# Quantum Measurement of Space-Time Events

Dorje C. Brody1,2 and Lane P. Hughston3 1Department of Mathematics, University
of Surrey, Guildford GU2 7XH, UK
2St Petersburg National Research University of Information Technologies,
Mechanics and Optics, St Petersburg 197101, Russia
3Department of Computing, Goldsmiths University of London, New Cross, London
SE14 6NW, UK

###### Abstract

The phase space of a relativistic system can be identified with the future
tube of complexified Minkowski space. As well as a complex structure and a
symplectic structure, the future tube, seen as an eight-dimensional real
manifold, is endowed with a natural positive-definite Riemannian metric that
accommodates the underlying geometry of the indefinite Minkowski space metric,
together with its symmetry group. A unitary representation of the 15-parameter
group of conformal transformations can then be constructed that acts upon the
Hilbert space of square-integrable holomorphic functions on the future tube.
These structures are enough to allow one to put forward a quantum theory of
phase-space events. In particular, a theory of quantum measurement can be
formulated in a relativistic setting, based on the use of positive operator
valued measures, for the detection of phase-space events, hence allowing one
to assign probabilities to the outcomes of joint space-time and four-momentum
measurements in a manifestly covariant framework. This leads to a localization
theorem for phase-space events in relativistic quantum theory, determined by
the associated Compton wavelength.

##  I Introduction

Starting with the pioneering work of Dirac Dirac , investigations of the
Hamiltonian formulation of space-time physics have been pursued by numerous
authors. One of the motivations behind such analysis has been that the
mathematical structures of phase-space formalisms can be highly amenable to a
quantum-mechanical description. The naive formulation of a relativistic phase
space as a kind of doubled-up Minkowski space with four position coordinates
and four momentum coordinates, while feasible in the classical theory, is not
satisfactory as a basis for relativistic quantum theory. Here we propose an
alternative approach in which the future tube of complexified Minkowski space
is taken to be the phase space of a relativistic system. Remarkably, this
phase space comes naturally equipped with both the symplectic structure and
the compatible Riemannian structure needed for the development of a fully
covariant relativistic quantum theory.

Let us write $\mathbb{M}$ for Minkowski space, by which we mean
$\mathds{R}^{4}$ equipped with the usual flat space-time metric $g_{ab}$ with
signature $(+,-,-,-)$. For the positions of points $x,y\in\mathbb{M}$ relative
to an origin in $\mathbb{M}$ we write $x^{a}$ and $y^{a}$, where
$a,b=0,1,2,3$. We say that $x$ and $y$ are time-like, space-like, or null
separated according to whether $g_{ab}(x^{a}-y^{a})(x^{b}-y^{b})$ is positive,
negative, or zero. In the time-like and null cases, the separation vector
$v^{a}=x^{a}-y^{a}$ is said to be future-pointing or past-pointing according
to whether $v^{0}$ is positive or negative. Then by complex Minkowski space
${\mathds{C}}\mathbb{M}$ we mean $\mathds{C}^{4}$ equipped with the same
metric tensor. The so-called future tube ${\Gamma}$ is the open submanifold of
${\mathds{C}}\mathbb{M}$ consisting of points that are of the form
$z^{a}=x^{a}-\mbox{$\rm i$}r^{a}$, where $r^{a}$ is time-like and future
pointing. Thus for all $z^{a}\in{\Gamma}$ it holds that $g_{ab}r^{a}r^{b}>0$
with $r^{0}>0$.

The future tube plays an important role in rigorous treatments of quantum
field theory. In particular, the Wightman functions – given by vacuum
expectations of field operators – are analytic in ${\Gamma}$, and one can
reconstruct the field theory from the data of these expectation values
Wightman 1956 ; Wightman 1960 ; Araki 1961 ; Streater and Wightman 1964 . The
future tube contains no real space-time points; however, the so-called
extended future tube, consisting of points attainable by the action of the
complex Lorentz group on ${\Gamma}$, contains real points, called Jost points.
One can then recover the field theory from the values of the Wightman
functions at Jost points Schweber . Complexified Minkowski space also plays an
important role in the Penrose twistor program Penrose 1967 ; Penrose 1968 ;
Penrose 1972 , as does the future tube. In twistor theory, the complex
projective space ${\mathds{C}}{\mathbb{P}}^{3}$ is divided into two parts, the
upper and lower half of ${\mathds{C}}{\mathbb{P}}^{3}$, separated by a five
real dimensional hypersurface $\mathbb{N}^{5}$ of null twistors. The points of
${\mathds{C}}\mathbb{M}$ correspond to complex projective lines in
${\mathds{C}}{\mathbb{P}}^{3}$. The points of ${\Gamma}$ correspond to lines
that lie entirely in the top half of ${\mathds{C}}{\mathbb{P}}^{3}$.

In both twistor theory and quantum field theory, the complexification of
Minkowski space, natural as it may be, is introduced primarily to enable one
to exploit the tools of complex analysis in relation to the positive frequency
condition on fields; and there is no direct physical significance attached as
such to the imaginary components of complex space-time points. Some form of
reality condition has to be brought into play to make the link to the physical
“real” spacetime.

From the view of the complex formulation of classical mechanics Mackey , it is
natural to ask whether the imaginary part of a point in ${\Gamma}$ is related
to the four momentum of a relativistic system. In what follows we offer an
affirmative answer to this question. This, in turn, allows us to construct a
Hilbert space of quantum states over the space-time phase space. The conformal
transformations of the underlying space-time phase space can then be
represented explicitly in terms of unitary operators acting on quantum states.
We then formulate a measurement postulate for the detection of the phase-space
location of a relativistic event by identifying the probability law for
measurement outcomes along with an appropriate post-measurement transformation
rule for the states. It is shown, in particular, that when the measurement
outcome yields a phase-space point, the state results in a coherent state of
the conformal group, centered at that point. The fact that coherent states are
the most localized states in the Hilbert space then leads to a localization
bound which shows for systems of short Compton wavelength that when an event
is detected to have occurred at a specific phase-space point, the resulting
state will be highly localized in phase space.

The problem of relativistic quantum measurement has been investigated by many
authors (see e.g. Landau Peierls 1931 ; Aharonov Albert 1981 ; Rovelli ; Peres
and references cited therein). It is often the case, however, that measurement
postulates of nonrelativistic quantum theory are used in a relativistic setup
to deduce implications of the postulates, which is unsatisfactory, for what is
required is a measurement postulate in a relativistic setup, as we propose
here.

##  II Relativistic mechanics

We begin by reviewing the cotangent bundle approach to relativistic mechanics
Todorov ; Woodhouse ; Marsden ; SSG . The phase space is taken to be the
cotangent bundle of Minkowski space, where the cotangent vectors in the fibre
over a point in Minkowski space are identified with the momentum four-vectors
that the particle might possess. The bundle is an eight-dimensional manifold
$T^{*}\mathbb{M}$, with base coordinates $x^{a}$ and fibre coordinates
$p_{a}$. We form the canonical one-form $\theta=p_{a}\mbox{$\rm d$}x^{a}$ on
$T^{*}\mathbb{M}$ along with its exterior derivative, the associated
symplectic form $\omega=\mbox{$\rm d$}p_{a}\wedge\mbox{$\rm d$}x^{a}$. Given a
smooth function $H:T^{*}\mathbb{M}\to\mathds{R}$ we then write Hamilton’s
equations for a dynamical trajectory

$\displaystyle s\in{\mathds{R}}^{+}\mapsto\\{x^{a}(s),p_{a}(s)\\}\in
T^{*}\mathbb{M}$ (1)

in the form

$\displaystyle\frac{\mbox{$\rm d$}x^{a}}{\mbox{$\rm d$}s}=\frac{\partial
H}{\partial p_{a}}\quad{\rm and}\quad\frac{\mbox{$\rm d$}p_{a}}{\mbox{$\rm
d$}s}=-\frac{\partial H}{\partial x^{a}}\,,$ (2)

and we call $H(x^{a},p_{a})$ the Hamiltonian function. As before, we let
$g_{ab}={\rm diag}(+1,-1,-1,-1)$ be the metric on the base space
${\mathbb{M}}$, which we can use to raise and lower indices on the fibre
elements as well. Then we can write $(x^{a},p^{a})=(x^{a},g^{ab}p_{b})$ and
put Hamilton’s equations in the more symmetrical form

$\displaystyle\frac{\mbox{$\rm d$}x^{a}}{\mbox{$\rm d$}s}=g^{ab}\frac{\partial
H}{\partial p^{b}}\quad{\rm and}\quad\frac{\mbox{$\rm d$}p^{a}}{\mbox{$\rm
d$}s}=-g^{ab}\frac{\partial H}{\partial x^{b}}\,.$ (3)

The cotangent-bundle approach works well for characterizing the dynamics of
typical mechanical systems in space-time. To convince oneself it suffices to
explore a few examples.

