# Optimal version of the fundamental theorem of chronogeometry

Michiya Mori  Graduate School of Mathematical Sciences, The University of
Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan; Interdisciplinary
Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, 2-1 Hirosawa,
Wako, Saitama, 351-0198, Japan<EMAIL_ADDRESS>and  Peter Šemrl
Institute of Mathematics, Physics, and Mechanics, Jadranska 19, SI-1000
Ljubljana, Slovenia; Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia<EMAIL_ADDRESS>lj.si

###### Abstract.

We study lightlikeness preserving mappings from the $4$-dimensional Minkowski
spacetime $\mathcal{M}_{4}$ to itself under no additional regularity
assumptions like continuity, surjectivity, or injectivity. We prove that such
a mapping $\phi$ satisfies one of the following three conditions.

  1. (1)

The mapping $\phi$ can be written as a composition of a Lorentz
transformation, a multiplication by a positive scalar, and a translation.

  2. (2)

There is an event $r\in\mathcal{M}_{4}$ such that
$\phi(\mathcal{M}_{4}\setminus\\{r\\})$ is contained in one light cone.

  3. (3)

There is a lightlike line $\ell$ such that
$\phi(\mathcal{M}_{4}\setminus\ell)$ is contained in another lightlike line.

Here, a line that is contained in some light cone in $\mathcal{M}_{4}$ is
called a lightlike line. We also give several similar results on mappings
defined on a certain subset of $\mathcal{M}_{4}$ or the compactification of
$\mathcal{M}_{4}$.

###### Key words and phrases:

fundamental theorem of chronogeometry; special relativity; coherency
preserving mapping

###### 2020 Mathematics Subject Classification:

Primary 83A05, Secondary 47B49, 51B20, 53C50.

The first author was supported by JSPS KAKENHI Grant Number 22K13934.

The second author was supported by grants J1-2454 and P1-0288 from ARIS,
Slovenia.

###### Contents

  1. I Introduction and preliminaries
    1. 1 Introduction
      1. 1.1 Concise description of our result
      2. 1.2 The Minkowski spacetime versus the space of hermitian matrices
      3. 1.3 Structure of the paper
      4. 1.4 Further research directions
    2. 2 The compactification ${\overline{H_{2}}}$ of $H_{2}$
      1. 2.1 Definition of ${\overline{H_{2}}}$
      2. 2.2 The relation between ${\overline{H_{2}}}$ and $U_{2}$
      3. 2.3 The relation between ${\overline{H_{2}}}$ and $\overline{\mathcal{M}_{4}}$
    3. 3 Basic concepts
      1. 3.1 Automorphisms
      2. 3.2 Bloch representation
      3. 3.3 Existence of standard automorphism with certain properties
      4. 3.4 Lines
      5. 3.5 Surfaces
      6. 3.6 Three points
      7. 3.7 Identity-type theorem
  2. II Main results and their proofs
    1. 4 Standard coherency preservers
      1. 4.1 Proof of Theorem A
      2. 4.2 Applications of Theorem A
    2. 5 Degenerate coherency preservers
      1. 5.1 Two types of degenerate coherency preservers
      2. 5.2 Applications of Theorem B
      3. 5.3 More concrete description of degenerate coherency preservers
    3. 6 Coherency preservers on a closed interval
      1. 6.1 The case $d(\varphi(A),\varphi(B))=2$
      2. 6.2 The case $d(\varphi(A),\varphi(B))=0$
      3. 6.3 The case $d(\varphi(A),\varphi(B))=1$

##  Part I Introduction and preliminaries

###  1\. Introduction

####  1.1. Concise description of our result

In this paper, we study the standard Minkowski spacetime $\mathcal{M}_{4}$ of
dimension $3+1$. In the mathematical foundations of special relativity, we
adopt the harmless normalization that the speed of light equals $1$. Recall
that two spacetime events
$r_{1}=(x_{1},y_{1},z_{1},t_{1}),r_{2}=(x_{2},y_{2},z_{2},t_{2})\in\mathcal{M}_{4}$
are _lightlike_ if

$(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}=(t_{2}-t_{1})^{2}.$

Hence, $r_{1}$ and $r_{2}$ are lightlike if the light signal can pass between
$r_{1}$ and $r_{2}$.

The Lorentz–Minkowski indefinite inner product on $\mathcal{M}_{4}$ is defined
by

$\langle r_{1},r_{2}\rangle=-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}+t_{1}t_{2}$

for a pair of spacetime events $r_{1}=(x_{1},y_{1},z_{1},t_{1})$,
$r_{2}=(x_{2},y_{2},z_{2},t_{2})\in\mathcal{M}_{4}$. Thus, $r_{1},r_{2}$ are
lightlike if and only if

$\langle r_{1}-r_{2},r_{1}-r_{2}\rangle=0.$

For a given spacetime event $r$, the set of all spacetime events $s$
satisfying $\langle s-r,s-r\rangle=0$, that is, the set of all spacetime
events $s$ such that $s$ and $r$ are lightlike, is called the _light cone_
with vertex $r$.

Recall that a _Lorentz matrix_ is a $4\times 4$ real matrix $Q$ satisfying
$Q^{t}MQ=M$, where

(1.1) $M=\left[\begin{matrix}-1&0&0&0\cr 0&-1&0&0\cr 0&0&-1&0\cr
0&0&0&1\cr\end{matrix}\right]$

and $Q^{t}$ denotes the transpose of $Q$. A mapping on ${\mathcal{M}}_{4}$ of
the form $r\mapsto Qr$ for some Lorentz matrix $Q$ is called a _Lorentz
transformation_. A Lorentz transformation is characterized as a linear mapping
$\phi\colon{\mathcal{M}}_{4}\to{\mathcal{M}}_{4}$ satisfying

$\langle\phi(r_{1}),\phi(r_{2})\rangle=\langle r_{1},r_{2}\rangle$

for every pair $r_{1},r_{2}\in{\mathcal{M}}_{4}$. A mapping on
${\mathcal{M}}_{4}$ of the form $r\mapsto Qr+a$ for some Lorentz matrix $Q$
and a spacetime event $a$ is called a _Poincaré transformation_.

A map $\phi\colon\mathcal{M}_{4}\to\mathcal{M}_{4}$ is said to _preserve
lightlikeness in both directions_ if it satisfies

(1.2) $\langle
r_{1}-r_{2},r_{1}-r_{2}\rangle=0\iff\langle\phi(r_{1})-\phi(r_{2}),\phi(r_{1})-\phi(r_{2})\rangle=0$

for every pair of spacetime events $r_{1},r_{2}\in\mathcal{M}_{4}$. Observe
that a bijection $\phi\colon\mathcal{M}_{4}\to\mathcal{M}_{4}$ preserves
lightlikeness in both directions if and only if the light cone with vertex $r$
is mapped by $\phi$ onto the light cone with vertex $\phi(r)$ for every
$r\in{\mathcal{M}}_{4}$. The following theorem is known as the _fundamental
theorem of chronogeometry_.

######  Theorem 1.1.

Every bijective map $\phi\colon\mathcal{M}_{4}\to\mathcal{M}_{4}$ satisfying
(1.2) for every pair of spacetime events $r_{1},r_{2}\in\mathcal{M}_{4}$, is
of the form

(1.3) $\phi(r)=c\,Qr+a,\ \ r\in{\mathcal{M}}_{4},$

for some positive real number $c$, some Lorentz matrix $Q$, and some spacetime
event $a$.

Note that linearity or continuity is not assumed in this theorem. Conversely,
it is easily seen that a mapping of the form (1.3) preserves lightlikeness in
both directions. According to [3], the fundamental theorem of chronogeometry
was first given by Alexandrov in 1949 [1] (see also [2], [4], [32]). For a
physical interpretation of this theorem, we refer to [24, p.691].

The fundamental theorem of chronogeometry has been improved in many ways. The
same conclusion as in Theorem 1.1 has been obtained under some weaker
assumptions, see [5], [7], [21], [22], [24], [25], [26], and [32]. Our aim is
to optimize the fundamental theorem of chronogeometry. We will assume no
regularity conditions like injectivity or surjectivity. The assumption of
preserving lightlikeness in both directions will be replaced by the following
weaker condition. A map $\phi\colon\mathcal{M}_{4}\to\mathcal{M}_{4}$ is said
to _preserve lightlikeness_ (or more precisely, it preserves lightlikeness in
one direction) if for every pair of spacetime events
$r_{1},r_{2}\in\mathcal{M}_{4}$ we have

$\langle
r_{1}-r_{2},r_{1}-r_{2}\rangle=0\Rightarrow\langle\phi(r_{1})-\phi(r_{2}),\phi(r_{1})-\phi(r_{2})\rangle=0.$

In other words, a mapping $\phi\colon\mathcal{M}_{4}\to\mathcal{M}_{4}$
preserves lightlikeness when the light cone with vertex $r$ is mapped by
$\phi$ to a (possibly proper) subset of the light cone with vertex $\phi(r)$
for every $r\in{\mathcal{M}}_{4}$. We study a general mapping
$\phi\colon\mathcal{M}_{4}\to\mathcal{M}_{4}$ that preserves lightlikeness in
one direction only. Under such a weak assumption, not all lightlikeness
preserving maps are of the form (1.3).

Now let us state one consequence of our main results. A _lightlike line_ is a
subset of $\mathcal{M}_{4}$ of the form

$\\{(x_{0}+tx,y_{0}+ty,z_{0}+tz,t)\,:\,t\in\mathbb{R}\\}$

for some $(x_{0},y_{0},z_{0}),(x,y,z)\in\mathbb{R}^{3}$ with
$x^{2}+y^{2}+z^{2}=1$. Observe that this is a subset of the light cone with
vertex $(x_{0},y_{0},z_{0},0)\in{\mathcal{M}}_{4}$.

######  Theorem 1.2.

Let $\phi\colon\mathcal{M}_{4}\to\mathcal{M}_{4}$ preserve lightlikeness in
one direction. Then one of the following holds.

  1. (1)

The mapping is of the form (1.3) for some positive real number $c$, some
Lorentz matrix $Q$, and some spacetime event $a$.

  2. (2)

There are events $r,r^{\prime}\in\mathcal{M}_{4}$ such that
$\phi(\mathcal{M}_{4}\setminus\\{r\\})$ is contained in the light cone with
vertex $r^{\prime}$.

  3. (3)

There are lightlike lines $\ell,\ell^{\prime}$ in $\mathcal{M}_{4}$ such that
$\phi(\mathcal{M}_{4}\setminus\ell)$ is contained in $\ell^{\prime}$.

In fact, we will give a result that applies to a mapping defined on a more
general subset of $\mathcal{M}_{4}$.

####  1.2. The Minkowski spacetime versus the space of hermitian matrices

There is another source of motivation from matrix theory to think of
lightlikeness preserving mappings. With $H_{2}$ we denote the set of all
$2\times 2$ complex hermitian matrices. To each spacetime event
$r=(x,y,z,t)\in\mathcal{M}_{4}$, we associate a $2\times 2$ hermitian matrix

(1.4) $\xi(r)=A=\left[\begin{matrix}t-z&x+iy\cr
x-iy&t+z\cr\end{matrix}\right]\in H_{2}.$

A straightforward computation shows that the spacetime events $r_{1},r_{2}$
are lightlike if and only if the associated matrices $A_{1}$ and $A_{2}$
satisfy

$\det(A_{2}-A_{1})=0.$

For a $2\times 2$ matrix, its determinant is zero if and only if it is either
the zero matrix or a matrix of rank one. Recall that two matrices
$A_{1},A_{2}$ are said to be _coherent_ , $A_{1}\sim A_{2}$, if

$\mathrm{rank}\,(A_{2}-A_{1})\leq 1.$

Hence, two spacetime events $r_{1},r_{2}$ are lightlike if and only if the
associated matrices $A_{1}$ and $A_{2}$ are coherent. There is a vast
literature on mappings on the space of matrices that preserve the coherency
relation or some similar relation that involves rank. See for example Hua’s
series of work [10, 11, 12, 13, 14, 15, 16, 17] and related results [19],
[27], [30]. This kind of results have many applications, for example in the
geometry of algebraic homogeneous spaces, see [6], and in the study of
symmetries of certain quantum structures, see [28], [29], and the references
therein.

By considering a mapping satisfying the assumption in Theorem 1.2, we get a
mapping $\varphi\colon H_{2}\to H_{2}$ satisfying

$\mathrm{rank}\,(A-B)\leq
1\Rightarrow\mathrm{rank}\,(\varphi(A)-\varphi(B))\leq 1,$

or equivalently,

(1.5) $\det(A-B)=0\Rightarrow\det(\varphi(A)-\varphi(B))=0$

for every pair $A,B\in H_{2}$. It is easily seen that this is also equivalent
to the condition

$\mathrm{rank}\,(\varphi(A)-\varphi(B))\leq\mathrm{rank}\,(A-B)$

for every pair $A,B\in H_{2}$. Therefore, by considering $H_{2}$ instead of
$\mathcal{M}_{4}$, we get at least three formulations of this condition.

This paper is basically formulated in terms of $2\times 2$ hermitian matrices
instead of the Minkowski spacetime. We avoid using results on the Minkowski
spacetime like Theorem 1.1. Therefore, the reader needs no prerequisite
knowledge about the geometry of Minkowski spacetime. Instead, throughout the
paper we freely use facts about $2\times 2$ matrices. We believe that in such
a way the results are more accessible for the general audience.

####  1.3. Structure of the paper

In the next section, we will introduce the space $\overline{H_{2}}\supset
H_{2}$. We extend the coherency relation $\sim$ on $H_{2}$ to
${\overline{H_{2}}}$, and show that the space $\overline{H_{2}}$ endowed with
the coherency relation can be identified with the $2\times 2$ unitary group
$U_{2}$ via the Cayley transform. We also show that ${\overline{H_{2}}}$ can
be identified with $\overline{\mathcal{M}_{4}}$, which appears in the
literature as a compactification of $\mathcal{M}_{4}$.

Let $\mathcal{A}\subset{\overline{H_{2}}}$. A mapping
$\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ is called a _coherency
preserver_ , or it preserves coherency (in one direction), if it satisfies

$A\sim B\Rightarrow\varphi(A)\sim\varphi(B)$

for every pair $A,B\in\mathcal{A}$. Instead of lightlikeness preserving
mappings on $\mathcal{M}_{4}$, we will think of the equivalent problem of
coherency preserving mappings on $H_{2}$. In fact, we will study mappings
defined on a subset of ${\overline{H_{2}}}$ rather than on $H_{2}$. An
advantage of working with ${\overline{H_{2}}}$ (or
$\overline{\mathcal{M}_{4}}$) rather than $H_{2}$ (or $\mathcal{M}_{4}$)
derives from the fact that there are more symmetries in ${\overline{H_{2}}}$
than in $H_{2}$.

Section 3 collects basic concepts concerning the coherency relation of
${\overline{H_{2}}}$. In Subsection 3.1 we introduce the concept of an
automorphism, that is, a bijective mapping of ${\overline{H_{2}}}$ that
preserves coherency in both directions. We define two special classes of
automorphisms: affine automorphisms and standard automorphisms (Definition
3.6). It is known that an automorphism is always standard. For the sake of
completeness, we give a proof of this fact in Subsection 4.2. We remark here
that the concept of standard automorphism coincides with that of conformal
transformation in the literature.

We then introduce basic notions like projections (together with its relation
to the Bloch representation, Subsection 3.2), lines (Subsection 3.4), and
surfaces (Subsection 3.5). We give some properties of such sets. We also study
the relative position of three points in ${\overline{H_{2}}}$ (Subsection
3.6). In the last subsection of that section (Subsection 3.7), we collect
results about the identity-type theorem in our setting.

The main part of our paper starts in Section 4. We study coherency preserving
mappings from a subset of ${\overline{H_{2}}}$ to ${\overline{H_{2}}}$. Such a
map is said to be standard if it extends to a standard automorphism of
${\overline{H_{2}}}$. The first main result is Theorem A, which gives a
sufficient condition for a coherency preserving mapping to be standard. To
demonstrate the potential of this theorem, we show that the classical version
of the fundamental theorem of chronogeometry can be obtained easily from
Theorem A (Subsection 4.2).

In Section 5, we first introduce two important types of coherency preserving
maps (Definitions 5.6, 5.8) in correspondence with the latter two items in
Theorem 1.2. The key result is Theorem B. Applications of Theorem B are given
in Subsection 5.2. We prove that every coherency preserver from $\mathcal{U}$
into ${\overline{H_{2}}}$ is either standard or of one of the two types if
$\mathcal{U}$ is either a matrix interval in $H_{2}$ or the whole space
${\overline{H_{2}}}$. This together with the identity-type theorem in
Subsection 3.7 shows that a coherency preserver defined on an open connected
subset of ${\overline{H_{2}}}$ is either standard or locally degenerate in a
certain sense (Theorem 5.18). We also demonstrate that Theorem 1.2 and the
main result in Lester’s article [21] can be obtained easily from our theorems.
In Subsection 5.3, we give a more concrete description of coherency preservers
of the two types. In particular, we study the case where the domain is either
$H_{2}$ or ${\overline{H_{2}}}$. A rather long proof of Theorem B is given in
the last section. The proof is split into three cases.

####  1.4. Further research directions

Before closing the current section, let us mention the possibility of
generalizing our work. In this paper, we study everything in the setting of
$4$-dimensional Minkowski spacetime or its compactification. Most of our
discussion, after translating the concepts in a suitable manner, is valid for
a mapping from (a subset of the compactification of) the Minkowski spacetime
of arbitrary dimension $\geq 5$ to itself, even though the picture in terms of
hermitian matrices does not make sense. In the $3$-dimensional case there is
at least one point where the argument in the 4-dimensional space cannot be
modified easily. More precisely, it is Lemma 6.24. We do not know whether the
same conclusions as in the 4-dimensional case hold in the 3-dimensional case.

There are further possible directions of research that seem highly
challenging. For example, what happens if we consider mappings on a space
endowed with a more general symmetric bilinear form instead of the
Lorentz–Minkowski indefinite inner product? Is it possible to give a more
general result for coherency preserving mappings on the space of hermitian
matrices of an arbitrary size? How about mappings satisfying (1.5)?

###  2\. The compactification ${\overline{H_{2}}}$ of $H_{2}$

Whenever appropriate, matrices will be identified with linear operators. We
use the symbol $I$ for the unit matrix and $0$ for the zero matrix. For any
complex matrix $A$, we denote by $A^{t}$ the transpose of $A$, by $A^{\ast}$
the conjugate transpose of $A$, and by $\mathrm{tr}\,A$ the trace of $A$. Let
$i,j$ be integers, $1\leq i,j\leq 2$. By $E_{ij}$ we denote the $2\times 2$
matrix whose all entries are zero but the $(i,j)$-entry which is equal to $1$.
Vectors in $\mathbb{C}^{2}$ will be represented by $2\times 1$ complex
matrices. Every $2\times 2$ complex matrix of rank one is of the form
$xy^{\ast}$ for some nonzero vectors $x,y\in\mathbb{C}^{2}$. A _projection_ is
a matrix $P$ satisfying $P=P^{2}=P^{*}$. If $x=y$ is a vector of norm one,
then $xx^{\ast}$ is a projection of rank one, and every projection of rank one
is of this form. Let $\\{e_{1},e_{2}\\}$ be the standard basis of
$\mathbb{C}^{2}$. Then $E_{ij}=e_{i}e_{j}^{\ast}$.

With $\leq$ we denote the usual partial order (Loewner order) on $H_{2}$. That
is, $A\leq B$ means that both eigenvalues of $B-A$ are at least $0$. For
$A,B\in H_{2}$ we will write $A<B$ if $B-A$ is a positive invertible matrix,
that is, both eigenvalues of $B-A$ are positive. Note that the order relation
$A<B$ can be interpreted as the following situation in special relativity
under the identification via the mapping $\xi$ as in (1.4): The spacetime
event corresponding to $A$ is in the past of that corresponding to $B$.

For a $2\times 2$ matrix $A$, $\sigma(A)$ denotes its spectrum, that is, the
set of eigenvalues of $A$. Let ${\mathcal{P}}$ denote the collection of all
rank one projections in $H_{2}$. For $P\in\mathcal{P}$, we write
$P^{\perp}:=I-P\in\mathcal{P}$. Let
${\overline{\mathbb{R}}}=\mathbb{R}\cup\\{\infty\\}$ denote the one-point
compactification of $\mathbb{R}$. Later in this paper, we will frequently use
the symbols $(a,\infty]:=(a,\infty)\cup\\{\infty\\}$,
$[a,\infty]:=[a,\infty)\cup\\{\infty\\}\subset{\overline{\mathbb{R}}}$ for
$a\in\mathbb{R}$.

In this section, we introduce a space ${\overline{H_{2}}}\supset H_{2}$ in a
somewhat intuitive manner that involves only $2\times 2$ hermitian matrices.
Then we endow the space ${\overline{H_{2}}}$ with the coherency relation
$\sim$ that extends the usual coherency relation in $H_{2}$. We will show that
the space ${\overline{H_{2}}}$ endowed with the coherency relation can be
identified via the Cayley transform with the $2\times 2$ unitary group
$U_{2}$, and also with the compactification $\overline{\mathcal{M}_{4}}$ of
$\mathcal{M}_{4}$.

