## Introduction

The species–area relationship, or SAR (Preston 1960, 1962; MacArthur & Wilson 1963; May 1975; Connor & McCoy 1979; Rosenzweig 1995; Tjorve 2003), characterizes the increase in the observed number of species with increasing sample area, and has been referred to as the closest thing to a law in ecology (Lawton 1999). The SAR has played a seminal role in understanding the generation and maintenance of biodiversity, and forms a crucial basis for estimates of extinction due to habitat loss (May *et al.* 1995; Thomas *et al.* 2004). A number of different shapes have been proposed for the relationship (Rosenzweig 1995; Tjorve 2003), but one of the most generally accepted SARs falls into three distinct phases, with the different phases applying as sample area is increased from local to continental scales (Preston 1960; Williams 1964; Brown 1995; Rosenzweig 1995; Hubbell 2001). This triphasic SAR has an inverted S shape (Williams 1964), so that there is a steep increase in species at very local scales, followed by levelling off at intermediate scales and an accelerating increase in species number with area at the very largest, continental scales. The intermediate phase has commanded particular attention, and it has been proposed that over these scales species number increases as a power of area sampled, following the power law curve introduced by Arrhenius (Arrhenius 1921). This power law behaviour has been identified across a broad range of geographical regions (Rosenzweig 1995; Drakare *et al.* 2006) and across the tree of life (Green *et al.* 2004; Horner-Devine *et al.* 2004; Green & Bohannan 2006), but the reasons for the ubiquity of the power law SAR, and the forces driving the value of its exponent have yet to be determined definitively from first principles.

One of the earliest approaches to understanding the SAR was introduced by Preston (1960, 1962), who demonstrated that if the distribution of species abundances followed a lognormal distribution, then the number of species present in a random sample increases as a power law with increasing sample size, with the power law exponent close to 0.25. May later considered a wider range of possible species abundance distributions than Preston (May 1975), but found that the exponent of this power law would still be within a narrow range, typically between 0.15 and 0.4, consistent with a wide range of empirical results. The weakness of this framework is that real communities tend to exhibit spatial clustering (Plotkin *et al.* 2000b), so that individuals are more likely to be found near their conspecifics, violating the assumption that a spatial sample is equivalent to a random sample. More recent top-down approaches have made a range of different assumptions for this spatial clustering (Harte *et al.* 1999; Martin & Goldenfeld 2006; Harte *et al.* 2009), to test its impact on the SAR, and one influential example is the assumption of self-similar spatial aggregation of individuals (Harte *et al.* 1999). However, spatial clustering appears not to be self-similar with sufficient generality (Plotkin *et al.* 2000a) to provide a universal explanation for the shape of the SAR.

An alternative strategy, avoiding *a priori* assumptions for the distribution of species abundances or the spatial clustering of individuals, is to model a community from the bottom-up. This means that we specify some mechanistic rules for the behaviour of individuals, and then see what macroecological patterns emerge. An example of this approach is the neutral biodiversity theory introduced by Hubbell (2001), building on earlier work (Watterson 1974; Caswell 1976), and extensively developed (Chave & Leigh 2002; Volkov *et al.* 2003; Chave 2004; Etienne 2005; Etienne *et al.* 2007; Rosindell & Cornell 2007; Aguiar *et al.* 2009; O'Dwyer *et al.* 2009) in recent years. Neutral communities are idealized approximations where patterns are assumed to be primarily driven by the effects of stochasticity, but the present lack of a neutral prediction for the SAR reflects an outstanding mathematical problem in theoretical ecology: the combination of stochastic dynamics with a continuous spatial landscape (Durrett & Levin 1994; Bolker & Pacala 1997). Progress in dealing with stochasticity in continuous space has been limited by the lack of a practically useful, flexible mathematical framework, with the consequence that it has not so far been possible to derive a theoretical, bottom-up prediction for the SAR.

Our goal is to overcome precisely this problem, and quantum field theory provides the perfect set of tools. Field theory was first developed as a model for particle physics (Schwinger 1958), where collisions of electrons and photons are expressed in terms of a theory of fluctuating electromagnetic fields. The same formalism has been applied to solve many-body problems in numerous fields, including the theory of phase transitions and critical phenomena, where the fields are reinterpreted as fluctuations in the density of a gas, or as fluctuations in the magnetization of a ferromagnetic material at a critical point. The central tool used to solve these problems is a moment generating functional, or partition function, which summarizes all the observable spatial patterns in these systems, and the challenge of solving a field theory is in solving for this partition function (Ryder 1996; Zinn-Justin 2002). Our key step is the introduction of a partition function for spatial ecology, illustrated conceptually in Fig. 1. Our methods follow earlier work in size-structured community assembly (O'Dwyer *et al.* 2009), and our biogeographical field theory provides a very general framework to make calculations for discrete individuals undergoing stochastic processes on a continuous spatial landscape. This flexibility also opens up the possibility for a more comprehensive understanding of spatial community assembly, with the potential to break neutrality and test which biological processes have the most impact on the macroscopic patterns we observe in nature.

In this article we begin by deriving a spatially explicit generalization of neutral biodiversity theory, on a spatially discrete, grid-like landscape, as a first step towards building a framework for spatial ecology on a continuous landscape. We then define the theory in continuous space by introducing a partition function, and we find that the partition function satisfies a functional differential equation, analogous to the Schwinger-Dyson equations of quantum field theory (Zinn-Justin 2002). Having derived the defining equation for our model, we solve for the SAR, and we also derive the expected total number of individuals as a function of area, and the turnover in species composition with spatial separation, relating these quantities to our prediction for the SAR. We conclude by discussing the significance of our results for predicting spatial patterns of biodiversity, and detail the ways in which our model can be generalized to integrate non-neutral approaches to community assembly.