The Construction Interpretation: a Conceptual Road to Quantum Gravity

The Construction Interpretation: a Conceptual Road to Quantum Gravity

Lucien Hardy
Perimeter Institute,
31 Caroline Street North,
Waterloo, Ontario N2L 2Y5, Canada
Abstract

In the first part of paper I propose the Construction Interpretation of the Quantum. The main point of this interpretation is that (unlike previous interpretations) it is not actually an interpretation but rather a methodology aimed to encourage a conceptually driven construction of a theory of Quantum Gravity. In so doing we will, I hope, resolve the ontological problems that come along with Quantum Theory. In the second part of this paper I offer a particular perspective on the theory of General Relativity and set out a path of seven steps and three elements corresponding (more-or-less) to the path that Einstein took to this theory. In the third part I review a particular operational approach to Quantum Field Theory I have been developing - the operator tensor formulation. The physicality condition (this ensures that probabilities are between 0 and 1 and that choices in the future cannot effect the past) is discussed in some detail in this part and also in an appendix. In the fourth part of the paper I set out one particular possible conceptually driven path to Quantum Gravity. This approach works by establishing seven steps and three elements for a theory of Quantum Gravity that are analogous to those of General Relativity. The holy grail in this approach is to generalize the physicality conditions from Quantum Field theory (which presume definite causal structure) to the new situation we find ourselves in in Quantum Gravity (where causal structure will be indefinite). Such conditions, in the present approach, would be analogous to the Einstein Field Equations.

Contents

1 Prelude

In 1687 Newton published his PhilosophiæNaturalis Principia Mathematica [55] in which he wrote down his law of universal gravitation

The force between two masses depends on the instantaneous distance, , between them. This was regarded as deeply unsatisfactory by Newton and others since it constitutes action at a distance without any mediating substance. There were numerous attempts to explain this gravitational attraction in mechanical terms. Netwon himself (in letters to Oldenburg and Boyle [12]) considered an aether model in which the attractive force was due to the decrease of pressure when fluid passes at speed (later the effect was formalized by Bernoulli). Descartes had a model [72] in which there are vortices in fine matter (basically an ether) which catch rough matter (such as the planets) up in their motion pushing this rough matter towards the center of the vortex. Another model (originally due to Nicolas Fatio de Duillier [87] and reinvented by George-Luis Le Sage (see Poincaré’s exposition [66]) among others) has a flux of particles with equal density in all directions except where some massive object causes a shadow. Any other massive object in this shadow will feel unequal pressure and experience a force towards the first mass.

How did these issues with Netwon’s law of universal gravitation actually get resolved? The answer came 228 years [30] later when Einstein successfully solved a problem (let us call it the problem of Relativistic Gravity) to find a theory that limits to Newtonian Gravity on the one hand and to special relativistic field theories (such as Maxwell’s electromagnetism) on the other

He called the theory of relativistic gravity he found General Relativity because it involved consideration of transformations between general coordinate systems. For speeds small compared with , General Relativity limits to Newtonian Gravity. For small masses it limits to the case of Special Relativistic Field Theory. It is important to note, however, that General Relativity fits in with neither the framework of Newtonian Gravity nor of Special Relativistic Field Theories. There are corrections to Newtonian Gravity that kick in for speeds close to that of light and to Special Relativistic Field Theories that kick in when gravitationally induced curvature is big. There is a certain even-handedness: General Relativity modifies both Newtonian Gravity and Special Relativistic Field Theories.

In the light of General Relativity we might ask what the correct interpretation of Newtonian Gravity is. If we take the limit of General Relativity the actual model we get (which agrees with the predictions of Newton’s gravitational theory) is something called the Newton-Cartan formalism [13]. In this formalism space and time separate out each having their own metrics. These metrics are the “stuff” that communicate the gravitational force between distant masses. The force is still instantaneous (which is fine as we have ) but now it can be regarded as a contact force (curvature in the metrics mediate the force).

The Newton-Cartan formalism is of academic interest but it is not the correct description of the world. It is, incidentally, difficult to imagine arriving at the Newton-Cartan picture without first having General Relativity. The deeper concept is curvature of spacetime - a radically new idea from the perspective of Newtonian physics. Einstein did not go through the intermediary of the Newton-Cartan formalism or any of the mechanical models of gravitation mentioned above to get to General Relativity. Rather, he followed a conceptual approach (which we will discuss in some detail later).

Many of the attempts to provide interpretations of Quantum Theory look to me like the mechanical models of gravitation. They are, at best, an attempt to guess the analogue of the Newton-Cartan formalism without first attempting to solve the deeper problem of Quantum Gravity. The problem of Quantum Gravity can be represented as follows.

The problem is to find a theory that limits to General Relativity on the one hand and Quantum Theory (in particular, Quantum Field Theory) on the other hand. This clearly has a similar shape to the problem of Relativistic Gravity. We should, then, be prepared to take an evenhanded approach to Quantum Gravity wherein both General Relativity and Quantum Field Theory are modified.

While there were many twists and turns along the way, Einstein eventually found the right path to understanding Gravity. This involved solving the problem of Relativistic Gravity. The purpose of this paper is to nudge researchers working on the foundations of Quantum Theory along a similar kind of path but, this time, guided by attempting to solve the problem of Quantum Gravity employing the conceptual tools developed in foundations.

Part I The Construction Interpretation

2 More than merely academic

In view of the preceding remarks, I wish to propose The Construction Interpretation of the Quantum. The Construction Interpretation is distinct from other interpretations in that it is not actually an interpretation at all. Rather, it is a methodology aimed at obtaining an interpretation through dedicating ourselves to a conceptually driven construction (hence the name) of a theory of Quantum Gravity.

The great danger facing existing interpretations is that they will overturned by progress in physics. In particular, we expect Quantum Theory (and, indeed, General Relativity) will eventually be replaced by a new theory - a theory of Quantum Gravity. It is possible, likely even, that this theory will be so different from Quantum Theory as to make existing interpretations redundant. They will still be of academic interest to historians of science but they will not be taken seriously by practising physicists as providing an actual description of the world. If Quantum Gravity requires a radical departure from existing theories, the most interpretations of Quantum Theory can aspire to is to be the correct limiting version of Quantum Gravity just as the Newton Cartan formalism is the correct limit of General Relativity.