Before we proceed, let us give one easy lemma. Let $H_{2}^{++}$ denote the set
of all positive invertible matrices in $H_{2}$, and let $H_{2}^{--}$ denote
the set of negative invertible matrices in $H_{2}$. Let $H_{2}^{+-}$ denote
the set of matrices in $H_{2}$ having one positive eigenvalue and one negative
eigenvalue.

######  Lemma 2.1.

Let $P,Q\in{\mathcal{P}}$ satisfy $P\neq Q$. Let $a,b\in\mathbb{R}$.

  * •

If $a,b>0$, then $aP+bQ\in H_{2}^{++}$.

  * •

If $a,b<0$, then $aP+bQ\in H_{2}^{--}$.

  * •

If $ab<0$, then $aP+bQ\in H_{2}^{+-}$.

More generally, if $a,b$ are nonzero complex numbers, then $aP+bQ$ is an
invertible matrix.

###### Proof.

There is no loss of generality in assuming that $P=E_{11}$. Then

$Q=\left[\begin{matrix}c&e^{it}\sqrt{c-c^{2}}\cr
e^{-it}\sqrt{c-c^{2}}&1-c\cr\end{matrix}\right]$

for some $0\leq c<1$ and some $t\in[0,2\pi)$. Now the verification of the
statement is easily done by the equation

$\det(aE_{11}+bQ)=(a+bc)\cdot b(1-c)-b^{2}(c-c^{2})=ab(1-c)$

and by looking at the $(1,1)$-entry of $aE_{11}+bQ$. ∎

####  2.1. Definition of ${\overline{H_{2}}}$

We consider the collection ${\overline{H_{2}}}$ of all formal sums
$aP+bP^{\perp}$ for $P\in{\mathcal{P}}$ and $a,b\in{\overline{\mathbb{R}}}$,
with the following rules:

  * •

$aP+bP^{\perp}=bP^{\perp}+aP$ for any $a,b\in{{\overline{\mathbb{R}}}}$ and
any $P\in{\mathcal{P}}$,

  * •

$aP+aP^{\perp}=aQ+aQ^{\perp}$ for any $a\in{{\overline{\mathbb{R}}}}$ and any
$P,Q\in{\mathcal{P}}$.

Then it is clear that the set $H_{2}$ embeds into ${\overline{H_{2}}}$ in a
natural manner. Using this embedding, we will always regard elements of
$H_{2}$ as elements of ${\overline{H_{2}}}$. Therefore, for $a\in\mathbb{R}$
and $P,Q\in{\mathcal{P}}$, we have $aP=aP+0P^{\perp}$ and
$aI=aP+aP^{\perp}=aQ+aQ^{\perp}$. We also use the symbols
${{\widehat{\infty}}}:=\infty P+\infty P^{\perp}=\infty Q+\infty Q^{\perp}$
and $\infty P:=\infty P+0P^{\perp}$. We extend the coherency relation $\sim$
on $H_{2}$ to ${\overline{H_{2}}}$ in the following way:

  * •

For $A,B\in H_{2}$, we define $\sim$ as before, i.e., $A\sim
B\iff\mathrm{rank}\,(B-A)\leq 1$.

  * •

If $a\in{\mathbb{R}}$, $A\in H_{2}$, and $P\in{\mathcal{P}}$, then $\infty
P+aP^{\perp}\sim A$ if and only if $P^{\perp}AP^{\perp}=aP^{\perp}$ (note that
the condition $P^{\perp}AP^{\perp}=aP^{\perp}$ is equivalent to
$\mathrm{tr}\,(P^{\perp}A)=a$).

  * •

If $a,b\in{\mathbb{R}}$ and $P,Q\in{\mathcal{P}}$, then $\infty
P+aP^{\perp}\sim\infty Q+bQ^{\perp}$ if and only if $P=Q$.

  * •

${{\widehat{\infty}}}\sim\infty P+aP^{\perp}$ for every
$a\in{{\overline{\mathbb{R}}}}$ and every $P\in{\mathcal{P}}$, and
${{\widehat{\infty}}}\not\sim A$ for every $A\in H_{2}$.

For $A,B\in{\overline{H_{2}}}$, we define $d(A,A)=0$, $d(A,B)=1$ if $A\neq
B\sim A$, and $d(A,B)=2$ if $A\not\sim B$. It is easily seen that $d$
satisfies the axioms of distance.

For $A\in{\overline{H_{2}}}$, the collection ${\mathcal{C}}_{A}$ of all
elements $B\in{\overline{H_{2}}}$ satisfying $A\sim B$ is called the _cone_
with vertex $A$. The following are easy to verify.

  * •

If $A\in H_{2}$, then

${\mathcal{C}}_{A}=\\{B\in H_{2}\,:\,A\sim B\\}\cup\\{\infty
P+\mathrm{tr}\,(P^{\perp}A)P^{\perp}\,:\,P\in{\mathcal{P}}\\}.$

  * •

If $P\in{\mathcal{P}}$ and $a\in{\mathbb{R}}$, then

${\mathcal{C}}_{\infty P+aP^{\perp}}=\\{\infty
P+bP^{\perp}\,:\,b\in{{\overline{\mathbb{R}}}}\\}\cup\\{B\in
H_{2}\,:\,P^{\perp}BP^{\perp}=aP^{\perp}\\}.$

  * •

${\mathcal{C}}_{{\widehat{\infty}}}={\overline{H_{2}}}\setminus H_{2}$.

####  2.2. The relation between ${\overline{H_{2}}}$ and $U_{2}$

In order to demonstrate that the above definition of $\sim$ is natural, we
will identify ${\overline{H_{2}}}$ with $U_{2}$, the group of all $2\times 2$
unitary matrices. The identification is given by the Cayley transform

(2.1) $f(aP+bP^{\perp})=f(a)P+f(b)P^{\perp},$

where $f\colon{{\overline{\mathbb{R}}}}\to\mathbb{T}$ is determined by

(2.2) $f(t)=\frac{t-i}{t+i},\ \ \ t\in\mathbb{R},\ \ \ \text{and}\ \ \
f(\infty)=1.$

Here, $\mathbb{T}$ denotes the group of all complex numbers of modulus one.
See also Uhlmann’s article [31].

The coherency relation on $U_{2}$ is defined in the same way as on the set of
hermitian matrices, that is, unitary matrices $U$ and $V$ are coherent if and
only if $\mathrm{rank}\,(U-V)\leq 1$. More generally, for an arbitrary pair of
$2\times 2$ complex matrices $A,B$, we say $A$ and $B$ are coherent and write
$A\sim B$ when $\mathrm{rank}\,(A-B)\leq 1$.

######  Lemma 2.2.

Let $A$ be a $2\times 2$ complex matrix of rank at most one. Then $A\sim I$
holds if and only if $\mathrm{tr}\,A=1$.

###### Proof.

By choosing a suitable orthonormal basis of $\mathbb{C}^{2}$, one may assume
that $A$ is of the form

$\left[\begin{matrix}\mathrm{tr}\,A&0\cr*&0\cr\end{matrix}\right].$

It is now easy to get the desired conclusion. ∎

It is clear that the map $A\mapsto f(A)$, $A\in{\overline{H_{2}}}$, is a
bijection of ${\overline{H_{2}}}$ onto $U_{2}$. Moreover, the image $f(H_{2})$
equals the set of all unitary matrices $U\in U_{2}$ with the property that
$1\not\in\sigma(U)$.

We will show that for any pair $A,B\in{\overline{H_{2}}}$ we have $A\sim B$ if
and only if $f(A)\sim f(B)$. We first note that

(2.3) ${t-i\over t+i}=1-{2i\over t+i}.$

Therefore, for any $A,B\in H_{2}$ we have

$\begin{split}f(A)-f(B)&=\left(I-2i(A+iI)^{-1}\right)-\left(I-2i(B+iI)^{-1}\right)\\\
&=-2i(A+iI)^{-1}+2i(B+iI)^{-1}\\\ &=2i(A+iI)^{-1}((A+iI)-(B+iI))(B+iI)^{-1}\\\
&=2i(A+iI)^{-1}(A-B)(B+iI)^{-1}.\end{split}$

Hence for $A,B\in H_{2}$, we have $A\sim B$ if and only if $f(A)\sim f(B)$.

Let us consider the case $A=\infty P+aP^{\perp}$ for some $a\in{\mathbb{R}}$
and $P\in{\mathcal{P}}$. Then $U:=f(A)=P+f(a)P^{\perp}$. Note that
$f(a)\in\mathbb{T}\setminus\\{1\\}$.

######  Claim 2.3.

Let $B\in{\overline{H_{2}}}$ and $V:=f(B)\in U_{2}$. Then $V$ is coherent to
$U$ if and only if one of the following holds:

  * •

$B\notin H_{2}$ and there is $\lambda\in\mathbb{T}$ such that $V=P+\lambda
P^{\perp}$.

  * •

$B\in H_{2}$ and $P^{\perp}BP^{\perp}=aP^{\perp}$.

###### Proof.

Note that $U\sim V$ is equivalent to $U-I\sim V-I$. Note also that
$U-I=(f(a)-1)P^{\perp}\neq 0$. Assume that $B\notin H_{2}$, or equivalently,
$1\in\sigma(V)$. Then $V-I$ is a scalar multiple of a rank one projection.
Observe that $V-I$ is a scalar multiple of $P^{\perp}$ if and only if there is
$\lambda\in\mathbb{T}$ such that $V=P+\lambda P^{\perp}$. If these conditions
hold, we clearly have $U\sim V$. If $V-I=bQ$ for some $P^{\perp}\neq
Q\in\mathcal{P}$ and $0\neq b\in\mathbb{C}$, then $V-U=bQ-(f(a)-1)P^{\perp}$
is invertible by Lemma 2.1. It follows that $U$ is not coherent to $V$ in this
case.

Assume that $B\in H_{2}$, or equivalently, $1\notin\sigma(V)$. We have
$f(a)-1=-2i(a+i)^{-1}$ and $V-I=-2i(B+iI)^{-1}$ by (2.3). Thus the condition
$U-I\sim V-I$ is equivalent to $(a+i)^{-1}P^{\perp}\sim(B+iI)^{-1}$, which is
in turn equivalent to $(a+i)^{-1}(B+iI)P^{\perp}\sim I$. By Lemma 2.2, this is
further equivalent to
$1=\mathrm{tr}\,((a+i)^{-1}(B+iI)P^{\perp})=(a+i)^{-1}(\mathrm{tr}\,(BP^{\perp})+i)$.
This leads to $\mathrm{tr}\,(BP^{\perp})=a$, which means
$P^{\perp}BP^{\perp}=aP^{\perp}$. ∎

This claim clearly implies that for $B\in{\overline{H_{2}}}$ we have $A\sim B$
if and only if $f(A)\sim f(B)$. In the case that $A={{\widehat{\infty}}}$ and
$B$ is any element of ${\overline{H_{2}}}$, we have $I=f(A)\sim f(B)$ if and
only if $1$ is an eigenvalue of $f(B)$, which is equivalent to $B=\infty
P+aP^{\perp}$ for some $P\in{\mathcal{P}}$ and some
$a\in{{\overline{\mathbb{R}}}}$. We have shown that the map
$f\colon{\overline{H_{2}}}\to U_{2}$ is an isomorphism with respect to the
coherency relation.

####  2.3. The relation between ${\overline{H_{2}}}$ and
$\overline{\mathcal{M}_{4}}$

The compactification $\overline{\mathcal{M}_{4}}$ of $\mathcal{M}_{4}$ is a
concept studied by both mathematicians and physicists, notably by R. Penrose.
Essentially the same space can be introduced in several ways, and it is called
by different names such as the conformal compactification, the conformal
Minkowski space, the compactified Minkowski space, etc. The space
$\overline{\mathcal{M}_{4}}$ is visualized with the so-called Penrose’s
diagram. Those readers who are familiar with this concept are encouraged to
think of a visual image to see what is going on in each of the arguments in
the subsequent sections. See for example [9, Section 5.1], [23] for more
information about the space $\overline{\mathcal{M}_{4}}$.

In this subsection, we first introduce the space $\overline{\mathcal{M}_{4}}$
in accordance with Lester’s article [20]. The space
$\overline{\mathcal{M}_{4}}$ is naturally endowed with a binary relation that
extends the lightlikeness relation in $\mathcal{M}_{4}$. We give a complete
proof of the fact that $\overline{\mathcal{M}_{4}}$ endowed with this binary
relation can be identified with ${\overline{H_{2}}}$ endowed with our
coherency relation. The results in the current subsection will not be used in
the rest of this paper, so those readers who are new to the space
$\overline{\mathcal{M}_{4}}$ may skip to Section 3.

Let us denote by $(\cdot\,,\cdot)$ the usual inner product on
$\mathbb{R}^{6}$. We further denote

$\overline{M}=\left[\begin{matrix}-1&0&0&0&0&0\cr 0&-1&0&0&0&0\cr
0&0&-1&0&0&0\cr 0&0&0&1&0&0\cr 0&0&0&0&0&-{1\over 2}\cr 0&0&0&0&-{1\over
2}&0\cr\end{matrix}\right].$

The symbol $\mathbb{P}(\mathbb{R}^{6})$ stands for the projective space over
$\mathbb{R}^{6}$, that is,

$\mathbb{P}(\mathbb{R}^{6})=\\{[X]\,:\,X\in\mathbb{R}^{6}\setminus\\{0\\}\\}.$

Here, $[X]$ denotes the one-dimensional subspace spanned by the nonzero vector
$X$. Next we introduce the symmetric bilinear form
$\langle\cdot,\cdot\rangle\colon\mathbb{R}^{6}\times\mathbb{R}^{6}\to\mathbb{R}$
defined by

$\langle X,Y\rangle=(\overline{M}X,Y),\ \ \ X,Y\in\mathbb{R}^{6},$

and the corresponding quadratic form $q\colon\mathbb{R}^{6}\to\mathbb{R}$
defined by

$q(X)=\langle X,X\rangle=(\overline{M}X,X),\ \ \ X\in\mathbb{R}^{6}.$

Note that the symbol $\langle\cdot,\cdot\rangle$ has been used before to
denote the Lorentz–Minkowski indefinite inner product on $\mathcal{M}_{4}$.
When using this symbol it will be always clear from the context which of the
two bilinear forms is on our mind. Clearly,

(2.4) $q(x,y,z,t,h,n)=-x^{2}-y^{2}-z^{2}+t^{2}-hn,\ \ \
(x,y,z,t,h,n)\in\mathbb{R}^{6}.$

For every nonzero $X\in\mathbb{R}^{6}$ and every nonzero real number $s$ we
have $q(X)=0$ if and only if $q(sX)=0$. We define

$\overline{\mathcal{M}_{4}}=\\{[X]\,:\,[X]\in\mathbb{P}(\mathbb{R}^{6})\,\
\text{and}\ \,q(X)=0\\}.$

We say that $[X],[Y]\in\overline{\mathcal{M}_{4}}$ are coherent, $[X]\sim[Y]$,
if $\langle X,Y\rangle=0$. The set $\overline{\mathcal{M}_{4}}$ equipped with
the coherency relation is called the conformal Minkowski space.

In the next step we will classify points in $\overline{\mathcal{M}_{4}}$ and
at the same time we will construct a bijective map $\xi$ mapping
$\overline{\mathcal{M}_{4}}$ onto ${\overline{H_{2}}}$ which preserves
coherency in both directions. Let $[X]=[(x,y,z,t,h,n)]$ be a point in
$\overline{\mathcal{M}_{4}}$. We will distinguish two possibilities.

We start with the possibility that $h\not=0$. All points in
$\overline{\mathcal{M}_{4}}$ with this property will be called finite points.
If $[X]=[(x,y,z,t,h,n)]$ is a finite point then we can assume with no loss of
generality that $h=1$. It follows from $q(X)=0$ that

$X=(x,y,z,t,1,-x^{2}-y^{2}-z^{2}+t^{2}).$

If the spacetime event $(x,y,z,t)\in\mathcal{M}_{4}$ is denoted by $r$, then
we can write shorter

$X=(r,1,\langle r,r\rangle).$

For such a point $[X]\in\overline{\mathcal{M}_{4}}$ we define

$\xi([X])=\left[\begin{matrix}t-z&x+iy\cr x-iy&t+z\cr\end{matrix}\right].$

Clearly, $\xi$ is a bijection of the set of all finite points in
$\overline{\mathcal{M}_{4}}$ onto $H_{2}$.

It is easy to see that the coherency relation on the set of finite points in
the conformal Minkowski space corresponds to the coherency relation on
$H_{2}$, that is, to the lightlikeness in Minkowski space. Indeed, let
$[X],[Y]\in\overline{\mathcal{M}_{4}}$ be finite points,

$X=(r_{1},1,\langle r_{1},r_{1}\rangle)$

and

$Y=(r_{2},1,\langle r_{2},r_{2}\rangle).$

Then we have

$\langle X,Y\rangle=\langle r_{1},r_{2}\rangle-{1\over 2}(\langle
r_{1},r_{1}\rangle+\langle r_{2},r_{2}\rangle)=-{1\over 2}\langle
r_{1}-r_{2},r_{1}-r_{2}\rangle.$

Therefore, the map $r\mapsto[(r,1,\langle r,r\rangle)]$ is an embedding of the
space $\mathcal{M}_{4}$ endowed with the lightlikeness relation into the space
$\overline{\mathcal{M}_{4}}$ endowed with the relation $\sim$.

When $h=0$ we further distinguish two possibilities. The first one is that
$x=y=z=t=0$. There is only one such point in the conformal Minkowski space
that will be denoted by $[e]$, where $e=(0,0,0,0,0,1)$. We define
$\xi([e])={{\widehat{\infty}}}$. It is clear that for
$[X]\in\overline{\mathcal{M}_{4}}$ we have $\langle X,e\rangle\not=0$ if and
only if $[X]$ is a finite point in $\overline{\mathcal{M}_{4}}$.

It remains to consider points $[X]=[(x,y,z,t,0,n)]$ with the property that at
least one of the coordinates $x,y,z,t$ is nonzero. Then because of $q(X)=0$
and (2.4) we have $t\not=0$ and without loss of generality we can assume that
$t=1/2$. So each such point $[X]$ can be represented by

(2.5) $X=\left(x,y,z,{1\over 2},0,a\right),$

where $x^{2}+y^{2}+z^{2}=1/4$, and we define

$\xi([X])=\xi\left(\left[\left(x,y,z,{1\over
2},0,a\right)\right]\right)=\infty P+aP^{\perp},$

where

$P=\left[\begin{matrix}{1\over 2}-z&x+iy\cr x-iy&{1\over
2}+z\cr\end{matrix}\right]\ \ \ \text{and}\ \ \
P^{\perp}=\left[\begin{matrix}{1\over 2}+z&-x-iy\cr-x+iy&{1\over
2}-z\cr\end{matrix}\right]$

is an orthogonal pair of projections of rank one.

We show that for every $X$ of the form (2.5) and for every
$[Y]\in\overline{\mathcal{M}_{4}}\setminus[e]$ we have
$[X]\sim[Y]\iff\xi([X])\sim\xi([Y])$. If both $X$ and $Y$ are of the form
(2.5),

$X=\left(x_{1},y_{1},z_{1},{1\over 2},0,a_{1}\right)=(r_{1},0,a_{1})$

and

$Y=\left(x_{2},y_{2},z_{2},{1\over 2},0,a_{2}\right)=(r_{2},0,a_{2})$

with $r_{j}=\left(x_{j},y_{j},z_{j},1/2\right)$, $j=1,2$, then $\langle
X,Y\rangle=0\iff\langle r_{1},r_{2}\rangle=0$ which by a straightforward
application of the Cauchy–Schwarz inequality for vectors in $\mathbb{R}^{3}$
is equivalent to $r_{1}=r_{2}$, that is, $\xi([X])\sim\xi([Y])$.

If

$X=\left(x_{1},y_{1},z_{1},{1\over 2},0,a_{1}\right)=(r_{1},0,a_{1})$

and $Y=(r_{2},1,n)$ with $r_{2}=(x_{2},y_{2},z_{2},t_{2})$ and $n=\langle
r_{2},r_{2}\rangle$, then $\langle X,Y\rangle=\langle
r_{1},r_{2}\rangle-(1/2)a_{1}$. On the other hand, $\xi([X])\sim\xi([Y])$ if
and only if $\mathrm{tr}\,(P^{\perp}A)=a_{1}$, where

$P^{\perp}=\left[\begin{matrix}{1\over 2}+z_{1}&-x_{1}-iy_{1}\cr-
x_{1}+iy_{1}&{1\over 2}-z_{1}\cr\end{matrix}\right]\ \ \ \text{and}\ \ \
A=\left[\begin{matrix}t_{2}-z_{2}&x_{2}+iy_{2}\cr
x_{2}-iy_{2}&t_{2}+z_{2}\cr\end{matrix}\right].$

A straightforward calculation shows that $[X]\sim[Y]\iff\xi([X])\sim\xi([Y])$
in this case as well.

