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The natural sciences, by definition, rely on experimental evidence to substantiate theoretical models. However, one is led to wonder whether the entirety of physics can be reconstructed from a few basic assumptions about the nature of processes in the universe. For, indeed, if most laws of physics can be expressed purely mathematically, can these laws not be expressed as derivations from more fundamental ideas about the universe? Could the laws of physics possibly be emergent from these underlying truths, and not fundamental in themselves?

This line of reasoning is similar to the mathematician’s way of arriving at theorems from axioms. Axioms are statements that are inherently assumed to be true, and the theorems are true within only that axiomatic system. Likewise, the axiomatic approach to physics treats the “fundamental” microscopic laws of the universe as axioms within the axiomatic system that is the universe, and the laws of physics are thus theorems that are emergent from these axioms.

The idea of physics from the axioms is Hilbert’s 6th problem—David Hilbert, in 1900, proposed 23 problems that he thought should guide the progress of 20th century mathematics. The 6th problem deals with establishing physics on a purely mathematical foundation so that results about the universe can be obtained without any experimentation.

First, we must note that physics involves building up approximate laws for the macroscopic scale from axioms for the microscopic world. Such a problem was explicitly named in Hilbert’s write-up of the sixth problem. Assuming a classical universe, we can axiomatize gasses to be made of billiard-ball like particles that follow Newton’s laws of physics, that are true simply because their predictions are in alignment with what actually happens in our simple classical universe.

Naturally, the interactions between these simple gaseous particles can be modelled using Newton’s laws—if there is a single particle involved, the system essentially becomes a perfectly elastically colliding billiard ball within a table, which is simple to computationally simulate. We shall return to this case shortly. When progressively more particles are added to this system, it becomes difficult to use Newton’s laws at each instant of time, which we assume moves forward at infinitesimally small discrete intervals, to find out the positions and velocities of particles after collisions. However, we still wish to make physical predictions of this system—on average, how fast are particles moving? How much energy is contained within the system? How will the system behave if a small hole is opened in the container? Naturally, the most accurate answer to this question is obtained by computing the trajectory of every single particle in the system, which is not very convenient (initially mathematically, and then progressively even computationally).

Instead, we find the need to make approximations to arrive at new laws of physics derived from these axioms. For instance, we can consider the use of average velocities to find the total energy or heat contained within the system—this approximate law is called the Boltzmann equation for gasses, and is valid for classical ideal gasses, and the law can naturally be experimentally verified because it stems from Newton’s laws of physics.

Similarly, we can make the assumption that the gas is not discrete but is rather continuous, and in doing so we can model the evolution of the continuous function representing the gas over time—this is much easier to do as we do not deal with piecewise trajectories and can instead employ the tools of vector calculus. In doing this, we arrive at the Navier-Stokes equations, which approximately predict the evolution of the gas with respect to time.

Neither of these laws is correct in the same way that a proof in mathematics is correct. A mathematical proof is not an approximation, but rather follows directly from the axioms rather than approximating trends. For instance, the proof of the infinitude of prime numbers directly uses the axioms of prime number theory, which can be recovered from just the proof or the result. On the contrary, the approximate laws of physics are what we observe as macroscopic entities, and it is close to impossible to recover the axioms (Newton’s laws of physics) from these approximate results.

There are of course exceptions, such as asymptotics and distribution theory. Gauss’s prime number theorem, for instance, describes the approximate distribution of prime numbers but is unable to generate new prime numbers. In some ways, approximations in physics are analogous to asymptotics, which explores the behaviour of functions as parameters approach extremes, akin to how physicists change the scale they are investigating to find new approximate laws. But the approximations are fundamentally different, because asymptotics is making concrete correct predictions about what happens when parameters approach extremes, while the physics approximations may lead to incorrect results. No laws of physics in statistical mechanics or thermodynamics can for this reason be considered axioms—they are all approximations of more fundamental laws of the universe.

This is one of the reasons why Hilbert’s 6th problem is so relevant—by focusing on building a system where the laws follow forth from the axioms, simply identifying the axioms can generate laws for much larger scales.

