Caspar Jacobs

A theory consists of a formalism and an interpretation. The formalism is just a piece of mathematics, and the interpretation tells us what that formalism means. In her book Interpreting Quantum Theories, Laura Ruetsche ([2011]) cashes out interpretation in terms of ‘what the world could be like […] if the theory were true’ (in the words of Bas van Fraassen’s [1991], p. 242). In slightly more formal terms, an interpretation maps each of the theory’s observables to a physical quantity, such as position or spin, and each of the theory’s states to a possible world. In this way, an interpretation delineates a space of worlds, which are called ‘physically possible’: they are ways the world could be if the theory were true.

This account of interpretation entails a distinction between initial conditions and laws of nature. In short, the laws of nature are whatever remains constant across all physically possible worlds. From the perspective of the theory in question, the laws are immutable. The initial conditions, meanwhile, are allowed to vary across the possible worlds. They concern particular matters of fact that are not already determined by the laws of nature—the particular spin or location of a particle, for example. Ruetsche dubs this view the ‘ideal of pristine interpretation’. Behind this ideal lies the thought that the laws of nature are philosophically interesting, whereas the initial conditions are only relevant to the special sciences (to ‘the astronomer, geographer, geologist, etc.’, as Ruetsche quotes Houtappel et al. ([1965], p. 596)).

However, Ruetsche’s ([2011]) book contains a sustained criticism of pristinism. In particular, Ruetsche argues that pristinism fails in the context of infinite-dimensional quantum theories, such as quantum field theory (QFT) and quantum statistical mechanics (QSM). Taking the latter as an example, Ruetsche argues that the occurrence of ‘unitarily inequivalent representations’ spells trouble for pristinism. Leaving the mathematical details aside, the problem Ruetsche pinpoints is that theories such as QSM allow for more than one space of physically possible worlds and no mathematically sound way to ‘glue’ these spaces together. This means that the interpreter of such a theory has to choose a space of possible worlds on the basis of ‘geographical’ considerations, such as the particular state of a system under study, or even the particular aims and interests of the scientists in question. But this blurs the distinction between laws and initial conditions: while some physical fact may appear to be a law from the perspective of a single space of possibilities—as it is constant across this space—that same fact may vary across distinct spaces. On her alternative, the coalescence approach:

[…] there can be an a posteriori, even a pragmatic, dimension to content specification, and […] physical possibility is not monolithic but kaleidoscopic. Instead of one possibility space pristinely associated with a theory from the outset, many different possibility spaces, keyed to and configured by the many settings in which the theory operates, pertain to it. (Ruetsche [2011], p. 147)

In my article, I argue that Ruetsche’s criticism of the pristine ideal is not limited to infinite-dimensional quantum theories, but equally applies to classical mechanics, classical statistical mechanics, and ordinary (non-infinite) quantum mechanics. On the one hand, this means that Ruetsche is mistaken in claiming that the pristine ideal fails specifically because of the mathematical nature of infinite-dimensional quantum theories. Instead, I claim, the pristine ideal was too simplistic from the start, far removed from the reality of physical interpretation. On the other hand, my article ultimately provides further support for Ruetsche’s coalescence approach. In that sense, it is a friendly amendment to her own work. But the version of the coalescence approach that I defend is a slightly attenuated form of Ruetsche’s own. In particular, it does not pose a threat to scientific realism, pace Ruetsche’s claims to the contrary.

In order to see the coalescence approach in action in a non-quantum context, consider classical particle mechanics. The space of possibilities for this theory is represented by a 6N-dimensional ‘phase space’, where N is the number of particles in the world. For each particle, this phase space specifies six observables: three positions (one for each of the three x, y, and z axes) and three velocities. For example, if the actual system of interest contains ten particles, then classical particle mechanics models this system via a sixty-dimensional phase space. However, if we consider one such phase space, it seems that the number of particles is fixed across each state. Put differently, each ‘point’ in a phase space represents the exact same number of particles. From the ideal of pristine interpretation, which identifies laws with whatever remains constant across the space of possibilities, it would follow that it is a law that the universe contains a certain number of particles. This conflicts with our intuitions that the world could have contained a bigger or smaller number of particles, without any change in the laws that govern these particles.

