Keto Derivatives of Group IV Organometalloids

Full text of DOI:10.1016/S0065-3055(08)60293-2

Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
Contents lists available at ScienceDirect
Studies in History and Philosophy
of Modern Physics
journal homepage: www.elsevier.com/locate/shpsb
Local reduction in physics
Joshua Rosaler
University of Minnesota, Center for Philosophy of Science, Heller Hall, 271 19th Ave S 55455, Minneapolis, MN, United States
a r t i c l e i n f o
a b s t r a c t
Article history:
A conventional wisdom about the progress of physics holds that successive theories wholly encompass
the domains of their predecessors through a process that is often called “reduction.” While certain
influential accounts of inter-theory reduction in physics take reduction to require a single “global”
derivation of one theory's laws from those of another, I show that global reductions are not available in
all cases where the conventional wisdom requires reduction to hold. However, I argue that a weaker
“local” form of reduction, which defines reduction between theories in terms of a more fundamental
notion of reduction between models of a single fixed system, is available in such cases and moreover
suffices to uphold the conventional wisdom. To illustrate the sort of fixed-system, inter-model reduction
that grounds inter-theoretic reduction on this picture, I specialize to a particular class of cases in which
both models are dynamical systems. I show that reduction in these cases is underwritten by a
mathematical relationship that follows a certain liberalized construal of Nagel/Schaffner reduction,
and support this claim with several examples. Moreover, I show that this broadly Nagelian analysis of
inter-model reduction encompasses several cases that are sometimes cited as instances of the
“physicist's” limit-based notion of reduction.
Received 6 May 2014
Received in revised form
29 January 2015
Accepted 23 February 2015
Available online 24 April 2015
Keywords:
Reduction
Local Reduction
Models
Dynamical systems
Nagel
Limits
& 2015 Elsevier Ltd. All rights reserved.
When citing this paper, please use the full journal title Studies in History and Philosophy of Modern Physics
1. Introduction
thermodynamics to statistical mechanics, and more. In order to
assess the truth of the conventional wisdom, however, it is
necessary to gain a more precise sense of what is needed in a
given case to show that one theory reduces to another.
According to the most commonly told story about the progress
of physics, successive theories in physics come ever closer to
revealing the true, fundamental nature of reality. This convergence
rests on the supposition that later theories bear a special relation-
ship to their predecessors often called “reduction,” which minimally
requires one theory to encompass the domain of application
of another. More specifically, the conventional wisdom tells us
that Newtonian mechanics “reduces to” special relativity,1 special
relativity to general relativity, classical mechanics to quantum
mechanics, quantum mechanics to relativistic quantum mechanics,
relativistic quantum mechanics to quantum field theory,
In his widely cited 1973 paper, Nickles distinguished two types
of approach to reduction in physics: first, the approach commonly
employed by philosophers, which originates in Ernest Nagel's
well-known account of reduction, and second, the approach
commonly employed by physicists that requires one theory to be
a “limit” or “limiting case” of another (Nickles, 1973). Since Nickles'
paper, these two accounts have tended to dominate philosophical
discussion concerning issues of the general methodology of
reduction in physics. As commonly presented, both strongly
suggest—and in some cases, state explicitly—that reduction
between theories in physics should rest on a single “global”
derivation of a high-level theory's laws from those of a low-level
theory. Here, I argue by means of a particular example that global
reduction is not always available in cases where the conventional
wisdom requires reduction to hold. However, I argue that it is
possible to a define a weaker “local” notion of reduction in physics
that suffices to uphold the conventional wisdom in that it suffices
to ensure the subsumption of one theory's domain by another. This
1
As Nickles noted several decades ago, two opposing conventions have arisen
in the literature on inter-theoretic reduction, one (employed most commonly in the
philosophical literature) that takes a less encompassing theory to “reduce to” a
more encompassing one, and the other (employed most commonly in the physics
literature) that takes the more encompassing theory to “reduce to” the less
encompassing one (Nickles, 1973). Here, I will adopt the first of these conventions.
It should also be noted that the two concepts of reduction that Nickles discusses in
his paper differ on more substantive points than this choice of convention, which I
discuss further below.
http://dx.doi.org/10.1016/j.shpsb.2015.02.004
1355-2198/& 2015 Elsevier Ltd. All rights reserved.

J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
55
notion of reduction is “local” in the sense that it permits the
reducing theory to account for the reduced theory's success
through numerous context-specific derivations that are relativized
to different systems in the high-level theory's domain. These
derivations concern the specific models that the theories use to
describe a single fixed system, rather than the theories as a whole.
This paper has two main goals, which are mutually supporting.
The first is to motivate and develop a local account of inter-
theoretic reduction in physics. Inter-theoretic reduction in physics,
understood minimally as the requirement that one theory sub-
sume the domain of another, does not require anything as strong
as global reduction directly between theories; local reduction
suffices, and moreover avoids difficulties that afflict global
approaches. I further argue that local reduction between theories
should be understood in terms of the more basic notion of
reduction between models of a single fixed system. The second
goal of the paper is to give an account of fixed-system, inter-model
reduction—on which this local account of inter-theoretic reduction
rests—in a specialized class of cases where both models of the
system in question are dynamical systems, and in particular to
show that such cases can be analyzed in terms of a certain local,
model-based adaptation of the Nagel/Schaffner approach to reduc-
tion. I further show that this Nagelian analysis of inter-model
reduction encompasses many cases that have been cited as
instances of physicists' limit-based notion of reduction, as well
as providing a more precise characterization of these cases than do
existing formulations of the limit-based approach.
The present analysis of reduction is given in two parts,
corresponding respectively to the two goals just described. Part I,
which consists of Sections 2 and 3, is largely non-technical and
concerns issues of general methodology. As suggested, its purpose
is to motivate and present a certain local, model-based approach
to inter-theoretic reduction and to explain how this strategy
avoids certain difficulties that afflict more global approaches. In
Section 2, I briefly review two approaches to reduction—global
Nagelian and global limit-based—that are often taken as the focus
of philosophical discussions on this topic, and highlight some of
their limitations. In Section 3, I sketch a local approach to inter-
theoretic reduction in physics that relies on the more basic notion
of fixed-system reduction between models and respond to one
major objection that such an approach is likely to elicit.
Part II, which consists of Sections 4–6, provides a detailed
technical analysis of fixed-system, inter-model reduction in a parti-
cular set of cases where both models of the system in question are
dynamical systems, as well as briefly discussing various possible
expansions of this analysis to a more comprehensive account of
fixed-system, inter-model reduction in physics. Section 4 describes a
general mathematical relationship between dynamical systems mod-
els that serves to underwrite many real instances of fixed-system,
inter-model reduction in physics. In a certain strong sense, this
mathematical relationship constitutes an application of the criteria
for Nagel/Schaffner reduction to the context of fixed-system, inter-
model reduction between dynamical systems models. Section 5
shows how this general relationship serves to characterize reduction
across a wide range of particular cases, and to subsume a number of
cases that are commonly cited as examples of the “physicist's” limit-
based notion of reduction. Section 6 briefly discusses possibilities for
extending and generalizing this strategy for inter-model reduction
beyond the set of cases discussed here: first, to an analysis of the
relationship between symmetries of the two models involved in a
reduction, and second, to an analysis of cases where one or both of
the models involved in the reduction is not a dynamical system but
some other kind of model (e.g., stochastic, non-dynamical, etc.).
The distinct portions of the analysis given in Parts I and II
complement each other in a number of important ways. Part I
serves to frame the analysis of reduction between dynamical
systems given in Part II within a more general picture of inter-
theoretic reduction and in particular to situate this analysis
relative to the two accounts of inter-theoretic reduction in physics
first distinguished by Nickles. By the same token, Part II provides a
concrete illustration of the sort of fixed-system, inter-model
reduction that is taken as the basis for the local approach to
inter-theoretic reduction described in Part I.
1.1. A few points of terminology
Before proceeding, it is worth taking a moment to clarify
several points of terminology.
Because debates about reduction are often frought with ambi-
guity as to what, precisely, is meant by reduction, I should clarify
my use of the term here. I do not attach my usage to any specific
account of reduction—for example, Nagelian, limit-based, New
Wave and functionalist approaches. Rather, I use it to designate a
certain general concept that, I take it, all, or most, of the many
specific accounts aim to make more precise. “Reduction,” then, is
taken to designate the general requirement that two descriptions
of the world “dovetail” in such a manner that one description
entirely encompasses the range of successful applications of the
other. That is, reduction on this usage requires subsumption of one
description's domain of applicability by the other, while the
specific sense in which the two descriptions “dovetail” in order
to achieve this is deliberately left vague, so as not to bias its usage
toward any particular account.
As Nickles has noted, the usage of the term “reduction” most
common among philosophers takes the less accurate and encom-
passing description in a reduction to “reduce to” the more accurate
and encompassing description, whereas the usage most common
among physicists takes the more accurate, encompassing descrip-
tion to “reduce to” the less accurate and encompassing descrip-
tion. In what follows, I will always adopt the philosopher's
convention, even when discussing the physicist's limit-based
notion of reduction, so that if theory T2 is a “limiting case” of T1,
we will say that T2 “reduces to” T1.
I will also reserve the term “high-level” to refer to the
description that is purportedly reduced and “low-level” to refer
to the description that purportedly does the reducing. This usage
generalizes another use of the “high-level/low-level” distinction,
which presupposes that the high-level description is in some
sense a coarse-graining of the low-level description, or that the
high-level description is in some sense “macro” and the low-level
description in some sense “micro.” Here, no such assumption is
made. For example, where the relation between Kepler's and
Newton's theories of planetary motion is concerned, Kepler's
theory would count on our usage as the “high-level” and Newton's
as the “low-level” theory even though Kepler's theory is not in any
normal sense a coarse-graining of Newton's. While some authors
have emphasized the distinction between “inter-level” reductions
(e.g., thermodynamics to statistical mechanics) and “intra-level”
reduction (e.g., Newtonian mechanics to special relativity, or
Kepler's to Newton's theory of planetary motion), the picture of
reduction presented here does not rely on this distinction and
treats both kinds of reduction on a par.2
Henceforth, when I speak of “Nagelian” reduction, the reader
should take this to refer specifically to the Nagel/Schaffner account
of reduction, which allows for approximative derivations rather
than requiring exact derivations. While Nagel/Schaffner reduction
is widely framed within a syntactic view of theories—as opposed
2
As with the term “reduce,” it is worth noting that one occasionally finds the
high-/low- distinction inverted, so that the “high-level” description is the more
encompassing and the “low-level” description the less encompassing of the two.

