by Jim Baggott
In truth, whilst Rovelli has always considered quantum mechanics to be a profoundly revolutionary theory, he has never considered it to be inconsistent or incomplete in any sense. Early in their collaboration, Rovelli and Smolin were joined in deep discussion by mathematical physicist Louis Crane. They tossed ideas around about the relation between elements of reality and the observer, the philosophy of Leibniz, and quantum mechanics. ‘The way I remember it,’ Smolin explained, ‘we each took the basic ideas and constructed theories which expressed the idea that quantum mechanics was relational.’5
In developing his theories of relativity, Einstein sought to banish from physics the entirely metaphysical concepts of absolute space and time. One consequence is that the observer is put firmly back into the picture, making measurements with rulers and clocks and performing observations inside the physical reality that is being examined, rather than from some unique perspective, from some kind of ‘God’s-eye view’ of the Universe. It’s worth noting that, by ‘observer’, we don’t necessarily mean a human observer. It’s enough that there be something (anything) with which a physical system can establish some kind of relationship. Instead of dealing with metaphysical things-in-themselves, we’re then dealing with things-in-relation-to-other-things. If the relation is with a human observer, then we can talk about things-as-they-appear. If the relation is with some apparatus, such as a ruler or a clock, then we can talk about things-as-they-are-measured.
The observer plays a fundamentally important role in quantum mechanics, too. Just as Einstein rejected the notion that there can be absolutes in space and time, so Rovelli chose to reject the notion that a quantum system exists in an absolute, observer-independent state. In other words, in relational quantum mechanics, we can discover nothing at all about the physical quantum-states-in-themselves. He wrote: ‘The thesis…. is that by abandoning such a notion (in favour of the weaker notion of state—and values of physical quantities—relative to something), quantum mechanics makes much more sense.’6
In making such an assertion, Rovelli isn’t rejecting the existence of an objective reality or the reality of ‘invisible’ entities such as electrons. He’s broadly accepting of Propositions #1 and #2. There are objective, independently existing things-in-them-selves—there are such things as electrons and they continue to exist when nobody’s looking at them or thinking about them. But, as Kant argued, we can’t discover anything about these. It only makes sense to talk about their quantum states and their properties when they establish a relationship with another system. This calls into question the viability of Proposition #3: quantum mechanics is about relations between things, not the real properties of real physical things independent of relation.
So, if the mathematical equations we use in quantum mechanics do not refer to the independently existing real physical states of quantum systems, to what do they then refer? Rovelli’s answer is that they refer to information about the quantum system derived from our experience of it.
Now, Rovelli’s relational interpretation does not in any way demand that we assign any special significance to the process of measurement. As far as he is concerned, measurement is just one among many different ways of establishing the relationships essential to quantum mechanics. But, of course, measurement is fundamental in that this is how we acquire knowledge about quantum systems. So let’s therefore try to understand this interpretation by walking step by step through a typical measurement process.
Let’s suppose, once again, that we prepare a quantum system which, based on our previous experience and understanding of the physics, can be in one of two possible states—↑ and ↓. Let’s further suppose that there is only one kind of particle involved, which we label as A. As we now know, the correct way to describe this system is in terms of a total wavefunction expressed as a superposition of the contributions of the wavefunction for A in the ↑ state and the wavefunction for A in the ↓ state:
Before we go on, let’s just consider where this expression has come from. We know from previous experiments that if we prepare the quantum system in just this way, our experience and understanding of the physics lead us to anticipate that it can be in one or the other state. Or, alternatively, if we establish a certain kind of relation between the quantum system and the device we use to prepare it, we anticipate that it can be in one or the other state. We also know that in order to make the right predictions about the future behaviour of the system, we need to represent these two states as a superposition which we call the total wavefunction. We use information from our previous experience of the physics to write the total wavefunction of the quantum system as a superposition of ↑ and ↓ states.
It’s important to be clear on my use of language here. I’m using words like ‘we’ and ‘information’, which can be taken to suggest that this is once again all about human observers and information about the results of measurements which, for example, might get recorded in a laboratory notebook. And again, this is not the intended meaning. Rovelli refers to ‘information’ very much in a physical sense, in a form that can be manifested in inanimate objects: ‘a pen on my table has information because it points in this or that direction. We do not need a human being, a cat, or a computer, to make use of this notion of information.’7
Now if the contributions from each of the wavefunctions in the superposition are equal, we can anticipate that in a subsequent measurement we will get either ↑ or ↓ with equal (50:50) probability. The outcomes are random: we have no way of knowing in advance which outcome we will get.
But what if, instead of measuring the ↑ or ↓ property of the system, we measure another property? Let’s call this + or −. We know, again from previous experience and our understanding of the physics, that a quantum system consisting of a set of particles A prepared exclusively in the ↑ state will yield both + and − with equal probability in a subsequent measurement. Similarly, a quantum system prepared exclusively in the ↓ state will yield both + and − with equal probability.