We have shown that ${\overline{H_{2}}}$ can be identified either with the
conformal Minkowski space, or with $U_{2}$. The subset
$H_{2}\subset{\overline{H_{2}}}$ corresponds to the set of all finite points
in $\overline{\mathcal{M}_{4}}$ that can be further identified with the
classical Minkowski space. Alternatively, the subset
$H_{2}\subset{\overline{H_{2}}}$ corresponds to the set of all unitary
matrices $U\in U_{2}$ with the property that $1\not\in\sigma(U)$.

###  3\. Basic concepts

In the current section, we introduce basic concepts in ${\overline{H_{2}}}$
and give their properties.

####  3.1. Automorphisms

An _automorphism_ of ${\overline{H_{2}}}$ is a bijective map
$\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ that preserves
coherency in both directions, i.e.,

(3.1) $A\sim B\iff\varphi(A)\sim\varphi(B)$

for every pair $A,B\in{\overline{H_{2}}}$. Let $c\in\\{-1,1\\}$, $S$ be an
invertible $2\times 2$ complex matrix, and $T\in H_{2}$. It is easily seen
that the mapping $\varphi\colon H_{2}\to H_{2}$ given by

(3.2) $\varphi(A)=cSAS^{\ast}+T$

for every $A\in H_{2}$; or

(3.3) $\varphi(A)=cSA^{t}S^{\ast}+T$

for every $A\in H_{2}$, is a bijection satisfying (3.1) for every pair $A,B\in
H_{2}$.

######  Remark 3.1.

It is known that a bijection $\varphi\colon H_{2}\to H_{2}$ satisfying (3.1)
for every pair $A,B\in H_{2}$ needs to be of the form (3.2) or (3.3). For the
sake of completeness, we will give a proof of this fact in Subsection 4.2,
which leads to our proof of the fundamental theorem of chronogeometry. See
also [18, Subsection 5.3].

In what follows, we show that a map $\varphi\colon H_{2}\to H_{2}$ of the form
(3.2) or (3.3) extends to an automorphism of ${\overline{H_{2}}}$. We first
introduce the rule $-\infty=\infty$ and define

$-(aP+bP^{\perp})=(-a)P+(-b)P^{\perp},\ \ \ P\in\mathcal{P},\
\,a,b\in{\overline{\mathbb{R}}}.$

Note that this is well-defined, and that if $A\in H_{2}$ then $-A$ defined in
this manner coincides with $-A$ as a matrix. It is easy to see that the map
$X\mapsto-X$, $X\in{\overline{H_{2}}}$, is a bijection of ${\overline{H_{2}}}$
onto itself and for every pair $X,Y\in{\overline{H_{2}}}$ we have

$X\sim Y\iff-X\sim-Y.$

Thus, the map $X\mapsto-X$, $X\in{\overline{H_{2}}}$, is an automorphism of
${\overline{H_{2}}}$. Clearly, this automorphism is the inverse of itself,
that is, $-(-X)=X$, $X\in{\overline{H_{2}}}$.

Secondly, let $B\in H_{2}$. We define

${{\widehat{\infty}}}+B={{\widehat{\infty}}}$

and

$(\infty P+aP^{\perp})+B=\infty P+(a+\mathrm{tr}\,(P^{\perp}B))P^{\perp},\ \ \
P\in\mathcal{P},\ \,a\in\mathbb{R}.$

Then $A+B$ is defined for every $A\in{\overline{H_{2}}}\setminus H_{2}$. For
$A\in H_{2}$, $A+B$ is defined as a matrix, as usual.

######  Claim 3.2.

The map $X\mapsto X+B$, $X\in{\overline{H_{2}}}$, is an automorphism of
${\overline{H_{2}}}$.

###### Proof.

We need to show that the map preserves coherency in both directions. We will
verify only the special case when $X\in H_{2}$ and $Y=\infty P+aP^{\perp}$ for
some real $a$ and some $P\in\mathcal{P}$. (The other cases are easier or
similar.) Then $X\sim Y$ if and only if $P^{\perp}XP^{\perp}=aP^{\perp}$ which
is equivalent to $\mathrm{tr}\,(P^{\perp}X)=a$.

On the other hand, $X+B\sim Y+B=\infty
P+(a+\mathrm{tr}\,(P^{\perp}B))P^{\perp}$ if and only if
$P^{\perp}(X+B)P^{\perp}=(a+\mathrm{tr}\,(P^{\perp}B))P^{\perp}$, which is
true if and only if
$\mathrm{tr}\,(P^{\perp}(X+B)P^{\perp})=a+\mathrm{tr}\,(P^{\perp}B)$. The last
equality is easily seen to be equivalent to $\mathrm{tr}\,(P^{\perp}X)=a$.

For the verification of bijectivity see the next paragraph. ∎

A mapping of the form $X\mapsto X+B$ will be called a _translation_. It is
trivial to see that the map $X\mapsto X+(-B)$, $X\in{\overline{H_{2}}}$, is
the inverse of this mapping. We will write shortly $A+(-B)=A-B$,
$A\in{\overline{H_{2}}}$. Clearly, we have $A-B=-(-A+B)$ for every
$A\in{\overline{H_{2}}}$ and every $B\in H_{2}$. We will occasionally write
$B+A$ instead of $A+B$ and $B-A$ instead of $-A+B$ for
$A\in{\overline{H_{2}}}$ and $B\in H_{2}$. For a subset
$\mathcal{A}\subset{\overline{H_{2}}}$ and $B\in H_{2}$, we will sometimes use
the symbols like $-\mathcal{A}=\\{-A\,:\,A\in\mathcal{A}\\}$ and
$\mathcal{A}+B=\\{A+B\,:\,A\in\mathcal{A}\\}$ etc.

Thirdly, we define

$(aP+bP^{\perp})^{t}=aP^{t}+b(P^{t})^{\perp},\ \ \ P\in\mathcal{P},\
\,a,b\in{\overline{\mathbb{R}}}.$

Note that this is well-defined, and that if $A\in H_{2}$ then $A^{t}$ defined
in this manner coincides with the usual transpose of the matrix. It is easy to
see that $X\mapsto X^{t}$, $X\in{\overline{H_{2}}}$, is an automorphism of
${\overline{H_{2}}}$ with $(X^{t})^{t}=X$ for every $X\in{\overline{H_{2}}}$.

Fourthly, let $S$ be any invertible $2\times 2$ complex matrix. If $P$ is any
projection of rank one, then $SPS^{\ast}$ is a positive rank one matrix.
Therefore,

(3.4) $Q={1\over\mathrm{tr}\,(SPS^{\ast})}\,SPS^{\ast}$

is a projection of rank one. We define

$S\,{{\widehat{\infty}}}\,S^{\ast}={{\widehat{\infty}}}$

and

$S(\infty P+aP^{\perp})S^{\ast}=\infty
Q+a\mathrm{tr}\,(Q^{\perp}SS^{\ast})\,Q^{\perp},\ \ \ P\in\mathcal{P},\
\,a\in\mathbb{R},$

where $Q$ is defined by (3.4). Then $SAS^{*}$ is defined for every
$A\in{\overline{H_{2}}}\setminus H_{2}$. For $A\in H_{2}$, $SAS^{*}$ is
defined as a matrix, as usual.

######  Claim 3.3.

The map $X\mapsto SXS^{\ast}$, $X\in{\overline{H_{2}}}$, is an automorphism of
${\overline{H_{2}}}$.

###### Proof.

We need to verify that for every pair $X,Y\in{\overline{H_{2}}}$ we have

(3.5) $X\sim Y\iff SXS^{\ast}\sim SYS^{\ast}.$

The verification of this equivalence is trivial in the following cases:

  * •

at least one of $X$ and $Y$ is equal to ${{\widehat{\infty}}}$,

  * •

both $X$ and $Y$ belong to $H_{2}$, and

  * •

both $X$ and $Y$ belong to ${\overline{H_{2}}}\setminus H_{2}$.

Thus, it remains to consider the case when $X\in H_{2}$ and $Y=\infty
P+aP^{\perp}$ for some real $a$ and some $P\in\mathcal{P}$. Then $X\sim Y$ if
and only if $\mathrm{tr}\,(P^{\perp}X)=a$. And we have $SXS^{\ast}\sim
SYS^{\ast}$ if and only if

$\mathrm{tr}\,(Q^{\perp}SXS^{\ast})=a\mathrm{tr}\,(Q^{\perp}SS^{\ast}),$

or equivalently,

$\mathrm{tr}\,(S^{\ast}Q^{\perp}SX)=a\mathrm{tr}\,(S^{\ast}Q^{\perp}S).$

Note that $S^{\ast}Q^{\perp}S$ is a positive matrix of rank one. We show that
the equality

$\mathrm{tr}\,(P^{\perp}X)={1\over\mathrm{tr}\,(S^{\ast}Q^{\perp}S)}\,\mathrm{tr}\,(S^{\ast}Q^{\perp}SX)$

holds for all $X\in H_{2}$. It suffices to show that the positive matrix
$S^{\ast}Q^{\perp}S$ of rank one is a (positive) scalar multiple of
$P^{\perp}$. This is easily seen by the following equation:

$S^{\ast}Q^{\perp}SP=S^{\ast}Q^{\perp}SPS^{\ast}(S^{\ast})^{-1}=S^{\ast}(Q^{\perp}SPS^{\ast})(S^{\ast})^{-1}=S^{\ast}\cdot
0\cdot(S^{\ast})^{-1}=0.$

For the verification of bijectivity see the next claim. ∎

######  Claim 3.4.

The inverse of the map $X\mapsto SXS^{\ast}$, $X\in{\overline{H_{2}}}$, is the
map $X\mapsto S^{-1}X(S^{-1})^{\ast}$, $X\in{\overline{H_{2}}}$.

###### Proof.

We know that (3.5) holds for every pair $X,Y\in{\overline{H_{2}}}$. In the
same way, we see that

$X\sim Y\iff S^{-1}X(S^{-1})^{\ast}\sim S^{-1}Y(S^{-1})^{\ast}$

for every pair $X,Y\in{\overline{H_{2}}}$. If we denote by $\varphi$ and
$\psi$ the maps from ${\overline{H_{2}}}$ to itself defined by
$\varphi(X)=SXS^{\ast}$ and $\psi(X)=S^{-1}X(S^{-1})^{\ast}$,
$X\in{\overline{H_{2}}}$, then we have

$X\sim Y\iff\psi(\varphi(X))\sim\psi(\varphi(Y))$

for every pair $X,Y\in{\overline{H_{2}}}$, and $\psi(\varphi(X))=X$ for every
$X\in H_{2}\cup\\{{{\widehat{\infty}}}\\}$. It is easy to see that for any
pair of points $X,Y\in{\overline{H_{2}}}\setminus H_{2}$ we have

(3.6) $\mathcal{C}_{X}\cap H_{2}=\mathcal{C}_{Y}\cap H_{2}\iff X=Y.$

It follows that $\psi(\varphi(X))=X$ for every $X\in{\overline{H_{2}}}$, and
in the same way we see that $\varphi(\psi(X))=X$ for every
$X\in{\overline{H_{2}}}$. Hence, the inverse of the automorphism $X\mapsto
SXS^{\ast}$ is $X\mapsto S^{-1}X(S^{-1})^{\ast}$. ∎

We introduce the rules $\infty^{-1}=0$, $0^{-1}=\infty$ in
${{\overline{\mathbb{R}}}}$. For $A\in{\overline{H_{2}}}$, we define $A^{-1}$
in the following manner:

$(aP+bP^{\perp})^{-1}=a^{-1}P+b^{-1}P^{\perp},\ \ \ P\in\mathcal{P},\
\,a,b\in{{\overline{\mathbb{R}}}}.$

Note that this is well-defined and compatible with the usual inverse of
matrices when $a,b\in{\mathbb{R}}\setminus\\{0\\}$.

######  Claim 3.5.

The map $X\mapsto X^{-1}$, $X\in{\overline{H_{2}}}$, is an automorphism of
${\overline{H_{2}}}$.

###### Proof.

Let us consider everything inside $U_{2}$. Let $f$ be defined as in (2.1) and
(2.2). For a unitary matrix $U\in U_{2}$, take
$a,b\in{{\overline{\mathbb{R}}}}$ and $P\in{\mathcal{P}}$ such that
$U=f(a)P+f(b)P^{\perp}$. Then $aP+bP^{\perp}$ is the corresponding point in
${\overline{H_{2}}}$, and $(aP+bP^{\perp})^{-1}=a^{-1}P+b^{-1}P^{\perp}$. It
follows that $f(a^{-1}P+b^{-1}P^{\perp})=f(a^{-1})P+f(b^{-1})P^{\perp}$. Since
(2.2) clearly implies $f(t^{-1})=-\overline{f(t)}$ for every
$t\in{{\overline{\mathbb{R}}}}$, we obtain

$f(a^{-1})P+f(b^{-1})P^{\perp}=-\overline{f(a)}P-\overline{f(b)}P^{\perp}=-U^{*}.$

Now, we obtain the desired conclusion as a consequence of the following fact.
For every pair of unitaries $U,V\in U_{2}$, we have $U\sim
V\iff-U^{*}\sim-V^{*}$. ∎

The automorphism $X\mapsto X^{-1}$ of ${\overline{H_{2}}}$ will be called the
_inversion_. The inverse mapping of the inversion is the inversion itself.

######  Definition 3.6.

An automorphism $\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ of the
form

$\varphi(X)=cSXS^{\ast}+B,\ \ \ X\in{\overline{H_{2}}};$

or

$\varphi(X)=cSX^{t}S^{\ast}+B,\ \ \ X\in{\overline{H_{2}}}$

for some $c\in\\{-1,1\\}$, some invertible $2\times 2$ complex matrix $S$, and
some $B\in H_{2}$ will be called an _affine automorphism_. We say that an
automorphism of ${\overline{H_{2}}}$ is _standard_ if it is written as a
composition of finitely many affine automorphisms and inversions.

It is easily seen that the set of standard automorphisms forms a group. It is
known that every automorphism of ${\overline{H_{2}}}$ is standard. For the
sake of completeness, we will give a proof of this fact in Subsection 4.2. We
also see the following.

######  Lemma 3.7.

The set of affine automorphisms forms a group.

###### Proof.

Let $\varphi_{1},\varphi_{2}$ be two affine automorphisms. We show that
$\varphi_{2}\circ\varphi_{1}$ is an affine automorphism. Let us consider the
case where $\varphi_{i}$ is of the form $X\mapsto S_{i}XS_{i}^{\ast}+B_{i}$
for some invertible $2\times 2$ complex matrix $S_{i}$, and some $B_{i}\in
H_{2}$, $i=1,2$. Let $\varphi$ be the affine automorphism
$X\mapsto(S_{2}S_{1})X(S_{2}S_{1})^{*}+S_{2}B_{1}S_{2}^{*}+B_{2}$. It is
easily seen that $\varphi^{-1}\circ\varphi_{2}\circ\varphi_{1}(X)=X$ for every
$X\in H_{2}$. This leads to
$\varphi^{-1}\circ\varphi_{2}\circ\varphi_{1}(X)=X$ for every
$X\in{\overline{H_{2}}}$ (as in the discussion in the proof of Claim 3.4),
hence we get $\varphi_{2}\circ\varphi_{1}=\varphi$. The other cases can be
considered in an analogous manner. Similarly, we may show that
$\varphi_{1}^{-1}$ is an affine automorphism. ∎

In fact, the class of affine automorphisms coincides with the group of
symmetries on $H_{2}$, that is, the group of all bijections on $H_{2}$ that
preserve coherency in both directions (Theorem 4.11 and Corollary 4.12).
Clearly, the class of standard automorphisms is bigger than that of affine
automorphisms. This is the reason why we introduce the space
${\overline{H_{2}}}$. If we work with $H_{2}$ together with symmetries on
$H_{2}$ instead of ${\overline{H_{2}}}$, then we cannot obtain most of the
results in the subsequent subsections (particularly in Subsections 3.3, 3.4,
3.5, 3.6).

####  3.2. Bloch representation

The symbol $\|\cdot\|$ denotes the usual operator norm. A matrix $A\in H_{2}$
of trace one with $0\leq A$ is called a _density matrix_. We first recall some
known facts about the Bloch representation of the set of all density matrices
in $H_{2}$. The reader will note that the idea is the same as in Introduction
where we have identified spacetime events with $2\times 2$ hermitian matrices.

It is easy to see that the map given by

(3.7) $\left[\begin{matrix}1/2-z&x+iy\cr
x-iy&1/2+z\cr\end{matrix}\right]\mapsto(x,y,z)\in\mathbb{R}^{3}$

is a bijection of the set of all density matrices in $H_{2}$ onto the closed
ball in $\mathbb{R}^{3}$ centered at the origin with radius $1/2$, which we
call the _Bloch ball_. The sphere of the Bloch ball is called the _Bloch
sphere_ , and we denote it by $\mathcal{S}^{2}$. Observe that the above
mapping sends the set ${\mathcal{P}}$ of $2\times 2$ projections of rank one
onto $\mathcal{S}^{2}$.

Let

$A=\left[\begin{matrix}1/2-z_{1}&x_{1}+iy_{1}\cr
x_{1}-iy_{1}&1/2+z_{1}\cr\end{matrix}\right]\ \ \ \text{and}\ \ \
B=\left[\begin{matrix}1/2-z_{2}&x_{2}+iy_{2}\cr
x_{2}-iy_{2}&1/2+z_{2}\cr\end{matrix}\right]$

be any density matrices and $u=(x_{1},y_{1},z_{1})$ and
$v=(x_{2},y_{2},z_{2})$ the corresponding points in the Bloch ball. Because
$A-B$ is a $2\times 2$ hermitian matrix with trace zero, we have

(3.8)
$\|A-B\|^{2}=-\det(A-B)=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2},$

that is, the norm distance between $A$ and $B$ is equal to the Euclidean
distance in the Bloch ball.

If in addition $A=P,B=Q$ are projections, then the right-hand side of (3.8) is
further equal to

$(u-v,u-v)=\frac{1}{4}+\frac{1}{4}-2(u,v)=\frac{1-\cos\alpha}{2}=\sin^{2}\frac{\alpha}{2},$

where $\alpha\in[0,\pi]$ is the angle between $u$ and $v$ in
$\mathcal{S}^{2}$, that is, $\cos\alpha=4(u,v)$. Therefore, $\arcsin\|P-Q\|$
equals $\alpha/2$, which is the geodesic distance between $u$ and $v$ in the
Bloch sphere (that is, the length of the shorter arc in the great circle
passing through $u$ and $v$). We define

(3.9) $d^{g}(P,Q)=\arcsin\|P-Q\|$

for $P,Q\in{\mathcal{P}}$. For us, one important fact about this distance
function is the following obvious property.

######  Lemma 3.8.

If $P,Q\in{\mathcal{P}}$ with $d^{g}(P,Q)=s$ and $n$ is a positive integer and
$t$ a real number such that $nt\geq s$, then there exist points
$P_{0}=P,P_{1},\ldots,P_{n}=Q$ in ${\mathcal{P}}$ such that
$d^{g}(P_{j-1},P_{j})\leq t$, $j=1,2,\ldots,n$.

###### Proof.

This is trivial by thinking about the geodesic distance in $\mathcal{S}^{2}$.
∎

The following theorem will be used in Section 4.

######  Theorem 3.9 (Special case of the non-bijective version of Wigner’s
theorem).

Let $\mu\colon\mathcal{P}\to\mathcal{P}$ be a mapping satisfying

(3.10) $\mathrm{tr}\,(\mu(P)\mu(Q))=\mathrm{tr}\,(PQ),\ \ \
P,Q\in\mathcal{P}.$

Then there exists a $2\times 2$ unitary matrix $U$ such that either

$\mu(P)=UPU^{\ast},\ \ P\in\mathcal{P},$

or

$\mu(P)=UP^{t}U^{\ast},\ \ P\in\mathcal{P}.$

See [8] for an elementary proof of this theorem.

######  Remark 3.10.

Since $(P-Q)^{2}$ is a scalar multiple of $I$, we get

$\|P-Q\|^{2}=\|(P-Q)^{2}\|=\frac{1}{2}\mathrm{tr}\,\left((P-Q)^{2}\right)=\frac{1}{2}\mathrm{tr}\,(P-PQ-
QP+Q)=1-\mathrm{tr}\,(PQ).$

From this, we see that (3.10) is equivalent to

$\|\varphi(P)-\varphi(Q)\|=\|P-Q\|,\ \ \ P,Q\in\mathcal{P}.$

We continue collecting some lemmas concerning ${\mathcal{P}}$ that will be
used in Section 4.

######  Lemma 3.11.

Let $P,Q\in\mathcal{P}$. Then

$\|P-Q\|^{2}+\|P^{\perp}-Q\|^{2}=1.$

###### Proof.

This is clear from (3.8) and the fact that $P$ and $P^{\perp}$ correspond to
antipodal points in the Bloch sphere. ∎

######  Lemma 3.12.

For $A\in{\overline{H_{2}}}$, the following are equivalent:

  * •

$A\sim 0$ and $A\sim I$,

  * •

$A\in\mathcal{P}$.

