Our Universe is vast – almost unfathomably so. Temporally, we believe the age of the Universe to be roughly 13.8 billion years. Spatially, the *observable universe* is a sphere 93 billion light-years across. What’s beyond this sphere is a mystery: the observable universe could be all there is, or it could be a tiny fraction of something that extends infinitely in all directions. Materially, the observable universe is thought to have the mass-equivalent of Hydrogen atoms. And this *regular *matter pales in proportion to both dark matter and dark energy.

With all of that space and time and *stuff *to fill them, the possibilities appear functionally endless. Suffice to say there’s a lot for nature to work with, and we can at least to some extent wrap our heads around the fact that from such possibilities could arise chemical and biological complexity. Once the organism is born, Evolution admits teeming diversity of life.

This picture is uncomplicated by the relative paucity of physical building blocks. To our knowledge there are only a handful of elementary particles, organized into what physicists call the* Standard Model*. This model, which describes three of the four fundamental forces (Electromagnetism, Strong and Weak Nuclear), boils everything down to twelve Fermions (massive particles that obey the Pauli Exclusion Principle), a few gauge Bosons (massless particles that mediate the fundamental forces), and the Higgs Boson. Each of these particles has its own set of distinguishing properties.

*Emergence* is the idea that the whole may behave differently than the collection of its parts. As the ever wise physicist P.W. Anderson put it, “More is different”. We learn to accept this premise early on in our education. In Chemistry, the primitive objects are physical: particles like electrons, protons and neutrons, the latter two of which are each composed of three quarks. From these particles derive a panoply of elements, each with its own chemical properties. In Biology the basic object of study is DNA, which contains the instructions for life and is responsible for its diversity therein. DNA is constructed from chains of nucleotides – themselves chemical compounds. Perhaps the most astounding thing about life is that all its variants are generated by different sequences of the four (!) nucleotides that comprise DNA.

Examples like these make us comfortable with the fact that great complexity can arise from immense simplicity. But they implicitly give us the impression that complexity is a hierarchy, in which primitives at one level result in complex behavior at the *next *level. By this logic, Chemistry is infinitely more complicated than Physics, and biological phenomena infinitely more so than chemical. On this view, the plenitude of elementary particles gives us little in the way of additional physical behavior; Hydrogen atoms collectively behave just like individual Hydrogen atoms, and all we have to work with are the defining properties of particles in the Standard Model.

This perspective is patently false, and in reality our Universe can in fact host quite exotic physical phenomena. To understand the sense in which these exotic phenomena exist, we need to define physical diversity. And more pressingly, we need to elaborate on the world that entertains this existence, i.e. the Universe.

Deriving from latin *unus *(one) and *versus *(transform), the word *universe *literally translates as “turned into one”. In the broadest sense, *universe *refers to “the totality of existing things”. The concept naturally arises in physics, where the physical Universe consists of all space, time, energy and matter. Thus, there is only *one *physical universe. Distinct from this physical Universe, the concept of a universe exists in mathematics, where it refers to “the collection of all objects one wishes to consider”. Unlike physics, mathematics is constructive. This means that infinitely many mathematical universes can exist.

This constructive freedom is part of what makes mathematics so useful in the description of physics. In our attempts to make sense of the world, physicists devise physical theories to describe certain aspects of the Universe. For our purposes, such physical theories consist of:

1. An ontology (the objects, i.e. what exists) 2. A universe (where they exist) 3. Rules for manipulating the objects (physical laws)

In so doing, they associate to the physical Universe a mathematical universe.

A quick clarification is in order: First of all, not all sets constitute valid mathematical universes. Take for example the set in Russell’s paradox: “The set of all sets that do not contain themself”, which quickly leads to logical contradiction. Furthermore, many mathematical universes bear no relation to the physical world. One might imagine naïvely that surely *some *mathematical universe would suffice to successfully represent our physical Universe, but it would likely be highly complex. Yet remarkably, much of physical reality has been faithfully represented by tremendously simple mathematical structures.

This simplicity of correspondence is simultaneously deeply profound and mysterious. To conclude his article *The Unreasonable Effectiveness of Mathematics in the Natural Sciences*, Nobel Laureate Eugene Wigner writes,

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it… will extend, for better or for worse, to our pleasure.”

