3 Dirac’s Equation and High-Energy Quantum Physics Without Relativity
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7 The Principles of Quantum Information Theory, Dirac’s Equation, and Locality…
The first point, in accordance with the program of this study, is the significance
of the fundamental principles in quantum theory, now including the principles of
quantum information, some of which recast and thus give a deeper meaning to previously established quantum principles. The task of understanding these principles
is far from being completed, and in some respects, it has barely begun to be undertaken. The fact that D’Ariano and Perinotti are able to give their principles as principles of quantum information a proper mathematical expression is important, and
doing so is a major accomplishment.
The second point is that Dirac’s equation could be derived by using principles of
information processing alone, without using the relativity principle, while, however,
using the locality principle. Although this derivation is only a starting point of high-
energy quantum theory, it has major implications for foundational thinking in
fundamental physics, in particular the possibility that, while the locality principle
retains its significance in large-scale physics, the principles of relativity may not. In
D’Ariano and Perinotti’s scheme:
The Dirac equation in three space dimensions emerges from the large-scale dynamics of the
minimum-dimension quantum cellular automaton [QCA], satisfying [linearity], unitarity,
locality, homogeneity, and discrete isotropy of interactions [without appealing to special
relativity]. The Dirac equation is recovered for small wave vector and inertial mass, whereas
Lorentz covariance is generally distorted in the ultrarelativistic limit [of very large wave-
vectors] (D’Ariano and Perinotti 2014, p. 1).
This scale is beyond “the usual Fermi scale of high-energy physics,” because
“the QCA as a microscopic mechanism for the emergent quantum field” is proposed
“as a framework to unify a hypothetical discrete Planck scale with the usual Fermi
scale of high-energy physics” (D’Ariano and Perinotti 2014, p. 1). The hypothetical
nature of some of the assumptions made and the theories cited by D’Ariano and
Perinotti is important and must be kept in mind. I shall return to this aspect of their
argument below.
Dirac’s equation might be seen as an enactment of the mathematical correspondence principle applied at a level beyond the Fermi scale.12 This enactment is complex, which should not come as a surprise. Even establishing, via Pauli’s spin theory,
that Schrödinger’s equation is the quantum-mechanical limit of Dirac’s equation
was nontrivial, especially at the time, and it was, again, crucial. In the present case,
one deals with a deeper level of quantum reality, or possibly a reality beyond the
quantum, although D’Ariano and Perinotti’s framework suggests, at least to this
author, that at stake is a certain general quantumness, reflected in the linearity and
unitarity of their framework at the large scale. The key principles of quantum theory
and their quantum-theoretical mathematical expressions and consequences are
clearly at work throughout—the probabilistic or statistical nature of the theory (the
QP/QS principle), locality, Hilbert spaces, operators, noncommutativity, the tensor
structure, symmetry, group representation, and so forth (D’Ariano and Perinotti
12
On the other hand, the article provides “an analytical description of the QCA for the narrow-band
states of quantum field theory in terms of a dispersive Schrödinger equation holding at all scales”
(D’Ariano and Perinotti 2014, pp. 1, 4).
7.3 Dirac’s Equation and High-Energy Quantum Physics Without Relativity
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2014, pp. 2–3). There is some important and implicative new mathematics as well,
such as that of “quasi-isometric embedding,” a profound recent development in
geometry, primarily due to M. Gromov and his geometric-group theory (D’Ariano
and Perinotti 2014, p. 3). The QCA is the quantum cellular automaton operative at
the large scale. The derivation of Dirac’s equation is achieved by finding Dirac’s
automata, which are “obtained by locally coupling Weyl automata” and give the
Dirac equation at the relativistic limit from the informational principles proposed,
all of these principles, rather than only linearity and unitarity, which are quantum in
character (D’Ariano and Perinotti 2014, pp. 5, 6–9). This is a reflection and recasting of the fact that Dirac’s spinors are composed of Weyl’s spinors.