Even at the fundamental scale, the approach to axiomatize physics can be used—for instance, quantum mechanics is based entirely on the laws of probability, which are mathematical in nature. In fact, the axioms of probability were used to prove Bell’s theorem, which is supposedly a proof that nature is fundamentally quantum and not classical, but only a theory of everything will tell us which one it really is.

General relativity, the other mainstream branch of physics, however, uses the axioms of differential geometry on a Lorentzian manifold, and when these axioms are applied to quantum field theory, there are multiple contradictions and inconsistencies because some symmetries used in QFT cannot be used on curved spacetime.

The idea is thus to begin viewing physics as an interpretation of mathematics—everything in the universe is a representation theory of some mathematical object, and the universe is a representation of a mathematical system with axioms, giving rise to behaviours that we know as the laws of physics.

Mathematical truths will stay true for all eternity, but laws of physics are by definition temporary—every so often, a law is falsified, and a new one replaces it. Thus, by axiomatizing physics, we can ensure that every possible law of physics can be generated from these axioms—this is, in some sense, the holy grail of physics—for instance, if we stumble upon the axioms of quantum gravity. But we have no way of knowing for sure that the axioms we take up are actually axioms and not laws in themselves. It may never be possible to have a complete set of axioms of physics, and until then, physicists must work with pseudo-axioms that work in specific situations—but, acknowledging their non-universality, keeps open the possibility that these axioms will predict nonsensical results and these predictions constantly need to be experimentally verified.

FIN.

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We previously discussed Noether’s theorem for continuous symmetries, and symmetry breakage seen in the various facets of this universe. We shall now explore an elegant mathematical formulation of symmetry in the context of conserved field quantities.

Previously, charge conservation was spoken of—when the phase of all complex numbers in the electrodynamic scalar potential are changed at the same rate, the phase difference between the complex numbers remains the same. Complex numbers are a convenient representation encoding a “real” and “imaginary” part that can be visualized on a 2D plane. Multiplication by a complex number of radius 1 is an effective rotation of another complex number by the angle encoded by the first one. That is,

If the phase difference between all members of the field is constant despite the initial rotation 𝜃, the laws of physics stay the same, leading to charge conservation. This is effectively a statement that a circle of radius 𝜌 n the complex plane looks the same despite any rotation. This set of symmetries is referred to as SU(1). This mathematical structure is called a “group,” which means that an associative binary operation between two elements of a set (in this case, complex multiplication between two elements of the unit complex circle) ensures that the result is still in the set (additionally, in a group, every element of the set has an inverse, and there is an identity element which leaves every element unchanged).

Groups happen to be extremely effective at encoding symmetries in a system, for the simple reason that the binary operation is associative and invertible, generalizing the notion of a “transformation.” This allows us to study what elements of the group are invariant under given transformations, corresponding to the definition of a physical symmetry. They may be used to concisely describe symmetric behaviour for certain conserved quantities with unintuitive invariance properties, such as weak hypercharge.

In the 1960s, new developments in technology and experiments with the new-found particle accelerators yielded, within the span of just a few years, the discovery of tens of new particles—it was, needless to say, difficult to keep track of what was going on: some of these particles were fundamental while others were not, and theoretical particle physics had not kept pace with the kinds of interactions required to keep some of the composite particles together. This was known as the “particle zoo” age. During this time, Murray Gell-Mann and Yuval Ne'eman proposed a classification scheme for particles that paved the way for the creation of the standard model.

A classification scheme in particle physics is unlike one in chemistry—the classifications are naturally more fundamental. Particles are not just grouped based on shared properties, but also based on their precise interactions, the values of conserved quantities, and so forth. Contrary to popular belief, the standard model of particle physics is not something like the periodic table, as it is usually portrayed—such visualizations fail to take into account the particle interactions and how they conserve particle statistics (or “properties”). In reality, the collection of all particles in the standard model forms a group, and quarks, leptons, gauge bosons, and scalar bosons form subgroups.

