This paper on the Uncertainty Principle has not been presented or published, but is drawn from my philosophical research into the Measurement Problem in quantum mechanics. A pdf version may be downloaded here, and my other papers on this subject are also available on my Academia.edu site.
The Relevance and Irrelevance of Heisenberg’s Uncertainty Principle for the Quantum Measurement Problem
Quantum mechanics is not something you would have guessed. The moment you juxtapose quantum mechanics and everyday experience, the mysteries of how the former relates to, much less explains, the latter seem to have no end. Scientists are predisposed to take the obviousness of the world for granted (rightfully so) while trying to explain and justify quantum mechanics. Many philosophers also take the obviousness of the world for granted (improperly so). But there are a few philosophers who have taken note that the very obviousness of the world is rather surprising. It’s surprising because that which is so obvious is at the same time so unobtrusive; it is so obvious it practically insists that we overlook it. Why does the world already make sense to us, at least in an unreflective way, the moment we turn our attention to it, before we’ve had a chance to formulate the first question about it? The child contends with and utilizes gravity long before its unceasing effects arouse curiosity. Upon a moment’s reflection, we can see that our first tentative intellectual steps toward understanding, like learning our first musical tune, are already upheld by a robust commitment to the consistency and congruity of sensuous experience. We enter the world with a basic commitment to the world, what Merleau-Ponty called “perceptual faith.”
Then why does quantum mechanics, the most empirically successful physical theory ever formulated, exhibit features that are inconsistent and incongruous with our understanding of the world, subverting our perceptual faith? Or do we exaggerate the strangeness of quantum mechanics? Is it in fact replete with everydayness? In my view, to “save the phenomena” of quantum mechanics from irrationalism, one must first recover the felt wonder of the world from its mundane unobtrusiveness. Perhaps by recovering the primordial sense of the world we would also find that quantum “mysteries” cohere with it. One of these quantum mysteries is the measurement problem. Before we turn to this problem specifically, we should briefly review the nature of measurement generally.
Measurement is a mathematical activity that constitutes the possibility of a thing not just being present in the here and now, but in a mode that is ever-present, identical for any subjective viewer. Thus, it is the primitive mathematical act. There are other such objectifying activities of a mathematical character (comparison, subordination, colligation), but measurement is that special activity whereby we are engaged concretely with a thing in the register of the sensible and we construe that thing in terms of a number. Measurement is of a higher order than the other basic mathematical activities of putting things side by side, or ordering them with respect to one another, or binding them together. These, too, involve us concretely with things, i.e., in the realm of the visible or the tangible. But measurement makes the number of a thing as definitive and provides the entry (some might say the escape) to a rarefied realm of intelligibility “beyond” the sensible.
The measurement problem in quantum mechanics
The broad foundational question about the connection between quantum theory and physical reality has an attenuated form in the so-called “measurement problem.” The problem is straightforward: when juxtaposing probabilistic expectations for experimental outcomes with real experimental experience, the world “shows a unique real datum: an actual fact.” One might be inclined to think that this is no more a mystery than how an actual number turns up when rolling a fair die. The difficulty is that the theory gives no account of how an actual unique datum comes to be realized at the end of an individual measurement (whereas ordinary classical mechanics tells how this works in the case of the die). Furthermore, when measurements are repeated under the same conditions, even repeated many times over, the disjunction between the theory and the actual outcome holds every time. “The status of actual facts in the theory remains nevertheless an open and troublesome question. Where does this uniqueness and even this existence come from? This is undoubtedly the main remaining problem of quantum mechanics
and the most formidable one.”1
Most scientists work under the presumption that doing more science will resolve this problem. But not all of them are willing to stride, like one of Arthur Koestler’s sleepwalkers, through such an admitted “fundamental obscurity.” John Bell is perhaps one of the more famous of these realists. He demanded that any interpretation of quantum mechanics meet the minimum condition of maintaining the Copernican perspective that displaced human beings from the center of the universe. Accordingly, he argued that concepts such as ‘observable’ and ‘measurement’ were “rather woolly,” and being anthropocentric, had no place in an authentic physical theory.