In this regard, the situation is parallel to deriving QFDT from a larger (than only
quantum) set of principles. Locality, homogeneity, and isotropy are additional
principles here, which allow one to dispense with the principle of relativistic
(Lorenz) invariance in deriving the equation. Locality merits a special attention,
both in view of its fundamental, irreducible role in this derivation and in general.
The locality requirement itself and the corresponding principle are defined by the
requirement that the cardinality of the set, Sg, of sites g', interacting with the site g
(both from a denumerable set G of systems, involved in the cellular automaton and
described by Fermionic field operators) is “uniformly bound over G, namely
Sg £ k < ¥ for every g” (D’Ariano and Perinotti 2014, p. 2). This concept or principle is more general and deeper than the principle of (special) relativity, defined the
Lorentz invariance, although relativity, special or general, conforms to the principle
of locality. The Lorentz invariance emerges from this locality principle at the corresponding (relativistic) limit and the standard (Fermi) scale of quantum field theory, and hence of Dirac’s equation. The significance of the locality requirement is,
however, more general and is critical to the informational QCA framework proposed by the authors. This requirement is consistent with the view of locality
adopted from the outset of this study, in part following D’Ariano and Perinotti: all
physical interactions and indeed everything that essentially defines phenomena,
information, and so forth is local. The principle has important connections to local
symmetries, specifically local gauge symmetries of QFT (those of the Yang-Mills
theory), for example, those of the color group SO(3), which corresponds to the local
symmetry whose gauging gives rise to quantum chromo dynamics (QCD). These
connections, far from sufficiently explored in literature, cannot, however, be
addressed here. In addition, as noted in Chap. 1, according to D’Ariano (private
communication), while falsifiability is a crucial (Popperian-like) requirement to be
satisfied by this framework, it is equally crucial that all falsifiability is local, which
may be seen as a principle in its own right.
The QCA as such is, again, an enactment of large-scale quantum dynamics, the
quantum character of which is manifest in and is defined by linearity and unitarity.
There are significant potential implications and benefits of this broader conception
beyond the fact the Dirac equation is the Fermi-scale limit of the theory:
The additional bonus of the automaton framework is that it also represents the canonical
solution to practically all issues in quantum field theory, such as all divergences and the
problem of particle localizability, all due to the continuum, infinite volume, and the
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7 The Principles of Quantum Information Theory, Dirac’s Equation, and Locality…
Hamiltonian description. ...Moreover the QCA is the ideal framework for a quantum theory
of gravity, being quantum ab initio (the QCA is not derivable by quantizing a classical
theory), and naturally incorporates the informational foundation for the holographic principle, a relevant feature of string theories (Zwiebach 2004; Becker et al. 2008) and the main
ingredient of the microscopic theories of gravity of (Jacobson 1995) and (Verlinde 2011).
Finally, a theory based on a QCA assumes no background, but only interacting quantum
systems, with space-time and mechanics as emergent phenomena.
The assumption of Planck-Scale discreteness has the consequence of breaking Lorentz
covariance along with all continuous symmetries: These are recovered at the Fermi scale in
the same way as in the doubly-special relativity of Amelino-Camelia (Amelino-Camelia
2002; Amelino-Camelia and Piran 2001), and in the deformed Lorentz symmetry of
Magueijo and Smolin (2002, 2003). Such Lorentz deformations have phenomenological
consequences, and possible experimental tests have been recently proposed by several
authors (Moyer 2012; Hogan 2012; Pikovski et al. 2012; Amelino-Camelia et al. 2009).
The deformed Lorentz group of the automaton has been preliminarily analyzed in (Bibeau-
Delisle et al. 2015). (D’Ariano and Perinotti 2014, p. 1)
It is beyond my scope to address these aspects of the program and their implications. They are exciting and promising but would also require an extended careful
analysis, including considering the works cited here. However, while the potential
significance of the proposed program for large-scale physics is thus made apparent,
the hypothetical nature of the theories just mentioned should be kept in mind as
well. To cite D’Ariano and Perinotti’s conclusion:
In conclusion, we remark that [the] Lorentz covariance is obeyed only in the ±relativistic limit
E
|k| ≪ 1, whereas the general covariance (corresponding to invariance of wk ) is a nonlinear
deformation of the Lorentz group, with additional invariants in the form of energy and distance scales (Bibeau-Delisle et al. 2015), as in the doubly-special relativity (AmelinoCamelia 2002; Amelino-Camelia and Piran 2001) and in the deformed Lorentz symmetry
(Magueijo and Smolin 2002; Magueijo and Smolin 2003), for which the automaton then
represents a concrete microscopic theory. Correspondingly, also [the] CPT symmetry of
Dirac’s QCA is broken in the ultrarelativistic scale. (D’Ariano and Perinotti 2014, p. 9)
The theories invoked here are highly hypothetical. Amelino-Camelia and Piran,
cited here, call a hypothetical suspension of the (Lorentz) relativistic invariance a
“drastic assumption,” albeit made to address certain puzzling data (Amelino-
Camelia and Piran 2001, p. 1). It may be somewhat less drastic insofar as the locality principle is preserved. In any event, it may pay off to be cautious, even though
and indeed because alternative proposals, some of which are based on alternative
proposal for using cellular automata in quantum theory (‘t Hooft 2014), are equally
hypothetical. However, that Dirac’s equation could be derived without using the
relativity (Lorentz-invariance) principle, in addition to being a major achievement
in its own right, again, manifests a promise and potential of the authors’ informational program for exploring fundamental physics on scales beyond those of quantum field theory.
Profound foundational questions are at stake. How these questions will be pursued and what role the principles of quantum informational theory will play in it are
difficult to predict. The present discussion does not aim to do so. It does suggest,
however, that fundamental physical principles may play a more significant role than
they have more recently in approaching these questions. Equally crucially and more
radically, it also suggests, with the help of both Dirac’s own and D’Ariano and
7 The Principles of Quantum Information Theory, Dirac’s Equation, and Locality…
263
Perinotti’s derivations of Dirac’s equation, that the fundamental principles of quantum physics might prove to be more crucial in this pursuit than those of relativity,
not inconceivably, general relativity included.13 Quantum principles may also prove
to be more crucial rather than those of general relativity in understanding gravity,
although, as explained above, some form of locality principle may be equally crucial, and homogeneity and isotropy are likely to be required as well, as very general
principles. In other words, while relativity may be abandoned on the large scale
(extending to the Planck scale), quantum theory is likely to survive, reflecting the
ultimately quantum character of nature. If so, and one cannot, once again, be certain, we may need yet new quantum principles, possibly conceived on lines of principles of information processing and QCA, even if developed, as these informational
principles are, by building on some among the older principles.
But then, this may be our best way to arrive at new principles in, to cite
Heisenberg’s title, “physics and beyond” (Heisenberg 1962). Fundamental physics
as understood here is always physics and beyond, not the least because it is also a
physics of the beyond. It is the physics of that which is beyond the reach of the physics available at a given moment of time, and beyond physics itself, for example, if
one adopts the RWR principle, which, in its strongest version, places the ultimate
constitution of nature beyond thought itself, but not beyond existence or reality.
This does not stop physics, quite the contrary, because, as I argued, in the spirit of
Copenhagen, throughout this study, this “beyond the reach of physics,” also makes
possible new physics, physics that would not be possible otherwise. Nature and
spirit, or rather nature, spirit, and technology, continue to work together.
13
Cf. also (D’Ariano 2010, 2012).
Chapter 8
Conclusion: The Question Concerning
Technology in Quantum Physics and Beyond
Abstract Chapter 7 suggests that the principles of quantum information theory,
implicitly at work, as I have argued, already in Heisenberg’s discovery of quantum
mechanics, may play a major role in the future of fundamental physics and principle
thinking there. Heisenberg’s work and quantum information theory also share their
“technological” nature in the broad sense of technology, as considered in Chap. 2
(Sect. 2.5). Indeed, which is the main theme of this conclusion, the question of
quantum information and “the question concerning technology,” in Heidegger’s
famous phrase, are deeply interconnected and may even be seen as one and the same
question. For, the physics of quantum information is also the physics of the relationships between experimental and mathematical technology. All technology is essentially linked to information. Indeed, technology may be best deﬁned as the means of
obtaining information about systems, in the case of quantum systems, in the RWRprinciple-based interpretations, as the means of generating information (classical in
nature) through the interaction between quantum systems and the technology of
measuring instruments. In this way, this conclusion brings together all of the main
themes and fundamental principles discussed in this book.
The discussion undertaken in Chap. 7 suggests that the principles of quantum information theory are likely to play a major role in the future of fundamental physics.
As I have argued in this study, they have implicitly done so beginning with
Heisenberg’s work, with which quantum information theory also shares its technological dimension in the broad sense of technology discussed in the end of Chap. 2 (Sect.
2.5) in conjunction with Heisenberg’s work. For, if “on the one hand experiments
are described by circuits resulting from the connection of physical devices, [and]
on the other hand each device in the circuit can have classical outcomes and the
theory provides the probability distribution of outcomes when the devices are connected to form closed circuits (that is, circuits that start with a preparation and end
with a measurement)” (Chiribella et al. 2011, p. 3), then the physics of quantum
information is also the physics of the relationships between experimental and
mathematical technologies. Technology, experimental, mathematical, or other, is
essentially linked to information. Indeed, it is arguably best deﬁned as the means
of obtaining information about systems. In the case of quantum systems, in nonrealist, RWR-principle-based, interpretations, technology (jointly experimental and
© Springer International Publishing Switzerland 2016
A. Plotnitsky, The Principles of Quantum Theory, From Planck’s
Quanta to the Higgs Boson, DOI 10.1007/978-3-319-32068-7_8
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Conclusion: The Question Concerning Technology in Quantum Physics and Beyond
mathematical) becomes the means of generating, creating, information (classical in
nature) through the interaction between quantum systems and the technology of
measuring instruments, thus bringing together the main principles of quantum theory, as considered in this book.
It is important to keep in mind that experimental technology is a broader concept
than that of measuring instruments. It would, for example, involve devices that
make it possible to use the measuring instruments, or still other devices. More generally, technology is a means of doing something and doing it more successfully
than previously, to get “from here to there,” as it were. In this sense, any technology
is the invention of new wheels, which is, however, not the same as reinventing the
wheel. Thus, the experimental technology of quantum physics enables us to understand how nature works at the ultimate level of its constitution, in this case, strictly
in the sense of what kind of effects this constitution produced upon measuring
devices, without being able to represent or even conceive of the character,
architecture of this constitution. This, however, is sufﬁcient to have QM, QFT, or
QFDT, and to use their technologies for obtaining information through quantum
phenomena, which quantum objects enable us to have, and to work with this information, as in quantum cryptography and computing. Quantum objects themselves
are not technology; it is something technology helps us to discover, understand,
work with, and so forth. However, they can become part of technology, beginning
with the quantum parts of measuring instruments through which parts these instruments interact with quantum objects, concerning which we obtain information.
Quantum objects can also become part of devices we use elsewhere, such as lasers,
electronic equipment, and MRI machines (this list is very long by now). Thus, the
discovery of the Higgs boson is the result of the joint workings of three
technologies:
1. The experimental technology of the Large Hadron Collider (LHC);
2. The mathematical technology of QFT (sometimes coupled to the technology of
philosophical thought);
3. Digital computer technology.
Any quantum event, I argue, is made possible through the joint workings of the
ﬁrst two technologies, with the third becoming increasingly more prominent.
Although the role of digital technology is one of the deﬁning aspects of contemporary physics, I can only mention it in passing. There is yet another technology (in
the present, extended sense) involved: that of science as a cultural project, which is,
however, a separate subject, which, too, can only be mentioned in passing here.1
As discussed at the end of Chap. 2, via Bohr’s argument concerning the role of
“mathematical instruments” in Heisenberg’s discovery of quantum mechanics, in its
use in physics, all, or in any event most, mathematics is technology in this broader
sense. But one could think of the technological functioning of mathematics even in
mathematics itself, insofar as certain mathematical tools, such as, say, homotopy or
1
See Chap. 1, note 5, especially Galison (1997) and Latour (1999), mentioned there, and for a
discussion of this subject by the present author (Plotnitsky 2002).
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Conclusion: The Question Concerning Technology in Quantum Physics and Beyond
267
cohomology groups, are technologies akin to measuring instruments in physics.
According to J-P. Marquis, who borrows the concept from physics and even speciﬁcally quantum physics “homotopy groups should be thought of as measuring instruments since they provide information about certain crucial aspects of spaces. … [T]hey
are epistemologically radically different from … transformation [symmetry] groups
of a space. ...Homotopy groups (and here we might as well mention homology and
cohomology groups) do not act on anything. The purpose of these geometric devices
is to classify spaces by their different homotopy [or cohomology] types.” By contrast, ﬁbrations (through which homotopy and cohomology groups are deﬁned) are
not “measuring instruments,” but rather “devices that make it possible to apply measuring instruments [such as cohomology and homotopy groups] and other devices”
(Marquis 2006, p. 259). In theoretical physics, speciﬁcally elementary particle
physics, symmetry groups are a mathematical technology. It follows, however, that
in parallel with experimental technology, the mathematical technology of physics or
even of mathematics itself is not only or even primarily a way or representing reality
(although it may be this, too) but is rather a way of experimenting with reality, and
thus also creating new realities, physical and mathematical.
Taking advantage of and bringing together both main meanings of the word
“experiment” (as a test and as an attempt at an innovative creation), one might argue
that, while not without help from nature, the practice of quantum physics is the ﬁrst
practice of physics or science that is both, jointly, fundamentally experimental and
fundamentally mathematical. That need not mean that this practice has no history;
quite the contrary, beginning, again, with Galileo’s and Newton’s work. Both were
also experimentalists, both in the conventional sense and in this sense under discussion at the moment, as were most other major ﬁgures of experimental physics or
relativity, who advanced and made possible both theories—Boyle, Huygens, Young,
Faraday, and Michelson are just a few major names. (The list of even major ﬁgures
becomes long from the late nineteenth century on.) Lagrange’s and Hamilton’s analytical mechanics, Maxwell’s electrodynamics, the thermodynamics of Maxwell,
Boltzmann, Gibbs, and Planck, and Einstein’s relativity are all major events of this
history of experimenting with mathematical models in physics. As I said at the outset of this study, the commitment itself to creative experimentation may well be, in
the language of (Kant’s) moral philosophy, the categorical imperative of all good
science. This is certainly a point on which classical and quantum physics converge:
creative experimentation, physical, mathematical, or philosophical is the categorical
imperative and the primary force of causality of both, whatever the nature of this
causality (a difﬁcult problem in its own right) may be. Nevertheless, this experimentation acquires a new form with quantum mechanics and then extends to higher
level quantum theories, and a new understanding of the nature of quantum phenomena and thus experimental quantum physics, as discussed in Sect. 2.5 of Chap. 2.
The practice of quantum physics is fundamentally experimental because we no
longer track, as we do in classical physics or relativity, the independent behavior of
the systems considered, and thus track what happens in any event, by however ingenious experiments. Instead we define what will happen in the experiments we perform, by how we experiment with nature by means of our experimental technology,
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even though and because we can only predict what will happen probabilistically or
statistically.2 Thus, in the double-slit experiment, the two alternative setups of the
experiment, whether we, respectively, can (by way of using one experimental device
or another) or cannot know, even in principle, through which slit each particle, say,
an electron, passes, we obtain two different outcomes of the statistical distributions
of the traces on the screen, with which each particle collides. Or, thus also giving a
rigorous physical and philosophical meaning to the uncertainty relations, we can set
up our apparatus so as to measure and correspondingly predict, again, probabilistically or statistically, either the position or the momentum of a given quantum object,
but never both together. Either case requires a separate experiment, incompatible
with the other, rather than merely representing an arbitrary selection of either type
of measurement within the same physical situation, by tracking either one of its
aspects or the other in the way we do in classical mechanics. There, this is possible
because we can, at least in principle, measure and assign simultaneously both quantities within the same experimental arrangement. In quantum physics we cannot.
This difference, as we have seen, was crucial to Bohr’s understanding of the situation, including in the EPR experiment:
In the phenomena concerned we are not dealing with an incomplete description characterized by the arbitrary picking out of different elements of physical reality at the cost of
[sacriﬁcing] other such elements [for a given quantum object], but with a rational discrimination between essentially different experimental arrangements and procedures which are
suited either for an unambiguous use of the idea of space location, or for a legitimate application of the conservation theorem of momentum. Any remaining appearance of arbitrariness concerns merely our freedom of handling the measuring instruments, characteristic of
the very idea of experiment. (Bohr 1935, p. 699; emphasis added)
Quantum physics, however, changes what experiments do: they define what will
or will not happen, in terms of probabilistic or statistical quantum causality, as
deﬁned in Chap. 5, rather than allowing it to follow what is bound to happen in any
event in accordance with classical causality.
By the same token, quantum physics is also fundamentally mathematical,
because the mathematical formalism of the theory is not in the service of tracking,
by way of a mathematical representation, what would have happened anyhow,
which would shape the formalism accordingly, but is in the service of predictions
required by our experiments. The mathematical formalism of quantum theory is
able to predict correctly the experimental data in question without offering a representation at all of the physical processes responsible for these data. Quantum
mechanics, at least in nonrealist, RWR-principle-based, interpretations, is strictly a
theory, a mathematical technology, of predictions concerning the outcomes of possible future experiments on the basis of previously performed experiments, both
deﬁned by our experimental technology. This is why quantum mechanics established radically new relationships between mathematics and physics, vis-à-vis
those, essentially realist in character, found in classical physics and relativity.
Indeed, it also follows that, as quantum physics experiments with nature by using
2
I am indebted to G. M. D’Ariano on this point.
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mathematical and experimental technology, quantum theory experiments with
mathematics itself, more so and more fundamentally than in classical physics or
relativity. This is because we invent, in the way Heisenberg or Dirac did, effective
mathematical schemes of whatever kind and however far they may be from our
general phenomenal intuition, rather than proceed by reﬁning mathematically our
phenomenal representations of nature, which process limits us in classical physics
or even (to some degree) in relativity.
As discussed in Chap. 2, Heisenberg’s thinking, thus, revolutionized the very
practice of theoretical physics, and it redeﬁned experimental physics as well, or
reﬂected what the practice of experimental physics had in fact already become by
that point. I brieﬂy recapitulate the nature of this transformation here. The practice
of experimental physics does not consist of tracking what happens or what would
have happened independently of our experimental technology, but in creating,
again, unavoidably creating, conﬁgurations of this technology. This situation
reﬂects the fact that what happens is unavoidably deﬁned by what kinds of experiments we perform, and how we affect quantum objects, rather than only by their
independent behavior, although their independent behavior does of course contribute to what happens. The practice of theoretical physics no longer consists in offering an idealized mathematical representation of quantum objects and their behavior,
which is impossible by the RWR principle. It consists in developing mathematical
machinery that is able to predict, in general (in accordance with what obtains in
experiments) probabilistically or statistically, the outcomes of (always discrete)
quantum events and of correlations between such events, observed in the conﬁgurations of experimental technology, which brings together the QD and QP/QS principles. The situation, as the analysis of QFT offered in Chap. 6 shows, acquires a
more complex form in quantum electrodynamics and quantum ﬁeld theory, and
experimental physics in the corresponding (high) energy regimes, in view of the PT
principle, which is added to the RWR, QD, and QP/QS principles of quantum
mechanics, or QFDT. This addition reﬂects the greater complexity of high-energy
physics as concerns the conﬁgurations of experimental technology deﬁning it; a
more complex nature of the mathematical formalism as the mathematical technology of the theory; and a more complex character of the quantum-ﬁeld-theoretical
predictions; and, thus, of the relationships between the mathematical and the experimental technologies in high-energy quantum physics.
Indeed, as suggested above, we can adopt this experimental-technological viewpoint and this quantum-theoretical model, thus also the RWR principle, in mathematics itself. Consider R. Langlands’s argument concerning the so-called Langlands
program, one of the most extraordinary endeavors of twentieth-century mathematics. A. Wyles’s proof of Fermat’s theorem was, for example, connected to the program. Langlands’s argument was inﬂuenced and even inspired by quantum
mechanics and quantum ﬁeld theory, and the role of inﬁnite-dimensional group representations there. According to Langlands:
The introduction of inﬁnite-dimensional representations entailed an abrupt transition in the
level of the discourse, from explicit examples to notions that were only described metaphorically, but at both levels we are dealing with a tissue of conjectures that cannot be
attacked frontally.
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The aesthetic tension between the immediate appeal of concrete facts and problems on
the one hand, and, on the other, their function as the vehicle to express and reveal not so
much universal laws as an entity of a different kind, of which these laws are the very mode
of being, is perhaps more widely acknowledged in physics, where it has long been accepted
that the notions needed to understand perceived reality may bear little resemblance to it,
than in mathematics, where oddly enough, especially among number theorists, conceptual
novelty has frequently been deprecated as a reluctance to face the concrete and a ﬂight from
it. Developments of the last half-century have matured us, as an examination of Gerd
Falting’s proof of the Mordell conjecture makes clear, … but there is a further stage to
reach.
It may be that we are hampered by the absence of a central unresolved difﬁculty and by
the extremely large number of currently inaccessible conjectures, at whose extent we have
hardly hinted. Some are thoroughly tested; others are in doubt, but they form a coherent
whole. What we do in the face of them, whether we search for speciﬁc or general theorems,
will be determined by our temperament or mood. For those who thrive on the interplay of
the abstract and the concrete, the principle of functoriality for the ﬁeld of rational numbers
and other ﬁelds of numbers has been particularly successful in suggesting problems that are
difﬁcult, that deepen our understanding, and yet before which we are not completely helpless. (Langlands 1990, p. 209)
One would need a proper discussion of the mathematical theories involved to
make apparent the mathematical and philosophical profundity of what is at stake in
this passage and in Langlands’s program itself, and such a discussion is beyond my
scope.3 A more general philosophical argument is sufﬁcient to make my main
point, which is as follows. The program and the way of pursuing it harmonize with
my argument in this book, applied to quantum theory, especially the point “that the
notions needed to understand perceived reality may bear little resemblance to it.”