In 1961, Gell-Mann and Ne'eman attempted a classification scheme for hadrons—composite particles that we now know to be made up of two quarks held together through strong interaction. Examples include protons and neutrons. The proposed classification scheme was called the “eightfold path.” Visually, when the hadrons’ strangeness (a measure of strong decay) and electric charge were plotted against each other, strange hexagonal and triangular patterns emerge. Rigorous mathematical exploration revealed that the two groups of mesons and the single group of baryons obeyed SU(3) symmetry. A few years later, this revealed a much more fundamental symmetry—the flavour symmetry, obeyed by the quark subgroup. This shows that the standard models’ symmetries supersede composite particle symmetries, despite both the quark group and hadron group being representation theories of SU(3).

However, the standard model is considered to be incomplete—there are certain situations in which symmetry breakage occurs, for instance—symmetry breakage usually may occur only when the laws of physics are broken. This leads to an interesting idea—that the laws of physics must change to account for and correct for the symmetry breakage that occurs, by predicting new theories that preserve the symmetry. Typically, this is done by reducing an explicit symmetry breakage scenario (symmetries are broken by violating the laws of physics) to spontaneous breakage scenarios (symmetries are broken without violating the laws of physics), effectively incorporating the special-case breakage into the new set of symmetries.

An instance of this was seen in the discovery of the Higgs boson in 2011. For instance, at extremely high temperatures, it was found that W and Z boson masses fluctuated significantly—this was a case of symmetry breakage, and was accounted for by the existence of a Higgs field that the bosons interacted with to gain a mass. The existence of the field was experimentally confirmed by the discovery of the Higgs boson, which is an excitation of the field.

This perspective of physics focuses on fundamental particles and particle categories as being irreducible representations of groups—this is what gives rise to interactions, and therefore must be all that matters. Thus, while the fundamental particles themselves obey the gauge symmetries in question, their macroscopic solutions may not—this is spontaneous symmetry breakage.

Another interesting symmetry that has recently come up is supersymmetry. Bosons are force-carriers, while fermions are the matter particles—each form their own group, with independent particle statistics. A supersymmetric theory is one where there is a symmetry set between the two groups—thus, the equations for boson interactions should be interchangeable with the equations for fermion interactions under a transformation leaving the universe unchanged. The first attempts at these supersymmetric theories were purely based on particle statistics, and were experimentally shown to be broken—they are of interest because if one takes space-time and the Poincaré symmetries into account when extending the group of fermions into that of bosons and vice versa, we may have a theory that accounts for the existence of dark matter and dark energy, in addition to possible predicting a quantum theory of gravity—such a supersymmetric theory would be a theory of everything candidate if empirically verified.

Symmetries are fundamental to the universe, and the progression of physics relies on that the universe, as an isolated system, must be invariant under specific transformations. The mathematical formulation of symmetries serves as a disturbing reminder that our universe may simply be a representation theory of some abstract mathematical structure. Stephen Hawking’s question remains unanswered: “What is it that breathes fire into the equations and makes a universe for them to describe?”

FIN.

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There is one fundamental property of the universe that has led to multiple discoveries about its nature, and highly effective tools that make accurate predictions about the universe—the existence of the Higgs boson, that of antimatter, or even the statement that the net energy in an isolated system cannot change—now, perhaps this property could be the key to unifying the forces of nature. This property is called “symmetry.” It sounds simple, but many of the fundamental observations about our universe stem from here.

A symmetry is formally defined as a transformation that leaves the state of a system unchanged. In the context of physics, this means that after a transformation is applied to an aspect of the universe, the laws of physics do not change. This may seem obvious. Let us take up a hypothetically empty region of space, where the value of every force field is zero. The laws of physics will be the same in this region, as they will be in another similar region, farther away—this means that the universe possesses a translational symmetry. No matter where we observe the universe from, it would seem, the laws of physics are unchanged. This translational invariance of the physical laws means that the universe is “homogeneous.”

Similarly, we may change the angular perspective from which we observe the universe. In the aforementioned hypothetically empty region of space, where the value of every force field is zero, let us shift the direction from which we view space. Surely enough, we see that the laws of physics hold in one direction as they do in any other. The laws of physics are rotationally symmetric—this is called the “isotropy” of space. Similarly, the universe is temporally translationally symmetric—as the local time increases, the laws of physics continue to change the same—they are not a function of time.