56
J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
to the semantic, model-theoretic view adopted here—and is often
taken to require global rather than local derivations, my use of the
label “Nagelian” here does not presuppose these characteristics.
Rather, what is taken to be constitutive of “Nagelian” reduction on
my usage is the general requirement that it be possible to derive,
on the basis of the low-level description and through the use of
bridge principles, approximate versions of the laws or constraints
or equations that serve to characterize the high-level description.
My usage does not presuppose any view as to whether theories are
understood syntactically or semantically—although the specific
local approach to reduction advocated here fits much more
naturally with a semantic view. Moreover, my usage does not
assume any commitments as to the particular nature of these
bridge principles—such as whether they are to be understood as
empirically established laws or as definitions—apart from their
role in enabling a translation or comparison between the frame-
works of the two descriptions in question. Generally speaking,
rather than taking the term “Nagelian” to designate a specific set
of precisely defined formal requirements for reduction, I use it
here to designate a certain broad strategy for reduction.
Since the local approach to inter-theoretic reduction that I
describe here is grounded in a certain account of inter-model
reduction, I should say something about what I take to be the
relationship between theories and models. For the purposes of this
discussion, it will suffice to note that the manner in which any
theory serves to successfully describe a physical system in its
domain is through some particular model of that theory. More-
over, the specification of any such model entails much narrower
commitments than those that serve to characterize the theory
itself. For example, specification of a particular quantum or
classical model of a material object's behavior requires commit-
ments to a particular form for the Hamiltonian (or force law or
Lagrangian), including particular values for quantities like mass
and charge, while the theories of quantum and classical mechanics
themselves are compatible with many functional forms of the
Hamiltonian and many values for these parameters. One of the
central points of my discussion here is that it is sometimes not just
the more general specifications that serve to characterize the high-
and low-level theories in question that are relevant to under-
writing the success of the high-level theory in a given case, but
also the narrower, more context-dependent specifications that
characterize the particular models of the two theories.
reduction as a special case. In fact, local reduction accommodates a
whole spectrum of cases varying in the extent of their “global-ness”:
for inter-theoretic reductions at the least global extreme of this
spectrum, a separate derivation is required for each separate system
in the domain of the high-level theory in order to effect the
necessary subsumption of domains; at the most global extreme, it
is possible to effect a fixed-system, inter-model reduction for every
system in the high-level theory's domain on the basis of a single
derivation that applies uniformly across all such systems (and so
does not depend essentially on details that characterize some
systems but not others). In cases of inter-theoretic reduction that
lie between these extremes, the derivations that underwrite fixed-
system, inter-model reduction will rely on general results and
mechanisms that apply across a wide range of systems, while also
requiring reference to system-specific details at certain points in the
derivation. An example of an inter-theoretic reduction at the global
extreme of the spectrum is the reduction of Kepler's theory of
planetary motion (which consists of models that obey Kepler's three
laws) to Newton's theory of gravitation. The domain of Kepler's
theory consists of the motions of the various planets around the Sun
(as well as other similar solar systems in the cosmos). Because the
same general derivation connects models of Kepler's theory to
models of Newton's theory irrespective of the particular planet in
this domain that is being represented, the reduction of Kepler's
theory to Newton's theory lies at the global end of the spectrum. On
the other hand, not all inter-theoretic reductions required by the
conventional “imperialist” view of the progress of physics are global
in this way, since in some cases separate derivations are required to
connect the models of the two theories for different systems in the
domain of the high-level theory, as I show explicitly in Section 3.1.
2. Nickles' two senses of reduction
Since Nickles' 1973 paper, most philosophical discussion about
the general methodology of reduction in physics has revolved around
the Nagelian and limit-based approaches. Conventional presentations
of both approaches treat inter-theory reduction as a matter of
deriving one theory from another (whether through Nagelian
deduction or a limiting process). This manner of framing the issue
strongly suggests that inter-theoretic reduction is being taken in
these presentations to be a matter of effecting a single derivation of
the high-level theory's laws from the low-level theory, rather than
many separate derivations specialized to different contexts. This in
turn suggests that it is the general assumptions characterizing the
two theories, rather than the more specific details characterizing the
theories' models of particular systems, that are relevant to showing
how the low-level theory serves to encompass the high-level
theory's domain of success. Thus, implicitly if not explicitly, these
approaches seem to demand a global rather than a local form of
reduction. While it is not impossible in some cases to interpret
conventional formulations of both approaches as allowing for many
local derivations, what can be said with relative definiteness is that
these formulations do not make the need for local derivations, or the
fact these local derivations concern specific models of the theories
rather than the theories themselves, at all explicit; and both of these
points, I take it, do bear emphasizing explicitly. Moreover, as I discuss
below, where the Nagel/Schaffner approach is concerned, the exten-
sive philosophical literature that takes multiple realizability to
preclude application of this approach does explicitly presuppose a
global construal of it.
Part I: local reduction in physics
In the literature on reduction across the sciences, and especially
in philosophy of mind, concerns about multiple realization have
led many philosophers to espouse a more “local” form of reduction
that allows a low-level description to account for a high-level
description's successes through many context-specific derivations
that employ context-specific bridge principles. As a whole, the
literature on reduction specifically within physics—which merits
focused attention in part because of the special mathematical
issues that it raises—has been slow to absorb this development, in
that discussions of the methodology of inter-theoretic reduction in
physics very often still presuppose a global understanding of
reduction. The goal of Part I of this article is to argue that insights
about local reduction that have arisen in the general philosophy of
science and philosophy of mind literatures ought to be imported
into the analysis of reduction specifically within physics, and to
suggest how this should be done.
In emphasizing a local approach to inter-theoretic reduction in
physics, I do not mean to suggest that it is not possible to effect
global reduction, or something approximating it, in some cases. Local
reduction is weaker than global reduction and so includes global
2.1. Nagel/Schaffner reduction
Reduction on the Nagel/Schaffner approach can be understood
as a three-step process, starting with the basic ingredients of a

J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
57
low-level theory Tl, a high-level theory Th, and a set of bridge
principles:
multiple realization occurs is to identify the relevant concept in
the high-level description with a disjunction of associated ele-
ments in the low-level description, and it is simply false in most
cases that such a disjunction will be a natural kind of the low-level
description (e.g., in the sense that it occurs naturally in the laws or
equations of the low-level theory, Bickle, 2013; Fodor, 1974). In this
manner, multiple realization typically precludes the existence of
bridge principles of the sort that are taken to be required by
Nagelian reduction on these formulations.
Implicitly, this sort of MR-based critique presupposes a global
interpretation of the Nagel/Schaffner approach, since its argument
assumes that bridge principles are required to identify an element
of the high-level description with the same “natural kind” element
of the low-level description across all contexts where the high-
level description applies. In cases of multiple realization, there
simply do not exist such global bridge principles. The unavail-
ability of these bridge principles, in turn, precludes the possibility
of the sort of global derivation required by global formulations of
Nagelian reduction.
1. Derive an “image” theory Thn for some restricted boundary or
initial conditions within the low level theory Tl. The “laws” of
the image theory Tnh take the same form as the laws of Th, but
relate quantities that are defined using the terms and concepts
of Tl.
2. Use bridge principles to replace terms in Tnh, which belong to
the vocabulary of the low-level theory, with corresponding
terms belonging to the high-level theory. This yields the
“analogue”3 theory T0 . Like the “laws” of Tnh, those of T0 have
h
the same form as thhe laws of Th, but employ the terms and
concepts of Th rather than of Tl.
3. If the analogue theory T0h is “strongly analogous” to the high-
level theory Th, the high-level theory has been reduced to Tl.
The “strong analogy” relation is sometimes also characterized
as approximate equality, close agreement, or good approxima-
tion and can be understood in any of these senses.
Note that reduction on the Nagel/Schaffner approach does not
require derivation of the high-level theory's laws themselves, but
rather of some suitable approximations to them. This is the key
point that distinguishes Nagel/Schaffner reduction from Nagel's
original approach. A further point worth noting is that there
remains some dispute as to whether the bridge principles of
Nagel/Schaffner reduction are empirically discovered laws or mere
definitions, and whether they are biconditional or merely condi-
tional relations. The popular term “bridge law,” which is often
used to designate bridge principles, suggests a clear bias toward
interpreting them as empirical laws; here, I use the term “bridge
principle” so as to avoid the suggestion of bias toward any
particular view on this matter.
2.3. Limit-based reduction
On the limit-based approach to reduction, a high-level theory
Th reduces to a low-level theory Tl if Th is a “limit” or “limiting
case” of Tl. Somewhat more precisely, Th reduces to Tl if there exists
some set of parameters fϵig defined within Tl such that
f lim Tl ¼ Th
ð1Þ
ϵ
i-0g
4
(Batterman, 2007; Nickles, 1973). Unlike Nagelian reduction, the
notion of a limit-based approach to reduction, as first explicitly
identified by Nickles, seems to arise not from any clear-cut
statement of the general conditions for this kind of reduction to
take place, but rather from a plethora of highly suggestive
mathematical results in physics, all of which involve or somehow
gesture at the use of mathematical limits. It also seems to arise
partially from a certain related manner of speaking that is often
employed in talk about inter-theory relations in physics, as
exemplified by references to the “nonrelativistic limit” of special
relativity, the “classical limit” of quantum mechanics, the “thermo-
dynamic limit” of statistical mechanics, the “geometric optics
limit” of wave optics, and so on. Various facets of this approach
have been explored by a number of authors, including Batterman
(2002), Berry (1994), Butterfield (2011a, 2011c), Ehlers (1986),
Rohrlich (1988), and Scheibe (1997, 1999), among others. Talk of
reduction in terms of limits also pervades discussion of inter-
theory relations in many physics textbooks and journals. Recently,
Norton has highlighted a role for limiting procedures in physics
outside the context of reduction - specifically, in distinguishing
between the activities of approximation and idealization and in
fending off potential confusions caused by conflation of the two
(Norton, 2012). It is clear that limits have an important role to play
in our understanding of inter-theory relations generally; however,
where reduction specifically is concerned, the vague and sche-
2.2. Difficulties with Nagelian reduction: multiple realization
For a comprehensive review of, and response to, the many
critiques that have been levelled against Nagelian approaches to
reduction, Dizadji-Bahmani et al.' (2010) article “Who's Afraid of
Nagelian Reduction?” is excellent. In the present context, I will
focus exclusively on critiques grounded in multiple realization
(MR) because such critiques have served as the major impetus—
particularly in the philosophy of mind and general philosophy of
science literatures—for a move to consider more local forms of
reduction both in the context of Nagelian and non-Nagelian
approaches.
As its name suggests, multiple realization occurs when a single
element of a high-level description (e.g., model, theory or whole
science) is realized by more than one element of some lower-level
description. A classic example from the philosophy of mind
literature is the high-level psychological property of pain, which
can be multiply realized in human brains, dog brains, badger
brains and so on, all of which have different low-level biological
(and physical) descriptions (Putnam, 1967). In the context of
Nagelian reduction, realization of an element in the high-level
theory by some element of the low-level theory is signified by
means of a bridge principle. On many formulations of the Nagel/
Schaffner approach, bridge principles are required to be
bi-conditional identity statements identifying natural kinds in
the high-level theory with natural kinds in the low-level theory.
But, as Fodor has argued, the best one can do in cases where
matic relation lim T ¼ T appears to be as close to a statement
fϵi-0g
l
h
of general criteria for limit-based reduction as has been given in
the literature.
2.4. Difficulties with the limit-based approach
Perhaps the most significant concern about the limit-based
approach to reduction is that, in spite of its mathematical nature, it
is extremely vague as a general characterization of the sort of
3
The labels “image” and “analogue” are not common to most presentations of
the Nagel/Schaffner approach. Nevertheless, the substantive points of procedure
described here are. See, for example, Batterman (2007) and Dizadji-Bahmani, Frigg,
& Hartmann (2010).
4
Note that if one has lim
T ¼ T , or lim
T ¼ T where 0oao1, one
i-ag
f
ϵi -1g
f
ϵ
l
h
l
h
can always redefine the parameters fϵig so that the limit approaches 0.

58
J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
relationship that it takes to underwrite reduction. The expression
lim
T ¼ T is not mathematically well-defined, as there is no
mind literature, were prompted to advocate for a more local
approach to reduction in which a lower-level description accounts
for the successes of a higher-level description through many
context-specific derivations employing context-specific bridge
principles, rather than through a single global derivation employ-
ing the same set of bridge principles for all systems in the high-
level description's domain. In particular, Kim has argued that
while multiple realization rules out “structure-independent”
reductions of psychology to physical science, it does allow for
“structure-specific” local reductions between these levels (Kim,
1992). Following Kim, Dizadji-Bahmani et al. also have advocated a
local response to anti-reductionist arguments from multiple rea-
lizability in the context of their wider defence of Nagelian reduc-
tion (Dizadji-Bahmani et al., 2010). Similar views can be found in
the work of Churchland (1986, chap. 7), Enc (1983), Hooker (1981),
Lewis (1969), Bickle (2013), and Schaffner (2012). The main goal of
the present analysis is to show explicitly how this sort of local
approach, which has been developed primarily in discussions
about reduction in philosophy of mind and general philosophy of
science, can be imported into methodological discussions of inter-
theoretic reduction specifically within physics.
fϵi-0g
l
h
precise or general definition of what it is for one theory to be a
limit or limiting case of another. Moreover, it is unclear from this
expression whether the prescription to take the limit is to be
understood literally or in some loosened sense. For example, if we
are to understand the claim that Newtonian mechanics is the limit
as vc-0 of special relativity literally, then the claim is patently false,
since the limit of special relativity as vc-0 is a theory in which
nothing moves, not Newtonian mechanics (assuming we take c,
the speed of light, to be constant, as it is for all real physical
systems). For this paradigmatic case to be regarded as an instance
of limit-based reduction, it seems necessary to adopt a more
liberal construal of the term “limit”—for example, by making use
v
c
of first- or higher-order approximations in Taylor expansions in
(where only the zero-th order term of such an expansion gives the
actual limit of this series as vc-0). On the other hand, in other
cases, including certain discussions of the thermodynamic limit of
statistical mechanics, in which the number of degrees of freedom
in a system is taken to infinity, the limiting process is interpreted
literally—for example, when it is pointed out that the only way to
recover the discontinuities of certain functions in thermodynamics
from statistical mechanics is to literally take the limit as the
number of degrees of freedom approaches infinity. Beyond the
points of vagueness already mentioned, it is not clear on this
approach which parts of Th and Tl must be related by these “limits”
3.1. The need for local reduction in physics: an illustrative example
An example, concerning the relationship between classical and
quantum mechanics, will serve to make my general point that global
reductions are not available in all cases where the conventional
wisdom requires reduction to occur, and that only a weaker (but still
highly non-trivial), more local form of reduction is available in such
cases. As we will see, the source of trouble for global approaches to
reduction in physics is that it is often not just the broad generalities
that characterize the theories—which are common to all models of a
given theory—but also specific details that differentiate models of the
same theory, that play a role in the low-level theory's account of why
the high-level theory works in a given case. In other words, system-
or context-specific details often play a necessary role in accounting
for the high-level theory's success on the basis of the low-level
theory in a given case, and this precludes the sort of generality
demanded by global forms of reduction.
The conventional wisdom about the progress of physics asserts
that quantum mechanics has strictly superseded quantum
mechanics in the sense that any system whose behavior can be
accurately modelled in classical mechanics also can be modelled,
and modelled more accurately, in quantum mechanics.5 Here,
I will argue that any demonstration to this effect could not
possibly, even in principle, take the form of a global reduction.
Consider two systems in the domain of classical mechanics, both
of which are described by the same classical model of a simple
harmonic oscillator: the first system is a mass on a spring; the
second is an electric charge of the same mass moving along a path
bored through the axis of a uniform spherical charge distribution
(one can show that in the second case, there will be a restoring
force on the charge that varies linearly with its distance from the
center of the sphere). Assume moreover that frictional/radiation
effects can be ignored in both systems, and that the linear
restoring force in both cases is characterized by the same effective
spring constant. Assuming that macroscopic bodies do belong to
in order for the relation lim T ¼ T to hold; presumably, not
fϵi-0g
l
h
just any limiting relation between any two parts of the theories
will do. Finally, limit-based approaches tend to differ on what
constraints, if any, should be placed on the parameters
ϵi—for
example, whether they are supposed to be dimensionless, or may
also be dimensionful constants of nature.
Given that this approach offers very little by way of precise
characterizations or clear commitments as to the nature of
reductive relations in physics, it seems possible to take many
cases that we might wish to characterize as successful reductions
and categorize them as instances of limit-based reduction. But if
existing formulations of the limit-based approach succeed at
accommodating many cases in physics, it is largely for the reason
that they tell us so little, and are so vague, about the general
requirements for reduction. It seems possible to count any reduc-
tion as an instance of limit-based reduction as long as the
reduction somehow incorporates a procedure that may liberally
be construed as “taking a limit.” The worry, then, is that this
approach, at least in its existing formulations, may give the false
impression of providing an authoritative, general account of
reduction in physics in spite of having offered little by way of
general insight beyond the assertion that, in cases of reduction,
something like a limit is somehow involved in the explanation of
the high-level theory's success on the basis of the low-level theory.
Recalling the focus here on comparing global and local forms of
reduction, I should point out that since the limit-based approach is
conventionally formulated in terms of limiting relations between
whole theories, it seems most natural to read this approach as a
global strategy for reduction. However, given the vagueness of
existing formulations of this approach, it does not appear that
anything precludes interpreting this relation on a more local basis.
Nevertheless, such a formulation is still vulnerable to worries
about precisely the sort of vagueness that allows for this vast
flexibility of interpretation.
5
There is a sense in which this particular case might appear to be less than an
ideal example, given the various added complexities brought about by the
notorious interpretational difficulties that afflict quantum mechanics, which seem
as though they should bear quite heavily on the relation between quantum and
classical descriptions of real physical systems. However, these difficulties do not
affect my central point here, as long as we are safe in assuming that the correct
interpretation of quantum mechanics, whatever that happens to be, is able to
underwrite the success of classical descriptions on a case-by-case basis.
3. Local reduction in physics
It was largely in response to critiques of global Nagel/Schaffner
reduction that a number of authors, mostly in the philosophy of