Example 1: Free particle. The Hamiltonian is taken to be

$\displaystyle H=[g_{ab}p^{a}p^{b}]^{1/2}.$ (4)

Then $H$ will be a constant of the motion which we identify as the mass $m$.
The phase space is foliated by surfaces of constant $H$, and as an initial
condition we choose $\\{x^{a}(0),p_{a}(0)\\}$ to lie on the surface $H=m$.
Hamilton’s equations (3) imply

$\displaystyle m{\dot{x}}^{a}=p^{a},\quad{\dot{p}}^{a}=0.$ (5)

The phase-space trajectory is given by

$\displaystyle x^{a}(s)=x^{a}(0)+s\,m^{-1}p_{a}(0),\quad p_{a}(s)=p_{a}(0),$
(6)

corresponding to a geodesic motion in Minkowski space subject to the specified
initial conditions.

Example 2: Charged particle in an electromagnetic field. Let the charge be $q$
and write $A^{a}(x)$ for the electromagnetic four-potential. With the familiar
minimal coupling, we extend the previous example by taking the Hamiltonian to
be of the form

$\displaystyle H=\left[g_{ab}(p^{a}-qA^{a})(p^{b}-qA^{b})\right]^{1/2}.$ (7)

We foliate the phase space with surfaces of constant $H$, identifying the
value of $H$ with the mass of the particle. Hamilton’s equations give

$\displaystyle
m{\dot{x}}^{a}=(p^{a}-qA^{a}),\quad{\dot{p}}^{a}=q{\dot{x}}_{c}\nabla^{a}A^{c},$
(8)

where $\nabla_{a}=\partial/\partial x^{a}$. Further differentiation leads to
the Lorentz force law

$\displaystyle{m\ddot{x}}^{a}=qF^{ab}\,{\dot{x}}_{b},\quad
F^{ab}=\nabla^{a}A^{b}-\nabla^{b}A^{a}.$ (9)

Example 3: Relativistic two-body problem with a force of mutual attraction.
Let us write $x^{a}$, $y^{a}$, $X^{a}$, $Y^{a}$ for the space-time positions
and momenta of the two particles, setting

$\displaystyle q^{a}=\mbox{$\textstyle\frac{1}{2}$}(x^{a}-y^{a}),$ (10)

along with

$\displaystyle P^{a}=X^{a}+Y^{a},\quad Q^{a}=X^{a}-Y^{a}.$ (11)

To model a central force we project $q^{a}$ onto the space-like hypersurface
orthogonal to the total momentum $P^{a}$ to measure the separation of the two
particles. Since $P^{a}$ is time-like, the resulting “internal” coordinate
$\xi^{a}$ defined by

$\displaystyle\xi^{a}=q^{a}-\frac{q_{c}P^{c}}{P_{c}P^{c}}\,P^{a}$ (12)

is space-like. Thus $\xi_{a}\xi^{a}\leq 0$ and for the potential we set

$\displaystyle V(\xi^{a})=\Phi(-\,\xi_{a}\xi^{a}),$ (13)

for some function $\Phi:{\mathds{R}}^{+}\backslash\\{0\\}\to{\mathds{R}}$ in
$\rm{C}^{1}(0,\infty)$. For example, for a harmonic oscillator we set
$\Phi(u)=ku$ where $k\in{\mathds{R}}^{+}$. For a Coulomb potential set
$\Phi(u)=-e_{1}e_{2}u^{-1/2}$, where $e_{1},e_{2}$ are the charges of the
particles. For a gravitational potential, set $\Phi(u)=Gm_{1}m_{2}u^{-1/2}$,
and so on. Consider now a pair of particles interacting via the potential $V$.
We write

$\displaystyle X^{2}=m_{1}^{\,2}+V,\quad Y^{2}=m_{2}^{\,2}+V,$ (14)

where $X^{2}=X_{a}X^{a}$ and $Y^{2}=Y_{a}Y^{a}$, and $m_{1}$, $m_{2}$ are the
rest masses. These conditions imply

$\displaystyle\mbox{$\textstyle\frac{1}{2}$}(P^{2}+Q^{2})-2V=m_{1}^{\,2}+m_{2}^{\,2},\quad
P_{c}Q^{c}=m_{1}^{\,2}-m_{2}^{\,2}.$ (15)

Hence for the Hamiltonian we set

$\displaystyle
H=\left(\mbox{$\textstyle\frac{1}{2}$}(P^{2}+Q^{2})-2V\right)^{1/2}.$ (16)

Since $H$ will be a constant of the motion, we choose the initial conditions
so that $\\{r^{a}(0),q^{a}(0)\\}$ lies on the surface

$\displaystyle H=[m_{1}^{\,2}+m_{2}^{\,2}]^{1/2}\,.$ (17)

Hamilton’s equations show that $P_{c}Q^{c}$ is also a constant of the motion,
so we set

$\displaystyle P_{c}Q^{c}=m_{1}^{\,2}-m_{2}^{\,2}\,,$ (18)

thus fixing the two masses. A calculation then shows that

$\displaystyle\frac{\mbox{$\rm d$}^{2}\xi^{a}}{\mbox{$\rm
d$}s^{2}}=-\frac{1}{m_{1}^{\,2}+m_{2}^{\,2}}\,\Phi^{\prime}(-\xi_{c}\xi^{c})\,\xi^{a},$
(19)

where $\Phi^{\prime}(u)=\mbox{$\rm d$}\Phi(u)/\mbox{$\rm d$}u$. Since the
right side is a function of $\xi^{a}$, we can solve for $\xi^{a}(s)$, which in
turn allows us to determine the phase-space trajectory. For example, in the
case of an oscillator, we have $\Phi^{\prime}=k$, so we obtain

$\displaystyle\xi^{a}(s)=\alpha^{a}\cos(\omega s)+\beta^{a}\sin(\omega s),$
(20)

where $\alpha^{a}$, $\beta^{a}$ are constant spacelike vectors such that
$\alpha^{a}=\xi^{a}(0)$ and $\omega\beta^{a}={\dot{\xi}}^{a}(0)$, with
$\omega^{2}=k/(m_{1}^{\,2}+m_{2}^{\,2})$.

Despite the merits of these examples, there are limitations to the
effectiveness of the cotangent bundle approach as a foundation for the theory
of relativistic dynamics. For a start, there is no intrinsic mechanism to
prevent the momentum from becoming space-like or past-pointing. This problem
can be avoided in specific examples, such as the ones above, but it is
undesirable that one should have to manage the situation on an ad hoc basis.
The cotangent bundle approach also poses problems when we look at field
theories, since the cotangent bundle does not admit a natural complex
structure. In particular, there is no general recipe for combining position
and momentum in a linear way, allowing one to write complex expressions of the
form $x^{a}+\mbox{$\rm i$}p^{a}$. Such variables arise in the quantization of
oscillators, but in that case there is a dimensional constant that allows one
to modify the expressions to produce terms of the same dimensionality. If the
constants of nature at ones disposal are the speed of light and Planck’s
constant, then one cannot convert a quantity with units of momentum to one
with units of length. Further, the interpretation of the parameter $s$ as a
proper time in the cotangent bundle approach is ambiguous when many particles
are involved.

What is the best way forward? Many authors have considered the problems
arising with relativistic phase spaces, both for classical theories and
quantum theories Currie ; Kaiser ; King ; Komar ; Rohrlich . Our approach
incorporates ideas drawn from all of these, and from geometric quantum
mechanics as well Kibble 1979 ; Page 1987 ; Gibbons 1992 ; Hughston 1995 ;
Ashtekar Schilling 1998 ; Brody Hughston 2001 ; Bengtsson Zyczkowski 2006 . We
also look closely at the role of probability in the course of our development
of a relativistic theory of quantum measurement based on the geometry of the
future tube.

