Spring 2024 · Vol. 53 No. 1 · pp. 51–64
Ambiguity and Uncertainty in Physics
He pondered on the greatness and the living presence of God, on the mystery of eternity in the future, and, even more strange, eternity in the past, on all the infinity manifest to his eyes and to his senses; and without seeking to comprehend the incomprehensible he contemplated these things. He pondered on the sublime conjunction of atoms that gives matter its substance; that reveals forces in discovering them, creates the separate within the whole, proportion within immensity, countless numbers within infinity; and through light gives birth to beauty. This conjunction, this ceaseless joining and disjoining, is life and death.
—Victor Hugo, Les Misérables 1
The chasm between our finite minds and the reality outside of ourselves is large.
Imagine hikers at the bottom of a mountain standing between them and their destination, or a corn flake at the bottom of a cereal bowl. To get from one side of the barrier to the other, the hikers or the cornflake {52} must have enough energy to get over the barrier or they are effectively trapped. In physics, we call this a “well.” During one of my lectures of a standard first-year introductory course on modern physics, I referred to an idealized “potential well” to describe how to model an electron bound to the nucleus of an atom. I had just finished introducing these same students to a class of mathematical objects called imaginary numbers, one of which is the square root of negative one. One of my students raised his hand to ask this question: “Is this well a real well or an imaginary well?” I smiled at what I interpreted as an expression of innocent curiosity but was also moved by the profundity in those words.
The gaping chasm between our finite minds and the reality outside of ourselves is large: “The reality that exists outside our minds is independent of the choice of words or concepts that we use to translate that reality into little manageable packets of thoughts.” 2
Physics is built on concepts and tools that we use to understand and describe the natural world in its most fundamental form. Its aim is to incorporate the greatest possible number of empirical facts by deduction from the fewest possible hypotheses. Its historical belief is that the physical world is out there, unfolding entirely independent of us, according to universal laws.
Imagining that the world is governed by universal and robust laws is what our minds do intuitively when deciding on a course of action in daily life. We imagine a set of possible actions, consider the likely consequences of each, and then choose the action we believe will most likely produce the results we desire. Whether we choose rightly or wrongly, we learn to connect certain actions with certain outcomes. By experimenting in this way, we come to at least a basic understanding of how the world works.
Similarly, physics concepts are generated by our imaginations to reflect our experience of the world as we perform very carefully designed, repeated measurements. At the same time they also serve as instruments for probing nature to confirm our ideas. Our concepts are subsequently used for generating new ideas and discovering new observational facts. However, since the theories we devise are mental in nature, they will never accurately describe the world as it is. Reality is too large and complex, and our minds are too limited. Even our eyes take in light as electrical signals which are then translated by our brains into an image that generates conscious sensations and emotions in a holistic way. What is “out there” in front of us is from the very beginning quite inaccessible to us. What we “see” in front of our eyes is the result of electrical signals in our brain.
I do not know what the first genuinely “scientific” thought was like, but I can imagine it crystallized in response to a gaze at the mysterious universe of which humans find themselves a part, a universe that seems {53} uncontrollable, random, quite possibly uncaring, yet regular and amenable to yielding fruitful answers to the questions such a universe naturally elicits.
Some scholars argue that the seeds of scientific thought and the overall success of the methodology of modern science is due to its Judeo-Christian theological underpinnings. Israelites believed in a God who created both heaven and earth, endowed matter with its properties, and continues to sustain the universe. As nuclear physicist Peter Hodgson says, “the essential presuppositions of science, that matter is good, orderly, rational, contingent, and open to the human mind, are all to be found in the Old Testament.” 3 Stanley Jaki elaborates: “Once more the Christian belief in the Creator allowed a break-through in thinking about nature. Only a truly transcendental Creator could be thought of as being powerful enough to create a nature with autonomous laws without his power over nature being thereby diminished. Once the basic among those laws were formulated science could develop on its own terms.” 4
The scientific endeavor has continued to flourish since its inception and has produced theories whose beauty rivals the work of Shakespeare, the music of Bach, and the artistry of Michelangelo. On the other hand, the questions and seeking have brought us a world with jet airplanes, genetic engineering, cellular phones, GPS, Curiosity Rovers, vaccines, therapeutic drugs, and much more. None of these developments would have been possible had the scientific method never been devised and applied systematically, had a firm belief not arisen that the world is uniform enough to provoke scientific questions.