J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
59
the domain of quantum mechanics, it is clear in this case that the
quantum mechanical models that serve to describe these two
physical systems are going to be radically different from each
other, and that the process of accounting for the success of the
classical harmonic oscillator model on the basis of these quantum
models is going to play out very differently in the two systems. In
the second case, the classical potential generated by the electric
field in the classical model will be the same potential that appears
in the Schrodinger equation of the underlying quantum model of
the charge's behavior. In the first case, the fact that one can
employ a harmonic oscillator potential in the classical model in
order to describe the motion of the block is something that needs
to be explained in terms of the complex microscopic constitution
of the spring—at the microscopic level, this potential will be wildly
fluctuating on the length scale of the atoms and molecules making
propositions. Second, it is not simply a two-place relation between
the theory Th and the theory Tl, but rather a three-place relation
between Th, Tl, and that class of real systems in the world that are
well-described by models of Th. Since reduction is understood here
to require that one description dovetail (in some appropriate
sense) with another specifically in those cases where the reduced
description works, some specification of the reduced description's
domain of success is required in order to assess whether any
dovetailing between the theories is sufficient to underwrite the
required subsumption; here, the relevant dovetailing between
theories may occur on a local level, between the different models
that they use to describe systems in the high-level theory's
domain. Third, reductionM, characterized broadly, requires that Ml
furnish at least as accurate a description of K as Mh does in those
cases where Mh successfully tracks (to within some margin of
approximation) K's behavior. Fourth, the precise nature of relation-
ship between models Mh and Ml that serves to underwrite
reductionM in a given case varies depending on the particular
mathematical form of these models. Fifth, given the emphasis here
on the distinction between theories and their individual models, it
is also important to distinguish between what is meant here by
the domain of a theory, on the one hand, and the domain of a
model that the theory uses to describe a particular physical
system. The domain of a single model of a theory, relative to some
physical system K, consists of the set of circumstances under
which the model accurately tracks the system's behavior. The
domain of a theory, as understood here, consists of the range of
physical systems whose behavior is well-described (under some
fairly robust range of circumstances) by some particular model of
the theory, as well as the set of circumstances under which each of
these models succeeds in tracking the behavior of the system that
it describes. Sixth, the account of reductionM that I develop in Part
II for the particular case of reductions between dynamical systems
models follows, in a certain strong but qualified sense, the
prescriptions of a localized Nagel/Schaffner approach. As I discuss
in Section 6.2, the broadly Nagelian character of reductionM in the
case of dynamical systems likely extends to cases of reductionM
involving other kinds of model as well. For this reason, the strategy
for inter-theory reduction in physics described here can be under-
stood as a local, model-based formulation of the Nagel/Schaffner
approach.
up the spring.
A global reduction of classical to quantum
mechanics is thus out of the question, since distinct derivations
of the classical model's success on the basis of the quantum model
are required for each of the two systems.
While global reduction between classical and quantum
mechanics fails in this case, there remains a non-trivial, and more
local, sense of inter-theoretic reduction in physics, which
I
describe in the next section, that has the potential to accommo-
date this sort of example. One thing that people might—and, I take
it, often do—mean by the claim that “classical mechanics reduces
to quantum mechanics” is that every physical system whose
behavior can be modelled in classical mechanics also can be
modelled in quantum mechanics at least as precisely. This manner
of construing the term “reduction” allows for the possibility that
for each system separately, there is a direct relationship between
the quantum and the classical model of that system that permits us
to understand why the classical model succeeds at describing the
system's behavior given that the quantum model is the more
detailed and accurate of the two descriptions. Arguments from
multiple realization present no reason for doubting that such
system-specific quantum-mechanical accounts of the success of
classical mechanics are available in such cases.
3.2. A local, model-based approach to inter-theoretic reduction
The example just discussed suggests a local picture of inter-
theoretic reduction in physics where reduction between theories
may consist of many local, fixed-system reductions between
models—which in turn may rely on assumptions specific to
individual systems—rather than requiring a single global deriva-
tion that references only the broad generalities that characterize
the theories as a whole. Let us provisionally suppose a notion of
reduction between two models of a single fixed system, reduc-
tionM, whereby the low-level model accounts for the success of the
high-level model at tracking the behavior of the system in
question; the nature of this relationship will be further illustrated
in the second half of this paper through the detailed analysis of
reductionM in a particular class of cases where both models are
dynamical systems. Let us designate our local concept of reduction
between theories reductionT, which we define as follows:
3.3. A worry about local reduction
The local approach to inter-theoretic reduction described in the
last section addresses worries about multiple realization (MR) by
allowing domain-relative, local derivations that employ context-
specific bridge principles. One concern about this sort of move,
however, is that in its acceptance of disjointed, local derivations of
multiply realized high-level regularities, this strategy foregoes a
certain legitimate demand for explanation of MR—that is, an
explanation that alleviates our sense of mystery as to how systems
with such disparate low-level descriptions can all give rise to the
same high-level regularity, presumably by identifying some salient
commonality among them.
The appropriate response to this sort of worry, I believe, begins
by distinguishing two problems concerning inter-theory relations:
first, the problem of demonstrating the sort of subsumption that I
take to define reduction, or more precisely of showing that any
system that can be accurately modelled in the high-level theory
can be modelled at least as accurately in the low-level theory;
second, the problem of explaining multiple realization. The
response then is simply to acknowledge that local approaches to
reduction address the first problem but not the second. Put more
bluntly, the response is simply that it is not the job of reduction, as
the notion is understood here, to explain how multiple realization
Criteria for Local Inter-Theoretic Reduction: Theory Th redu-
cesT to theory Tl iff for every system K in the domain of Th—that
is, for every system K whose behavior is accurately represented
by some model Mh of Th—there exists a model Ml of Tl also
representing K such that Mh reducesM to Ml.
There are several crucial points to note about this local, model-
based concept of inter-theoretic reduction. First, it presupposes a
semantic understanding of theories as families of models, rather
than a syntactic view which treats theories as axiomatic sets of

60
J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
comes about. Given the vast differences in the mathematical and
conceptual frameworks of theories between which reduction is
purported to hold, the task of understanding precisely how one
and the same physical system can consistently be described by
models of both theories simultaneously poses a significant enough
challenge that it deserves not to be conflated with the task of
explaining how multiple realization comes about. Reduction
already has enough on its plate, so to speak. Indeed, there are
many who doubt, and some, including pluralists like Cartwright
and Dupre, who outright deny that it is possible to effect the sort
of subsumption required by local approaches to reduction—that is,
who deny the possibility of the sort of theoretical “imperialism”
that seeks to include macroscopic systems like the moon within
the domain of quantum mechanics, or thermodynamic systems
within the domain of statistical mechanics (Cartwright, 1999;
Dupre, 1993). Local reduction facilitates such imperialism, but
without providing the sort of explanation of multiple realization
that is sometimes called for.
Brownian motion, or of GRW quantum mechanics), or non-
dynamical (e.g., the Ideal Gas Model), or where there is no global
splitting of the space of possible solutions of the model into state
spaces at different times. I also suggest how various features of DS
reduction might be extended to provide an analysis of the
relationship between symmetries of different models of the same
system. I leave it to future work to give a more detailed analysis of
reductionM in such contexts. However, I argue in Section 7 that the
strategy for reduction in these other cases is also likely to be
Nagelian in the broad, liberalized sense described in Section 1.1. As
was noted in the Introduction, one important general aspect of the
approach to reduction adopted here is to begin narrowly by
considering reduction in relatively specialized contexts and then
consider what generalities can be drawn across ever wider ranges
of cases, rather than demanding a high level of generality at the
outset.
4. Fixed-system reduction between dynamical systems models
Having made these remarks, it is worth pointing out that a
good deal of important philosophical analysis has been done on
the question of how to explain multiple realization in physics. In
particular, Batterman has noted that the phenomenon of univers-
ality in physics just is multiple realization, and that in physics one
does find explanations of universality in the form of renormaliza-
tion group analyses (Batterman, 2000). However, I emphasize
again that this sort of issue is distinct from (though closely related
to) the issue of how to effect the sort of theoretical subsumption
required by reduction, and that it is only this latter issue that I seek
to address here.
Dynamical systems models occur widely throughout physics: in
Hamiltonian models of non-relativistic and relativistic classical
mechanics and non-relativistic and relativistic classical field the-
ory, in Schrodinger picture models of non-relativistic quantum
mechanics, relativistic quantum mechanics and quantum field
theory, and in heat diffusion equations in thermodynamics, to
name a few cases.
The notion of a dynamical system has been defined in a
number of distinct ways that vary both in generality and in their
specific requirements. However, all definitions capture some
variation on the intuitive notion of a deterministic rule that
governs the time evolution of a point in some mathematical space,
which in real-world applications usually represents the state of a
physical system. More formally, a dynamical system is a triple
M ¼ ðS; T ; DÞ, where the “state space” S is a set, the “timeset” T ꢀ R
is an additive semigroup, and the “evolution operator” D : S ꢁ
T -S is a one-parameter group of transformations of S satisfying
the properties Dðx; 0Þ ¼ x and DðDðx; t1Þ; t2Þ ¼ Dðx; t1 þt2Þ for all xAS
and t1; t2 AT (Broer & Takens, 2010). For our purposes here, I will
further assume that S is a differentiable manifold endowed with a
norm, that T ¼ R, that D is continuous, and that D is an invertible
function of x for each time t. In dynamical systems that occur in
physics, the evolution operator D is usually determined by some
set of first-order differential equations of motion:
Part II: the case of dynamical
systems
My purpose in the second half of this paper is to illustrate more
precisely, through a specific class of cases, what is meant by “fixed-
system, inter-model reduction” (“reductionM”) in the local picture
of inter-theoretic reduction (“reductionT”) spelled out in Section
3.2. In the class of cases that I consider here, both models of the
system in question are dynamical systems models and, moreover,
are assumed to share a common time parameter. In Section 4, I
describe a general mathematical relationship, which I call “dyna-
mical systems reduction,” “DS reduction,” or “reductionDS,” that
provides a template for reduction between dynamical systems
models across a wide range of cases, as I show explicitly with
several examples in Section 5. Moreover, I show in Section 4.4 that
there is a strong sense in which DS reduction follows the broad
dx
dt
¼ f ðxÞ;
ð2Þ
where f(x) is a continuous function of x. The functions f and D are
related by the equation ∂ Dðx0; tÞ ¼ f ðDðx0; tÞÞ. That is, Dðx0; tÞ for
∂t
prescriptions of Nagel/Schaffner reduction, albeit in
a form
fixed x0 specifies a solution to Eq. (2).
adapted to the specialized context of fixed-system reduction
between dynamical systems models. Part of the significance of
reductionDS is that it provides an example of a general mathema-
tical relationship between models whereby one model can
improve on the scope, detail and accuracy of another model in
its description of a physical system, fully encompassing the latter
model's domain of application. It also serves to illustrate how
models employing radically different mathematical and concep-
tual frameworks can both consistently (within some margin of
approximation) and accurately describe the behavior of one and
the same physical system.
4.1. ReductionM for dynamical systems
The analysis of reduction between dynamical systems provided
in this section draws on the work of several authors. All explore
variations on the notion that reduction between dynamical
descriptions of
a system requires commutation between the
operation of dynamical evolution and some other operation, such
as coarse-graining, that maps elements of the low-level state space
into elements in the high-level state space. More precisely, the
thought is that application of the low-level dynamics for time t
followed by application of the mapping from the low- to the high-
level state space should approximately equal the result of first
applying this mapping and then applying the high-level dynamics
for time t. Wallace explores variations of this idea in the context of
an investigation into the quantum-mechanical (specifically,
As I argue in Section 6, certain central features of reductionDS
can likely be carried over to cases of reductionM in which the
models in question do not meet the preconditions for reductionDS
—
for example, in cases of reduction between dynamical systems
models that do not share a time parameter, or in cases where one
or both of the models Ml and Mh is stochastic (e.g., models of