##  III Back to the future tube

That there is an appropriate map from the cotangent bundle to the future tube
is not immediately apparent, but a dimensional argument will lead the way. In
order for us to be able to regard $x^{a}-\mbox{$\rm i$}r^{a}$ as a complex
phase-space variable in a relativistic context we shall require $r^{a}$ to
have units of inverse momentum. Then if we multiply $r^{a}$ by Planck’s
constant we obtain a vector with units of position that can be combined with
$x^{a}$. Specifically, we consider the Kelvin inversion

$\displaystyle r^{a}=\hbar p^{a}/(p_{c}p^{c}),\quad p^{a}=\hbar
r^{a}/(r^{a}r_{a}).$ (21)

This transformation maps the cone of time-like future-pointing Minkowski space
vectors into itself. Thus we have

$\displaystyle\hbar\,g^{ab}\frac{\partial}{\partial
p^{b}}=r_{c}r^{c}\left(g^{ab}-\frac{2\,r^{a}r^{b}}{r_{c}r^{c}}\right)\frac{\partial}{\partial
r^{b}}$ (22)

and

$\displaystyle\frac{1}{\hbar}\,g_{ab}\frac{\mbox{$\rm d$}p^{a}}{\mbox{$\rm
d$}s}=\frac{1}{r_{c}r^{c}}\left(g_{ab}-\frac{2\,r_{a}r_{b}}{r_{c}r^{c}}\right)\frac{\mbox{$\rm
d$}r^{a}}{\mbox{$\rm d$}s}.$ (23)

Now define a symmetric tensor $h_{ab}$ with inverse $k^{ab}$ by setting

$\displaystyle
h_{ab}=-\frac{1}{r_{c}r^{c}}\left(g_{ab}-\frac{2\,r_{a}r_{b}}{r_{c}r^{c}}\right),\,\,\,\,k^{ab}=-r_{c}r^{c}\left(g^{ab}-\frac{2\,r^{a}r^{b}}{r_{c}r^{c}}\right).$
(24)

Then $k^{ab}\,h_{bc}=\delta^{a}_{c}$, and a straightforward calculation shows
that Hamilton’s equations on the future tube take the form

$\displaystyle\hbar\,\frac{\mbox{$\rm d$}x^{a}}{\mbox{$\rm
d$}s}=-k^{ab}\frac{\partial H}{\partial r^{b}}\quad{\rm
and}\quad\hbar\,\frac{\mbox{$\rm d$}r^{a}}{\mbox{$\rm
d$}s}=k^{ab}\frac{\partial H}{\partial x^{b}}.$ (25)

That the signs in (25) are reversed in comparison with (3) is an artefact of
the convention that defines the future tube by points of the form
$x^{a}-\mbox{$\rm i$}r^{a}$ with $r^{a}$ time-like and future-pointing. But
what is not so obvious, and comes perhaps as a surprise, is that the quadratic
form $h_{ab}$ is positive definite, thus defining a Riemannian metric on the
future tube, given by

$\displaystyle\mbox{$\rm d$}s^{2}=h_{ab}\,(\mbox{$\rm d$}x^{a}\,\mbox{$\rm
d$}x^{b}+\mbox{$\rm d$}r^{a}\,\mbox{$\rm d$}r^{b}).$ (26)

As a consequence we see that the arc-length along a smooth curve can be taken
as a canonical parametrization of the phase-space trajectory. In particular,
in situations where two or more particles are interacting, the phase space of
the system as a whole can be taken to be the product of the phase spaces of
the individual systems, with an overall positive definite metric, thus leading
to a natural way of synchronizing the dynamics of the constituents.

##  IV Relativistic phase-space geometry

As a number of authors have pointed out, there are several distinct but
ultimately equivalent ways of arriving at the geometrical structure of the
future tube Uhlmann1 ; KN ; Uhlmann3 ; Ruhl ; Ruhl2 ; Odzijewicz ;
Odzijewicz_1976 ; Carey 1977 ; Coquereau 1990 ; Vladimirov ; Gibbons .
Building on these works, we pursue here an alternative approach to the
geometry of ${\Gamma}$ that ties in naturally with quantum measurement theory.
We begin with the Hilbert space ${\mathcal{H}}={L}^{2}(\Gamma,{\mathcal{O}})$
of square-integrable holomorphic functions on the future tube. If we let $f,g$
be elements of ${\mathcal{H}}$, then for their inner product we write

$\displaystyle\langle\bar{g}\,|f\rangle=\int_{\Gamma}\\!f(z)\,{\bar{g}}({\bar{z}})\,\mbox{$\rm
d$}\mu_{z},$ (27)

where

$\displaystyle\mbox{$\rm d$}\mu_{z}=\frac{1}{16}\,\mbox{$\rm
d$}^{4}z\,\mbox{$\rm d$}^{4}{\bar{z}}$ (28)

denotes the usual Lebesgue measure on $\Gamma$. The fact that such functions
constitute a Hilbert space is nontrivial, for it is not immediately obvious
that any Cauchy sequence in ${L}^{2}(\Gamma,{\cal O})$ converges to an element
of ${L}^{2}(\Gamma,{\mathcal{O}})$. That such convergence holds follows as a
consequence of a well-known bound Krantz , which states that for any compact
subset $Q\subset\Gamma$ there exists a constant $C_{Q}$ such that for all
$\phi\in{L}^{2}(\Gamma,{\cal O})$ we have

$\displaystyle\sup_{z\in Q}|\phi(z)|\leq C_{Q}\,||\phi||,$ (29)

where

$\displaystyle||\phi||=\left(\int_{\Gamma}\\!\phi(z)\,{\bar{\phi}}({\bar{z}})\,\mbox{$\rm
d$}\mu_{z}\right)^{1/2}.$ (30)

Now let $\\{\phi^{n}\\}_{n\in\mathds{N}}$ be an orthonormal basis for
${\mathcal{H}}$ so that

$\displaystyle\int_{\Gamma}\\!\phi^{n}(z)\,{\bar{\phi}}_{m}({\bar{z}})\,\mbox{$\rm
d$}\mu_{z}=\delta^{n}_{m}.$ (31)

We introduce the Bergman kernel Meschkowski ; Bergman  on $\Gamma$ by setting

$\displaystyle
K(z,{\bar{w}})=\sum_{n}\,{\bar{\phi}}_{n}({\bar{w}})\,\phi^{n}(z),$ (32)

which is independent of the choice of orthonormal basis. Thus $K(z,{\bar{w}})$
is holomorphic in $z$ and antiholomorphic in $w$, and for any holomorphic
function $f\in{\cal H}$ we evidently have

$\displaystyle\int_{\Gamma}\\!K(z,{\bar{w}})\,f(w)\,\mbox{$\rm
d$}\mu_{w}=f(z).$ (33)

We thus see that the Bergman kernel acts as an identity operator or
reproducing kernel on ${\cal H}$. In particular, for all $x,y,z\in\Gamma$ we
have the identity

$\displaystyle\int_{y}K(x,{\bar{y}})\,K(y,{\bar{z}})\,\mbox{$\rm
d$}\mu_{y}=K(x,{\bar{z}}).$ (34)

Now consider a smooth curve

$\displaystyle\gamma:\sigma\in[0,1]\mapsto w_{\sigma}\in\Gamma.$ (35)

For each value of the parameter $\sigma$ the function
$\psi_{\sigma}:\Gamma\to{\mathds{C}}$ defined by

$\displaystyle\psi_{\sigma}(z)=K(z,{\bar{w}}_{\sigma})$ (36)

is holomorphic and square integrable. It follows that $\psi_{\sigma}$
describes a curve in ${\mathcal{H}}$ as $\sigma$ varies, so we can work out
the length along $\gamma$ by use of the Fubini-Study metric KN ; Page 1987  :

$\displaystyle\mbox{$\rm d$}s^{2}=\frac{\int\mbox{$\rm
d$}{\psi}_{\sigma}(z)\,\mbox{$\rm
d$}{\bar{\psi}}_{\sigma}({\bar{z}})\,\mbox{$\rm
d$}\mu_{z}}{\int{\psi}_{\sigma}(z)\,{\bar{\psi}}_{\sigma}({\bar{z}})\,\mbox{$\rm
d$}\mu_{z}}-\frac{\left|\,\int\mbox{$\rm
d$}{\psi}_{\sigma}(z)\,{\bar{\psi}}_{\sigma}({\bar{z}})\,\mbox{$\rm
d$}\mu_{z}\,\right|^{\,2}}{\left(\,\int{\psi}_{\sigma}(z)\,{\bar{\psi}}_{\sigma}({\bar{z}})\,\mbox{$\rm
d$}\mu_{z}\,\right)^{\,2}}\,.$ (37)

A calculation then shows that

$\displaystyle\mbox{$\rm d$}s^{2}=\frac{\partial^{2}\log
K(w,\bar{w})}{\partial w^{a}\,\partial\bar{w}^{b}}\,\mbox{$\rm
d$}w^{a}\,\mbox{$\rm d$}{\bar{w}}^{b}.$ (38)

Thus, the Fubini-Study metric on $\mathcal{H}$ induces a Kähler metric on
$\Gamma$. This is the so-called Bergman metric Bergman . The ideas of Bergman
kernel and the associated metric can be elucidated by considering an
elementary example of a bounded domain ${\mathds{C}}^{+}$ of ${\mathds{C}}$
defined by $\mbox{$\rm i$}(z-{\bar{z}})>0$. Writing $z=x-\mbox{$\rm i$}r$ the
domain can be expressed as the half plane for which $r>0$. Thus
${\mathds{C}}^{+}$ can be thought of as representing one-dimensional future
tube, i.e. phase space in zero space dimensions, with only time and energy. An
example of a set of orthonormal functions on ${\mathds{C}}^{+}$ is given by

$\displaystyle\phi^{n}(z)=2\mbox{$\rm i$}\sqrt{\frac{n}{\pi}}\frac{(\mbox{$\rm
i$}-z)^{n-1}}{(\mbox{$\rm i$}+z)^{n+1}},$ (39)

from which one can easily verify that the corresponding Bergman kernel takes
the form

