Spacetime topology from the tomographic histories approach I: Non-relativistic Case

Abstract

The tomographic histories approach is presented. As an inverse problem, we recover in an operational way the effective topology of the extended configuration space of a system. This means that from a series of experiments we get a set of points corresponding to events. The difference between effective and actual topology is drawn. We deduce the topology of the extended configuration space of a non-relativistic system, using certain concepts from the consistent histories approach to Quantum Mechanics, such as the notion of a record. A few remarks about the case of a relativistic system, preparing the ground for a forthcoming paper sequel to this, are made in the end.

Full text

Page 1

arXiv:gr-qc/0506088v2 18 Jul 2006
Glafka 2004: Spacetime topology from the
tomographic histories approach I: Non-relativistic
Case
Ioannis Raptis∗, Petros Wallden†and Rom` an R. Zapatrin‡
Abstract
The tomographic histories approach is presented. As an inverse problem, we
recover in an operational way the effective topology of the extended configuration
space of a system. This means that from a series of experiments we get a set
of points corresponding to events. The difference between effective and actual
topology is drawn. We deduce the topology of the extended configuration space
of a non-relativistic system, using certain concepts from the consistent histories
approach to Quantum Mechanics, such as the notion of a record. A few remarks
about the case of a relativistic system, preparing the ground for a forthcoming
paper sequel to this, are made in the end.
1 Introduction with Motivational Remarks
In the standard formulation of relativity theory, the spacetime topology is a priori
fixed by the theorist to that of a continuous manifold; hence, it is not an observable
entity. Only the metric structure is traditionally supposed to be dynamically variable.
With the exception of Wheeler’s celebrated, but largely heuristic, spacetime foam sce-
nario [21], there is no well developed theory in which the spacetime topology can be
regarded as a dynamical variable proper, with quantum traits built into the theory
from the very start. However, one may try to consider idealized situations where cer-
tain topological features are represented as quantum variables that can in principle
be observed and measured [2]. Even in General Relativity (GR), where no variable
quantity is supposed to be quantum—i.e.,subject to coherent quantum superpositions
and associated uncertainty in its determinations, we need histories (e.g.,material par-
ticles’ causal geodesic trajectories) to actually define the topology of spacetime. This
is because the concept of neighborhood turns out to be something which someone (:an
observer), located at some point in spacetime, deduces for regions that belong to her
causal past. Similarly, the concept of distance can be established only if information
∗EU Marie Curie Reintegration Postdoctoral Research Fellow, Algebra and Geometry Section,
Department of Mathematics, University of Athens, Panepistimioupolis, Athens 157 84, Greece; and
Visiting Researcher, Theoretical Physics Group, Blackett Laboratory, Imperial College of Science,
Technology and Medicine, Prince Consort Road, South Kensington, London SW7 2BZ, UK; e-mail:
i.raptis@ic.ac.uk
†Theoretical Physics Group, Blackett Laboratory, Imperial College of Science, Technology
and Medicine, Prince Consort Road, South Kensington, London SW7 2BZ, UK; e-mail:
ros.wallden@imperial.ac.uk
‡Department of Information Science, The State Russian Museum, Inzenernaya 4, 191186, St.
Petersburg, Russia; e-mail: zapatrin@rusmuseum.ru
pet-