J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
61
error
δ over timescale τ for initial conditions xh0 within some
domain of states dh ꢀ Sh. Then define the relation reductionDS as
follows (Fig. 1):
Mh reducesDS to Ml iff there exists a differentiable function B :
Sl-Sh that does not depend explicitly on time and a nonempty
subset dl ꢀ Sl in the domain of B such that dh ꢀ BðdlÞ, and for any
xl Adl,
jBðDlðxl; tÞÞꢂDhðBðxlÞ; tÞjh o2
δ
;
ð3Þ
or more concisely,
BðDlðxl; tÞ ꢃ DhðBðxlÞ; tÞ;
ð4Þ
for all 0rtr
The notation j○jh designates the norm on Sh, and the margin of
error 2
is taken to be implicit in the ꢃ of (4). Note that the
τ.
δ
expression BðDlðxl; tÞÞ on the left-hand side of (4) represents the
trajectory on Sh induced by the low-level dynamics Dl via the
function B, while the expression DhðBðxlÞ; tÞ on the right-hand side
of (4) represents the trajectory prescribed by the high-level
dynamics for the initial condition BðxlÞASh. The mathematical
relationship between Mh and Ml specified by reductionDS thus
requires that the function B—let us call it a “bridge map”—
commute with the operation of time evolution, where the time
evolution is specified by the low-level dynamics if it comes before
application of the bridge map and by the high-level dynamics if it
comes after application of the bridge map. The requirement that
dh ꢀ BðdlÞ is imposed for the purpose of ensuring that all possible
trajectories of Mh that would accurately track the behavior of the
relevant degrees of freedom of K are well-approximated by
trajectories induced via B by the low-level dynamics. In fact, in
cases of reduction one expects the trajectory BðDlðxl; tÞÞ to track the
behavior of these degrees of freedom more accurately than the
trajectory DhðBðxlÞ; tÞ if the low-level model Ml is the more accurate
Fig. 1. ReductionDS requires the existence of a time-independent “bridge map” B
from the state space of the low-level model, Sl, to that of the high-level model, Sh,
that approximately commutes with the operation of dynamical evolution for initial
states xl in a certain domain dl ꢀ Sl. This amounts to the requirement that the
trajectories on Sh induced via
B by the low-level dynamics approximate the
trajectories prescribed by the corresponding high-level dynamics.
Everettian) arrow of time (Wallace, 2012).6 Giunti discusses this
requirement in the context of reduction between dynamical
systems generally, although the specific criteria that he imposes
are too strong to apply to very many, if any, realistic cases in
physics (Giunti, 2006). Yoshimi discusses a similar approach in the
context of a general discussion about supervenience, with a view
toward applications in philosophy of mind (Yoshimi, 2012). Butter-
field suggests a formulation of this commutation condition that is
quite similar to (but formulated independently of) Yoshimi's, with
a view to applications in physics; however, he points out a number
of ways in which this condition may need to be weakened in order
to be applied to realistic cases in physics (Butterfield, 2011b). The
formulation of the commutation-based requirement for reduction
that I describe here differs from those of Wallace, Yoshimi and
Butterfield in that it takes the high-level and the low-level models
to be specified independently of each other, whereas on these other
formulations the high-level dynamics are specifically defined as
the dynamics induced by the low-level dynamics through some
coarse-graining procedure. Giunti's formulation, like the one
suggested here, does take the high- and low-level models to be
independently specified, though requires the operation connecting
the state spaces of the models specifically to be an injection and
requires the commutation to be exact; both of these requirements
preclude application to most realistic cases in physics. As I show in
the next section, the formulation of the requirement that I present
here is refined in such a way as to allow for broad application
specifically to inter-model reduction in physics.
Assume, then, that two distinct dynamical systems models
Mh ¼ ðR; Sh; DhÞ and Ml ¼ ðR; Sl; DlÞ, which share a time parameter t,
both serve to describe the same physical system K, where the
subscripts "h" and "l" designate the "high" and "low"-level models
and their components. (This is the case, for example, with non-
relativistic and relativistic models of a proton, or between non-
relativistic quantum-mechanical and relativistic quantum-field-
theoretic models of an electron, or between classical and quantum
mechanical models of the center of mass of a golf ball, and in a
wide range of other instances.) Assume, moreover, that Mh
succeeds at tracking the dynamical behavior of some subset of
the degrees of freedom characterizing system K within margin of
of the two descriptions. That is, we expect that over timescale
BðDlðxl; tÞÞ tracks the behavior of the relevant physical degrees of
freedom of K to within a margin of error such that . This in
turn requires that the trajectories DhðBðxlÞ; tÞ and BðDlðxl; tÞÞ agree
with each other to within a margin of 2 . Thus, the physical
τ,
γ
γoδ
δ
property of system K that is represented by xh in the high-level
model is represented in the low-level model by BðxlÞ, so that there
is a strong sense in which the high-level state and the function of
the low-level state associated with the bridge map co-refer in the
range of contexts where both successfully track the behavior of the
system.7
I should clarify here that reductionDS is not being presented as a
necessary or sufficient condition for reductionM in cases where two
dynamical systems sharing a time parameter are purported to
describe the same physical system (although I suspect that it does
serve to characterize most such instances). It is not being put
forward as a sufficient condition because, if the quantity BðxlÞ in
the low-level model is supposed to stand in as the low-level
model's surrogate for xh, one should expect it to mimic xh's
behavior in ways other than through its dynamical evolution—
for example, through its symmetry transformation properties. This
point is discussed further in Section 6. ReductionDS is also not being
put forward as a necessary condition in order to leave open the
possibility that there might be other ways in which dynamical
systems models that share a time parameter and describe the
same physical system may relate to each other so as to facilitate
the kind of domain subsumption required of reduction.8 Rather,
7
This claim should be interpreted with the understanding that these quan-
tities are not exactly equal and that they only serve to approximate the true
behavior of the relevant physical degrees of freedom in K.
6
8
It was also Wallace who first introduced me, in conversation, to this way of
Assuming that the Everett interpretation is an empirically viable formulation
thinking about reduction.
of quantum mechanics, the reduction of classical Newtonian models to quantum

62
J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
the relation specified by reductionDS is being suggested here as a
template for reduction that is useful for the treatment of a wide
range of examples in this class of cases. Moreover, it suggests a
promising line of approach to reduction in some inter-model
relations that are not as yet well-understood (e.g., between models
of non-relativistic quantum mechanics and interacting quantum
field theory, both of which can be formulated as dynamical
systems using the Schrodinger picture) as well as suggesting
extensions to inter-model reduction involving other kinds
of model.
circumstances where the high-level model “works” at tracking the
behavior of the system K. And, of course, the question as to what,
precisely, these circumstances are is an empirical one requiring
reference to the system itself.
The criteria for DS reduction reflect a more general approach to
reduction in which reduction is, in a certain sense, regarded not as
a two place relation between high- and low-level descriptions, but
as a three-place relation between high-level description, low-level
description and that portion of the physical world that both
represent. This follows naturally from the particular construal of
“reduction” adopted here, which requires that the low-level
description subsume the domain of application of the high-level
description; in order to determine what this domain of application
is, it is necessary to consider not only the descriptions themselves,
but the quality and scope of agreement between the high-level
description and the portion of the world it is supposed to
represent.
4.2. Reduction as a three-place relation
I wish to address a concern about DS reduction that may occur
to some readers: specifically, that given any two dynamical
systems models in which the low-level state space has at least
the same cardinality as the high-level one, the criteria for DS
reduction will be satisfied trivially if we allow
large, and and dh to be sufficiently small. This concern can be
addressed by highlighting the fact that not just any values of δ, τ
and dh are consistent with the criteria for DS reduction; rather,
these parameters are constrained by the nature of the fit between
the model Mh and the behavior of the system K itself. Recall that
δ to be sufficiently
τ
4.3. “Differential” conditions for DS reduction
Assume that the dynamics of the high- and low-level models
are specified by first-order differential equations of the form (2),
dxh
dxl
dt
so that
¼ f hðx; tÞ and ¼ f lðx; tÞ.10 Then it is possible to state a
the parameter
which Mh tracks the behavior of K;
δ
serves to characterize the degree of accuracy with
characterizes the timescales
strongerd,t“differential” version of (4) that requires the low-level
quantity BðxlÞ to approximately satisfy the high-level equations of
motion. Making the variable substitution x0h ꢄ BðxlÞ, we can write
this requirement compactly as
τ
over which it does so; dh characterizes the set of initial states for
which Mh succeeds in tracking the system's behavior within this
timescale and margin of error. Determining the values of these
parameters is therefore an empirical matter, to be assessed on the
basis of the observed fit between the high-level model and
system.9 Only once we have fixed these parameters—or ranges of
possible values for them—does the task of assessing whether the
formal requirements of DS reduction are met for a given pair of
models become a question purely of mathematics. Moreover, once
these parameters have been set, the criteria for DS reduction
represent a highly non-trivial constraint on the mathematical
relationship between the two models.
It should be emphasized here that the two models Mh and Ml
by themselves do not provide us with sufficient information to
assess whether the requirements for DS reduction are met
between them. A third element—namely, the system itself, and
more specifically a comparison of the system's behavior with the
behavior prescribed by Mh—is needed in order to assess whether
the criteria for DS reduction are satisfied. This should come as no
surprise since reductionM requires that the low-level model dove-
tail structurally with the high-level model only in those
dx0h
dt
ꢃ f hðx0h; tÞ
ð5Þ
or more explicitly as
dBðxlðtÞÞ
ꢃ f hðBðxlðtÞÞ; tÞ:
dt
ð6Þ
If this condition holds over timescale
τ
and the margin of error
characterizing the approximate equality in (6) is
(4) will be satisfied over timescale and within margin of error
As we will see in Section 5, in typical cases, the
η, then condition
τ
2
δ .
ητ 11
ꢄ
approximate equality (6) will continue to hold only as long as
xl(t) remains in some restricted domain dl0 ꢀ Sl. Also, since the
validity of the relation (6) hinges on the behavior of BðxlðtÞÞ, which
in turn depends on the behavior of xl(t), which in turn depends on
the low-level dynamical equations of motion, the condition (6)
should be deduced from the low-level dynamics, together with the
restriction xl Adl0.
4.4. DS reduction as a special case of local Nagel/Schaffner reduction
(footnote continued)
Recall that global Nagel/Schaffner reduction, described in Section 2,
Everettian models in the context of certain classically behaving systems will require
reference to the branching structure of the quantum state in Everett's theory. In
this case, as in the cases considered here, reduction concerns two dynamical
systems models (a classical Hamiltonian model and a model specifying the unitary
evolution of the quantum state) sharing a time parameter. However, unlike the
cases considered here, the quantum-mechanical quantity that co-refers with the
phase space point in the classical model is not a function of the whole low-level
state, but only of some particular component associated with one particular branch
of that state that emerges dynamically through decoherence (where the branch in
turn corresponds to one particular “world” in the Everettian Many Worlds picture).
Reductions such as this, in which the low-level model describes a multiverse that
emerges from the dynamics, are not included in the present analysis and require
separate treatment.
distinguishes four “theories”: a low-level theory Tl, a high-level theory
Th, an “image” theory Thn, and an “analogue” theory T0 . Tnh is
h
formulated in terms of the concepts of Tl and deduced directly from
Tl, typically for some restricted boundary or initial conditions. T0 is
then obtained from Tnh by straightforward bridge principle substhitu-
tion, and is formulated in terms of the concepts of Th. If the reduction
is successful, T0h will be “strongly analogous” to Th, where strong
analogy signifies approximate agreement of some sort.
DS reduction, if effected through proof of the differential condition
described in Section 4.3, follows essentially the same series of steps,
but with models rather than theories. By analogy with the four
“theories” of global Nagel/Schaffner reduction, in DS reduction one
9
Having said this, it is also important to acknowledge that it may not always
be a simple matter to assess the precise parameters of agreement between our
models and the systems they represent. Nevertheless, it is often possible at the very
least to place empirically determined bounds on δ, τ and dh. For example, we know
10
that current models of quantum field theory will likely cease to track the dynamical
behavior of systems they represent for initial states of momentum near the Planck
scale. We do not expect any successor to QFT to dovetail with the theory's
prescriptions for the behavior of states in this domain.
Where, recall, ∂ Dhðxh0; tÞ ¼ f hðDhðxh0; tÞÞ and ∂ Dlðxl0; tÞ ¼ f lðDlðxl0; tÞÞ.
∂t
∂t
11
To see this, integrate both sides of (6) from t¼0 up to any time t less than τ,
employing the substitution f hðBðxlðtÞÞÞ ¼ ∂∂tDhðBðxl0Þ; tÞ, the fact that BðDlðxl0; 0ÞÞ ¼
Bðxl0Þ ¼ DhðBðxl0Þ; 0Þ, and the fact that xlðtÞ ¼ Dlðxl0; tÞ.