$\displaystyle
K_{{\mathds{C}}^{+}}(z,\bar{w})=\sum_{n=1}^{\infty}\phi^{n}(z)\,{\bar{\phi}_{n}(\bar{w})}=\frac{1}{\pi(z-{\bar{w}})^{2}}.$
(40)

The associated Bergman metric is then the usual hyperbolic metric of the half
plane. Note that under the map $w=f(z)$ given by the Cayley transform

$\displaystyle f(z)=\mbox{$\rm i$}\left(\frac{1-z}{1+z}\right),$ (41)

points of ${\mathds{C}}^{+}$ are mapped invertibly to points of the unit disk
$|w|<1$, which is the so-called Poincaré disk ${\mathbb{D}}$. The orthonormal
basis elements are then mapped to

$\displaystyle\phi_{n}(w)=\sqrt{\frac{n}{\pi}}\,w^{n-1},$ (42)

and a short calculation shows that the kernel function becomes

$\displaystyle K_{\mathbb{D}}(z,\bar{w})=\frac{1}{\pi(1-z{\bar{w}})^{2}}\,.$
(43)

In the case of the future tube, the Bergman kernel can be worked out
explicitly, and we have

$\displaystyle K(z,{\bar{w}})=\left(\frac{2^{3}\cdot
4!}{\,\pi^{4}}\right)\frac{1}{[g_{ab}(z^{a}-{\bar{w}}^{a})(z^{b}-{\bar{w}}^{b})]^{4}}\,.$
(44)

Note that similar to case of the half plane ${\mathds{C}}^{+}$, the future
tube can be mapped to a “unit ball” domain ${\mathbb{B}}$, which acts as the
higher-dimensional analogue of the Poincaré disk ${\mathbb{D}}$. Specifically,
if we write points of $\Gamma$ in the $2\times 2$ matrix form

$\displaystyle{\hat{Z}}=z^{0}{\hat{I}}+z^{1}{\hat{\sigma}}_{x}+z^{2}{\hat{\sigma}}_{y}+z^{3}{\hat{\sigma}}_{z}\,,$
(45)

where ${\hat{I}}$ denotes the $2\times 2$ identity matrix and
${\hat{\sigma}}_{x}$, ${\hat{\sigma}}_{y}$, ${\hat{\sigma}}_{z}$ are the Pauli
matrices, the future tube is defined by the condition that

$\displaystyle{\hat{I}}-{\hat{Z}}^{\dagger}{\hat{Z}}>0,$ (46)

where the inequality here means that the left side is a positive-definite
matrix. The Cayley transform of the future tube in this representation is
given by

$\displaystyle W=\mbox{$\rm
i$}({\hat{I}}-{\hat{Z}})\,({\hat{I}}+{\hat{Z}})^{-1},$ (47)

which is the analogue of (41) for the half plane. With this transformation in
mind, the numerical factor $V_{4}=\pi^{4}/(2^{3}\cdot 4!)$ appearing in (44)
can then be seen as arising from the Euclidean volume of the ball domain Hua .
In the literature of the future tube, it is common to adopt the
characterization in terms of ${\mathbb{B}}$, since this allows for a somewhat
simpler treatment of the group-theoretic analysis associated with conformal
transformations Uhlmann1 ; KN ; Uhlmann3 ; Odzijewicz_1976 ; Ruhl ; Ruhl2 ;
Carey 1977 . However, for in the consideration of quantum theory we find it
more transparent to work directly with the standard characterization of the
future tube as the domain $\Gamma$ in the complexified Minkowski space. In
particular, for its Bergman metric, substitution of (44) into (38) gives

$\displaystyle\frac{\partial^{2}\log K(z,\bar{z})}{\partial
z^{a}\,\partial\bar{z}^{b}}=h_{ab},$ (48)

where $h_{ab}$ turns out to be none other than the metric (24) that we
introduced earlier using the Kelvin transformation. Since a Bergman metric is
fully determined by the complex analytic structure of the underlying domain,
it follows that $h_{ab}$ admits the symmetry group of $\Gamma$, which is the
15-parameter conformal group of Minkowski space. These phase-space symmetries
are generated by Hamiltonian flows on $\Gamma$.

##  V Quantum states

Going forward, now let $u,v,w,x,y,z$ denote points of $\Gamma$. Having
introduced the Hilbert space ${\cal H}$ of holomorphic functions on $\Gamma$
we are in a position to build a quantum theory. A general state will be a
density matrix $\rho(y,{\bar{z}})\in{L}^{2}(\Gamma,{\cal
O})\times{L}^{2}(\Gamma,\bar{\cal O})$. For such a state, we require the
following: (a) that $\rho(y,{\bar{z}})=\bar{\rho}({\bar{z}},y)$, (b) that
$\rho(y,{\bar{z}})$ should be positive, that is to say

$\displaystyle\int\\!\bar{\alpha}(\bar{y})\,\rho(y,{\bar{z}})\,\alpha(z)\,\mbox{$\rm
d$}\mu_{y}\,\mbox{$\rm d$}\mu_{z}\geq 0$ (49)

for $\alpha(z)\in{L}^{2}(\Gamma,{\cal O})$, and (c) that it should have unit
trace,

$\displaystyle\int\\!K(z,{\bar{y}})\,\rho(y,{\bar{z}})\,\mbox{$\rm
d$}\mu_{y}\mbox{$\rm d$}\mu_{z}=\int\\!\rho(z,{\bar{z}})\,\mbox{$\rm
d$}\mu_{z}=1.$ (50)

A state is then said to be pure if
$\rho(y,{\bar{z}})=\xi(y)\,\bar{\xi}(\bar{z})$ for some holomorphic function
$\xi\in{L}^{2}(\Gamma,{\cal O})$ with unit norm. We observe that for both pure
and mixed states the “diagonal” function $\rho(z,{\bar{z}})$ takes the form of
a probability density on $\Gamma$. That a probability density function on
phase space arises naturally in the present context is significant, since the
construction of such densities in configuration-space models for relativistic
quantum mechanics is known to be problematic.

The interpretation of a density matrix is that it represents the quantum state
of a relativistic event. Such an event is accompanied by position and momentum
data. The fact that wave functions are holomorphic then prohibits the
possibility that they can be concentrated with arbitrarily high precision in a
given region of phase space. This follows from the fundamental inequality
(29). Many aspects of the theory can be understood as being analogous to the
Bargmann-Segal construction in nonrelativistic quantum mechanics Bargmann ;
Segal .

##  VI Space-time transformations

It is natural to enquire how space-time transformations are represented in
${\mathcal{H}}$. That is, we are interested in constructing a unitary
representation of the action of the Poincaré group $\mathcal{P}(\Gamma)$ on
$\Gamma$. Such transformations are of the form

$\displaystyle P:z^{a}\mapsto L^{a}_{b}\,z^{b}+B^{a}$ (51)

where $B^{a}$ is a real four vector and $L^{a}_{b}$ satisfies

$\displaystyle g_{ab}L^{a}_{c}L^{b}_{d}=g_{cd}\,.$ (52)

A family of unitary operators $\\{\hat{U}_{P},P\in\mathcal{P}(\Gamma)\\}$
generating Poincaré transformations through the action
$\hat{U}_{P}:\mathcal{H}\to\mathcal{H}$ can then be seen to take the form

$\displaystyle{U(x,\bar{y})}=K(L^{a}_{b}\,x^{b}+B^{a},{\bar{y}}^{a}),$ (53)

where $K(x,\bar{y})$ is the Bergman kernel (44). To see that the operator
${\hat{U}}$ thus defined is unitary, it suffices to show that
${\hat{U}}{\hat{U}}^{\dagger}=\hat{\mathds{1}}$, where on account of (32) the
identity operator $\hat{\mathds{1}}$ here in the phase-space coordinate
representation is given by the kernel function. Then by (53) we have

$\displaystyle{\bar{U}}(y,\bar{x})=K(y^{a},L^{a}_{b}\,{\bar{x}}^{b}+B^{a}),$
(54)

and it follows from (33) and (52) that

$\displaystyle\int\\!U(x,{\bar{y}})\,{\bar{U}}(y,{\bar{z}})\,\mbox{$\rm
d$}\mu_{y}$ $\displaystyle=$
$\displaystyle\int\\!K(Lx+B,{\bar{y}})\,K(y,L{\bar{z}}+B)\,\mbox{$\rm
d$}\mu_{y}$ (55) $\displaystyle=$ $\displaystyle K(Lx+B,L{\bar{z}}+B)$
$\displaystyle=$ $\displaystyle\left(\frac{2^{3}\cdot
4!}{\,\pi^{4}}\right)\frac{1}{[g_{ab}(L^{a}_{c}\,x^{c}-L^{a}_{c}\,{\bar{z}}^{c})(L^{b}_{d}\,x^{d}-L^{b}_{d}\,{\bar{z}}^{d})]^{4}}$
$\displaystyle=$ $\displaystyle K(x,{\bar{z}}),$

as desired. It is also apparent that
${\hat{U}}_{P^{\prime}}{\hat{U}}_{P}={\hat{U}}_{P^{\prime}P}$ for all
$P,P^{\prime}\in{\mathcal{P}}(\Gamma)$, so we conclude that (53) gives a
unitary representation of the Poincaré group on the Hilbert space
${\mathcal{H}}={L}^{2}(\Gamma,{\cal O})$.

