## Abstract

The relationship whereby one physical theory encompasses the domain of empirical validity of another is widely known as “reduction.” Elsewhere I have argued that one influential methodology for showing that one physical theory reduces to another, associated with the so-called “Bronstein cube” of theories, rests on an oversimplified and excessively vague characterization of the mathematical relationship between theories that typically underpins reduction. I offer what I claim is a more precise characterization of this relationship, which here is based on a more basic notion of reduction between distinct models (one from each theory) of a single physical system. Reduction between two such models, I claim, rests on a particular type of approximation relationship between group actions over the models’ state spaces, characterized by a particular function between the model state spaces and a particular subset of the more encompassing model’s state space. Within this approach, I show formally in what sense and under what conditions reduction is transitive, so that reduction of a model 1 to another model 2 and reduction of model 2 to a third model 3 entails direct reduction of model 1 to model 3. Building on this analysis, I consider cases in which reduction of a model 1 to a model 3 may be effected via distinct intermediate models 2a and 2b, and motivate a set of formal consistency requirements between distinct “reduction paths” having the same models as their “end points”. These constraints are explicitly shown to hold in the reduction of a model of non-relativistic classical mechanics (model 1) to a model of relativistic quantum mechanics (model 3), which may be effected by a composite reduction that proceeds either via a model of non-relativistic quantum mechanics (model 2a) or a model of relativistic classical mechanics (model 2b). I offer some brief speculations as to whether and how this sort of consistency requirement might serve to constrain the reductions relating other theories and models, including the relationship that the Standard Model and general relativity must bear to any viable unification of these frameworks.

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## Notes

There exist conflicting conventions as to the precise use of the term “reduction.” On one usage, it is the more encompassing theory that “reduces to” the less encompassing theory; such uses reflect the general connotation of reduction that suggests simplification, as in “6/4 reduces to 3/2.” On another set of usages, it is rather the less encompassing theory that “reduces to” the more encompassing theory; such uses reflect the alternative connotation of reduction as subsumption into a more general framework, as in the claim that “chemistry reduces to physics” or “thermodynamics reduces to statistical mechanics.” We will adopt the latter connotation here, so that the less encompassing description is understood to reduce to the more encompassing description. For further discussion of the distinction between these uses of the term “reduce,” see Nickles’ [21].

This point is also emphasized in Crowther [5].

Instead of “A reduces to B” one sometimes says that “B reduces A.”

Theories of “fundamental physics” here are understood to include non-statistical mechanical and non-thermodynamic theories, such as classical mechanics, quantum mechanics, relativistic quantum mechanics, quantum field theory, and quantum gravity. While many of the claims about reduction defended here also apply in the context of thermodynamics and statistical mechanics, these theories also introduce novel probabilistic aspects that demand special treatment.

Property A (e.g., a macrostate) supervenes on property B (e.g., a microstate) if and only if there can be no difference in A without a difference in A. The value of property B uniquely determines the value of property A while the reverse is typically not the case.

“Physically realistic” here indicates that the trajectory approximates the behavior of the real physical degrees of freedom described by the model to within some specified error tolerance.

The difference between \(x_{h}(\tau )\) and \(B(x_{l}(\tau ))\) is required to be less than \(2\delta _{emp}\) rather than \(\delta _{emp}\) because if the \(x_{h}(\tau )\) and \(B(x_{l}(\tau ))\) both approximate

*K*’s behavior within error bound \(\delta _{emp}\), then they may differ from each other by at most \(2\delta _{emp}\).As suggested above, many features of the approach to reduction described here can be extended straightforwardly to reductions involving models with stochastic dynamics, with certain important modifications to accommodate the probabilistic nature of the models. For example, approximate equality of induced and high-level state space trajectories is replaced by approximate equality with high likelihood. The understanding of reduction in terms of instantiation may also be extended to non-dynamical models such as the Ideal Gas model although we do not explore this here.

For example, the deterministic equations of classical mechanics are only known to describe the trajectory of Saturn’s moon Hyperion over certain limited timescales, beyond which classical predictability is lost due to quantum and classically chaotic effects. See for example [32] and [33] for further details. Quantum models should only be required to recover classical trajectories over timescales for which these classical trajectories furnish an accurate representation of the system in question.