J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
63
has a low-level model Ml, a high-level model Mh, an “image model”
of factors affecting the timescale on which (6) holds. Where
necessary, proof of (6) is deferred to the Appendix.
Mnh and an “analogue model” M0 . The image model Mnh is formulated
h
using elements of the model Ml—that is, in terms of mathematical
structures defined on Ml's state space—and can be deduced from Ml
solely on the basis of a restriction to a particular domain of states in Sl.
Its dynamics are given by the relation,
5.1. CM/QM
Let the high-level model Mh be a model of classical mechanics
whose state space is some N-particle, 6N-dimensional phase space
Sh ꢄ ΓN and whose dynamics Dh are given by the solutions to the
“Image Model” Dynamics:
d
dt
ꢀ
ꢁ
ꢀ
ꢁ
BðxlðtÞÞ ꢃ f hðBðxlðtÞÞ; tÞ
ð7Þ
dX dP
∂H
∂P
∂H
P2
Hamilton equations
;
¼
; ꢂ∂X , with H ¼ 2M þVðXÞ. In the
dt dt
ꢀ
∂H
which holds for xl in some domain d0l ꢀ Sl. Note that this relation
approximately takes the same form as the high-level equation of
motion dxh=dt ¼ f hðxh; tÞ, but with xh replaced by its counterpart
BðxlÞ in the low-level model. Recall from Section 4.3 that satisfac-
tion of the image model dynamics within appropriate margins
suffices to ensure satisfaction of the condition (4). The analogue
model M0h is then obtained from the image model through
Bridge Map Substitution:
notation of Section 4.3, we have xh ꢄ ðX; PÞ and f hðxhÞ ꢄ
;
∂P
ꢀ
ꢁ
∂H
∂V
ꢂ∂XÞ ¼ MP ; ꢂ∂X
.
Let the low-level model Ml be a model of non-relativistic
quantum mechanics whose state space is some N-particle Hilbert
space Sl ꢄ HN and whose dynamics Dl are given by the solutions to
2
^
the Schrodinger equation i∂j ¼ Hj
ψ
〉, with H ¼ 2M þVðXÞ. In the
ψ
ψ
∂t
〉
P
^
^
^
^
notation of Section 4.3, we have xl ꢄ j
〉 and f lðxlÞ ꢄ ꢂiHj
ψ〉.
Consider the function B : HN-ΓN from the low-level to the
high-level state space that maps a quantum state into the phase
space point associated with the expectation values of position and
momentum in this state:
x0h ꢄ BðxlÞ
ð8Þ
and the analogue dynamics are specified by the relation:
“Analogue Model” Dynamics:
dx0h
dt
^
^
^
^
BðxlÞ ꢄ ð〈
Consider further the domain of narrow wave packet states:
d0l ꢄ fj
〉AHN j j 〉 ¼ jq; p〉 for some q; pAΓNg;
ψ
jX j
ψ
〉; 〈
ψ
jP j
ψ
〉Þ ¼ ð〈X〉; 〈P〉Þ:
ð11Þ
ꢃ f hðx0h; tÞ:
ð9Þ
Note that this dynamical equation is identical to the high-level
equation of motion, apart from the approximate nature of the given
equality. The domain of applicability of this equation within Sh is the
image domain Bðd0lÞ. Note that the expression BðxlÞ, which occurs in
the image model, is an expression built from structures within the low
level model Ml—in this sense, the image model is formulated in the
mathematical “language” of the low-level model. On the other hand,
the more condensed notation of the analogue model conceals the
detailed construction of x0h from quantities in the low-level model Ml,
regarding x0h simply as a point in Sh rather than as a quantity
constructed from elements of Ml; in this sense, one may view the
analogue model as formulated in the mathematical “language” of the
high-level model. For reduction to take place, the analogue model Mh0
must be “strongly analogous” to the high level model Mh. In DS
reduction, the relationship of strong analogy is unambiguous, and
specifically requires that
ψ
ψ
ð12Þ
where jq; p〉 denotes a narrow wave packet with average position q
and average momentum p, and the required standard of narrow-
ness in X is defined relative to the scale of spatial variation of the
potential V. On the basis of the low-level Schrodinger dynamics,
one can show that for j
this particular case, so that the quantity associated with the bridge
ψ
〉 in d0l, the relation (6) holds relative to
map approximately satisfies the high-level Hamilton equations:
!
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
d
dt
∂H
∂P
∂H
∂X
^
^
ð〈
ψ
jX j
ψ
〉; 〈
ψ
jPj
ψ
〉Þ ꢃ
; ꢂ
ψ〉
^
^
^
^
ψ ψ
〈
ψ
j X j
ψ
〉;〈
ψ
j P j
〈
ψ
j X j
ψ
〉;〈
j P j
〉
!
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
P
M
∂VðXÞ
∂X
¼
; ꢂ
;
ð13Þ
^
^
〈X〉
〈P〉
or, more compactly, employing the variable substitutions
‘Strong Analogy’:
ðX0; P0Þ ꢄ ð〈
ψ
jX j
ψ
〉; 〈
ψ
jP j
ψ
〉Þ,
^
^
jx0hðtÞꢂxhðtÞjh o2
δ
80rtr
τ
;
ð10Þ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
!
ꢃ
ꢄ
ꢂ
ꢂ
ꢂ
d
∂H
∂P
∂H
∂X
P
M
∂VðXÞ
∂X
ðX0; P0Þ ꢃ
; ꢂ
¼
; ꢂ
:
ð14Þ
where again is the reduction timescale and the measure of
τ
X0;P0
X0;P0
P0
X0
dt
approximation between the analogue and high-level dynamics is
provided by the norm j○jh on the high-level state space. Note that
this “strong analogy” claim is just the condition (3) rewritten using
bridge map substitution xh0 ðtÞ ꢄ BðDlðxl0; tÞÞ and the definition
xhðtÞ ꢄ DhðBðxl0Þ; tÞ.
This claim follows straightforwardly from Ehrenfest's theorem,
^
^
^
^
∂X
which states that
from the well-known result that for wave packets whose position-
space width is narrow by comparison with the characteristic
¼ ꢂ〈∂VðXÞ〉, from the fact that
¼
MP , and
d〈P〉
d〈X〉
dt
dt
^
ꢃ ꢂ∂VðXÞj
(Merzbacher,
∂X 〈X〉
d〈P〉
length scales on which V(X) varies,
^
dt
5. Examples of DS reduction
1998).
On timescales where wave packets remain in d0l, the relation
(13) ensures that quantum expectation values of position and
momentum approximately satisfy the classical dynamics of the
high-level model, and that (4) is satisfied. These timescales in turn
will depend on two general factors: (1) the maximum spread in
position and momentum of wave packets allowed by the defini-
tion of d0l, which in turn will depend on the required accuracy of
the approximation in (13) and on the scale of spatial variation of
the potential V; (2) the dynamics of the low-level model, and in
particular the mass M and the strength of chaotic effects asso-
ciated with the Hamiltonian H. In general, the larger M is, the
slower wave packets in HN will tend to spread; moreover, it is
typically the case that the smaller the Lyapunov exponent char-
acterizing chaotic divergences of phase space trajectories
In this section, I show that the conditions for DS reduction are
satisfied by a range of different model pairs. I do so specifically by
showing that the relation (6) holds for a certain choice of bridge
map and a certain domain of states in the low-level state space,
offering qualitative remarks concerning the timescale over which
the relation (6) continues to hold. My presentation of each example
will follow a common outline: (a) specification of the state spaces
and dynamical equations of the two models; (b) specification of the
bridge map between the models; (c) specification of the domain dl0
where Eq. (6) holds between the models; (d) statement of the
condition (6) for the particular pair of models being considered;
(e) restatement of this condition, in the form (5), which more
directly resembles the high-level dynamical equation; (f) discussion

64
J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
according to the classical dynamics of the system, the slower the
the domain of low-momentum 4-spinors:
rate of spreading of wave packets in HN (Zurek & Paz, 1995).12
(
)
Z
μ
X
μ
m
3
d0l ¼ ψaðxÞAHDjψaðxÞ ¼
d k
iðkÞuai ðkÞeꢂikx
;
51
;
~
ψ
0
i ¼ 1;2
ð16Þ
where uai ðkÞ are positive energy eigenstates of the Dirac Hamilto-
5.2. NRQM/RQM
Let the high-level model Mh be a model of nonrelativistic
quantum mechanics of a spin-1/2 particle whose state space is
nian indexed by momentum k and the spin i, and the upper limit
μ
on the momentum integral imposes the restriction to low-
momentum states. On the basis of the low-level Dirac dynamics,
one can show that for ψaðxÞAdl0, the relation (6) holds for the given
bridge map and models, so that BðxlÞ in this case approximately
satisfies the high-level Pauli equation:
13
the Hilbert space of 2-spinors Sh ꢄ HP and whose dynamics Dh
are given by the solutions to the Pauli equation, i∂∂tϕαðx; tÞ ¼
αβ
2m σ ꢅ ðp ꢂqAðxÞÞꢆ þqVðxÞg ϕβðx; tÞ, where ϕαðx; tÞ are 2-spinor
2
1
^
f
½
functions on 3-D space,
α
;
β
¼ 1; 2, σi are the Pauli matrices, m is
the particle's mass, q is its electromagnetic charge, and A(x) and V
ꢇ
ꢈ
(x) are fixed background electromagnetic potentials.14 In the
αβ
ꢅ
ꢆ
∂
∂t
1
2m
α
β
2
eimtPa ψaðx; tÞ ꢃ ꢂi
½
σ
ꢅ ðꢂi∇ꢂqAðxÞÞꢆ þqVðxÞ
ðeimtPa ψaðx; tÞÞ
notation of Section 4.3, we have xh ꢄ ϕαðx; tÞ and f ðxhÞ ꢄ
h
αβ
2m σ ꢅ ðp ꢂqAðxÞÞꢆ þqVðxÞg ϕβðx; tÞ.15
2
1
^
ꢂif
½
ð17Þ
Let the low-level model Ml be a model of relativistic quantum
mechanics for a spin-1/2 particle whose state space is the Dirac
Hilbert space of 4-spinors Sl ꢄ HD and whose dynamics Dl are
or, more compactly, employing the variable substitution
ϕ0αðx; tÞ ꢄ eimtPa ψaðx; tÞ,
β
given by the solutions to the Dirac equation i∂∂tψaðx; tÞ ¼ p½
α
ꢅ
ꢇ
ꢈ
αβ
!
∂
∂t
1
2m
ϕ0αðx; tÞ ꢃ ꢂi
½
σ
ꢅ ðꢂi∇ꢂqAðxÞÞꢆ þqVðxÞ
ϕ0βðxÞ
ð18Þ
ac
2
ðꢂi∇ꢂq A ðxÞÞþ
β
mþqVðxÞꢆ ψcðx; tÞ, where a ¼ 1; 2; 3; 4 and like-
wise for b and c, where repeated spinor indices have been
summed over, αi and are the Dirac matrices, m is the mass of
the particle, q is its electromagnetic charge, and A(x) and V(x) are
β
for all spatial positions x (not to be confused with the high-level
state xh or the low-level state xl). Proof of this relation can be
found in Holland and Brown (2003), or in most good textbooks on
relativistic quantum mechanics, such as Merzbacher (1998). The
timescales on which ψaðx; tÞ remains in d0l will depend primarily on
the choice of background fields A(x) and V(x); the domain d0l will
be preserved as long as these background fields do not transfer
significant amounts of momentum (that is, on the order of m) to
the spinor field.
fixed background electromagnetic potentials. In the notation of
!
Section 4.3, we have xl ꢄ ψaðx; tÞ and f lðxlÞ ꢄ ꢂi½
α
ꢅ ðꢂi∇ꢂq A ðxÞÞ
16
ac
þ
β
mþqVðxÞꢆ ψcðx; tÞ.
Consider the function B : HD-HP from the low-level to the
high-level state space given by
α
BðxlÞ ꢄ eimtPa ψaðx; tÞ;
ð15Þ
α
where Pa is the projector onto the subspace of the 4-spinor space
corresponding to the upper two components of any spinor.
Because of the time-dependent factor eimt, this bridge map may
seem to violate the requirement that bridge maps not depend
explicitly on time. However, the violation is only apparent since
the state space is in fact the projective Hilbert space, and multi-
plication of a Hilbert space vector in either the high-or low-level
model by an overall phase (whether time-dependent or not) does
not affect the projective representation of the state. Now consider
5.3. NRQM/RQFT
Let the high-level model Mh be model of N free spinless
particles in non-relativistic quantum mechanics whose state space
is some N-particle Hilbert space Sh ꢄ HN and whose dynamics Dh
are given by the solutions to the non-relativistic Schrodinger
equation for N free particles all with mass m: i∂∂tψðx1; …; xN; tÞ ¼
P
N
1
ꢂ
xh ꢄ
∇
2i ψðx1; …; xN; tÞ. In the notation of Section 4.3, we have
i ¼ 1
2m
P
N
2i ψðx1; …; xN; tÞ.17
1
ψ
ðx1; …; xN; tÞ and f hðxhÞ ꢄ i
∇
i ¼ 1
2m
12
I should note that the low-level model in this case will not suffice to model
Let the low-level model Ml be a model of a free massive scalar
quantum field in relativistic quantum field theory whose state
space is the free-particle Fock space of Klein–Gordon quantum
field theory, Sl ꢄ FKG, and whose dynamics Dl are given by the
macroscopic systems like the center of mass of a planet or a baseball, since any
realistic quantum model of such a system must take account of the system's
interaction with its environment and, in particular, the effects of decoherence (the
reduction between quantum and classical models of such systems is a more
intricate matter, which I will not consider here). However, in smaller systems like
large molecules, effects of decoherence can often be neglected and the evolution of
the molecule's center of mass in some cases can be accurately described by the
unitary evolution of a pure state (“bucky ball” interference experiments with large
molecules have shown this explicitly, as discussed, e.g., in Schlosshauer, 2008, chap.
6). The relatively large mass of such molecules by comparison with smaller
particles like electrons, protons and neutrons slows the rate at which pure state
wave packets spread, thereby allowing narrow wave packets to maintain their
approximately classical evolutions over longer timescales in these systems. On such
timescales, both the classical and quantum models are adequate to describe the
motion of the molecule's center of mass to within a certain reasonable margin
of error.
solutions to the Schrodinger equation for a free Klein–Gordon
R
3
2
2
^
^
quantum field theory, i∂∂tj
Ψ
〉 ¼
d x½
ðxÞþð∇
ϕ
ðxÞÞ þm2
ϕ
2ðxÞꢆ
1
2
^
π
^
j
Ψ
〉, where
ϕ
ðxÞ is the Hilbert space operator associated with the
^
π
scalar field, and
ðxÞ the Hilbert space operator associated with its
^
^
π
conjugate momentum. Using the usual expansions of
ϕ
ðxÞ and ðxÞ
†
^
^
in terms of creation and annihilation operators ak and ak,the
Schrodinger equation can be re-written, i∂∂tj
R
d3k
^† ^
Ψ
〉 ¼ ð
Ekakak þ
ð2π
Þ3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
CÞj
Ψ
〉, where Ek ꢄ k2 þm2 and C signifies a constant (associated
13
Where the “P” is for “Pauli.”
Unless explicitly stated otherwise, I employ the Einstein summation con-
14
with the vacuum energy) that diverges with the cutoff of
vention over repeated indices throughout.
the theory. In the notation of Section 4.3, we have xl ꢄ j
Ψ〉
15
The reader should note that for purposes of concision, I am abusing notation
2ðxÞꢆj
Ψ
〉 ¼ ½
R
R
^
^
d3k
3
2
here, since the state xh is not properly described by the components ϕαðx; tÞ, but
and
f lðxlÞ ꢄ ꢂi12 d x½
ðxÞþð∇
ϕ
ðxÞÞ þm2
ϕ
2
^
π
ð2
π
Þ3
R
P
α
α
rather by the expression xh ꢄ jϕ〉 ¼
dxϕαðx; tÞ jx 〉, where jx 〉 is a state of
α
^† ^
ðEkakak þCÞꢆj
Ψ〉.
position x and spin
α
(say, in the z-direction). Likewise, the expression for f hðxhÞ
employs a similar abuse of notation, as does my notation describing the low-level
relativistic Dirac model.
16
17
As in the high-level model, the reader should note the abuse of notation for
I employ the same abuse of notation here described in the footnotes of the
purposes of concision.
previous section.