More generally, a representation of the full 15-parameter conformal group can
also be identified by use of the kernel function. To see this, we consider
first the four-dimensional subgroup ${\mathcal{S}}(\Gamma)$ of special
conformal transformations, given by

$\displaystyle
S:z^{a}\mapsto\frac{z^{a}+z^{2}\lambda^{a}}{1+2\lambda\\!\cdot\\!z+\lambda^{2}z^{2}},$
(56)

where $\lambda^{a}$ is a real four-vector, and we write
$\lambda\\!\cdot\\!z=g_{ab}\lambda^{a}z^{b}$,
$\lambda^{2}=g_{ab}\lambda^{a}\lambda^{b}$, and $z^{2}=g_{ab}z^{a}z^{b}$. The
transformation (56) is obtained by applying an inversion $z^{a}\mapsto
z^{a}/(g_{bc}z^{b}z^{c})$, then shifting the result by $\lambda^{a}$, and
finally applying a further inversion. The unitary operator ${\hat{U}}_{S}$
generating such a transformation on states in ${\mathcal{H}}$ is given by

$\displaystyle{U}(x,\bar{y})=\frac{1}{(1+2\lambda\\!\cdot\\!x+\lambda^{2}x^{2})^{4}}\,K\left(\frac{x^{a}+x^{2}\lambda^{a}}{1+2\lambda\\!\cdot\\!x+\lambda^{2}x^{2}},{\bar{y}}\right).$
(57)

To see that ${\hat{U}}_{S}$ defines a unitary representation of the group
${\mathcal{S}}(\Gamma)$ on ${\mathcal{H}}$, we let $S\in{\mathcal{S}}(\Gamma)$
be parameterized by $\lambda^{a}$ and $S^{\prime}\in{\mathcal{S}}(\Gamma)$ be
parameterized by $\mu^{a}$, and consider the action of
${\hat{U}}_{S^{\prime}}{\hat{U}}_{S}$ on a generic state
$\psi(x)\in{\mathcal{H}}$. A calculation gives

$\displaystyle{\hat{U}}_{S^{\prime}}{\hat{U}}_{S}\,\psi(x)=\frac{1}{(1+2\nu\\!\cdot\\!x+\nu^{2}x^{2})^{4}}\,\psi\left(\frac{x^{a}+x^{2}\nu^{a}}{1+2\nu\\!\cdot\\!x+\nu^{2}x^{2}}\right),$
(58)

where $\nu^{a}=\lambda^{a}+\mu^{a}$, and it follows that
${\hat{U}}_{S^{\prime}}\,{\hat{U}}_{S}={\hat{U}}_{S^{\prime}\,S}$. The
remaining component of the conformal group is the one-parameter dilatation
group $\mathcal{D}(\Gamma)$, which consists of transformations of the form
$D:z^{a}\mapsto\Lambda z^{a}$, where $\Lambda$ is a strictly positive real
number. It should be apparent that the unitary operator generating a
dilatation with parameter $\Lambda$ is

$\displaystyle{U}(x,\bar{y})=\Lambda^{4}\,K(\Lambda x^{a},{\bar{y}}^{a}).$
(59)

The corresponding action of a dilatation on a state $\psi(z)\in{\mathcal{H}}$
is thus $D:\psi(z)\mapsto\Lambda^{4}\,\psi(\Lambda z)$, and we see that (59)
defines a unitary representation of the dilatation group on
${L}^{2}(\Gamma,{\cal O})$.

The idea that we exploit in arriving at explicit phase-space representations
for unitary operators is the fact that on a Hilbert space with a reproducing
kernel, every operator admits an integral representation Meschkowski . Thus,
for example, the generator of the space-time translation $z^{a}\to
z^{a}+b^{a}$, where $b^{a}$ is a real four-vector, is given by
$b^{a}{\hat{P}}_{a}$, where ${\hat{P}}_{a}=\mbox{$\rm
i$}\hbar\partial/\partial x^{a}$ is the four-momentum operator; but the four
momentum operator ${\hat{P}}_{a}$ admits the following phase-space
representation:

$\displaystyle P_{a}(z,{\bar{w}})=-\frac{\,3\cdot 2^{9}\,\hbar\,\mbox{$\rm
i$}\,}{\,\pi^{4}}\frac{g_{ab}(z^{b}-{\bar{w}}^{b})}{[g_{ab}(z^{a}-{\bar{w}}^{a})(z^{b}-{\bar{w}}^{b})]^{5}}.$
(60)

##  VII Quantum measurements

To make sense of the notion of quantum detection in a relativistic setting we
need positive operator-valued measures DL ; Davies ; Holevo . In the present
context a POVM can be formed by taking a collection $\Phi$ of positive
operators $\\{\phi_{A}(y,{\bar{z}})\\}_{A\in{\cal B}}$ on phase space labelled
by elements of the Borel $\sigma$-algebra ${\cal B}(\Gamma)$ generated by the
open subsets of $\Gamma$. We require $\Phi$ to have the following properties:
(a) $\phi_{A}(y,{\bar{z}})$ is positive for each $A\in{\cal B}$, (b)
$\phi_{\Gamma}(y,{\bar{z}})=K(y,{\bar{z}})$, and (c) for any countable
collection of disjoint sets $\\{A_{n}\\}_{n\in\mathds{N}}$ in ${\cal B}$ with
union $A=\cup_{n\in\mathds{N}}\,A_{n}$ it holds that

$\displaystyle\phi_{A}(x,{\bar{z}})=\sum_{n}\phi_{A_{n}}(x,{\bar{z}})\,.$ (61)

We consider now a measurement operation appropriate for detecting the location
of an event in phase space. The POVM is defined by

$\displaystyle\phi_{A}(x,{\bar{z}})=\int_{y\in
A}\\!K(x,{\bar{y}})\,K(y,{\bar{z}})\,\mbox{$\rm d$}\mu_{y},\quad A\in{\cal
B}.$ (62)

The recorded outcome of such a measurement will be a measurable set $A$ in
phase space: for instance, the detection of a particle in a certain space-time
region, accompanied by a four-momentum taking values in a certain range. One
can have in mind, for example, the detection of a cosmic ray. By (33) and
(62), the probability that the event lies in the set $A\in{\cal B}$ is

$\displaystyle{\mathbb{P}}(A)=\int\\!\phi_{A}(y,{\bar{z}})\,\rho(z,{\bar{y}})\,\mbox{$\rm
d$}\mu_{y}\,\mbox{$\rm d$}\mu_{z}=\int_{z\in
A}\\!\rho(z,{\bar{z}})\,\mbox{$\rm d$}\mu_{z}.$ (63)

We see, in particular, in accordance with our earlier discussion, that
$\rho(z,{\bar{z}})$ is the probability density for the outcome, and hence that
the expectation value of any measurable function $F:\Gamma\to\mathds{R}$ is
given by the integral

$\displaystyle{\mathbb{E}}[F]=\int\\!\\!F(z,{\bar{z}})\,\rho(z,{\bar{z}})\,\mbox{$\rm
d$}\mu_{z},$ (64)

which is well-defined and finite providing that

$\displaystyle\int\\!\\!\big{|}\,F(z,{\bar{z}})\,\big{|}\,\rho(z,{\bar{z}})\,\mbox{$\rm
d$}\mu_{z}<\infty.$ (65)

Once a measurement has been performed and the outcome recorded, the state of
the system changes. To model this we require a transformation operator of the
Krauss type Krauss 1971 ; Krauss 1983  :

$\displaystyle T_{A}(u,v,{\bar{x}},{\bar{y}})=\int\limits_{w\in
A}\\!\\!\frac{K(u,{\bar{w}})\,K(v,{\bar{w}})\,K(w,{\bar{x}})\,K(w,{\bar{y}})}{K(w,{\bar{w}})}\,\mbox{$\rm
d$}\mu_{w}.$ (66)

One can verify directly that the partial trace of the state transformation
operator generates the POVM. That is, we have

$\displaystyle\int T_{A}(x,y,\bar{y},\bar{z})\,\mbox{$\rm
d$}\mu_{y}=\phi_{A}(x,\bar{z})$ (67)

for each $A\in{\cal B}$. Now suppose that the system is initially in the state
$\rho_{\rm in}(y,{\bar{v}})$. Then after the measurement we find that

