Open Systems and Measurement in Relativistic Quantum Theory: Proceedings of the Workshop Held at the Istituto Italiano per gli Studi Filosofici Naples, April 3–4, 1998

232 Pages·1999·11.35 MB·English

by  Heinz-Peter Breuer

State Vector Reduction in Relativistic
Quantum Mechanics: An Introduction
Heinz-Peter Breuer and Francesco Petruccione
Fakult£t fiir Physik, Universit~t Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg
i. Br., Germany
Abstract. An introductory outline of measurements in relativistic quantum theory
is given. Following the ideas of Aharonov and Albert, the apparent paradoxes of
causality for nonlocal measurements and of the instantaneity of the state vector
reduction are discussed. A relativistically covariant prescription for measurements of
local and nonlocal observables is presented. The selective measurement of nonlocal
observables is formulated in terms of the state vectors of the quantum object and
of the corresponding quantum probe. Finally, some general conclusions concerning
the constraints imposed by causality on the measurability of operators and states
are drawn.
1 Introduction
In 1932 John von Neumann ]1[ introduced the following assumption about the
reduction of the wave function within the orthodox statistical interpretation
of quantum mechanics: As a consequence of the measurement of one (or more)
observables the state vector characterizing the measured system undergoes an
instantaneous change, a discontinuous quantum jump; after the measurement
the system is described by one of the possible eigenfunctions of the measured
observables.
It is the aim of this introductory article to address some paradoxes aris-
ing in the application of this 'standard' reduction postulate to relativistic
quantum mechanics and to show how these apparent problems can be solved.
Furthermore, we will indicate directions of actual and future research.
Applying naively the reduction postulate to relativistic quantum mechan-
ics one is immediately confronted with two apparent paradoxes. These para-
doxes are easily made explicit by considering two simple physical situations
which have been discussed in detail by Aharonov and Albert [2].
The paradox of causality for nonlocal measurements: Let a particle
be originally localized in a certain spacetime region A. At a certain time tl a
measurement of the momentum of the particle is performed. As a result of the
measurement the particle will be in a momentum eigenstate for times t > tl;
as a consequence of the measurement it will have a non-vanishing amplitude
in the whole 3-space. In other words: Apparently, the measurement of the
momentum distributes the particle instantaneously over the whole space (see
Fig. 1). Namely, when the location of the particle is measured at a time
2 Heinz-Peter Breuer and Francesco Petruccione
around t -- tl + e, the particle is found in a spacetime region B with a
probability which is different from zero:
P(B) > 0. (1)
On the other hand, the spacetime regions A and B can be chosen such that
each pair of points AX E A and xB E B is separated by a spacelike distance:
(XB--XA) 2-(xB-xA)~(xB-xA)~ <0 for all XB E B, AX EA. (2)
xO-- t
tl -~- E
tl
A
¢#0
Fig. 1. A particle is initially confined to some spacetime region A. A measurement
of the momentum at time tl redistributes the probability amplitude over all 3-space.
In this way information could be transferred from A to B with a speed greater than
the speed of light.
Thus, the causality principle is violated here; had we not measured the
momentum at time t = tl, then clearly P(B) = 0. It seems that through the
State Vector Reduction in Relativistic Quantum Mechanics 3
measurement of momentum, information can be carried from A to B with a
speed greater than the speed of light.
The paradox of the instantaneity of the state vector reduction"
This problem is related to the relativity of instantaneous events, i.e. to the
nonexistence of an absolute time in relativity theory. If the state vector re-
duction takes place instantaneously in an inertial system, in another inertial
system it will not be instantaneous. Again we consider an example. Let a free
particle be prepared in a momentum eigenstate at time t -- -oo. At time t -- 0
the same particle is found at x = 0 by some detector which is placed there
(see Fig. 2). The detector is assumed to interact locally with the particle. Ac-
Lorentz transformation
,, "-S;:" ,~;S +"
Fig. 2. Instantaneous state reduction following a local measurement. The state
reduction occurs along an equal-time hypersurface t = const in one frame (left). In
a moving inertial frame this surface is no longer an equal-time hypersurface (right).
cording to the reduction postulate at the time of measurement t = 0 the state
of the particle will change instantaneously from a momentum eigenstate to a
position eigenstate. The paradox of this situation is the following: Although
the local position measurement should be described by some local, covariant
interaction between object and measurement device, the above prescription
for an instantaneous state vector reduction is obviously not covariant. Thus,
the following question arises naturally: How does a covariant prescription for
state vector reduction look like in relativistic quantum theory?
