2}$.

(32) Lemma. $\tilde{M}\subset\hat{M}$.
Proof. Fix $\mathfrak{x}\in\tilde{M}$. Note first that $\mathfrak{x}\not\in
M^{\perp}$, as $\mathfrak{x}+\mathfrak{w}\in M$ for some not spacelike
$\mathfrak{w}$. Now let $\mathfrak{y}\in M^{\perp}$. Then
$\mathfrak{z}:=\mathfrak{x}-\mathfrak{y}\neq 0$. Obviously, it suffices to
show $\mathfrak{z}^{\cdot 2}<0$. Assume the contrary. Then there is
$s\in\mathbb{R}$ with $\mathfrak{x}-s\mathfrak{z}\in M$. This implies the
contradiction $0>(\mathfrak{x}-s\mathfrak{z}-\mathfrak{y})^{\cdot
2}=(1-s)^{2}\mathfrak{z}^{\cdot 2}\geq 0$.∎

Thus, by (11.2), one has $M\subset\tilde{M}=\tilde{\tilde{M}}\subset\hat{M}$,
whence $M=\tilde{M}$ if $M=\hat{M}$. But $M=\tilde{M}$ does not imply that $M$
is causally complete. A simple example is
$M:=\mathbb{R}^{4}\setminus\mathbb{R}\mathfrak{e}$ for
$\mathfrak{e}=(1,0,0,0)$ with $M=\tilde{M}$ and $\hat{M}=\mathbb{R}^{4}$.
(Indeed, $M=\tilde{M}$ is obvious. Next note
$M^{\perp}\subset\mathbb{R}\mathfrak{e}$, since generally $M^{\perp}\cap
M=\emptyset$, and $s\mathfrak{e}\not\perp(2+s,0,0,1)$. Hence
$M^{\perp}=\emptyset$, whence $\hat{M}=\mathbb{R}^{4}$.) Another example is
$M$ in (11.2). For the computations we anticipate (11.6). Furthermore,
$M:=\tilde{P}$ for $P$ from (11.5) is a particularly interesting example as
$P$ is maximal spacelike but not a base.

(33) Example for a non causally complete set of determinacy. Let $M:=K\cup L$
for $K:=\\{\mathfrak{x}:-1\leq x_{0}-x_{3}\leq 1,-3\leq x_{0}+x_{3}\leq-1\\}$,
$L:=\\{\mathfrak{x}:-1\leq x_{0}-x_{3}\leq 1,1\leq x_{0}+x_{3}\leq 3\\}$.
$K,L$ are causally complete by _( 11.1)(b)_. One has $M=\tilde{M}$ and
$\hat{M}=\\{\mathfrak{x}:-1\leq x_{0}-x_{3}\leq 1,-3\leq x_{0}+x_{3}\leq
3\\}$, whence
$\\{\mathfrak{x}:-1<x_{0}-x_{3}<1,-1<x_{0}+x_{3}<1\\}\subset\hat{M}\setminus\tilde{M}$.
Proof. First compute $\hat{M}=M^{\perp\perp}$. As $M=K\cup L=\big{(}h(\leq
1)\cap k(\geq-3)\cap h(\geq-1)\cap k(\leq-1)\big{)}\cup\big{(}h(\leq 1)\cap
k(\geq 1)\cap h(\geq-1)\cap k(\leq 3)\big{)}$, by (11.6)(ii) one has
$M^{\perp}=\big{(}\big{(}h(>1)\cap k(<-3)\big{)}\cup\big{(}h(<-1)\cap
k(>-1)\big{)}\big{)}\cap\big{(}\big{(}h(>1)\cap
k(<1)\big{)}\cup\big{(}h(<-1)\cap k(>3)\big{)}\big{)}=\big{(}h(>1)\cap
k(<-3)\cap h(>1)\cap k(<1)\big{)}\cup\big{(}h(>1)\cap k(<-3)\cap h(<-1)\cap
k(>3)\big{)}\cup\big{(}h(<-1)\cap k(>-1)\cap h(>1)\cap
k(<1)\big{)}\cup\big{(}h(<-1)\cap k(>-1)\cap h(<-1)\cap
k(>3)\big{)}=\big{(}h(>1)\cap k(<-3)\big{)}\cup\big{(}h(<-1)\cap
k(>3)\big{)}$, and further, $M^{\perp\perp}=h(\leq 1)\cap k(\geq-3)\cap
h(\geq-1)\cap k(\leq 3)$ as asserted.
We turn to $\tilde{M}$. Recall $M\subset\tilde{M}\subset\hat{M}$. One easily
checks $M\cap\big{(}k(>-1)\cap k(<1)\big{)}=\emptyset$ and $\hat{M}\setminus