J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
65
Consider the function B : FKG-HN from the low-level to the
high-level state space given by
5.4. NM/SR
Let the high-level model Mh be
a
model of Newtonian
BðxlÞ ꢄ eiðNmþCÞt〈0j
ϕ
ðx1Þ⋯
ϕ
ðxNÞj
Ψ
〉:
ð19Þ
^
^
mechanics whose state space is the non-relativistic phase space
Sh ꢄ ΓNR of a single massive charged particle in a background
electromagnetic field and whose dynamics Dh are given by the
solutions to the corresponding non-relativistic Hamilton equa-
As in the previous example, the requirement that the bridge map
does not depend explicitly on time is preserved despite the
exponential eiðNmþCÞt since it is projective Hilbert space rather
than the Hilbert space itself that represents possible physical
states. Now, consider the domain of states in the low-level space
consisting of N-particle states of momenta k1,…,kN much less than
m:
ꢅ
ꢆ
ꢀ
ꢁ
ðP ꢂqAðXÞÞ2
2M
∂H
dX dP
tions,
;
¼
NR; ꢂ∂HNR , with HNR
¼
þq ðXÞ, where q
ϕ
∂P
∂X
dt dt
is the particle's charge, M its mass, and
ϕ
ðXÞ and A(X) external
electrostatic and magnetic vector potentials, respectively. In the
ꢅ
ꢆ
∂H
notation of Section 4.3, xh ꢄ ðX; PÞ and f hðxhÞ ꢄ
NR; ꢂ∂HNR
¼
∂P
∂X
ꢇ
Z
ꢅ
ꢆ
μ
μ
m
†
kN
†
d3k1⋯d3kN
ðk1; …; kNÞa ⋯ak j0〉; 51g:
ð20Þ
P ꢂqA; qðP ꢂqA Þ ∂X ꢂq∂X . Using a number of vector identities and
∂
ϕ
∂Ai
d0l ¼
j
Ψ
〉AHKGjj
Ψ
〉 ¼
i
i
^
^
~
ψ
1
M
M
0
the definitions of the electric and magnetic fields in terms of the
On the basis of the low-level free RQFT dynamics, one can show
〉 in d0l, the relation (6) holds relative to this particular
potentials
ϕ and A, one can show that, combined, these two
that for j
Ψ
equations yield the non-relativistic Lorentz Force Law:
case, so that,
d2X
M
¼ qEþqv ꢁ B.
dt2
ꢅ
ꢆ
∂
∂t
i
eiðNmþCÞt〈0j
ϕ
ðx1Þ⋯
ϕ
ðxNÞj
Ψ〉
^
^
Let the low-level model Ml be a model of relativistic classical
mechanics whose state space is the relativistic phase space
Sl ¼ ΓSR of a single relativistic massive particle in a background
electromagnetic field and whose dynamics Dl are given by the
N
X
1
ꢅ
ꢆ
2
i
eiðNmþCÞt〈0j
ϕ
ðx1Þ⋯
ϕ
ðxNÞj
Ψ
〉
ð21Þ
^
^
ꢃ ꢂ
∇
i ¼ 1 2m
solutions to the corresponding relativistic Hamilton equations,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
∂HSR
∂HSR
∂x
2
dx
dt
dX
or, more compactly, employing the variable substitution
¼
,
¼ ꢂ
, with HSR
¼
ðpꢂqAÞ þM þq
ϕ, where q is the
∂p dt
^
^
ψ0ðx1; …; xNÞ ꢄ eiðNmþCÞt〈0j
ϕ
ðx1Þ⋯
ϕ
ðxNÞj
Ψ〉,
particle's charge, M its mass, and
ϕðxÞ and A(x) external electro-
static and magnetic vector potentials, respectively. In the notation
N
X
∂
∂t
1
0ðx1; …; xNÞ ꢃ ꢂ
∇
2i ψ0ðx1; …; xNÞ:
ð22Þ
of
Section
4.3,
we
have
∂Ai
xh ꢄ ðx; pÞ
and
i
ψ
ꢅ
ꢆ
ꢅ
ꢆ
i ¼ 1 2m
p ꢂqA; qðp ꢂqA Þ ∂x ꢂq∂x . Using a number of
∂
ϕ
∂H
SR; ꢂ∂HSR
¼
i
∂p
∂x
γ
M
i γ
M
f hðxhÞ ꢄ
The proof of this relation is given in the Appendix. If j
Ψ
〉 begins in
vector identities and the definitions of the electric and magnetic
d0l at t¼0, the relation (21) will be preserved for all time since
there are no interactions that might result in a change of particle
number or a transfer of momentum between particles in the low-
level state space.18
fields in terms of the potentials
ϕ and A, one can show that,
combined, these two equations yield the relativistic Lorentz Force
d
Law:
ð
dt γMvÞ ¼ qEþqv ꢁ B.
Consider the function B : ΓSR
-ΓNR from the low-level to the
As an extension of this example, one might consider the
reduction of the non-relativistic model considered here to a model
high-level state space that identifies the point (X,P) in ΓNR with the
point (x,p) in
BðxlÞ ꢄ ðx; pÞ:
Γ
SR, so that ðX; PÞ ꢄ ðx; pÞ:
of interacting relativistic quantum field th3eory—for example,
4
^
^
massive scalar quantum field theory with a
ϕ
or
ϕ
interaction.
ð23Þ
In such a case, we would wish to underwrite the non-relativistic
quantum description of N low-momentum particles widely sepa-
rated in space (for we do not expect them to act like free particles
when they are close together) on the basis of the interacting
quantum field theory. This is a substantially more difficult task
than in the case of free quantum field theory, for the question of
precisely how to identify those QFT states of the interacting theory
that behave as N-particle states, or of whether one can even do
this, remains a matter of some dispute in the foundations of
quantum field theory (Fraser, 2011; Wallace, 2011). Nevertheless,
one expects that the reduction, if it can be effected, will follow the
same basic pattern as in the case discussed here, except that the
domain of states in the QFT model for which the bridge map
function approximately satisfies the non-relativistic equation of
the high-level model will likely impose a restriction not only to
states of low momentum, but also to states in which the N
particles are in some quantifiable sense widely separated in space
(so that they do not interact).
Note that the bridge map in this case is trivial in a sense since it
maps the relativistic phase space point into the non-relativistic
phase space point with the same position and momentum (though
it should be noted that the dynamics of the two models will relate
momentum and velocity differently). Now consider the domain
ꢇ
ꢈ
pꢂqAðxÞ
d0l ꢄ ðx; pÞAΓSRj ¼
51
;
ð24Þ
γ
m
where v ¼ pꢂqA is the velocity of the particle (this follows from the
γ
well-known mrelationship between relativistic canonical momen-
tum and velocity, p ¼
γ
mvþqAðxÞ).19
On the basis of the low-level relativistic dynamics, one can
show that for (x,p) in d0l, the relation (6) holds relative to this
particular case, so that the quantity associated with the bridge
map (namely, position and momentum in the relativistic phase
space) approximately satisfies the nonrelativistic Hamilton equa-
tions:
18
!
However, it is important to note that scalar quantum field theories such as
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
d
dt
∂HNR
∂P
∂HNR
∂X
the one considered here are more a pedagogical tool for illustrating the basic
principles of quantum field theory than they are a description of any real physical
system that would also be well-described by a model of non-relativistic quantum
mechanics. Thus, it is unlikely that there is any real system K to which both models
apply. However, the example is nevertheless instructive in that it does serve to
illustrate the sort of inter-model relation that satisfies the requirements of DS
reduction. Moreover, this sort of reduction between quantum-mechanical and
quantum-field-theoretic models might be extended to more realistic QFT models.
ðx; pÞ ꢃ
; ꢂ
ð25Þ
x;p
x;p
19
Note that I have set c¼1. Were we to include factors of c in our analysis, we
would see that the domain restriction amounts to the requirement that vc 51.