Some of the questions related to the first paradox have already been dis-
cussed by Landau and Peierls in 1931 [3]. They concluded that all nonlocal
quantities, like the momentum operator, cannot be observables in relativistic
quantum theories. This seems to be a very severe restriction if one con-
4 Heinz-Peter Breuer and Francesco Petruccione
siders that the momentum is a very important quantity, e.g. for scattering
experiments, and that the canonical quantization of field theories is usually
performed in the momentum representation.
In a certain sense the important work of Bohr and Rosenfeld 'Zur Frage
der Mefibarkeit der elektromagnetischen FeldgrSi]en' [4], which was published
in 1933, can be understood as an argumentation against Landau and Peierls.
Bohr and Rosenfeld show how the operators of the electromagnetic field can
be measured, and that the uncertainity relations for the field operators are in-
deed satisfied. The essential difference to the reasoning of Landau and Peierls
is that Bohr and Rosenfeld assume the measuring apparatus to be classical,
and that, therefore, its atomistic structure can be neglected completely. Thus,
we see that one of the central questions of state vector reduction in relativis-
tic quantum mechanics will be: What can and what cannot be measured in
relativistic quantum mechanics?
The second paradox clearly has its origin in the fact that the standard
reduction postulate is definitely not Lorentz covariant. Accordingly, the sec-
ond fundamental question of relativistic state vector reduction is: What is
the relativistic covariant prescription of state vector reduction?
Such a prescription in the spirit of Landau and Peierls has beeen given
by Hellwig and Kraus [5]. They assume that the relativistic state reduction
for a local measurement has to occur along the backward ligh t cone of the
measurement event (see Fig. 3). As Hellwig and Kraus have shown their pre-
scription is manifestly Lorentz-covariant and leads to the correct predictions
for all probabilities of local measurements. However, this state vector reduc-
tion prescription fails for the measurement of nonlocal observables [2]. This
can be seen easily with the help of Fig. 3. If the particle has been prepared
at time t = -~ in a momentum eigenstate, a further measurement of the
momentum at time t = 0-e, i.e. at a time prior to the position measurement,
will simply affirm the state of the system. At time t = 0 - c the particle is
in a momentum eigenstate and not in the state indicated in Fig. 3, which is
concentrated along and within the backward light cone. Hellwig and Kraus
will not consider this criticism to be a problem, simply because, in the sense
of Landau and Peierls, the measurability of momentum at time t = 0 - e is
regarded as impossible. However, there is indeed a problem. Aharonov and
Albert have shown [2, ]6 that it is possible to measure and prepare nonlocal
observables. We will discuss this in detail in Sect. 3. There we will also ex-
plain why causality is not violated in such measurement schemes. Herewith
the reduction postulate of Hellwig and Kraus is invalidated.
The second paradox will be studied in detail in Sect. 2. Among the many
attempts to circumvent this problem we want to mention just the work of Co-
hen and Hiley [7]. These authors suggest to perform the state vector reduction
instantaneously in a so-called preferred Lorentz frame. This approach does
not seem to be particularly attractive. The introduction of a preferred coordi-
nate system in a certain sense seems to be a regression to a pre-relativistic age.
State Vector Reduction in Relativistic Quantum Mechanics 5
/
Fig. 3. Illustration of the relativistic state vector reduction postulate of Hellwig
and Kraus and its invalidation by Aharonov and Albert.
In particular, such a prescription implies the reintroduction of the concept
of an absolute time, which is essentially determined by the inertial system
in which the state vector reduction is instantaneous, i.e. in which it occurs
along the hypersurfaces t = const. There might be cases for which a preferred
frame can be introduced in a plausible way (e.g. considering a particle decay-
ing into two other particles, one could choose the rest frame of the common
center of mass), however there does not seem to be the possibility to achieve
this satisfactorily in more general eases. Consider for example an electron,
whose position we want to measure. If the reduction is to be performed in-
stantaneously in the rest system of the electron, we have to determine the
momentum of the electron; however the measurement of position leaves the
momentum completely undetermined. A continuous measurement of the po-
sition of the electron is therefore impossible.
