Brief note on concurrent spacetime foam

In quantum gravity, the notion of a spacetime event (point) breaks down nearby the Planck scale. We would like to replace the notion of a point with the notion of a “smeared” point, or more precisely, an open set of concurrent processes. The spacetime foam would be represented by a coherent superposition of sets of processes that act concurrently, and the reason that they act concurrently lies in the fact that they superpose each other at certain extent. (Note that such a superposition is not in space, because the processes themselves should represent a relational framework in which spacetime emerges at the classical level). It is a superposition constrained by causality. Causality should somehow represent the local “shared resource” among the coherent set of processes. That is why they act concurrently: they are causally constrained. A quantum state of a volume spacetime can only evolve as the result of the combined act of the fundamental processes that compete for a common resource: causality itself.

Each set of concurrent processes can be represented by a ditopological[*] space, and each ditopological space is partially mapped into some other because of the local causality constraint. The ditopological spaces and their common (causal) superpositions maps (the abstract shared resources) should have a combined effect of correlations and anti-correlations among processes (that is, allowed and forbidden regions in the ditopological spaces at their common superpositions), and these correlations/anti-correlations should have a correspondence with, for instance, a discrete spectrum of the volume operator, expected intuitively for a quantum spacetime foam (and calculated precisely in LQG).

[*]Some references

Corfield’s post on Computer Science and Physics

David Corfield over at n-Category Cafe posts on concurrency.

First steps in Quantum Concurrency

Let us attempt a somewhat primitive reinterpretation of a quantum state as a set of processes.

You can see a process as a function that gives some output from a given input. It’s somewhat like an unitary operator. A given particle state evolves to another as a combined result of a set of many processing, “internally active elements”. Imagine a qubit. A set of quantum states encompassing any possible state between |0> and |1> is here what I call a set of “fundamental processes”.

Internally, a lot of activity is happening to the qubit. The processes are not simply independent agents, but active elements which share finite mutual “resources”. I’m not sure at the present stage what to make of these “shared resources” in such a reinterpreted quantum theory (in LQG, they could be seen as the edges of a spin network: nodes *share* common edges representing spin, so the evolution of spin network states could be seen as a result of how these processes act and share resources, and in the present case is to maintain gauge invariance). In other words, the processes do not act completely independently, but are concurrent in the sense that they need to access some shared resources in order to evolve.

I’m not sure what to do with when you observe the system in such a framework.

Imagine now a n-dimensional space in which every orthogonal axis represents a process. Every point along an axis is a representation of a given quantum state (input) evolving to another (output). For instance, in one axis you could set up the following “scheduling”:

|0> -> |1/sqrt(2)> -> |1> -> |0> -> …

in another axis, this one:

|1/sqrt(2)> -> |1> -> |0> -> |1/sqrt(2)> -> |1> -> …

and so on, so you see there is a quite a large number of possible schedules for the qubit. But, say, two processes could not be at the same “time” sharing the same resources: this translates to some constraint that represents the forbidden usage of (or action upon) the same common resource by different processes which are competing at the same “time” (here, “time” is also to be interpreted in some partially ordered sense).

All possible scheduling (histories) of each of the processes form, combined, a directed topological manifold that encodes *all* the possible histories of a given particle. “Directed” in the sense that there is a local partial order structure imposed on the manifold as the quantum states evolve (a direction of “time”). A point in this manifold represents a superposition of processes (state functions). But since the processes that evolve quantum states must share common resources, there are forbidden regions on the manifold because the processes cannot access the same resource at the same “time”. So there are natural constraints that must be obeyed, and these constraints determine a topological, typical signature of the quantum system in question.

These constraints (that actually forbid the system to go into some kind of “deadlock”) could be seen as correlations/anti-correlations between quantum states, thus providing an interpretation of why energy levels of a harmonic oscillator are quantized, for instance.

There is an emergent field joining topology and concurrency theory — “di”topology — that study these various ditopological manifolds, which carry this extra structure (“di”rection).

It turns out that the idea seems to be easier to grasp when you include gravity. The reason comes from the fact that the scheduling of concurrent processes can be described, as I said, in topological terms by a manifold with a local partial order, a ditopological manifold. And pictorially, spin networks (or spin foams, for what is worth) seem adequate to fit this idea in a more immediate sense because of causality issues.

But that is another story.

Ubiquitous Concurrency

In our world, concurrency is everywhere. It is so ubiquitous and ordinary in our daily lives that we really do not give any conscious importance to it.

It is clear that, in a classical setting, Nature operates its constituents concurrently. Animals and plants can change, interact, evolve, disturb and be disturbed in a completely concurrent manner. For instance, living creatures can operate many sensory organs or chemical reactions independently. Inanimate objects can be acted upon concurrently and physical phenomena also present concurrent behaviour. For example, when a ball falls in a gravitational field, as it changes its kinetic energy, many other processes are allowed to happen along its worldline (e.g., it may fall burning, etc).

All this seems ridiculously obvious. In engineering and computational design problems, the obviously concurrent behaviour of the external world, including how information is processed and transmitted, must be worked out and anticipated quite explicitly. Sequential modeling can be viewed as a special case of concurrent modeling, and it is much easier to implement. But as computational facilities evolved, a shift of paradigm from sequential to concurrent modeling started to gain importance, and we humans started to have to “think” and develop techniques to deal with concurrency. We were very used to the sequential paradigm for some time.

Physics, on the other hand, seems to have taken for granted the concurrent behaviour of the world as an obvious consequence of dynamical laws, which purpose is to describe how a given configuration of self-interacting constituents evolves, given a time span, to another one.

Indeed, the parallel evolution of a given set of independent physical elements is a very uninteresting case of concurrency, so it is no surprise that physics never considered this as a fundamental problem at all. But when the elements must interact with each other, things get really interesting. In a concurrent language, we would say that the elements would have to share some “common resource” (like a mediating fundamental field). That is when concurrency is also interesting in its whole complexity.

Is the quantum world concurrent as well? I would say yes, but in my vision, something fundamental about how nature operates its concurrent aspects emerges in the quantum regime.

If concurrency is some deeply inherent property of the world, then it is not a consequence of dynamics, but the fundamental cause of it.

I believe that quantum gravity is the best instance in which such a change of paradigm can be worked out. Quantum gravity should emerge from some internal, concurrent description of the state space.

I will opportunely post on why I believe this is the case.