As discussed in my
last post, Lee Smolin concluded that the most promising models of quantum gravity include causality as a fundamental feature. In the next couple of posts, I’ll outline some notes from my reading regarding how this idea has been developed in several research programs. Below are brief notes on causal set theory and quantum causal histories. As always, I will make mistakes in my efforts to summarize some of the material, so please check out the papers themselves if you have interest.

Causal Sets

A very nice paper (actually lecture notes) by Rafael Sorkin was helpful in summarizing causal set theory and its application to spacetime (hat tip: Dan Christensen’s page of quantum gravity links).

As he tells the story, the idea of seeing the causal order of relativistic spacetime as its most fundamental aspect was present in the earliest days of Einstein’s theory. There were early efforts which made some headway in managing to recover the geometry of a flat four-dimensional spacetime from nothing more than the underlying point set and a timelike vector among points. The fact that the points in the manifold constitute a continuum proved to be an obstacle to this methodology. But of course the development of quantum mechanics gave good independent motivation to consider discrete models of spacetime geometry. In a discrete model, causal order plus the counting of the discrete volumes offers the potential of recovering spacetime geometry.

According to Sorkin: “The causal set idea is, in essence, nothing more than an attempt to combine the twin ideas of discreteness and order to produce a structure on which a theory of quantum gravity can be based.”

The basic idea of a causal set (or “causet”) is easy enough for a layperson to understand. Sorkin defines (p.6) a causal set as a locally finite ordered set: A set endowed with a binary relation possessing 3 properties – transitivity, irreflexivity, and local finiteness (which implies discreteness). The combination of transitivity and irreflexivity rules out cycles where the timelike vector can loop back to its beginning. The set could be depicted as a graph (with the elements as vertices and the relations as edges) or as a matrix, or it can be helpful to think of it as a “family tree”. The relation between elements is one of “precedes” or “lies to the past of”, etc.

Sorkin goes on to summarize work which looks to recover the elements of spacetime by analyzing the kinematics of a causal set. As an example, he says it is known that the length of the longest chain “provides a good measure of the proper time (geodesic length) between any two causally related elements of a causet that can be approximated by a region of Minkowski space.” He discusses how to reconstruct other constituents of Minkowski space (M4) from its causal order and volume-elements. He then talks about recovering more geometric information, such as dimensionality. This work seemed to be less definitive in its results.

Next, he discusses routes toward causal set dynamics, and hopefully a quantum causal set dynamics. Sorkin describes an idea where you create a classical stochastic evolution of the set in a (global) time direction, and modify the results to create a quantum dynamics (this part I don’t yet understand).

Some cosmological applications are discussed, the most exciting of which (mentioned by Smolin in his book) is a correct order of magnitude prediction for the cosmological constant which emerges from the model. Given these results, Smolin expects the causal set research program to be an ongoing part of the background-independent approaches to quantum gravity.

Quantum Causal Histories

Causal Set theory is discrete, but at its roots not distinctively quantum (as I read it).

Quantum Causal Histories (QCH) is a model which welds quantum features to the causal elements. As described by Fotini Markopoulou (her home page is reached by clicking through to "faculty" then her name on this Perimeter Institute link) in this neat little 6-page review, the idea is to “quantize” the causal structure by attaching Hilbert spaces to the events of a causal set. “These can be thought of as elementary Planck-scale [quantum] systems that interact and evolve by rules that give rise to a discrete causal history.

Actually, she discusses the fact that the finite dimensional Hilbert spaces, following the rules of quantum mechanics, would not in general respect local causality if attached to events. Instead she says one should attach them to the causal relations (edges on a graph), with operators put on the events (nodes or vertices). Then the quantum system evolution respects local causality. There is an intuition here also that an event denotes a change, and so fits with the notion of an operator. Note also when you link quantum systems together like this, there is not a global Hilbert space or wavefunction for the whole system: if these building blocks are built up into a cosmological model, there would be no wavefunction for the universe, but a collection of local ones. By the same token, there is no observer of a global quantum system outside the universe, all the observers on the inside.

There are different ways to link the spaces up (different kinds of graphs). One way is to use the spin networks, as in loop quantum gravity. When spin networks are used in a model of the causal evolution of quantum spatial geometry, the nodes of the spin network graph are the events in a causal set. Markopoulou details this notion and other ways QCH can be modeled. I should note that in later papers, the QCH structure appeared to be refined further; for instance this paper refers to the substituting of matrix algebras of operators for the Hilbert spaces.

