by Jim Baggott
Einstein’s realist interpretation of the wavefunction was very different to Schrödinger’s. If quantum probability is, after all, a statistical probability born of ignorance, then there must exist a further underlying reality that we are ignorant of, just as atomic motions underlie the temperature and pressure of a gas. This was Einstein’s point: as this underlying reality makes no appearance in quantum mechanics, then the theory cannot be considered to be complete. Einstein wouldn’t be drawn on precisely what he thought this underlying reality might be, and the EPR paper concludes with the comment ‘we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.’13 I’ll have more to say about this in a later chapter.
Schrödinger’s interpretation of the wavefunction couldn’t possibly be right, and Einstein sought to persuade him of this in a letter dated 8 August 1935. In this letter Einstein asked him to imagine a charge of gunpowder that, at any time over a year, may spontaneously explode. At the beginning of the year, the gunpowder is described by a wavefunction. But how should we describe the situation through the course of the year? Until we look to see what’s happened, we would have to regard the wavefunction as a superposition of the wavefunctions corresponding to an explosion, and to a non-explosion. He wrote:14
Through no art of interpretation can this [wavefunction] be turned into an adequate description of a real state of affairs; [for] in reality there is just no intermediary between exploded and not-exploded.
Schrödinger eventually relented, and came to share Einstein’s views. But the gunpowder experiment had set him thinking. As there is nothing in the mathematical formulation of quantum mechanics that accounts for the collapse of the wavefunction, then why assume this happens at the quantum level? Why not imagine that entanglement reaches all the way up the measurement chain, to the classical apparatus itself? In a reply dated 19 August he outlined another thought experiment that would become eternally enduring:15
Contained in a steel chamber is a Geigercounter prepared with a tiny amount of uranium, so small that in the next hour it is just as probable to expect one atomic decay as none. An amplified relay provides that the first atomic decay shatters a small bottle of prussic acid. This and—cruelly—a cat is also trapped in the steel chamber. According to the [wavefunction] for the total system, after an hour, sit venia verbo [pardon the phrase], the living and dead cat are smeared out in equal measure.
This is the famous paradox of Schrödinger’s cat.
Einstein was in complete agreement. A total wavefunction consisting of contributions from the wavefunctions of a live and dead cat is surely a fiction. Better to try to interpret the wavefunction realistically in terms of statistics. If the experiment is duplicated, the laboratory filled with hundreds of chambers each containing a cat, then after an hour we predict that in a certain number of these (predicted as a probability derived from the wavefunction), the cat will be dead.* The Geiger counter in each box clicks or doesn’t click. If it clicks, the relay is activated, the prussic acid is released and the cat is killed. If it doesn’t click, the cat survives. Nowhere in this experiment is a cat ever suspended in some kind of peculiar purgatory.
Schrödinger intended the cat paradox as a rather tongue-in-cheek dig at the apparent incompleteness of quantum mechanics, rather than a direct challenge to the Copenhagen interpretation. It does not seem to have elicited any kind of formal response from Bohr. Schrödinger wrote to Bohr on 13 October 1935 to tell him that he found his response to the challenge posed by EPR to be somewhat unsatisfactory. Surely, he argued, Bohr was overlooking the possibility that future scientific developments might undermine the assertion that the measuring apparatus must always be treated using classical physics. Bohr replied briefly that, if they were to serve as measuring instruments, then they simply could not be of quantum dimensions.
The community of physicists had in any case moved on by this time, and probably had little appetite for an endless philosophical debate that, in the view of the majority, had already been satisfactorily addressed by Bohr.
In the meantime, the ‘Copenhagener Geist’ had become formalized and enshrined in the very mathematical structure of quantum mechanics. The theory had emerged from a profoundly messy sequence of discoveries that had involved underlying violence to the mathematics, more than a few unjustified assumptions, and occasional conceptual leaps of faith. In its early years, the passage across the Sea of Representation had been difficult. Then there was the challenge of reconciling the two distinctly different approaches to quantum mechanics that had been developed by Heisenberg and by Schrödinger, which Schrödinger himself had demonstrated to be entirely mathematically equivalent.
In the late 1920s, Paul Dirac and John von Neumann separately sought to put the theory into some sort of order by establishing a single, formally consistent mathematical structure for quantum mechanics. Their approaches were summarized in two books: Dirac’s The Principles of Quantum Mechanics was first published in 1930, and von Neumann’s Mathematical Foundations of Quantum Mechanics was published in German in 1932. Their approaches were somewhat different, and von Neumann was critical of some aspects of Dirac’s mathematics, but from these emerged the structure that is taught to students today.
Von Neumann had been a student of the great mathematician David Hilbert who, in a lecture delivered to the International Congress of Mathematicians in Paris in 1900, had outlined a long list of key problems that he believed would occupy the next generation of leading mathematicians. This list has become known as Hilbert’s problems. The sixth of these concerns the mathematical treatment of physics. Hilbert argued that an important goal for future mathematicians would be to treat the physical sciences in the same manner as geometry. This means grounding physics in a set of axioms.