6 Heinz-Peter Breuer and Francesco Petruccione
2 State Vector Reduction for Local Measurements
Let us now concentrate our attention on the paradox of the instantaneity of
state vector reduction. We begin by considering local measurements. Essen-
tially there are two fundamental possibilities to circumvent this paradox.
First, one can stick to the usual concept of a wave function ¢(t, x) as a
function on the spacetime continuum. If one wants to construct a Lorentz
covariant prescription for the state vector reduction, one has to identify some
Lorentz invariant hypersurface (such as the backward light cone) along which
the reduction takes place, that is along which the dynamics of the wave
function is not continuous.
There is a second possibility which has been first proposed by Aharonov
and Albert [8]. This possibility consists in the assumption that the state vec-
tor reduction takes place instantaneously in all inertial flames. Formulated
in a more precise way this means the following: Let us assume that a local
measurement takes place in the spacetime point P. Then the state vector
reduction takes place along all spacelike hypersur/aces a, which pass through
this point P .1 In the following we will denote this set of surfaces by ~p (see
Fig. 4). The set of all flat, spacelike hypersurfaces in Minkowski space is then
given by
U )3(
Obviously, this construction leads to a Lorentz covariant prescription of the
state vector reduction, because the set ~p is Lorentz invariant.
Each surface a E ~p appears as an 'equal-time' hypersurface in some
reference frame. Thus ~'p can be identified with the set of all local observers
which go through the point P. In other words we can identify Ep with the
set of all worldlines y = y(~-) (parametrized by their proper time 7) passing
the point P,
y(0) = P, (4)
with a timelike 4-velocity n,
n-- YTy~d r=0 , n,n" = 1. (5)
Given such a path, the hypersurface which is defined by
a = a(n,P) = {x e ]R41n(x- P) = 0} (6)
represents an 'equal-time' hypersurface T = 0 for the corresponding local
observer (in the instantaneous rest frame K' of this observer we have = ~ n
1 Aharonov and Albert have formulated this prescription for arbitrary, not neces-
sarily flat, spacelike hypersurfaces. For our present needs, however, it is sufficient
to consider only the subset of flat hypersurfaces, which is closed under Lorentz
transformations.
State Vector Reduction in Relativistic Quantum Mechanics 7
a Ep
X 1
Fig. 4. The covariant state reduction postulate of Aharonov and Albert for a local
measurement at the spacetime point P.
(1, 0, 0, 0) and the equation for a becomes simply x °~ - ~ =t const). Hence
we see that ~p is a 3-dimensional manifold, which is parametrized uniquely
by a timelike unit normal vector n.
In this way the reduction prescription of Aharonov and Albert appears to
be completely natural. In agreement with the theory of relativity all instan-
taneous observers are physically equivalent. Regardless from the meaning we
want to attribute to the state vector reduction as a real physical process,
the practical scope of the state vector reduction is to describe in a consistent
way how to exploit the information gained in a certain measurement for the
prediction of the probabilities of future events. It should describe how the
result of a certain measurement determines the following state of the quan-
tum mechanical system. The question to be answered is: How does a certain
observer in his frame o/ reference use the information gained in the mea-
surement to describe future states of the system? In this formulation the set
~p appears very naturally and the paradox of the instantaneity of the state
vector reduction loses all aspects of a violation of covariance.
8 Heinz-Peter Breuer and Francesco Petruccione
The above prescription for the reduction of the state vector does have an
important consequence. Namely, the wave function is no longer a function
on the spacetime continuum. In order to explain this important point we
suppose that in the infinite past an electron has been prepared in a state
which is given by a superposition of two wave packets )/(1) and )/(2) localized
at xl and x2, respectively,
¢(x) )7(
= x(1)(x) +
In the following we shall neglect, for simplicity, the extension as well as the
spreading of these wave packets. At the spacetime point P a position mea-
surement is performed with the result that the electron is at P. Given such
a situation we may consider two spacelike hypersurfaces al and a2 which
intersect at the spacetime point Q (see Fig. 5). Both hypersurfaces appear as
equal-time hypersurfaces in appropriately chosen coordinate frames K1 and
Ks, that is there are observers 1O and 02 at rest in K1 and/(2, respectively,
such that al is an equal-time hypersurface for O1, and a2 is an equal-time
hypersurface for O2. The important difference between both observers is that
for 02 the measurement has already taken place, whereas for O1 it has not.