We could be driven to despair. If future developments in physics are likely to completely overturn all our hard work then what is the point in pursuing a conceptual understanding of Quantum Theory at all? Few physicists wish to dedicate themselves to merely providing amusement to future historians. The point of The Construction Interpretation is to take the noble instincts that lead us to attempt understand Quantum Theory in conceptual terms and re-purpose them to the problem of constructing a theory of Quantum Gravity. The hope (though, admittedly, this could turn out to be in vain) is that the correct theory of Quantum Gravity will (like General Relativity) suggest its own ontology and so we will, naturally, arrive at an interpretation. Thinking about foundational issues in the light of the concrete problem of constructing a theory of Quantum Gravity will, I hope, change our perspective in a useful and productive way. If we are genuinely interested in obtaining the best understanding of Quantum Theory then this strategy is, I suggest, more likely to lead to progress than presenting interpretations of Quantum Theory as complete and finished works.

3 The tenets

No respectable interpretation of the Quantum (even a non-interpretation like the present one) is without basic tenets. So, without further ado, here are the five tenets of The Construction Interpretation.

Driven.

The Construction Interpretation is driven by the intent to construct a theory of Quantum Gravity. As such, it is opportunistic in that it will adopt and abandon ideas (and, possibly, readopt them) as serves this goal.

Conceptual.

The Construction Interpretation re-purposes concepts and principles taken from the foundations of Quantum Theory and the foundations of General Relativity for the primary purpose of theory construction.

Operational and ontological.

Operational methodologies relating to actions and observations of agents are valid as are ontological methodologies which attempt to make assertions about what the underlying reality is.

Frameworks and principles.

The Construction Interpretation uses mathematical frameworks based on operational and/or ontological ideas as a place to impose different concepts and principles to get traction on the construction of a new physical theory.

Non-interpretation.

The Construction Interpretation is not actually an interpretation, but rather, a methodology aimed at finding a theory of Quantum Gravity. As such, the Construction Interpretation is pluralistic - it encourages many different conceptually driven approaches.

My hope is that a theory of Quantum Gravity, once so constructed, will suggest its own interpretation (just as General Relativity does as discussed in Sec. 8).

In Parts II and III I will present my own takes on General Relativity and Quantum Field Theory respectively. This is done in preparation for Part IV which presents a possible road to a theory of Quantum Gravity. In accord with the pluralistic exhortations of the last tenet above, I want to emphasize that this is only one possibility and other conceptually driven approaches are encouraged.

Part II General Relativity

4 The problem of Relativistic Gravity

In the early part of the last century physicists faced the problem of Relativistic Gravity. As discussed in the prelude, this is to find a theory that gives rise to Newtonian Gravity on the one hand and Special Relativistic Field Theory (such as described by Maxwell’s equations) on the other under appropriate limits.

This problem was solved in the years leading up to 1915 by Albert Einstein in the form of the theory of General Relativity. Unlike Quantum Theory, there is a clear and widely agreed up on interpretation of General Relativity (which we will discuss in Sec. 7). General Relativity represents a radical new take on reality (in some ways more radical than that of Quantum Theory). Here I will clarify what, exactly, the theory of General Relativity along with its interpretation is. Also, I will provide a simplified sketch of the conceptual steps that Einstein took to arrive at the theory.

5 Einstein’s path to General Relativity

In setting up the framework for General Relativity Einstein followed a particular path which I will schematize in steps elements. The seven steps take us to a framework for physical theories (that of tensor fields defined on a manifold satisfying local field equations).

The three elements (discussed in more detail in Sec. 7) single out General Relativity (one particular theory that might be formulated in this framework).

We will see that there is an analogous path to a possible framework for Quantum Gravity. Analogy is a potentially powerful source of inspiration in theory construction. To be in a good position to understand the analogous steps I propose, I will outline these steps and elements in the next two Sections. It is worth saying that the above path is intended to be schematic. It is an oversimplification of all the twists and turns in the actual historical path that Einstein took.

6 Seven steps to a framework

The equivalence principle states that the laws of physics are the same locally in any freely falling frame of reference. Einstein saw that the equivalence principle implies that we cannot set up a global inertial reference frame as in Special Relativity. This forces the use of general coordinates. These general coordinates can be thought of as charting some part of a manifold, . To cover all of the manifold we may need multiple such charts. These are the first three steps.

Once we have a coordinate system we introduce locality by working with quantities defined at each point. These are the fields corresponding to the various physical quantities under consideration. The laws of physics are expressed by demanding that these fields satisfy field equations. This takes us over steps four and five.

The sixth step concerns a particular way of defining fields - namely using using local tensor fields based on tangent space. It is worth elaborating a little here to point out that these are not the most general field theoretic objects we might define on a manifold. Consider the following

  1. The tensor fields used in General Relativity can be expressed with respect to a basis choice for each index, , at each point, . The basis choice used corresponds to the (flat) tangent space which approximates the manifold at this point. This is achieved by using partial derivative operators, , to span the tangent space. However, we could imagine setting up a local basis corresponding to a curved (rather than flat) approximation to the manifold at each point. We would do this by using higher derivatives for our basis (such objects are studied and go by the name of ncJets).

  2. The tensor fields used in General Relativity are local: they are defined at each point, . One could imagine more general objects that were defined at two points, or, indeed, defined at even more points, or even a continuum of points. In quantum theory, entangled states can be thought of as corresponding to such multipoint tensors. Even in classical probability theory, correlations can be represented by multipoint tensors [42].

Following these first six steps we have a pretty clear picture emerging. Physics written in terms of these local tensor fields based on tangent spaces. Equations in terms of such tensor fields give local physics in the weak sense that they specify relationships between physical quantities at each point . As an aside, it is worth mentioning that General Relativity also has a stronger locality property (we will call this signal locality) in that it is not possible to signal faster than the speed of light. It inherits the signal locality property from Special Relativity. There is, however, more to the matter than this as we need to be sure that the non-linear equations of General Relativity do not introduce some faster-than-light signalling possibilities[35]. In the theory of partial differential equations, this has to do with the properties of hyperbolic partial differential equations.

There is, in general, no global inertial reference frame. Hence, there is no special choice of coordinate system - all coordinate systems are equally good. It is reasonable, then, to demand that the equations of the theory take the same form in any coordinate system. This (the seventh step) is the principle of general covariance:

The principle of general covariance: The laws of physics should be written in such a way that they work for any coordinate system and so that they take the same form in any coordinate system.

This is the seventh step and puts in place the mathematical and conceptual framework within with the theory of General Relativity is formulated.

7 The three elements of General Relativity

Having arrived at a general framework following the above seven steps now I want to provide the extra ingredients that specify General Relativity. I do this by giving three elements. These provide a statement of what General Relativity is within this general framework.