So, what do we do now?
Remember from Chapter 1 that there is no such thing as the ‘right’ wavefunction. We’re perfectly at liberty to choose a form for the total wavefunction that’s most appropriate for the specific problem we’re trying to solve. What we need is a different superposition. Instead of ↑ and ↓, we need a superposition of + and −.
And we can do this fairly easily. I won’t distract you with the details. Suffice to say that we use the information from our previous experience of the behaviour of quantum systems prepared exclusively in the ↑ and ↓ states to deduce that
I’ll admit this looks like we’ve simply substituted the wavefunctions for ↑ and ↓ with the wavefunctions for + and −. But trust me when I tell you that, no matter what it looks like, this isn’t the case. There are some pretty rigorous mathematical rules that we must follow when we make this kind of change. It might help to know that the states ↑ and ↓ and + and − are often referred to as basis states, and what we’ve done therefore is change the basis of the representation of the total wavefunction. There is really no such thing as the ‘right’ or ‘preferred’ basis. We use information from our previous experience of the physics to change the total wavefunction to whatever basis is relevant to the problem we’re looking to solve. In this case we change to a superposition of the measurement states + and −.
As before, the contributions from the wavefunctions for + and − are equal, so we can anticipate that in a subsequent measurement we will get either + or − with equal (50:50) probability. Once again, we have no way of knowing in advance which outcome we will get.
Take it from me that this is all perfectly correct. We know (again from experience) that in a series of measurements on identically prepared systems, we’re likely to get a random sequence of results such as +, −, +, +, +, −, −, +,…. . Although all laboratory measurements are subject to experimental errors, we also know that after making a statistically significant number of measurements, we’ll find
that we got + 50% of the time, and − 50% of the time.
What just happened here?
Rovelli argues that we simply use the wavefunction as a convenient way of coding our information about the quantum system. ‘The [wavefunction] that we associate with a system…. is therefore, first of all, just a coding of the outcome of these previous interactions with [the system].’8 We do this as a way of using information derived from previous experience to make predictions for the future behaviour of the system in measurements yet to be performed. The coded information allows us to make predictions about relationships that have yet to be formed.
In other words, the wavefunction isn’t real, in the sense of Proposition #3. It is not a base concept. It does not represent the real state of the quantum system. ‘In [relational quantum mechanics] the quantum state is not interpreted realistically, but the position of the electron when it hits the screen is…. an element of reality (although relative to the screen).’9 The wavefunction is merely a convenient device that allows us to connect past and future.
Rejecting Proposition #3 in quantum mechanics frees us from all kinds of apparent contradictions. When we form a total wavefunction as a superposition of two possibilities, we’re simply acknowledging that from previous experience we know to expect that the quantum system will produce outcomes such as ↑ or ↓, or + or −, depending on the type of measurements we’re going to make. The superposition is one of information, and not real, independently existing physical states.
If the wavefunction is just coded information, then it is not required to conform to any physical laws or mechanical processes. Information isn’t ‘local’ or ‘non-local’. In itself it isn’t constrained by Einstein’s special theory of relativity (though any attempt to communicate this information will be so constrained). Information can change instantaneously. A wavefunction which consists only of information is not obliged to undergo some kind of discontinuous, physical collapse. As Rovelli explains: ‘This change is unproblematic, for the same reason for which my information about China changes discontinuously any time I read an article about China in the newspaper.’10
This is no more mysterious than the referee tossing a coin at the beginning of a soccer or tennis match. If we felt the need, we could code the outcomes of this procedure as a superposition of ‘heads’ and ‘tails’. The coin spins through the air and lands on the ground, and we get the result ‘heads’. We believe that the two outcome possibilities persist on either side of the coin throughout, but as we’re ignorant of the precise mechanics of the toss and the motion of the coin through the air we resort to probabilities. We don’t tend to declare that these two possibilities ‘collapse’ to one outcome as the coin interacts with the ground, although we could, in principle.
In Rovelli’s relational interpretation of quantum mechanics, we may understand the mechanics reasonably precisely within the limits imposed by the uncertainty principle, but we lose sight of the outcome possibilities, as we can say nothing at all about the independently existing quantum states until they have established a relation with another system. In terms of a quantum coin toss, it is as though we can anticipate the mechanical motions of the coin through the air and the number of spins it will make, but now the sides of the coin no longer exist independently except in relation to their interaction with the ground. We resort to probabilities because we’re ignorant of the sides-in-themselves. We can only know the sides-on-the-ground.
Let’s push this a little further. Like every material thing in the Universe, the device we use to measure the + or − state of the quantum system is also made of ‘invisible’ quantum objects, such as atoms consisting of quarks and electrons. Suppose we connect this device to a gauge with a readout and a pointer. If the system is measured to be in the + state, the pointer points to the left. If it is measured to be in the − state, the pointer points to the right.
What is the correct quantum-mechanical description of this situation?
