# Symmetry Redux: Conservation Laws

## Prelude

One of the first topics I ever posted about here on Silhouette of Science is Symmetry. I was fascinated by the manifold ways, both obvious and subtle, that Symmetry pervades our world. In that post I touched upon the presence of Symmetry in processes biological, chemical and physical, and the striking absence of Symmetry in a select few instances. I appealed to cellular automata as an example of simplicity spawning complexity. And lastly I yearned for some deeper symmetry, or a more fundamental rule from which the apparent breaking of symmetries could be explained.

In the intervening years that fascination has only grown, at times blossoming into a full-fledged obsession. I come back time and again to the same concepts, wrestling with the same questions. Now, at least, I have the language to better voice them:

Why does our Universe break symmetries?

What would a ‘deeper’ symmetry look like?

Why do we yearn for Symmetry if the Universe is not Symmetric?

The last question has been the most troubling. It hints at an unenviable paradox that seems to plague our pursuit of science.

In this blog post, I want to offer the seedlings of a potential resolution. I believe that Symmetry is an ideal worth yearning (and fighting) for, and that the Universe is indeed Symmetric. The problem lies in our assumptions about what Symmetry means in physics. As I argue below, the so-called “symmetries” of physics are mathematical constructs, and not physical symmetries. When viewed as physical symmetries, these mathematical constructs run counter to the essence of Symmetry. The related notion of “conservation laws” more closely captures the sentiment of Symmetry, enabling us to take a more optimistic stance on both scientific progress and the nature of our Universe.

## Impressions of Symmetry

Symmetry is (almost) everywhere. In some form or another, it is central to countless disciplines, from art, music, architecture and aesthetics, to physics, mathematics and philosophy. Its ubiquity is part of its allure, and points to its fundamentality as an ideal. At the same time, these many guises make near impossible the task of finding one all-encompassing definition. Symmetry manifests in so many ways that it means something different to the physicist than it does to the artist.

We all bring biases to the table when discussing such abstract concepts. As a physicist by training, I’ve struggled to disentangle the mathematical ‘symmetry’ in physics from the concept of Symmetry, which exists outside of physics, philosophy, or any discipline for that matter. Nevertheless, it is important to recognize the ways in which particular disciplines engage with the concept. In other words, how closely do these various impressions of Symmetry capture the essence of the concept? In what ways do these impressions allow us to probe aspects of Symmetry, and in what ways are they merely distortions?

In this post, I intend to ask these questions about the impression of Symmetry in physics. But how can we evaluate the fidelity of projections when we can’t agree on the edifice itself? While Symmetry may elude definition, productive discourse requires that we start somewhere. To this end, we will ground ourselves in philosophy – not because philosophical perspectives ring any more true than others, but rather because we can trace our modern conceptions of symmetry back to the ancient Greek philosophers.

The word “symmetry” derives from the Greek sun, meaning “with” or “together”, and metron for “measure”. When combined into sunmetria, they jointly signify commensurability; a harmonious relation of parts to whole; unity. These relations – in arrangement, composition and proportionality – implied notions of patternand regularity, notions which became integral to the idea of symmetry itself. Moreover, from the onset Symmetry was intimately connected to ideals of beauty.

Beauty, commensurability, harmony and regularity are all complex concepts in their own right, about which seas of ink can be and have been spilt. Furthermore, Symmetry is not simply their sum or union. For instance, whatever Symmetry is, we can agree that a circle exhibits the property in some way. But it isn’t necessarily true that a circle is beautiful. Some people may find it aesthetically pleasing; others may not. It’s easy to come up with examples which push on our understanding of Symmetry and force us to interrogate our beliefs. A suit jacket with a pocket square clearly breaks bilateral symmetry, yet one could easily argue that the parts still remain in relation to the whole, that proportionality and unity are preserved, and that composition and aesthetic quality are enhanced. A rich ecosystem full of flora and fauna obeys no obvious pattern, but can one really say that the parts are not in harmony? That the trees and the tree-dwellers, the insects and the animals cannot be taken as one cohesive whole, which displays some form of unity?

Some of this difficulty in developing an objective and foolproof definition of Symmetry derives from the ambiguity (and often inherent subjectivity) of concepts like beautyandcommensurability. If Symmetry exists as a concept independent of human beings, then it should not be influenced by evolutionary or cultural prejudices.

Mathematics attempts to do away with these ambiguities via rigorous definition. The first notions of symmetry in mathematics – dating back as far as Euclid – were purely geometrical; a square rotated by 90 degrees looks the same as the original square; a circle mirrored across any line that cuts through the center looks the same as the original circle.

Over time the mathematical impression of Symmetry has leaned increasingly heavily into the “relation of parts to whole”, as the geometric notions were generalized and subsumed into the language of Group Theory. In this language, there are objects (the parts that make up the whole), and transformations (the operations on and among those objects). A “symmetry” is then a transformation on the objects that leaves the whole unchanged, or invariant.

### Symmetry in Physics: Noether and Conservation Laws

Symmetry plays a variety of roles in physics. Symmetry principles are often used to constrain allowed forms for dynamical laws; symmetry arguments aid in the solving of problems by enabling one to deduce final state from initial symmetries; spontaneous symmetry-breaking, in which a solution to the equations of motion breaks a symmetry obeyed by the equations. Accordingly, there are many different aspects to the physical impression of symmetry, some well-understood and oft-acknowledged, others implicit and unspoken.