Before this, however, we need to define what a diffeomorphism is. A diffeomorphism is a smooth invertible map, , that maps each point on a manifold to a new point, . If we cover the manifold with coordinates, then a diffeomorphism corresponds to an active coordinate transformation (a diffeomorphism is, then, the abstract version of a general coordinate transformation).

The three elements of General Relativity consist of a prescription, an addendum, and an interpretation:

A prescription

to convert special relativistic matter field equations into diffeomorphism invariant general relativistic matter field equations by taking the following steps

This technique is often called called minimal substitution. Here the indices correspond to a global inertial reference frame while the indices correspond to a general coordinate system. Note that the term “matter” refers to all physical fields (electromagnetic, fluid, …) other than the gravitational field, .

An addendum.

We now have an extra ten real variables in the symmetric tensor, , so we need an extra ten equations

These are the Einstein field equations (also diffeomorphism invariant). There are ten of them since and are symmetric. All well and good it would seem. There is, however, a surprising twist in the tale. It follows from the mathematical identity, , that only six of the ten Einstein field equations are independent. This is not a problem because of the last element …

An interpretation.

A solution is given by specifying the fields (consisting of matter fields and the metric) at each point, , on a manifold,

where this must satisfy the field equations (matter plus Einstein) at every point . The interpretation is that beables are those properties of solutions that are invariant under general diffeomorphisms. Thus, is a beable if and only if

where is the solution obtained by applying a diffeomorphism to (see [45]).

Here I have adopted John Bell’s term, beables, [10] for the properties that are taken to be ontologically real. The minimal substitution technique fails to provide a unique prescription when we have higher than first derivatives (as partial derivatives commute while covariant derivatives do not). Maxwell’s equations and perfect fluids involve only first derivatives so there is a substantial part of physics where this ambiguity is not an issue.

The need for the first element follows from the equivalence principle for two reasons: (i) because it tells us we have to go general coordinates as outlined above and (ii) also because it tells us that there is a local inertial reference frame at every point (a freely falling one) where we have Special Relativistic physics locally - this is ensured by using the covariant derivative which is equal to the partial derivative in a local inertial reference frame. The need for the second element follows from the first element because we need extra field equations (beyond the matter field equations) so we can solve for the have ten real fields in the metric. The third element follows from the second as Einstein’s field equations only provide six equations (we will discuss the interpretation in more detail in Sec. 8.

We can see very clearly how General Relativity solves the problem of Relativistic Gravity in the these elements. In particular, note that the first element says that the special relativistic matter field equations are changed when we go to General Relativity but in such a way that we recover Special Relativity when the effects of gravity can be ignored (when spacetime is flat and there exists a global inertial reference frame). The second element shows us how Newton’s law of gravity is changed. Einstein’s field equations give Newton’s law of universal gravitation (in the form of Poison’s equation for the potential) in the limit of weak gravitational fields. Thus, General Relativity is even handed with respect to the two less fundamental theories. We should, I anticipate, expect the same sort of evenhandedness for Quantum Gravity.

8 How GR suggests its own interpretation

It is important not to let the Einstein field equations, beautiful as they are, distract us from the bigger picture. The bigger picture is that provided above: first we need the framework then we have the prescription, addendum (Einstein’s field equations), and interpretation. This bigger picture amounts to a radical new view of reality. Let us look now at the interpretation. Even after providing Einstein’s field equations, we have four fewer field equations than real fields. Further, these field equations are all invariant under diffeomorphisms. These two facts go hand in hand because a diffeomorphism is described by four equations. This is clear if we return to using general coordinates. Then we have four equations describing the transformation. We can find solutions, , for the field equations. Further, if we act on any solution with a diffeomorphism, we will find another solution, , that is also consistent with the field equations (since these field equations are invariant under diffeomorphisms). We can argue that these two situations are physically equivalent as follows (this is Einstein’s hole argument [78, 56]). First, consider a compact region, , for which we want to solve the field equations. Assume that, everywhere outside this region, we know all the fields. Thus, we should have sufficient boundary information to determine the physical situation inside . However, for any solution, , we find (consistent with this boundary information) there exists an infinity of solutions where the diffeomorphism, , is different from the identity only in . These solutions have the same fields outside . This means that no matter how much boundary information we provide for there are still an infinity of solutions to the field equations differing inside . This leaves us with two choices. Either there is a breakdown of determinism (in the strong sense that boundary conditions fail, even probabilistically, to determine physical properties) or these different solutions correspond to the same physical situation. We have to choose the latter if we want to have a sensible physical theory. Thus, General Relativity suggests its own ontological interpretation. It is worth noting that these diffeomorphisms leave unchanged certain key properties such as whether two fluid blobs intersect or not. Diffeomorphisms also leave unchanged the distance between between pairs of such blob intersection events. The familiar world of tables, chairs and so on lives amongst the beables of the theory since such objects can be understood in terms of scalars (such as matter densities) taking values in coincidence along with distances between such coincidences.

Part III Quantum Field Theory

9 Preliminaries

If we take the limit of Quantum Gravity to a case where spacetime is fixed, then we expect to obtain some form of Quantum Field Theory. Now QFT is a very big subject which I could not hope to cover in this paper. Rather, I will give my own take on the subject in the form of what I call Operator Tensor Quantum Field Theory [45]. This approach brings out the operational aspects and is being developed with the kind of program discussed in this paper in mind.

I originally proposed the idea for operator tensor QFT in 2011 (see Sec. 16 of [43]) and developed the idea in [45]. The closest other approach to this is that of Oeckle. He initiated his general boundary formalism for QFT in 2003 [57]. In its original incarnation Oeckle’s formalism puts states (represented as elements of a Hilbert space) on the boundaries of arbitrary regions of space-time. This is a pure state formalism linear in quantum amplitudes. In 201???, motivated by work in the GPT approach, he developed the positive formalism [58] in which positive operators are associated with regions of space time. This, like the operator tensor approach, is linear in probabilities. Though the details differ, there is much similarity in spirit between the positive formalism and the operator tensor approach to Quantum Field Theory.

10 Operator tensor Quantum Theory

There is not agreement as to the correct ontological interpretation of Quantum Theory. However, there is broad agreement on the operational interpretation. Here I will summarize the operator tensor formulation of Quantum Theory [43, 44] as a prelude to doing the QFT case.

