## Path Algebras in Quantum Causal Theory

This Saturday I am presenting a talk on the role of path algebras in quantum causal theory at the Conference on Ordered Algebraic Structures at Louisiana State University.

Below are my slides:

Path Algebras in Quantum Causal Theory

The slides have a lot of nice graphics, and even if some of the material is unfamiliar, they should give you a nice overview of what quantum causal theory is about.

You’d be hard-pressed to find another half-hour math talk with Newton, Cauchy, Riemann, Einstein, Schrödinger, Wheeler, Feynman, Grothendieck, Hawking, Sorkin, Connes, and Malament in it!

## On the Axioms of Causal Set Theory

The following paper, which I recently posted on the arXiv, outlines many of my ideas about quantum causal theory.

On the Axioms of Causal Set Theory

## Quantum Circuit Graphs

The following paper presents five projects on connections among graph theory, relativity, and quantum computing, suitable for undergraduates.  These projects are currently being used in the quantum information seminar at Louisiana State University.  Pending appropriate support, I hope to produce detailed slide presentations of these projects in the near future.

Quantum Circuit Graphs

## Order-Theoretic Representation Theory

Group representation theory lies at the heart of modern physics as the mathematical expression of symmetry, remaining perhaps the most promising vehicle for initial progress beyond relativity and the standard model.  Covariance in relativity is expressed locally in terms of the Poincaré group of symmetries of four-dimensional Minkowski spacetime.   This spacetime model is taken for granted in the quantum field theory underlying the standard model, thereby constraining particle states to correspond to representations of the Poincaré group.  The gauge theories describing electromagnetism, the weak interaction, and the strong interaction are based on the representation theory of the gauge groups U(1), SU(2), and SU(3), respectively.   Despite the success of group representation theory in fundamental physics, the need for more general notions is now becoming apparent.

Nonmanifold models of spacetime microstructure arising in quantum gravity require different interpretations of covariance, based on structures such as partial orders.  The properties of these structures alter the constraints on quantum states.  Gauge theory also takes a different form in this context.  Modification of group representation theory in such models is often viewed in a negative sense, as “Lorentz invariance violation,” or “covariance breaking.”  However, such modifications should be pursued positively as a refinement, not abandonment, of the representation-theoretic tradition.

Group representation theory generalizes in interesting ways in the context of pure causal theory.   Before discussing this, I will briefly review how elements of the Poincaré group P may be applied to Minkowski spacetime M.   From the active viewpoint, an element of P induces an automorphism of M, which rearranges its elements.  From the passive viewpoint, an element of P induces a change of coordinates on M, which merely rearranges physically insignificant labels on its elements.  The active and passive viewpoints are virtually interchangeable in this context, but they are completely different in more general settings.

The relativity of simultaneity provides a good context for understanding the passive viewpoint.  Given two causally unrelated events in M, a coordinate system may be chosen in which either event precedes the other. Coordinate systems may therefore be viewed as refinements of the causal order on M.   Each element of P exchanges one refinement for another, so these elements may be identified with special ordered pairs of refinements of the causal order on M.

Now replace Minkowski space with a partially ordered set G.   An automorphism of G is an order-preserving bijection.   Meanwhile, a refinement of G is an inclusion of G into a finer order; i.e., an order with the same elements but more relations.   A pair of refinements of G is equivalent to an appropriate pair of inclusions from G into two finer orders.   The important point is that such a pair of inclusions is in general totally different from an automorphism of G!

The reason why the active and passive viewpoints seem so similar in the context of Minkowski space M is because operations on M are usually described in terms of some coordinate system.   This is true, in particular, for automorphisms of M, which are usually described by specifying the coordinates of an arbitrary element and its automorphic image.    This relates a pair of coordinate systems to each automorphism, thereby connecting the active and passive viewpoints.   However, this relationship is merely an artifact of the special structure of M.  It survives to some degree for general Lie group actions on manifolds, but usually disappears in the absence of manifold structure.

The question then is, which viewpoint is the “right” one in general, the active viewpoint or the passive viewpoint?  The answer is both!  The active viewpoint generalizes to the theory of automorphisms, while the passive viewpoint generalizes to the theory of augmented structures over a fixed base.   Both theories are important in physics, but in different ways.   In particular, theories involving spacetime microstructure generally do not have exact symmetries on large scales, unless these symmetries are imposed artificially.  Automorphisms in such theories will generally leave large-scale structures fixed, while permuting small-scale structures locally and independently.   By contrast, such theories will often exhibit pairs of “refinements” closely approximating any given coordinate transformation on an appropriate “smoothing.”  I will not attempt to make these notions more precise at present.