$\displaystyle\rho_{\rm out}(u,{\bar{x}})=\frac{\int
T_{A}(u,v,{\bar{x}},{\bar{y}})\,\rho_{\rm in}(y,{\bar{v}})\,\mbox{$\rm
d$}\mu_{v}\,\mbox{$\rm d$}\mu_{y}}{\int
T_{A}(z,v,{\bar{z}},{\bar{y}})\,\rho_{\rm in}(y,{\bar{v}})\,\mbox{$\rm
d$}\mu_{v}\,\mbox{$\rm d$}\mu_{y}\,\mbox{$\rm d$}\mu_{z}},$ (68)

which represents the transformed state that results when the measurement
determines that the phase-space event lies in the set $A\in{\cal B}$.
Substituting (66) in (68), and making use of the reproducing property (33), we
deduce that

$\displaystyle\rho_{\rm out}(u,{\bar{x}})=\frac{\int_{z\in
A}\Psi_{z}(u,\bar{x})\,\rho_{\rm in}(z,\bar{z})\,\mbox{$\rm
d$}\mu_{z}}{\int_{z\in A}\rho_{\rm in}(z,\bar{z})\,\mbox{$\rm d$}\mu_{z}},$
(69)

where

$\displaystyle\Psi_{z}(u,\bar{x})=\frac{K(u,{\bar{z}})\,K(z,{\bar{x}})}{K(z,{\bar{z}})}$
(70)

is the density matrix associated with the normalized wave function

$\displaystyle\psi_{z}(u)=\frac{K(u,{\bar{z}})}{\,\,[K(z,{\bar{z}})]^{1/2}}.$
(71)

Then in the limit that the recorded outcome shrinks to a phase-space point, we
find that

$\displaystyle\rho_{\rm out}(u,{\bar{x}})=\Psi_{z}(u,\bar{x}).$ (72)

In what follows we shall refer to a pure state of the form (71) as a coherent
state with focus $z$. We observe, in particular, that by virtue of (44) we
have

$\displaystyle\psi_{z}(u)=\frac{8\sqrt{3}}{\pi^{2}}\,\frac{[g_{ab}(z^{a}-{\bar{z}}^{a})(z^{b}-{\bar{z}}^{b})]^{2}}{[g_{ab}(u^{a}-{\bar{z}}^{a})(u^{b}-{\bar{z}}^{b})]^{4}}.$
(73)

We can also refer to a pure density matrix of the form (70) as a coherent
state, and we note that the family of such density matrices satisfies a
completeness relation of the form

$\displaystyle\int_{z}\Psi_{z}(x,{\bar{y}})\,K(z,{\bar{z}})\,\mbox{$\rm
d$}\mu_{z}=K(x,{\bar{y}}).$ (74)

That such a relation should hold is characteristic of the properties of
coherent states Peremolov 1972 ; Gazeau 2014  and follows from the fundamental
identity (34), as does the structure of the POVM given by (62). It is
interesting then to note that the coherent states arising in the present
context are in one-to-one correspondence with the so-called “elementary
states” that arise in the theory of zero rest mass fields Penrose 1972 .

The foregoing analysis shows that when the measurement apparatus detects that
an event has taken place in a region $A$ of phase space, the output state will
in general be a mixed state, given by the weighted average of the coherent
state $\Psi_{z}(u,\bar{x})$ over $z\in A$ with respect to the renormalized
density

$\displaystyle\rho_{A}(z,\bar{z})=\frac{\rho_{\rm in}(z,\bar{z})}{\int_{y\in
A}\rho_{\rm in}(y,\bar{y})\,\mbox{$\rm d$}\mu_{y}}.$ (75)

If, however, the record shows a specific phase-space point $z$ as the result,
then the output density matrix will be the coherent state
$\Psi_{z}(u,\bar{x})$ parameterized by $z$. At the other extreme, if the
measurement is performed but the outcome is not recorded, then the focus is
smeared over the whole of the phase space, representing a decoherence effect,
and we obtain

$\displaystyle\rho_{\rm out}(u,{\bar{x}})=\int\Psi_{z}(u,\bar{x})\,\rho_{\rm
in}(z,\bar{z})\,\mbox{$\rm d$}\mu_{z}.$ (76)

##  VIII Properties of coherent states

On the matter of the interpretation of the coherent states, we remark that the
family of coherent states $\\{\psi_{z}(u)\\}$ parameterized by $z\in\Gamma$ is
Poincaré invariant in the sense that under the unitary transformation (53) one
has

$\displaystyle{\hat{U}}\psi_{z}(u)=\psi_{w}(u),$ (77)

where $w^{a}=L^{a}_{b}\,z^{b}-B^{a}$. In other words, the action of the
unitary representation of the Poincaré transformation on a coherent state
$\psi_{z}(u)$ focussed at $z\in\Gamma$ is the coherent state $\psi_{w}(u)$
focussed at $w$, where $w$ is the result of the corresponding inverse Poincaré
transformation on $z$.

More generally, one can show that manifold of coherent states is invariant
under the action of the 15-parameter conformal group. To see this, consider
first the dilatation group. From (59) we see that
${\hat{U}}\psi_{z}(u)=\psi_{w}(u)$, where $w^{a}=\Lambda^{-1}z^{a}$. Thus, the
action of the dilatation on a coherent state $\psi_{z}(u)$ focussed at the
phase-space point $z$ is a coherent state $\psi_{w}(u)$ focussed at $w$, where
$w$ is the result of the corresponding inverse dilatation on $z$.

The action of the special conformal transformations on a coherent state is a
little more subtle. Writing ${\hat{U}}_{\lambda}$ for the unitary operator
(57), we find that

$\displaystyle{\hat{U}}_{\lambda}\psi_{z}(u)=\frac{8\sqrt{3}}{\pi^{2}}\,\frac{[(z-{\bar{z}})\\!\cdot\\!(z-{\bar{z}})]^{2}}{(1+2\lambda\\!\cdot\\!u+\lambda^{2}u^{2})^{4}}\left[\left(\frac{u+u^{2}\lambda}{1+2\lambda\cdot
u+\lambda^{2}u^{2}}-{\bar{z}}\right)\\!\cdot\\!\left(\frac{u+u^{2}\lambda}{1+2\lambda\cdot
u+\lambda^{2}u^{2}}-{\bar{z}}\right)\right]^{-4}\\!.$ (78)

A calculation then gives

$\displaystyle\left(\frac{u+u^{2}\lambda}{1+2\lambda\cdot
u+\lambda^{2}u^{2}}-{\bar{z}}\right)\\!\cdot\\!\left(\frac{u+u^{2}\lambda}{1+2\lambda\cdot
u+\lambda^{2}u^{2}}-{\bar{z}}\right)=\frac{1-2\lambda\\!\cdot\\!{\bar{z}}+\lambda^{2}{\bar{z}}^{2}}{1+2\lambda\\!\cdot\\!u+\lambda^{2}u^{2}}\left[(u-{\bar{w}})\\!\cdot\\!(u-{\bar{w}})\right],$
(79)

where

$\displaystyle
w^{a}=\frac{z^{a}-z^{2}\lambda^{a}}{1-2\lambda\\!\cdot\\!z+\lambda^{2}z^{2}}.$
(80)

Thus the terms involving $(1+2\lambda\\!\cdot\\!u+\lambda^{2}u^{2})$ appearing
in (78) cancel, and we obtain

$\displaystyle{\hat{U}}_{\lambda}\psi_{z}(u)=\frac{8\sqrt{3}}{\pi^{2}}\,\frac{[(z-{\bar{z}})\\!\cdot\\!(z-{\bar{z}})]^{2}}{(1-2\lambda\\!\cdot\\!{\bar{z}}+\lambda^{2}{\bar{z}}^{2})^{4}}\frac{1}{[(u-{\bar{w}})\\!\cdot\\!(u-{\bar{w}})]^{4}}.$
(81)

In fact, the right side of (81) is a phase-shifted version of the coherent
state $\psi_{w}(u)$ centred at $w$:

$\displaystyle{\hat{U}}_{\lambda}\psi_{z}(u)=\mbox{$\rm e$}^{{\rm
i}\theta}\,\frac{8\sqrt{3}}{\pi^{2}}\,\frac{[(w-{\bar{w}})\\!\cdot\\!(w-{\bar{w}})]^{2}}{[(u-{\bar{w}})\\!\cdot\\!(u-{\bar{w}})]^{4}},$
(82)

where the phase shift $\theta$ is given by

$\displaystyle\theta=\frac{1}{2\mbox{$\rm
i$}}\log\left[\frac{1-2\lambda\\!\cdot\\!z+\lambda^{2}z^{2}}{1-2\lambda\\!\cdot\\!{\bar{z}}+\lambda^{2}{\bar{z}}^{2}}\right]=\frac{1}{2\mbox{$\rm
i$}}\log\left[\frac{z^{2}{\bar{w}}^{2}}{{\bar{z}}^{2}w^{2}}\right].$ (83)

In other words, the result of the action of a special conformal transformation
on a coherent state $\psi_{z}(u)$ focussed at $z$ is a coherent state
$\psi_{w}(u)$ focussed at $w$ with a phase shift $\theta$, where $w$ is the
result of the action of the corresponding inverse special conformal
transformation on $z$. Since the physical state of a system is defined up to
an overall phase (or, equivalently, the phase factor drops out if we consider
the action of a conformal transformation on a pure-state density matrix), we
deduce that the manifold of coherent states is invariant under the
15-parameter conformal group.