In this approach, an operation is represented by a box with some number of quantum systems going in the bottom and some number coming out the top (so we can think of time as flowing up the page)

We can also represent an operation symbolically as shown on the right. Then subscripts represent inputs and superscripts represent outputs. Different types of quantum system (electrons, photons, protons, …) are represented by different labels, , , , …. An operation will have a setting, , corresponding to the positions knobs and suchlike on the associated apparatus. This apparatus will also have outcomes (corresponding to different lights flashing). Associated with a given operation will be some given set of outcomes, . A complete set of operations

is a set having a given setting and whose outcome sets are disjoint with union equal to the set of all possible outcomes on this apparatus.

We can wire operations together to form circuits (if no open wires are left over) or fragments (of circuits) if some are. For example

is a fragment. In the symbolic notation we use repeated subscripts (one raised one lowered) with an integer label for the given wire.

Associated with an operation is an Hermitian operator (which we can represent diagrammatically or symbolically).

This operator acts on a Hilbert space

as determined by the subscripts and superscripts.

We can wire together operators. Then wires correspond to the partial trace over the appropriate part of the Hilbert space. To understand this, consider the three expressions

The first expression is equal to the trace of the product of the two given operators (a real number). The second expression is equal to the trace of the product of the first two operators (a real number) times the third operator. To evaluate the third expression can be evaluated by expanding out as a sum of products (like ) then evaluating for each term. If we have a circuit (with no open wires) then we obtain a real number.

We will say that operations correspond to operators under some mapping if, under this mapping, the probability for any circuit comprised of operations is equal to the real number obtained by replacing these operations in this circuit with operators under this mapping.

There are constraints on operators in order that (i) probabilities are between 0 and 1 and (ii) causality is satisfied (so that it is not possible to communicate backward in time). These physicality [43, 44, 45] constraints are

(1)

where denotes taking the partial transpose over the associated part of the Hilbert space and is the identity operator acting on (these physicality constraints relate to the Pavia causality conditions [16]). Note that the second of these physicality conditions is time asymmetric. A complete set of physical operators,

is a set of physical operators satisfying the normalization condition

We can now give the following simple axiom for Quantum Theory

Axiom for QT: All complete sets of operations correspond to complete sets of physical operators and vice versa.

This succinct statement captures the content of Quantum Theory. In particular, as shown in [43, 44], it follows that preparations are associated with positive operators with trace less than or equal to one (density matrices), evolution is associated with completely positive trace non-increasing maps, and measurement is associated with positive operator valued measures.

11 Operator Tensor Quantum Field Theory

First consider the following discrete case. We have qubits traveling to the left and to the right (time goes up the page).

At each vertex, , in this diagram we assume we have an operation (as discussed above) having setting and outcome (where labels the vertices). Now consider a region, (consisting of a collection of vertices) such as

(2)

If we let the distance between the qubits tend to zero (while keeping fixed) then the qubit trajectories become denser and denser inside this region. As we go over to the continuous case the vertices are labeled by where is a manifold (or at least, part of a manifold covering the points of interest). For the example shown this will be a dimensional manifold but we can construct grids that limit to more general dimensional manifolds. In this continuous limit the setting field, , and the outcome field, , become continuous fields.

We now have the following elements.

Manifold.

We have a given manifold (or region of a manifold), , covering the spacetime points, , of interest.

Metric and time direction fields.

We assume that the manifold is covered by a fixed metric, , and also a fixed time direction field, , that points in the forward light cone at each point . We assume there are no closed forward pointing timelike paths in with respect to and . We need the time direction field as the physicality conditions are time asymmetric. There is a certain gauge freedom in defining the time direction field. The field is physically equivalent to if it points into the same light cone at each point .

Typing surfaces.

A typing surface is represented by

where is a surface (having dimension one less than ) and is a vector that is not anywhere embedded in the surface (it’s job is to single out one side or the other of this surface). Typing surfaces are used to bound all or part of a region such as . The direction is a conventional choice. As is clear from considering the discrete situation shown in (2) above, sections of the boundary may correspond to quantum systems being inputted, outputted, or a mixture of both. We define

We use the notation

for composite typing surfaces. We allow these typing surfaces to meet at their boundaries. Consequently, some points may appear twice with different values of .

Setting and outcome fields.

We can chose settings, , at point and have outcome, , at .

Operations.

An operation, , is specified with respect to some region, , as follows

where

and

and the typing surface is . The typing surface may bound all of or it may bound only part of (for example, we could have absorbing detectors or preparation devices included in the part of the boundary not covered by the typing surface). We can also represent operations diagrammatically

We use a circle rather than a rectangle to represent these operations to emphasize that they are different from the operations considered in Sec. 10. Here an arrow pointing out is conventional and does not have to correspond to systems progressing forward in time. We say that

constitutes a complete set of operations if the associated outcome sets are disjoint and their union is the set of all possible outcomes for region .

Operators.

We can associate operator tensors with a region . To do this, first we associate Hilbert spaces with typing surfaces

where the () will pertain to a time direction. An operator tensor is given by (in diagrammatic and symbolic notation)

and is an element of - the space of Hermitian operators acting on .

Composition.

We can wire operations together for form bigger operations. For example

If there are no open wires then we have a circuit.

Physicality.

We require, when we replace operations by operators in a circuit, that we get a number between 0 and 1 for the probability. We also require that we cannot signal into the past. These two requirements are satisfied if certain physicality properties are satisfied. These physicality conditions are more complicated than in than for the operations (represented by rectangles) of Quantum Theory as discussed in Sec. 10 for two reasons. First, the causal structure for an operation in QFT (represented by a circle) is different. In particular, for a QFT operation, it is possible for an output to be fed forward back into the same operation. This can be made clear by looking at the discrete situation in (2). Second, we have a continuous rather than discrete situation. We will say that an operator, , pertaining to region is physical if it satisfies the condition

I will describe this condition in some detail in the appendix (though there still remains more work to be done in understanding how physicality plays out in QFT). We say that

represent a complete set of physical operators if each operator in the set is physical and they satisfy the normalization condition

(see [45] for clarification).

Quantum Field Theory.

We can now state an axiom for QFT

Axiom for QFT: All complete sets of operations correspond to complete sets of physical operators and vice versa.

This succinct statement captures the essence of Quantum Field Theory in the operator tensor formulation.

Part IV A conceptual road to Quantum Gravity

I will now outline one approach to finding a theory of Quantum Gravity. Before I begin, I should emphasize once again that The Construction Interpretation is pluralistic in that it explicitly encourages a multitude of different conceptually driven approaches to the problem of Quantum Gravity in accord with the tenets given above. There is value, however, in outlining a particular approach if only to illustrate the general ideas of the Construction Interpretation.