## Causal Atomic Resolution

If spacetime degenerates into a nonmanifold structure near the Planck-scale, the number of fundamental elements involved in even the smallest-scale interactions currently observable might be very large: possibly on the order of Avogadro’s number.   This statement applies not only to pure causal theory, but also to dynamical triangulations, spin-foams in loop quantum gravity, and various other models of fundamental spacetime structure.  Organizing the information associated with such complex interactions is an important preamble to making detailed calculations and predictions.

Causal atomic resolution is an attractive way of organizing such information in the context of pure causal theory.   A causal atom is a special type of subgraph of a causal graph, that contains the intersection of its past and future.   An example is the Alexandrov subset given by intersecting the past of one element with the future of another.   Alexandrov subsets are generalizations of intervals in linearly ordered sets, and causal atoms are a further generalization.  They are also analogous to convex subsets in Euclidean space.  A causal atomic decomposition of a causal graph is a simplified graph whose vertices are causal atoms of the original graph.   A causal atomic resolution is a sequence of causal atomic decompositions that progressively simplify the original graph by combining causal atoms at each level of the sequence into larger atoms at the next level.   Causal atomic resolution imposes a fractal structure on causal graphs.   It organizes information in a manner analogous to multiresolution analysis in the theory of wavelets in functional analysis.

Causal atomic resolution is similar to the theory of categorification in abstract algebra.   Categorification replaces set-theoretic constructs with category-theoretic constructs, and vice versa (decategorification).   For instance, objects may be promoted to categories, or categories demoted to objects.  Elements may be promoted to objects, or objects demoted to elements.  Morphisms may be promoted to functors, or functors demoted to morphisms, and so on.  Categorification may be generalized in a graph-theoretic setting.  Every category possesses an underlying directed graph, but not every directed graph is a category.   A general system of “objects” and “morphisms” without the category-theoretic requirements of composition, associativity, and identity may be called a graphegory, and the graphegoric generalization of categorifcation may be called graphication.  Causal atomic resolution is a special case of graphication, in which elements are promoted to directed graphs, rather than objects of some other graphegory.

In the quantum causal theory of spacetime microstructure, causal atomic resolution may be combined with power space and path integral methods to carry the theory in promising and unexplored directions.   This approach features holism, self-similarity, and iterated structure.  Graphication appears here in two different ways: as the origin of quantum-theoretic structure, and as a computational tool for organizing information at different scales.

Causal atomic resolution has the potential for important applications to existing areas of computer science, economics, biology, physics, and mathematics.  These areas include computational complexity, network theory, neuroscience, self-organization, quantum information theory, and of course quantum gravity, abstract algebra, and graph theory.   The potential cross-pollination between the concrete and the abstract afforded by causal atomic resolution makes it an interesting topic for further study.

## Causal Theory in Power Space

George Ellis and a few other physicists have recently introduced the idea of top-down causation in fundamental physics.  In the context of classical spacetime, the hypothesis of top-down causation is that causal relationships among subsets of spacetime are not completely reducible to causal relations among their constituent events.

Causal set theory, by contrast, is purely bottom-up at the classical level.   In causal set theory, causality is modeled as an irreflexive, acyclic, locally finite binary relation on a set, whose elements are viewed as spacetime events.  Since causal structure alone is not sufficient to recover metric structure in the continuum limit, measure-theoretic data is also supplied in the form of a discrete constant measure. Rafael Sorkin, the founder of causal set theory, summarizes these two ingredients with the phrase “order plus number equals geometry.”   This statement is a special case of what I call the causal metric hypothesis.

If the hypothesis of top-down causation is true, then causal structure exists not at the level of spacetime itself, but at the level of the power set of spacetime, i.e., the set whose elements are subsets of spacetime.  Causal graphs may still be employed to describe such a structure, but the vertices now correspond to families of events, rather than only single events.  A possible concern is violation of locality, but this is not an insurmountable problem; it merely requires appropriate constraints among high-level and low-level relations.  I call the resulting theory causal theory in power space.

I will say a few words about why this idea is physically attractive.  First, the arguments in favor of top-down causation are rather convincing in their own right.  Second, one of the principal difficulties for pure causal theory is its parsimony; it is not clear that it contains enough structure to recover established physics.  Top-down causation, employed as I described above in terms of power-set relations, provides “extra structure without extra hypotheses.”   The types of mathematical objects and the general techniques are unchanged; it is only their interpretation that is generalized.  In particular, the causal metric hypothesis still applies, although not in the form “order plus number equals geometry.”