In response, one might suggest that we ‘glue’ the phase spaces for each N together to create one massive phase space, which contains points for each state for a variable number of particles. The problem with this suggestion is that it violates a principle of parsimony: for any given system, this ‘universal’ phase space contains many observables that are simply irrelevant. For example, if we study a universe with ten particles, then it doesn’t make sense to ask what the velocity of the twelfth particle in the y-direction is. This is why physicists in practice never work with such a monstrous phase space—indeed, the very idea of such a space seems slightly horrifying.

The alternative offered by the coalescence approach is to allow interpreters a degree of flexibility. Which phase space is appropriate depends on the number of degrees of freedom of the actual system under study, but the choice of one phase space does not mean that all other spaces are immediately unphysical. Instead, we can imagine the collection of all classical phase spaces as a reservoir of possibility spaces, from which physicists choose one as particularly relevant to the situation at hand. In Ruetsche’s ([2003], p. 1340) words, ‘other […] states aren’t impossible; they’re simply possibilities more remote from the present application of the theory’. This approach has consequences for the nature of lawhood. In fact, we allow laws of varying strength. Whatever is true across a single phase space most relevant to the application at hand is law-like in a narrower sense, whereas whatever is true across all phase spaces of possible interest is a law in a much stronger sense. Which notion of lawhood to use depends on the circumstances. For example, if we are interested in conservation laws (such as the conservation of mass or of the number of particles), it is not so crazy to consider the number of particles in the universe as a fixed law. But if we want to know whether space is an independent substance—and hence whether empty space could exist—it is appropriate to consider phase spaces of varying N. The strength of the coalescence approach is that it allows us the flexibility to use the same theory for both circumstances.

Finally, let’s return to the issue of scientific realism, briefly mentioned above. Ruetsche argues that the coalescence approach stymies the no miracles argument (NMA) for realism. The NMA states that a theory’s empirical virtues warrant our belief in its (approximate) truth. Ruetsche’s claim is that the NMA requires that all of a theory’s virtues accrue to a single interpretation, whereas on the coalescence approach different interpretations may display different virtues. I respond that this depends on how one construes the coalescence approach. I distinguish between a ‘modest’ and a ‘radical’ version, and argue that neither my case studies nor Ruetsche’s own require any recourse to the latter. But the modest version of the coalescence approach can sustain the NMA, as follows: instead of thinking of different choices of phase space as different interpretations of the theory, we think of these as different applications of the same theory. Put differently, an interpretation consists not of a single space of possibilities, but of a whole array of spaces, indexed to particular circumstances. Ruetsche uses the metaphor of a Swiss army knife, which seems to capture the spirit of the modest coalescence approach. The idea is that each blade of the knife represents a particular phase space, and which blade is used depends on the circumstances. But the Swiss knife as a whole corresponds to the theory, so it is not the case that each application of the knife requires a different interpretation. Rather, the multiple applications are already there, within the theory. The aim of the (modest) coalescence approach is to allow scientists to realize their theory’s full potential.

Listen to the audio essay


Jacobs, C. [2023]: ‘The Coalescence Approach to Inequivalent Representation: Pre-QM Parallels’, British Journal for the Philosophy of Science, 74, doi: 10.1086/715108

Caspar Jacobs
University of Pittsburgh


Houtappel, R. M. F., van Dam, H. and Wigner, E. P.  [1965]: ‘The Conceptual Basis and Use of the Geometric Invariance Principles’, Reviews of Modern Physics, 37, pp. 595–632.

Ruetsche, L. [2003]: ‘A Matter of Degree: Putting Unitary Inequivalence to Work’, Philosophy of Science, 70, pp. 1329–42.

Ruetsche, L. [2011]: Interpreting Quantum Theories, Oxford: Oxford University Press.

van Fraassen, B. [1991]: Quantum Mechanics: An Empiricist View, Oxford: Oxford University Press.

© The Author (2021)


Jacobs, C. [2023]: ‘The Coalescence Approach to Inequivalent Representation: Pre-QM Parallels’, British Journal for the Philosophy of Science, 74, doi: 10.1086/715108