66
J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
or, employing the variable substitutions ðX0; P0Þ ꢄ ðx; pÞ,
master equation, thereby also satisfying the condition (6):
!
^
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
d TrEρSE
^
^
^
^
^
i
ꢃ ½HS; TrE
ρ
SEꢆꢂi
Λ
½X; ½X; TrE
ρ
SEꢆꢆ;
ð28Þ
d
dt
∂HNR
∂P
∂HNR
∂X
ðX0; P0Þ ꢃ
; ꢂ
:
ð26Þ
dt
X0;P0
X0;P0
or, more compactly, employing the variable substitution
ρS ꢄ TrEðj
0
^
ψ
〉〈
ψ
jÞ,
The proof of this relation is straightforward and simply invokes the
1
0
p
ffiffiffiffiffiffiffiffiffi
fact that for v51,
γ
ꢄ
ꢃ 1. The relation (25) will continue to
^
d
ρS
1ꢂv2
ꢃ ½HS; ρ0Sꢆꢂi
Λ
½X; ½X; ρ0Sꢆꢆ:
ð29Þ
^
^
^
^
^
i
hold as long as (x,p) remains in d0l—that is, as long as v51. The
timescales on which this holds, in turn, are likely to depend on the
initial value of (x,p) in d0l and on the strength and form of the fields
dt
For a derivation of this equation and a discussion of the various
assumptions on which it relies, see Schlosshauer (2008, chap. 4).
As with the matter of specifying the domain d0l, the matter of
specifying precisely the timescale on which this relation holds is
more complicated than in the other examples considered; how-
ever, existing analyses strongly suggest that the high-level equa-
tions should hold to good approximation over macroscopically
long timescales, as observed in experimental applications of
density matrix master equations like the one considered here.
ϕ
ðxÞ and A(x).
5.5. Open-subsystem NRQM/closed-system NRQM
Let the high-level model Mh be a quantum-mechanical model
of an open system S whose state space Sh ¼ DðHSÞ is the space of
density operators on S's Hilbert space HS, and whose dynamics Dh
is given by solutions to the density matrix master equation
5.6. “Macro” CM/“Micro” CM
^
ρS
d
^
^
^
^
^
i dt ¼ ½HS; ρSꢆꢂi
Λ
½X; ½X; ρSꢆꢆ. In the notation of Section 4.3, we have
^
^
^
^
^
^
xh ꢄ ρS and f hðxhÞ ꢄ ꢂi½HS; ρSꢆꢂ
Λ
½X; ½X; ρSꢆꢆ. Let us assume here
Let the high-level model Mh be a non-relativistic model of
classical mechanics prescribing the Newtonian evolution of some
N centers of mass (e.g., of planets in the solar system) interacting
via some time-independent potential V dependent only on the
distance between the centers of mass (e.g., a Newtonian gravita-
tional potential). The state space will be a 6N-dimensional phase
space Sh ꢄ Γmacro and the dynamics Dh will be given by the
that the density matrix and its dynamics are defined indepen-
dently of and without reference to any lower-level pure state
description.20
Let the low-level model Ml be a model of non-relativistic
quantum mechanics in the closed system SE, whose state space
Sl ꢄ HS ꢇ HE is the tensor product space of S's Hilbert space and
the Hilbert space of its environment E; let the dynamics Dl of the
model be given by solutions to the Schrodinger equation
ꢀ
ꢁ
ꢀ
ꢁ
ꢁ
∂Hmacro
∂P
dX dP
solutions to the Hamilton equations
;
¼
; ꢂ∂Hmacro , with
dt dt
∂X
Pi2
2Mi
P
P
N
Hmacro
¼
þ1=2 iajVðXi ꢂXjÞ. In the notation of Section
iddtj
ψ
〉 ¼ ðHS ꢇ IE þIS ꢇ HE þHIÞj
system Hamiltonian HS, the environment Hamiltonian HE, and the
ψ
〉 for the closed system SE; the
^
^
^
^
^
i ¼ 1
ꢅ
ꢀ
P
PN
N
∂Hmacro
∂P
; ꢂ∂Hmacro
¼
1 ; …; M ; ꢂ∂X
∂
^
^
4.3, xh ꢄ ðX; PÞ and f hðxhÞ ꢄ
ja1VðX1 ꢂXjÞ; …; ꢂ
∂X
M1
1
P
P
∂
^
jaNVðXN ꢂXjÞÞ.
interaction Hamiltonian HI will not be precisely specified here, as
∂XN
the following remarks apply for a wide range of forms for these
operators and a wide range of systems. For examples of detailed
models, like the well-known Caldeira–Leggett model, in which a
precise form for these Hamiltonians is specified, as well as for a
detailed derivation of an approximate high-level density matrix
master equation of the form specified in Mh from these models,
the reader may wish to consult Schlosshauer (2008, chaps. 4 & 5)
and Joos et al. (2003, chap. 3) . In the notation of Section 4.3, we
Let the low-level model Ml also be a non-relativistic model of
classical mechanics whose state space Sl ꢄ Γmicro is the phase
space of some n point particles, where n4N, and whose dynamics
ꢅ
ꢆ
dp
dx
Dl are given by the solutions to the Hamilton equations
;
¼
dt dt
ꢅ
ꢆ
pi2
P
P
n
micro; ꢂ∂Hmicro , with Hmicro
notation
¼
4.3,
2m þ1=2 iajvðxi ꢂxjÞ. In the
∂H
i ¼ 1
∂p
∂x
i
of
Section
we
have
p
xh ꢄ ðX; PÞ
and
ꢅ
ꢆ
ꢅ
P
p
∂H
micro; ꢂ∂Hmicro
1 ; …; mn ; ꢂ∂x ja1vðx1 ꢂxjÞ; …;
∂
f hðxhÞ ꢄ
^
^
^
∂p
∂x
m1
N
1
have in this case that xl ꢄ j
ψ
〉
and f lðxlÞ ꢄ ꢂiðHS ꢇ IE þIS
P
∂
ψ
〉.21
ꢂ∂x janvðxN ꢂxjÞÞ.
^
^
ꢇHE þHIÞj
n
Now, consider a grouping of the n particles of the low-level
model into N sets, with each set corresponding to those particles
contained in one of the bodies whose center of mass is described
by the high-level model. Let n1 be the number of particles
contained in the first body, n2 the number in the second, and so
on up to nN in the Nth, where n1 þn2 þ⋯þnN ¼ n. Given this
Now, consider the function B : HS ꢇ HE-DðHSÞ from the low-
level to the high-level state space given by taking the trace over
the environment of the pure-state density operator associated
with the state j
ψ
〉AHS ꢇ HE:
BðxlÞ ꢄ TrEðj 〉〈
ψ
ψ
jÞ:
ð27Þ
grouping, consider the function B : Γmicro
level to the high-level state space given by
-
Γmacro from the low-
The relevant domain d0 ꢀ HS ꢇ HE in this particular case is harder
to specify than in thel earlier examples. However, it is generally
assumed to involve the restriction to states in which the tempera-
ture of the environment is high in comparison to certain other
relevant energy scales characterizing the system and environ-
ment; see Schlosshauer (2008, chap. 5).
!
P
P
ni
nN
X
X
m
x
m
x
α1 α1
αN αN
n1
nN
α1 ¼ 1
αN ¼ 1
BðxlÞ ꢄ
P
; …;
P
;
p
; …;
p
:
α1
αN
n1
nN
αN ¼ 1
α1 ¼ 1
αN ¼ 1
m
m
αN
α1
αi ¼ 1
ð30Þ
P
n
i
m
x
α
i
α
i
α
¼
i
1
For notational convenience, define X0i ꢄ
and P0i ꢄ
i
On the basis of the low-level Schrodinger dynamics for SE—
assuming any among a range of suitable low-level Hamiltonians—
P
n
m
1
α
i
α
¼
i
P
P
ni
p
. Moreover, assume that Mi ¼
m
and VðXi ꢂXjÞ ¼
αi
it is widely thought that one can show that for j
ψ〉 in some
jÞ associated with the
αi
αi
αi ¼ 1
restricted domain d0l, the quantity TrEðj
ψ
〉〈
ψ
ninj vðXi ꢂXjÞ, where Mi and V are the masses and potential
bridge map approximately satisfies the high-level density matrix
function of the high-level model. Now consider the domain
d0l ¼ fðx; pÞAΓmicrojjx ꢂX0j5Lv 8irirN and 81rαi rnig;
ð31Þ
αi
i
20
That is, let us forget momentarily about the association between the density
where Lv characterizes the typical length scale over which v varies,
determined by v's derivatives (see the Appendix for a more precise
definition of LV). On the basis of the low-level “micro” Hamilton
matrix of S and the trace over the environment of the pure state density matrix of
some larger system SE.
21
Thanks to David Wallace for this example.

J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
67
dynamics, one can show that for (x,p) in d0l, the relation (6) holds
relative to this particular case, so that the quantity associated with
the bridge map approximately satisfies the high-level “macro”
Hamilton equations, written out schematically as
translations, reflections and Galilean boosts. Similarly, regarding
the relation between non-relativistic and relativistic models of
quantum mechanics considered in Section 5.2, it can be shown
that the action of a Lorentz boost on the 4-spinor space of the
relativistic model induces a corresponding Galilean boost on the 2-
spinors of the non-relativistic model when acting within the
domain of low-momentum 4-spinors (Holland & Brown, 2003).
In the relation between the N-particle model of NRQM and the
model of free scalar quantum field theory discussed in Section 5.3,
spatial translations, rotations and reflections acting on the state of
the RQFT model induce corresponding transformations in the non-
relativistic model; it would be interesting to see whether Lorentz
boosts within the range of low-momentum N-particle states in the
QFT state space likewise induce approximate Galilean transforma-
tions on the state in the NRQM model. In the example of Section
5.4 concerning the relationship between Newtonian and relativis-
tic models of classical mechanics, rotations, translations, and
reflections on the relativistic phase space all induce corresponding
transformations on the non-relativistic phase space; likewise,
Lorentz boosts acting within the domain of low-velocity states of
the relativistic phase space induce transformations that approx-
imate corresponding Galilean transformations on the non-
relativistic phase space. In the example of Section 5.5, a rotation
on the pure state of the total closed system SE should induce a
corresponding rotation on the reduced density matrix describing
S, and likewise for translations and reflections. In the example of
Section 5.6, it is relatively straightforward to see that rotations,
translations, reflections and Galilean boosts acting on the micro-
level state space all induce corresponding transformations on the
macro-level state space.
!
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢃ
ꢄ
d
dt
∂Hmacro
∂P
∂Hmacro
∂X
P0
M
∂VðX0Þ
∂X0
ðX0; P0Þ ꢃ
; ꢂ
¼
; ꢂ
:
ð32Þ
X0;P0
X0;P0
Proof of this relation is given in the Appendix.22 The timescale on
which this relation holds will depend on timescales on which the
low-level initial condition ðx0; p0Þ remains in d0l as it evolves
according to the low-level dynamics.
5.7. A note on limit-based approaches to reduction
It is worth noting here that a number of the examples
discussed in this section—in particular, those discussed in
Sections 5.2–5.6—can in a certain sense be cited as instances of
limit-based reduction. All of these cases involve domain restric-
tions of the form
ϵ51 for some dimensionless parameter ϵ,
which, although they are not strictly speaking mathematical limits,
call to mind the sort of thing people often mean when they talk of
one theory's being a limit or limiting case of another. I only wish to
point out here that the relations (4) and (6) offer a far more precise
characterization of the common structure that these various
reductions possess, as well as of the structure they possess in
common with other cases, than does the claim that one model is a
limit or limiting case of another—a claim that, as we have seen,
may be interpreted in any number of ways.
In all of these cases, it is also natural to ask whether the bridge
map respects the composition properties of the various symmetry
transformations—that is, whether the transformation induced via
the bridge map by the composition of two symmetry transforma-
tions in the low-level model approximates the composition of the
corresponding transformations in the high-level model. Without
checking this explicitly in each case, it seems prima facie likely that
this condition (with appropriate qualifications) should hold in the
model pairs discussed in the above examples. What the prelimin-
ary considerations of this subsection all suggest is that the
relationship between BðxlÞ and xh is in fact much richer and more
comprehensive than is suggested merely by the similarity of
dynamical behavior discussed in Sections 4 and 5. Detailed
elaboration of these claims is postponed to a future article.
6. Extending DS reduction
In this section, I consider possible extensions of DS reduction,
first to include an analysis of the relations between symmetries of
different dynamical systems models and then to an analysis of the
relation reductionM outside the class of cases to which reductionDS
applies.
6.1. DS reduction and symmetry
Given that BðxlÞ and xh serve to represent the same physical
degrees of freedom (albeit within the context of different models),
it is natural to ask whether these two quantities share other
physically relevant transformation properties besides their dyna-
mical evolution—in particular, their transformation properties
under certain symmetry operations. More specifically, one can
inquire as to whether the sort of approximate commutation
condition that holds between the bridge map and the dynamical
evolution of the two models extends to the symmetries of the
models, so that if ThðxhÞ is a symmetry of the high-level model,
there exists some symmetry TlðxlÞ of the low-level model such that
for states xl in some subset d″l ꢀ d0l ꢀ Sl,
6.2. Extending DS reduction to other types of model
It is natural to ask which aspects of DS reduction can be extended
to fixed-system, inter-model reduction—what I have called reduc-
tionM—in cases outside the class of model pairs considered here—for
example, in cases where one or both of the models is not a dynamical
system, or where the two dynamical systems do not naturally share a
time parameter. It seems that at least two aspects of DS reduction are
likely to carry over to these cases: the use of bridge maps and
domain specificity. It also seems that the following general Nagelian
strategy employed in DS reduction may also apply in these other
cases: namely, to prove on the basis of the low-level model's
constraints (dynamical or otherwise), together with a restriction to
a certain subset in the space of the low-level model, that the quantity
associated with the bridge map approximately satisfies the con-
straints of the high-level model; this amounts to deriving a certain
“image” of the high-level model's constraints within the framework
of the low-level model; one can then employ bridge map substitu-
tions to write down an “analogue” form of these relations, which is
“strongly analogous” to the constraints imposed by the high-level
model. As in the case of DS reduction, one can interpret “strong
ThðBðxlÞÞ ꢃ BðTlðxlÞÞ:
ð33Þ
We can perform a cursory initial examination of the examples
considered in Section 5 to check whether such a condition holds in
these cases. Consider first the case presented in Section 5.1,
concerning the relation between quantum and classical dynamical
systems models; it is straightforward to see that rotations of the
quantum state in Hilbert space will induce rotations in the phase
space through the bridge map given there, and likewise for
22
Note that variable substitutions have already been incorporated into this
statement of condition (6), as writing it out explicitly in terms of the variables of
the low-level model would be more cumbersome than illuminating in this case.