In calculations, it turns out to be convenient in many situations to work with
Fourier transforms. In fact, there are some remarkable identities that turn
out to be useful in this connection. The Fourier transform of an element
$\psi\in{L}^{2}(\Gamma,{\cal O})$ is defined by

$\displaystyle\Psi(p^{a})=\int_{\Gamma}\\!\exp({{\rm
i}p_{a}\bar{z}^{a}})\,\psi(z^{a})\,\mbox{$\rm d$}\mu_{z}\,.$ (84)

The inverse Fourier transform is then given by

$\displaystyle\psi(z^{a})=\frac{1}{8\pi^{5}}\int_{V^{+}}\\!\exp({{-\rm
i}p_{a}{z}^{a}})\,\big{(}p_{b}p^{b}\big{)}^{2}\,\Psi(p^{a})\,\mbox{$\rm
d$}^{4}p,$ (85)

where the integration is over the interior of the forward cone defined by

$\displaystyle V^{+}=\\{p^{a}:p_{a}p^{a}>0,p^{0}>0\\}.$ (86)

The argument for (85) can be sketched as follows. Let $I(z^{a})$ denote the
outcome of the integral appearing on the right-hand side of (85). If we
substitute (84) into the formula for $I(z^{a})$, we get

$\displaystyle I(z^{a})=\frac{1}{8\pi^{5}}\int_{V^{+}}\\!\exp({{-\rm
i}p_{a}{z}^{a}})\,\big{(}p_{b}p^{b}\big{)}^{2}\int_{\Gamma}\exp({{\rm
i}p_{a}\bar{w}^{a}})\,\psi(w^{a})\,\mbox{$\rm d$}\mu_{w}\,\mbox{$\rm
d$}^{4}p.$ (87)

Then, reversing the order of integration, we have

$\displaystyle
I(z^{a})=\frac{1}{8\pi^{5}}\int_{\Gamma}\,\left[\int_{V^{+}}\\!\exp({{-\rm
i}p_{a}{z}^{a}})\,\big{(}p_{b}p^{b}\big{)}^{2}\exp({{\rm
i}p_{a}\bar{w}^{a}})\,\mbox{$\rm d$}^{4}p\right]\,\psi(w^{a})\,\mbox{$\rm
d$}\mu_{w}.$ (88)

Now, the inner integral can be carried out explicitly, and we obtain

$\displaystyle\frac{1}{8\pi^{5}}\int_{V^{+}}\\!\exp({{-\rm
i}p_{a}{z}^{a}})\,\big{(}p_{b}p^{b}\big{)}^{2}\exp({{\rm
i}p_{a}\bar{w}^{a}})\,\mbox{$\rm d$}^{4}p=K(z,\bar{w}),$ (89)

where the Bergman kernel is defined as in (44). An application of the
reproducing property (33) then shows that $I(z^{a})=\psi(z^{a})$, and thus we
obtain (85), the Fourier inversion formula.

Alternatively, suppose that we are given a map $\Psi:V^{+}\to{\mathds{C}}$ on
the positive cone such that

$\displaystyle\int_{V^{+}}\big{(}p_{b}p^{b}\big{)}^{2}\,\big{|}\Psi(p^{a})\big{|}^{2}\,\mbox{$\rm
d$}^{4}p<\infty,$ (90)

and we define a holomorphic function $\psi:\Gamma\to{\mathds{C}}$ by use of
(85). Let $J(p^{a})$ denote the outcome of the integral appearing on the right
side of (84). Then for $J(p^{a})$ we obtain

$\displaystyle J(p^{a})$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{5}}\int_{\Gamma}\\!\exp({{\rm
i}p_{a}\bar{z}^{a}})\int_{V^{+}}\\!\exp({{-\rm
i}q_{a}{z}^{a}})\,\big{(}q_{b}q^{b}\big{)}^{2}\,\Psi(q^{a})\,\mbox{$\rm
d$}^{4}q\,\mbox{$\rm d$}\mu_{z}$ (91) $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{5}}\int_{V^{+}}\left[\,\int_{\Gamma}\\!\exp({{\rm
i}p_{a}\bar{z}^{a}}{{-\rm i}q_{a}{z}^{a}})\,\mbox{$\rm
d$}\mu_{z}\,\right]\big{(}q_{b}q^{b}\big{)}^{2}\,\Psi(q^{a})\,\mbox{$\rm
d$}^{4}q.$

Let us consider the inner integration first. Writing $z^{a}=x^{a}-\mbox{$\rm
i$}r^{a}$ with $x^{a}$ real, and with $r^{a}$ timelike and future pointing, we
have

$\displaystyle{\mbox{$\rm i$}}p_{a}\bar{z}^{a}-{\mbox{$\rm
i$}}q_{a}{z}^{a}={\mbox{$\rm i$}}(p_{a}-q_{a})x^{a}-(p_{a}+q_{a})r^{a}.$ (92)

Since $\mbox{$\rm d$}\mu_{u}=\mbox{$\rm d$}^{4}x\,\mbox{$\rm d$}^{4}r$, we
find that the $x$-integration over Minkowski space gives a delta function.
Thus setting $\xi_{a}=p_{a}+q_{a}$ we have

$\displaystyle\int_{\Gamma}\\!\exp({{\mbox{$\rm i$}}p_{a}\bar{z}^{a}}{{-\rm
i}q_{a}{z}^{a}})\,\mbox{$\rm
d$}\mu_{z}=(2\pi)^{4}\,\delta^{4}(p^{a}-q^{a})\int_{V^{+}}\exp(-\xi_{a}r^{a})\,\mbox{$\rm
d$}^{4}r\,.$ (93)

For the $r$-integration, we can pass to spherical coordinates. Then if we set
$\xi^{2}=(\xi^{1})^{2}+(\xi^{2})^{2}+(\xi^{3})^{2}$ and
$R^{2}=(r^{1})^{2}+(r^{2})^{2}+(r^{3})^{2}$, because $r^{a}$ is time-like we
find

$\displaystyle\int_{V^{+}}\exp(-\xi_{a}r^{a})\,\mbox{$\rm
d$}^{4}r=\int\limits_{r^{0}=0}^{\infty}\int\limits_{R=0}^{r^{0}}\int\limits_{\theta=0}^{\pi}\int\limits_{\phi=0}^{2\pi}\exp\left({-\xi_{0}r^{0}+\xi
R\cos\theta}\right)\,R^{2}\mbox{$\rm d$}r^{0}\mbox{$\rm
d$}R\sin\theta\,\mbox{$\rm d$}\theta\,\mbox{$\rm d$}\phi\,.$ (94)

Because $r^{a}$ is future pointing, the integrals can now be performed
explicitly to give

$\displaystyle\int_{V^{+}}\exp(-\xi_{a}r^{a})\,\mbox{$\rm
d$}^{4}r=\frac{8\pi}{(\xi_{a}\xi^{a})^{2}}\,,$ (95)

from which it follows that

$\displaystyle\int_{\Gamma}\\!\exp({{\mbox{$\rm
i$}}p_{a}\bar{z}^{a}}{{-\mbox{$\rm i$}}q_{a}{z}^{a}})\,\mbox{$\rm
d$}\mu_{z}=\frac{8\pi^{5}}{(q_{a}q^{a})^{2}}\,\delta^{4}(p^{a}-q^{a}).$ (96)

Inserting this expression back into (91) we immediately see that the result of
the integral is $\Psi(p^{a})$.

By a similar argument it follows from (89) that if $\Psi(p)$ is the Fourier
transform of a square-integrable holomorphic function $\psi(z)$, and $\Phi(p)$
is the Fourier transform of a square-integrable holomorphic function
$\phi(z)$, then we have a Parseval identity of the form

$\displaystyle\int_{\Gamma}\,\,\psi(z^{a})\,\bar{\phi}(\bar{z}^{a})\,\mbox{$\rm
d$}\mu_{z}=\frac{1}{8\pi^{5}}\int_{V^{+}}\Psi(p)\,(p_{a}p^{a})^{2}\,{\bar{\Phi}(p)}\,\mbox{$\rm
d$}^{4}p.$ (97)

In the case of a coherent state $\psi_{z}(u)$ with focus $z\in\Gamma$, a
calculation shows that its Fourier transform is given by

$\displaystyle\Psi_{z}(p^{a})=\frac{8\sqrt{3}}{\pi^{2}}\,\big{[}\,g_{ab}(z^{a}-{\bar{z}}^{a})(z^{b}-{\bar{z}}^{b})\,\big{]}^{2}\,\exp({{\rm
i}\bar{z}_{a}p^{a}})\,.$ (98)

Now writing $z^{a}=x^{a}-\mbox{$\rm i$}r^{a}$, we have
$z^{a}-{\bar{z}}^{a}=-2\mbox{$\rm i$}r^{a}$, so

$\displaystyle\Psi_{z}(p^{a})=\frac{2^{7}\\!\sqrt{3}}{\pi^{2}}\,\big{(}r_{a}r^{a}\big{)}^{2}\,\mbox{$\rm
e$}^{-r_{a}p^{a}}\,\mbox{$\rm e$}^{{\rm i}x_{a}p^{a}}.$ (99)

This relation shows that as $z^{a}$ varies the Fourier component
$\Psi_{z}(p^{a})$ behaves like a plane wave in Minkowski space that has been
extended into the future tube, but is damped exponentially for large $r^{a}$.
We notice, in particular, that when $r^{a}$ is large, corresponding to the
case where focal point lies in a low-mass region of $\Gamma$, the damping of
the high-energy Fourier components is significant.