At first glance, these may seem trivial, obvious facts about the universe. As we shall see shortly, these are neither facts, nor are they trivial. Let us eliminate the triviality first. Every physical system may be represented by a function called the “Lagrangian” ℒ. It depends, classically, on the position, acceleration, and local time of a system Quantum mechanically, there are no discrete objects, so it depends on certain properties of quantum fields, such as their energy density.

The exact mathematical structure is not necessary to be known at this stage1, but its properties emerge from the fact that a quantity known as action, 𝒮, is the integral over ℒ of a system with respect to time. 𝒮 has the property that in any motion, the change in this quantity will be minimal. A ball, for instance, when dropped from a certain height, may take any path it chooses to the ground. Yet, it falls in a straight line rather than along any other path that is still compatible with the laws of physics (the laws of physics are “path-invariant”)—this is to minimize 𝒮. By manipulation of the principle of least action, one may explore the consequences of the homogeneity of space and time, and the isotropy of space.

Let us consider, first, the effect of the homogeneity of time. If the laws of physics do not change with time, then, logically, the quantity ℒ that characterizes the physical isolated system is not time-dependent. Exploration reveals that a quantity in the closed system remains constant with time—we call this quantity “energy,” and say that the homogeneity of time leads to the principle that the net energy of a closed system does not change with time. This is the principle of energy conservation. This may be intuitively understood as this: for a system to evolve the same way at each instant in time, some aspect of that system needs to remain the same, lest the system change differently with each instant in time—we call this aspect “energy.”

A similar argument can be made for the homogeneity of space—manipulation of the aforementioned Euler-Lagrange equations reveals that a quantity called “momentum” remains unchanged—the homogeneity of space leads to the conservation of momentum. If a ball is thrown through a null-force field region of space, the translational symmetry means that at each position the ball is at, the same laws of physics are valid; the ball evolves in the same way at each point, thereby proceeding in uniform linear motion. Likewise, from the isotropy of space emerges the conservation of angular momentum.

Here, we see that associated with each symmetry of the universe that we have previously discussed, there is a conservation law. Thus, the conservation laws themselves are not fundamental—they hold for only as long as the universe remains unchanged under the corresponding symmetric transformation. This idea that a global symmetry of the universe corresponds to a conservation law is documented in “Noether’s theorem”—it was proved by Emmy Noether in 1915, and published in 1918. William Jack Thompson states it as follows2:

“If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.” —William Jack Thompson

As long as the existence of the symmetry is shown, the conservation laws follow from therein. We have thus far explored symmetry and conservation laws from a classically applicable perspective. Let us consider a symmetry that has no classical analogue, to see the power of reasoning that Noether’s theorem provides about the world.

Every force has a potential energy field associated with it. This potential energy field is a scalar field, which means you only need numbers to describe the potential energy, rather than a direction as well. In the case of the electromagnetic force, the associated force field is a complex scalar field called the “electrodynamic scalar potential”—a field filled with complex numbers. Complex numbers have the property of “initial phase.” Here, we observe another symmetry—by changing the phase of all complex numbers that make up the field, one observes that the laws of electrodynamics do not change. This is a special kind of translational symmetry—it refers to translation in phase space. Once more, applying Noether’s theorem, and manipulating the Euler-Lagrange equations defined in electrodynamic fields, one sees that a quantity is conserved—we call it “charge.” This makes sense, because the laws of physics must be phase-invariant—charge does not depend on phase, the same way that energy did not depend on time, momentum did not depend on position, and angular momentum did not depend on the orbital angle.

In a similar manner, we obtain another symmetry associated with the strong nuclear force. Where electrodynamics associates an electric charge as characterizing the type and strength of interaction, the strong nuclear force (i.e., that holds nucleons and nuclei together) associates a “colour charge,” that comes in three types—red, green, and blue (rather than electric charge’s + and -). Going with the metaphor of colour, the potential field associated with the strong force is called the “chromodynamic potential field”—once again, translating the colour potential along the red, green, and blue axes in phase space (shifting the chromatic hue of the strong energy), followed by application of Noether’s theorem results in the conservation of colour charge.