Consequently, both observers assign different amplitudes to one and the same
objective spacetime point Q, that is, ¢(Q) on al is different from ¢(Q) on
0" 5.
The conclusion is that ¢ is no longer a function of the spacetime coordinates:
The value of ¢ at a spacetime point depends, in general, on the hypersurface
crossing this point and to which it is associated. Thus, the wave function
becomes a function on the set of spacelike hypersurfaces and we may write
¢ = ¢(a, x), (8)
where x runs over the points of the hypersurface a. For example, we have at
the spacetime point Q of Fig. 5:
¢(a~, Q) = x(2)(Q) ~ ¢(a2, Q) = 0. (9)
To illustrate further the consequences of this concept, we shall discuss an-
other example which also stems from Aharanov and Albert. At time t = -co
we prepare a particle to be in the state ¢1, which is a superposition of three
wave packets localized respectively at xl, x2 and x3. At time tl we measure
in xl and we find that the particle is not in xl. At time t2 a measurement at
x2 is performed and it turns out that the particle is not at x2. It is assumed
that the events (tl, xl) and (t2, x2) have a spacelike distance. Then there is
an inertial system K in which t2 > tl and the observer associated with it
ascribes the 'history' depicted in Fig. 6 to the wave function.
There is also another inertial frame of an observer B who sees the 'history'
of the wave function in his coordinate system K ~ in the following way (see
Fig. 7; this picture is drawn in the coordinate system K of the observer A).
For the observer B the measurement at x2 takes place before the measurement
at xl: His 'equal time' hypersurface crosses the measurement event at (t2, x2)
State Vector Reduction in Relativistic Quantum Mechanics 9
t-O
D
1 X
QJ
17(
27(
I
Xl X2
Fig. 5. Illustration of the fact that the state vector ceases to be a function on
spacetime if local measurements are taken into account.
before the measurement event at (tl, xl). The state ¢~ is obtained from the
state ¢1 by a unitary representation U(A) of a Lorentz transformation A
which maps al to a~,
¢~ = U(A)¢I. (10)
In the same way one gets ¢~ from ¢3 through a Lorentz transformation
¢~ = U(A)¢3. (11)
The same is true also for the components ]xi), whose superposition we have
considered
)~xI = U(A)lxi). (12)
However, the important point is that this is not true for the 'intermediate'
state ¢2. Namely, ¢~ is not the Lorentz transform of ¢2; these states are
not related by a unitary representation of some Lorentz transformation. The
processes formulated by the observers A and B are thus completely different.
Nevertheless, covariance is satisfied here. Since we have introduced the
wave function as a functional on the hypersurfaces, the processes described
01 Heinz-Peter Breuer and Francesco Petruccione
0-3 ¢3 = Ix3)
?
t2
¢2 = Ix2> + Ix3)
0- 2 •
I
I
tl - I
- - ?
¢1 = Ix1> + Iz2> + Ix3>
1-0 •
I I
I I
| I D
Xl X2 X3 1 X
Fig. 6. A system subjected to a sequence of state vector reductions as seen from
an observer A in a reference frame K in which t2 > tl.
in Figs. 6 and 7 are really different! Covariance requires only the equivalence
of all inertial systems, hence the independence of all physical statements from
a special coordinate system. And exactly this point has been made obvious
in what we have discussed above: Performing a Lorentz transformation from
an inertial system to another one has to trans]orm the states of the quantum
system as well as the observer who ascribes a history of state vectors to his
equal-time hypersur/aces.
3 Preparation and Measurement
of Nonlocal Observables
Having discussed the reduction of the state vector of a system under arbitrary
local measurements we now turn our attention to the problems arising in the
context of the measurement of nonlocal observables. In order to illustrate
the problem we consider a system of two spin-l/2 particles [2, 6, 8], which
are situated at two fixed points xl and x2 in a fixed coordinate system. The
Hilbert space of the spin states is 4-dimensional and it is spanned by the
states (we adopt the notation used in [2]):