– Eugene Wigner

Some have taken this connection so far as to speculate that our external physical reality *is *a mathematical structure – a conjecture referred to as the Mathematical Universe Hypothesis (MUH). Seeing as we are interested in characterizing physical diversity, we will only consider the mathematical structures of classical physics and Quantum Mechanics. By understanding what is *possible* in the mathematical universes of these theories (which are both of course approximations), we will lower bound the physical diversity of our Universe.

In Classical Mechanics, most physical properties can be satisfactorily represented by a single number at each moment in time. For instance, we live in three dimensions, so the position of a single object is given by three real-valued numbers – one for each dimension. The same is true of velocity, momentum, and acceleration. These values can take on any real value, and are allowed to change continuously. When multiple rigid bodies (classical objects) are considered, they can interact with each other, as they do via gravitational attraction or electromagnetic Coulomb-repulsion. The specific values attained for certain properties of the objects can now be inter-dependent in time, but importantly the complexity of the mathematical structure only grows linearly in the number of objects. This means that the mathematical structure is just as constrained in the case of completely disjoint, non-interacting particles as in the case of strongly-interacting particles.

Quantum Mechanics provides a more flexible mathematical framework by abstracting the *state *of a system away from the particular measured values of observable quantities. Rather, the state of a system is a vector in a complex-valued space called *Hilbert space*. This freedom allows for representing possibilities like *superposition*, in which a quantum system is partially occupying multiple abstract states at once. Contrasting with the classical case, the mathematical structure grows exponentially with the number of inter-dependent (entangled) particles.

When comparing the mathematical universes of classical and quantum physics, we see that the quantum framework provides much greater flexibility and freedom – especially in the case of many strongly interacting particles.

Naturally, the following concern arises: It is clear that the mathematical universe of classical physics does not capture all of the physics we observe at small scales, and that the universe of quantum mechanics is *sufficient *for this purpose. However, it is not immediately obvious that all of the added freedom is *necessary*. In other words, just because we posit a mathematical universe doesn’t mean that real-world physics can reach all possible points in that universe.

So how much of the universe can quantum states fill? In theory the rules for manipulating quantum states (our allowed operations) give us the power to densely fill the mathematical universe. And as Nobel Laureate Frank Wilczek puts it,

“The spontaneous activity of quantum systems explores all consistent possibilities… Nature, in her abundance, provides materials to embody all theoretically consistent possibilities.”

– Frank Wilczek

In practice there are many complications, including the fragility of quantum states, short coherence times, and challenges of performing many-body operations. That being said, even just the ground states of quantum systems, which tend to lie in a minuscule ‘corner’ of Hilbert space, contain interesting physics that goes beyond classical possibilities.

But what exactly does that additional mathematical structure provide in the way of physical diversity? What is physically possible? Even attempting here to give a complete account of the exotic physics would be a fool’s errand. So let’s take for granted that the values of the fundamental constants are fixed, and that space-time is flat. One reasonable way of characterizing the complexity of potential physical behavior is through classifying the ways in which particles can act *collectively*. In particular, it is instructive to look at the different phases of matter, or the ways systems can *order*, in the sense that particles in the same chunk of material exhibit correlations with other particles in the same chunk of material. These correlations can be in spin, charge, position, or something else entirely.

Classically, the only phase of matter are solid, liquid, gas, and plasma. The first three pertain to positional constituent correlations. The latter is an ionized gas. Classical models of ferromagnetic and paramagnetic ordering exist, although these are really quantum phenomena.

It is also informative to look at what happens when matter in these phases is given a slight nudge, or *perturbed*. When a solid is pushed, the particles on the surface are displaced inward, bringing them closer to their neighbors in the lattice. The extra force felt on the second plane of particles effectively pushes them further in that direction. This in turn brings them closer to the next plane of particles. In this manner, the perturbation makes its way through the chunk of material until it reaches the opposite side, where it reflects and propagates in the reverse direction. The particles vibrate in synchrony, and the *excitation* resulting from slightly perturbing the solid has a collective wave-like nature. In liquids and gases, the relative lack of positional order leads to more complicated perturbation dynamics, including chaotic phenomena like turbulent flow. Yet even still, the relative scope of classical possibilities is quite limited.

Quantum mechanics makes things much more interesting: strong correlations between elementary particles can lead to exotic new phases of matter with drastically different behavior.