All the symmetries that we observed earlier were purely continuous—we shall now observe some symmetries that are purely discrete. This distinction is similar to that between a line and the Koch curve—as we zoom into a line, it appears the same in every zoom level. However, the Koch curve only looks the same as the macroscopic structure after set zoom levels. A discrete symmetry present in the universe is “CPT-symmetry”—charge conjugation, parity, and time reversal are symmetries when applied in that order.

Charge conjugation refers to the idea that the laws of physics will remain unchanged if all positive and negative electrodynamic charges in the universe are swapped. This means that the chirality of particles (and, thus, handedness, “left- and right-handed”) remains unchanged. Everything appears to be compatible with the laws of physics until one realizes that all neutrinos are left-handed, whereas all antineutrinos are right-handed—no one knows why.3

This means that charge conjugation symmetry applied to a left-handed neutrino will result in a left-handed antineutrino, which has not been observed because if it does exist, it does not interact through any known force. So C-symmetry is not a symmetry because we have observed a violation.

What about P-symmetry? P-symmetry refers to parity transformation symmetry—parity is a transformation under which the position of particles is reflected about a certain axis or point. Invariance of the laws of physics under parity transformation would mean that no force is dependent on the chirality of a particle, because if the parity changes, spin will be conserved; thus the chirality will be inverted. Unfortunately, in 1956, Chien-Shiung Wu established experimentally that parity symmetry is violated by the weak force, which depends on chirality. She did this by showing that a certain nuclear reaction produces varying amounts of gamma rays (photons) depending on the direction of a polarizing field. If the universe obeyed parity symmetry, a mirror image of our world, where left and right are exchanged, would be indistinguishable, physically, from our own. But parity, it would seem, is not conserved, unless P-symmetry is violated in a similar manner from the mirror world, which is not empirically verifiable.

Time reversal (T-symmetry) is not even possible in principle, because of the second law of thermodynamics, which tells us that entropy must increase with time—if the universe was symmetric under time reversal, entropy would increase with the progression of time in the opposite direction, which is a direct contradiction of the laws of physics. There is a reason that the arrow of time points on way and not the other; this is because of the second law of thermodynamics, and the non-equivalence of the application of different quantum measurements. A measurement made before a change of the wave function is different from a measurement made after this change; thus, there is a fundamental order established in the universe that “time” arises from.

C-, P-, and T- symmetry have been individually eliminated; what about CP symmetry? Surely this would solve all our problems. Under C-symmetry, a left-handed neutrino becomes a left-handed antineutrino, which the standard model does not confirm the existence of. But under P-symmetry, the left-handed antineutrino would become a right-handed antineutrino, which leaves the laws of physics unchanged. However, CP-symmetry also means that we should perceive antimatter and matter in the universe in equal proportion in the universe, which we do not. There was clearly a violation of CP-symmetry in the early universe, causing the annihilation of most of the antimatter, leaving us with a mostly-matter universe. The reason for this is unknown, and it is called the “baryogenesis problem.” Moreover, CP violation occurs in certain weak interactions. However, under CPT-symmetry, the role of matter and antimatter is swapped, thereby leaving us in a universe that is unchanged except for that time is now flowing in the opposite direction, left and right have exchanged roles, and positive and negative electric charge have been swapped—the laws of physics are unchanged, and there is no conflict. This is also why antimatter is sometimes referred to as “matter that moves backward in time”—under CPT-symmetry, matter’s and antimatter’s roles may be swapped only if one flows backward in time, under time reversal. As CPT-symmetry is discrete, not continuous, Noether’s theorem does not apply to it, and the conserved quantity, CPT, is not measurable physically.

We have seen that all conservation laws are associated with a symmetry (except CPT, which are discrete; Noether’s theorem does not apply directly to it). What would it mean if these conservation laws did not hold in a particular closed system? We have yet to observe the formalism for symmetry, and its relation to the standard model, and how global non-symmetries can be represented as a local symmetry, enabling us to make powerful predictions in particle physics. The tale of the universe’s inelegance does not end here.

TO BE CONTINUED.

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