Axioms are self-evident truths that are assumed without proof, and represent the foundations of the mathematical structure that is derived from them. The proof of the axioms then lies in the consistency of the structure and the truth of the theorems that can be deduced from it. Hilbert’s axiomatic method represented an almost pathological drive to eliminate any form of intuitive reasoning from the mathematics, arguing that the subject was far too important for its truths to be anything less than ‘hard-wired’. Applied to physics, this demand for mathematical rigour and consistency inevitably resulted in a rather disconcerting increase in obscure symbolism and abstraction. In his review of Dirac’s Principles, Pauli warned that Dirac’s abstract formalism and focus on mathematics at the expense of physics held ‘a certain danger that the theory will escape from reality’.16 It became nearly impossible for anyone of average intelligence but without formal training in mathematics or logic fully to comprehend aspects of modern physics.
Actually, you’re already quite familiar with the first few of the axioms of quantum mechanics from your reading of Chapter 1. We start with:
Axiom #1: The state of a quantum mechanical system is completely defined by its wavefunction.
In other words, quantum mechanics is mathematically complete, as indeed it must be if it is to serve its purpose as a foundational theory of physics. So much for Einstein. I call this the ‘nothing to see here’ axiom.
Axiom #2: Observables are represented in quantum theory by a specific class of mathematical operators.
Again, I don’t propose to give you the details of what is meant here by ‘specific class’. All you really need to know is that these operators are particularly suited to the task of extracting the values of observables from the wavefunction. I think of this as the ‘right set of keys’ axiom. To get at the observables, such as momentum and energy, we need to unlock the box represented by the wavefunction. Different observables require different keys drawn from the right set.
Axiom #3: The average value of an observable is given by the expectation value of its corresponding operator.
This tells us how to use the keys. I think of it as the ‘open the box’ axiom. It is the recipe we
use to get at the observables themselves.
If quantum mechanics is to be a useful predictive theory of physics, we obviously need to know how to use it:
Axiom #4: The probability that a measurement will yield a particular outcome is derived from the square of the corresponding wavefunction.*
This is known as the ‘Born rule’.† Or, if you prefer, you can think of this as the ‘What might we get?’ axiom. Note that when we apply this to a quantum superposition with two or more possible outcomes, it doesn’t say what we will get in any individual measurement.
There is one further axiom in the main framework, related to the way we anticipate that the wavefunction will change in time:
Axiom #5: In a closed system with no external influences, the wavefunction evolves in time according to the time-dependent Schrödinger equation.
This means that, once established, the wavefunction evolves in a predictably deterministic and continuous manner, its properties at one moment determined entirely by its properties the moment before. Think of the electron wavefunction evolving smoothly in time as it passes through two slits, forming a ‘wavefront’ which alternates between high and low or zero amplitude as a result of interference, as shown in Figure 5a. This is the ‘how it gets from here to there’ axiom.
There is no place here for the kind of discontinuity we associate with the process of measurement. As von Neumann understood, accepting Axiom #5 forces us to adopt a further (but related) axiom in which we assume that a wavefunction representing a superposition of many measurement possibilities collapses to give a single outcome.
Of course, we never had to do anything like this in classical mechanics.
Now, Euclid’s geometric axioms are concerned with the properties of straight lines, circles, and right angles and, I would contend, meet the criterion of self-evident truth. But there’s nothing particularly self-evident about the axioms of quantum mechanics. I guess this is hardly surprising. The formulation of quantum mechanics is as abstract and obscure as Euclidean geometry is familiar.
These axioms leave entirely open the question of the reality or otherwise of the wavefunction—after all, this is mathematics, not philosophy. But I think it’s helpful to note just how many of the Copenhagen interpretation’s basic tenets became absorbed into the axiomatic structure of quantum mechanics. Just as empirical facts can never be free of some theory needed to interpret them, so theory can never be completely free of the metaphysical preconceptions that assisted at its birth. The standard mathematical formulation of quantum mechanics is not an entirely neutral witness to the debate that would follow.
This is the great game of theories. Let’s now see how physicists have played this game for the past ninety years or so.
* And the incident will likely be followed by a visit from animal welfare authorities.
* Again, just to be clear, recall from Chapter 1 that we actually use the modulus-square of the wavefunction.
† Strictly speaking, the Born rule relates to the probability of finding an associated particle at a specific position in space. However, much the same manipulations are involved in deducing the probabilities of obtaining specific measurement outcomes from the square of the total wavefunction. What is important here is that we obtain probabilities from the squares of the wavefunctions involved, so for the sake of simplicity I will continue to call this the ‘Born rule’.