So, within the community of scientific practitioners we see fundamental disagreement over what the measurement problem means and what are the conditions for its solution and explanation. How shall we get our bearings? What is the convergent perceptual setting upon which theorists diverge conceptually? Specific examples are easy to find because the problem exists for any kind of quantum measurement, indeed all quantum measurements, whether the system is as simple as a single photon exciting an atom or as complex as the highly energetic experiments in giant accelerators searching for new particles. It seems that some measurements within this range of events should fail to have determinate outcomes. But this flies in the face of manifest perceptual experience. So how is the transition from indeterminate states to determinate states effected?
Quantum mechanics provides a set of causal principles which describe and predict the mechanics of a quantum system. The functional cornerstone of these principles is the unitary transformation postulate, which describes how one state at some initial time evolves into another state at some later time. The problem is that the foundational deterministic equation arising from this postulate (the Schrödinger equation) seems to exclude the possibility of a measurement ever occurring. So theorists add a separate principle of measurement, which requires a rupture to the smooth, linear evolution of the quantum system. This postulate requires the theorist to “project” what was a potentially determinate value onto an actually determinate value. It is this projection postulate that has sustained the most attention and criticism because it introduces physically incomprehensible notions like “collapse of the wave function” or “reduction of the state vector.” It is an admittedly irrational worm in an otherwise lovely apple.
What are we to make of a (purportedly) sensible thing that seems to have no definite place or position? If ‘to be’ means ‘to be there’, then how are we to understand the ‘there’ of a photon or an electron that is described by a wave that propagates everywhere? (Heidegger’s meditation on nearness and annihilation in the opening section of “The Thing” is appropriate here.) Furthermore, how are we to understand an object that is not indifferent to acts of observation or measurement? How must we transform our classical view of measurement, that a pre-formed reality is open to human observation while yet remaining uninfluenced by actual measurements, in the light of quantum mechanics where measurement is an intrinsically invasive procedure?
A phenomenological analysis of the measurement problem would involve at least five questions:
- Why has measurement become a “problem” in quantum mechanics? What is the true source of the trouble?
- How do the entities investigated “phenomenalize?” How do they emerge into the register of the sensible, the visible?
- How does measurement of quantum entities and processes relate to phenomenalization?
- What is the role, if any, of human involvement, e.g., perception, in measurement?
- What is the role, if any, of human involvement in the thing measured? What does the measurement problem teach us about the (non)sensible thing?
In this paper, I shall limit our considerations to the first question. Let us begin by setting to the side any predetermination of the “reality” or “existence” of the entities in question. We need to minimize the influence of our natural predispositions to talk about and think about atoms, electrons and photons as if they were ordinary things like tables and chairs, an equivalence which is manifestly not the case. My phenomenological feint is not proposed in order to answer the same questions taken up elsewhere, only now from a “phenomenological” perspective, whatever that might mean to the hearer. What I do hope to elucidate is the nature of the watershed in physics between realists and anti-realists, its genealogy, and other possibilities that might be envisioned.
In order to do justice to the task of concrete research, I begin from an atypical beginning than most philosophical research on the subject of quantum mechanics. I want to gain some understanding of how the measurement problem ever became a problem at all. This is not meant in the sense of a question of empirical history. Rather, we will need to undertake some conceptual archeology. The measurement problem, characterized as an interaction between an observer and something observed, suffers from the obscurity created by the entrenched conceptual doublets of modernist metaphysics (nature-man, mind-body, self-other, subjectobject, constituting agent-constituted thing, etc.). The first task then, is to clarify the interaction
at the root of the measurement problem on a basis that is not so conceptually hamstrung. It is true that there are many claims in the scientific and philosophical literature that the measurement problem has been solved (or that it is merely a pseudo-problem), but the proposed “solutions” entail other nonrealistic consequences (e.g., nonlocality); and, while these insights are philosophically suggestive, so far, solutions to the measurement problem have merely transposed the original problem into a different register with the same metaphysical precommitments.