###### Proof.

If $A\in\mathcal{P}$, then obviously $A\sim 0$ and $A\sim I$. If $A\sim 0$,
then $A=tP$ for some $P\in\mathcal{P}$ and some
$t\in{{\overline{\mathbb{R}}}}$. Clearly, $tP\sim I$ if and only if $t=1$.
This completes the proof. ∎

For $A,B\in{\overline{H_{2}}}$ with $d(A,B)=2$, let $\mathcal{S}_{A,B}$ denote
the set $\mathcal{C}_{A}\cap\mathcal{C}_{B}$.

######  Lemma 3.13.

Let $0<a\leq 1$ and $P\in\mathcal{P}$. Then

(3.11)
$\mathcal{S}_{(1-a)P,P+aP^{\perp}}=\\{(1-a)P+aR\,:\,R\in\mathcal{P}\\}.$

###### Proof.

It is an easy consequence of Lemma 3.12 that

$\mathcal{S}_{0,aI}=\\{aR\,:\,R\in\mathcal{P}\\}.$

For any $A,B\in H_{2}$ with $d(A,B)=2$ we have

$\mathcal{S}_{A+(1-a)P,B+(1-a)P}=\\{X+(1-a)P\,:\,X\in\mathcal{S}_{A,B}\\}.$

Combining the two equalities with $A=0$, $B=aI$, we get (3.11). ∎

######  Lemma 3.14.

Let $P,Q\in\mathcal{P}$ and $a$ be a positive real number, $a<1/2$. Then the
intersubsection

$\\{(1-a)P+aR\,:\,R\in\mathcal{P}\\}\,\cap\,\\{(1-a)Q+aR\,:\,R\in\mathcal{P}\\}$

is nonempty if and only if

$\|P-Q\|\leq{a\over 1-a}.$

###### Proof.

Since the mapping (3.7) is a restriction of an affine mapping, we see that the
sets $\\{(1-a)P+aR\,:\,R\in\mathcal{P}\\}$ and
$\\{(1-a)Q+aR\,:\,R\in\mathcal{P}\\}$ are sent by this mapping to spheres with
radius $a/2$ that are both inscribed in $\mathcal{S}^{2}$. Thus we need to
verify that if $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ are spheres of radius
$a/2$ that are inscribed in $\mathcal{S}^{2}$ with contact points
$P,Q\in\mathcal{S}^{2}$, respectively, then $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$ intersect if and only if the Euclidean distance between $P$
and $Q$ is at most $a/(1-a)$. The proof of this claim is fairly easy and we
omit it. ∎

####  3.3. Existence of standard automorphism with certain properties

######  Lemma 3.15.

Let $A,B\in{\overline{H_{2}}}$. If $d(A,B)=2$, then there is a standard
automorphism $\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ such that
$\varphi(A)=0$ and $\varphi(B)={{\widehat{\infty}}}$.

###### Proof.

There exists a standard automorphism
$\psi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ such that
$\psi(B)={{\widehat{\infty}}}$. Indeed, if $B\in H_{2}$ then take the map
$X\mapsto(X-B)^{-1}$. If $B=aP+\infty P^{\perp}$ with $P\in{\mathcal{P}}$,
$a\in\mathbb{R}$, then take the map $X\mapsto((X+(1-a)P)^{-1}-P)^{-1}$. If
$B={\widehat{\infty}}$, let $\psi$ be the identity mapping. From
$d(\psi(A),\psi(B))=d(A,B)=2$, we deduce that $\psi(A)\in H_{2}$. Hence, the
standard automorphism

$X\mapsto\psi(X)-\psi(A),\ \ \ X\in{\overline{H_{2}}},$

has the desired property. ∎

######  Lemma 3.16.

Let $A,B\in{\overline{H_{2}}}$ with $d(A,B)=2$. Then for any
$X,Y\in\mathcal{S}_{A,B}$ we have either $X=Y$, or $d(X,Y)=2$.

###### Proof.

Applying Lemma 3.15, we see that there is no loss of generality in assuming
that $A=0$ and $B={\widehat{\infty}}$. In that case, we have
${\mathcal{S}}_{A,B}=\\{\infty P\,:\,P\in{\mathcal{P}}\\}$, and the desired
conclusion follows trivially. ∎

######  Lemma 3.17.

Let $A,B\in{\overline{H_{2}}}$. If $d(A,B)=1$, then there is a standard
automorphism $\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ satisfying

$\varphi(A)=0\ \ \text{and}\ \ \varphi(B)=E_{11}=\left[\begin{matrix}1&0\cr
0&0\cr\end{matrix}\right].$

###### Proof.

Without loss of generality, we may assume that $A=0$. Then $B=sP$ for some
rank one projection $P$ and some nonzero $s\in{{\overline{\mathbb{R}}}}$. If
$s\not=\infty$, then we can find an invertible $2\times 2$ matrix $T$ such
that $T(sP)T^{\ast}=\pm E_{11}$ and we are done. In the case that $s=\infty$,
we can find a standard automorphism $\varphi_{1}$ of ${\overline{H_{2}}}$ such
that $\varphi_{1}(0)=0$ and $\varphi_{1}({{\widehat{\infty}}})=I$. Because
$\varphi_{1}(\infty P)\sim\varphi_{1}(0)=0$ and $\varphi_{1}(\infty
P)\sim\varphi_{1}({{\widehat{\infty}}})=I$, Lemma 3.12 implies that
$\varphi_{1}(B)=Q$ for some rank one projection $Q$. Multiplying $\varphi_{1}$
with an appropriate unitary similarity, we get a standard automorphism
$\varphi$ with the desired properties. ∎

Let $A,B\in H_{2}$ and $A\leq B$. We denote by $[A,B]\subset H_{2}$ the matrix
interval

$[A,B]=\\{X\in H_{2}\,:\,A\leq X\leq B\\}.$

Further, let $(A,B)=\\{X\in H_{2}\,:\,A<X<B\\}$ and $[A,B)=\\{X\in
H_{2}\,:\,A\leq X<B\\}$, $(A,B]=\\{X\in H_{2}\,:\,A<X\leq B\\}$ when $A<B$. In
the language of special relativity, these correspond to the set of spacetime
events in the future of a fixed event and in the past of another event,
possibly with suitable boundary points.

######  Lemma 3.18.

Let $A,B,C,D\in H_{2}$ satisfy $A<B$ and $C<D$. Then there is an affine
automorphism $\psi$ of ${\overline{H_{2}}}$ satisfying $\psi([A,B])=[C,D]$ and

$X\leq Y\iff\psi(X)\leq\psi(Y),\quad X<Y\iff\psi(X)<\psi(Y)$

for any pair $X,Y\in H_{2}$.

###### Proof.

Let us define the affine automorphism $\psi_{1}$ by

$\psi_{1}(X)=(B-A)^{1/2}X(B-A)^{1/2}+A,\ \ X\in{\overline{H_{2}}}.$

Then we have $\psi_{1}([0,I])=[A,B]$ and

$X\leq Y\iff\psi_{1}(X)\leq\psi_{1}(Y),\quad X<Y\iff\psi_{1}(X)<\psi_{1}(Y)$

for any pair $X,Y\in H_{2}$. Similarly, we define the affine automorphism
$\psi_{2}$ by

$\psi_{2}(X)=(D-C)^{1/2}X(D-C)^{1/2}+C,\ \ X\in{\overline{H_{2}}}.$

We see that the affine automorphism $\psi_{2}\circ\psi_{1}^{-1}$ satisfies the
desired properties. ∎

######  Corollary 3.19.

Let $A,B\in H_{2}$ satisfy $A<B$. Then $\mathcal{S}_{A,B}\subset[A,B]\subset
H_{2}$.

###### Proof.

By the preceding lemma, it suffices to consider the case $A=0$ and $B=I$. In
this case, we have $\mathcal{S}_{0,I}={\mathcal{P}}\subset[0,I]\subset H_{2}$,
as desired. ∎

######  Lemma 3.20.

Let $C\in(0,I)$. There is a standard automorphism $\psi$ of
${\overline{H_{2}}}$ such that

$\psi([0,I])=[0,I],\ \ \psi(0)=0,\ \ \psi(I)=I,\ \ \text{and}\ \
\psi((1/2)I)=C.$

###### Proof.

Consider the standard automorphisms $\psi_{1}$ of ${\overline{H_{2}}}$ defined
by $\psi_{1}(X)=X^{-1}-I$, $X\in{\overline{H_{2}}}$. It is easily seen by the
definition of the mapping that

$\psi_{1}(0)={\widehat{\infty}},\ \ \psi_{1}(I)=0,\ \ \psi_{1}((1/2)I)=I,$
$\psi_{1}([0,I])={\overline{H_{2}^{+}}}:=\\{aP+bP^{\perp}\,:\,a,b\in[0,\infty],\,P\in\mathcal{P}\\},$

and that $\psi_{1}(C)$ is a positive invertible matrix in $H_{2}$. Consider
another standard automorphism $\psi_{2}$ defined by
$\psi_{2}(X)=\psi_{1}(C)^{1/2}X\psi_{1}(C)^{1/2}$, $X\in{\overline{H_{2}}}$.
It satisfies

$\psi_{2}(0)=0,\ \ \psi_{2}({\widehat{\infty}})={\widehat{\infty}},\ \
\psi_{2}(I)=\psi_{1}(C),\ \ \text{and}\ \
\psi_{2}({\overline{H_{2}^{+}}})={\overline{H_{2}^{+}}}.$

It follows that the composition $\psi_{1}^{-1}\circ\psi_{2}\circ\psi_{1}$
satisfies the desired property. ∎

The above two lemmas clearly imply the following.

######  Corollary 3.21.

Let $A,B,C,A^{\prime},B^{\prime},C^{\prime}\in H_{2}$ satisfy $A<C<B$ and
$A^{\prime}<C^{\prime}<B^{\prime}$. Then there is a standard automorphism
$\psi$ of ${\overline{H_{2}}}$ satisfying

$\psi([A,B])=[A^{\prime},B^{\prime}],\ \ \psi(A)=A^{\prime},\ \
\psi(B)=B^{\prime},\ \ \text{and}\ \ \psi(C)=C^{\prime}.$

####  3.4. Lines

A subset $\mathcal{A}\subset{\overline{H_{2}}}$ is said to be coherent if
$A\sim B$ for every pair $A,B\in\mathcal{A}$. A maximal coherent set is called
a _line_. Clearly, if $\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$
is an automorphism and $\ell\subset{\overline{H_{2}}}$ is a line, then
$\varphi(\ell)$ is a line as well.

######  Lemma 3.22.

Let $a,b\in{{\overline{\mathbb{R}}}}\setminus\\{0\\}$ and $P,Q\in\mathcal{P}$.
Then $aP\sim bQ$ if and only if $P=Q$.

###### Proof.

It is clear that $P=Q$ implies $aP\sim bQ$. Let us prove that $aP\sim bQ$
implies $P=Q$. Lemma 2.1 verifies this when
$a,b\in\mathbb{R}\setminus\\{0\\}$. Assume that
$b\in\mathbb{R}\setminus\\{0\\}$ and $\infty P\sim bQ$. Then we have
$bP^{\perp}QP^{\perp}=0$, thus $P=Q$. Similarly,
$a\in\mathbb{R}\setminus\\{0\\}$ and $aP\sim\infty Q$ imply $P=Q$. Finally, it
is trivial that $\infty P=\infty Q$ implies $P=Q$. ∎

######  Lemma 3.23.

Let $\ell$ be a line. Then either there exist $A\in H_{2}$ and
$P\in\mathcal{P}$ such that

$\ell=\\{aP+A\,:\,a\in{\overline{\mathbb{R}}}\\}$

or there exists $P\in\mathcal{P}$ such that

$\ell=\\{\infty P+aP^{\perp}\,:\,a\in{{\overline{\mathbb{R}}}}\\}.$

###### Proof.

We first consider the possibility that there exists $A\in\ell\cap H_{2}$. Then
$\ell^{\prime}=\ell-A=\\{X-A\,:\,X\in\ell\\}$ is a line that contains $0$. We
know that $\mathcal{C}_{0}=\\{aP\,:\,P\in\mathcal{P},\
\,a\in{{\overline{\mathbb{R}}}}\\}$. Since $\ell^{\prime}$ is a maximal
coherent set, the preceding lemma implies that
$\ell^{\prime}=\\{aP\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ for some
$P\in\mathcal{P}$. Hence

$\ell=\ell^{\prime}+A=\\{aP+A\,:\,a\in{\overline{\mathbb{R}}}\\}.$

The remaining possibility is that $\ell\subset{\overline{H_{2}}}\setminus
H_{2}$. Since $\ell$ is a maximal coherent set, there exists a point in $\ell$
that is different from ${{\widehat{\infty}}}$. Thus, there exists
$P\in\mathcal{P}$ and $a_{0}\in\mathbb{R}$ such that $\infty
P+a_{0}P^{\perp}\in\ell$. The set of all points in
${\overline{H_{2}}}\setminus H_{2}$ that are coherent to $\infty
P+a_{0}P^{\perp}$ is $\\{\infty
P+aP^{\perp}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$. It is easy to see that
this is a maximal coherent set. Hence this set is equal to $\ell$. ∎

######  Lemma 3.24.

Let $A,B\in{\overline{H_{2}}}$ and $d(A,B)=1$. Then there exists exactly one
line $\ell$ in ${\overline{H_{2}}}$ such that $A,B\in\ell$, which equals
${\mathcal{C}}_{A}\cap{\mathcal{C}}_{B}$.

###### Proof.

Let $\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ be an automorphism.
Then there exists exactly one line passing through $A$ and $B$ if and only if
there exists exactly one line passing through $\varphi(A)$ and $\varphi(B)$.
Thus, by Lemma 3.17, with no loss of generality we may assume that $A=0$ and
$B=E_{11}$. In that case, it is obvious from Lemma 3.22 that
${\mathcal{C}}_{A}\cap{\mathcal{C}}_{B}=\\{aE_{11}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$
is the unique line that contains both $A$ and $B$. ∎

######  Lemma 3.25.

Let $\ell$ be a line in ${\overline{H_{2}}}$ and $A$ a point in
${\overline{H_{2}}}$ such that $A\not\in\ell$. Then there is exactly one point
on the line $\ell$ that is coherent to $A$.

###### Proof.

With no loss of generality, we may assume that $A={\widehat{\infty}}$. From
$A={\widehat{\infty}}\not\in\ell$ and Lemma 3.23, we see that there are $B\in
H_{2}$ and $P\in\mathcal{P}$ such that

$\ell=\\{aP+B\,:\,a\in{\overline{\mathbb{R}}}\\},$

and then clearly, $\infty P+B=\infty P+\mathrm{tr}\,(P^{\perp}B)P^{\perp}$ is
the unique point on $\ell$ that is coherent to $A={\widehat{\infty}}$. ∎

For $A,B\in{\overline{H_{2}}}$ with $d(A,B)=1$, we denote by $\ell_{A,B}$ the
unique line passing through $A$ and $B$. Note that by Lemma 3.24 we have
$\ell_{A,B}=\mathcal{C}_{A}\cap\mathcal{C}_{B}$. Denoting
$\mathcal{C}_{A}\cap\mathcal{C}_{B}$ with two different symbols (depending on
the distance between $A$ and $B$) makes sense because the sets
$\mathcal{S}_{A,B}$ and $\ell_{A,B}$ are of completely different shapes.

######  Remark 3.26.

We will quite often use the following observation. Assume that
$A,B,C\in{\overline{H_{2}}}$ are points such that $d(A,B)=1$ and $d(A,C)=2$.
Then there is a unique point $D\in{\overline{H_{2}}}$ that is coherent to $A$
and $B$ and $C$. Indeed, let us denote by $\ell$ the unique line passing
through $A$ and $B$. Then, clearly $C\not\in\ell$ and there is a unique
$D\in\ell$ such that $D\sim C$.

######  Corollary 3.27.

Let $s,t$ be nonzero distinct real numbers and $Q\in\mathcal{P}$. Assume that
$A\in{\overline{H_{2}}}$ satisfies $A\sim sI$, $A\sim tI$, and $A\sim sQ$.
Then $A=sQ+tQ^{\perp}$.

###### Proof.

Since the set $\\{sQ,sI,A\\}$ is coherent, it is contained in the line
$\ell_{sQ,sI}$, thus we have $A=sQ+aQ^{\perp}$ for some
$a\in{{\overline{\mathbb{R}}}}$. By $A\sim tI$ we obtain $a=t$. ∎

######  Corollary 3.28.

Let $\ell_{1},\ell_{2},\ell_{3}$ be pairwise different lines in
${\overline{H_{2}}}$. Assume that $\ell_{1}\cap\ell_{2}\not=\emptyset$,
$\ell_{1}\cap\ell_{3}\not=\emptyset$, and
$\ell_{2}\cap\ell_{3}\not=\emptyset$. Then there exists
$A\in{\overline{H_{2}}}$ such that

$\ell_{1}\cap\ell_{2}=\ell_{1}\cap\ell_{3}=\ell_{2}\cap\ell_{3}=\\{A\\}.$

###### Proof.

By Lemma 3.24, the intersubsection of any two of the above three lines is a
singleton. Denote $\ell_{1}\cap\ell_{2}=\\{A\\}$,
$\ell_{1}\cap\ell_{3}=\\{B\\}$, and $\ell_{2}\cap\ell_{3}=\\{C\\}$. We need to
show that $A=B=C$. Assume on the contrary that this is not true, say
$A\not=B$. Then $A,B$ are two distinct points on $\ell_{1}$ and $C$ is
coherent to both of them. By Lemma 3.25, we have $C\in\ell_{1}$, too. Thus

$C\in\ell_{1}\cap\ell_{2}\cap\ell_{3}=(\ell_{1}\cap\ell_{2})\cap(\ell_{1}\cap\ell_{3})=\\{A\\}\cap\\{B\\}=\emptyset,$

a contradiction. ∎

####  3.5. Surfaces

######  Lemma 3.29.

Let $\ell_{1}$ and $\ell_{2}$ be distinct lines in ${\overline{H_{2}}}$.

  * •

If $\ell_{1}\cap\ell_{2}\not=\emptyset$, then there exists a standard
automorphism $\varphi$ of ${\overline{H_{2}}}$ such that

$\varphi(\ell_{1})=\\{aE_{11}\,:\,a\in{{\overline{\mathbb{R}}}}\\}\ \ \
\text{and}\ \ \
\varphi(\ell_{2})=\\{aE_{22}\,:\,a\in{{\overline{\mathbb{R}}}}\\}.$

  * •

If $\ell_{1}\cap\ell_{2}=\emptyset$, then there exists a standard automorphism
$\varphi$ of ${\overline{H_{2}}}$ such that

$\varphi(\ell_{1})=\\{aE_{11}\,:\,a\in{{\overline{\mathbb{R}}}}\\}\ \ \
\text{and}\ \ \ \varphi(\ell_{2})=\\{aE_{11}+\infty
E_{22}\,:\,a\in{{\overline{\mathbb{R}}}}\\}.$

###### Proof.

Assume first that $\ell_{1}\cap\ell_{2}\not=\emptyset$. Then, by Lemma 3.24,
the intersubsection of these two lines is a singleton. With no loss of
generality, we may assume that $\ell_{1}\cap\ell_{2}=\\{0\\}$. Then there are
$P,Q\in\mathcal{P}$ with $P\not=Q$ and
$\ell_{1}=\\{aP\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ and
$\ell_{2}=\\{aQ\,:\,a\in{{\overline{\mathbb{R}}}}\\}$. If $u,v$ are unit
vectors spanning the image of $P$ and the image of $Q$, respectively, then
they are linearly independent and there exists an invertible linear operator
$S\colon\mathbb{C}^{2}\to\mathbb{C}^{2}$ such that $Su=e_{1}$ and $Sv=e_{2}$.
It follows that

$SPS^{\ast}=Suu^{*}S^{*}=(Su)(Su)^{*}=e_{1}e_{1}^{*}=E_{11}$

and similarly, $SQS^{\ast}=E_{22}$. Let $\psi$ denote the automorphism
$X\mapsto SXS^{*}$. Then $\psi(\ell_{1})$ is a line that contains $0$ and
$E_{11}$, hence
$\psi(\ell_{1})=\\{aE_{11}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$. Similarly,
$\psi(\ell_{2})=\\{aE_{22}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$.

Assume next that $\ell_{1}\cap\ell_{2}=\emptyset$. Since
$\ell_{1}\cup\ell_{2}$ is clearly not a coherent set, by Lemma 3.15 we can
assume that $0\in\ell_{1}$ and ${\widehat{\infty}}\in\ell_{2}$. It follows
that there are $P,Q\in\mathcal{P}$ such that
$\ell_{1}=\\{aP\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ and $\ell_{2}=\\{\infty
Q+aQ^{\perp}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$. We have $P\not=Q$, since
otherwise $\infty P=\infty Q$ would belong to the intersubsection of
$\ell_{1}$ and $\ell_{2}$. We see that there exists an invertible $2\times 2$
matrix such that $SPS^{\ast}=E_{11}$ and $SQS^{\ast}=E_{22}$. Let $\psi$
denote the automorphism $X\mapsto SXS^{*}$. Then
$\psi(\ell_{1})=\\{aE_{11}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ as before.
Since $\psi({\widehat{\infty}})={\widehat{\infty}}$ and $\psi(\infty Q)=\infty
E_{22}$, we get $\psi(\ell_{2})=\\{aE_{11}+\infty
E_{22}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$. ∎

######  Definition 3.30.

Let $\ell_{1}$ and $\ell_{2}$ be lines in ${\overline{H_{2}}}$ with
$\ell_{1}\cap\ell_{2}=\emptyset$. A _surface_ along $\ell_{1}$ and $\ell_{2}$
is the union of all lines $\ell\subset{\overline{H_{2}}}$ satisfying
$\ell\cap\ell_{1}\not=\emptyset$ and $\ell\cap\ell_{2}\not=\emptyset$.