Considering the accomplishments of the scientific endeavor, it is not overstating matters to say that a scientific understanding of the world provides humanity with a certain kind of truth. Indeed, approaching the world with a scientific attitude enables us to grasp its beauty in ways that purely visual perception cannot and also helps us find “explanations that transform the world,” a phrase that physicist David Deutsch has used as the subtitle of his book, The Beginning of Infinity. 5
But approaching the natural world with a scientific eye also requires much humility. Richard Feynman, a Nobel Prize winner and one of the most original thinkers in twentieth-century physics, emphasized that “Scientific knowledge is a body of statements of varying degrees of certainty—some most unsure, some nearly sure, but none absolutely certain.” 6
Feynman’s assertion sums up the nature of the statements we make about our claims to knowledge and truth in physics. In this article, it is my goal to describe the nature of ambiguity and uncertainty in physics by illustrating the ways in which we are confronted by it in our pursuit of {54} knowledge and by outlining some fundamental theories in physics—their underlying motivations, limits, and power.
THE NATURE OF AMBIGUITY AND UNCERTAINTY IN PHYSICS
One of the primary goals of physics is to produce physical theories that allow the successful prediction of future outcomes of carefully constructed measurements. The better the theory is at making successful predictions of experimental outcomes, the more it is believed to be a reasonably accurate description of how matter behaves. A ubiquitous source of uncertainty in physics is measurement. To what degree of precision do we know a measurable property of an object in quantitative terms? Common properties of objects include position, velocity, mass, charge, etc. For example, the National Institute of Standards and Technology (NIST) reports the charge of an electron to be 1.602 176 565 × 10-19 C ± 0.000 000 035 × 10-19 C. That is, there is an associated uncertainty in this number because we are unable to measure to infinite precision. But that is neither interesting nor unexpected!
The second type of uncertainty is ignorance due to an inability to track in enough detail the complexity of the interactions of a system. For example, Newton’s Laws were monumental in the history of physics, as they overturned the much older Aristotelian physics that had held sway for centuries and revolutionized the way we think about motion. We can use these laws to predict and describe the trajectory of an object in motion, like a basketball, provided it is not moving too quickly, is not too heavy, nor too tiny. Starting with an appropriate coordinate system, it is possible to record a position of the ball at each point in time, and then infer the velocity (the rate of change of position with time). We call position and velocity objective properties of the basketball. The notion of force is added to account for our sense that there is a cause for the observed change of velocity from point to point—that is, the acceleration. This notion of force also appeals to our sense of a “push” or “pull.”
According to Newton’s second law, if we know what all the forces on the ball are and what their strengths and directions are at any moment, the resulting trajectory of the ball is perfectly determined—we know exactly where it will land. However, in the real world, we do not have perfect knowledge of all the forces that will act on that ball at every point on its trajectory. A gust of wind may arise, the air may have a different density from one point to another, etc., and because of our ignorance our predictions soon become approximate. We are reduced to speaking of probabilities, even though the laws of motion are applicable. {55}
Due to the complex and plentiful sources of interactions of macroscopic objects with their physical environments, most of what we work with in classical physics is unpredictable to some degree. This source of unpredictability is considered to be epistemological. For better predictive power, corrections can be made to describe a real basketball moving through air. Thus, according to classical physics, all motion is determined in principle (even though it is difficult to calculate for many but the simplest of systems), and when we use probabilities we are using them to express our ignorance of the various influences that may cause deviation from our prediction.
The philosophical notion of determinism has caused much unsettledness in human hearts since Newton’s theories generated a belief in wider society of a clockwork universe that looked as though it was wound up like a watch and left to unwind with no possibility of intervention (even though Newton himself did not believe this). Newton’s mechanistic universe has influenced philosophers with a materialist bent to denounce their belief in a purposeful universe, to the point of arguing that free will is an illusion.
It is important to note that the notion of determinism in physics is grounded in a methodological presupposition that reflects the belief physicists have in the uniformity of natural law and the attendant assumption that physical processes occur in a regular and lawful way. That is, determinism arises from classical theory, not from experimental fact. Peter Hodgson 7 provides an illustrative example. A specific causal sequence such as the sun having risen every morning for thousands of years is not in itself proof that it will do so in exactly the same manner, ad infinitum. It just makes it highly likely. This belief that certain natural events will consistently repeat cannot itself be proven by scientific methods; it is a metaphysical perspective that is brought to the table when a scientist begins a systematic study of regularities.
