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II Downward Causation and the Sciences

The Use of Downward Causation in Condensed Matter Physics

Stewart J. Clark and Tom Lancaster

Introduction

When atoms are brought together to form a solid, electrons interact quantum mechanically and new states of matter emerge. These include ordered states, new excitations and unusual “topological” objects (Altland and Simons 2006; Chaikin and Lubensky 1995). Modern condensed matter physics is the investigation of this exotic, emergent world of quantum matter and provides an insight into the Universe just as fundamental as the study of elementary particles or black holes. Downward causation is invoked less in physics than in the philosophy of physics, although, in the debate surrounding emergence, it is often raised as a possible explanation for the condensation of complex states of aggregative matter (Ellis 2012).

In the context of condensed matter physics, an explanation based on downward causation might be framed as follows: microscopic constituents of matter interact at a lower level quantum mechanically via a relatively simple set of interactions, such as the Coulomb repulsion of similarly charged electrons in the presence of the oppositely charged atomic nuclei. It is, at present (at the very least), an impossible task to follow the details of all of these interactions and their consequences, because in a macroscopic sample (say, a gram or so in weight) it would typically involve keeping track of N и 1023 particles and their pairwise interactions with all of the other particles. However, it is often found that the behaviour of the system can be well described by variables that result from averaging over the behaviour of fine-grained degrees of freedom such as the particles’ momenta or magnetic moments.1 This so-called course- gaining routine of averaging over degrees of freedom that change fairly slowly in space and time results in the macroscopically defined variables operating at a higher level. (A more technical definition of course graining is replacement of microscopic degrees of freedom by average variables on an expanded length scale, see Chaikin and Lubensky (1995).) This leads to the consideration of two levels of behaviour: the lower, microscopic level at which electrons operate and the upper, macroscopic level, operating on an expanded length scale, at which we make many of our measurements. In physics, the coarse-grained variables are often viewed as causally interacting downwards on individual microscopic constituents of the matter. This may be described in terms of providing boundary conditions or constraints (such as the walls of a container containing the atoms of a gas) or more directly (such as the effective field theories described below). This is the physics of downward causation and we examine it in this essay, giving several examples from the field of condensed matter physics.

The general theme of our examples is as follows: calculating the properties of a macroscopic system often involves coupling a microscopic subsystem with a macroscopic reservoir, and this result in fields or other boundary conditions that dictate behaviour. By identifying this as downward causation, we are, in the sense stressed by Blundell (2016), tracing out a causal path for our own convenience in describing and understanding the complex behaviour of many-particle systems. Moreover, we shall see below that the causal links we find are more complicated than those that would simply imply a top-down process. However, with this in mind, we believe that the scheme that we describe here, which does not invoke any mysterious higher-level physics, does cast light on the use and limitations of notions of downwards causation in physics, particularly in the emergence debate.

Below, we describe in detail some common examples of how these ideas are applied and we discuss their use in identifying novel behaviour in condensed matter, their shortcomings and their place in a hierarchical series of approximations known as a perturbation series. We then turn to density functional theory, which is a first principles method of describing real systems of condensed matter, whose practical implementation requires many concepts involved in the discussion of downwards causation.

 
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