##  IX Phase-space localization

With a view to getting a better understanding of the degree of localization in
phase space that might be achievable in such a detection experiment, let us
consider properties of the coherent states in more detail.

For each choice of the focal point $z\in\Gamma$, the associated coherent state
is represented by the normalized wave function $\psi_{z}(u)$. Now, if
$|\phi\rangle\in L^{2}(\mathcal{H},\mathcal{O})$ is any other normalized
state, we have the Cauchy-Schwartz inequality

$\displaystyle\langle\,\psi_{z}\,|\,\phi\,\rangle\,\langle\,\phi\,|\,\psi_{z}\,\rangle\leq
1.$ (100)

It then follows immediately from (33), (71) and (100) that

$\displaystyle\phi(z)\,\bar{\phi}(\bar{z})\leq K(z,\bar{z})$ (101)

for all $z\in\Gamma$, which shows that the amplitude of any pure state at a
phase-space point $z$ is bounded by the amplitude at $z$ of the coherent state
with focus $z$.

Hence the coherent states are the most sharply peaked states, and the peak of
a coherent state occurs at its focus. It thus makes sense that if the
measurement outcome takes the form of a specific point in phase space, then
the transformed state should be peaked as much as possible at that point, and
hence a coherent state with that peak, as we obtained in (72).

An interesting physical interpretation of the inequality (101) can be deduced
if we write ${\rm Im}\,(z^{a})=-r^{a}$ and make use of the phase-space
correspondence (21). It follows from (44) that

$\displaystyle
K(z,{\bar{z}})=\frac{3}{\,4\,\pi^{4}\,\hbar^{8}}\left(g_{ab}\,p^{a}p^{b}\right)^{4}.$
(102)

Then if we let

$\displaystyle M_{z}=\left(g_{ab}\,p^{a}p^{b}\right)^{1/2}$ (103)

denote the mass associated with the phase-space point $z^{a}$, we can write
(101) in the form of a localization bound on the probability density. In
particular, we obtain

$\displaystyle\rho(z,\bar{z})\leq\frac{3}{\,4\,\pi^{4}\,\hbar^{8}}\,M_{z}^{\,8}\,,$
(104)

which shows that states cannot be localized very sharply in regions of phase
space with low mass, but that for higher mass a much greater degree of
localization can be achieved.

We can thus think of (104) as a localization theorem for relativistic quantum
theory. Suppose that a phase-space event of a relativistic system is
characterized by a pure state $\phi(z)$. Then the probability of detecting the
event is determined by the normalized density function

$\displaystyle\rho(z,{\bar{z}})=\phi(z)\,\bar{\phi}(\bar{z})\,.$ (105)

The localization theorem shows that the maximum value that the density
function can take, at any given point in the eight-dimensional phase space, is

$\displaystyle\rho_{\rm
max}(z,\bar{z})=\frac{3}{\,4\,\pi^{4}\,\lambdabar_{z}^{\,8}}\,,$ (106)

where $\lambdabar_{z}$ denotes the reduced Compton wavelength associated with
the phase-space point $z$. Now, if an event is detected to have occurred at a
specific phase-space point $z$, then the resulting output wave function will
be given by the corresponding coherent state, for which the bound in (104) is
saturated. Since the density function $\rho(z,{\bar{z}})$ has to integrate to
unity over the phase space, one is thus able to conclude that for a system of
short Compton wavelength the event will be highly localized in phase space. It
follows that we can view the coherent states as representing in some sense the
most “classical” type of state that can be formed over the relativistic phase
space.

To gain further intuition about the nature of localization, let us consider an
example in which the state of the system is given by a holomorphic wave
function $|\,\xi\rangle$ which we take to be normalized, and an
experimentalist wishes to determine whether a localized event at a phase space
point $z$ can be detected. Since this is a “yes-no” type of question, a
projective measurement is appropriate, and accordingly we consider the
projection operator

$\displaystyle\hat{\Pi}_{z}=|\psi_{z}\rangle\,\langle\psi_{z}|\,,$ (107)

where $|\psi_{z}\rangle$ denotes a normalized coherent state focussed at the
point $z$. Here as an aid to intuition we introduce the usual bra-ket notation
and we add “hats” to operators. Then $|\,\xi\rangle$ can be split uniquely
into a part that is localized at $z$ and a part that is orthogonal to
$|\psi_{z}\rangle$, so

$\displaystyle|\,\xi\rangle=|\psi_{z}\rangle\,\langle\psi_{z}|\,\xi\rangle\,+(\hat{\mathds{1}}-\hat{\Pi}_{z})|\,\xi\rangle\,.$
(108)

In fact, it is not difficult to show that if a holomorphic function
$\theta(u)\in L^{2}(\mathcal{H},\mathcal{O})$ is orthogonal to a coherent
state $\psi_{z}(u)$ with focal point $z$, then $\theta(u)$ vanishes at the
focal point. Thus, we can say in a meaningful sense that any state that is
orthogonal to a coherent sate with focus $z$ is delocalized away from the
focal point.

If the outcome of the projective measurement is affirmative, then, by the
usual Lüders-type rules for projective measurements Luders 1951 , the
transformed state will be the localized state $\psi_{z}(u)$. Otherwise, we
obtain the delocalized wave function given by the uniquely determined relative
state. In particular, if the initial state is given by the holomorphic
function $\xi(u)$ then the outcome of a projective measurement based on the
projection operator associated with the coherent state $|\psi_{z}\rangle$ will
be affirmative with probability

$\displaystyle
p=\langle\psi_{z}|\,\xi\,\rangle\langle\,\xi|\,\psi_{z}\rangle=\frac{\,\xi(z)\,\bar{\xi}(\bar{z})}{K(z,\bar{z})}.$
(109)

Now suppose that the wave function $|\xi\rangle$ is itself a coherent state,
centred at the phase-space point $w=x-\mbox{$\rm i$}r$. Then we have
$|\xi\rangle=|\psi_{w}\rangle$, and the probability $p$ of obtaining a “yes”
outcome in a projective measurement involving the projection operator
${\hat{\Pi}}_{z}$ with $z=x^{\prime}-\mbox{$\rm i$}r^{\prime}$ is given by
$|\langle\psi_{w}|\psi_{z}\rangle|^{2}$. A calculation shows that

$\displaystyle
p=\left[\,\frac{(w-{\bar{w}})\cdot(w-{\bar{w}})\,\,(z-{\bar{z}})\cdot(z-{\bar{z}})}{(w-{\bar{z}})\cdot(w-{\bar{z}})\,\,(z-{\bar{w}})\cdot(z-{\bar{w}})}\,\right]^{\,4},$
(110)

which can be viewed as a cross ratio between the four points $w^{a}$, $z^{a}$,
$\bar{w}^{a}$ and $\bar{z}^{a}$. In fact, it is well known that the transition
probability between two pure states in nonrelativistic quantum mechanics
admits an interpretation as a cross ratio between points in complex projective
space Brody Hughston 2001 .

But in the present context what is surprising is that the cross ratio involves
points in complex Minkowski space. It is a straightforward exercise to verify
that the cross ratio is conformally invariant and hence a fortiori Poincaré
invariant.

##  X Discussion

In summary, we have shown that the future tube possesses a phase-space
geometry appropriate both for (a) formulation of a consistent Hamiltonian
mechanics for relativistic systems, and (b) construction of a quantum theory
of space-time events. In particular, the Hilbert space of square-integrable
holomorphic functions on the future tube can be interpreted as the pure state
space of relativistic quantum mechanics. The resulting structure is rich
enough to allow for the development of a manifestly covariant theory of
measurement for the detection of phase-space events. The theory incorporates a
natural transformation rule for the quantum state after the measurement, a
concept that has hitherto been lacking in relativistic quantum theory. We are
also able to gain some understanding of the extent to which relativistic
events can be localized. An upper bound for the phase-space probability
density can be determined, which is inversely proportional to the eighth power
of the Compton wavelength. The upper bound is achieved at any given point in
phase space by the probability density associated with the phase-space
coherent state that has its focal point at that point.

Acknowledgements. DCB acknowledges support from the Russian Science
Foundation, grant 20-11-20226. We thank M. Hoban for helpful discussions.

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