Duncan 2013 -- The Conceptual Framework of Quantum Field Theory

Created: 11 Jun 2013

Modified: 23 Jul 2020

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Motivation

I’m always interested in the conceptual structure of scientific theories, because in the social sciences we’re not good at putting theory together yet. We mainly borrow and adapt, but there are reasons why a specific theory performs well in a specific discipline to explain specific phenomena. In particular, I’m interested in how physical theory combines dynamical processes (which generate change through time in system state), with constraints that winnow down the possible number of models and theories to those which actually describe the world. And how physical theories often display different observable behavior when observed at different length or time scales.

All of these are issues we face in constructing evolutionary theory for human cultural variation.

Why quantum field theory? Because field theories are useful in that they describe the summed contributions of many locally interacting processes, to observables that can be measured at points in space and time. Field theories also describe the dynamics of the entire ensemble, given those local interactions. And quantum field theories have clear mechanisms by which observational scale is incorporated into models to give “effective theories” at particular length/energy scales. So despite its mathematical complexities, QFT is an excellent paradigm of the type of formal theory structure that might be useful for us in the evolutionary social sciences. It’s worth noting that Dodd and Ferguson (2009) employed this approach to population and epidemiological models, which is what stimulated my thinking here.

Duncan’s book is unlike most QFT text in focusing specifically on the conceptual structure of the theory, rather than just teaching the models and methods, or the history of its development.

Components of QFT

Duncan notes that QFT is a combination of three “ingredients”:

1. Dynamical theory – in this case, quantum mechanics. Specifies how to describe the state of objects in the world, how to interpret the formalisms describing their state and changes in that state, and how state can change.
2. Symmetries – in this case, Lorentz invariance and various gauge symmetries. The role of symmetries is to explain why some state and evolution functions are allowable in a theory, as opposed to the infinity of possible functions. The symmetries of the Poincaré group, for example, generate the plethora of field equations one actually encounters (e.g., Dirac, Klein-Gordon), and provide the ingredients for constructing “allowable” Hamiltonians for the dynamical evolution of a system.
   
3. Clustering or locality – in this case, the insensitivity of local processes to distant parts of the environment. QFT follows “Einstein causality,” in which causal influence can only travel within a light cone, and thus local processes are limited, and do not have to take into account global system state for every calculation.

Clustering is both a precondition for successful experiments, since we can never measure everything about the world, and a constraint on possible models. Duncan points out that although #1 and #2 above are widely considered to be the “ingredients” of modern fundamental physics, it’s really #3 that sharpens down the possible infinity of models consistent with these (i.e., all the “unphysical” models), to a single “physical” model which agrees with experiment.

Applications

Why am I fascinated by this – the conceptual structure of a theory very different than what we study in the evolutionary biology of culture?

Part of it lies in the obvious parallels between some of the mathematical constructs. Evolutionary theory is an n-body interacting system, even if our simplest deterministic models in undergraduate population genetics do not reveal this. Evolution displays critical behavior, phase transitions, frustration and thus spin-glass-like behavior, and can sometimes be formulated in models straight out of statistical mechanics (and increasingly is).

But the deeper reason is that simple models of evolution (and cultural transmission) are too simple – too diffusionist – to capture the richness of behavior and the structure we see in our empirical data. What we lack are constraints that give us complexity. Perhaps the ways in which we interact in realistic social networks give us symmetries that constrain how the diffusion of traits can happen. Perhaps the modular structure of cultural information, skills, and objects give us symmetries that constrain how the diffusion of traits can happen. Taken together, perhaps these two sets of symmetries break up the otherwise smooth – like concentric ripples on a pond after throwing a stone – and wave-like diffusion models, and yield instead the complex patterns we see in real populations.

Perhaps. There are certainly other ways to talk about this issue of theory structure, and those with less of a “physics fascination” than I will certainly prefer them. I find physical theory a rich source of ideas, not about content, but about the conceptual structures we might use as we get beyond the “toy models” of our early explorations and try to build increasingly realistic theory about the evolution of culture.