######  Lemma 3.31.

The surface along the lines $\\{aE_{11}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$
and $\\{\infty E_{22}+aE_{11}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ is

$\\{aE_{11}+bE_{22}\,:\,a,b\in{{\overline{\mathbb{R}}}}\\}.$

###### Proof.

Denote $\ell_{1}=\\{aE_{11}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ and
$\ell_{2}=\\{\infty E_{22}+aE_{11}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$. For
every $a\in{{\overline{\mathbb{R}}}}$, the point $\infty E_{22}+aE_{11}$ is
the unique point on $\ell_{2}$ that is coherent to $aE_{11}\in\ell_{1}$.
Hence, there is exactly one line passing through $aE_{11}$ that intersects
$\ell_{2}$ and this line is
$\\{aE_{11}+bE_{22}\,:\,b\in{{\overline{\mathbb{R}}}}\\}$. Consequently, the
union of all lines that intersect $\ell_{1}$ and $\ell_{2}$ is
$\\{aE_{11}+bE_{22}\,:\,a,b\in{{\overline{\mathbb{R}}}}\\}$. ∎

We will call the set

(3.12) $\\{aE_{11}+bE_{22}\,:\,a,b\in{{\overline{\mathbb{R}}}}\\}$

the _diagonal surface_. A direct consequence of Lemmas 3.29 and 3.31 is that
any surface is a $\varphi$-image of the diagonal surface for some standard
automorphism $\varphi$ of ${\overline{H_{2}}}$.

######  Corollary 3.32.

Let $\Pi\subset{\overline{H_{2}}}$ be a surface.

  * •

Assume that $A,B\in\Pi$ satisfy $d(A,B)=1$. Then the line $\ell_{A,B}$ passing
through $A$ and $B$ is contained in $\Pi$.

  * •

Assume that $\ell_{1}$ and $\ell_{2}$ are lines contained in $\Pi$ and
$\ell_{1}\cap\ell_{2}=\emptyset$. Then $\Pi$ is the surface along $\ell_{1}$
and $\ell_{2}$.

###### Proof.

Without loss of generality, we may assume that $\Pi$ is the diagonal surface.
Then the verification of both statements is easy. ∎

######  Lemma 3.33.

Let $Q\in\mathcal{P}$. Assume that $\Pi\subset{\overline{H_{2}}}$ is a surface
containing $0$, $(1/2)I$, $I$, and $Q$. Then

$\Pi=\\{aQ+bQ^{\perp}\,:\,a,b\in{{\overline{\mathbb{R}}}}\\}.$

###### Proof.

We have $0,Q\in\Pi$ and therefore, by Corollary 3.32 the line
$\\{aQ\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ is contained in $\Pi$. In
particular, $(1/2)Q\in\Pi$. It follows that the line
$\ell_{1}:=\\{Q+aQ^{\perp}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ passing
through $Q$ and $I$, and the line
$\ell_{2}:=\\{(1/2)Q+aQ^{\perp}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ passing
through $(1/2)Q$ and $(1/2)I$, are both contained in $\Pi$. Observe that the
surface $\\{aQ+bQ^{\perp}\,:\,a,b\in{{\overline{\mathbb{R}}}}\\}$ also
contains $\ell_{1}$ and $\ell_{2}$. Since $\ell_{1}\cap\ell_{2}=\emptyset$,
the second item of Corollary 3.32 leads to the desired conclusion. ∎

Let $P\in\mathcal{P}$ and consider the set

(3.13) ${\square}_{P}=\\{aP+bP^{\perp}\,:\,a,b\in[0,1]\\}.$

It is contained in the surface
$\\{aP+bP^{\perp}\,:\,a,b\in{{\overline{\mathbb{R}}}}\\}$.

######  Lemma 3.34.

Let $P\in\mathcal{P}$ and $A,B\in{\square}_{P}$ satisfy $d(A,B)=1$. Then every
point of ${\square}_{P}$ is coherent to some point of
$\ell_{A,B}\cap{\square}_{P}$.

###### Proof.

By the assumption, we see that one of the following holds. Either there are
$p,q,r\in[0,1]$ with $A=pP+qP^{\perp}$, $B=pP+rP^{\perp}$, and $q\neq r$, or
there are $p,q,r\in[0,1]$ with $A=pP+qP^{\perp}$, $B=rP+qP^{\perp}$, and
$p\neq r$. We have
$\ell_{A,B}=\\{pP+tP^{\perp}\,:\,t\in{{\overline{\mathbb{R}}}}\\}$ in the
former case and
$\ell_{A,B}=\\{tP+qP^{\perp}\,:\,t\in{{\overline{\mathbb{R}}}}\\}$ in the
latter case. It is now clear to see that the desired conclusion holds. ∎

######  Lemma 3.35.

Let $P\in\mathcal{P}$ and assume that
$\varphi\colon{\square}_{P}\to{\overline{H_{2}}}$ is a coherency preserver.
Then $\varphi({\square}_{P})$ is contained in a cone or in a surface.

###### Proof.

The first possibility we will consider is that there exists $a\in[0,1]$ such
that the line segment $s_{a}:=\\{aP+bP^{\perp}\,:\,b\in[0,1]\\}$ is mapped to
a singleton $\\{A\\}$. Then clearly, $\varphi({\square}_{P})$ is contained in
the cone with vertex $A$.

So, we may assume from now on that for each line segment $s_{a}$, $0\leq a\leq
1$, there is a unique line $\ell_{a}\subset{\overline{H_{2}}}$ such that
$\varphi(s_{a})\subset\ell_{a}$.

Assume next that there is a pair $a_{1},a_{2}\in[0,1]$ such that
$\ell_{a_{1}}\cap\ell_{a_{2}}=\emptyset$. We will show that in this case
$\varphi({\square}_{P})$ is contained in the surface along $\ell_{a_{1}}$ and
$\ell_{a_{2}}$. We observe that for any line segment
$s^{\prime}_{b}:=\\{aP+bP^{\perp}\,:\,a\in[0,1]\\}$, $0\leq b\leq 1$, there is
a line $\ell^{\prime}_{b}\subset{\overline{H_{2}}}$ such that
$\varphi(s^{\prime}_{b})\subset\ell^{\prime}_{b}$. Moreover, each
$\ell^{\prime}_{b}$, $0\leq b\leq 1$, intersects both $\ell_{a_{1}}$ and
$\ell_{a_{2}}$, as desired.

It remains to consider the case that for every pair $a_{1},a_{2}\in[0,1]$
either $\ell_{a_{1}}=\ell_{a_{2}}$, or $\ell_{a_{1}}$ and $\ell_{a_{2}}$
intersect in one point. The desired conclusion holds trivially if
$\ell_{a_{1}}=\ell_{a_{2}}$ for every pair $a_{1},a_{2}\in[0,1]$. So, the
final possibility we need to consider is that there are $a_{1},a_{2}\in[0,1]$
such that $\ell_{a_{1}}$ and $\ell_{a_{2}}$ intersect in one point $A$. Let
$0\leq a\leq 1$. If $\ell_{a}\in\\{\ell_{a_{1}},\ell_{a_{2}}\\}$, then we have
$A\in\ell_{a}$. If $\ell_{a}\notin\\{\ell_{a_{1}},\ell_{a_{2}}\\}$, then
Corollary 3.28 shows that $A\in\ell_{a}$. Thus, we get
$\varphi({\square}_{P})\subset{\mathcal{C}}_{A}$ in this case. ∎

We know that a surface is determined by any pair of lines that belong to the
surface and do not intersect. The situation is different for pairs of lines
that do intersect. To see this, we need the following lemma.

######  Lemma 3.36.

Let $P,Q\in{\mathcal{P}}$ be distinct elements. Then the points $0,\infty
P,\infty Q,{\widehat{\infty}}$ are contained in no single cone, and are
contained in the unique surface

(3.14) $\Pi_{P,Q}:=\\{aP+bQ\,:\,a,b\in{\mathbb{R}}\\}\cup\ell_{\infty
P,{\widehat{\infty}}}\cup\ell_{\infty Q,{\widehat{\infty}}}.$

###### Proof.

To show that these points are contained in no single cone, assume that
$A\in{\overline{H_{2}}}$ is coherent to these points. Since $0\sim
A\sim{\widehat{\infty}}$, we see that $A=\infty R$ for some
$R\in{\mathcal{P}}$. We also have $\infty P\sim A=\infty R\sim\infty Q$, which
implies $P=R=Q$, contradicting the assumption.

Let $\ell_{1}$ be the line passing through $0$ and $\infty P$, and $\ell_{2}$
the line passing through $\infty Q$ and ${\widehat{\infty}}$. Then
$\ell_{1}=\\{aP\,:\,a\in{{\overline{\mathbb{R}}}}\\}$ and $\ell_{2}=\\{\infty
Q+aQ^{\perp}\,:\,a\in{{\overline{\mathbb{R}}}}\\}$. For each $t\in\mathbb{R}$,
the unique point in $\ell_{2}$ that is coherent to $tP$ is $\infty Q+tP=\infty
Q+t\mathrm{tr}\,(PQ^{\perp})Q^{\perp}$. Using Lemma 3.23, we easily see that
the line passing through $tP$ and $\infty Q+tP$ is
$\\{sQ+tP\,:\,s\in{\overline{\mathbb{R}}}\\}=\\{tP+sQ\,:\,s\in\mathbb{R}\\}\cup\\{\infty
Q+t\mathrm{tr}\,(PQ^{\perp})Q^{\perp}\\}$. The unique point in $\ell_{2}$ that
is coherent to $\infty P$ is ${\widehat{\infty}}$. This completes the proof. ∎

Consider two lines $\ell_{1}=\ell_{0,\infty E_{11}},\ell_{2}=\ell_{\infty
E_{11},{\widehat{\infty}}}$ in the diagonal surface. There are infinitely many
surfaces that contain both $\ell_{1}$ and $\ell_{2}$. Indeed, for every
$P\in\mathcal{P}\setminus\\{E_{11}\\}$, we have $0,\infty
E_{11},{\widehat{\infty}}\in\Pi_{E_{11},P}$, which leads to
$\ell_{1}\cup\ell_{2}\subset\Pi_{E_{11},P}$.

####  3.6. Three points

Next, we will be interested in a relative position of any three points
$A_{1},A_{2},A_{3}\in{\overline{H_{2}}}$ with
$d(A_{1},A_{2})=d(A_{1},A_{3})=d(A_{2},A_{3})=2$.

######  Lemma 3.37.

Let $A_{1},A_{2},A_{3}\in{\overline{H_{2}}}$ satisfy
$d(A_{1},A_{2})=d(A_{1},A_{3})=d(A_{2},A_{3})=2$. Then one and only one of the
following holds.

  * •

There exists a standard automorphism
$\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ with the property that

$\varphi(A_{1})=0,\ \varphi(A_{2})={\widehat{\infty}},\ \ \text{and}\ \
\varphi(A_{3})=I.$

  * •

There exists a standard automorphism
$\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ with the property that

$\varphi(A_{1})=0,\ \ \varphi(A_{2})={\widehat{\infty}},\ \ \text{and}\ \
\varphi(A_{3})=J:=\left[\begin{matrix}1&0\cr 0&-1\cr\end{matrix}\right].$

###### Proof.

By Lemma 3.15, there is a standard automorphism
$\psi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ with the property that
$\psi(A_{1})=0$ and $\psi(A_{2})={\widehat{\infty}}$. By
$d(\psi(A_{3}),{\widehat{\infty}})=d(\psi(A_{3}),\psi(A_{2}))=2$, we get
$\psi(A_{3})\in H_{2}$. Because
$d(\psi(A_{3}),0)=d(\psi(A_{3}),\psi(A_{1}))=2$, the $2\times 2$ hermitian
matrix $\psi(A_{3})$ is invertible. In the case that $\psi(A_{3})<0$, we
replace the automorphism $\psi$ by the automorphism $X\mapsto-\psi(X)$,
$X\in{\overline{H_{2}}}$. Hence, we have either

$\psi(A_{1})=0,\ \ \psi(A_{2})={\widehat{\infty}},\ \ \text{and}\ \
\psi(A_{3})\in H_{2}^{++},$

or

$\psi(A_{1})=0,\ \ \psi(A_{2})={\widehat{\infty}},\ \ \text{and}\ \
\psi(A_{3})\in H_{2}^{+-}.$

In the first case, we can find an invertible $2\times 2$ complex matrix $S$
such that $S\psi(A_{3})S^{\ast}=I$. In the second case, we can find an
invertible $2\times 2$ complex matrix $S$ such that $S\psi(A_{3})S^{\ast}=J$.
Then the standard automorphism $\varphi(X)=S\psi(X)S^{\ast}$,
$X\in{\overline{H_{2}}}$, has one of the above two desired properties.

It remains to verify that one cannot find an automorphism of
${\overline{H_{2}}}$ that sends $0$, ${{\widehat{\infty}}}$, and $I$, to $0$,
${{\widehat{\infty}}}$, and $J$, respectively. This is a consequence of the
following lemma. ∎

######  Lemma 3.38.

We have

(3.15)
${\mathcal{C}}_{0}\cap{\mathcal{C}}_{{\widehat{\infty}}}\cap{\mathcal{C}}_{I}=\emptyset$

but

(3.16)
${\mathcal{C}}_{0}\cap{\mathcal{C}}_{{\widehat{\infty}}}\cap{\mathcal{C}}_{J}=\\{\infty
P\,:\,P^{\perp}JP^{\perp}=0\\}\neq\emptyset.$

Thus there is no automorphism $\varphi$ of ${\overline{H_{2}}}$ with

$\varphi(0)=0,\ \ \varphi({\widehat{\infty}})={\widehat{\infty}},\ \
\text{and}\ \ \varphi(I)=J.$

###### Proof.

Every element of ${\mathcal{C}}_{0}\cap{\mathcal{C}}_{{\widehat{\infty}}}$ is
of the form $\infty P$ for some $P\in{\mathcal{P}}$. It is clear that $\infty
P\notin{\mathcal{C}}_{I}$, so (3.15) is established. We have
$P^{\perp}JP^{\perp}=0$ for

$P=\frac{1}{2}\left[\begin{matrix}1&1\cr
1&1\cr\end{matrix}\right]\in{\mathcal{P}},$

thus we obtain (3.16). ∎

######  Definition 3.39.

A triple $A_{1},A_{2},A_{3}\in{\overline{H_{2}}}$ with
$d(A_{1},A_{2})=d(A_{1},A_{3})=d(A_{2},A_{3})=2$ is said to be in a _timelike_
position if the first option of Lemma 3.37 holds, and it is in a _spacelike_
position if the second option holds.

######  Remark 3.40.

Based on the above lemma, we should have spoken of ordered triples of elements
in ${\overline{H_{2}}}$, but actually there is no need to do so. Namely, there
exist automorphisms $\tau,\sigma$ of ${\overline{H_{2}}}$ such that

$\tau(0)=I,\ \ \ \tau(I)=0,\ \ \ \text{and}\ \ \
\tau({{\widehat{\infty}}})={{\widehat{\infty}}},$

and

$\sigma(0)=I,\ \ \ \sigma(I)={{\widehat{\infty}}},\ \ \ \text{and}\ \ \
\sigma({{\widehat{\infty}}})=0.$

These are

$\tau(X)=I-X\ \ \ \text{and}\ \ \ \sigma(X)=(I-X)^{-1},\ \ \
X\in{\overline{H_{2}}}.$

Putting $J$ in place of $I$, we obtain similar mappings for spacelike triples.

######  Example 3.41.

The triple $0,(1/2)I,I$ is in a timelike position. Indeed, the automorphism
$\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ given by
$\varphi(X)=-I+(I-X)^{-1}$ satisfies

$\varphi(0)=0,\ \ \varphi((1/2)I)=I,\ \ \text{and}\ \
\varphi(I)={{\widehat{\infty}}}.$

On the other hand, the triple $0,(1/2)J,J$ is in a spacelike position. Indeed,
we have

$\varphi(0)=0,\ \ \varphi((1/2)J)=J,\ \ \text{and}\ \
\varphi(J)={{\widehat{\infty}}},$

where the automorphism $\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$
is given by $\varphi(X)=-J+(J-X)^{-1}$, $X\in{\overline{H_{2}}}$.

######  Remark 3.42.

Let us recall that the triple $0,(1/2)I,I$ corresponds to spacetime events
$(0,0,0,0)$, $(0,0,0,1/2)$, and $(0,0,0,1)$, respectively. Thus, we have the
origin $(0,0,0)$ of the space $\mathbb{R}^{3}$ at three different moments,
$t_{1}=0$, $t_{2}=1/2$, and $t_{3}=1$. On the other hand, the triple
$0,(1/2)J,J$ corresponds to spacetime events $(0,0,0,0)$, $(0,0,-1/2,0)$, and
$(0,0,-1,0)$, respectively. Thus we have three different points on the line
through the origin of $\mathbb{R}^{3}$ all of them considered at the moment
$t=0$. This motivates our choice of terminology above.

######  Example 3.43.

For an invertible $A\in H_{2}$, the triple $0,{\widehat{\infty}},A$ is in a
timelike position if and only if $A\in H_{2}^{++}\cup H_{2}^{--}$. To show
this, consider the image of these points by a suitable affine automorphism
sending $0$ to $0$.

For $P\in{\mathcal{P}}$, the triple $(1/2)P,(1/2)P^{\perp},I$ is in a
spacelike position. Indeed, we have
$\mathcal{C}_{(1/2)P}\cap\mathcal{C}_{(1/2)P^{\perp}}\cap\mathcal{C}_{I}\neq\emptyset$.
A concrete description of the left-hand side follows.

######  Lemma 3.44.

Let $P\in\mathcal{P}$. Then

$\mathcal{C}_{(1/2)P}\cap\mathcal{C}_{(1/2)P^{\perp}}\cap\mathcal{C}_{I}=\\{Q+(1/3)Q^{\perp}\,:\,Q\in\mathcal{P},\,\mathrm{tr}\,(PQ)=1/2\\}.$

Note that for any two projections $P,Q\in\mathcal{P}$, we have
$\mathrm{tr}\,(PQ)=1/2$ if and only if $\mathrm{tr}\,(PQ^{\perp})=1/2$.

###### Proof.

Without loss of generality, we may assume that $P=E_{11}$. Then the set
$\\{Q\in\mathcal{P}\,:\,\mathrm{tr}\,(PQ)=1/2\\}$ is equal to

$\left\\{{1\over
2}\left[\begin{matrix}{1}&z\cr\overline{z}&{1}\cr\end{matrix}\right]\
:\,z\in{\mathbb{T}}\right\\}.$

Clearly, for every $z\in\mathbb{T}$ we have

$\left({1\over
2}\left[\begin{matrix}{1}&z\cr\overline{z}&{1}\cr\end{matrix}\right]\right)^{\perp}={1\over
2}\left[\begin{matrix}{1}&-z\cr-\overline{z}&{1}\cr\end{matrix}\right]$

and consequently,

$\left\\{Q+{1\over 3}Q^{\perp}\,:\,Q\in\mathcal{P},\,\mathrm{tr}\,(PQ)={1\over
2}\right\\}=\left\\{{1\over
3}\left[\begin{matrix}{2}&z\cr\overline{z}&{2}\cr\end{matrix}\right]\
:\,z\in\mathbb{T}\right\\}.$

In the next step, we will verify that

$\mathcal{B}:=\mathcal{C}_{(1/2)E_{11}}\cap\mathcal{C}_{{(1/2)}E_{22}}\cap\mathcal{C}_{I}\subset
H_{2}.$

Clearly, ${{\widehat{\infty}}}\not\in\mathcal{B}$. If $\infty
R+bR^{\perp}\in\mathcal{B}$ for some real number $b$ and some
$R\in\mathcal{P}$, then from $\infty R+bR^{\perp}\sim I$ we conclude that
$b=1$. But then $\infty R+R^{\perp}\sim(1/2)E_{11}$ implies
$(1/2)R^{\perp}E_{11}R^{\perp}=R^{\perp}$. Considering the trace, we get a
contradiction.