There is considerable precedent, at least in mathematics, for this type of generalization.  For example, Grothendieck’s approach to algebraic geometry involves “higher-dimensional points” corresponding to subvarieties of algebraic varieties, and the explicit consideration of these points leads to the theory of algebraic schemes, which has considerable advantages.   In particular, the scheme structure on a variety is consistent with the variety structure, but contains useful additional information.  Similarly, higher-level causal structure, suitably constrained, is consistent with lower-level causal structure, but contains important additional information that might be essential in recovering established physics.   The holistic aspects of causal theory in power space might aid in establishing the shape/causal duality discussed in my previous post.

## Shape/Causal Duality?

Shape dynamics, invented by Julian Barbour, is a theory of spacetime that applies an action principle, called Jacobi’s principle, and a procedure called best matching, to a configuration space of conformal 3-manifolds, called shape space, to recover the Hamiltonian formulation of general relativity, called the ADM formalism.   Shape dynamics is a “space-first” theory, with time arising in a secondary fashion via Jacobi’s principle.   The ADM formalism produces a spacelike foliation of spacetime, whose leaves are parameterized by the “time coordinate.”

Causal theory, invented by Rafael Sorkin, is a “time-first” theory. Time is viewed as a proxy for causality, which is taken to be fundamental.  Spatial separation is viewed, as in relativity, as indicative of causal disjunction, and is taken to be secondary.  Several different versions of causal theory exist, of which the best-known “pure causal” version is Rafael Sorkin’s causal set theory.   Dynamical triangulation is another well-known “causal” theory, but is really a hybrid theory with both spatial and temporal structure built in at the fundamental level.   My own version of causal theory is summarized here.   It is based on the causal metric hypothesis, which states that the fundamental structure of spacetime (and optimistically, all of physics!) is determined, up to scale, by causal relations.  Sorkin’s motto “order plus number equals geometry” may be regarded as a special case of the causal metric hypothesis.

Shape dynamics and causal theory both rely heavily on binary relations.   In shape dynamics, the relations are symmetric, and are viewed as encoding spatial structure.   In causal theory, the relations are antisymmetric, and are viewed as encoding causal (i.e., temporal) structure.   The following question occurred to me while discussing these two theories with Daniel Alves, Lawrence Crowell, Sean Gryb, and Flavio Mercati during the 2012 FQXi essay contest: is there a shape/causal duality principle relating appropriate versions of shape dynamics and causal theory? Of course, a continuum theory like standard shape dynamics is not directly comparable to a locally finite theory like causal set theory, but these theories can be reformulated in comparable contexts.

Shape dynamics and causal theory have seemingly complementary features; each takes to be fundamental what the other takes to be secondary.   If some version of shape/causal duality exists, neither idea need be at fault.   Between the two theories, I tend to be biased in favor of causal theory, largely on philosophical grounds.  While I see no good reason to regard spatial relations as fundamental, I view causality as a very compelling foundational principle.   However, shape dynamics does have certain advantages.  Julian Barbour tells me that just as “nothing comes from nothing,” so “not much comes from not much,” and causal theory is “not much” in his view.   It is true that the structure of causal theory is very parsimonious, and to propose recovering known physics thereby requires considerable optimism.  Of course, these are only two of many theories, and both may be incorrect or inadequate.  However, a shape/causal duality would lend greater credence to both theories, while opening up interesting lines of research.

## Recent Ideas

The following are some new ideas about physics.    I have most of these ideas partially written up, but I am so busy with other work at present that it may be some time before I can put them into a respectable form.  Until then, I offer them as food for thought.   To avoid making this post too long, I will give only very brief summaries of these ideas.   Subsequent posts will treat them one at a time, in somewhat more detail.

1. Shape/Causal Duality

Shape dynamics, invented by Julian Barbour, is a theory of spacetime that applies an action principle and a procedure called best matching to a configuration space of conformal 3-manifolds to recover the Hamiltonian formulation of general relativity. Shape dynamics is a “space-first” theory, with time arising in a secondary fashion. Causal theory, pioneered by Rafael Sorkin, is a “time-first” theory. Time is viewed as a proxy for causality, which is taken to be fundamental.  Spatial separation is viewed as indicative of causal disjunction, and is taken to be secondary.  Shape dynamics is based on symmetric relations between elements, while causal theory is based on antisymmetric relations. The following question occurred to me while discussing these two theories with Daniel Alves, Lawrence Crowell, Sean Gryb, and Flavio Mercati during the 2012 FQXi essay contest: is there a shape/causal duality principle relating appropriate versions of shape dynamics and causal theory? Continuum theories such as standard shape dynamics are not directly comparable to locally finite theories such as causal set theory, but these theories can be reformulated in comparable contexts. Shape/causal duality would lend greater scope and credence to both theories, while opening up interesting lines of research.