12 The problem of Quantum Gravity

As discussed in the prelude, the problem of Quantum Gravity is to find a theory that gives rise to General Relativity on the one hand and to Quantum Field Theory on the other under appropriate limits.

At least we must recover the parts of these theories that have been empirically verified. While a theory of Quantum Gravity must have these limits, the conceptual and mathematical formalism could be very different from either General Relativity or Quantum Theory. Indeed, a theory of Quantum Gravity could be as different from either General Relativity or Quantum Theory as, for example, General Relativity is from Newtonian Mechanics. There may be more than one candidate theory of Quantum Gravity that has these limits. If so, we might try to distinguish between these alternative theories on the grounds of simplicity, elegance, or conceptual coherence. In the end, of course, experiment is the final arbitrator. A theory of Quantum Gravity would be especially interesting if it has features that are genuinely new and cannot be understood in solely General Relativistic or Quantum terms.

13 Path to a framework for Quantum Gravity

We can attempt to plot a path with steps elements for Quantum Gravity which is analogous to the path for General Relativity outlined earlier. In this path indefinite causal structure (inferred from some facts about General Relativity and Quantum Theory as we will outline) plays an analogous role to the fact that there is no global inertial reference frame (inferred from some facts about falling bodies) in the General Relativity path outlined above. The path proposed here is analogous to that for General Relativity outlined earlier rather than being the same path.

In brief the path I propose consists, first, of the following 7 steps to bring us to a general mathematical framework for Quantum Gravity

Proceeding by analogy with Einstein’s path to General Relativity, we can write down 3 additional steps that might help us get to an actual theory of Quantum Gravity formulated within this framework. These are

I will now outline in some detail the steps in this path insofar as they are clear to me (the last three steps being particularly opaque). We will indicate which element is being considered at the beginning of the relevant section along with which element this is analogous to in the path to General Relativity outlined in Sec. 5

It is worth mentioning at the outset that indefinite causal structure raises big challenges for our usual way of doing physics. In particular, we cannot consider a state on a spacelike hypersurface and then evolve this in time. We need a different way to do physics.

14 Step one: Some features of GR and QT

Dynamical causal structure (from GR) and indefiniteness (from QT)

analogous to

Equivalence principle

To gain clues as to what kind of theory might limit to GR and QT we should examine the conceptual structure of these less fundamental theories. In so doing we see that they are each conservative and radical, though in complementary respects as summarized in the following table

GR QT
conservative deterministic fixed causal structure
radical dynamical causal structure probabilistic

General Relativity is conservative in that, as discussed above, if we specify sufficient boundary information the physical situation is determined. On the other hand, Quantum Theory is radical in that it is inherently probabilistic. Quantum Theory is conservative in that it operates on a fixed causal background structure. On the other hand, General Relativity is radical in that the causal structure (given by the metric) is dynamically determined by the equations of the theory.

15 Step two: Indefinite causal structure

Indefinite causal structure

analogous to

No global inertial reference frame

It seems that a theory of Quantum Gravity would need to take the radical road in both cases. Thus, it would be a probabilistic theory with dynamical causal structure. But, actually, it would likely be even more radical for the following reason. In Quantum Theory, quantities subject to variation are also subject to quantum indefiniteness (or “no-matter-of-the-factness”). If a particle can go through the left or the right slit then there is no-matter-of-the-fact as to which slit it goes through. If causal structure is dynamical then we expect there to be situations in which there is no matter-of-the-fact as to what the causal structure is. This is indefinite causal structure. For example it could be the case that, for example, there is no matter-of-the-fact as to whether the separation between some particular two events (identified in operational terms perhaps) is space-like or time-like.

It is worth pointing out that the equivalence principle plays another role in the construction of General Relativity. This is to motivate replacing partial derivatives by covariant derivatives (as discussed in Sec. 7). Though this takes us outside the main narrative, it is worth asking whether there is an analogue to this here. One radical possibility, which I will discuss in Sec. refsec:firstelement, is that we can find a quantum frame of reference in which the indefinite causal structure is transformed away locally (but not globally).

16 Step three: Compositional space

Compositional space

analogous to

General coordinates

16.1 Need for a new compositional space

The world is a big place and we need a way to break it up (conceptually) into smaller pieces if we are to make sense of it. We can then join these smaller pieces back together to get the bigger picture. We will call the space we use to do this the compositional space. Usually this role is played by spacetime.

A standard way of breaking the world up into such smaller pieces is to evolve (in time) a state defined on a spacelike hypersurface. We can think of the infinitesimal timesteps as defining small regions of spacetime (narrow in time while being wide in space). This approach is blocked if we have indefinite causal structure since then we cannot, in general, talk about spacelike hypersurfaces.

A more general approach is to break spacetime up arbitrarily into small regions and then have rules at the boundaries of these regions for how they regions fit together. Indeed, we can think of local field equations (more-or-less) in this way - they relate local derivatives and thus provide constraints on what can happen in an infinitesimal region around any point. This approach is also blocked if we have indefinite causal structure as we will now discuss.

In General Relativity spacetime is represented by points in a manifold. Consider a region of this manifold. There are no beables corresponding to this region as there is no diffeomorphism invariant function that only looks at the fields in . This is because, if we apply a diffeomorphism, then the values for the fields that were in will be replaced with values for these fields coming from some other region. There is, then, no non-trivial function of the fields in that is invariant under diffeomorphisms. Reality is “non-stick” so far as the manifold is concerned. If we want the compositional space to be physical then the manifold is not fit for this role.

We could try to fudge this by introducing a gauge fixing. This is done by introducing four additional field equations, having no physical content themselves, but providing enough extra field equations to fix the solution. Not only do these four additional field equations have no physical content, they also obscure the actual physical content of the solution (as it is not clear which features of the solution are a consequence of this gauge fixing and which are real physical features). Nevertheless, this strategy is mathematically possible and is usually adopted when presenting actual solutions.

The gauge fixing fudge no longer makes sense when we introduce indefinite causal structure. In fact, even if we have something like a probabilistic ignorance of the classical variety, we can see that gauge fixing is not adequate and we need an alternative compositional space. In the classical probabilistic case, we will have probabilistic contributions from many different solutions. These different solutions will live on different manifolds, (where is labeling the different solutions), and these may even be topologically different. Then any region, , of our compositional space would have to correspond to a region, , of each of the manifolds in this mixture. Without some principle telling us how the subregions, , correspond for the different , there is no sensible way of doing this. Indefinite causal structure only makes this worse. Then we do not have a classical mixture of underlying solutions and it would appear that we are completely blocked from using the usual manifold of General Relativity for our compositional space.