Since the development of Newton’s mechanics, two major “corrections” have been made: Einstein’s Theory of Relativity (which accounts for the behaviour of objects at speeds near the speed of light and with large masses) and quantum mechanics (behavior of particles with very small masses), which will be discussed in what follows. Within the frameworks of both, (classical) Newtonian theory is recovered as a special case when applying it so slow-moving objects, for which it remains appropriate and accurate.
MODELS AND REALITY: TWO MODELS OF GRAVITATION
When scientists approach their work, they come to it with basic beliefs about their world that influence the way they think about and probe the object of their study. A scientist’s first act in trying to conceive the working {56} of the universe is both bold and humble. The act of writing down an equation or envisioning a concept such as a force or energy to initiate the process of theory-building necessarily involves imposing a mental model onto a physical thing. For example, when you envision a perfect circle you are imagining an idealized object that has no replica in nature. A constructed house always differs in some way from the plan drafted by its architect, and yet the architect’s imaginative vision as expressed in the blueprint must precede the building of the house.
The better a theory describes and accounts for all observed data, the more merit is attributed to it. But its merit can also be judged in a variety of ways. First, on the nature of a scientific theory itself, American philosopher of science F. S. C. Northrop stressed that “this means that the theory of physics is neither a mere description of experimental facts nor something deducible from such a description; instead, as Einstein emphasized, the physical scientist only arrives at his theory by speculative means.” 8
Conversely, it means that once a theory is constructed, further experimentation is interpreted through the lens of that theory. For example, when the Higgs boson (the particle that gives mass to fundamental particles) was discovered in July 2012, it was detected as an excitation of a sensor that was interpreted contextually to be the Higgs boson. It wasn’t as if the Higgs boson just walked up to us and said hello like a new species of alligator might (if an alligator could indeed greet us in the English language, but I digress). In other words, detection of subatomic particles is always an inference or indirect observation.
Secondly, a dominant source of ambiguity in physics stems from an interpretation of what our carefully constructed theories mean, or in other words, what the physical picture represented by a theory is. For example, are the concepts that we invoke to categorize the world (such as energy and fields) real things that exist independent of our minds, or are they mere calculational devices? What is the status of properties? These questions are regularly debated by philosophers of science, but they also influence physicists in the development of new and revised theories.
Philosophers of science outline two basic categories of scientific thinking: realism and anti-realism. Most scientists fall somewhere between these two extremes. Einstein and Newton were realists while Niels Bohr, a significant contributor to the development of quantum theory, was an anti-realist, at least when it came to quantum mechanical theory.
Bas van Fraassen describes a realist as one who subscribes to the view that “Science aims to give us, in its theories, a literally true story of what the world is like; and acceptance of a scientific theory involves the belief that it is true.” 9 That is, the realist position is that our theories really do approximate the reality of the world as it is. This belief takes seriously {57} the classical principle of adaequatio mentis et rei, the correspondence between mind and the thing.
The anti-realist position can take on a variety of forms, but the one most pertinent to the study of physics is instrumentalism, a philosophical view that ideas and scientific theories are only instruments of action and that their usefulness determines their truth. Thus, objects of knowledge are only pragmatic tools. If a theory can successfully predict measurement outcomes, it is true for practical use. Broad speculation on what picture of reality the theory provides is regarded with suspicion because there is no way to prove it experimentally.
It is the tension between these two extremes that often drives progress in the construction of a physical theory. I will use the examples of gravitation and quantum mechanics to illustrate.
You may have heard the story of how an apple hitting Newton’s head changed our perception of the cosmos. The scientific community went from believing that the laws governing the heavens and those governing the earth were entirely different, until Newton persuaded us that they were the same. Newton recognized that the moon moving in orbit around the Earth was a result of the same law that caused the apple to fall on his head from the tree above.
For three centuries, Newton’s Law of gravitation was the most powerful one we had to explain the tendency of apples to fall to the earth and moons to orbit planets. Despite its success, it had an aura of mystery about it. How does one mass (such as the Earth) detect the presence of another one (the Sun) millions of miles away? According to Newton’s model, if the Sun suddenly disappeared, the Earth would immediately go out of orbit, with no time delay between the two events.
Newton was satisfied that “gravity really does exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and our sea.” 10 His rival, Gottfried Wilhelm Leibniz, criticized the strangeness of the postulated gravitational force by exclaiming that he “revived the occult qualities of the school with the idea of attraction.” 11
Nevertheless, its strength, as Newton insisted, was that it worked for practical applications and provided an explanation for many observations about the cosmos. For example, using Newton’s Law of gravitation, the orbits of all the planets could be calculated and predicted to high precision (except for Mercury). In the twentieth century, astronauts were sent to the moon based on our understanding of Newton’s laws.