Heisenberg’s uncertainty principle and the Pythagorean root of quantum mechanics
Why has measurement become a problem in quantum mechanics? Because it, more than the other presumed problems of quantum physics, is the problem of foundations. It is a philosophical problem posed by physics. Other features of quantum mechanics which have incited much philosophical reflection are not our real concern, even though they sometimes are falsely associated with the measurement problem. Chief of these is Heisenberg’s famous “Uncertainty Principle.” This tenet of quantum mechanics, which to many is so closely associated with the inherent “mystery” of quantum theory, is not, as it turns out, relevant to quantum mechanics per se. It is quite simply not a discovery or determination unique to quantum mechanics. It tells us little to nothing about the concrete aspects of microscopic phenomena and our involvement with them (despite breathless claims to the contrary in some popularizations of quantum mechanics, beginning with Heisenberg himself). The “Uncertainty” (better: “Indeterminacy”) Principle is a mathematical artifact created by a precommitment to economical priorities in the interest of simplifying calculation or computability, not from measurement disturbances. The Heisenberg indeterminacy relation takes two forms:
Δp Δq > h/4π (1)
ΔE Δt > h/4π (2)
They express the variance of two canonically conjugate2 quantities: momentum and position in the first case and energy and time in the second case (Heisenberg’s original derivation published in 1927 describes an electron moving in empty space). The right side of the inequalities is a constant, with Planck’s constant, h, in the numerator (6.626 x 10-34 J • s). This indeterminacy principle is as ubiquitous as potsherds in the mathematical sciences. Let us examine the ways in which it appears in different guises in the theoretical and applied sciences and attempt to trace its genealogy.
In my undergraduate days in electrical engineering, I toiled long hours on signal analysis. Be it an osprey call or an FM radio transmission, all signals have two elementary features, irreducible (though transformable) to one another: time and frequency. In the case of the bird song, you can listen as the signal varies in intensity through time. You can also hear, at any given moment, the pitch or pitches of the signal, its frequency component. You cannot hear all of the frequencies simultaneously, just like you cannot see all of the colors in ordinary light; you need a tool to break up the complex signal into its components. For light, we use a prism; for signals (more precisely: for the functional representation of a signal), we use the Fourier transform. When you transform a signal from the time domain into the frequency domain you transform a signal into a spectrum with harmonics at different frequencies and different magnitudes (amplitudes). Now, a signal generated “naturally,” or “in the wild,” is sloppy; the tones aren’t pure or perfect, they’re “noisy.” The clicks, chirps, chattering, or other interruptions to the subject of the signal (speaking musically) are not the features that we want to stand out; quite the contrary, we want to filter them out so that the subject stands out more clearly.
In the representation of the signal, there is always some spread or variance from where the frequency is centered, the value around which it is concentrated (in statistics, this is the expectation value). This is where the indeterminacy relation enters: there is always a minimum degree of divergence between the two spreads, between the time variance and the frequency variance, and that divergence is expressed as an inequality:
s x S ≥ 1/16π2, (3)
where s is the time signal variance and S is the Fourier transform or frequency signal variance.
What does this particular mathematical expression mean? The purer (or clearer or more defined) the time signal, the fuzzier is the frequency signal. And vice versa: the clearer the spectrum, the more indistinct the time signal. Note well: the indeterminacy (or “fuzziness”) is
not an aspect of the actual phenomenon as it is experientially manifest (e.g., the osprey call that I hear); it is a result of the abstract analysis we have applied to the signal, which is a functional representation of the phenomenon (in the case of the bird song, a representation of something audible). In other words, the mathematized expression of the bird song re-presents an irrevocable distortion of the original phenomenon, the song as it is sung or heard. But how did we generate this mathematical artifact? Hidden within the function we applied to the signal to determine the variances s and S is a simplification: it is linear. But the original signal to which we applied the function is nonlinear—there is harmonic distortion, frequency compression, clipping—and vastly more complex than we would prefer or manage for calculative purposes. So, for economic reasons, we make a simplification, we make the math more convenient. Note well: other interests shape and guide us, practical interests, according to which we discard features or elements of the phenomenological totality for the sake of aesthetic, pragmatic or other considerations.
Now, it is no accident that the Heisenberg indeterminacy equations (1) and (2) have more than a family resemblance to the signal variance inequality (3). Structurally, they express the same relation: the product of two spreads or variances on the left side which is greater than or equal to some constant value. And, just as was the case with (3), we must keep in mind that (1) and (2) are also functional representations of abstract concepts; i.e., the indeterminacy of ‘position’ or ‘momentum’ spread expressed by the equations is not a feature of a concrete phenomenon. Furthermore, these are also ideal operations: the resolution of one variable can be varied infinitely with corresponding deterioration or improvement in the quantitative determinacy of the companion variable without any implication that some real sound in the world approximated by one of the variations is itself sensuously indeterminate. Where theorists too often go astray is in the common assumption (since Galileo) that mathematical phenomena transparently and unproblematically map onto or correspond with the phenomena encountered and engaged in experiential manifestness, “in the wild,” if you will; that our neat, cultivated idealities must have some positive ontological status, either in themselves or as the only “true” representation of some concrete phenomenon. Obviously, this selective perspective or eidos of bird songs, electromagnetic waves and electrons means that the way in which we are going to contrive these as objects and signify the world itself as object is as rigorously representable by linear means.