Page 2

(:causal signals, or actual travelling material particles) is (causally) transmitted from
one point to another. All in all, the causal nexus of the world determines both its
topological and metric structures.
On the other hand, an interesting feature of quantum mechanics is that we may
be able to make and verify statements about topology from a single-time case, as long
as we are allowed to repeat experiments (and in principle we are allowed to do that
indefinitely, if only in a theoretical, idealized, ‘gedanken’/theoretical fashion) in order
to get the relative frequencies. In the classical (i.e.,non-quantum mechanical) case,
one-time measurements do not give any information about global properties, such as
the background topology.
Having said that, a remarkable consequence of quantum mechanics is that the
wavefunction is a non-local entity, so that we may be in a position to deduce topo-
logical properties of the background, provided that we have enough repetitions of the
experiment to reconstruct the relative frequencies. Thus, instead of saying that the
wavefunction is a square integrable function on a topological space and use this to
deduce probabilities about experimental outcomes (:events), we hereby propose to do
the converse. We start from probabilities and the continuity assumption for events,
and from this information we derive the structure of the topological space in which
these events are supposed to happen. A word of caution is due here: the continuity as-
sumption is normally taken to presuppose a topology—for how else can one talk about
a continuous wavefunction? Well, and here is the crux of the inverse scenario: our
assumption is that the wavefunction must be continuous with respect to the topology
to be deduced from the relative frequencies of events. In other words, the (continuity
of the) wavefuction is born with the topology being deduced.
In what follows, we do the same for the 4-dimensional case and recover ‘spacetime’.
We should point out that no matter that we talk about space-time, we are still in
the non-relativistic regime. We just speak of space points labelled by their ‘absolute’,
Galilean time of occurrence. The relativistic case will be considered in a forthcoming
paper [20].
One could say that histories are still needed to define global properties, such as the
topology; however, here we maintain instead that they are needed in order to extract
the form of the wavefunction. Of course, prima facie one can counter our arguments
by holding that the assumption about (complete) knowledge of the wavefunction im-
mediately leads to EPR-type of paradoxes. Our retort is that EPR-phenomena do not
arise in our setting, while causality is rescued by the fact that we need classical com-
munication to recover the full (:complete) state, as for example in various (quantum)
teleportation scenarios—see, e.g.,Aharonov and Vaidman [1].
Let us outline the contents of the present paper. In the first part we introduce
the consistent histories approach whereby we are given a configuration space for the
system, its full Hamiltonian (including interactions), as well as the initial conditions
(generally speaking, ‘exosystemic’ parameters of the problem traditionally supposed to
be determined by an experimenter external to the experimentee—the physical system
under experimental focus), and from these we calculate the probabilities for histories
to occur. In our inverse—alias, ‘tomographic’—approach, we are given the sets of ob-
served histories together with their relative frequencies, and from these we reconstruct
(some of) the parameters of the problem, with no allusion to external/internal systemic
distinctions, as befits the histories approach. Then, certain issues about topology and
the character of various possible indeterminacies of the derived topology that are in-
volved in our approach are highlighted.
The main part of the paper follows, where we present what we are able to recover
and how we do that. In this paper we specifically develop the non-relativistic case and

Page 3

focus on what can be said about topology using the set of histories alone, and also
what needs some further measurements in order to be ‘sharply’ determined. Finally,
we illustrate all this by virtue of two toy-models. The first is our variant of the usual
double-slit experiment, both when the particle is detected at the slit, and when it is
not. The second is an example of an environment involving a ‘bath of sensors’.
But before we delve into the paper, we feel that the new term ‘tomography’ ought
to be further explained; otherwise, there is no reason to have it only for the sake of
fancy neologisms and lexiplacy. We believe that its use can be justified on the following
semantic grounds: experiments and their records may be thought of as ‘cuts’ (:‘τoµ´ ǫς
in Greek) incurred on the quantum system.1
tional measurement-slices’ and their relative frequencies of occurrence, we ‘retro-write’
(:‘redraw’, or ‘reconstruct retrodictorily’ so to speak)—as it were, ‘after the fact’—the
(spacetime) topology. Moreover, in Greek, the verb ‘to write’ (or ‘to draw’, generically
speaking) is γρ´ αφω. Hence ‘tomo-graphy’:2we are re(tro)sketching spacetime topology
from ‘experimental cuts’ exercised on the quantum system(!) All in all, this etymologi-
cal dissection of the word ‘tomography’ accords with the title of the paper: “spacetime
topology (derived, or effectively re-sketched) from ‘tomographic’, inverse histories”.
From (the results of) these ‘observa-
2Histories and inverse histories approach
The decoherent histories approach to quantum mechanics deals with the kind of ques-
tions that may be asked about a closed system, without the assumption of wavefunction
collapse (upon measurement). It tells us, in a non-instrumentalist way, under what
conditions we may meaningfully talk about statements concerning histories of our sys-
tem, by using ordinary logic. This approach was mainly developed by Gell-Mann and
Hartle [4, 5, 3, 9, 10, 6, 11], and it was largely inspired by the original work of Griffiths
[7] and Omn` es [14, 15, 16, 17, 18, 19].
In this section, after we briefly recall useful rudiments of the standard histories
scheme, we introduce its ‘converse’ theoretical scenario that interests us presently: the
inverse histories approach. Pictorially, the two schemes are related as follows:
the
Hamiltonian
configuration
space
choice of
measurement
basis
choice of
precision
observed
frequencies
-
?
Standard histories approach
Inverse histories approach
1Recall the Heisenberg ‘schnitts’ (German for ‘cuts’) in the standard Copenhagean quantum theory.
2In Greek, ‘τoµo-γραφ´ ια:=‘slice-wise writing/skethching/drawing’).