However, new developments in physics at the end of the nineteenth century—James Clerk Maxwell’s development of electromagnetic field theory and its implied limit to the speed at which light could travel—paved {58} the way for the supplanting of Newton’s theory by Einstein’s theory of general relativity, a theory that revolutionized how we viewed space. In this theory, the presence of mass distorts measurements of both space and time in such a way that spacetime itself curves. The concept of gravitational force is altogether removed (Figure 1). An approximate analogywould be that of a rubber sheet, pulled taut and fastened to a rigid frame. Gravitating matter pushes on the sheet, and the rubber surface curves by an amount dictated by the mass and its location, and by the properties of the rubber itself. It is a difficult concept to grasp; attempts to visualize it in this way don’t quite suffice.
Figure 1. An object such as a planet thus moves from point A to point B along the lines of curved space produced by a large mass.
Is this really the way the universe is? The predictions of planetary orbits that Einstein’s general theory yields are almost no different from Newton’s, except for Mercury’s orbit, whose precession was now accounted for. Furthermore, other effects predicted by the general theory of relativity, such as that light should bend due to the curvature of space around a massive object such as the Sun have been verified and are necessarily accounted for in technological devices such as GPS. If that were not the case, relying on these devices would get us lost in a hurry.
ULTIMATE UNCERTAINTY IN MEASUREMENT AND QUANTUM THEORY
The mechanics of Newton recognizes uncertainty, but the underlying metaphysical principle in classical mechanics is that our inability to predict future outcomes rests on our ignorance of the many complex interactions that could affect those outcomes. Statistics became a substitute for knowledge and physicists built it into their theories.
However, in 1905 this conception of the universe was challenged. New observations of nature at the scale of the atom challenged the physics {59} community. Werner Heisenberg asked whether nature could “possibly be as absurd as it seemed to us in these atomic experiments.” 12
To understand the nature of the conundrum that physicists faced, imagine a brook in which water waves flow between two rocks that obstruct and alter the motion of the waves. As they move around the obstruction, the waves bend—a phenomenon known as diffraction (Figure 2(a)). If this experiment is set up in a controlled way in a ripple tank, such that the space between the crests of water waves is constant, an interference pattern emerges (Figure 2(b)): alternating bands of waves that reinforce each other or cancel each other out so that water rises higher in some places and lower in others.
The nature of light has mystified human beings from time immemorial. Shimmering off a lake, bending around doorways, warming your hand, or radiating through stained glass windows: light seemed to have its own unique nature, altogether different from matter.
Figure 2. (a) Diagram showing how wavefronts that propagate toward a barrier with two small openings will flare out at the openings. Beyond the openings, some portions of the wave will reinforce (Max), and some portions will cancel (Min). (b) Interference pattern for water: regions of reinforcement are white and cancellation blue. (c) Interference pattern for red laser light. Regions of reinforcement are red, and cancellation is black. For both water waves and light, the resulting patterns match the diagram in (a).
In 1704, Newton postulated that light consisted of small “corpuscles” or particles that streamed through the air. In 1799, Young’s mimicry of the water experiment with light by shining it on two very narrow slits, resulted in the formation of a similar interference pattern, as shown in Figure 2(c). Thus, Young concluded, light must be a wave. A century later, the photoelectric (1905) and Compton effect (1923) were discovered. In {60} the photoelectric effect experiment, light of various wavelengths (color) was shone on a metallic surface to free electrons from their respective atoms in the metal. The surprising result was that the energy carried away by each free electron after its interaction with light depended only on its wavelength (or frequency), not on the amplitude (brightness) of the light. If light was behaving like a classical wave (like water) in this experiment, a low-wavelength wave of large enough amplitude would be enough to dislodge an electron. But increasing the brightness of the light was ineffective in dislodging an electron. In this experiment, light appeared to be behaving as something particle-like instead of wave-like. Einstein named these quantized bundles of light photons. The Compton effect showed further evidence that light behaves like quantized bundles due to their ability to bounce off electrons and other particles, all the while conserving both energy and momentum—something that classical billiard and bowling balls do in collisions.
The results of these experiments showed that the nature of light is ambiguous. Two alternate models of light were equally effective at describing its nature, depending on which experiment was being performed: shining light on a pair of narrow slits or bouncing it off electrons.