Let me give another example to reinforce my earlier claim about oft-overlooked simplifications of linearity. One of the most popular mathematical expressions formulated in the twentieth century is E=mc2. This is Einstein’s famous mass-energy equivalence formulation, a follow-up to his original Special Relativity theory of 1917. The relation between energy E and mass m is modified by a constant of proportionality, c2. But, there are an infinite number of nonlinear terms on the right side of the equation that are not shown that make a more precise determination of the desired variable (either E or m) far less manageable. Exactitude is sacrificed for elegance. It is no wonder that the trade is sought given the high value placed on an objective sense of balance (viz., laws of conservation) and completeness and totality of representation. Thus, the determination of a number by measurement does not entail that precision or exactitude of quantity is the desired aim.
Modern natural science finds the pragmatic principle of “for all practical purposes” indispensable. Analyses are condensed or abridged without noticeably relinquishing control of prediction, planning or common standards of measurement. Some conscientious theorists are uneasy with this pragmatic incursion into quantificational matters because they can find no rational basis for calling a halt to what they already know is rational, viz., the mathematical rigor of the formulation and the certitude of the calculative operations. Can we know in advance the value at which we’ve reached the threshold of mathematical “control”? Why or why not? If so, can we state or specify this a priori as clearly and distinctly as the mathematical certitude derived from it? If not, is there anything from “nature” other than experimental repetition or a posteriori empirical operations that we can point to as a basis for our decision to interrupt the infinite iterations that unfold before us? These are the questions that need to be asked and that constitute the real philosophical import of the Heisenberg indeterminacy principle.3 But, historically speaking, we could have asked these questions before quantum mechanics was formulated.
Heisenberg indeterminacy relations, both quantum and classical, arise because the world is just too complex, or, speaking mathematically, “nonlinear,” for our practical purposes. In the process of idealization from nonlinear to linear, and abstraction from phenomena “in the wild” to their more docile, cultivated mathematical representations, we simplify the representations of concrete phenomena so that we can perform linear math on them. We find ourselves in a forest out of a Brothers Grimm tale and in order to make sense of it, we raze, prune, trim, flatten, and straighten all the wildness out of it until we have a tame, formal, English garden. The “higher order” terms are ignored as Rococo excesses of nature. This sweeping approximation requires
the insertion of an estimated value into what is manifest (the “knowns”), a straightening of crooked curves and wiggles, a smoothing of rough terrain, i.e., an idealization. No matter how disheveled the crown of a tree, one can always determine smoothness by arbitrarily narrowing
the focus to a smaller region. What must always be borne in mind is that the “global” view of the tree manifestly differs from the linear, smooth, local view. I am not inferring that we cannot thereby mathematicize the global phenomenon; I merely wish to point out that we ought to avoid recklessly transferring the “good fit” of a linear formulation from a local level to a nonlinear holistic level.
