Infinitesimal idealization, easy road nominalism, and fractional quantum statistics

Abstract

It has been recently debated whether there exists a so-called “easy road” to nominalism. In this essay, I attempt to fill a lacuna in the debate by making a connection with the literature on infinite and infinitesimal idealization in science through an example from mathematical physics that has been largely ignored by philosophers. Specifically, by appealing to John Norton’s distinction between idealization and approximation, I argue that the phenomena of fractional quantum statistics bears negatively on Mary Leng’s proposed path to easy road nominalism, thereby partially defending Mark Colyvan’s claim that there is no easy road to nominalism.

Appendix: The configuration space approach to fractional quantum statistics

Let the configuration space of a particle in d-dimensions be ${\mathbb {R}}^{d}$ (where ${\mathbb {R}}$ is the set of real numbers) so that the position of the particles is given by an element of the space $x\in {\mathbb {R}}^{d}$. Consider N such identical particles. The configuration space approach argues that the appropriate configuration space Q for N identical particles is not the Cartesian product of the single particle spaces, ${\mathbb {R}}^{Nd}\equiv {\mathbb {R}}^{d}\times \cdots \times {\mathbb {R}}^{d}$ (N times), as one would expect if the particles were not identical. Instead, since the particles are identical, configurations that differ only by a permutation of particles ought to correspond to the same physical state. For the simplest case $d=1N=2$, this means that the two configurations $\left( {x_1,x_2 } \right) =\left( {1,2} \right)$ and $\left( {x_1,x_2 } \right) =\left( {2,1} \right)$ actually represent the same state, and so we must divide out such permuted configurations. In other words, we move from the entire space ${\mathbb {R}}^{Nd}={\mathbb {R}}^{2*1}={\mathbb {R}}^{2}$ of the two-dimensional plane to consider only half the plane. In the context of the general case, this corresponds to considering the quotient (or identification) space that one attains by diving out the permutation group $S_N$: $Q=\frac{{\mathbb {R}}^{d}}{S_N }$.

Next, we’ll want to excise the set $\Delta$ of diagonal points in ${\mathbb {R}}^{Nd}$, which represent all points where the particles coincide. For the $d=1N=2$ case, this means that we must excise points of the sort: $\left( {x_1,x_2 } \right) =\left( {x,x} \right)$, where $x_1 =x_2.$ If we do not excise coincidence points (known as singular points) the resulting configuration space will not have a structure rich enough to represent fermions (or anyons).

We are left with a configuration space for N identical particle in d dimensions, in which we have divided out the action of the permutation group and excluded diagonal points: $\frac{{\mathbb {R}}^{d}-\Delta }{S_N }$. The exclusion of the diagonal point implies that the space in not simply connected and I denote this new configuration space with $\mathop {\tilde{Q}}$. We must now undertake QM on a multiply connected space. In order to do so I will follow Morandi (1992, pp. 114–144) closely.

Ordinary QM (in the Schrodinger picture) on simply connected regions represents the states of physical systems by complex wave functions $\Psi$ which are elements of a Hilbert space ${\mathcal {H}}$ of square-integrable functions on the configuration space Q over the field of complex numbers ${\mathbb {C}}$: $\Psi \in \mathcal{H}=L_{\mathbb {C}}^2 (Q)$, where ${\mathcal {H}}^{N}=L_{\mathbb {C}}^2 \left( Q \right) \times \cdots \times L_{\mathbb {C}}^2 \left( Q \right) \cong L_{\mathbb {C}}^2 \left( {Q^{N}} \right)$ for N identical particles. In order to extend ordinary QM to multiply connected regions we need to appeal the topological notion of a universal covering spaceQ of $\mathop {\tilde{Q}}$. That is to say, for any topological space, including ones that are not simply connected such as $\mathop {\tilde{Q}}$, we can construct a universal covering space Q that is simply connected (so that ordinary QM applies) with a covering projection map: $\pi {:}\,Q\rightarrow \mathop {\tilde{Q}}$ (where $q\in Q$ and $\mathop {\tilde{q}}\in \mathop {\tilde{Q}})$.