Probing nature ever more deeply revealed more surprises. Louis de Broglie, a French theoretical physicist, invoked the timeless “nature loves symmetry” principle and posed the question: If light has both wave and particle-like properties, then might entities we envision as being particle-like also have wave properties? That is, might atomic particles such as electrons, protons, and neutrons also act as waves?
This idea was bold and was tested by passing a beam of electrons through a crystal of nickel, which mimicked the double-slits in Young’s light experiment. If electrons behave as particles, then passing electrons through two open slits should produce the result in Figure 3(a), such as we would get if macroscopic marbles were passing through appropriate-sized slits. But what was observed is in fact an interference pattern, like that for light (Figure 3(b)). And not only was this pattern obtained after sending a whole stream of electrons towards the slits at once, but also when the pattern was built one electron at a time, demonstrating that electrons colliding with each other en masse was not the reason for the pattern. The results suggest that the electron acts in response to the presence of the other slit to produce interference. But how can it behave as if it went through both slits if it is a classical particle?
Furthermore, if an attempt is made to detect which slit the electron goes through by bouncing photons off it, the interference pattern is destroyed, and the electron once more behaves like a classical particle would. The measurement itself has the effect of disturbing the system and changing {61} the outcome from what would have happened if it had been left alone. The ensemble of experimental results reveals an inherent ambiguity about the nature of light and microscopic particles.
Figure 3. (a) What we expect from the double-slit experiment if electrons act like particles. (b) The double-slit experiment using electrons.
The intense debate in the wake of these experiments produced quantum theory, a mathematical description of quantum particles that accurately predicts the outcomes of quantum experiments. The theory describes atomic systems in a radically different conceptual framework than would ever have been possible in classical terms. It has been extremely successful to date, producing technological applications and describing chemical bonding.
The biggest change from classical physics is that in quantum physics, a subatomic particle is described by a mathematical entity called a wavefunction (due to Erwin Schrödinger) that changes with time. The wavefunction represents all the possible positions the electron can be at, and when the electron is detected at a screen, it is said to collapse.
The equation that describes how the wavefunction changes with time is deterministic in the sense that it will evolve the same way every time an experiment is repeated (the overall pattern observed at the screen is repeatable). The randomness only appears when an observation of a single electron is made, and the associated mechanism of wavefunction collapse is something unexplained within the theory. Unlike uncertainty in classical physics, which is the result of a lack of knowledge, the exact location of an electron’s appearance upon measurement is unpredictable in principle.
Furthermore, for basketballs, cats, people, and everything bigger than atoms, Schrödinger’s equation describing the motion of particles transforms into Newton’s equation of motion as a limiting case, as it should—the classical world still behaves according to Newtonian mechanics.
The wavefunction we use to describe quantum systems is not generally believed to be a result of our ignorance but is built into the system of the world itself. However, the more mysterious aspect of quantum mechanics {62} is not the indeterminism of future events, but the fact that the past trajectory is also indeterministic. The intuitive concept of a “trajectory” itself is undefined, because until a measurement is performed, there is just an evolving wavefunction.
Although the quantum mechanical formalism is powerful and widely accepted, there are about a dozen contending ways of interpreting what the theory means (from Bohm’s hidden variable interpretation 13 to Everett’s many-worlds interpretation 14), each interpretation being grounded in substantial theoretical support. Many books have been written about the deep philosophical and physical concerns that each one brings to the table. A physicist’s choice of interpretation is largely a matter of personal philosophical prejudice because there is yet no further experimental data to force one interpretation to eliminate any others.
THE COPENHAGEN INTERPRETATION OF QUANTUM THEORY
The Copenhagen interpretation of quantum theory has gained the widest acceptance of physicists, is considered to be physics’ orthodox stance, and is the one I will focus on here. This interpretation aligns most closely with the philosophical doctrine of positivism, which asserts that the only meaningful statements we can make about a physical system are those that can be verified by experiment and measurement. A positivist would insist that the aim of science is only to describe and predict. Speculations that cannot be confirmed by further experimental testing are considered irrelevant to the physical description. 15 At heart, the Copenhagen interpretation is rooted in a pragmatism that can be expressed in the adage, “If it works, it’s true.”
In this context, the wavefunction is interpreted to be a mathematical machine that manufactures probabilities only; it collapses when a measurement is made. But the collapse is understood to be a bookkeeping procedure rather than a real physical process.