It is worth emphasizing here that relation (3) bears a close resemblance in certain respects to the requirements for reduction proposed by Ernest Nagel and Kenneth Schaffner, for whom reduction required that it be possible to logically deduce approximate versions of the laws of the reduced theory from those of the reducing theory via the use of “bridge laws” [20, 29]. The approach here is in some ways similar in spirit to Nagel/Schaffner approach in showing that, by virtue, of the low-level model’s equations of motion, \(B(x_{l}))\) approximately satisfies the high-level model’s equations of motion. However, unlike their approach, the requirements for reduction here are formalized mathematically within the specific context of group actions over state space manifolds. Moreover, reduction here concerns relations between two specific models of a single fixed system, rather than between entire theories as in the Nagel/Schaffner approach. For recent discussion of Nagel and Schaffer’s approach to reduction, see [1, 7, 29].

Physical symmetries are understood here as those that map between physically distinct states when the symmetries are interpreted as active transformations, while gauge symmetries only map between redundant representations of a single physical state and never between physically distinct states.

Note that despite the subatomic nature of the charged particles that are collided in accelerators like the LHC, it is the

*classical*Lorentz Force Law, rather than a quantum equation, that is used to guide their motions.Assuming that effects of decoherence can be ignored—which is realistic for such small systems—the dynamics of the charged particle can be modeled by a purely unitary dynamics to a good approximation.

See, e.g., [12] for proof and discussion of this result.

See, e.g., [24], Chap. 2 for further discussion of this well-known result.

The reduction of Newtonian gravity to general relativity has been examined extensively in the work of Ehlers and Rohrlich [9, 23]. I leave it to future work to assess the compatibility of Ehlers’ and Rohrlich’s treatments with the methodology described here. However, it is possible to offer a preliminary sketch of how an analysis of this reduction within the present framework would proceed. The high-level state space would be the classical phase space of N particles and the high-level model’s dynamics would be specified by a Hamiltonian with Newtonian gravitational potential. The low-level model could be formulated in terms of a 3 + 1 splitting of spacetime (assuming global hyperbolicity), so that its state space consists of configurations of the 3-metric and corresponding conjugate momenta, together with the configurations and conjugate momenta of any matter degrees of freedom. The dynamics of such a model would be given in terms of a Hamiltonian formulation of Einstein’s field equations coupled with some evolution equations for the matter degrees of freedom. The state space domain of such a reduction would consist of 3-metric configurations and momenta consistent with a weak-field approximation (i.e., close to the flat Minkowski metric), and of states of the matter degrees of freedom in which the kinetic energy is much smaller than the rest energy. The bridge function would map the low-level state into the high-level state space by mapping the coordinates and momenta of the matter degrees of freedom into a locally inertial coordinate system. While the gravitational degrees of freedom are dynamical in the low-level model, in the high-level Newtonian model they are not, so that there are no gravitational degrees of freedom represented in the high-level Newtonian model’s state space. Instead, the influence of the gravitational degrees of freedom described by the low-level model manifest their influence within the high-level model by serving to determine the form Hamiltonian of the high-level model.

See [24], Chap. 1 for a more formal proof of this claim.

Within the recent philosophical literature, the metaphysical basis for and implications of such a reductionist view, including careful analysis of what it means for one theory to be more “fundamental” than another, is examined extensively in the work of Ladyman, French, Saatsi, McKenzie, and many others [13, 18, 19, 28].

Butterfield employs this distinction, between behavior in the limit, and on the way to the limit, in order to propose a reconciliation between ostensibly clashing concepts of reduction and emergence in physics [2].

For recent extensive discussion of non-relativistic limits, including in the context of gravitational theories, see [11].

This presumably would be some quantum field theoretic Schrodinger equation for the Standard Model quantum state.

See, Huggett and Vistarini’s [16], and sources therein, for discussion of the role of coherent states in the relationship between string theory and classical general relativity.

The relationship between GR and SM cannot be characterized as a case of reduction since it is not true that the domain of either is contained in that of the other. However, as we discuss here, their domains overlap. On this overlap domain, which includes the domain of CED, we expect a weaker relationship - what Crowther has called “correspondence” - to hold. Correspondence requires distinct theories to approximately agree in cases where their domains of validity overlap. As Crowther notes, reduction is a special case of correspondence in which the overlap of the theories’ domains is the entire domain of one of the theories [5].

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## Acknowledgements

This work was supported by the DFG, Grant FOR 2063.

The author gratefully acknowledges support from the DFG

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Rosaler, J. The Geometry of Reduction: Compound Reduction and Overlapping State Space Domains.
*Found Phys* **49**, 1111–1142 (2019). https://doi.org/10.1007/s10701-019-00299-3

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DOI: https://doi.org/10.1007/s10701-019-00299-3