Symmetry and Identity

This is a guest post by Kenneth Shinozuka.

Everything is changing all the time, even though many of the objects in the world around us appear to be totally still. As the philosopher Heraclitus said over two millennia ago, “Everything gives way and nothing stays fixed … You cannot step twice in the same river.”

The leaves change color. Buildings decay. Your body grows old.

Yet most of us subscribe to the idea that there is a stable identity that underlies all of this metamorphosis. A leaf that is now red isn’t, we believe, a separate entity from the one that was originally green. We don’t think that someone changes into a different person if he swaps out his outfit or dyes his hair to another color. In fact, we believe that you keep the same identity throughout your entire life, even though your appearance will change so much that it might be impossible for someone else to recognize you based on how you looked when you were many decades younger. In other words, identity is a feature that persists through the changes brought on by time.

Many of us believe that an object can retain its identity even when it undergoes far more dramatic changes. For example, the age-old Ship of Theseus thought experiment asks whether a ship remains the same object after all of its components have been replaced. A lot of us are inclined to believe that it does, since the new ship, though comprised of an entirely different set of planks, looks no different from the previous one.

But questions about identity become much more complex once we move beyond this simple case, and some of these complexities take us to the unstable world of quantum mechanics, where nothing is easily distinguishable.

Identity as a relational pattern

In the case of the Ship of Theseus, what if the wooden planks were replaced with an entirely different material,  like say steel plates? This is where people will begin to disagree. Some might claim that the ship has changed into a different thing once its material composition has fundamentally changed. Others would argue that the new ship is the same as the old one since the arrangement of its constituent parts has remained intact. What’s more important to identity: what an object is made of, or its overall structure?

Interestingly, this variation on the Ship of Theseus thought experiment isn’t as polarizing when  translated to human bodies. We wouldn’t consider somebody a different person if he were to lose his arm and substitute it with a bionic limb. If a friend is physically scarred by a tragic accident — to the point where she looks totally different — those who didn’t know her well might mistake her for someone else. Yet the people who are close to her would still refer to her by the same name. Finally, someone who totally erases his memory with a neural implant (say, in order to forget a traumatic incident) could be identified instantly by other people even though he might not recognize himself as the same person; his entire conception of self would probably get deleted as well.

This third and most striking example illustrates a crucial feature of our notion of human identity. You experience your entire life as a single unity because you have an uninterrupted stream of consciousness that lasts all the way from birth to death (save for when you sleep or, God forbid, find yourself in a coma). Because your experience of yourself flows seamlessly through time, you don’t consider yourself to be a different person as time evolves. Your inextricable relation to your conscious self-perception renders your identity constant throughout your life, barring any huge act of intervention like a memory implant. At the risk of making generalizations that would raise the problem of other minds, we are capable of discerning people because we unconsciously associate them with a mental image that we have of them from the past, with a particular narrative history that we have built around them. Changes in a person’s appearance — his material makeup — may render him unrecognizable from one moment to the next, but the set of relationships that unify these various guises together remains the same across time. So if we replace the Ship of Theseus with a human, then the answer to the question I posed above is very clear — the relations between the building blocks of an object lie at the core of what it is, not the building blocks themselves.

To elaborate on this idea that an object is a relationship between its parts, it is useful to turn to notions of identity in physics.

Quantum mechanics and identity

Our contemporary understanding of physics seems to corroborate this notion of identity as a relational pattern. In fact, quantum mechanics deeply challenges the idea that any building blocks even exist at the most fundamental level of reality, at least in a way that our minds can understand. As Bertrand Russell put it in his Outline of Philosophy,

“The main point for the philosopher in the modern theory is the disappearance of matter as a ‘thing.’ It has been replaced by emanations from a locality … All sorts of events happen in the physical world, but tables and chairs, the sun and the moon, and even our daily bread, have become pale abstractions, mere laws exhibited in the successions of events which radiate from certain regions.”

Why such a radical departure from the way we normally think about matter? Very outdated, but conventionally-minded, models of the atom depict an electron orbiting a nucleus in a similar way that the Earth revolves around the sun. More up-to-date models, such as those taught in high school textbooks, describe the position of an electron within an atom as a “cloud” that specifies the probability of finding it at any given area in space. But in fact, quantum mechanics is unclear not only about where an electron could be located in space, but also about whether the electron even has a position in the first place (at least, that is, in a way that we can comprehend). The famed Heiseinberg uncertainty principle states that we can never know both a quantum particle’s momentum and its position simultaneously, and furthermore, due to the effects of quantum superposition, a particle can be in two places at once.

Since an object’s location is central to our understanding of its individuality, the nonlocality embedded in quantum mechanics appears to overthrow traditional notions of identity.