Page 4

2.1The HPO version of the standard histories approach
The formulation of the standard histories scenario that we follow presently is due to
Isham et al.(e.g.,see [12, 13]), and it is called HPO (History Projection Operator)
approach. It consists of a space of histories UP, which is the space of all possible
histories of the closed system in question, and a space of decoherence functionals D.
Parenthetically, the space of histories is usually assumed to be a tensor product of
copies of the standard QM Hilbert space. Two histories are called disjoint, write
α ⊥ β, if the realization of the one excludes the other. Two disjoint histories can be
combined to form a third one γ = α ∨ β (for α ⊥ β). A complete set of histories is a
set {αi} such that αi⊥ αj
(∀αi,αj,i ?= j), and α1∨ α2∨ ... ∨ αi... = 1
A decoherence functional is a complex valued function d : UP × UP −→ C with
the following properties:
a) Hermiticity: d(α,β) = d∗(β,α)
b) Normalization: d(1,1) = 1
c) Positivity: d(α,α) ≥ 0
d) Additivity: d(α,β ⊕ γ) = d(α,β) + d(α,γ) for any β ⊥ γ
A complete set of histories {αi} is said to obey the decoherence condition,
i.e.,d(αi,αj) = δijp(αi) while p(αi) is interpreted as the probability for that history
to occur within the context of this complete set.
The decoherence functional encodes the initial condition as well as the evolution of
the system. Here we should also note that the topology of the space-time is presupposed
when we group histories into complete sets, i.e.,in collections of partitions of unity.
In standard QM, histories correspond to time ordered strings of projections and to
combination of these when they are disjoint. An important issue here is the relation
between decoherence and records. Namely, it can be shown that if a set of histories
decoheres, there exists a set of projection operators on the final time that are perfectly
correlated with these histories and vice versa.3These projections are called records. It
is this concept that figures mainly in our approach (e.g.,see Halliwell [8]).
To sum things up, in the standard histories approach
• The system is given, as well as its environment. The latter is represented by
prescribing initial conditions and in some cases final conditions.
• The space, its topological structure in particular, is presupposed.
• The interactions are given in terms of the decoherence functional, which encodes
the dynamical information. For the complete dynamics, the full Hamiltonian
must be known.
2.2Tomographic histories approach
In our approach things are different, as we solve the inverse problem. While in standard
histories one is given the Hamiltonian, initial conditions, as well as the space on which
they are defined, and the aim is to predict probabilities for histories, we do the opposite
thing. We make repetitions to get the frequencies for different records. Then, by
making certain assumptions about these records, namely, that they are nothing but
records of events, we recover the topological structure of the underlying configuration
3This is the case for a pure initial state, and we restrict ourselves to it.

Page 5

space. This means that from a set of events, with no other structure presupposed (:a
priori imposed from outside the system), we end up with a causal set representing the
discretized version of the extended configuration space of the system in question.
The extended configuration space that we get will be an ‘effective’ one, and in a
sense it accounts for certain properties of the Hamiltonian, such as interactions with
other objects not controlled by the experimenter. For instance, the latter could be
some kind of ‘repulsive’ field that prohibits the system to go somewhere (:in a region
of its configuration space), which can then be recovered as a hole (:a dynamically
inaccessible region) in that space.
To compare the two approaches, let us review for a moment the standard histories
approach where the decoherence functional, as well as the space of histories, are given.
For these, one is assumed to be given the initial conditions, the configuration space,
and the Hamiltonian of the system in focus—i.e.,generally speaking, the parameters
of the system. When we are able to perform multiple runs of the experiment and we
choose a decohering set of histories, the decoherence functional yields the probability
for each history to occur, which, in turn, corresponds to the history’s relative frequency
with respect to the set chosen.
Having the same Hamiltonian and the same initial conditions, we may consider
another decohering set of histories, not necessarily compatible with the previous one,
for which again the probabilities can be calculated. This is in broad terms what the
usual histories approach accomplishes.
We on the other hand will be tackling the inverse problem. The essence of our
approach is the following. Since we can carry out our experiment sufficiently many
times, we have access to the following two things—the set of possible histories and
the relative frequencies for each history to occur for every initial state. From this
we recover the parameters of the experiment, namely, the effective topology of the
extended configuration space.
One thing to highlight here is what corresponds to a decoherent set of histories in
our inverse scenario. It is one particular partition of unity of the record space. Our
freedom of choosing a particular basis in which to measure things will in general give
different decoherent sets than had we chosen a different one (:different basis, different
decoherent sets). Note also that since we consider histories operationalistically, we
always deal with histories that are contained in a decoherent set, namely, the set that
corresponds to the set of records that we choose to analyze.
In our setup we shall assume that the records capture the spatio-temporal properties
(of the system in focus). This means that the histories are coarse-grained trajectories
of the system, belonging to a space whose topological properties we ultimately wish
to deduce. We shall then claim that the whole concept of spacetime, as a background
structure, does not make sense in finer-grained situations. In this way, all the histo-
ries are single-valued on our discretized version of ‘effective spacetime’. One should
note here that we may still have histories that have the particle in a superposition of
different position eigenstates, but only if the latter are ‘finer’ than the degree of our
coarse-graining. With the coarse-graining we effectively identify (i.e.,we group into an
‘equivalence class’ of some sort) the points that we cannot distinguish operationally,
with the resulting equivalence class of ‘operationally indistinguishable points’ corre-
sponding to a ‘blown up’, ‘fat point’ in our discretized version of ‘effective spacetime’.