Since all the physical information in QM is contained in the squared modulus of the wave function (i.e. the probability density) $\left| {\Psi }\left( q \right) \right| ^{2}$, we will want this quantity to be projectable down to $\mathop {\tilde{Q}}$ for any $q\in Q$ in the sense that $\left| {{\Psi }\left( q \right) } \right| ^{2}$ will depend only on a point $\mathop {\tilde{q}}=\pi \left( q \right)$ in the multiply connected configuration space $\mathop {\tilde{Q}}.$ If this condition is satisfied we say that we have a projectable quantum mechanics, and we will have succeeded in extending ordinary QM to a multiply connected region. To that effect, consider an arbitrary point $\mathop {\tilde{q}}\in \mathop {\tilde{Q}}$, let $\mathop {\tilde{C}}$ be a closed curve beginning and ending at $\mathop {\tilde{q}}$, and let [$\mathop {\tilde{C}}$] be the corresponding (first) homotopy class of curves based at $\mathop {\tilde{q}}$ so that [$\mathop {\tilde{C}}$]$\in \pi _1 \left( {\mathop {\tilde{Q}} ,\mathop {\tilde{q}}} \right)$.³³ Further, let $q=\pi ^{-1}\left( {\mathop {\tilde{q}}} \right)$ be any point in the fiber over $\mathop {\tilde{q}}$. The homotopy lifting theorem says that all curves $\mathop {\tilde{C}}$ in [$\mathop {\tilde{C}}$] are lifted to a curve C in Q beginning at q and ending at some point $q^{\prime }$ that is also in the fiber over $\mathop {\tilde{q}}$ (so that $\pi \left( {{q}'} \right) ) = \mathop {\tilde{q}}$ and $q^{\prime }\in Q)$.³⁴ Denote this result by $q^{\prime }=\left[ \mathop {\tilde{C}} \right] \cdot q$ and recall that $\gamma$ represents a phase factor (although, as of now, we have placed no constraints on the form of $\gamma )$. We then have the following two central theorems (Morandi 1992, pp. 119–120):

Theorem 1

Projectable Quantum Mechanics are obtained if and only if the wave functions on the universal covering space obey the boundary conditions:

$$\begin{aligned} \Psi \left( {\left[ {\mathop {\tilde{C}} } \right] \cdot q} \right) =\gamma \left( {\left[ {\mathop {\tilde{C}}} \right] } \right) \Psi \left( \hbox {q} \right) \,\hbox {for}\,\hbox {all}\,q\in Q\\ \hbox {Where} \left| {\gamma \left( {\left[ {\mathop {\tilde{C}}} \right] } \right) } \right| =1\,\hbox {for}\,\hbox {all}\,\left[ {\mathop {\tilde{C}}} \right] \in \pi _1 \left( {\mathop {\tilde{Q}}} \right) \end{aligned}$$

This means that wave functions on the universal covering space at different points (e.g. q and $q^{\prime })$, but on the same fiber above some point in the multiply connected configuration space (e.g. $\mathop {\tilde{q}})$, can differ at most by a phase factor of modulus 1. Moreover, since the universal covering space is simply connected, the wave functions must be single valued. We then get:

Theorem 2

The map $\gamma {:}\,\pi _1 \left( {\mathop {\tilde{Q}}} \right) \rightarrow U\left( 1 \right)$ by $\left[ {\mathop {\tilde{C}} } \right] \rightarrow \gamma \left( {\left[ {\mathop {\tilde{C}}} \right] } \right)$ is a one-dimensional unitary representation of $\pi _1 \left( {\mathop {\tilde{Q}}} \right)$.

Accordingly, we get our main result: in order to ascertain what type of phase factor is gained by a wave function when it is permuted—with the corresponding available quantum statistics—we must enquire into the one-dimensional unitary representation of the fundamental group of the configuration space of the system. Recall that in the case of N identical particles in d dimensions we have:

$$\begin{aligned} \mathop {\tilde{Q}} =\frac{{\mathbb {R}}^{d}-\Delta }{S_N } \end{aligned}$$

It has been shown by Artin (1947), Fadell and Neuwirth (1962), and Fox and Neuwirth (1962) that the fundamental group for the two- and three-dimensional cases are given by:

$$\begin{aligned} \pi _{1} \left( {\mathop {\tilde{Q}}} \right) =B_N \hbox { for } d=2\\ \pi _{1} \left( {\mathop {\tilde{Q}}} \right) =S_N \hbox { for } d=3 \end{aligned}$$

where $S_N$ is the permutation group and $B_N$ is the braid group. Moreover, as stated previously, the one-dimensional representation of $S_N$ is $\gamma =\pm \,1,+\,1$ for bosons and $-\,1$ for fermions, while the one-dimensional representation of $B_N$ is $\gamma _{\left( \theta \right) } =e^{i\theta }$ where $0\le \theta \le 2\pi$ so that the exchange phase can take on a continuous range of factors allowing for bosons, fermions and anyons.