In spite of these bizarre properties, many physicists, particularly when relating their research to the public, attempt to depict quantum particles with metaphors that liken them to the macro objects of our everyday experience. This representation of quantum physics is inevitable because macro objects are so deeply interwoven into our conception of reality that we can’t seem to wrap our minds around anything else. Indeed, quantum physics becomes much more comprehensible when we represent it in terms of macro objects like fields, even though such a description mischaracterizes many of its complexities.

As such, it is very expedient for us to conceive of quantum physics in terms of fields rather than particles, for example. The Heisenberg uncertainty principle lends itself to a very accessible interpretation once we start to conceptualize an electron as a field. In this analogy, the field is compared to a lake in which someone has dropped a pebble. Knowing the location of a quantum particle is like knowing the point where the pebble has struck the surface of the lake. Since the excitation in the field will move out in all directions, determining the velocity of the particle is similar to measuring the speed of the resulting waves from the pebble. Therefore, observing the pebble gives information about the position of the disturbance in the lake, but not its velocity, and vice versa if we were to measure the waves instead. Similarly, viewing the excitation in a quantum field as either wave or particle inherently captures limited information, hence the uncertainty principle. The true identity of the quantum particle, according to this conception, is a field, since neither the wave nor the particle could be propagating in all directions at the same time.

Indeed, as described in a paper by physicist Art Hobson,  There are no particles, there are only fields, an accelerating observer in a vacuum will, in what is known as the Unruh effect, observe particles (quanta), whereas one that is traveling at constant velocity will not. In a particle-centric mindset, a single quantum would have to manifest different properties to different observers. Yet if we treat the quantum as a field, then the Unruh effect will no longer seem so paradoxical. There is only one field, but the acceleration of one person transforms the vacuum fluctuations in such a way that the other observer sees only emptiness.

Quantum mechanics, as it is commonly taught to high school students, encourages us to think that the electron has a “wave-particle duality”; it behaves like a particle in certain contexts and as a wave in others, so neither term is sufficient to capture the totality of its behavior. Yet trying to understand the electron as both wave and particle is bewildering for obvious reasons, the foremost of which is that the fundamental identity of the electron would vary by environment and circumstance.

The electron qua field is both an internally consistent concept and something that persists through change; it aligns with the metaphysical conception of identity that I laid out earlier.

Unfortunately, while this conception of quantum particles as fields is very easy to understand, it is ultimately wrongheaded. Quantum particles exhibit the Heisenberg uncertainty principle as a result of their wave functions, not the underlying fields from which they arise. Furthermore, a particle is not a ripple in a quantum field in the same way that a pebble triggers a ripple on the surface of a pond. It takes a great deal of sophisticated mathematics, rather than a quick and convenient visual metaphor, to truly account for the utterly non-classical relationship between particles and fields. Accurately describing the identity of a quantum particle without making at least a passing reference to abstract mathematical structures is an impossible task. The foremost problem with Hobson’s paper is that, fundamentally, there are no particles, waves, or fields in quantum reality – merely phenomena that are particle-like, wave-like, or field-like depending on the circumstances.

The identity of subatomic “particles,” therefore, is not reducible to anything material. After all, as I wrote earlier, we typically conceive of an object’s identity as a feature that persists through the changes brought on by time, but everything that is material at the quantum level is utterly transient. Particles like quarks and leptons are known to rapidly transform into one another through the exchange of gauge bosons. Certain mesons, a class of subatomic particles that consist of a quark and an antiquark, decay within a matter of 10-24 seconds. Neutrinos, incredibly small particles that lack electric charge, appear to change into muon and tau neutrinos as they travel from the Sun and collide with Earth’s atmosphere. Deep within the apparent nothingness of space is a “quantum foam” that, as Eric Perlman of the Florida Institute of Technology explains, consists of virtual particles “quadrillions of times smaller than atomic nuclei and last for infinitesimal fractions of a second.”

From particles to patterns

The challenge that quantum reality poses to our conventional understanding of reality calls for an entirely different understanding of identity, one that doesn’t ground the essence of an object in any material substrate. Indeed, many elementary particles do not have a stable identity that is independent of the underlying mathematical structure that describes its behavior. What we call a quantum particle is really just a manifestation of what is known as a Hilbert space, which, as a generalization of Euclidean space to infinite dimensions, totally defies non-mathematical intuition.

In particular, there are physical symmetries on Hilbert spaces that are maintained even when the identities of particles change amidst the chaos of quantum reality. Before I move on, let me offer some background on what constitutes symmetry in physics. A thing is symmetrical, as Richard Feynman very broadly defined it, “if there is something we can do to it so that after we have done it, it looks the same as it did before.” There are several different classes of symmetries, including global symmetries that hold at all points in spacetime and local symmetries that hold at only some points. Laws of physics are globally symmetrical, such that that an experiment performed at a different time or location in space (with all the relevant environmental features transposed accordingly) will yield the same results, and it should come as no surprise that each conservation law in physics is associated with a symmetry. Most of the symmetries that occur on the level of particle physics are internal symmetries, which means they don’t involve transformations on spacetime. As a result, they aren’t subject to straightforward interpretation, unlike, for example, the rotational symmetry of a circle. Suffice it to say that the symmetries in quantum field theory are algebraic structures described by something known as group theory.