In superconductors for example, electrons (elementary fermions) pair up and form bosons called Cooper pairs, which no longer obey the Pauli Exclusion Principle, subsequently allowing the electrons (still paired) to flow without resistance.

In one-dimensional electron gases, the phenomenon of *spin-charge separation* occurs: while the elementary particles (electrons) have both spin and charge, the low-energy excitations are found to carry either spin or charge, but not both.

Perhaps even stranger is the Fractional Quantum Hall Effect (FQHE), in which the conductance plateaus in fractions of the original electron charge. Another way of phrasing the difference between bosons and fermions is in term os their exchange statistics. All particles of the same type, e.g. all electrons, are identical. However, when two electrons are interchanged, their combined quantum state changes, and it only reverts to its original state when they are swapped back. For bosons, the combined quantum state is oblivious to swaps. In other words, it takes two fermion swaps but only a single boson swap to recoup a combined quantum state. The FQHE gives rise to effective particles that have fractional charge in units of the fundamental electron charge, *and *exhibit non-fermionic, non-bosonic exchange statistics: The number of swaps that leaves the quantum state invariant can in principle be *anything*! For this reason, the new particles are called *anyons*.

These examples typify the concept of the *quasiparticle *in condensed matter – an effective particle that describes the emergent behavior of microscopically complicated systems. Not all quasiparticles provide tremendous insight beyond classical physics: According to the wave-particle duality of quantum theory, the wave-like collective vibrations of a lattice also take on characteristics of a particle – a quasiparticle known as a phonon. However, as collective modes, quasiparticles *can* exhibit physical phenomena distinct from that of the elementary particles.

Practically, quasiparticles might be incredibly difficult to harness as building blocks, and ontologically they are not as fundamental as the elementary particles. But within the hierarchy of the sciences, elementary and quasi-particles sit on the same level. More is different indeed.

Just how different can the allowed phenomena be? Classical and quantum theory are both approximations to physical reality. The mathematical universe required to capture the physics of our Universe must be larger than that of quantum theory, whether that freedom lies in six hidden spatial dimensions or somewhere else entirely. Surely that extra ‘space’ will give additional physics. Yet in one deep sense, the possibilities are quite limited.

*Universality* posits that certain qualities are shared by all entities – a concept which is ubiquitous in philosophy, religion, and science. A universal computer, for example, is capable of simulating any other computer efficiently. In physics, universality manifests in the study of phase transitions. Second order phase transitions entail systems continuously transitioning from order to disorder as one parameter or degree of freedom is varied.

The classical 2d Ising model for instance – a grid of ‘spins’ (each of which can be either up or down) interacting with nearest neighbors – exhibits a ferromagnet-to-paramagnet as a function of temperature. At zero temperature, the state with all spins aligned (all up, or all down) is infinitely more likely than all other possible states. As we increase temperature, states with all but a few spins aligned become more likely. This means that the probability of *fluctuations*, or deviations from the ground state. Increasing the temperature further, fluctuations begin to occur on larger and larger length-scales – a block of consecutive overturned spins becomes likely for larger and larger . In the disordered (paramagnetic) phase, all configurations are equally favorable. Approaching the phase transition, we observe that fluctuations on all length-scales become equally likely, and the system becomes *scale-invariant. *

This phenomenon is in no way unique to the Ising model. In fact, the scale-invariance of fluctuations at a phase transition point is universal! Moreover, regardless of the underlying order or the variable parameter, the behavior of these fluctuations is characterized by one of a few discrete classes, called *universality classes*.

One might think that the emergence of such restrictive behavior results from the simplicity of the mathematical universe. I emphasize that universality of these phase transitions is *physical *in origin, and that the meager number of universality classes is a direct consequence of scale invariance, which is a *physical *symmetry constraint. The set of universality classes does not grow when we expand from the classical to the quantum; in fact, the universality class of any – dimensional quantum system is precisely that of the – dimensional classical system. String theory or additional physical dimensions would not expand this set.

What do we learn from all of this? By considering phases of matter as a form of physical diversity, we see that the mathematical universe plays a substantial role in enabling complexity, and more mathematically flexible physical theories might inform us of even broader diversity. Universal qualities of the physical world, on the other hand, fundamentally limit the potential for variety in physical behavior.

I conclude by revisiting Anderson:

“the ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.”

–P.W. Anderson

Phases of matter are not the only form of physical diversity. Who knows where it may appear.