16.2 Beables as axes for compositional space

A better strategy is, instead, to use some beables to constitute this compositional space. This could be obtained by taking some nominated (ordered) set of beables,

and letting these form the axes of our space. In a particular run of the experiment only some points in this compositional space would happen (we will illustrate this in the case of General Relativity below). This is radically different from using a manifold where we have fields at every point.

What considerations can motivate the choice of a subset, , of beables? One idea is that they should correspond to our direct experiences of the world. Thus, we can set up an operational space in which the axes are things we directly experience (at some level of description). Regions of operational space would correspond to regions of the world as identified by our observations. Such a choice would be contingent on how we look at the world. Different creatures may nominate different beables for their operational space. Our experience with Quantum Theory demonstrates that this kind of contingency is not a problem - or at least it is not an obstacle for doing calculations. In Quantum Theory we can nominate which degrees of freedom correspond to measurement outcomes putting the Heisenberg cut between the classical and quantum worlds wherever is convenient. As long as sufficient decoherence has occurred, it makes no difference for all practical purposes where we put this split. We should be able to do the same for Quantum Gravity. This pleasing blending of considerations from General Relativity along with our intuitions from Quantum Theory strongly suggest setting up an operational space for Quantum Gravity.

16.3 An object in compositional space

We want to break the world up into smaller pieces in compositional space. We will call these smaller pieces objects. An object, , is specified as follows

The region refers to a certain region of compositional space. We will adopt the convention of associating object (in \mathsf font) with region (i.e. denoted by the same uppercase letter but in \mathtt font).

The settings are things we may choose in region . These could set by adjusting knobs, or they could correspond to imposing some external field (a magnetic field for example).

The happenings specifies some particular thing that may happen. This may be described at an ontological level or an operational level. In the operational case we consider only beables that correspond to things we observe. In this case we will refer to happenings as outcomes (these might correspond to the red light on a box flashing for example). In the ontological case the happenings could include beables we do not have direct observational access to (traditionally called “hidden variables” in the Quantum Foundations literature).

Finally, we have handles. These tell us how this object can be joined to other objects. We will discuss this later.

In the case that our object refers to happenings that are outcomes we will call it an operation.

Operational theories allow us to make calculations for composite operations.

16.4 Operational space and operations in GR

What kind of beables should we nominate to constitute our operational space? Here I will discuss the choice of operational space advocated in [45] for General Relativity. This approach borrows from the work of Westman and Sonego [82, 83] (though they did not have the same operational motivations as here). Consider an (ordered) set of scalars,

We can think of a space whose axes are the . This is our operational space. Each scalar, , is calculated from the tensor fields, (these are the matter and metric fields as discussed in Sec. 7), by forming a quantity in which all the indices are summed over. For example, we might have where is a current, where is a density, and so on.

Now consider a solution, , to the field equations of General Relativity (as discussed in Sec. 7). For each , we can calculate from the fields, , at in this solution. In this way, the point, , gives rise to a point, , in operational space. When we repeat this for all points, , we get a surface, , in operational space as shown in Fig. 1.

{Compose}

00 \CxaxisS_10,021 \CyaxisS_20,021 \CzaxisS_30,021 \csurface[0.7]0,01 \ccuboid[-0.05]7,33744.73.834.2 \ccuboid[0.05]4,537442.72.23.1 \thispointGS20, 7, 11 \thispointG26, 9, 10 \pointingarrowG-180GS0 \csymbolalt[1,0]Γ \thispointAO8.5,9.44 \thispointA10,15 \pointingarrow[above right]A-140AO70 \csymbolaltA \thispointBO5.5,11.44 \thispointB7,17 \pointingarrow[above right]B-140BO70 \csymbolaltB \thispointGAS7.6, 5.7 \thispointGA-6, 4 \pointingarrow[left]GA0GAS-150 \csymbolalt[-75,0]Γ_A=A∩Γ \thispointGBS4.6, 8 \thispointGB-9, 6 \pointingarrow[left]GB0GBS-150 \csymbolalt[-75,0]Γ_B=B∩Γ \Cxaxis[ultra thin, dashed]S_10,021 \Cyaxis[ultra thin, dashed]S_20,021 \Czaxis[ultra thin, dashed]S_30,021

Figure 1: The surface in operational space. Also shown are two regions and .

Under a diffeomorphism mapping to , scalars transform as . Hence, the property of some set of scalars having certain values in coincidence is unaffected by a diffeomorphism. This means that, if we apply a diffeomorphism to the solution so we have and go through the exercise of plotting again then we get the same surface ( does not move under a diffeomorphism). Hence, , is a beable.

We can consider regions of operational space such as and shown in Fig. 1. We have, now, a space that can be used for compositional purposes. It is an operational space if we assert that our basic observations are coincidences in the values of the set of nominated scalars.

We can define operations for General Relativity as follows.

Here is a region of operational space as illustrated in Fig. 1, are settings (it is beyond the scope of this paper to describe how these can be implemented in General Relativity - see [45] for details), and is the outcome (it is what we see in region ). Finally, , are the handles. Here the handles are given by specifying boundary surfaces for the region . Thus, may be the surface where and meet. We also equip these boundary surfaces with a (conventional) choice of direction. If the direction points away from the region associated with the operation we write it on the left of the comma and if it points towards the enclosed region we write it on the right. So, in our example points away from while and point towards . We include the handles as superscripts and subscripts in our notation for the operation. This helps with writing composite objects. For example, we may have

for the composite region shown in Fig. 1 whose components are joined at . We can represent objects using diagrammatic notation instead. Thus we can write as

Then we can write the composite object in Fig. 1 as

The diagrammatic notation is equivalent to the symbolic notation.

In [45] this approach is used to provide an operational formulation of General Relativity. In the first place a possibilistic formulation is given. In this composite operations can be converted into calculations that say whether a particular configuration is possible or not. This enables us to think about General Relativity in operational terms (kind of like thinking about the theory from a users perspective who is inside the world rather than outside it). In the second place we provide a probabilistic formulation for those cases where we have probabilistic ignorance.