I’m about to get into a lot of heady physics, so I’ll frame it in the context of an intuitive visual metaphor, even though I think such analogies are generally poor at encapsulating the “meaning” of quantum mechanics. Symmetry is like a mirror that reflects a single image. When the mirror is broken, the image splits into separate fragments. However, while there are multiple fragments, each one is an instantiation of the same reality. In the Beauty and the Beast scene pictured below, a shattered mirror creates several different reflections, but they are all of the same face. Similarly, our physical reality is essentially a broken symmetry, and the subatomic particles that constitute it are manifestations of the same underlying entity.

Figure 2. Belle staring at a broken mirror in Beauty and the Beast (1991). Image from here.

Here are the specifics: in the 1930s, Werner Heisenberg noticed that, aside from their differences in electric charge, protons and neutrons are virtually indistinguishable; they have, for instance, very similar masses. As such, he posited that a deeper symmetry underlies the proton and neutron. In particular, he discovered that the symmetry in the interactions between protons and neutrons, also known as the strong force, are invariant under the action of one of the aforementioned algebraic groups. Protons and neutrons, as such, are nothing more than up and down states of a single entity that displays an internal symmetry known as the isospin, which, critically, doesn’t get altered when particles change into new ones. It is important to mention that the isospin is not a spin in physical space but rather one in an abstract space with no associated angular momentum, demonstrating that the symmetry belongs to a deeper layer of reality. Though the identities of the individual particles appear to be conditional on the observer’s frame of reference, the symmetry in which they participate remains constant, no matter the various transformations that may be applied to them. In that sense, the symmetry is the objective reality, since, by definition, it persists through change.

Furthermore, as the philosopher of science Kerry McKenzie writes, “knowledge of the symmetries associated with a law can also allow us to describe the particles whose behavior is described by that law.” This is a startling observation, because it indicates that the existence of the particle is derived from the symmetry of the particle and not the other way around. We tend to think that the elementary objects of the physical world aggregate like building blocks to form a larger structure, one in which those objects are related to each other through their relative positions and other features, but our most contemporary understanding of particle physics subverts this conventional worldview. Rather, the identity and individuality of the objects in our world are contingent on the underlying structure that relates them to one another, even though it would seem that the converse is true: that a relation between things is conditional on the existence of the things that are being linked together.

This seemingly radical view – that structure is the ultimate physical reality – is known in philosophical circles as ontic structural realism (OSR).

OSR in its most extreme form might seem implausible because, as many of its critics argue, there are “no relations without relata.” There can be no symmetries, it would appear, unless there are some objects to be symmetrically related to each other in the first place. Yet as the philosopher of science Aharon Kantarovich notes, a hypothetical universe that is deprived of all hadrons (recall that these are composite particles that are comprised of quarks) is nonetheless still governed by internal symmetries, since it is the symmetry that determines the kinds of hadrons that can be produced when, say, two photons collide with each other.¹ The symmetry is therefore more fundamental than the particles themselves because it is impossible for the particles to exist without the underlying symmetry, yet possible for the symmetry to exist without the particles.

The essence of an individual particle, then, is totally recast by OSR. As Steven French, one of the core developers of the theory, claims, “the [fundamental] elements themselves, regarded as individuals, have only a heuristic role in allowing for the introduction of the structures which then carry the ontological weight.” In fact, all that an elementary particle is – its essence, as McKenzie points out – is an irreducible representation, or “irrep,” of the symmetry associated with the family/species to which that particle belongs.

In 1939, the Nobel Prize-winning mathematician Eugene Wigner found that the irreps of a fundamental symmetry in particle physics, the Poincaré invariance, should have masses and spin within a particular range. Since all the particles in the Standard Model² ended up having exactly these properties, Wigner’s analysis enabled physicists to predict the existence of most of the elementary particles by identifying them with irreps of symmetry groups, long before they were confirmed through experiment. As a pair of theoretical physicists wrote, “Ever since the fundamental paper of Wigner on the irreducible representations of the Poincare group, it has been a (perhaps implicit) definition in physics that an elementary particle ‘is’ an irreducible representation of the group, G, of symmetries of nature.”