This short paper concludes with references to work underway (this was a 1999 paper) to use QCH as a structure for a quantum gravity model. I will come back to QCH as part of a subsequent post with my notes on a much more recent paper by Markopoulou featuring work mentioned in Smolin’s book.

Causal Sets

A very nice paper (actually lecture notes) by Rafael Sorkin was helpful in summarizing causal set theory and its application to spacetime (hat tip: Dan Christensen’s page of quantum gravity links).

As he tells the story, the idea of seeing the causal order of relativistic spacetime as its most fundamental aspect was present in the earliest days of Einstein’s theory. There were early efforts which made some headway in managing to recover the geometry of a flat four-dimensional spacetime from nothing more than the underlying point set and a timelike vector among points. The fact that the points in the manifold constitute a continuum proved to be an obstacle to this methodology. But of course the development of quantum mechanics gave good independent motivation to consider discrete models of spacetime geometry. In a discrete model, causal order plus the counting of the discrete volumes offers the potential of recovering spacetime geometry.

According to Sorkin: “The causal set idea is, in essence, nothing more than an attempt to combine the twin ideas of discreteness and order to produce a structure on which a theory of quantum gravity can be based.”

The basic idea of a causal set (or “causet”) is easy enough for a layperson to understand. Sorkin defines (p.6) a causal set as a locally finite ordered set: A set endowed with a binary relation possessing 3 properties – transitivity, irreflexivity, and local finiteness (which implies discreteness). The combination of transitivity and irreflexivity rules out cycles where the timelike vector can loop back to its beginning. The set could be depicted as a graph (with the elements as vertices and the relations as edges) or as a matrix, or it can be helpful to think of it as a “family tree”. The relation between elements is one of “precedes” or “lies to the past of”, etc.

Sorkin goes on to summarize work which looks to recover the elements of spacetime by analyzing the kinematics of a causal set. As an example, he says it is known that the length of the longest chain “provides a good measure of the proper time (geodesic length) between any two causally related elements of a causet that can be approximated by a region of Minkowski space.” He discusses how to reconstruct other constituents of Minkowski space (M4) from its causal order and volume-elements. He then talks about recovering more geometric information, such as dimensionality. This work seemed to be less definitive in its results.

Next, he discusses routes toward causal set dynamics, and hopefully a quantum causal set dynamics. Sorkin describes an idea where you create a classical stochastic evolution of the set in a (global) time direction, and modify the results to create a quantum dynamics (this part I don’t yet understand).

Some cosmological applications are discussed, the most exciting of which (mentioned by Smolin in his book) is a correct order of magnitude prediction for the cosmological constant which emerges from the model. Given these results, Smolin expects the causal set research program to be an ongoing part of the background-independent approaches to quantum gravity.

Quantum Causal Histories

Causal Set theory is discrete, but at its roots not distinctively quantum (as I read it).

Quantum Causal Histories (QCH) is a model which welds quantum features to the causal elements. As described by Fotini Markopoulou (her home page is reached by clicking through to "faculty" then her name on this Perimeter Institute link) in this neat little 6-page review, the idea is to “quantize” the causal structure by attaching Hilbert spaces to the events of a causal set. “These can be thought of as elementary Planck-scale [quantum] systems that interact and evolve by rules that give rise to a discrete causal history.

Actually, she discusses the fact that the finite dimensional Hilbert spaces, following the rules of quantum mechanics, would not in general respect local causality if attached to events. Instead she says one should attach them to the causal relations (edges on a graph), with operators put on the events (nodes or vertices). Then the quantum system evolution respects local causality. There is an intuition here also that an event denotes a change, and so fits with the notion of an operator. Note also when you link quantum systems together like this, there is not a global Hilbert space or wavefunction for the whole system: if these building blocks are built up into a cosmological model, there would be no wavefunction for the universe, but a collection of local ones. By the same token, there is no observer of a global quantum system outside the universe, all the observers on the inside.

There are different ways to link the spaces up (different kinds of graphs). One way is to use the spin networks, as in loop quantum gravity. When spin networks are used in a model of the causal evolution of quantum spatial geometry, the nodes of the spin network graph are the events in a causal set. Markopoulou details this notion and other ways QCH can be modeled. I should note that in later papers, the QCH structure appeared to be refined further; for instance this paper refers to the substituting of matrix algebras of operators for the Hilbert spaces.

This short paper concludes with references to work underway (this was a 1999 paper) to use QCH as a structure for a quantum gravity model. I will come back to QCH as part of a subsequent post with my notes on a much more recent paper by Markopoulou featuring work mentioned in Smolin’s book.