Thus, $\mathcal{B}\subset H_{2}$. Let $t,s$ be real numbers and $\alpha$ a
complex number. Then

$A=\left[\begin{matrix}t&\alpha\cr\overline{\alpha}&s\cr\end{matrix}\right]\in\mathcal{B}$

holds if and only if

$\det(A-(1/2)E_{11})=\det(A-(1/2)E_{22})=\det(A-I)=0,$

which is further equivalent to

$ts-{1\over 2}s=|\alpha|^{2},\ \ \ ts-{1\over 2}t=|\alpha|^{2},\ \ \
\text{and}\ \ \ (t-1)(s-1)=|\alpha|^{2}.$

Real numbers $t,s$ and a complex number $\alpha$ satisfy the above system of
equations if and only if $t=s=2/3$ and $|\alpha|=1/3$. Hence,
$A\in\mathcal{B}$ if and only if

$A={1\over
3}\left[\begin{matrix}{2}&z\cr\overline{z}&{2}\cr\end{matrix}\right]$

for some $z\in\mathbb{T}$. ∎

######  Lemma 3.45.

Let $A,B\in H_{2}$ satisfy $A<B$. Then $X\in H_{2}$ is coherent to some
element of $\mathcal{S}_{A,B}$ if and only if

(3.17) $X\notin(A,B),\ \ X\leq B,\ \ \text{and}\ \ X\not<A,$

or

(3.18) $X\notin(A,B),\ \ X\geq A,\ \ \text{and}\ \ X\not>B$

holds.

###### Proof.

By Lemma 3.18, we may assume that $A=0$ and $B=I$ without loss of generality.
Then we have $\mathcal{S}_{A,B}=\mathcal{S}_{0,I}=\mathcal{P}$. In that case,
it is easy to check that (3.17) or (3.18) holds if and only if one of the
eigenvalues of $X$ is in $[0,1]$ and the other is in
$(-\infty,0]\cup[1,\infty)$. We need to verify that $X$ satisfies this
condition if and only if $X$ is coherent to some element of $\mathcal{P}$.

Assume first that $X\sim P$ for some $P\in\mathcal{P}$, that is, $X=P+R$ for
some $R\in H_{2}$ of rank at most one. If $R\geq 0$, then $P\leq X$ and thus
the larger eigenvalue of $X$ is at least $1$. On the other hand, for a unit
vector $y\in\mathbb{C}^{2}$ that is perpendicular to the range of $R$, we have
$(Xy,y)=((P+R)y,y)=(Py,y)\in[0,1]$. This together with $X\geq 0$ implies that
the smaller eigenvalue of $X$ is in $[0,1]$. Similarly, $R\leq 0$ implies that
one of the eigenvalues of $X$ is in $[0,1]$ and the other is in $(-\infty,0]$.

To prove the other direction, assume that one of the eigenvalues of $X$ is in
$[0,1]$ and the other is in $(-\infty,0]\cup[1,\infty)$. Then
$X=aQ+bQ^{\perp}$ for some $Q\in{\mathcal{P}}$, $a\in[0,1]$, and
$b\in(-\infty,0]\cup[1,\infty)$. It follows that
$\det(X-Q)\cdot\det(X-Q^{\perp})\leq 0$. Combining this with the fact that
${\mathcal{P}}$ is connected, we see that there is $P\in{\mathcal{P}}$
satisfying $\det(X-P)=0$, as desired. ∎

######  Corollary 3.46.

Let $A,B\in H_{2}\subset{\overline{H_{2}}}$ satisfy $A<B$. Let $X\in H_{2}$ be
a matrix with $d(A,X)=d(B,X)=2$. Then the triple $A,B,X$ is in a spacelike
position if and only if (3.17) or (3.18) holds.

From this corollary, we see for example that a triple $A,B,C\in H_{2}$ with
$A<C<B$ is in a timelike position.

######  Corollary 3.47.

Let $A\in H_{2}\cap\mathcal{S}_{0,J}$. Then the triple $(1/2)A,(1/2)(J-A),J$
is in a timelike position.

###### Proof.

From $0\sim A\sim J$, we get $0\sim J-A\sim J$. It follows that there are
projections $P,Q\in{\mathcal{P}}$ and $a,b\in\mathbb{R}$ such that $A=aP$ and
$J-A=bQ$. We get $aP+bQ=J$. By the definition of $J$, we see that $P\neq Q$
and $0={\rm tr}\,J={\rm tr}\,(aP+bQ)=a+b$. Thus, $a=-b$. If $a<0$, then
$aI\leq aP\leq aP+bQ=J$, and hence, $a\leq-1<1\leq b$. Similarly, if $a>0$,
then $b\leq-1<1\leq a$.

Assume that $a\leq-1<1\leq b$. Then we have $A=aP<bQ=J-A$ by Lemma 2.1.
Moreover, since $J-(1/2)A=(1/2)aP+bQ$ and $J-(1/2)(J-A)=aP+(1/2)bQ$, Lemma 2.1
also implies that $J-(1/2)A,J-(1/2)(J-A)\in H_{2}^{+-}$. Thus the preceding
corollary implies that the triple $(1/2)A,(1/2)(J-A),J$ is in a timelike
position. Similarly, if $b\leq-1<1\leq a$, then we see that the triple
$(1/2)(J-A),(1/2)A,J$ is in a timelike position. ∎

####  3.7. Identity-type theorem

The set $U_{2}$ of $2\times 2$ unitaries is endowed with the usual topology as
a closed subset of the finite-dimensional vector space. We endow
${\overline{H_{2}}}$ with the topology that comes from the identification with
$U_{2}$ as in Subsection 2.2. From the discussion there, it is easy to see
that $H_{2}\subset{\overline{H_{2}}}$ is open and that the relative topology
given to the subset $H_{2}\subset{\overline{H_{2}}}$ coincides with the usual
topology of $H_{2}$.

######  Lemma 3.48.

For any pair $A,B\in{\overline{H_{2}}}$, there is a homeomorphic automorphism
$\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ satisfying
$\varphi(A)=B$.

###### Proof.

This can be easily seen by considering $U_{2}$ instead of
${\overline{H_{2}}}$. Indeed, if $V,V^{\prime}\in U_{2}$, then the mapping
$\varphi\colon U_{2}\to U_{2}$ defined by $U\mapsto V^{\prime}V^{*}U$, $U\in
U_{2}$, is clearly a homeomorphism satisfying $\varphi(V)=V^{\prime}$ and

$U\sim U^{\prime}\iff\varphi(U)\sim\varphi(U^{\prime})$

for every pair $U,U^{\prime}\in U_{2}$. ∎

######  Remark 3.49.

In fact, it turns out that every automorphism of ${\overline{H_{2}}}$ is a
homeomorphism, but we will not need such a general fact.

######  Lemma 3.50.

The interior of a cone in ${\overline{H_{2}}}$ is empty. Let
$\ell\subset{\overline{H_{2}}}$ be a line and
$\mathcal{U}\subset{\overline{H_{2}}}$ an open subset. If
$\ell\cap\mathcal{U}$ is nonempty, then it has infinitely many points.

###### Proof.

Observe that a unitary $U\in U_{2}$ is coherent to $I$ if and only if one of
the eigenvalues of $U$ is $1$. It is easily seen that the set of all such
unitaries has empty interior in $U_{2}$. This together with Lemma 3.48 proves
the first statement. Let us show the second statement. By Lemma 3.48 again, it
suffices to consider the case $0\in\ell\cap\mathcal{U}$. Then
$\ell=\\{tP\,:\,t\in{\overline{\mathbb{R}}}\\}$ for some $P\in{\mathcal{P}}$,
and $H_{2}\cap\mathcal{U}$ is an open neighborhood of $0$ in $H_{2}$. It
follows that $\ell\cap\mathcal{U}$ has infinitely many points. ∎

######  Lemma 3.51.

Let $\mathcal{U}\subset\mathcal{A}\subset{\overline{H_{2}}}$ and assume that
$\mathcal{U}$ is open in ${\overline{H_{2}}}$. Let
$\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ be a coherency preserving map.
Assume that $\varphi(X)=X$ for every $X\in\mathcal{U}$. If $A\in\mathcal{A}$
satisfies $\mathcal{C}_{A}\cap\mathcal{U}\neq\emptyset$, then $\varphi(A)=A$.

###### Proof.

Let $A\in\mathcal{A}$ satisfy $\mathcal{C}_{A}\cap\mathcal{U}\neq\emptyset$.
Since $\mathcal{U}$ is open and the interior of a line is empty, we may take a
pair of distinct lines $\ell_{1}$, $\ell_{2}$ passing through $A$ such that
$\ell_{1}\cap\mathcal{U}$ and $\ell_{2}\cap\mathcal{U}$ are nonempty. Then, by
the openness of $\mathcal{U}$ again we see that $\ell_{1}\cap\mathcal{U}$ and
$\ell_{2}\cap\mathcal{U}$ have infinitely many points. Observe that
$\varphi(\ell_{1}\cap\mathcal{A})$ and $\varphi(\ell_{2}\cap\mathcal{A})$ are
coherent sets containing
$\varphi(\ell_{1}\cap\mathcal{U})=\ell_{1}\cap\mathcal{U}$ and
$\varphi(\ell_{2}\cap\mathcal{U})=\ell_{2}\cap\mathcal{U}$, respectively. It
follows that $\varphi(\ell_{1}\cap\mathcal{A})\subset\ell_{1}$ and
$\varphi(\ell_{2}\cap\mathcal{A})\subset\ell_{2}$. Thus we conclude that

$\varphi(A)\in\varphi(\ell_{1}\cap\mathcal{A})\cap\varphi(\ell_{2}\cap\mathcal{A})\subset\ell_{1}\cap\ell_{2}=\\{A\\}.$

∎

######  Lemma 3.52.

Let $A,B,C\in H_{2}$ satisfy $A<B<C$. Let
$\varphi\colon[A,C]\to{\overline{H_{2}}}$ be a coherency preserver. Assume
that $\varphi(X)=X$ for every $X\in[A,B]$. Then $\varphi(X)=X$ for every
$X\in[A,C]$.

###### Proof.

By Lemma 3.18, we may, and we will assume that $A=0$ and $C=I$ without loss of
generality. Hence, we are considering $B\in H_{2}$ with $0<B<I$ and a
coherency preserver $\varphi\colon[0,I]\to{\overline{H_{2}}}$ satisfying
$\varphi(X)=X$ for every $X\in[0,B]$. Let $t_{0}$ be the smaller eigenvalue of
$B$. Then $0<t_{0}<1$.

We consider the subset $\mathcal{U}_{1}:=(0,t_{0}I)\subset[0,I]$ that is open
in ${\overline{H_{2}}}$. Note that $\mathcal{U}_{1}\subset[0,B]$. If one of
the eigenvalues of a matrix $A\in[0,I]$ lies in $(0,t_{0})$, then
$\mathcal{C}_{A}\cap\mathcal{U}_{1}$ is clearly nonempty, so Lemma 3.51
implies $\varphi(A)=A$. Next we define the open subset $\mathcal{U}_{2}$ of
${\overline{H_{2}}}$ to be the set formed of all matrices $X$ in $(0,I)$ such
that $\sigma(X)\cap(0,t_{0})\neq\emptyset$. We already know that
$\varphi(X)=X$ for every $X\in\mathcal{U}_{2}$. If
$A\in[0,I]\setminus(\\{0,I\\}\cup\mathcal{P})$, then it is easy to see that
$\mathcal{C}_{A}\cap\mathcal{U}_{2}$ is nonempty, thus we obtain
$\varphi(A)=A$ again by Lemma 3.51. We know $\varphi(0)=0$ from the
assumption. From $\varphi(P+tP^{\perp})=P+tP^{\perp}$ for every $t\in(0,1)$
and $P\in\mathcal{P}$, it clearly follows that $\varphi(I)=I$. Finally, for
$P\in\mathcal{P}$ we know that
$\varphi(P)\in{\mathcal{S}}_{\varphi(0),\varphi(I)}=\mathcal{P}$ and
$\varphi(P)\sim\varphi((1/2)P)=(1/2)P$, and therefore $\varphi(P)=P$. ∎

######  Corollary 3.53.

Let $A,B,C\in H_{2}$ with $A<B<C$. Let
$\varphi\colon[A,C]\to{\overline{H_{2}}}$ be a coherency preserver. Assume
that $\varphi(X)=X$ for every $X\in[B,C]$. Then $\varphi(X)=X$ for every
$X\in[A,C]$.

###### Proof.

We define a coherency preserver
$\varphi_{1}\colon[0,C-A]\to{\overline{H_{2}}}$ by

$\varphi_{1}(X)=C-\varphi(C-X),\ \ \ X\in[0,C-A].$

It is straightforward to check that the restriction of $\varphi_{1}$ to
$[0,C-B]$ is the identity map. By Lemma 3.52, $\varphi_{1}$ is the identity on
the whole matrix interval $[0,C-A]$, which is equivalent to $\varphi(X)=X$ for
every $X\in[A,C]$. ∎

In the proof of the next corollary, we will use the following simple fact: Let
$\mathcal{U}\subset H_{2}$ be an open subset and $A\in\mathcal{U}$. Then there
exist $B,C\in H_{2}$ such that $B<A<C$ and $[B,C]\subset\mathcal{U}$. Indeed,
we only need to observe that for each $\varepsilon>0$ we have $[A-\varepsilon
I,A+\varepsilon I]=\\{D\in H_{2}\,:\,\|D-A\|\leq\varepsilon\\}$.

######  Corollary 3.54.

Let $A,B\in H_{2}$ with $A<B$. Let $\varphi\colon[A,B]\to{\overline{H_{2}}}$
be a coherency preserver. Assume that the set
$\\{X\in[A,B]\,:\,\varphi(X)=X\\}$ as a subset of $H_{2}$ has nonempty
interior. Then $\varphi(X)=X$ for every $X\in[A,B]$.

###### Proof.

By the assumption together with the remark above, we may take $C,D\in(A,B)$
with $C<D$ such that $\varphi(X)=X$ for every $X\in[C,D]$. By Lemma 3.52,
$\varphi$ is the identity on the matrix interval $[C,B]$. By Corollary 3.53,
$\varphi$ is the identity on the whole matrix interval $[A,B]$. ∎

######  Corollary 3.55.

Let $A,B\in H_{2}$ with $A<B$. Let $\varphi\colon[A,B]\to{\overline{H_{2}}}$
be a coherency preserver. Assume that there exists an automorphism $\psi$ of
${\overline{H_{2}}}$ such that the set
$\\{X\in[A,B]\,:\,\varphi(X)=\psi(X)\\}$ as a subset of $H_{2}$ has nonempty
interior. Then $\varphi(X)=\psi(X)$ for every $X\in[A,B]$.

###### Proof.

After introducing a new coherency preserver defined by

$X\mapsto\psi^{-1}(\varphi(X)),\ \ \ X\in[A,B],$

we may assume with no loss of generality that $\psi$ is the identity. Then the
desired conclusion follows directly from Corollary 3.54. ∎

We continue with an identity-type theorem.

######  Theorem 3.56.

Let $\mathcal{A}$ be a nonempty connected open subset of ${\overline{H_{2}}}$
and let $\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ be a coherency
preserving map. Let $\mathcal{D}$ be a nonempty open subset of $\mathcal{A}$.
If an automorphism $\psi$ of ${\overline{H_{2}}}$ satisfies
$\varphi(X)=\psi(X)$ for every $X\in\mathcal{D}$, then $\varphi(X)=\psi(X)$
for every $X\in\mathcal{A}$.

###### Proof.

We denote by $\mathcal{L}\subset\mathcal{A}$ the set of all points
$A\in\mathcal{A}$ with the property that $\varphi(X)=\psi(X)$ for every $X$ in
some neighborhood of $A$. It is trivial to see that
$\mathcal{D}\subset\mathcal{L}$ and that $\mathcal{L}$ is an open subset of
$\mathcal{A}$. It remains to prove that $\mathcal{L}$ is a closed subset of
$\mathcal{A}$.

Hence, assume that $A\in\mathcal{A}$ and that there exists a sequence
$A_{n}\in\mathcal{L}$, $n\geq 1$, such that $\lim A_{n}=A$. By Lemma 3.48,
there is a homeomorphic automorphism
$\psi_{1}\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ satisfying
$\psi_{1}(A)=0$. Thus $\psi_{1}(A_{n})$ tends to $0$ as $n\to\infty$. Let
$\varepsilon$ be a positive number such that $[-\varepsilon I,\varepsilon
I]\subset\psi_{1}(\mathcal{A})$. There exists a positive integer $m$ such that
$\psi_{1}(A_{m})\in(-\varepsilon I,\varepsilon I)$. By Corollary 3.55, we have
$\varphi\circ\psi_{1}^{-1}(X)=\psi\circ\psi_{1}^{-1}(X)$ for every
$X\in[-\varepsilon I,\varepsilon I]$. Thus, we get $A\in\mathcal{L}$. This
completes the proof. ∎

######  Remark 3.57.

The assumption of openness in Theorem 3.56 is essential. For example, since
$tI\not\sim sI$ for distinct $s,t\in\mathbb{R}$, we see that any mapping from
$\\{tI\,:\,t\in\mathbb{R}\\}$ to ${\overline{H_{2}}}$ preserves coherency even
when $\varphi(tI)=tI$ for every $t>0$.

The assumption of connectedness is also essential. For example, consider the
mapping $\varphi\colon H_{2}^{++}\cup H_{2}^{--}\to H_{2}$ defined by

$\varphi(X)=X,\ \ X\in H_{2}^{++},\ \ \text{and}\ \
\varphi(H_{2}^{--})=\\{0\\}.$

Since $X\not\sim Y$ for any $X\in H_{2}^{++}$ and $Y\in H_{2}^{--}$, we see
that $\varphi$ preserves coherency. The identity mapping on
${\overline{H_{2}}}$ is an automorphism that extends $\varphi|_{H_{2}^{++}}$,
but $\varphi$ is not the identity mapping on $H_{2}^{--}$.

We close this section with two lemmas of the nature of the identity theorem.

######  Lemma 3.58.

Let $A,B\in{\overline{H_{2}}}$ satisfy $d(A,B)=2$. If $C\in{\overline{H_{2}}}$
has the property $C\sim X$ for every $X\in\mathcal{S}_{A,B}$ then
$C\in\\{A,B\\}$.

###### Proof.

Without loss of generality, we may assume $A=0$ and $B={\widehat{\infty}}$. In
this case, we have $\mathcal{S}_{A,B}=\\{\infty P\,:\,P\in\mathcal{P}\\}$, and
it is easy to get to the desired conclusion. ∎

######  Lemma 3.59.

Let $A\in H_{2}$, $0<A\leq I$, and $B\in{\overline{H_{2}}}$. Assume that for
every $P\in\mathcal{P}$ and every real $s\in[0,1]$ we have

$A\sim sP\Rightarrow B\sim sP.$

Then $B=A$ or $B=0$.

###### Proof.

By Lemma 3.58, it suffices to show that
$\mathcal{S}_{0,A}\subset\\{sP\,:\,s\in[0,1],P\in\mathcal{P}\\}$. Using the
standard automorphism $X\mapsto A^{1/2}XA^{1/2}$, $X\in{\overline{H_{2}}}$, we
see

$\mathcal{S}_{0,A}=\\{A^{1/2}PA^{1/2}\,:\,P\in\mathcal{S}_{0,I}\\}.$

Observe that for $P\in\mathcal{S}_{0,I}=\mathcal{P}$ the norm of the rank-one
positive operator $A^{1/2}PA^{1/2}$ is at most $1$, which leads to the desired
claim. ∎

##  Part II Main results and their proofs

###  4\. Standard coherency preservers

Let $\mathcal{A}$ be a subset of ${\overline{H_{2}}}$ and
$\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ a map. We say that $\varphi$
is a _standard coherency preserver_ if $\varphi$ extends to a standard
automorphism of ${\overline{H_{2}}}$. The main result of the current section
is the following one.

######  Theorem A.

Let $\mathcal{A}$ be a connected open subset of ${\overline{H_{2}}}$ and
$\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ a coherency preserving map.
Assume that there exist $A,B,C\in\mathcal{A}\cap H_{2}$ such that

  * •

$A<C<B$,

  * •

$[A,B]\subset\mathcal{A}$, and

  * •

the triple $\varphi(A),\varphi(C),\varphi(B)$ is in a timelike position.

Then $\varphi$ is standard.

####  4.1. Proof of Theorem A

We first show the following proposition.

######  Proposition 4.1.

Let $\varphi\colon[0,I]\to{\overline{H_{2}}}$ be a coherency preserver
satisfying

(4.1) $\varphi(0)=0,\ \ \ \varphi((1/2)I)=(1/2)I,\ \ \ \text{and}\ \ \
\varphi(I)=I.$

Then there exists a unitary $U\in U_{2}$ satisfying either

$\varphi(X)=UXU^{*},\ \ \ X\in[0,I],$

or

$\varphi(X)=UX^{t}U^{*},\ \ \ X\in[0,I].$

To prove this, we assume that $\varphi\colon[0,I]\to{\overline{H_{2}}}$ is a
coherency preserver satisfying (4.1). By Lemma 3.12, we have
$\varphi(\mathcal{P})\subset\mathcal{P}$. Lemma 3.12 also shows that the
triple $0,(1/2)I,I$ is not contained in any cone.