16.5 Operational space in QG

The preceding example of operational space pertained to General Relativity. However, we can expect that the basic idea will pass over into Quantum Gravity. In Quantum Theory we nominate some set of quantities to correspond to outcomes of our measurements. These quantities are described by variables from classical physics (such as the position of pointers on measurement apparatuses). Likewise, we can expect that there will be something like measurement outcomes in Quantum Gravity and that these will be described by variables from the classical theory (General Relativity in this case). We propose that these classical quantities are coincidences in the values of scalars. These scalars may be related to objects in the underlying theory of Quantum Gravity in a different way than they are in the classical theory.

It is possible to imagine other choices of operational space. For example, we could relate the axes of of our operational space to quantities measured by some given set of instruments. If these instruments have physical extension in space and time (such as measuring rods and clocks) then they will not be simply related to scalar coincidences.

An alternative approach I pursued in [40, 41] is to define the operational space directly from recorded data. In that case data was imagined to be stored on cards. A region of operational space corresponded to a subset of all possible cards.

17 Step four: Formalism locality

Formalism locality

analogous to

Local physics

17.1 A desideratum

We want to use a physical theory to make predictions. Imagine we are interested in predictions pertaining to some region in our compositional space. How are we to proceed? In theories having a fixed causal structure we typically refer to this causal structure in doing calculations. For example, imagine in such case we are interested in predictions pertaining to a region . We can find a bigger region, (such that ), having the properties that (i) has an initial spacelike hypersurface and (ii) the backward light cone of every point in is partitioned by this initial spacelike hypersurface. In this case, we can evolve a state defined on the initial spacelike hypersurface through all of . In this way, we can make predictions about . There are two less than satisfactory aspects to this. First, we need to have a fixed causal structure. Second, we need to consider points outside of in order to make predictions about . If we do not have fixed causal structure then how are we to proceed? Then we cannot even find an appropriate region to apply this technique. Clearly a different idea is needed.

First let us state a desideratum. We want to formulate physical theories (whether they have definite causal structure or not) such that the formalism has the following property.

Formalism locality: A formulation of a physical theory is said to have the property of formalism locality if it is such that all predictions the theory can make pertaining to some given region (of the compositional space) can be made by referring only to mathematical objects (which we will call generalized states) in the theory pertaining to this region.

Clearly this dictates against using the above approach of finding a larger causally well formed region that encloses the given region.

17.2 Prediction heralding

At first sight it appears that there is an insurmountable obstacle to formulating a theory in a formalism local way. We cannot use the notion of an initial state to take account of all external influences. In other words, we cannot use causal structure as an aid to screening off against influences outside the region of interest. When we have indefinite causal structure there is a sense in which regions are fundamentally open. Consider trying to make a prediction of the general form

Here and are outcomes that may happen in region and and are some settings we choose in region . Our predictions could be probabilistic. In this case we might, for example, be interested in the relative probability of seeing outcome with setting verses seeing outcome with setting (this makes most sense when as then we can just divide the number of times happens by the number of times happens). Or our predictions could be possibilistic. We might be interested in whether it is possible to have outcome with setting given that it is possible to have outcome with setting . Rather than probabilistic or possibilistic, we might have some other kind of kind of prediction.

The problem is that the theory may simply fail to be able to make a prediction. This might be the case if, for example, external influences can effect what happens inside since then, whatever prediction we write down, an external adversary could send in some external influence to make our prediction wrong. When we have definite causal structure we can, under certain circumstances we can guard against this. For example we can have a region such as region above (with an initial spacelike hypersurface with all other points in its domain of dependance) and we let and correspond to conditions that effectively fix the initial state. Under these circumstances we can then make predictions about the probability or possibility of with settings . This is a very particular type of situation. Generically, regions will not have this kind of special shape. Further, if we have indefinite causal structure then we cannot, in general, even set up regions like this nor can we have an initial state. In this case we need a different tactic to guard against meddling external adversaries. The way forward is to give up on always being able to make predictions and them mathematise the question as to whether a prediction is possible. External adversaries will be able to mess up some predictions we might try to make but not necessarily all of them. We need a mathematical criterion telling us when we can make predictions. We will call this prediction heralding. If this criterion is satisfied then we need a calculation saying what the actual prediction is. This two stage process

seems to be the only way to implement formalism locality. If we have indefinite causal structure then this way of doing local physics is pretty much forced on us.

17.3 Causaloid implementation of prediction heralding

How can we actually implement prediction heralding? I first discussed how to do this in the context of the causaloid framework [40, 41]. The same approach appears in subsequent papers setting out the duotensor framework [42], the operator tensor formulation of Quantum Theory [43, 44], and the operational formulation of General Relativity [45]. Let me here outline how this works in the causaloid framework. The causaloid framework is a General Probabilistic Theory (GPT) approach for indefinite causal structure. First imagine that, for a sufficiently big region, , we are able to make sensible probabilistic predictions so that all probabilities of the form

(3)

have values in the theory. This assumption can be dropped later once we have the mathematical framework in place. Now consider a region . We can write (3) as

(4)

Here we can think of as the state “prepared” in when we have outcomes and settings in region . The entries of the vector the probabilities

for a special fiducial set of “measurements” in associated with outcome setting pairs where labels these fiducial measurements. We put inverted commas round “prepared” and “measurements” since there need not be any sense in which is before - these are just words whose usefulness derives from our familiarity with them in other situations. This fiducial set of measurements is selected so that we can write the probability in (4) as a dot product as shown. We can always do this since, in the worst case, we could have a fiducial measurement for every possible and then the vector would just have a 1 in the corresponding position and 0’s everywhere else. In general, we expect the probabilities to be related to each other so that we can choose a smaller set of fiducial measurements. We assume that we choose a minimal set (in the sense that no set measurements of smaller rank can constitute a fiducial set). This set up is a standard trick in the GPT approach (see [39] for example). There is a duality between and vectors. We could equally well have set this up with a for and an for . These objects (whether in the or form) are generalized states.

Now we can see how to implement formalism locality with a two stage calculation. Let the thing we want to calculate be the relative probability

(5)

pertaining to region . Something we can calculate (pertaining to region ) is

We have the extra arguments and pertaining to region so this prediction is not of the form in (5). If we are to have formalism locality our predictions must be independent of and . This means the prediction must be independent of . This will be true if and only if the following condition holds

(6)

This is the prediction heralding condition. In the case the condition does hold then

(7)

where is given by

(8)

This shows how to implement formalism locality in probabilistic theories. In possibilistic theories a similar strategy can be pursued (see [45] for an example).