Yet symmetry can often be broken in the universe, often through a spontaneous process, and it’s because of this process that we’re able to distinguish between very similar subatomic particles. One of the most classic examples of spontaneous symmetry breaking is the Higgs mechanism, which is associated with the famed and highly reported Higgs boson and field.³ The primary significance of the Higgs boson/field lies in its manifestation of a particular kind of symmetry breaking.⁴ In fact, the discovery of the Higgs boson/field was primarily motivated by a mathematical inconsistency that relates to a symmetry between two of the four fundamental forces in nature: the electromagnetic force and the weak force. At particularly high temperatures, these two forces are unified into a single one, the electroweak interaction. Yet quantum field theory could not explain how the quantum of the electromagnetic field, the photon, and that of the weak field, the W and Z bosons, could both be contained within a symmetrical state. Since it turns out that the photon is massless and the two bosons are massive, Higgs’ quest to sort out this mathematical incongruity also ended up answering questions about the origins of mass. The Higgs field, it was suggested, is responsible for spontaneously breaking the symmetry associated with the electroweak force, and this process consequently produces the photon and the W and Z bosons.

While protons would carry mass in a world without the Higgs mechanism, electrons would be totally massless, thereby precluding all interaction between atoms and chemistry itself. In that sense, symmetry breaking is crucial to our very existence; as Marie Curie stated in the late 1800s, the process is responsible for nothing less than the creation of phenomena. Indeed, a totally symmetric universe must look the same no matter where an observer is located or oriented in space, which is only possible, as quantum field theory predicted, if there isn’t any mass whatsoever. Additionally, as the physicist Dave Goldberg explains, symmetry requires that all matter in the universe is accompanied by antimatter, and indeed, particles always emerge out of the quantum vacuum in quark-antiquark pairs. Yet these pairs never exist for more than a very fleeting period of time because they necessarily annihilate each other. Therefore, at some point immediately after the Big Bang, there must have been a slight imbalance in the quantity of matter and antimatter; otherwise, there wouldn’t be anything at all!

The qualities that symmetry breaking produces, such as mass and charge, are the ones that we typically associate with the identity of a particle, since we don’t expect them to change when the particle undergoes transformation.

Assuming that its identity is determined by particle-like properties, however, presupposes that it behaves as a particle in all circumstances. As I noted earlier, this macro level approach is misguided because what we refer to as quantum “particles” can act like waves or fields in different contexts and frames of reference. Protons and neutrons aren’t particles so much as they are states of an underlying symmetry. The essences of these and other subatomic particles are not their constituent material components but instead the mathematical structure that they arise from. It does not make sense to claim that a system is defined by the fact that it is comprised of electrons, since they are totally indistinguishable from one another. Rather, as some structural realists have argued persuasively, it is determined by the arrangement of these constituent electrons.

As long as the overall structure of subatomic particles, observed at a macroscopic level, remains the same, then the name or label that we attach to the macro object – like “table” – doesn’t change either.

Granted, quantum effects are usually irrelevant to our measurement of the macro world. Particles like bosons, which are more than just round spheres of matter since they also carry forces, have no place in our classical model of the world, which revolves around simple, Lego-like building blocks. But even at the macro level, we change considerably over time. Your body will quite literally not be the same in five decades as it is now. In the face of all this transformation, how do we recognize ourselves as the same person in one moment as we do in another? The answer is that we don’t think of ourselves as material entities but instead as patterns that remain consistent across time. The arrangements of atoms that we inhabit in our lifetimes are in constant flux, but the relationship between those various arrangements, which is the stream of consciousness that weaves your life together, persists through time.

There is no “I” outside of time, but there is an “I” with a past, present, and future.

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[1] Note that, in Kantarovich’s proposed thought experiment, the universe isn’t totally “particle-less” since he allows for the existence of leptons. But this distinction doesn’t really matter.

[2] The Standard Model is physicists’ best model for classifying the elementary particles of the universe and describing three of the four fundamental forces (strong, weak, and electromagnetic).

[3] Peter Higgs and several other physicists proposed in the early 1960s that space is permeated with the quantum Higgs field, which creates the resistance that particles encounter whenever they try to move. Highly complex interactions between this field and the bosons, described by the Higgs mechanism, gave mass to the latter. However, it is impossible to directly observe the Higgs field, since it is invisible and hypothesized to be totally massless. The only way to empirically validate the existence of the field is to collide subatomic particles at remarkably high velocities, which will cause a Higgs boson to flick out of the field in the same way that part of a wall will get chipped off whenever someone throws a rock at it. Since all quantum fields are associated with a certain particle (known simply as the quantum), evidence of the Higgs boson essentially serves as confirmation of the Higgs field. When the Large Hadron Collider at CERN finally detected the Higgs boson in the summer of 2012, physicists finally had, as cosmologist Michael Turner put it, an explanation for how mass came about in the physical universe.

[4] The masses of nuclei – or, more accurately, the quarks that constitute hadrons like protons or neutrons – are indeed attributable to their interactions with the Higgs field. But the three quarks in a hadron constitute only 0.2% of its mass, so it turns out that the Higgs field actually plays a relatively insignificant role in accounting for mass in space.