The coevent formulation of quantum theory
Abstract
Understanding quantum theory has been a subject of debate from its birth. Many different formulations and interpretations have been proposed. Here we examine a recent novel formulation, namely the coevents formulation. It is a histories formulation and has as starting point the Feynman path integral and the decoherence functional. The new ontology turns out to be that of a coarsegrained history. We start with a quantum measure defined on the space of histories, and the existence of zero covers rules out singlehistory as potential reality (the Kochen Specker theorem casted in histories form is a special case of a zero cover). We see that allowing coarsegrained histories as potential realities avoids the previous paradoxes, maintains deductive noncontextual logic (alas nonBoolean) and gives rise to a unique classical domain. Moreover, we can recover the probabilistic predictions of quantum theory with the use of the Cournot’s principle. This formulation, being both a realist formulation and based on histories, is well suited conceptually for the purposes of quantum gravity and cosmology.

1. University of Athens, Physics Department, Nuclear & Particle Physics Section, Panepistimiopolis 15771, Ilissia Athens, Greece

2. Technological Educational Institute of Chalkida, General Science Department, Psahna34400, Greece

Email: petros.wallden@gmail.com, pwallden@phys.uoa.gr
1 Motivation
Quantum theory is undoubtedly one of the most successful theories. The understanding and interpretation of quantum theory, has been a subject of debate from the birth of the theory until now. There is no unique, widely accepted interpretation of quantum theory. In the recent years, due to the developments in the field of quantum information and the search of a quantum theory of gravity and cosmology, there is a new wave of interest for the foundations of quantum theory. This contribution, is a review of a novel formulation of quantum theory namely the “coevents formulation”. The motivation for such a formulation, is twofold.
First is the need for a realist interpretation of quantum theory. The standard view of quantum theory, is that of an instrumentalist. Avoids to refer to the actual ontology, and makes assertions only related with some idealised measurements carried out by an external observer. However, particularly for the field of quantum cosmology where one treats the full universe as a quantum system, the need for an interpretation that does not require an external observer becomes vital. In this sense the need for a realist interpretation, becomes more than a philosophical investigation.
Second is the need for a formulation that treats space and time in equal footing. In constructing a quantum theory of gravity, one is led to the observation that while space and time are totally different objects in quantum theory, they are moreorless the same in general relativity. This leads to a tension that has both technical and philosophical difficulties (e.g. see the problem of time [problemoftime]). We can claim that better suited for the purpose of quantum gravity should be a formulation of quantum theory that is based on full histories of the system (such as in the Feynman path integral) and not a canonical formulation (that relies on the Hamiltonian and uses onemomentoftime propositions). The latter may not even be applicable for certain approaches to quantum gravity such as the causal sets.
The coevents formulation was introduced by Rafael Sorkin in [coevent1] following his earlier work in [coevent2] (see section LABEL:3.0). It is a realist interpretation, that is based on histories. In particular, the starting point is the Feynman amplitudes for histories, and interpreting those amplitudes in an observerfree, contextindependent way is the aim of the formulation. In section 2 we will give the standard histories view for classical physics, state which is the importance of precluded sets and stress which is the problem in adopting a “naive” realist view in quantum physics. In section LABEL:3.0 we will introduce the coevents formulation, giving two different (but equivalent) views. In section LABEL:4.0 we will explore (briefly) certain developments of the formulation. In particular we will define what a classical domain is, show that there exist a unique, contextindependent such domain and point out that this is not the case for the consistent histories approach\@footnotemark\@footnotetextThe approach is also known as decoherent histories. (section LABEL:4.1). We will give an account of how probabilistic predictions arise (section LABEL:4.2), and point out the role and the status of the initial state in the formulation (section LABEL:4.3). In section LABEL:5.0 we will summarise and conclude.
2 Histories, classical physics and quantum measure
We are taking a histories view. Here we will review classical physics casted in histories perspective and stressing certain important features that become important in quantum physics.
2.1 General Structure
In describing physics with histories, there are three important mathematical structures.
(1) The first, is the space of all finest grained descriptions . In probability theory, it is called sample space, while in histories formulation, is called history space. The histories space, in classical physics is the space of potential realities. However this picture does not carry over in quantum theory, as we will see, where potential realities are no longer finegrained descriptions.
Each element of this space , corresponds to a full description of the system, specifying every detail and property. For example, a fine grained history gives the exact position of the system along with the specification of any internal degree of freedom, for every moment of time. For a single nonrelativistic particle, would be the space of all trajectories in the physical space.
Along with , we define the space , which consists of all the subsets and the Boolean algebra associated with them (called events algebra), where the addition is defined as the symmetric difference between subsets and the multiplication given by the intersection of subsets . Each subset is called event, as in probability theory. Note, that all physical questions that one can ask, correspond to one of those subsets. If, for example, one wishes to ask “was the system at the region at time ?”, it corresponds to the subset defined as , i.e. all histories that the system at time is in the region .