Part II
Playing the Game
5
Quantum Mechanics is Complete
So Just Shut Up and Calculate
The View from Scylla: The Legacy of Copenhagen, Relational Quantum Mechanics, and the Role of Information
The correspondence between Einstein and Schrödinger shows that, already by the summer of 1935, the Copenhagen interpretation had become the orthodoxy. It was already the default way in which physicists were meant to think about quantum mechanics. Schrödinger was pleased that EPR had ‘publicly called the dogmatic quantum mechanics to account’.1 Einstein referred to the Copenhagen interpretation as ‘Talmudic’; a ‘religious’ philosophy that is to be interpreted only through its qualified priests, who insist on its essential truth, and who countenance no rivals.2
The philosopher Karl Popper called it a schism:3
One remarkable aspect of these discussions was the development of a split in physics. Something emerged which may be fairly described as a quantum orthodoxy: a kind of party, or school, or group, led by Niels Bohr, with the very active support of Heisenberg and Pauli; less active sympathizers were Max Born and P[ascual] Jordan and perhaps even Dirac. In other words, all the greatest names in atomic theory belonged to it, except two great men who strongly and consistently dissented: Albert Einstein and Erwin Schrödinger.
This is all really quite unfortunate. As the interpretation was forged through an uneasy alliance between Bohr and Heisenberg, brokered by Pauli, it was always going to be something of a compromise. For many physical scientists who—even today—routinely make use of quantum mechanics without worrying overmuch about what it means, the thought that there might be ‘nothing to see here’ is not particularly troubling. But for those (admittedly fewer) scientists who prefer to dig a little more deeply before committing to an anti-realist interpretation like this, it’s fair to say that Copenhagen fails to satisfy. For every question it appears to answer, more questions go unanswered. The ambiguity and confusion it engenders, when combined with the almost pathological levels of mathematical abstraction that were first introduced into the theory by Dirac and von Neumann, render quantum mechanics virtually incomprehensible, even to many scientists.
Bohr’s insistence on the principle of complementarity meant that discussions about interpretation quickly devolved into discussions about the inadequacy of the language we use to describe quantum systems and the relationship between these and the classical apparatus used to perform measurements on them. It also drew a line between the quantum and classical worlds that seems entirely arbitrary—just where is the quantum world supposed to end and the classical world begin? At the level of atoms and molecules? Or cats? The physicist John Bell called this the ‘shifty split’.4
At the time I’m sure this must have seemed perfectly reasonable. Neither Bohr nor Heisenberg could have possibly anticipated the abilities of talented experimentalists, and the development of apparatus of extraordinary subtlety and sophistication that would be brought to bear on these questions, forty or fifty years later. As we will see in subsequent chapters, later generations of physicists would be able to make measurements on quantum systems of a kind undreamt in the philosophy of the Danish priesthood.
Complementarity and the limitations of classical apparatus—central planks in the Copenhagen interpretation—were simply not future-proof.
But these are really distractions. They are actually inessential to Bohr’s core argument, which says that the quantum world is inaccessible. Unlike the classical world, which we had come to understand to be perfectly accessible—with its base concepts, such as momentum and energy, sitting right ‘on the surface’ of the equations—the quantum world lies beyond our reach. This is the principal metaphysical preconception that lurks at the heart of the Copenhagen interpretation. This preconception says that in quantum physics we have run up against a fundamental limit. We have hit the boundary that distinguishes the metaphysical things-in-themselves from the empirical things-as-they-appear.
What then happens if we forgo these additional preconceptions? What happens if we accept that our interpretation is not constrained by our classical language descriptions or limited by the nature of our measuring apparatus? All we need to do is separate our experience of quantum physics from the way we choose to represent it, in whatever language we deem to be appropriate. In other words, instead of being really rather ambiguous about Proposition #3, we come off the fence and reject it outright. Less Bohr, more Heisenberg.
And this is what contemporary theorist Carlo Rovelli has done.
Rovelli has spen
t his entire career as a theoretical physicist in pursuit of a quantum theory of gravity. In essence, this is about finding a way to bring together the two most successful foundational theories of physics—quantum mechanics and Einstein’s general theory of relativity. The former describes the physics of the very small. In the form of various quantum field theories, it underpins the current standard model of particle physics. The discovery of the Higgs boson at CERN in Geneva in 2012 was only the most recent of the standard model’s many triumphs. The latter is essentially a theory of space and time. It describes how mass–energy causes spacetime to curve, leading to the phenomenon we call gravity. The general theory of relativity is the basis for the current standard Big Bang model of the Universe, and the detection of gravitational waves in 2015—ripples in spacetime caused by violent events such as the merging of black holes—is only the most recent of this theory’s many triumphs.
As Lee Smolin explained in his book Three Roads to Quantum Gravity, published in 2000, there are potentially three approaches that can be taken. You can start with quantum mechanics and seek to constrain it so that it meets the stringent requirements of general relativity. Or you can start with general relativity and find a way to ‘quantize’ this—yielding a theory in which space and time are themselves quantum in nature. Or you can start over, seeking a new theory which has both quantum mechanics and general relativity as limiting forms.
Rovelli and Smolin are numbered among the chief architects of a theory called loop quantum gravity, forged by taking the road from general relativity.*
It goes without saying that if quantum mechanics is to be a foundation on which an elaborate quantum theory of gravity is to be built, a complete lack of clarity on the question of its interpretation and meaning is extremely unhelpful. Interestingly, both Rovelli and Smolin have sought clarity for themselves, but the interpretation of quantum mechanics is one of two problems in contemporary physics that they disagree on (the other is the reality of time).