The implication is that we know nothing about what happens between one measurement of the system and the next. Instead of simply being ignorant of the position and momentum of an electron, the Copenhagen interpretation asserts that a well-defined momentum and position simply do not exist for the electron until it is observed. Only when a measurement is made can properties be ascribed to the particle.
In this way, an observer seemingly creates the reality observed, and consequently, the physical situation prior to measurement, whatever it was, is affected. Niels Bohr, the main proponent of this interpretation, often recited the commandment, “Thou shalt not talk of the atomic world in itself,” because the Copenhagen interpretation denies ontological status {63} to atoms. Heisenberg emphasized that “atoms or elementary particles themselves are not real; they form a world of potentialities and possibilities rather than one of things or facts.”
Although it is very compelling and natural to speak of and create visual models of microscopic particles as if they were akin to classical marbles, in the context of this interpretation, we can count only the response of our macroscopic, classical measuring apparatus as real.
Problems for the interpretation include its inability to make a clear distinction between microscopic and macroscopic systems. At what point does a measuring apparatus become macroscopic? 16 Neither can it convincingly account for the instantaneous collapse that is said to occur when an observation is made. 17 Collapse is an ad hoc post-measurement addition to the theory that holds little physical meaning.
The Copenhagen interpretation also requires belief that the observer creates reality, which means that physics ultimately encounters consciousness, another part of the interpretation that poses serious philosophical issues. 18
A more general source of uncertainty is that it provides a dearth of coherence when it comes to such fundamental issues about our universe as the status of properties, and the relation between objects and their properties, let alone the status of a truly objective world that exists independently of the way we choose to view it. How does our mentality arise through the physical structures of the brain in which it seems to be embedded?
CONCLUSION
Physical models can be powerful ways of organizing experimental data into a framework that describes our experience in a more complete and robust way. At the same time, a new physical theory can aid in interfacing with the world with technological devices. The methodology of science, however, does not permit us to answer questions about our universe that go beyond its scope. One of my insightful first-year physics students reminded the class that no matter what science reveals about the universe, why the laws are the way they are is a persistent mystery.
What we can say for certain is, in the words of Joseph Silk, “Humility in the face of the persistent great unknowns is the true philosophy that modern physics has to offer.” 19 {64}
NOTES
- Victor Hugo, Les Misérables, trans. Norman Denny (London: Penguin, 1012), 67.
- Milton A. Rothman, Discovering the Natural Laws: The Experimental Basis of Physics (New York: Dover, 1989), 47.
- Peter E. Hodgson, Theology and Modern Physics (Aldershot, Hants, England: Ashgate, 2005), 224.
- Stanley Jaki, Christ and Science (Royal Oak, MI: Real View Books, 2000), 23.
- David Deutsch, The Beginning of Infinity: Explanations that Transform the World (London: Penguin, 2011).
- Richard Feynman, What Do You Care What Other People Think? (Toronto: Penguin, 2007).
- Hodgson, Theology and Modern Physics, 77.
- F. S. C. Northrop, introduction to Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science (New York: Harper & Row, 1962), 3.
- Bas van Fraassen, “Arguments Concerning Scientific Realism,” in Scientific Knowledge, Basic Issues in the Philosophy of Science, ed. Janet A. Kourany (Belmont, CA: Wadsworth Publishing, 1998), 357.
- Isaac Newton, The Principia (1686), tr. I. B. Cohen and A. Whitman (Berkeley: University of California Press, 1999), 943.
- Leibniz, quoted by Hodgson, Theology and Modern Physics, 65.
- Heisenberg, Physics and Philosophy, 42.
- David Bohm, Wholeness and the Implicate Order (London; New York: Routledge, 1980.
- Hugh Everett III, “Relative State Formulation of Quantum Mechanics,” Reviews of Modern Physics 29, no. 3 (1957): 454–62.
- Hans C. Ohanian, Principles of Quantum Mechanics (New Jersey: Prentice Hall, 1990), 345.
- Alistair Rae, Quantum Physics: Illusion or Reality? (Cambridge: Cambridge University Press, 2004), 89.
- Ohanian, Principles of Quantum Mechanics, 359.
- Bruce Rosenblum and Fred Kuttner, Quantum Enigma: Physics Encounters Consciousness, 2nd ed. (New York: Oxford University Press, 2011); see also Rae, Quantum Physics.
- Martin Rees, quoting Joseph Silk, Before the Beginning (Reading, MA: Addison-Wesley, 1997), 6.