(2) The second structure, is the space of truth values (for which we will use the notation ), and the algebra associated with them. In the case of classical physics, the truth values are simply the two elements set (or simply ), and the Boolean algebra, called the truth values algebra, associated with them. We have . It is possible to use as truth values a more general algebra.
(3) The third structure, is the space of possible truth valuations maps (). These maps assign a truth value, to each of the possible questions (events). We shall use the notation for this space. However, to be able to reason, we need to be able to make deductions, in other words to be able to obtain the truth value of some event from the truth values of some other events. The most strict condition, that holds in classical physics, is that the allowed truth valuation maps are homomorphisms between the events algebra and the truth values algebra . In other words we require the map to obey:
(a) Multiplicativity
(1) 
(b) Additivity (Linearity)
(2) 
An important observation, is that maps that are homomorphisms\@footnotemark\@footnotetextIn the rest paper, when we refer to homomorphisms or homomorphic maps, it should be understood as homomorphisms between the event algebra and the truth values algebra ., are in a onetoone correspondence with single elements of the space of histories . In particular, maps that are the characteristic function for a particular history , are homomorphisms between the Boolean event algebra and the Boolean truth values algebra.
(3) 
Moreover, all homomorphisms are of this type. There exists one homomorphic map for each one of the (single) elements of . Due to the onetoone correspondence of homomorphic maps and single elements , we could adopt a dual view, and state that classical reality is a homomorphic map between the event algebra and the truth values algebra . To sum up, for classical physics, potential realties can be viewed either as single histories elements of or as homomorphic maps . The potential realities are further restricted by the dynamics. As we will see later in section LABEL:3.0, the ontology in quantum physics changes, but the existence of a dual picture of reality as an event or as a map is maintained.
We have explored, so far, the “kinematic” part of the theory. We have not mentioned the dynamics (Hamiltonian), or initial condition. In classical deterministic physics, given the initial state and the dynamics, we know all the evolution of the system, and thus the full history. In other words, we know which history from the space of histories is actually realised. Given the initial state and dynamics, there is only one possible potential reality , the one that is actually realised. This is the meaning of determinism. Obtaining predictions, becomes trivial, since an event is true if and only if it contains the one realised history. We can define a classical measure on , such that and . This gives the probability that an event occurs, which is always either one or zero. This simple measure, coincides with the truth valuation , of the single history that is realised, but should not be confused, since the analogy holds only for deterministic classical physics.
In classical stochastic physics, the picture is different. We are not given which history is actually realised, but given the initial state and dynamics, we can obtain a classical measure (nontrivial this time) on the space of subsets of . The measure of each event, corresponds to the probability that this event occurs. The measure is no longer related with the valuations . It is important to note here, that the actual realised reality in stochastic physics, will still be one fine grained history element of . Since our theory is no longer deterministic, the space of potential such histories has many elements and in particular (for finite histories space) all fine grained histories that have nonzero measure, . The role of the measurement in stochastic classical physics, is restricting further the potential realities. For example, before tossing a coin, two outcome were possible, but after performing the measurement and looking the outcome, it is determined wether the outcome was heads or tails. In histories formulation, the latter corresponds to the experimenter restricting the set of possible histories in the universe to those that are compatible with his new observation. To sum up, in stochastic classical physics, the ontology remains the same but the mechanism to obtain predictions changes. We will see more on probabilistic predictions for closed classical or quantum systems, in section LABEL:4.2 after having introduced the coevents formulation.
2.2 Precluded sets
As we have seen, the set of potential realities, both in deterministic and stochastic classical physics, is determined by the classical measure that is defined on , which in its term is fixed by the initial state and dynamics. More specifically, for finite , it is the measure zero sets that determine which are the allowed potential realities/histories \@footnotemark\@footnotetextFor infinite more care is needed, and definition of measurable sets is required. However, most of the arguments carry over.. We define precluded event to be an event such that its measure is zero . The precluded events do not occur. If an event is precluded, any subset of , should also be ruled out. The latter is a vital condition that needs to be satisfied, if one wishes to make any deductive argument\@footnotemark\@footnotetextThis is related with the modus ponens rule of inference. E.g. “if all humans have two hands” and “Plato is human”, we wish to be able to deduce that “Plato has two hands”.. In classical physics, this condition is guaranteed by the fact that we require the maps/elements of , to be homomorphisms of the Boolean algebras. Simply requiring the condition leads directly to . While this discussion seems trivial for classical physics, it will become apparent that it is important for quantum physics.