68
J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
analogy” in a precise sense if the high-level model includes specifica-
Appendix A. Proof of Eq. (21)
tion of a norm on the high-level space.
A general state j 〉AFKG in the low-level model can be
Ψ
expanded in the form
Z
1
X
†
kN
†
jΨ〉 ¼ ψ0j0〉þ
d3k1⋯d3knψnðk1; …; knÞa ⋯ak j0〉
ðA:1Þ
^
^
~
1
7. Conclusion
n ¼ 1
^
^
N
~
where ψ0 ꢄ 〈0j
Ψ
〉
and ψnðk1; …; knÞ ꢄ 〈0ja ⋯ak
j
Ψ
〉. The QFT
k1
I have argued against a general, if not always explicit, tendency
in the literature on inter-theory relations in physics to conceive of
inter-theoretic reduction as an exclusively global affair—that is, as
primarily a matter of deriving, in a completely general way that is
insensitive to the particularities of different systems, the laws of
one-theory from those of another. Taking a cue from certain
authors primarily in the philosophy of mind literature, I have
advocated a more local approach to inter-theory reduction in
physics that is sensitive to such particularities. More precisely, I
have argued that such an approach should focus not on derivations
directly between theories, but on derivations that relate the
particular models that the high- and low-level theories use to
describe individual systems in the high-level theory's domain. I
have shown that in the particular class of cases where both models
of the system in question are dynamical systems, this sort of fixed-
system reduction between models follows the main strategic
prescriptions of Nagel/Schaffner reduction (suitably adapted to
the context at hand). I have also suggested in a preliminary way
how these prescriptions might be extended to fixed-system inter-
model reduction in other kinds of cases. Within the specialized
context of reduction between dynamical systems, I have shown
that a particular mathematical relationship—which follows the
general outlines of Nagel/Schaffner reduction—precisely charac-
terizes the manner in which the low-level model underwrites the
success of the high-level model across a wide range of cases.
Moreover, I have shown that while a number of these cases also
might be cited as instances of limit-based reduction, the criteria
for DS reduction offer a much more precise and comprehensive
characterization of these cases than does the vague claim that one
model is a “limit” or “limiting case” of another.23 While there is
still much work to be done in elaborating the local, model-based
approach to inter-theoretic reduction in physics described here—in
particular, in examining the relationship between the symmetries
of different models, and in extending this approach to other
model-types—I hope that the preceding analysis has served to
underscore the value of analyzing reduction first in specific cases
and then expanding outward to see what generalities can be
drawn, rather than demanding an unrealistically high level of
generality at the outset.
Schrodinger equation for the low-level model straightforwardly
entails that
∂
∂t
i
i
ψ0 ¼ Cψ0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
∂
∂t
2
ψnðk1; …; kn; tÞ ¼ ðC þ jk1j þm2
~
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
~
þ⋯þ jknj þm Þψnðk1; …; kn; tÞ 8n: ðA:2Þ
If we now restrict the state j
Ψ
〉 to lie in the domain d0 of low-
momentum N-particle states defined by Eq. (20), then we may
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
make the approximation
jkij þm2 ꢃ mþ21mki . Within this
domain and under this approximation, Eq. (A.2) yields
∂
∂t
i
ψ0 ¼ 0
ꢃ
ꢄ
∂
∂t
1
2m
1
2m
~
i
ψnðk1; …; kN; tÞ ꢃ C þNmþ
k21 þ⋯þ
k2N ψNðk1; …; kN; tÞ:
~
ðA:3Þ
We can return the position representation ψnðx1; …; xN; tÞ of N-
particle
states
using
the
this,
expansion
we get
R
d3k
3 2Ek
1
ikx
ꢂikx
†
^
^
^
ϕ
ðxÞ ¼
ðake þa e
Þ.
Doing
ð2πÞ
^
^
ψnðx1; …; xN; tÞ ꢄ 〈0j
ϕ
ðx1Þ⋯
ϕ
ðxNÞj 〉 ¼
Ψ
R
d3k1
d3kN
1
1
N x
ψNðk1; …; kN; tÞeik x⋯eik
. Together with
1
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ~
⋯
⋯
ð2
π
Þ3 ð2
π
Þ3 2Ek
2EkN
1
this expression, (A.3) entails
∂
∂t
^
^
i
〈0j
ϕ
ðx1Þ⋯
ϕ
ðxNÞj
Ψ〉
ꢃ
ꢄ
1
1
2m
2
N
∇21 ꢂ⋯ꢂ
∇
〈0j
ϕ
ðx1Þ⋯
ϕ
ðxNÞj
Ψ
〉
ðA:4Þ
ðA:5Þ
^
^
ꢃ
C þNmꢂ
2m
This in turn entails that
ꢅ
ꢃ ꢂ
ꢆ
∂
∂t
i
eiðNmþCÞt〈0j
ϕ
ðx1Þ⋯
ϕ
ðxNÞj
Ψ〉
^
^
N
X
1
ꢅ
ꢆ
2
i
eiðNmþCÞt〈0j
ϕ
ðx1Þ⋯
ϕ
ðxNÞj
Ψ
〉 :
^
^
∇
i ¼ 1 2m
Appendix B. Proof of Eq. (32)
I prove the X- and P-components of the relation (32) separately.
For the X-components,
Acknowledgments
!
P
P
dX0i
dt dt
Pi0
m
x
αi αi
p
αi
αi
d
I would like to thank David Wallace, Simon Saunders, Christo-
pher Timpson, Jeremy Butterfield, John Norton and an anonymous
referee for invaluable comments on earlier drafts of this work, and
Jos Uffink, Cian Dorr and David Albert for helpful discussions.
Thanks also to audiences in Munich, Pittsburgh, Amersfoort and
Minnesota, where earlier versions of this article were presented.
This work was supported by the University of Pittsburgh's Center
for Philosophy of Science.
αi
P
P
¼
¼
¼
;
ðB:1Þ
m
m
αi
Mi0
αi
αi
αi
where in the second equality I have used the low-level Hamilton
p
dx
α
α
dti
m
i
equation
¼
. For the P-components
α
i
!
dP0i
dt dt
X
X
X
d
∂
∂x
¼
p
¼ ꢂ
vðx ꢂx Þ
αi
α
j
αi
αi
αi
αi
α
j aαi
X X
∂
ꢃ ꢂ
vðX0i ꢂX0jÞ
ðB:2Þ
∂X0i
αi
α
j aαi
23
Whether it is possible to give a more precise meaning to such a claim that
0
1
A
extends across the range of examples considered here is not something I have the
space to comment on here. Suffice it to say that no existing formulations of the
limit-based approach succeed in avoiding the vagueness that has grounded my
critique of this approach here.
X
∂
¼ ꢂ
∂X0i
0
0
@
ninjvðXi ꢂXjÞ
jai

J. Rosaler / Studies in History and Philosophy of Modern Physics 50 (2015) 54–69
69
0
@
1
A
Churchland, Patricia S. (1986). Neurophilosophy: Towards a unified theory of the, MIT
Press, Cambridge, Massachusetts.
Dizadji-Bahmani, F., Frigg, R., & Hartmann, S. (2010). Who's afraid of Nagelian
X
∂
0
0
¼ ꢂ
VðXi ꢂXjÞ
:
ðB:3Þ
∂X0i
jai
reduction? Erkenntnis, 73(3), 393–412.
Dupre, John (1993). The disorder of things: Metaphysical foundations of the disunity of
science. Cambridge, Massachusetts: Harvard University Press.
Ehlers, J. (1986). On limit relations between and approximate explanations of
physical theories. In Logic, methodology and philosophy of science, VII. Elsevier
Science, Amsterdam.
In going from the first to the second line I have used the low-level
P
dp
α
∂
dt i
Hamilton equation,
the second to third line, I have used the definition x ꢄ X0 þr
¼ ꢂ∂x
vðx ꢂx Þ. In going from
αi
α
j
α
j aαi
α
i
αi
αi
i
∂v
∂v
∂2v
i
Enc, Berent (1983). In defense of identity theory. Journal of Philosophy, 80, 279–298.
0
0
;X0 þr
0;Xj
0;Xj
0
and the fact that
jX þr
ꢃ ∂x jX
þ∂x
jX
r
þ
αi
α
α
j
∂x
∂x
α
i
i
α
α
α
i
i
j
i
i
i
Fodor, Jerry (1974). Special science: Or the disunity of science as
a working
∂
j
2v
∂x ∂x
hypothesis. Synthese, 28, 97–115.
0;Xj
0
jX
r
þ⋯, so that if jr j is sufficiently small, all but the
αj
αi
i
α
α
i
Fraser, Doreen (2011). How to take particle physics seriously: A further defence of
axiomatic quantum field theory. Studies in History and Philosophy of Science Part
B: Studies in History and Philosophy of Modern Physics, 42(2), 126–135.
Giunti, M. (2006). Emulation, reduction, and emergence in dynamical systems. 〈http://
philsci-archive.pitt.edu/2682/〉.
lowest order terms in the expansion can be ignored and
∂v
∂v
0
0
jX þr ꢃ ∂x jX . Define Lv to be the length scale (determined by
∂x
α
α
i
i
v's derivatives) such that all higher-order terms in the expansion
Holland, Peter,
& Brown, Harvey R. (2003). The non-relativistic limits of the
can be neglected when jr j5Lv
8
αi. We can then rewrite
αi
Maxwell and Dirac equations: The role of Galilean and gauge invariance. Studies
In History and Philosophy of Science Part B: Studies In History and Philosophy of
Modern Physics, 34(2), 161–187.
∂vðX0i ꢂXjÞ
∂X0i
ÞjX as
. In going from the third line to the fourth, I
∂vðx ꢂx
α
α
i
j
0
∂x
α
i
have used the fact that under the approximation made in the
previous step, vðX0i ꢂXj0Þ does not vary with changes in the index αi
for fixed i, or in αj for fixed j, so that
Hooker, C. A. (1981). Towards
a general theory of reduction. Part iii: Cross-
categorical reductions. Dialogue, 20, 496–529.
P
Joos, E., Zeh, D., Kiefer, C., Giulini, D., Kupsch, J.,
& Stamatescu, I.-O. (2003).
ni
αi ¼ 1
Decoherence and the appearance of a classical world in quantum theory (2nd
ed.). Berlin: Springer.
Kim, Jaegwon (1992). Multiple realization and the metaphysics of reduction.
Philosophy and Phenomenological Research, 1–26.
Lewis, David (1969). Review of art, mind, and religion. Journal of Philosophy, 66,
23–35.
P
P
P
ni
vðX0i ꢂXj0Þ ¼ nivðX0i ꢂX0jÞ and
vðX0i ꢂX0jÞ ¼
∂
∂
∂X0i
αi ¼ 1 ∂X0i
α
j aαj
jai
njnivðX0i ꢂX0jÞ. In the final step, I have used the assumption
VðXi ꢂXjÞ ¼ ninj vðXi ꢂXjÞ.
Merzbacher, Eugen (1998). Quantum mechanics (3rd ed.). John Wiley and Sons, Inc,
New York.
References
Nickles, T. (1973). Two concepts of intertheoretic reduction. The Journal of
Philosophy, 181–201.
Norton, J. (2012). Approximation and Idealization: Why the Difference Matters.
Batterman, R. (2002). The devil in the details: Asymptotic reasoning in explanation
reduction and emergence. New York: Oxford University Press.
Batterman, R. (2007). Intertheory relations in physics. In Stanford encyclopedia of
philosophy.
Batterman, Robert W. (2000). Multiple realizability and universality. The British
Journal for the Philosophy of Science, 51(1), 115–145.
Berry, M. (1994). Asymptotics, singularities and the reduction of theories. In Brian
Skyrms Dag Prawitz, & Dag Westerståhl (Eds.), Logic, methodology and philoso-
phy of science, IX: Proceedings of the ninth international congress of logic,
methodology and philosophy of science, Uppsala, Sweden, August 7–14, 1991
(Studies in Logic and Foundations of Mathematics) (Vol. 134, pp. 597–607).
Bickle John (2013). Multiple realizability. In The Stanford encyclopedia of philosophy.
Broer, Henk, & Takens, Floris (2010). Dynamical systems and chaos, Vol. 172. Utrecht:
Springer.
Butterfield, J. (2011a). Emergence, reduction and supervenience: A varied landscape.
Foundations of Physics, 41(6), 920–959.
Butterfield, J. (2011b). Laws, causation and dynamics at different levels. Interface
Focus (Royal Society London), 1, 1–14.
Butterfield, Jeremy (2011c). Less is different: Emergence and reduction reconciled.
Foundations of Physics, 41(6), 1065–1135.
Cartwright, N. (1999). The dappled world: A study of the boundaries of science.
Cambridge, England: Cambridge University Press.
Philosophy of Science, 79(2), 207–232.
Putnam, Hilary (1967). Psychological predicates. Art, Mind, and Religion, 1, 37–48.
Rohrlich, F. (1988). Pluralistic ontology and theory reduction in the physical
sciences. The British Journal for the Philosophy of Science, 39, 295–312.
Schaffner, K. F. (2012). Ernest Nagel and reduction. Journal of Philosophy, 109(8/9).
Scheibe, E. (1997). Die Reduktion Physikalischer Theorien, Teil I, Grundlagen und
Elementare Theorie, Vol. 1. Berlin: Springer.
Scheibe, E. (1999). Die Reduktion Physikalischer Theorien, Teil II, Inkommensurabilitat
und Grenzfallreduktion. Berlin: Springer.
Schlosshauer, M. A. (2008). Decoherence: And the quantum-to-classical transition.
Heidelberg: Springer.
Wallace, D. (2012). The emergent multiverse: Quantum theory according to the Everett
interpretation. Oxford: Oxford University Press.
Wallace, David (2011). Taking particle physics seriously: A critique of the algebraic
approach to quantum field theory. Studies in History and Philosophy of Science
Part B: Studies in History and Philosophy of Modern Physics, 42(2), 116–125.
Yoshimi, J. (2012). Supervenience, dynamical systems theory, and non-reductive
physicalism. The British Journal for the Philosophy of Science, 63(2), 373–398.
Zurek, W. H., & Paz, J. P. (1995). Quantum chaos: A decoherent definition. Physica D:
Nonlinear Phenomena, 83(1), 300–308.