We fix $P\in\mathcal{P}$. Since $0,(1/2)I,I\in\varphi({\square}_{P})$, where
${\square}_{P}$ is defined as in (3.13), Lemma 3.35 yields that
$\varphi({\square}_{P})$ is contained in some surface. By Lemma 3.33, we have

$\varphi({\square}_{P})\subset\\{a\varphi(P)+b\varphi(P)^{\perp}\,:\,a,b\in{{\overline{\mathbb{R}}}}\\}.$

######  Claim 4.2.

The equation

(4.2) $\varphi(aP+bP^{\perp})=a\varphi(P)+b\varphi(P)^{\perp}$

holds for any $a,b\in\\{0,1/2,1\\}$.

###### Proof.

We have $\varphi((1/2)P)\sim\varphi((1/2)I)=(1/2)I$ and
$\varphi((1/2)P)\sim\varphi(0)=0$, and by a simple modification of Lemma 3.12
we see that $\varphi((1/2)P)=(1/2)Q$ for some $Q\in\mathcal{P}$. But
$\varphi((1/2)P)\sim\varphi(P)\in\mathcal{P}$ and therefore,

$\varphi((1/2)P)=(1/2)\varphi(P).$

We continue with $\varphi((1/2)P+P^{\perp})$. By

$(1/2)P+P^{\perp}\sim(1/2)I,I,(1/2)P,$

we get

$\varphi((1/2)P+P^{\perp})\sim(1/2)I,I,(1/2)\varphi(P).$

Using Corollary 3.27, we conclude that

$\varphi((1/2)P+P^{\perp})=(1/2)\varphi(P)+\varphi(P)^{\perp}.$

Now,

$P^{\perp}\sim 0,I,(1/2)P+P^{\perp}$

yields

$\varphi(P^{\perp})\sim 0,I,(1/2)\varphi(P)+\varphi(P)^{\perp},$

and thus

$\varphi(P^{\perp})=\varphi(P)^{\perp}.$

Using the same arguments as above, we see that

$\varphi((1/2)P^{\perp})=(1/2)\varphi(P)^{\perp}\ \ \ \text{and}\ \ \
\varphi(P+(1/2)P^{\perp})=\varphi(P)+(1/2)\varphi(P)^{\perp}.$

∎

######  Claim 4.3.

The equation (4.2) holds for any $a,b\in\\{0,1/3,1\\}$.

###### Proof.

Take $Q\in\mathcal{P}$ with $\mathrm{tr}\,(PQ)=1/2$. Claim 4.2 implies that

$\varphi((1/2)Q)=(1/2)\varphi(Q),\ \
\varphi((1/2)Q^{\perp})=(1/2)\varphi(Q)^{\perp}.$

By Lemma 3.44, we have

$P+(1/3)P^{\perp}\in\mathcal{C}_{(1/2)Q}\cap\mathcal{C}_{(1/2)Q^{\perp}}\cap\mathcal{C}_{I}.$

Thus we obtain

(4.3)
$\begin{split}\varphi(P+(1/3)P^{\perp})&\in\mathcal{C}_{(1/2)\varphi(Q)}\cap\mathcal{C}_{(1/2)\varphi(Q)^{\perp}}\cap\mathcal{C}_{I}\\\
&=\\{R+(1/3)R^{\perp}\,:\,R\in\mathcal{P},\,\mathrm{tr}\,(\varphi(Q)R)=1/2\\}.\end{split}$

Because $P+(1/3)P^{\perp}\in\ell_{P,I}$, we have

(4.4)
$\varphi(P+(1/3)P^{\perp})\in\ell_{\varphi(P),\varphi(I)}=\ell_{\varphi(P),I}=\\{\varphi(P)+a\varphi(P)^{\perp}\,:\,a\in{{\overline{\mathbb{R}}}}\\}.$

It follows from (4.3) and (4.4) that

$\varphi(P+(1/3)P^{\perp})=\varphi(P)+(1/3)\varphi(P)^{\perp},$

and similarly,

$\varphi(P^{\perp}+(1/3)P)=\varphi(P)^{\perp}+(1/3)\varphi(P).$

Further, $\varphi((1/3)P)\in\ell_{0,\varphi(P)}$ and
$\varphi((1/3)P)\sim\varphi(P)^{\perp}+(1/3)\varphi(P)$. It follows that

$\varphi((1/3)P)=(1/3)\varphi(P)\ \ \ \text{and similarly,}\ \ \
\varphi((1/3)P^{\perp})=(1/3)\varphi(P)^{\perp}.$

Finally, $\varphi((1/3)I)$ belongs to the lines

$\ell_{(1/3)\varphi(P),(1/3)\varphi(P)+\varphi(P)^{\perp}}\not=\ell_{(1/3)\varphi(P)^{\perp},(1/3)\varphi(P)^{\perp}+\varphi(P)}$

which intersect in the point $(1/3)I$. Thus, $\varphi((1/3)I)=(1/3)I$. ∎

In what follows, we sometimes make use of the coherency preserving map
$\varphi_{1}\colon[0,I]\to{\overline{H_{2}}}$ defined by
$\varphi_{1}(X)=I-\varphi(I-X)$, $X\in[0,I]$. Clearly,

$\varphi_{1}(0)=0,\ \ \varphi_{1}((1/2)I)=(1/2)I,\ \ \varphi_{1}(I)=I,\ \
\text{and}\ \ \varphi_{1}(P)=\varphi(P).$

######  Claim 4.4.

The equation (4.2) holds for any $a,b\in\\{0,1/3,2/3,1\\}$.

###### Proof.

Applying Claim 4.3 to the mapping $\varphi_{1}$ in place of $\varphi$, we have
(4.2) for every $a,b\in\\{0,2/3,1\\}$.

Let $a,b\in\\{0,1/3,2/3,1\\}$. We have shown that (4.2) holds when one of the
numbers $a,b$ equals $0$ or $1$. In the remaining cases, we have
$\\{aP+bP^{\perp}\\}=\ell_{aP,aP+P^{\perp}}\cap\ell_{bP^{\perp},P+bP^{\perp}}$.
Thus

$\begin{split}\varphi(aP+bP^{\perp})&\in\ell_{\varphi(aP),\varphi(aP+P^{\perp})}\cap\ell_{\varphi(bP^{\perp}),\varphi(P+bP^{\perp})}\\\
&=\ell_{a\varphi(P),a\varphi(P)+\varphi(P)^{\perp}}\cap\ell_{b\varphi(P)^{\perp},\varphi(P)+b\varphi(P)^{\perp}}\\\
&=\\{a\varphi(P)+b\varphi(P)^{\perp}\\},\end{split}$

and consequently, $\varphi(aP+bP^{\perp})=a\varphi(P)+b\varphi(P)^{\perp}$. ∎

######  Claim 4.5.

For every positive integer $n$, the equation (4.2) holds for any

$a,b\in\left\\{0,{1\over 3}\left({2\over 3}\right)^{n-1},\left({2\over
3}\right)^{n}\right\\}.$

###### Proof.

The case $n=1$ is true by the previous claim. So, assume that the above is
true for some positive integer $n$. We introduce a new coherency preserving
map $\tau\colon[0,I]\to{\overline{H_{2}}}$ by

$\tau(X)=\left(3/2\right)^{n}\varphi\left(\left(2/3\right)^{n}X\right),\ \ \
X\in[0,I].$

Using the induction hypothesis, we see that

$\tau(0)=0,\ \ \ \tau((1/2)I)=(1/2)I,\ \ \ \text{and}\ \ \ \tau(I)=I.$

Moreover, $\tau(P)=\varphi(P)$. But then we already know that

$\tau(a^{\prime}P+b^{\prime}P^{\perp})=a^{\prime}\varphi(P)+b^{\prime}\varphi(P)^{\perp}$

for any $a^{\prime},b^{\prime}\in\\{0,1/3,2/3\\}$. This immediately implies
that (4.2) holds for any

$a,b\in\left\\{0,{1\over 3}\left({2\over 3}\right)^{n},\left({2\over
3}\right)^{n+1}\right\\},$

as desired. ∎

######  Claim 4.6.

For every positive integer $n$, we have

(4.5)
$\varphi\left(\left(1-\left(\frac{2}{3}\right)^{n}\right)P\right)=\left(1-\left(\frac{2}{3}\right)^{n}\right)\varphi(P)$

and

(4.6)
$\varphi\left(P+\left(\frac{2}{3}\right)^{n}P^{\perp}\right)=\varphi(P)+\left(\frac{2}{3}\right)^{n}\varphi(P)^{\perp}.$

###### Proof.

We know that

(4.7) $\varphi((2/3)^{n}P)=(2/3)^{n}\varphi(P)\ \ \ \text{and}\ \ \
\varphi((2/3)^{n}P^{\perp})=(2/3)^{n}\varphi(P)^{\perp}$

for every positive integer $n$.

Using (4.7) for the coherency preserving map $\varphi_{1}$, we arrive at

$\varphi(P+(1-(2/3)^{n})P^{\perp})=\varphi(P)+(1-(2/3)^{n})\varphi(P)^{\perp}$

and

$\varphi((1-(2/3)^{n})P+P^{\perp})=(1-(2/3)^{n})\varphi(P)+\varphi(P)^{\perp}$

for every positive integer $n$.

We now use the fact that

$\varphi((1-(2/3)^{n})P)\sim
0,\,\varphi(P),\,\varphi((1-(2/3)^{n})P+P^{\perp})$

to deduce that

$\varphi((1-(2/3)^{n})P)=(1-(2/3)^{n})\varphi(P)$

and similarly,

$\varphi((1-(2/3)^{n})P^{\perp})=(1-(2/3)^{n})\varphi(P)^{\perp}$

for every positive integer $n$.

The last equation remains true if we replace $\varphi$ by $\varphi_{1}$. This
yields

$\varphi(P+(2/3)^{n}P^{\perp})=\varphi(P)+(2/3)^{n}\varphi(P)^{\perp}$

for every positive integer $n$. ∎

######  Claim 4.7.

There is a $2\times 2$ unitary matrix $U\in U_{2}$ such that

$\text{either}\ \ \varphi(P)=UPU^{\ast},\ \ P\in\mathcal{P},\ \ \text{or}\ \
\varphi(P)=UP^{t}U^{\ast},\ \ P\in\mathcal{P}.$

###### Proof.

Let $a$ be a positive real number $<1/2$. Using Lemmas 3.13 and 3.14, we see
that a pair $P,Q\in\mathcal{P}$ satisfies

$([0,I]\supset)\,\,\mathcal{S}_{(1-a)P,P+aP^{\perp}}\cap\mathcal{S}_{(1-a)Q,Q+aQ^{\perp}}\not=\emptyset$

if and only if

$\|P-Q\|\leq{a\over 1-a}.$

Therefore, the two equations (4.5) and (4.6) imply that for any positive
integer $n>2$, we have

$\|P-Q\|\leq{(2/3)^{n}\over
1-(2/3)^{n}}\Rightarrow\|\varphi(P)-\varphi(Q)\|\leq{(2/3)^{n}\over
1-(2/3)^{n}}.$

Applying (3.9), we conclude that there exists a sequence of positive real
numbers $a_{n}$, $n\geq 1$, such that $\lim a_{n}=0$ and

$d^{g}(P,Q)\leq a_{n}\Rightarrow d^{g}(\varphi(P),\varphi(Q))\leq a_{n}$

for every positive integer $n$.

Let $P,Q\in\mathcal{P}$, $P\not=Q$, and $\varepsilon>0$. Take a positive
integer $n$ such that $a_{n}<\varepsilon$. Find a positive integer $k$ such
that

$d^{g}(P,Q)\leq ka_{n}<d^{g}(P,Q)+\varepsilon.$

By Lemma 3.8, we can find rank one projections
$P_{0}=P,P_{1},\ldots,P_{k-1},P_{k}=Q$ such that $d^{g}(P_{j-1},P_{j})\leq
a_{n}$. But then

$\begin{split}d^{g}(\varphi(P),\varphi(Q))&\leq
d^{g}(\varphi(P_{0}),\varphi(P_{1}))+\cdots+d^{g}(\varphi(P_{k-1}),\varphi(P_{k}))\\\
&\leq ka_{n}<d^{g}(P,Q)+\varepsilon.\end{split}$

Therefore,

$d^{g}(\varphi(P),\varphi(Q))\leq d^{g}(P,Q),\ \ \ P,Q\in\mathcal{P}.$

Applying (3.9) once more, we conclude that

$\|\varphi(P)-\varphi(Q)\|\leq\|P-Q\|,\ \ \ P,Q\in\mathcal{P}.$

Finally, it follows from Lemma 3.11 and
$\varphi(P^{\perp})=\varphi(P)^{\perp}$ that

$\|\varphi(P)-\varphi(Q)\|=\|P-Q\|,\ \ \ P,Q\in\mathcal{P}.$

Now, Wigner’s theorem (Theorem 3.9, see also Remark 3.10) completes the proof.
∎

Hence, after composing $\varphi$ with the standard automorphism of
${\overline{H_{2}}}$ given by $X\mapsto U^{\ast}XU$, and possibly with the
standard automorphism of ${\overline{H_{2}}}$ given by $X\mapsto X^{t}$, we
get to the situation where

$\varphi(P)=P,\ \ \ P\in\mathcal{P}.$

The following claim finally completes the proof of Proposition 4.1.

######  Claim 4.8.

Under the above situation, we have $\varphi(A)=A$ for every $A\in[0,I]$.

###### Proof.

Fix $P\in\mathcal{P}$ and a real number $t$, $1/3\leq t\leq 1$. By $0\sim
tP\sim P$, $\varphi(0)=0$, and $\varphi(P)=P$, there exists
$s\in{{\overline{\mathbb{R}}}}$ such that $\varphi(tP)=sP$. It follows from
Claim 4.3 that for every $Q\in\mathcal{P}$ we have

$\varphi((1/3)Q+Q^{\perp})=(1/3)Q+Q^{\perp}.$

Further,

$(1/3)Q+Q^{\perp}-tP=((1/3)Q+Q^{\perp})\,(I-(3Q+Q^{\perp})(tP)).$

Therefore, by Lemma 2.2, the condition $(1/3)Q+Q^{\perp}\sim tP$ holds if and
only if

$1=\mathrm{tr}\,((3Q+Q^{\perp})(tP))=\mathrm{tr}\,((2Q+I)(tP))=t(1+2\mathrm{tr}\,(QP)).$

Because $(1-t)/(2t)\in[0,1]$, we can find a projection $Q\in\mathcal{P}$ such
that $\mathrm{tr}\,(QP)=(1-t)/(2t)$, and then by the above,
$(1/3)Q+Q^{\perp}\sim tP$. It follows that $(1/3)Q+Q^{\perp}\sim sP$, which
further yields that $t=s$.

Hence, for every real $t$, $1/3\leq t\leq 1$, and every $P\in\mathcal{P}$, we
have $\varphi(tP)=tP$. Using the coherency preserving map $\varphi_{1}\colon
X\mapsto I-\varphi(I-X)$, we obtain $\varphi(tP+P^{\perp})=tP+P^{\perp}$ for
every real $t$, $0\leq t\leq 2/3$, and every $P\in\mathcal{P}$. Since
$\varphi(tP)\sim 0,\varphi(P),\varphi(tP+P^{\perp})$, we get $\varphi(tP)=tP$
when $0\leq t\leq 2/3$. We conclude that

$\varphi(tP)=tP,\ \ \ P\in\mathcal{P},\ \,t\in[0,1].$

Let now $A$ be any element of $[0,I]$ of rank two. Applying Lemma 3.59 with
$B=\varphi(A)$, we see that $\varphi(A)=A$ or $\varphi(A)=0$. We now use the
obtained result for the map $\varphi_{1}\colon X\mapsto I-\varphi(I-X)$ to
conclude that $\varphi(A)\in\\{A,I\\}$ for every $A\in[0,I]$. Hence we obtain
$\varphi(A)=A$ for every $A\in[0,I]$. ∎

######  Corollary 4.9.

Let $\varphi:[0,I]\to{\overline{H_{2}}}$ be a coherency preserver with the
property that $\varphi(0)$, $\varphi((1/2)I)$, and $\varphi(I)$ are in a
timelike position. Then $\varphi$ is standard.

###### Proof.

This statement is a straightforward consequence of Proposition 4.1 and Example
3.41. ∎

###### Proof of Theorem A.

Suppose that $\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ satisfies the
assumption of Theorem A. By Corollary 3.21, there is a standard automorphism
$\psi_{1}$ of ${\overline{H_{2}}}$ satisfying

$\psi_{1}([0,I])=[A,B],\ \ \psi_{1}(0)=A,\ \ \psi_{1}(I)=B,\ \ \text{and}\ \
\psi_{1}((1/2)I)=C.$

It follows that the restriction of the coherency preserver
$\varphi\circ\psi_{1}$ to $[0,I]$ satisfies the assumption of Corollary 4.9.
Thus, Corollary 4.9 and Theorem 3.56 imply that $\varphi\circ\psi_{1}$ is
standard, which further shows that $\varphi$ is standard, too. ∎

####  4.2. Applications of Theorem A

We begin with a useful lemma.

######  Lemma 4.10.

Let $\mathcal{U}$ be an open connected subset of $H_{2}$. For any pair
$X,Y\in\mathcal{U}$, there are $X_{1}=X,X_{2},\ldots,X_{n}=Y\in\mathcal{U}$
such that $X_{1}\sim X_{2}\sim\cdots\sim X_{n}$.

###### Proof.

For $X,Y\in\mathcal{U}$, we write $X\approx Y$ if there are
$X_{1}=X,X_{2},\ldots,X_{n}=Y\in\mathcal{U}$ such that $X_{1}\sim
X_{2}\sim\cdots\sim X_{n}$. It is clear that $\approx$ is an equivalence
relation.

First, let us consider the case where $\mathcal{U}$ is an open interval. When
$X<Y$, then we have $\mathcal{S}_{X,Y}\subset\mathcal{U}$ by Corollary 3.19.
We may take any point $Z\in\mathcal{S}_{X,Y}\subset\mathcal{U}$, and then
$X\sim Z\sim Y$. Thus $X\approx Y$. If $X,Y$ are arbitrary, then we may take
$Z\in\mathcal{U}$ such that $X<Z$, $Y<Z$, hence we obtain $X\approx Z\approx
Y$. Thus the proof is complete when $\mathcal{U}$ is an open interval.

Let $\mathcal{U}$ be an arbitrary open connected subset of $H_{2}$. Fix
$X_{0}\in\mathcal{U}$, and let $\mathcal{U}_{0}$ denote the collection of
$Y\in\mathcal{U}$ with $X_{0}\approx Y$. Then the preceding paragraph shows
that $\mathcal{U}_{0}$ and $\mathcal{U}\setminus\mathcal{U}_{0}$ are open.
Since $\mathcal{U}$ is connected, we obtain $\mathcal{U}_{0}=\mathcal{U}$. ∎

Let us prove the fundamental theorem of chronogeometry as a consequence of the
preceding subsection.

######  Theorem 4.11.

Let $\varphi\colon H_{2}\to H_{2}$ be a bijective map satisfying

$A\sim B\iff\varphi(A)\sim\varphi(B)$

for every pair $A,B\in H_{2}$. Then $\varphi$ extends to an affine
automorphism.

###### Proof.

After composing $\varphi$ with a translation, we can assume with no loss of
generality that $\varphi(0)=0$. As before, we define
$d(A,B)=\mathrm{rank}\,(A-B)\in\\{0,1,2\\}$, $A,B\in H_{2}$. It is clear that
we have $d(A,B)=d(\varphi(A),\varphi(B))$, $A,B\in H_{2}$. The set $H_{2}^{2}$
of invertible matrices in $H_{2}$ is formed of $A\in H_{2}$ with the property
$d(A,0)=2$. Thus we get

$\varphi(H_{2}^{2})=H_{2}^{2}.$

Recall that $H_{2}^{2}$ is the disjoint union of three sets
$H_{2}^{++},H_{2}^{--}$, and $H_{2}^{+-}$. We first claim that

$\varphi(H_{2}^{++}\cup H_{2}^{--})=H_{2}^{++}\cup H_{2}^{--}.$

The above equality follows directly from the following characterization of the
set $H_{2}^{++}\cup H_{2}^{--}$: An invertible matrix $A\in H_{2}$ belongs to
$H_{2}^{++}\cup H_{2}^{--}$ if and only if for each line $\ell$ passing
through zero there exists $B\in\ell\cap H_{2}$ such that $A\sim B$. Indeed,
let $A\in H_{2}^{++}$ and $\ell=\\{aP\,:\,a\in{{\overline{\mathbb{R}}}}\\}$
where $P\in\mathcal{P}$. Then $A^{-1}>0$ and therefore
$\mathrm{tr}\,(A^{-1}P)=\mathrm{tr}\,(PA^{-1}P)>0$. It follows that there
exists exactly one (positive) $a\in\mathbb{R}$ such that
$\mathrm{tr}\,(A^{-1}(aP))=1$, or equivalently, $A\sim aP$ (see Lemma 2.2). In
the same way we treat the case when $A\in H_{2}^{--}$. So, assume finally that
$A\in H_{2}^{+-}$. Then $A^{-1}$ is again in $H_{2}^{+-}$. Therefore, there
exists a unit vector $x\in\mathbb{C}^{2}$ such that $x^{\ast}A^{-1}x=0$,
yielding that $\mathrm{tr}\,(A^{-1}Q)=0$, where $Q=xx^{\ast}$. It follows from
Lemma 2.2 that there is no point in $\\{aQ\,:\,a\in\mathbb{R}\\}$ that is
coherent to $A$.