17.4 Replace trade-craft with calculation

In standard formulations of physical theories there is a certain trade-craft to knowing what quantities we can make predictions for. As physicists, we generally have a good intuition for this without fully appreciating that we are that we are only considering a tiny subset of possible predictions for which questions can be posed. This trade-craft invariably involves looking at the causal structure operating in the background. The idea here is to liberate ourselves from this intuition led approach and consider predictions of any sort. Then we have a mathematical criterion for prediction heralding. This leads to formalism locality and the two stage calculation process and, indeed, this is pretty much forced on us by indefinite causal structure. However, even when we have definite causal structure it may be argued that the approach argued for here is superior. In particular, this approach is democratic with regard to what types of predictions it will attempt to make.

17.5 A fallacy

One fallacy regarding the formalism local approach here is that it is much worse than the “standard” approach as predictions are only heralded in the very non-generic situation in which two vectors are parallel. To respond to this let me first point out that any situation in the standard approach to physics where we can make a prediction will, when translated into these terms, necessarily have these two vectors parallel (since this was a necessary condition for the heralding of a prediction). Hence the non-generic issue is just as much of a problem for the standard approach. But, secondly, within the approach advocated here we can relax the prediction heralding condition (this is not open to us in the standard approach). For example, if the two vectors are not quite parallel then, while we cannot give an exact prediction for the relative probability in (5), we can put some bounds on this prediction. For an outline of how to do this see [53].

18 Step five: Laws given by correspondence map

Laws given by correspondence map

analogous to

Laws expressed by field equations

The way a physical law is expressed depends on the framework within which the theory is given. In General Relativity we provide the laws of physics by giving a set of field equations that constrain the fields. In operational formulations the laws are given by providing a map from operations (that describe the operational aspects of the real world) to generalized states (that are used in calculations to make predictions)

If we denote the operation associated with region by and the generalized state by then the correspondence map is

To specify the laws we need to provide the equations that give this map.

The next principle will concern composite regions and how to combine generalized states for them. Thus, it is sufficient to express the laws (by giving correspondence maps) for small regions since then we can obtain generalized states for larger regions. We could choose some set of small (but finite) regions for this purpose. Or we might choose to take the limit as these regions become infinitesimal. In the latter case we only require a correspondence map for infinitesimally small regions. Schematically, we can represent this for each point, , in compositional space

and now the law is given by . How to implement this infinitesimal schema will be the subject of further research.

19 Step six: Boundary mediated compositional description

Boundary mediated compositional description

analogous to

Local tensor fields based on tangent space

How do we describe a composite object? In basic terms we should provide a list of the components and the relationships between these components. A relationship can exist between any subset of the components. For example, in a composite object consisting of component objects , , , and , there could in principle be relationships between all pairs, relationships between all triples, and a relationship between all four components. The description of this object would be represented by a hypergraph having all possible hyperedges. It could be the case, however, that some relationships are implied by other relationships. In particular, consider an object whose component objects correspond to a number of regions that border one another. We can represent this by a graph in which edges correspond to common boundaries. For example

(9)

In the graph on the left we only specify relationships between regions that border one another. We also include open boundaries (such as in the above example) since we may consider joining this composite object to other objects. The arrows indicate that we can associate a conventional choice of direction with the boundary surfaces (they do not indicate the flow of time). We can use symbolic (rather than diagrammatic) notation to represent this graph

In this case, subscripts correspond to boundaries with the direction pointing in and superscripts to boundaries with the direction pointing out.

We will call this boundary mediated compositional description. The point is that we adopt this form of description for composite objects. This matters when we adopt the principle of general compositionality.

20 Step seven: The principle of general compositionality

Principle of general compositionality

analogous to

Principle of general covariance

20.1 The principle

We can break a big region of compositional space into smaller component regions in many different ways. In the absence of definite causal structure, no particular way of breaking up compositional space is preferred. Objects are associated with regions of compositional space. Consequently, we need to write the laws of physics so that they work for any decomposition of an object into component objects. Furthermore, in the absence of definite causal structure, the structure of these calculations can only be informed by the compositional description of these objects. Hence, it is reasonable to assume that the calculation has the same compositional structure. These considerations lead to the following:

The principle of general compositionality: The laws of physics should be written in such a way that they work for any decomposition an object into components and so that the calculations for any such decomposition have the same compositional structure as the compositional description of the object for this decomposition.

It is understood that the compositional description used here is the boundary mediated compositional description outlined in Sec. 19. This principle is deliberately written in similar language to Einstein’s principle of general covariance: the laws of physics should be written in such a way that they work to any coordinate system and so that they take the same form in any coordinate system. The motivation for the principle of general compositionality (from the absence of definite causal structure) is strongly analogous to the motivation for Einstein’s principle (from the absence of a global inertial reference frame (as outlined in Sec. 6)).

20.2 Correspondence

In Sec. 18 we saw how laws are given by providing a correspondence map from operations to generalized states. In Sec. 19 we adopted a boundary mediated compositional description. To formulate the laws of physics in accord with the principle of general compositionality we need, then, to construe our generalized states in such a way that they are related to the boundary mediated descriptions of objects. We need a correspondence principle of the form

from objects to generalized states. This takes us beyond the causaloid formalism discussed in Sec. 17.3. We will discuss two ways of so representing generalized states. First, by a duotensor [42] and second by an operator tensor [43]. Both the duotensor and operator tensor are linear in discussed in Sec. 17.3. However, crucially, they are expressed with respect to boundary information - they provide a way to understand composition that is missing in the causaloid formalism.

For general probability theories having the property of tomographic locality the generalized state can be represented by a duotensor

Something similar can be done for general possibilistic theories (having a similar tomographic locality property) [45]. In possibilistic theories we calculate whether something is possible (which we give value 1) or impossible (which we give value 0). A duotensor is like a tensor but with a bit more structure. We can have two positions for each superscript and each subscript in symbolic notation. In diagrammatic notation we represent these two positions by using black or white dots. Physically, the duotensor with all black dots corresponds to the probability when we putting fiducial objects (labeled by to ) on each boundary. In the case of Quantum Theory we can use these duotensors to linearly weight fiducial operators to construct operator tensors having the form

As discussed in Sec. 10, an operator tensor is an Hermitean operator acting on a Hilbert space determined by the labels on the legs ( in this case).

20.3 Implementation

Once we have generalized states of the above form we can do a calculation. We have already learned that, in order to make predictions for some region, we need the relevant generalized states. So we need to know how to calculate the generalized state for a composite operation. Let us consider the example in (9). In the duotensor formalism the generalized state associated with this composite operation is given by replacing each operation with the corresponding duotensor: