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2 - The Map versus the Territory

Published online by Cambridge University Press:  22 April 2022

Ruth E. Kastner
University of Maryland, Baltimore


This chapter provides a brief overview of the history of the development of quantum theory, with a critical focus on the antirealist tradition inaugurated by Niels Bohr. The distinction between “principle theories” and “constructive theories” is discussed, and it is noted that quantum mechanics is a “principle theory.” It is argued that quantum theory is amenable to a fully realist interpretation provided we let go of the demand that reality be classically picturable.

The Transactional Interpretation of Quantum Mechanics
A Relativistic Treatment
, pp. 26 - 43
Publisher: Cambridge University Press
Print publication year: 2022

In this chapter, I consider some general issues of interpretive methodology, to present to the reader the motivation behind the new TI. I then argue in favor of a realist approach as opposed to an instrumental one.Footnote 1

First, I should note that I offer an interpretive reformulation of what MacKinnon (Reference MacKinnon2005) calls a “functioning,” or informal theory: nonrelativistic quantum mechanics and its extension into the relativistic domain via quantum field theory. Since functioning theories are often inherently “untidy” (in either a mathematical or conceptual sense or both), philosophers of physics often engage in “rational reconstruction” of theories in order to render them more logically self-consistent in the hopes that the resulting formal theory will better lend itself to an unambiguous interpretation.Footnote 2 However, MacKinnon observes that in general, history does not support the notion that such recast, formalized theories lead to robust ontological insights. He instead characterizes the interpretive task as one of “find[ing] a way of relating philosophical questions about epistemology and ontology to functioning physical theories, rather than idealized constructions” (p. 4). That, in a nutshell, is the aim of the present work, although I believe that ultimately the model proposed herein is significantly more “tidy,” ontologically self-consistent, and formally unified than conventional approaches to quantum theory.

2.1 The Irony of Quantum Theory

The original inception of quantum theory and the course of its subsequent evolution contain a deep irony. To appreciate this irony, we first need to revisit a bit of history.

2.1.1 Heisenberg’s Breakthrough

A major breakthrough in quantum theory was achieved in 1925 through a decision by German physicist Werner Heisenberg to let go of certain preconceived metaphysical assumptions about the nature and behavior of matter: specifically, that we could picture electrons as little particles – corpuscles in the Greek (Democritan) conception – orbiting an atomic nucleus. Facing a theoretical impasse in accounting for atomic phenomena, he renounced these classical anschaulich (German for “picturable”)Footnote 3 assumptions and retained only observable quantities such as energy differences and radiation frequencies, which could be measured and recorded as hard data. These he entered into arrays which he sardonically termed “laundry lists,” and which his then-teacher Max Born would soon realize were matrices (arrays of numbers in a form well known in mathematics). Thus was born Heisenberg’s “matrix mechanics” version of the theory, which successfully predicted the experimental (spectral) data arising from observations of the hydrogen atom. Subsequent development would eventually lead to a powerful, empirically successful theory which could be expressed in different forms (probably the best known being the Schrödinger wave mechanics, based on Erwin Schrödinger’s celebrated equation), and whose formal structure was described, as von Neumann had first noticed, by an abstract mathematical space called Hilbert space.

2.1.2 Bohr’s Antirealism

However (as observed in Chapter 1), nearly a century later, researchers are still deeply puzzled about how to interpret the theory, in the sense of understanding what it says about reality (if anything). Most physicists and philosophers of physics are aware that Heisenberg’s breakthrough came as a result of renouncing his preconceived metaphysical assumptions, and many of them (including, most notably, Heisenberg’s fellow quantum theory founder Niels Bohr) have taken from this fact what I believe is the wrong lesson: they have renounced realism with regard to quantum theory. That is, the idea that there was some understandable, underlying physical reality described by quantum theory tended to be viewed suspiciously, as a misguided impulse to drag in metaphysical baggage that Heisenberg’s approach had discredited as inappropriate methodology. Probably nobody says this more emphatically than Neils Bohr: “There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.”Footnote 4

The above sentence by Bohr presupposes that nature can only be talked about using classical concepts, that is, the very “picturable” notions that Heisenberg had renounced in order to arrive at his matrix formulation of quantum theory. In effect, Bohr presumed that physics cannot “say how Nature is,” even though quantum theory, as a formal structure, may be doing just that, albeit not in the traditionally picturable manner. Bohr’s positivistic prohibition on “finding out how nature is” was not necessarily heeded by everyone, but it had, at the very least, a chilling effect on interpretive inquiry.Footnote 5

In particular, Bohr’s legacy is alive and well among many practicing physicists, whose job it is to calculate experimental predictions and analyze results, and who tend to regard efforts by philosophers of physics to “find out how nature is” to be a misguided waste of time. Many of them approach interpretational puzzles of quantum theory from the kind of deflationary, “debunking” view alluded to at the end of the previous chapter. Of course, nobody is to be faulted for choosing not to be realist about physical theory, especially when it is not in their job description to do so. But the main thesis of this work is that, contra Bohr, it is perfectly reasonable to be realist about the subject matter of quantum theory, and that it is perfectly possible to “find out how nature is,” as long as we don’t expect it to be “classically anschaulich” and are willing to entertain some new and unexpected ideas of how nature might be (analogous to the strange specter of energy having to be “quantized,” which led to Max Planck’s successful derivation of the blackbody radiation spectrum).Footnote 6

2.1.3 Einstein’s Realism and a Further Irony

Einstein, as is well known, completely disagreed with Bohr’s approach. His motivation was, in his own words, to “know God’s thoughts.”Footnote 7 Yet, ironically, a similar antirealist tendency has recently arisen based on the methodology Einstein used in formulating his theory of special relativity. Einstein famously arrived at his theory by thinking in terms of what someone could actually measure with (idealized) rigid rods and clocks, and concluded that one needed to renounce certain metaphysical notions about space and time: in particular, Newton’s view that space and time are absolute, immutable “containers” for events. What is less often remembered is that Einstein also used formal theoretical assumptions: in particular, he demanded the invariance of electromagnetism, requiring that the theory not be dependent on an observer’s state of motion. But the prevailing message of relativity came to be that there is no such thing as absolute simultaneity or absolute lengths of objects and that these concepts were metaphysical ballast to be jettisoned. Einstein’s renunciation of such absolute metaphysical concepts is often amplified, like Heisenberg’s renunciation of the trajectory concept, into a universal doctrine that any notion of an underlying (i.e., subempirical) reality is to be eschewed.

However, not only is this an inappropriate lesson to take from these theoretical achievements; it is not even consistently applied: most researchers (and especially physicists) continue to be thoroughgoing realists about spacetime, viewing it as a fundamental substantive “container” or backdrop which not only underlies all possible theoretical models but which even has causal powers to “steer” particles on trajectories.Footnote 8 And one must note the additional irony that the notion of “trajectory” persists, despite the widespread view that fundamental reality should not be considered “picturable.”Footnote 9

2.1.4 Theory Construction versus Theory Interpretation

The point generally overlooked in the trend described above is that theory formulation/discovery is an entirely different process from that of theory interpretation. We need to distinguish between (1) the valid point that preconceived metaphysical assumptions can serve as a barrier to theory invention or discovery, especially when a successful new theory cannot be based on such assumptions, and (2) realist interpretation of an existing empirically successful theory as a way of discovering new features of reality uncovered by that theory. The deep irony of quantum theory, I suggest, is that its discovery was made possible by the renunciation of a then-realist approach and attendant metaphysical baggage; yet when interpretationally queried from a realist perspective in the proper way, quantum theory can open the way to an entirely new and richer understanding of physical reality: a strange new kind of model that we could not have discovered without first letting go of inappropriately classical metaphysical concepts. In making this claim, I invite the reader to reflect on the insightful quote at the end of Chapter 1, by the late Jeeva Anandan.

2.2 “Constructive” versus “Principle” Theories

What do I mean by querying a theory “in the proper way”? In order to address this, I first need to review an important distinction in theory type: “constructive” versus “principle” theories. Simply put, a constructive theory is one based on a model. A famous example is the kinetic theory of gases, which represents the behavior of gases in terms of small, impenetrable spheres in collision with one another and the walls of their container. By applying known physical laws to this model, James Clerk Maxwell and Ludwig Boltzmann were able to deduce the large-scale thermodynamic behavior of gases; for example, Boyle’s Law relating temperature, pressure, and volume (PV = nRT).Footnote 10 Such a “constructive” theory is powerful and illuminating because it allows us to understand the “nuts and bolts” of what is really going on at a level beyond ordinary experience, that is, beneath the phenomenal level of appearance. That is what Einstein meant when he talked about wanting to “know God’s thoughts.” He didn’t just want to know about how God’s creation appears and to be able to analyze, classify, and predict those appearances; he wanted to know how it all works beneath the merely phenomenal level, “to boldly go” where Bohr summarily pronounced that nobody should be able, nor wish, to go.Footnote 11

In contrast, a “principle” approach to theory development lacks a physical model. It starts from an abstract principle or principles that serve to constrain the form that the theory can take, and then fits the theory, with the help of mathematical consistency and basic physical laws such as energy conservation, to empirical observation. Relativity was a principle theory, and Einstein was very dissatisfied with this aspect of it. He felt that only a constructive theory, with its attendant illuminating model, provided genuine insight into “how nature really is.” Similarly, quantum mechanics was a principle theory, as we can see by the fact that Heisenberg had to explicitly jettison the models he was trying to work with (i.e., his erroneous metaphysical pictures of how atoms behaved), and to work only with empirical observations that served to constrain the form of the theory. Before that, Planck used a purely mathematical trick – summing over discrete energy levels instead of assuming energy was a continuously variable quantity – to obtain the correct empirical result for blackbody radiation (see, e.g., Eisberg and Resnick, Reference Eisberg and Resnick1974, section 1.1 and especially p. 14 for a clear account of how this phenomenon presented a fatal problem for classical electromagnetism and forced the invention of quantum theory). His desperate resort to this tactic led to the discovery of Planck’s constant, h, the fundamental physical constant which characterizes the smallest unit of action (units of energy times time or momentum times length). Thus his approach to the discovery of the new theory was principle-based (i.e., using formal mathematical considerations), not model-based.

2.3 Bohr’s Kantian Orthodoxy

Now, as noted above, Bohr was perfectly content with the idea that quantum mechanics was a principle theory. He assumed from the way that the theory was arrived at – by rejecting a model that didn’t work – that there can be no model for quantum theory, that is, no way of picturing “how nature is.” In other words, Bohr elevated the fact that one cannot apply classical model-making to a nonclassical realm into a broad-brush policy that, at the quantum level, one should not try to find models of any kind. He assumed that if one cannot have a classical model, there can be no model, and that quantum theory represents the end of the scientific search for understanding of the physical world in a realist sense: that is, independently of how we happen to be looking at it.

Put differently, he assumed that classical modeling is equivalent to giving a realist account of micro-reality and that one therefore cannot give such an account. This formulation is rebutted eloquently by Ernan McMullin, who wrote:

[I]maginability must not be made the test for ontology. The realist claim is that the scientist is discovering the structures of the world; it is not required in addition that these structures be imaginable in the categories of the macroworld.

(as quoted in Ladyman, Reference Ladyman2009)

Bohr’s formulation depended heavily on appealing to phenomenal and epistemic notions, such as the fact that in order for scientists to communicate their results, they had to be able to talk about pointer readings, thereby working with classical phenomena and speaking in “classical language.” This is true, of course – it is the means by which all physical theories are tested and corroborated. However, it does not follow from limitations on scientists’ interactive verbal language requirements that the mathematical structure of the theory has no real, objective referent. Bohr simply jumped to an unwarranted conclusion in this regard, based on his tacit assumption that any reality describable by a physical theory must be classical in nature.Footnote 12 As observed by McMullin, the interpretational question is whether the theory’s formal content (e.g., the Schrödinger equation and its solutions) has some physical referent, regardless of whether or not that physical referent can be directly observed (or described in macroscopic, classical terms).

At this point it is useful to acknowledge a distinct similarity between Bohr’s thought and the work of German philosopher Immanuel Kant. Kant proposed that reality has two fundamental aspects: (1) the world of appearance and (2) the “thing-in-itself” (or “noumenon”), which he held was unknowable.Footnote 13 For an accessible introduction to the problem of gaining knowledge of the “thing-in-itself,” the reader is encouraged to consult chapter 1 of Bertrand Russell’s The Problems of Philosophy, in which the author considers an ordinary table and presents a convincing case that the table itself, apart from any perception of it, is a deeply mysterious object, “if it exists at all.” (For an updated version of this epistemological puzzle, see Section 7.5.) Kant also proposed that there are “categories of experience” that make knowledge of the world of appearance possible, and which are the only means through which knowledge is constructed.Footnote 14 Knowledge, for Kant, was only about item (1), the world of appearance; recall that part of the definition of (2), the thing-in-itself or underlying reality, was that it was intrinsically unknowable. Among the “categories of experience” were concepts like space, time, and causality. In particular, Kant proclaimed that Euclidean space was an a priori category of understanding, meaning a necessary concept behind any knowable phenomenon – an assertion which, it should be noted, has since been decisively falsified by relativity’s non-Euclidean accounts of spacetime.

Bohr seems to have assumed, much like Kant, that all knowledge obtained by way of physical theories applies only to the world of appearance and that the “classical modes of description” are required for all knowledge. So Bohr’s “classical modes of description” play the same role as Kant’s “categories of experience.” Bohr, in essence, proclaimed that while quantum theory might have placed us just at the doorstep of the “noumenal” realm, the nature of the theory required that we could not gain knowledge about it and that, moreover (as a “normative” principle echoed by modern day “Qbists”), it would be scientifically and methodologically unsound to think that we should try to do so, as reflected in his previous quote. By “abstract quantum mechanical description,” Bohr preemptively denied that the formalism could be referring to anything physically real, thus effectively relegating it to a linguistic or computational device. As noted above, this assumption can and should be questioned.

It has often been pointed out (e.g., by Bohr and Heisenberg) that in general there can be no mechanistic, deterministic account of individual microscopic events. This fact is often referred to in terms of “quantum jumps” that cannot be predicted, even in principle. Yet a realist understanding of micro-reality need not take the form of a detailed mechanical account of an individual event – the entity that remains elusive to causal description, as Anton Zeilinger notes.Footnote 15 To assume, like Bohr, that a realist understanding must be in terms of the usual “classical,” causal account is to unnecessarily limit ourselves to a pseudo-Kantian “category of experience.” Many of these have already been shown to be obsolete by scientific advance, as noted above. The new realist understanding may not be in terms of causal, mechanistic processes. It may instead encompass a fundamental indeterminism at the heart of nature, but one which is well defined in terms of the conditions under which it occurs – in contrast to prevailing orthodox interpretations which suffer from an ill-defined micro/macro “cut“ (as discussed in Section 1.3.4). The new understanding offered here is a rational account, in the sense of being well-defined and self-consistent, even while it lacks certain features, such as determinism and mechanism, that have been traditionally assumed to be requirements for an acceptable scientific account of phenomena.

Thus, as alluded to above, Bohr’s famous conclusion that “It is wrong to think that the task of physics is to find out how nature is” is a logical fallacy. It simply does not follow logically that the failure of classical model-making entails that no model of any kind is possible.Footnote 16 As noted above, one may regard Bohr as making the same kind of mistake as Kant when the latter presumed that there can be no knowledge of a realm that is not based on a Euclidean space. While it may be true, as a matter of contingent fact, that there is no adequate model, there is no reason that a failure of a particular sort of inappropriate model should be turned into a general prohibition against modeling. On the contrary, a principle theory can provide truly groundbreaking insights into new aspects of reality: it can ultimately lead us to a new kind of model, one so utterly different from how we are used to thinking about reality that we could not have approached it directly, “from the ground up” so to speak, but had to arrive at it through an indirect route, “top down,” as Heisenberg did. This is the insight expressed by Jeeva Anandan, quoted at the end of Chapter 1.

2.4 The Proper Way to Interpret a “Principle” Theory

So, what is the “proper way” to interpret such a principle theory, one that was developed without reference to any model? To answer this question, let’s turn to a famous dictum by Bryce DeWitt, who presented it as the essential motivation for his development of the Everett interpretation into what became known as the many worlds interpretation:Footnote 17 “The mathematical formalism of the quantum theory is capable of yielding its own interpretation” (DeWitt, Reference DeWitt1970).

I take this to mean that the formalism resulting from whatever methodology was needed to develop an empirically successful theory – especially a principle theory like quantum mechanics, which was not based on prior construction of a model – has features that may well point to heretofore hidden or unnoticed features of reality. A perfect case in point, again, is Planck’s stumbling upon quantized energy because his empirically successful quantitative theory said so, not because he wanted it that way. Since the features of an empirically successful principle theory are (apparently) not something we could have thought of unaided, they are not available to us as a possible model, and we (like Heisenberg) have to proceed without their help, “groping in the dark,” so to speak, aided only by previously established physical principles, mathematical consistency, and empirical data to guide us to the form of the theory.

Heisenberg, in choosing to “listen to reality” by renouncing his previous unhelpful metaphysical assumptions, wrote down the “laundry list” formalism (matrix mechanics) that turned out to be a useful instrument for predicting observations arising from the microscopic systems he was studying. But, as argued above, it does not logically follow that all there is to reality is those abstract “laundry lists.” A possibly useful analogy here is a map to some buried treasure: Heisenberg, through his choice to adopt a Zen-like “beginner’s mind” approach to the phenomena under study, stopped listening to his own ineffective ideas and began to listen to the message of reality instead, as encoded in the phenomena. Thus, he was able to “hear” what reality was trying to tell him by writing down what became a useful “map.” The realist impulse that underlies and motivates all fundamental scientific advance is to acknowledge that there is some reason, however obscure, that such a theoretical “map” allows us to predict phenomena. Unless we wish to believe in miracles or coincidence as the explanation for the success of a theory like quantum mechanics, or to deny that theory success even needs explaining, which is to retreat from the deepest aspects of the scientific and philosophical mission, we are obligated to acknowledge that the “map” reflects something about reality – however utterly new and unfamiliar.

Another analogy for the inspiration leading to a successful principle theory is in the realm of psychology and interpersonal relationships. Successful mediators know that conflicts can be resolved when the parties are helped to let go of their own preconceived notions, desires, or requirements for the other person and start to listen to what the other person is saying. More broadly, a socially effective person has the ability to be receptive to the messages from their environment and the flexibility to adapt to the meaning of the messages, that is, to let go of preconceived notions about “how things should be” and to behave in ways that are more appropriate and fruitful. But they don’t conclude from that that there is nothing further to be learned about that other person or situation, or that there is nothing beyond those messages they heard which allowed them to behave more effectively. A new way of behaving is more “fruitful” because there is something there yielding fruit. Heisenberg’s approach exemplifies, albeit in a different context, the behavior of a successful person in social relationships. He stopped presuming and started listening, and was able to write down a very useful “map.” We should not mistakenly conclude from his methodological success that there is no more to reality than that map.

So, as will be developed in later chapters, the proper way to interpret the theory is to “listen” carefully to its unexpected mathematical features. A crucial step was made by Max Born, who linked the absolute square of the Schrödinger wave functionFootnote 18 to something empirical, if only statistical: this quantity could be seen to function as the probability of observing the associated property when one conducted a measurement of the system. His finding became known as the “Born Rule,” and it is the fundamental empirical link between quantum theory and the world of phenomena. As noted in the previous chapter, in most prevailing interpretations, the Born Rule either is simply assumed as part of the mathematical machinery that does not merit or require explicit interpretation or is given a pragmatic, “for all practical purposes” (FAPP) account which, in my view, fails to do it justice as the crucial link between theory and concrete experience. The Born Rule constitutes a deep mystery for all prevailing interpretations; there would appear to be no straightforward ontological (i.e., non-epistemic, nonstatistical) explanation for it in any interpretation other than TI.Footnote 19

2.5 Heisenberg’s Hint: A New Metaphysical Category

Heisenberg took a further step in “listening” to quantum theory when he made the following statement: “Atoms and the elementary particles themselves are not real; they form a world of potentialities or possibilities rather than things of the facts.”Footnote 20 This assertion was based on the fact that quantum systems such as atoms are generally described by quantum states with a list of possible outcomes, and yet only one of those can be realized upon measurement. I think that he was on to something here, except that I would adjust his characterization of quantum systems as follows: they are real, but not actual. In his terms, they are something not quite actual; they are “potentialities” or “possibilities.” Thus my proposal is that quantum mechanics instructs us that we need a new metaphysical category: something more concrete than the merely abstract (or mental), but less concrete than, in Heisenberg’s terms, “facts” or observable phenomena.Footnote 21 The list of possible outcomes in the theory is just that: a list of possible ways that things could be, where only one actually becomes a “fact.” This proposal is directly analogous to Planck’s proposal, in view of the inescapable formal features of his theory, that energy is quantized.

The distinction between a quantum possibility and a fact is clarified in a comment that Heisenberg made later in his life (and will be further clarified in Chapter 7):

The probability wave of Bohr, Kramers, Slater … was a quantitative version of the old concept of “potentia” in Aristotelian philosophy. It introduced something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality.

(Heisenberg, Reference Heisenberg2007, p. 15)

So, Heisenberg had arrived at a new kind of metaphysical understanding, a “picture,” if you will, of the reality described by quantum theory. However, in view of his ambivalence about it – he was a practicing physicist, after all, and expected models to be based on “things of the facts”– he did not pursue this insight as a viable description of the underlying reality described by quantum theory. Among my goals in this work is to essentially pick up where he left off (this ontological exploration begins in Chapter 4).

A further important aspect of “listening to the formalism” of quantum theory is to acknowledge its time-symmetric (or at least “advanced”) aspects. Specifically, it cannot be overemphasized – since the fact is habitually neglected – that advanced (time-reversed) states necessarily enter into any calculation needed to obtain empirical content (i.e., probabilities for outcomes of measurements, or expectation values for the values of measured observables). Indeed, this overlooked fact is so important that I will elevate it to an interpretational maxim for any realist interpretation:

Maxim: Mathematical operations of a theory which are necessary to obtain correspondence of the theory with observation merit a specific (exact) ontological interpretation.

This proposed maxim no doubt requires some elucidation. For one thing, TI’s rival “purist” interpretation (i.e., the collection of approaches constituting the so-called many worlds interpretation based on Hugh Everett’s proposal of Reference Everett1957) does not adhere to it. As alluded to earlier, MWI addresses the Born Rule by epistemological or statistically approximate methods: by arguing, via decision theory, that a rational observer would choose to bet on outcomes obeying the Born Rule; by arguing that Everettian worlds violating the Born Rule have approximately zero measure; and so on. Similarly, the Bohm theory proposes that the distributions of Bohmian particles closely approximate that specified by the Born Rule. Now, in the absence of any mathematical property of the basic theory that could provide an unambiguous ontological basis for the Born Rule, such approximate and/or ad hoc approaches might be justified. But the theory does possess a specific mathematical object that can provide an exact ontological basis for the Born Rule: the set of advanced solutions which, under TI, are confirmation waves arising from the ubiquitous absorption processes neglected in other interpretations. Since absorption processes are physically present whenever there is a detection (the latter being a requirement for an observation), the advanced solution is the obvious mathematical entity to interpret as a component of the ontological basis for the Born Rule.

Despite the counterintuitive aspects of advanced states, I believe that truly hearing what the formalism is saying means taking seriously the idea that it describes something with advanced (as opposed to the usual retarded) qualities. This is where, in my view, TI improves upon Everettian interpretations which try to approach the formalism from a receptive, “purist” point of view, but which fail to notice that the advanced states are a crucial part of the theory with physical content that should not be neglected.

The transactional conceptual picture represents a parallel to that of Einstein’s conceptual unification of the instrumental and pragmatic prerelativistic quasi-theories, as described by Zeilinger (Reference Zeilinger and Ketvel1996):

It so happened that almost all relativistic equations which appear in Einstein’s publication of 1905 were known already before …, mainly through Lorentz, Fitzgerald and Poincaré – simply as an attempt to interpret experimental data quantitatively. But only Einstein created the conceptual foundations, from which, together with the constancy of the velocity of light, the equations of the theory of relativity arise. He did this by introducing the principle of relativity, which asserts that the laws of physics must be the same in all inertial systems. I maintain that it is this very fact of the existence of such a fundamental principle on which the theory is built which is the reason for the observation that we do not see a multitude of interpretations of the theory of relativity.

(p. 2)

The Born Rule equating the probability of a particular result to the square of the wave function is one of the equations allowing quantitative interpretation of experimental data in quantum theory, just as the Lorentz contraction allowed quantitative empirical correspondence in prerelativistic theories. The current multitude of competing “mainstream” interpretations of quantum theory (among these the Bohmian theory, ad hoc “spontaneous collapse,” approaches, MWI) are all different ways of providing approximate, pragmatic, after-the-fact justifications for the Born Rule and the conditions of its application – showing that its use is consistent with the rest of the theory in some limit – rather than an explanation for how it arises naturally from the theory. In contrast, the conceptual picture of a transactional process is what allows the operational equation of the Born Rule to arise from the theoretical formalism, just as Einstein’s postulates allow the Lorentz contraction to emerge as a natural consequence.

2.6 Ernst Mach: Visionary/Reactionary

I digress slightly here to discuss Ernst Mach, a prominent figure in nineteenth-century physics, because he probably exemplifies more than anyone else the irony discussed in this chapter. He exemplified, on the one hand, the virtue of humble submission and obedience to nature’s empirical messages and, on the other hand, the philosophical mistake of assuming that those empirical phenomena are all there is or that knowledge cannot, or should not, go beyond them. As a strict empiricist, Mach insisted that all knowledge is based on sensation or observation – a position that of course confines any empiricist to knowledge about the world of appearance only. Yet it does not follow that the only thing that exists is appearances, as noted earlier. Here I endorse von Weizsaecker’s dictum that “What is observed certainly exists; about what is not observed we are still free to make suitable assumptions. We use that freedom to avoid paradoxes.”Footnote 22 (Descartes has more pungent remarks for the strict empiricist, as we will see shortly.)

Thus, while I agree with Mach’s eliminativistFootnote 23 account of spacetime as fundamentally based on comparisons (i.e., I adopt a relational view of spacetime), that does not mean that the interpretation of all physical theories which were discovered through the application of mathematical analysis to observations must be limited to subjective sensations, as Mach unnecessarily (and I believe mistakenly) concludes in the last clause below:

[W]e do not measure mere space; we require a material standard of measurement, and with this the whole system of manifold sensations is brought back again. It is only intuitional sense-presentations that can lead to the formulation of the equations of physics, and it is precisely in such presentations that the interpretation of these equations consists.

(Mach, Reference Mach1914; emphasis added)

Thus, Mach’s justified insistence that theory construction be grounded in observation slides unjustifiably into categorical antirealism about possible unobservable entities pointed to by those theories. As noted previously, this is a logical and methodological error, unambiguously revealed as such when Mach’s refusal to entertain the existence of atoms – because they were unobservable – was shown to have been on the wrong side of scientific progress. One can acknowledge that perhaps what we think of as “spacetime” can be understood in terms of the ordering of sensations (also known as material objects), but it does not logically follow that there is nothing more to reality than those sensations. The ordering we discover can be seen as an objective property of reality insofar as all our observations conform to it and it cannot be altered by purely subjective means (i.e., by imagining or desiring it to be different). Thus, objective reality may be something real, even if not directly observable, which is capable of giving rise to sensations (i.e., observations or actualized events). The unjustified assumption that because our knowledge of reality is derived largely from sensation, our interpretation of theories and our understanding of reality must be limited to accounts of sensation, is subjected to rather harsh criticism by Descartes in his Treatise on Light. I quote generously here, as Descartes takes a while to establish his point:

[T]he spaces where we sense nothing are filled with the same matter, and contain at least as much of that matter, as those occupied by the bodies that we sense. Thus, for example, when a vessel is full of gold or lead, it nonetheless contains no more matter than when we think it is empty. This may well seem strange to many whose [powers of] reasoning do not extend beyond their fingertips and who think there is nothing in the world except what they touch. But when you have considered for a bit what makes us sense a body or not sense it, I am sure you will find nothing incredible in the above. For you will know clearly that, far from all the things around us being sensible, it is on the contrary those that are there most of the time that can be sensed the least, and those that are always there that can never be sensed at all.

The heat of our heart is quite great, but we do not feel it because it is always there. The weight of our body is not small, but it does not discomfort us. We do not even feel the weight of our clothes because we are accustomed to wearing them. The reason for this is clear enough; for it is certain that we cannot sense any body unless it is the cause of some change in our sensory organs, i.e. unless it moves in some way the small parts of the matter of which those organs are composed. The objects that are not always present can well do this, provided only that they have force enough; for, if they corrupt something there while they act, that can be repaired afterward by nature, when they are no longer acting. But if those that continually touch us ever had the power to produce any change in our senses, and to move any parts of their matter, in order to move them they had perforce to separate them entirely from the others at the beginning of our life, and thus they can have left there only those that completely resist their action and by means of which they cannot be sensed in any way. Whence you see that it is no wonder that there are many spaces about us in which we sense no body, even though they contain bodies no less than those in which we sense them the most.

(Descartes, Reference Descartes1664, chapter 4; emphasis added)

Thus (in admittedly uncharitable language), Descartes argues that it is a mistake to assume that nothing exists beyond what we sense, as our material senses can only detect change, not entities that are always present or that are incapable of activating our sense organs. It is widely supposed that Descartes’ metaphysics, which postulated a dynamic plenum rather than a void underlying observable matter, was a quaint piece of “moribund metaphysics” (to use Van Fraassen’s term)Footnote 24 that was largely discredited by Newton’s theories. Yet, arguably, Descartes can now be seen as having presaged the development of relativistic quantum theory, which has taught us that what Newton thought of as the “void” is far from empty.Footnote 25 So we would do well to reacquaint ourselves with Descartes’ views on scientific methodology. We should also consider the von Weizsaecker quote above that “what is observed certainly exists; about what is not observed we are still free to make suitable assumptions.” Such a “suitable assumption,” as remarked earlier, was the existence of atoms. So despite Mach’s insights into the importance of recognizing how our knowledge is obtained largely through sensation, he refused to countenance a crucial theoretical construct – the atom – that crucially led to major scientific breakthroughs. The lesson, I suggest, is to acknowledge that we should not let metaphysical preconceptions get in the way of observations and theory construction based on those observations, but we should not uncritically assume from the success of that approach that, as Descartes says, “there is nothing in the world except what [we] touch.”

2.7 Quantum Theory and the Noumenal Realm

So what can be gained by exploring the possibility that certain aspects of the quantum formalism typically thought to have only operational significance (e.g., dual states or bracs, denoted as Ψ) may indeed have ontological significance? Recall Zeilinger’s observation that the “individual event” remains resistant to causal description, along with similar observations by the founders of quantum theory. For example, according to Jammer (Reference Jammer1993), Bohr referred to such events, such as the inherently unpredictable transitions of electrons in atoms from one stationary state to another,Footnote 26 as “transcending the frame of space and time.”Footnote 27 As discussed earlier in this chapter, Bohr regarded spacetime concepts (indeed, all “classical” concepts) as prerequisites for the endeavor of gaining physical knowledge of the world; thus, he explicitly restricted what counted as legitimate knowledge to that of the world of appearance, in Kantian terms. Yet the significance of his quoted remark is that it clearly implies there are real physical events which transcend the boundaries of the observable universe. For surely Bohr has to acknowledge that stationary states were instantiated in nature and that transitions between them did occur, as this much is empirically corroborated.

Recall that Bohr insisted that physics concerns “what we can say about nature.” But what is the “we” in this context? Is it ordinary language? Or is it the mathematical language of our best theories? If the former, obviously it is very difficult, if not impossible, to talk about events which “transcend the frame of space and time.” But even Bohr implicitly admitted, as noted above, that such events occur. Indeed, the very theory he helped invent is what led him to make this observation. Does that not, then, mean that the formal aspects of physical theory can point to heretofore unknown aspects of physical reality, however difficult it might be to talk about them – that physics can be more than what we can “say about Nature” in ordinary, classically anschaulich terms?

I believe that the answer to that question is “yes.” The fact that quantum theory, in Bohr’s words, seems to point to entities and/or processes “transcending the frame of space and time” means that quantum theory can reasonably be thought of as (at least in part) a theory about the noumenal realm.Footnote 28 That is, since concepts like space and time are considered vital for gaining and communicating knowledge about the world of appearance, processes that transcend those concepts must be processes belonging to the noumenal realm, which transcends the world of appearance. Therefore, I claim that the truly revolutionary message of quantum theory is not that we should stop asking questions about the nature of reality; on the contrary, the message is that quantum theory is offering a new and strange kind of answer about an aspect of reality traditionally pronounced “off limits” by Kant and those (like Bohr) who subscribe to the notion that physical theory can only be about the world of appearance. That this methodological restriction should be abandoned is supported by Bohr’s own comment about certain quantum processes “transcending space and time,” which, contrary to his other pronouncements, unambiguously testifies to knowledge gained from quantum theory concerning the possible existence of a realm transcending space and time.

Indeed, as Einstein and others have noted, there appears to be a deep and significant connection between certain mathematical objects and physical reality – were that not the case, the whole field of theoretical physics would be without power or purpose in providing an account of the empirical realm. There is ample precedent for entities and procedures that seem purely formal and abstract turning out to have concrete physical relevance. For example, in the words of Freeman Dyson, the mathematicians of the nineteenth century

had discovered that the theory of functions became far deeper and more powerful when it was extended from real to complex numbers. But they always thought of complex numbers as an artificial construction, invented by human mathematicians as a useful and elegant abstraction from real life. It never entered their heads that this artificial number system that they had invented was in fact the ground on which atoms move. They never imagined that nature had got there first.

2.8 Science as the Endeavor to Understand Reality

As argued in the foregoing, I believe that quantum theory can present us with a new kind of understanding of nature, based on a wholly new kind of model, if we listen carefully and open-mindedly to what the formalism is saying. I take such a new understanding of nature afforded by a theory as an “explanation” of the empirical phenomena in the domain of the theory. However, for those who demand that a model be constructed out of actual, “things of the facts” (by this I mean ordinary, causal, “classical” facts as referred to by the oft-used term “local realism”), there can of course be no such “explanation,” as nearly a century of determined attempts has revealed. The failure of classical model-making has been well established and has largely been answered by a turn to strict empiricism and even frank instrumentalism by many researchers who assume, with Bohr, that all models must be classical. Empiricist approaches are essentially Bohrian in character, denying that the job of science is to “understand how nature is” and rejecting the whole idea of model construction as a misguided “demand for explanation” that need not be met (cf. Van Fraassen, Reference van Fraassen1991, p. 372).Footnote 29 In this perspective, it is seen as virtuous to renounce explanation in science, and a sign of enlightened wisdom to content ourselves with classifying and predicting phenomena. But, as argued above, this position does not follow logically from the failure of inappropriate mechanistic, deterministic, local (classical) models, and it is at odds with arguably the most important and exciting aspect of the scientific mission: the discovery of previously unseen and unknown aspects of reality (a case in point being the atom and its constituents). If we reconceptualize the process of modeling in light of quantum theory, perhaps we can find a new and more fruitful means of discovery.


1 This chapter primarily critiques instrumentalist views; however, many so-called realist approaches to quantum theory contain unacknowledged instrumentalist or positivist-flavored assumptions about what the term “reality” means (such as “real” equals “empirically detectable”), so the discussion herein is relevant to those as well.

2 As an example of this “untidiness,” nonrelativistic QM and its relativistic extension might well be considered two different functioning theories, yet clearly they must describe the same reality and therefore presumably must be parts of a larger theory. A point of contact is found in Zee’s observation (Zee, Reference Zee2010, p. 19) that nonrelativistic quantum mechanics can be obtained in the Lagrangian formulation as a 0+1-dimensional quantum field theory.

3 The term anschaulich presupposes that “picturable” means the usual classical picture of corpuscles following determinate trajectories. This assumption is contested in the present account: physical processes could be “picturable” in terms of an entirely different kind of picture.

4 As quoted in Petersen (Reference Petersen1963).

5 Some of Bohr’s most famous pronouncements about the meaning and implications of quantum theory depend heavily on optional metaphysical and/or epistemological claims treated by him as obligatory, or are simply self-contradictory (for some examples, see Kastner, Reference Kastner, Kastner, Jeknic-Dugic and Jaroszkiewicz2016b).

6 Planck had introduced a discrete sum of finite energy chunks as a calculational device only. When he tried to take the limit of the sum as the size of the chunks approached zero, he got back the old – wrong – expression. The chunks had to be of finite size in order to get the correct prediction.

7 “I want to know God’s thoughts. The rest is details.” Widely attributed to Einstein.

8 The commonplace notion that spacetime has causal power to steer particles is subject to sustained and cogent criticism by Harvey Brown (Reference Brown2002).

9 For example, many discussions of the “two-slit” experiment and similar experiments, in which the state of a single quantum is placed into a superposition by a half-silvered mirror or other means, are centered around so-called which-way information. This term is heavily laden with the presumption of a determinate trajectory: surely, if one talks about “which-way information,” one tacitly assumes that the entity under study went either one (spacetime) way or the other, that is, pursued a trajectory. So, even though perhaps not always intended, its use smuggles in a supposedly renounced classical metaphysical picture.

10 A comprehensive and very readable account of this scientific episode is found in Brush (Reference Brush1976).

11 With apologies to Gene Roddenberry.

12 For a detailed critique of Bohr’s unnecessary jump to instrumentalism about quantum theory, see Kastner (Reference Kastner2017a).

13 Kant often used the “thing-in-itself” interchangeably with the term “noumenon,” a Greek term which translates roughly as “object of the mind.” Kant’s division is very similar in structure to Plato’s division, as the reader will recall from Chapter 1.

14 Kant’s ideas discussed here were presented in his Critique of Pure Reason (Reference Kant1996).

15 Zeilinger (Reference Zeilinger2005). However, Zeilinger’s definition of “realism” is what I would call “actualism”; see Chapter 8 and Kastner (Reference Kastner2019c, chapter 5).

16 I recognize that Bohr adduced Kantian epistemological reasons for his prohibition against modeling in quantum theory, but I reject those as well. Specifically, it will be argued later in this chapter that the promise of quantum theory is to give us a glimpse of the “noumenal” realm, so I will be rejecting the Kantian claim that all knowledge must be restricted to phenomena.

17 As mentioned in Chapter 1, it is my view that MWI advocates overlook part of the formalism (advanced solutions).

18 More generally, the probability is the square of the projection of the quantum state onto a particular classically observable property, for example, position or momentum.

19 As noted in Chapter 1, Bohmians claim that the Born Rule is obtained as the statistical distribution of particle positions. But this is only for the so-called equilibrium state of the subquantum level (i.e., the level of determinate positions). Since the Bohmian theory allows for the particle position distribution to deviate from the Born Rule, it is a different theory from quantum mechanics. Even if one viewed the “non-equilibrium” state as improbable or even impossible, the account is only statistical, which I view as a weaker kind of physical explanation. A further challenge for the Bohmian account is that particles are continually created and destroyed in the relativistic regime, which would seem to increase the likelihood of distributions that might deviate from the “equilibrium” configuration needed for its empirical equivalence to standard QM. Everettians give an epistemologically based account of the Born Rule which must refer to the knowledge of an observer.

20 Heisenberg (Reference Heisenberg1958, p. 186).

21 This proposal is explored in some detail in Kastner et al. (Reference Kastner, Kauffman and Epperson2018).

22 Private communication, first quoted in Cramer (Reference Cramer1986).

23 A term meaning that the concept under study does not correspond to an independently existing entity or substance.

24 For example, Van Fraassen (Reference van Fraassen2004, p. 3).

25 For example, there is continual particle/antiparticle creation arising from the vacuum. Overall, an astonishing amount of activity goes on in so-called empty space.

26 Stationary states are states whose wave functions do not change with time. An atom’s discrete energy levels correspond to such states.

27 As quoted in Jammer (Reference Jammer1993, p. 189).

28 More precisely, that the domain of quantum theory includes the noumenal realm as a component.

29 Moreover, Van Fraassen (Reference van Fraassen1991, p. 24) conflates the possible existence of “randomness” with “no explanation” in passages such as this, addressing specific outcomes or asymmetries with no apparent antecedent cause: “[Pierre] Curie’s putative principle [that “an asymmetry can only come from an asymmetry”] betokens only a thirst for hidden variables, for hidden structure that will explain, will answer why? – and nature may simply reject the question.” In this regard, there may not be a causal, determinate, mechanistic account, but that doesn’t mean that there can be no account of relevant and interesting additional structure, so the pursuing of such an account is not merely evidence of a futile “thirst for hidden variables.” For instance, there is no deterministic account of how one ground state is selected from among many possible ones in spontaneous symmetry breaking, yet one can certainly give an account of the process of symmetry breaking in terms of an additional structure which sets the stage for the circumstance of symmetry breaking. This point is addressed in Chapter 4.

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  • The Map versus the Territory
  • Ruth E. Kastner, University of Maryland, Baltimore
  • Book: The Transactional Interpretation of Quantum Mechanics
  • Online publication: 22 April 2022
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  • The Map versus the Territory
  • Ruth E. Kastner, University of Maryland, Baltimore
  • Book: The Transactional Interpretation of Quantum Mechanics
  • Online publication: 22 April 2022
  • Chapter DOI:
Available formats