A final interesting remark, concerning precluded sets, is that one can fully reconstruct all probabilities using only the set of precluded events provided that he has identical copies of the system. Heuristically, any distribution of outcomes of the identical copies, that differs from the one given by probability, has small chance of occurring, and as the number of copies tends to infinity, this chance of occurring also tends to zero. Technically, let an event and independent copies. There exists a unique number such that the following event to be precluded ( is indicator function)
(4) 
See for example [preclusionprobability]. In this sense, all the content of relative frequency interpretation of probability, is included in the precluded sets. Moreover, the precluded sets can be used to recover some predictions via the Cournot principle (see later), even in the absence of many identically prepared copies. This is one more reason why, we choose to give specific importance to precluded events.
2.3 Quantum measure
The picture described above, cannot be (fully) carried over to quantum theory. The histories space and its subsets/events remains the same. The main difference arises, mathematically, from the fact that we no longer have a classical measure on but rather a quantum measure, which we will shortly define. Given an initial condition and dynamics the quantum measure is totally fixed. The initial condition and the dynamics can be given either in form of some initial condition on a path integral along with an action , or as an initial wavefunction in a Hilbert space along with a Hamiltonian operator.
To define the quantum measure (which was first done by Sorkin in [quantummeasure]) we need to introduce amplitudes to histories. Starting from Feynman’s path integral approach, one can assign an amplitude (complex number) to each history.
(5) 
which depends on the initial state and on the dynamics of the system encoded in the action . Obtaining the transition amplitudes from to , is done by summing through all the paths obeying the initial and final condition, i.e.
(6) 
The mod square of this amplitude is the transition probability. One can extend this to any event and proceed to define a quantum measure :
(7) 
where the initial state and a final time function is added.
Alternatively, one can take a Hamiltonian view, and obtain the quantum measure in the following way. Define the class operator
(8) 
Where is the unitary evolution operator that relates to the Hamiltonian via , and is the subspace that history lied at time . This class operator corresponds to the history that the system is at at and at at etc. The projection operators can be at any subspace of the Hilbert space. The quantum measure is then defined to be
(9) 
For cases where there is a well defined time, such as in nonrelativistic quantum mechanics, the two definitions are equivalent and one uses the more convenient one. For finite moments of time histories, the operator expression is usually easier to deal with\@footnotemark\@footnotetextIntroducing more moments of time is simply a fine graining of the previous histories..
Note, that the quantum measure is closely related with the decoherence functional [DecoherentHistories], since it arises from the diagonal parts of the latter.
Since the quantum measure is a nonnegative function that is also normalised (), one could be tempted to interpret the quantum measure as probability. However this is not possible, due to interference. The additivity condition of probabilities\@footnotemark\@footnotetextAdditivity of disjoint regions of the sample space., is not satisfied:
(10) 
However a weaker condition holds that shows that there is no threepaths interference, that cannot be deduced from pairwise interference:
(11) 
In other worlds, the quantum measure is fully determined once we know the quantum measure of single histories and of pairs of histories. The specific expression is given for example in [surya2010]:
(12) 
Recently, experiments [threeslits] have confirmed that indeed the condition of eq. (11) holds in nature.
To fully characterise a histories theory, we need the following triplet: The histories space, the events algebra and the quantum measure .
A partition of the histories space , is a collection of events such that for all and . It is an exhaustive and exclusive collection. Each event of the partition is called cell. A coarsegraining of a histories theory is defined in the following way:

The cells of a partition are the elements of the coarsegrained histories space.

The Boolean algebra generated by addition and multiplication of these cells , is the coarse grained event algebra.

The quantum measure defined on them should be compatible with the fine grained quantum measure, i.e.
2.4 The trouble with quantum theory
It is already clear, that since the quantum measure cannot be understood as probability, the picture of classical stochastic physics presented in the previous section might need modification. However, the main reason why one cannot interpret the quantum measure in the same way as the classical measure, appears from consideration of the precluded events. Note that from now on, when we mention precluded events we mean that they have quantum measure zero.
In many cases (see below for examples), one can find a collection of precluded events , where , that also cover the full histories space , i.e. that . We will call such collection of precluded events as a zero cover. Note, that since in general , this collection does not constitute a partition.
However, as we mentioned earlier, in order to be able to construct deductive arguments, the truth value of any subset of a precluded set has to be “False”. Having the full covered with precluded sets, imply that no single history can have a truth value “True”. This also leads to no homomorphic map existing.
We will first consider a simplified example, with a threeslits interference experiment. Consider a point at the screen, where crossing slit A destructively interferes with crossing through slit B and slit B destructively interferes with slit C, i.e. , however (see Figure LABEL:3_slits).