Well, that depends on whether we look at the pointer. Until we take a look to see which way it’s pointing, the correct representation for the total wavefunction is now something like
We’ve seen something like this before. The original quantum system and measuring device have become entangled with the gauge. Before we look to see which way the pointer went, the correct summary of the information that is available takes the form of yet another superposition.
We can go on like this forever, it seems, and this was precisely the point that Schrödinger was making with his famous cat paradox. If we rig the gauge such that pointing right kills a cat placed inside a closed box, then we’ve further entangled the quantum system, the original measuring device, the gauge, and the cat, giving
To discover the state of the cat we must introduce yet another device (me or you) capable of lifting the lid of the box and looking.
Our instinct is to insist that, surely, Schrödinger’s cat must already be either dead or alive before we lift the lid. But Rovelli just shrugs his shoulders. For sure, we can speculate about the physical state of the cat before the ‘act of measurement’ but we cannot escape a simple truth: we cannot know the state of the cat until we establish a relationship with it, by lifting the lid, and looking.
Our mistake is to think that the superposition represents the cat’s physical state—that the poor cat exists in some kind of purgatory—rather than simply representing a summary of our information about the situation. Lifting the lid doesn’t collapse the wavefunction in some physical sense, dragging the cat from purgatory into a state of deadness or aliveness. There is no physical collapse. The only thing that changes when we lift the lid is the state of our knowledge of the cat and, as Rovelli says, this is unproblematic.
So let’s now have a bit of fun. Alice and Bob are experimental physicists studying foundational aspects of quantum mechanics in a laboratory. Bob is running late, so Alice performs a measurement in his absence. She looks at the pointer and observes that this has moved to the left, , which signals the measurement outcome A+. She writes this down in her notebook as ‘+’ (if she had observed the outcome she would have written ‘–’). We denote these results as Alice+ and Alice–. As far as she is concerned, the state of the quantum system is definitely +, relative to Alice. She concludes that the quantum system is now in the state given by the wavefunction
Bob is now buttonholed in the corridor by his research supervisor, who wants to know what the state of the quantum system is that Alice has just experimented on. This might seem a bit unfair, as Bob has no way of knowing, but he does know his quantum mechanics, and draws on his knowledge of the system under study (which now includes Alice) and explains that the state is given by the wavefunction
According to Bob, Alice and the result she wrote in her notebook are now entangled in the total wavefunction. He goes on to inform his supervisor that there’s a 50% probability that Alice will have observed the outcome + (pointer on the left, ‘+’ in her notebook) and a 50% probability she observed − (pointer on the right, ‘–’ in her notebook). This might all seem perfectly reasonable—Bob can’t possibly know the outcome of the measurement because he wasn’t in the laboratory at the time the measurement was made. But if he now opens the door of the laboratory and asks Alice what outcome she got, then as far as Bob is concerned this constitutes a ‘measurement’ involving the total wavefunction in which Alice was entangled.
Before he opens the door, Alice and Bob ascribe different states (different wavefunctions) to the quantum system, leading Rovelli to conclude that ‘[i]n quantum mechanics different observers may give different accounts of the same sequence of events.’11
This would seem to make no sense at all if the wavefunctions are assumed to be physically real.
This logic can be extended without much difficulty to the situation envisaged by EPR, involving two entangled particles, A and B, and the quantum states ↑ and ↓. We know from Chapter 4 that the total wavefunction for such a system is given by
Let’s presume that the two particles move apart, A moving to the left and B moving to the right. We wait until they move some long distance apart such that they are no longer in causal contact, meaning that no physical influence or information having physical consequences can pass from one to the other in the time available.* We make measurements in two separate laboratories.* In the laboratory over on the left, Alice observes that particle A is measured to be in an ↑ state.
Now, because she knows how the original quantum system was prepared, she can speculate that particle B must be in a ↓ state, but at the instant that particle A is observed she personally cannot know the state of B, because she hasn’t established a relation with it. Likewise, in the laboratory over on the right, Bob observes that particle B is measured to be in a ↓ state, but can only speculate that particle A must therefore be ↑.
For this situation to change a further interaction is required which could involve Alice and Bob communicating with each other to share their results. Or perhaps they both share their results with a third observer—let’s call him Charles—who concludes from this that the states of the particles are indeed correlated—A is ↑ and B is ↓. They conclude that, as a result of making a measurement on either A or B, the total wavefunction collapsed to give the outcome A↑B↓. They proceed to scratch their heads as they ponder on the non-locality of the total wavefunction and the spooky action at a distance implied by quantum mechanics.
But Rovelli argues that this is the wrong way to think about what’s happening here. All that’s really changed through this sequence is the nature of the information available to Alice, Bob, and Charles. When Alice makes her measurement, she establishes a relation with particle A. Likewise, when Bob makes his measurement, he establishes a relation with particle B which is completely independent of Alice’s relation with A.