According to the nonrealist argumentation offered by this study, such notions,
more radically, bear no resemblance to this reality at all, a reality that in the ﬁrst
place cannot be perceived, but is only assumed, inferred from realities that we can
perceive. (Langlands would likely have agreed with this last qualiﬁcation.) Besides,
in accordance with the spirit of “fragmentation without wholeness” advocated in
this study, I am not certain about the existence of “the whole” in this case, even
though Langlands’s program has reached a ﬁrmer level of deﬁnition and cohesion
since this statement was made (in 1990). Rather we deal with a heterogeneous and
yet interactive engagement of theories and concepts, including differently if interactively deﬁned concepts, sometimes different while having the same name, such
as space or number. It is an intriguing question, again, beyond my scope here,
whether one can speak of complementarity in Bohr’s sense in mathematics.
However, one can speak of a fragmentation without a whole that this fragmentation “fragments.”
The differences between the two types of “objects”—material in physics and
mental in mathematics—remain important, of course. In particular, even when one
assumes the RWR principle, there still exists something in nature, idealized as
quantum objects, that is responsible for the observed quantum phenomena. The
meaning of the existence of mathematical objects as such is an entirely different
3
See (Bernstein and Gerbard 2003) and (Gelbart 1984) for a more accessible account.
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Conclusion: The Question Concerning Technology in Quantum Physics and Beyond
271
question, which has been debated since Plato, whose ghost still overshadows this
debate, but especially in modern mathematics, following Cantor’s set theory (Gray
2008). This question cannot be addressed here, and it would be presumptuous to
offer any general thoughts concerning this subject apart from engaging a great deal
of philosophical and technical literature concerning it. One could nevertheless argue
for analogues, such as those suggested by Langlands, of the quantum-mechanical
situation in mathematics. Such analogues arise when one needs to rely on the properties of a more accessible mathematical object, or one type of such objects, in order
to understand another mathematical object, or another type of object which is ultimately inaccessible or even ultimately an inconceivable mathematical object, which
can nevertheless be indirectly consistently deﬁned (again, leaving aside the question
of its existence). I now use “understanding” in the sense of establishing rigorous
connections (similar to those between mathematical formalism and quantum objects
in quantum mechanics or quantum ﬁeld theory), rather than merely in terms of more
or less loose analogies or metaphors. These connections become possible if both
types of objects can be linked in a particular way, as happens, for example, in the
case of Langlands’s program. Riemann’s work with the functions of complex variables by means of certain properties of these functions (such as their singularities),
rather than by means of deﬁning them by explicit formulas, is a remarkable early
example of this kind of situation. In fact, however, much modern mathematics operates under these conditions (whether the practitioners subscribe to this view or not),
or at least, as in quantum mechanics or quantum ﬁeld theory, this type of interpretation of modern mathematics or all mathematics as ultimately technological is possible. In any event, luckily for us, this technological approach works in mathematics,
just as it works in quantum physics, where, again, both nature and experimental
technology help us.
In this respect, while modernist mathematics may by deﬁned, following Gray in
his Plato’s Ghost (Gray 2008), by its divorce from physics and the (nonmathematical) world, this technological mathematics, which is often the same mathematics
technically, may be deﬁned by its divorce from mathematical representation of
mathematical reality, in this case, inaccessible by deﬁnition. This may be seen as the
ﬁnal break with Platonism. As such, it may in fact reconnect itself with the world,
as it does in relativity and, more radically, in quantum theory, as Gray acknowledges, via Einstein in the case of relativity (Gray 2008, p. 324, n. 28). But Gray
misses the RWR-principle-based thinking found in quantum theory, which connects
physical reality without realism with mathematics without realism. He misses
Heisenberg’s and Bohr’s modernist revolution in physics and its radical nature, a
revolution that also took place in modernist mathematics.
This thinking in mathematics and physics does, however, retain something, perhaps the most important thing, from Plato—from the spirit of Plato—rather than the
ghost of Plato (the difference between the two words is both inﬁnitesimal and inﬁnite). Plato’s thought, too, may be less Platonist than it might appear and than it
might have appeared even to Plato himself. This thinking retains the essential, shaping role of the movement of thought, a movement or a ﬂight that drives this technology of mathematics and physics. Even though, unlike in Plato (perhaps!), such