Monday, October 09, 2006

Emerging from the Noise

In “Towards Gravity from the Quantum”, Fotini Markopoulou (her homepage can be reached here if you click through on 'Faculty' then her name) describes work on a new approach to quantum gravity. This effort differs from other background independent approaches in that instead of seeking to quantize spacetime geometry, one starts with a microscopic description of a pre-spacetime quantum theory. Then, from this foundational theory, emerge “dynamically-selected excitations, that is coherent degrees of freedom which survive the microscopic evolution to dominate our scales.” The filter which enables emergence is borrowed from a process studied in quantum information theory. The plan, finally, is to define spacetime in terms of the interactions of these emergent excitations.

The pre-spacetime microscopic theory uses the formalism of quantum causal histories (“QCH” – which I introduced at the end of my prior post). QCH is “a locally finite directed network (graph) of finite-dimensional quantum systems.” There have been several ways QCH has been used to approach quantum gravity, which are reviewed in the paper. The emphasis is on this new approach which develops it as a “quantum information processor, which can be used as a pre-spacetime theory.”

In QCH (described in section 1.2 of the paper), the geometric paths followed in the graphs are constrained to be finite. Then quantum systems are associated with the graph. As I mentioned in the last post, early efforts attached Hilbert spaces to the relations/edges on the graphs. More recently, QCH uses a description of the evolution of an open quantum system called a “completely positive map” (alternatively, a “quantum channel”) to define the relations/edges between the vertices, which are (finite dimensional) Hilbert spaces and/or matrix algebra of operators acting on a Hilbert space.

So, for every edge on the graph, there is a completely positive map or quantum channel (an evolution from one vertex/Hilbert space to another).

I will skip over some of the details of this construction, but it looks like the other important part is to impose constraints on the evolution that preserve local causality in the spirit of the original causal set theory. Specifically, a source set for a given path(s) maps uniquely to a range map, thus imposing local causality when the edges are viewed as causal relata.

So, we have in QCH a (relatively) simple structure of open quantum systems, which form a local causal network.

Markopoulou describes several ways this structure has been used. It can be the basis of a discrete algebraic quantum field theory (section 1.3). More to the point, another way is to use it to create a path toward quantum gravity by taking quantum superpositions of the geometries defined by the model (section 1.4). This methodology has been part of the development of spin foam models as well as Causal Dynamic Triangulations (“CDT” -- which will be the subject of my next post). To make a causal spin foam model, you adapt QCH to spin network graphs, which are used in loop quantum gravity. Then you obtain a path integral of the superpositions of all constrained members of the set.

She discusses next the problems spin foam models have had recovering spacetime at lower energy scales (she notes briefly that the CDT model has had more success due to its unique features). The background independence and difficulty of implementing dynamics makes it difficult to apply coarse-graining techniques used elsewhere to recover a sensible low energy physics from these models.

The idea she wants to focus on builds on the idea that instead of summing over quantum geometry and trying to coarse-grain directly, one can first look for long-range propagating degrees of freedom that arise from the quantum systems and look to reconstruct the geometry from these.

She borrows from quantum information processing theory the notion of a “noiseless subsystem” in quantum error correction: a subsystem protected from the noise, usually thanks to symmetries of the noise (section 1.5). The analogy is that the “noise” is simply the fundamental microscopic evolution and the existence of a noiseless subsystem means a coherent excitation protected from the microscopic evolution. So we split the paths (quantum channels) into subsets A and B, where B is noiseless. B is an emergent subsystem (similar to the idea of a “decoherence –free” subsystem). I didn’t follow all the formalism describing noiseless subsystems. But I infer it’s a topic well discussed in quantum information theory.

Next, she investigates the idea that we have an emergent spacetime if these emergent coherent excitations behave as though they are in a spacetime. This subset of protected degrees of freedom (or coherent excitations) and their interactions will need to be invariant under Poincare transformations.

In preparation for the next section she works through some formalism to show more clearly how the sum over causal histories does include some of these coherent excitations. These turn out to be “braidings” of graph edges which are unaffected by the noise of evolution.

In section 1.6, Markopoulou describes her model whereby QCH is a pre-spacetime quantum information processor from which degrees of freedom emerge; interactions of these are theorized to be the events of our spacetime. (There are no separate gravitational degrees of freedom to be quantized).

She argues this model demonstrates a deeper form of background independence compared to other theories since the microscopic geometric degrees of freedom don’t survive as part of the description of emergent spacetime

The QCH quantum channels (graph edges) referred to before should now be viewed as information flows between quantum systems (vertices) with no reference to having spatio-temporal attributes at all.

Then, one analyzes the emergent coherent degrees of freedom (noiseless subsystems) and their interactions. She proposes that these can constitute an emergent Minkowskian spacetime if they are Poincare invariant at the relevant scale.

Important to this possibility is that the noiseless subsystems are not localized, they exhibit a global symmetry which allows them to be emergent at larger scales. They constitute their own “macro-locality”, which is unrelated to the original microlocality of the QCH graphs. Markopoulou outlines the promise and possible shortcomings of this approach, and says more is work is underway to develop these ideas. Importantly, she and her collaborators have not gotten gravity (Einstein’s equations) back out yet. On the other hand, when he mentioned this work in his book, Lee Smolin seemed excited by the idea that the emergent “particles” from this kind of approach might have the potential to lead to particle theory in addition to gravity (most background independent approaches to quantum gravity set aside the problem of matter fields at least initially, in their pursuit of gravity).

She finishes with a note on time:
Just as the emergent locality has nothing to do with the fundamental micro-locality, time and causality will also be unrelated macro vs. micro. So, the theory “puts” in time at the micro-level (via its causality constraints), but emergent spacetime will have no preferred time slice –as required in general relativity.

I think this distinction between microscopic and macroscopic time has interesting implications for thinking about causality. It is suggestive that a theory like Markopoulou’s implies that while causality is fundamental at the local level, the macroscopic “laws of physics” are emergent regularities.

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