Observe that if $A,B\in H_{2}^{++}\cup H_{2}^{--}$ and $A\sim B$, then either
$A,B\in H_{2}^{++}$, or $A,B\in H_{2}^{--}$. Moreover, $H_{2}^{++}$ and
$H_{2}^{--}$ are open and connected sets. Thus Lemma 4.10 implies that

$\varphi(H_{2}^{++})=H_{2}^{++}\ \ \ \text{and}\ \ \
\varphi(H_{2}^{--})=H_{2}^{--}\,;$

or

$\varphi(H_{2}^{++})=H_{2}^{--}\ \ \ \text{and}\ \ \
\varphi(H_{2}^{--})=H_{2}^{++}\,.$

Therefore, we get $\varphi(-I)<0=\varphi(0)<\varphi(I)$ or
$\varphi(-I)>0=\varphi(0)>\varphi(I)$. Hence Corollary 3.46 implies that the
triple $\varphi(-I),\varphi(0),\varphi(I)$ is in a timelike position. Theorem
A shows that $\varphi$ extends to an automorphism $\hat{\varphi}$ of
${\overline{H_{2}}}$. Note that $\hat{\varphi}(H_{2})=H_{2}$ implies
$\hat{\varphi}({\widehat{\infty}})={\widehat{\infty}}$. It remains to show
that $\hat{\varphi}$ is an affine automorphism. By composing $\hat{\varphi}$
with an appropriate affine automorphism of ${\overline{H_{2}}}$, we may assume
$\hat{\varphi}(0)=0$ and $\hat{\varphi}(I)=I$ without loss of generality.
(Recall that the composition of affine automorphisms is again an affine
automorphism by Lemma 3.7.)

Then the automorphism $\psi\colon X\mapsto X^{-1}-I$ of ${\overline{H_{2}}}$
satisfies

$\psi^{-1}\circ\hat{\varphi}\circ\psi(0)=\psi^{-1}\circ\hat{\varphi}({\widehat{\infty}})=\psi^{-1}({\widehat{\infty}})=0,$
$\psi^{-1}\circ\hat{\varphi}\circ\psi(I)=\psi^{-1}\circ\hat{\varphi}(0)=\psi^{-1}(0)=I,\
\ \text{and}$
$\psi^{-1}\circ\hat{\varphi}\circ\psi((1/2)I)=\psi^{-1}\circ\hat{\varphi}(I)=\psi^{-1}(I)=(1/2)I.$

Now we apply Proposition 4.1. We see that there is a $2\times 2$ unitary
matrix $U$ such that $\psi^{-1}\circ\hat{\varphi}\circ\psi(X)=UXU^{*}$ holds
for every $X\in[0,I]$, or
$\psi^{-1}\circ\hat{\varphi}\circ\psi(X)=UX^{t}U^{*}$ holds for every
$X\in[0,I]$. By the identity-type theorem (Theorem 3.56), the same equality
holds for every $X\in{\overline{H_{2}}}$. It is easy to see that
$\psi(UXU^{*})=U\psi(X)U^{*}$ and $\psi(X^{t})=\psi(X)^{t}$ for every
$X\in{\overline{H_{2}}}$. It follows that $\hat{\varphi}(X)=UXU^{*}$ holds for
every $X\in{\overline{H_{2}}}$, or $\hat{\varphi}(X)=UX^{t}U^{*}$ holds for
every $X\in{\overline{H_{2}}}$. ∎

###### Proof of Theorem 1.1.

Assume that $\phi\colon{\mathcal{M}}_{4}\to{\mathcal{M}}_{4}$ is a bijection
that preserves lightlikeness in both directions. We need to show that $\phi$
is of the form (1.3). We may assume $\phi(0)=0$ without loss of generality.
Let $\xi\colon\mathcal{M}_{4}\to H_{2}$ be the mapping as in (1.4). Then the
mapping $\varphi:=\xi\circ\phi\circ\xi^{-1}\colon H_{2}\to H_{2}$ clearly
satisfies the assumption of Theorem 4.11. It follows that $\varphi$ extends to
an affine automorphism. In particular, we see that $\varphi$ is linear. It
follows that $\phi=\xi^{-1}\circ\varphi\circ\xi$ is also linear, that is,
$\phi(r)=Sr$ for some real $4\times 4$ matrix $S$. Our task is to show that
$S=cQ$ for some positive real number $c$ and some Lorentz matrix $Q$. We know
that for every $r\in\mathcal{M}_{4}$ we have

$\langle r,r\rangle=0\iff\langle Sr,Sr\rangle=0,$

that is,

$(Mr,r)=0\iff(S^{t}MSr,r)=0.$

Here, $(\cdot,\cdot)$ denotes the standard inner product on the space of all
real quadruples, and $M$ is as in (1.1). Choosing suitable spacetime events
$r$ (say, $r=(\cos\gamma,\sin\gamma,0,1),\ldots$) and applying the fact that
$S^{t}MS$ is symmetric, it is trivial to deduce that

$S^{t}MS=dM$

for some real number $d$. The matrix $S^{t}MS$ has the same inertia as $M$,
that is, it has three negative eigenvalues and one positive eigenvalue. It
follows that $d$ is positive, and consequently,

$\left({1\over\sqrt{d}}S\right)^{t}\,M\,\left({1\over\sqrt{d}}S\right)=M,$

as desired. ∎

######  Corollary 4.12.

Every automorphism of ${\overline{H_{2}}}$ is standard. An automorphism
$\varphi$ of ${\overline{H_{2}}}$ is an affine automorphism if and only if
$\varphi({{\widehat{\infty}}})={{\widehat{\infty}}}$.

###### Proof.

Let $\varphi$ be an automorphism of ${\overline{H_{2}}}$. Since the triple
$0,I,(1/2)I$ is timelike (Example 3.41), we see that the triple
$\varphi(0),\varphi(I),\varphi((1/2)I)$ is also timelike. Thus Theorem A
together with Theorem 3.56 implies that $\varphi$ is standard. If in addition
$\varphi({\widehat{\infty}})={\widehat{\infty}}$, then we get
$\varphi(H_{2})=H_{2}$. Thus Theorems 4.11 and 3.56 show that $\varphi$ is an
affine automorphism. Conversely, it is clear that an affine automorphism
$\varphi$ satisfies $\varphi({{\widehat{\infty}}})={{\widehat{\infty}}}$. ∎

Therefore, we no longer need to make the distinction between an automorphism
and a standard automorphism.

######  Corollary 4.13.

Let $A,B\in H_{2}$. An automorphism $\varphi$ of ${\overline{H_{2}}}$
satisfies $\varphi(A)=B$ if and only if there is an affine automorphism $\psi$
satisfying

$\varphi(X)=\psi((X-A)^{-1})^{-1}+B$

for every $X\in{\overline{H_{2}}}$.

###### Proof.

If $\psi$ is an affine automorphism, then

$\psi((A-A)^{-1})^{-1}+B=\psi({\widehat{\infty}})^{-1}+B={\widehat{\infty}}^{-1}+B=B.$

Conversely, assume that an automorphism $\varphi$ of ${\overline{H_{2}}}$
satisfies $\varphi(A)=B$. Define two automorphisms $\varphi_{1},\varphi_{2}$
by $\varphi_{1}(X):=X^{-1}+A$, $\varphi_{2}(X):=(X-B)^{-1}$,
$X\in{\overline{H_{2}}}$. Then $\psi:=\varphi_{2}\circ\varphi\circ\varphi_{1}$
is an automorphism. By $\varphi(A)=B$, we get
$\psi({\widehat{\infty}})={\widehat{\infty}}$, or equivalently, $\psi$ is an
affine automorphism. We get

$\varphi(X)=\varphi_{2}^{-1}\circ\psi\circ\varphi_{1}^{-1}(X)=\psi((X-A)^{-1})^{-1}+B$

for every $X\in{\overline{H_{2}}}$, as desired. ∎

In particular, if $A=B=0$, then $\varphi$ is of the form
$\varphi(X)=\psi(X^{-1})^{-1}$. We may also see that a general automorphism
can be expressed as the composition of only a few affine automorphisms and
inversions. To show this, let $\varphi$ be an automorphism of
${\overline{H_{2}}}$. From the fact that the cone with vertex
${\widehat{\infty}}$ is mapped by $\varphi$ onto some cone, it is easy to see
that there is $A\in H_{2}$ such that $B:=\varphi(A)\in H_{2}$. Thus the
preceding corollary shows that $\varphi$ is expressed as the composition of
two inversions and three affine automorphisms (two of which are translations).

###  5\. Degenerate coherency preservers

####  5.1. Two types of degenerate coherency preservers

We first give some examples of coherency preservers that are not standard.
This is to motivate the definition of degenerate coherency preservers. Those
who want to get to the main result immediately may look at Definitions 5.6,
5.8, 5.9, and Theorem B, then skip to Subsection 5.2.

For $n\in\\{0,1,2\\}$, let $H_{2}^{n}$ denote the set of matrices in $H_{2}$
of rank $n$. Observe that $H_{2}^{2}$ is the set of all invertible ($=$
nonsingular) matrices in $H_{2}$.

######  Example 5.1.

Obviously, any mapping from $\mathcal{A}\subset{\overline{H_{2}}}$ into a line
$\ell$ is a coherency preserver. As a special case, let $P$ be any projection
of rank one, and define $\varphi\colon H_{2}\to{\overline{H_{2}}}$ by

$\varphi(A)=(\mathrm{tr}\,A)P,\ \ \ A\in H_{2}.$

This is a coherency preserver, and it is of interest to observe that $\varphi$
is linear.

######  Example 5.2.

Another simple example is the map $\varphi\colon H_{2}\to{\overline{H_{2}}}$
defined by

$\varphi(A)=A,\ \ A\in H_{2}\setminus H_{2}^{2}$

and

(5.1) $\varphi(A)=0,\ \ A\in H_{2}^{2}.$

A slight modification gives a much “wilder” example. We choose any function
$\eta\colon\mathcal{P}\to\mathcal{P}$ and for every rank one projection $P$ we
choose a function $f_{P}\colon\mathbb{R}\to{{\overline{\mathbb{R}}}}$
satisfying $f_{P}(0)=0$. Define $\varphi\colon H_{2}\to{\overline{H_{2}}}$ by

$\varphi(tP)=f_{P}(t)\eta(P),\ \ (t,P)\in\mathbb{R}\times{\mathcal{P}}$

and (5.1). It is easily seen from Lemma 3.22 that any such $\varphi$ preserves
coherency. If $\eta$ and all functions $f_{P}$ are chosen to be surjective,
then the image of $\varphi$ is the whole cone $\mathcal{C}_{0}$.

Observe that the range of a coherency preserver in the above example is
contained in one cone. We will give the general form of such coherency
preservers in Proposition 5.21. After some more reflection, one may encounter
an example of a coherency preserver whose range slightly protrudes one cone.
Here is such an example:

######  Example 5.3.

Let $f\colon\mathcal{P}\to\mathcal{P}$ be any mapping, and define
$\varphi\colon{\overline{H_{2}}}\to{\overline{H_{2}}}$ by

$\varphi(0)=0,\ \
\varphi({\overline{H_{2}}}\setminus{\mathcal{C}}_{0})=\\{I\\},$

and

$\varphi(tP)=f(P),\ \ P\in{\mathcal{P}},\ \
t\in{\overline{\mathbb{R}}}\setminus\\{0\\}.$

It is easy and left to the reader to show that $\varphi$ is a coherency
preserver. Observe that
$\varphi({\overline{H_{2}}})=f({\mathcal{P}})\cup\\{0,I\\}$ is not necessarily
contained in one cone.

Note that
$\varphi({\overline{H_{2}}}\setminus\\{0\\})=f({\mathcal{P}})\cup\\{I\\}$ is
contained in one cone $\mathcal{C}_{I}$ in this example. Let us give another
example, which is seemingly more complicated.

######  Example 5.4.

We denote by $\mathcal{L}\subset H_{2}^{1}$ the set of all rank one matrices
that are of the form

$t\,\left[\begin{matrix}s&i\sqrt{s(1-s)}\cr-i\sqrt{s(1-s)}&1-s\end{matrix}\right]$

for some nonzero real $t$ and some real $s$, $0\leq s\leq 1$. We set

$\mathcal{K}=\left\\{\left[\begin{matrix}0&p\cr p&0\end{matrix}\right]\
:\,p\in\mathbb{R}\setminus\\{0\\}\right\\}\subset H_{2}^{2}.$

Let $g\colon\mathbb{R}\setminus\\{0\\}\to{{\overline{\mathbb{R}}}}$ and
$\eta\colon[0,1]\to\mathcal{P}$ be any maps. We define $\varphi_{1}\colon
H_{2}\to{\overline{H_{2}}}$ by

$\varphi_{1}(0)=0,\ \
\varphi_{1}(H_{2}^{1}\setminus\mathcal{L})=\\{E_{11}\\},\ \
\varphi_{1}(H_{2}^{2}\setminus\mathcal{K})=\\{I\\},$
$\varphi_{1}\left(t\,\left[\begin{matrix}s&i\sqrt{s(1-s)}\cr-i\sqrt{s(1-s)}&1-s\end{matrix}\right]\right)=\eta(s),\
\ (t,s)\in(\mathbb{R}\setminus\\{0\\})\times[0,1],$

and

$\varphi_{1}\left(\left[\begin{matrix}0&p\cr
p&0\end{matrix}\right]\right)=g(p)E_{22}+E_{11},\ \
p\in\mathbb{R}\setminus\\{0\\}.$

Then $\varphi_{1}$ is a coherency preserver.

###### Proof.

Assume that $A,B\in H_{2}$ are coherent. Observe that rank one matrices are
mapped to rank one matrices. Thus, if one of $A$ or $B$ is the zero matrix,
then the other one is of rank at most one, and therefore
$\varphi_{1}(A)\sim\varphi_{1}(B)$. The next case we will treat is that both
$A$ and $B$ are of rank one, that is, $A=tP$ and $B=sP$ for some rank one
projection $P$ and some nonzero scalars $t,s$. Then clearly,
$\varphi_{1}(A)=\varphi_{1}(B)$. We continue by treating the case that both
$A$ and $B$ are invertible. Using the fact that $\varphi_{1}$ maps the set of
invertible matrices into the line

$\ell=\left\\{tE_{22}+E_{11}\,:\,t\in{{\overline{\mathbb{R}}}}\right\\},$

we see that $\varphi_{1}(A)\sim\varphi_{1}(B)$ in this case, as well.

It remains to consider the case when one of $A,B$, say $A$, is of rank one,
and $B$ is of rank two. Then clearly, $\varphi_{1}(A)\in\mathcal{P}$. If
$B\not\in\mathcal{K}$, then $\varphi_{1}(B)=I$ and
$\varphi_{1}(A)\sim\varphi_{1}(B)$. If $B\in\mathcal{K}$, then Lemma 5.5 below
implies $A\not\in\mathcal{L}$, which yields $\varphi_{1}(A)=E_{11}$.
Therefore, we have $\varphi_{1}(A)\sim\varphi_{1}(B)$ in this last case, too.
∎

######  Lemma 5.5.

Let $p,t$ be nonzero real numbers and let $s$ be a real number, $0\leq s\leq
1$. Then the matrices

$A=t\,\left[\begin{matrix}s&i\sqrt{s(1-s)}\cr-i\sqrt{s(1-s)}&1-s\end{matrix}\right]\in\mathcal{L}\
\ \ \text{and}\ \ \ B=\left[\begin{matrix}0&p\cr
p&0\end{matrix}\right]\in\mathcal{K}$

are not coherent.

###### Proof.

Observe that $A$ is of rank one while $B$ is invertible. We have
$B-A=B(I-B^{-1}A)$. It is straightforward to verify that $B^{-1}A$ is a trace
zero complex matrix and since it is of rank one, it is a square-zero matrix.
Consequently, $I-B^{-1}A$ is invertible, which further yields that $B-A$ is of
rank two, as desired. ∎

For $\varphi_{1}$ in Example 5.4, we see that the image
$\varphi_{1}(H_{2}\setminus\\{0\\})$ is contained in one cone
$\mathcal{C}_{I}$. Thus, for each of the above examples of coherency
preservers $\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$, there are
$A\in\mathcal{A}$, $B\in\varphi(\mathcal{A})$ with the property
$\varphi(\mathcal{A}\setminus\\{A\\})\subset\mathcal{C}_{B}$. We give a name
to this class of coherency preservers.

######  Definition 5.6.

A coherency preserver $\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ defined
on a subset $\mathcal{A}$ of ${\overline{H_{2}}}$ is of _type $(\mathcal{C})$_
if there exist $A\in\mathcal{A}$ and $B\in\varphi(\mathcal{A})$ such that
$\varphi(\mathcal{A}\setminus\\{A\\})\subset\mathcal{C}_{B}$.

Is every non-standard coherency preserver of type $(\mathcal{C})$? The answer
is no.

######  Example 5.7.

Let $f\colon\mathbb{R}\to{{\overline{\mathbb{R}}}}$ be any function. Let us
define a map $\varphi_{2}\colon H_{2}\to{\overline{H_{2}}}$ in the following
way. We start by defining

$\varphi_{2}(tE_{11})=f(t)E_{11}$

for every $t\in\mathbb{R}$. If a matrix $A\in H_{2}\setminus\ell_{0,E_{11}}$
is coherent to $tE_{11}$ for some real number $t$, then such a $t$ is uniquely
determined. In this case, we set

$\varphi_{2}(A)=E_{22}+f(t)E_{11}.$

In the remaining case that $d(A,tE_{11})=2$ for all $t\in\mathbb{R}$
(equivalently, $A$ is of rank two and the $(2,2)$-entry of $A$ is zero), we
define

$\varphi_{2}(A)=E_{22}.$

Then $\varphi_{2}$ is a coherency preserving map.

###### Proof.

We need to show that $\varphi_{2}(A)\not\sim\varphi_{2}(B)$ implies $A\not\sim
B$. Observe that $\varphi(H_{2}\cap\ell_{0,E_{11}})\subset\ell_{0,E_{11}}$ and
$\varphi(H_{2}\setminus\ell_{0,E_{11}})\subset\ell_{E_{22},I}$. Therefore,
$\varphi_{2}(A)\not\sim\varphi_{2}(B)$ implies that one of the two matrices
$A,B$ is in $H_{2}\cap\ell_{0,E_{11}}$ and the other is in
$H_{2}\setminus\ell_{0,E_{11}}$. In that case, it is obvious from the
definition of $\varphi_{2}$ that $\varphi_{2}(A)\not\sim\varphi_{2}(B)$
implies $A\not\sim B$. ∎

Observe that $\varphi_{2}$ defined in this manner is not necessarily of type
$(\mathcal{C})$. However, it is a coherency preserver of type $(\ell)$ in the
following sense.

######  Definition 5.8.

A coherency preserver $\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ defined
on a subset $\mathcal{A}$ of ${\overline{H_{2}}}$ is of _type $(\ell)$_ if
there are lines $\ell,\ell^{\prime}$ such that
$\varphi(\mathcal{A}\setminus\ell)\subset\ell^{\prime}$.

Note that the two types $(\mathcal{C})$, $(\ell)$ of coherency preservers are
not disjoint. For example, any mapping
$\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ whose range is contained in a
single line is a coherency preserver of both types $(\mathcal{C})$ and
$(\ell)$. Note also that in both types the range of $\varphi$ is contained in
a “small” subset of ${\overline{H_{2}}}$. Indeed, if $\varphi$ is of type
$(\mathcal{C})$, then its range is contained in a union of a point and a cone;
and if $\varphi$ is of type $(\ell)$, then its range is contained in a union
of two lines.

It will turn out that the two types of coherency preservers play a dominant
role in the class of non-standard coherency preservers, so let us give a name
to the union of these two classes.

######  Definition 5.9.

A coherency preserver $\varphi\colon\mathcal{A}\to{\overline{H_{2}}}$ defined
on a subset $\mathcal{A}$ of ${\overline{H_{2}}}$ is _degenerate_ if it is of
type $(\mathcal{C})$ or of type $(\ell)$.

Here we present the key result of the current section.

######  Theorem B.

Let $A,B\in H_{2}$ satisfy $A<B$ and $\varphi\colon[A,B]\to{\overline{H_{2}}}$
be a coherency preserver that is not standard. Then $\varphi$ is degenerate.

We will give its lengthy proof in Section 6. Before that, we give applications
of Theorem B in the next subsection. It will turn out that a coherency
preserver is either standard or degenerate if it is defined on a subset that
looks like a matrix interval.

However, it is not true that a non-standard coherency preserver defined on,
say, an open connected subset of ${\overline{H_{2}}}$ is always degenerate in
the sense of Definition 5.9. We give some examples. Let $\mathcal{U}\subset
H_{2}$ be the set formed of all matrices $X$ in $H_{2}$ such that the