We need to trace the ancestry of indeterminacy relations still further, for we have not yet reached their origin. Indeterminacy relations are found throughout the mathematical sciences, both classical and quantum. The quantum indeterminacy inequalities (1) and (2), and my chosen example of a classical indeterminacy inequality (3), are both representations of abstract objects: variance in ‘position’, ‘energy’, ‘frequency’, etc. Once objectified, these conceptual abstractions can be thought of in some (abstract) space. How are they related to one another in this abstract space? The fundamental mathematical activities (e.g., comparison, subordination, colligation, measurement) are not available to us in a non-sensible register, so we require a higher order analysis. First, we represent magnitudes by fixing arbitrary points A and B in an abstract space. Arbitrary vectors, A and B, can then be drawn with lines from a common origin to the two points. To complete their relation, construct an orthogonal (perpendicular) projection of one line to be superimposed on the other line. This projection creates a right-triangle relationship and leads to the “normal” equations of least-squares curve fitting. This projected right triangle contains a Heisenberg indeterminacy relation, the Cauchy-Schwarz inequality, which relates the lengths of the two vectors (the product of their norms) to the absolute value of the inner (or dot) product (also called the ‘correlation’) between them:
lA x lB ≥ |A ● B|, (4)or‹x, y›2 ≤ ‹x, x› ● ‹y, y› (5)
in generalized bra-ket notation. The Heisenberg indeterminacy principle just is the quantum mechanical expression of the Cauchy-Schwarz inequality. But recall how we generated the Cauchy-Schwarz inequality: two abstract straight lines and the formation of a right triangle. This procedure allows us to “normalize” unfixed vectors and simplify the “fitness” of an unknown quantity given a minimum number of known quantities. The paradigmatic example of determining an unknown value in light of two known values is the solution of the length of the side of a right triangle or the deflection of one of its unknown angles. All of the mathematical sciences, including both quantum and classical physics, insofar as they utilize or impose the constraint of linearity, contain an indeterminacy relation whose common ancestry can be traced to the Pythagorean theorem.
So, the mathematical formalism of quantum mechanics—its abstract “objects” (operators) and “space” (Hilbert space)—finds its roots not only in the algebraization of geometry begun by Descartes (quantum mechanics makes extensive use of the linear algebra generalized from analytic geometry) but also in the humble beginnings of Pythagorean and Euclidean geometry. Indeterminacy relations ultimately rest on the ubiquitous Pythagorean theorem. Underlying the modern use of the Pythagorean theorem is a notion of problem-solving and optimality whereby an unknown path is inferred from known components. The theorem depends on orthogonality conditions whereby two abstract objects intersect as if they were the legs of a right triangle. The orthogonality conditions permit the easiest way to find a trend in a scattering of data points and filter some of the noise from your car radio. The application of the Pythagorean theorem outside the realm of pure geometry, the finding of an optimum direction or value, the simplest interpolation, the easiest or least calculation, all indicate the supremacy of a principle of economy. But that is most certainly not a Pythagorean or Platonic principle. A philosophical reorientation was required to make it possible to have an interest in simplifying a problem for
Linearization is achieved by application of the Pythagorean theorem and it enables us to focus our efforts on the elements of a system that matter for calculative control, on the determination of manageable parts. This is precisely the approach taken in quantum mechanics. The initial appearance of subatomic entities and electromagnetic radiation as classically wavy phenomena allowed theorists to study them using well-understood and relatively simple concepts of linear wave mathematics: reflection, diffraction, interference, intensity, frequency, periodicity, superimposition. Formally, quantum mechanics is not about “things in the world” but about swarms of “linear operators” in a cosmos of matter waves. But how are we to understand the necessary interface between these classically derived concepts, these abstract objects, this
abstract space, and the perceived world of lived experience where these formulations are confirmed, the empirical manifold, the world of manifest perceptual experience?
Linear mathematics spawns indeterminacy relations. So, our philosophical interest is spurred not by indeterminacy relations per se, but by their origin in linearity assumptions. If a system is nonlinear, the parts of the linearized subsystem do not add up to the whole. The behavior of groups cannot be sufficiently understood as the accumulation of their components’ behaviors. Even if the mathematical artifacts of linearity assumptions, the indeterminacy relations, are somehow transferable or superimposable on phenomena, then it is possible to ask: how can it be that a measuring instrument (or a measurer, for that matter), which is a big, complex chunk of material, is a reliable guide for studying the finest divisions of matter? This question remains highly controversial and the analysis of indeterminacy relations can carry us no further. We must seek out the question where it is questionable, not in the register of the intelligible entities of mathematical operations, but in the register of the sensible. This is the central problematic for further phenomenological research on the measurement problem.
- The Interpretation of Quantum Mechanics. Roland Omnès. (Princeton, NJ: Princeton University Press, 1994), 60
- ‘Canonically conjugate’ variables are “quantities that are not independent of each other,” i.e., they have some relation such that one is irreducible to the other.
- It is on these foundational questions that you find commendable philosophical sensitivity on the part of physics theorists in the scientific literature.
- An excellent review of this tectonic shift is found in David Lachterman’s The Ethics of Geometry: A Genealogy of
Modernity. See also Marc Richir’s excellent review of this book.