2. Causal Theory in Power Space

George Ellis and a few other physicists have recently introduced the idea of top-down causation in fundamental physics.  In the context of classical spacetime, the hypothesis of top-down causation is that causal relationships among subsets of spacetime are not completely reducible to causal relations among their constituent events.  Sorkin’s causal set theory, by contrast, is purely bottom-up at the classical level.   In causal set theory, causality is modeled as an irreflexive, acyclic, locally finite binary relation on a set, whose elements are viewed as spacetime events.   However, if the hypothesis of top-down causation is true, then causal structure exists not at the level of spacetime itself, but at the level of the power set of spacetime.  Causal graphs may still be employed to describe such a structure, but the vertices now correspond to families of events.   This approach is attractive both because of the convincing arguments for top-down causation, and because power-space relations add potentially useful structure without changing the types of mathematical objects and techniques involved in the theory.   There is considerable precedent for this type of generalization, such as Grothendieck’s scheme-theoretic approach to algebraic geometry.  The holistic aspects of causal theory in power space might also aid in establishing shape/causal duality.

3. Causal Atomic Resolution

If spacetime manifold structure degenerates near the Planck-scale, the number of fundamental spacetime elements involved in observable interactions might be very large.  Organizing the information associated with such interactions is an important preamble to calculation and prediction.  Causal atomic resolution is an attractive way of organizing such information in the context of pure causal theory.  A causal atomic resolution is a sequence of causal atomic decompositions that progressively simplify the original graph by combining “causal atoms” at each level of the sequence into larger atoms at the next level, thereby imposing a fractal structure.  Causal atomic resolution is similar to the theory of categorification in abstract algebra, in which objects correspond to categories at different “algebraic levels.” Categorification may be generalized in a graph-theoretic setting, with causal atomic resolution as a special case.  In quantum causal theory, causal atomic resolution may be combined with power space and path integral methods to carry the theory in promising and unexplored directions.  Causal atomic resolution also has the potential for important applications to existing areas of computer science, economics, biology, physics, and mathematics.

4. Order-Theoretic Representation Theory

Group representation theory lies at the heart of modern physics as the mathematical expression of symmetry.  Covariance in relativity is expressed locally in terms of the Poincare group of symmetries of four-dimensional Minkowski spacetime.   The same group constrains particle states in the standard model.  The gauge theories describing electromagnetism, the weak interaction, and the strong interaction are also based on group representation theory.  However, more general notions are needed.  Nonmanifold models of spacetime microstructure arising in quantum gravity require different interpretations of covariance, based on structures such as partial orders.  The properties of these structures alter the constraints on quantum states.  Gauge theory also takes a different form in this context.  Representation theory has interesting analogues in the context of pure causal theory.   The active viewpoint on group actions is replaced by the theory of order automorphisms, while the passive viewpoint is replaced by the theory of refinements.  Both theories are interesting and important.

This image illustrates the behavior of a particular quantum circuit.  To be precise, this circuit is an iterated quantum controlled NOT gate (QCNOT) with a twist of 45.5 degrees in the Hopf fiber containing the identity.   This means that one of the input qubits, which may be taken to be the control qubit for definiteness, is subjected to a particular unitary transformation $\Theta$ before being fed into QCNOT. $\Theta$ should really be thought of as an element of the projective special unitary group $PSU(2),$ but we can be slightly redundant and view it as an element of $SU(2).$ If we identify $SU(2).$ with the three-dimensional sphere $S^3,$ then we may use the Hopf fibration $S^3\rightarrow S^2$ to twist (i.e., rotate) a given transformation in its Hopf fiber, which is a circle.   In this case, $\Theta$ is given by twisting the identity matrix $I_2$ by $45.5$ degrees in its Hopf fiber.

The image was created using the software Fractal Domains.  It shows a portion of the complex plane, viewed as a coordinate chart on the Riemann-Bloch sphere, which may be identified with the space of single-qubit states.   The center of the chart is at the origin, and the top right corner is at roughly $4.85+i.$  For each single-qubit state $z,$, the pair $(Uz,z)$ is fed into the QCNOT gate and the procedure is iterated using the target output qubit.   Note that although we cannot create a single device to duplicate every possible qubit state $z,$ because of the no-cloning theorem, we can still choose any particular state we like and create as many copies as desired.  The colors represent different orbits of a representative region of the plane.  The chaotic behavior arises from the nonlinearity of the composite operation $z\mapsto(Uz,z)\mapsto \mbox{QCNOT}(Uz,z).$