Without a doubt the most significant and deeply-entrenched pillar of the physical impression takes direct inspiration from mathematics. In physics, conservation laws strictly forbid the possibility of change in certain properties over time. The relationship between symmetries and conservation laws, first elucidated by mathematician Emmy Noether, derived principally from the idea of invariance. These results, codified in the eponymously named ‘Noether’s Theorems’, prove that physically, any continuous symmetry is also accompanied by a conservation law (and a corresponding conserved quantity), and conversely that conserved quantities are associated with continuous symmetries. For example, rotational symmetry is associated with conservation of angular momentum.

The tremendous success of Noether’s Theorems can be attributed in large part to their simultaneous simplicity and generality. They apply equally to any closed system, from a spring or a pendulum, to the entire universe; and they apply equally to spacetime symmetries and to internal or hidden symmetries, like the symmetry associated with the electromagnetic force in the Standard Model, which is responsible for conservation of electric charge.

This broad applicability means Noether’s Theorems are invoked by both the theoretical particle physicist, coming up with new theories about elementary interactions, and the undergraduate physics student, solving an elementary homework problem. The enduring impact these theorems have had on the interpretation of symmetry in physics has been accordingly colossal.

Ironically, even though Noether’s Theorems are themselves explicitly symmetric(!!!), stating that

Symmetry <——————> Conservation Law, the generally accepted interpretation is that symmetries explain conservation laws. Perhaps much of this attitude can be chalked up to the historical progression of the field. Conservation of energy and momentum were well-established far before the associated space-time symmetries were understood. The idea that conservation laws were explainable from deeper principles has been deeply ingrained since at least the dawn of Classical Mechanics. Newton’s third law – that for every reaction, there is an equal and opposite reaction – was used to ‘derive’ conservation of momentum, endowing conservation laws with relatively little explanatory power. Lagrange and Hamilton perpetuated this position in their own frameworks for mechanics. Where Newtonian mechanics is formulated in terms of forces, Lagrangian and Hamiltonian mechanics express equivalent statements via alternative (and more general) dynamical principles which extend more easily to quantum and relativistic mechanics. Both Lagrange and Hamilton insisted that their dynamical principles explained conservation laws.

Given this tradition of conservation laws needing to be explained, it makes sense that Noether’s Theorems, which systematized and clarified the role of symmetries in dynamical laws, could be used to justify a high modal status for symmetries. And while discrete physical symmetries (like the reversal of time and inversion of space) and their associated conserved quantities do not fit neatly into Noether’s framework, we ascribe a similarly exalted status to these symmetries.

## Meta-Laws

How deserved is this status for ‘symmetries’ in physics? And moreover, how closely does this physical impression of Symmetry capture the essence of the concept?

Before we answer these questions, we need to clarify what symmetries we are interested in.

After all, not all manifestations of symmetry in physics are created equal. And we wish to consider only manifestations that carry philosophical weight.

To exemplify this discrepancy in philosophical import, we will look at a few different instantiations of ‘symmetry’ in one model system within the context of Classical Mechanics. Take as our system a ‘universe’ comprised of only two spherical bodies, for instance a planet and its moon. On the physical impression, symmetries are related to invariances of the physics under transformations. One such transformation is a rotation of the planet about its center (and similarly for the moon about its center); rotating by any angle, the system looks exactly the same. This is due to the spherical symmetry of the planet (and of the moon).

But there is also another, subtler symmetry that this system exhibits – rotation of the entire system by an arbitrary angle. This latter transformation can be viewed in two equivalent ways, both of which require the notion of spatial coordinates and distances. Suppose that – for the sake of making predictions about the physics problem at hand – we lay out a grid of coordinates in space. For instance, we could conveniently set the x-axis to coincide with the line containing both the planet and its moon. To perform the desired rotation, one could imagine picking a point in space and, keeping the coordinates fixed, rotating the physical bodies about that point. Alternatively, keeping the bodies fixed and re-labeling coordinates achieves the same ends. Why, you may ask, are the these two perspectives equivalent? Because the coordinates we set down in the first place were entirely arbitrary! The coordinates do not correspond to anything physical – all that matters is the distance between bodies, which is preserved under these rotations of all of space.

While the example above is by all means contrived, it points to a profound distinction. In the first case, the invariance is associated with the spherical symmetry of the planet (resp. the moon). It is purely geometrical in origin, and applies only situationally. If the planet had been mountainous or otherwise departed from its purported spherical form, rotational invariance about its center would no longer hold. In the second case, the rotational invariance would obtain regardless of the shape or composition of the two bodies. In fact, in Classical Mechanics it would hold more generally for any closed system, so long as we rotated all objects in that system about the same point.

This is because the latter invariance is associated with deeper assertions about the nature of space – namely that space is isotropic (it looks the same in any direction). This regularity of space is reflected in the conservation of angular momentum in any closed system. Classical Mechanics also makes other key assertions about space and time. Key among these is the statement that space is homogeneous (it has the same properties at every point). In other words, if we shift all of the objects in any closed system 5 meters to the left, the physics should not change. This second regularity in space is related to the conservation of momentum in any closed system. And given that momentum translates objects linearly in space, and angular momentum rotates objects in space, momentum and angular momentum are said to be the generators of spatial translation and rotation, respectively. Energy plays a similar role with regard to the regularity of time: conservation of energy is intimately related to the statement that energy is the generator of translation in time.

These latter symmetries have a greater variety of necessity than the former, as they hold under much more general conditions.In the framework of Noether’s Theorems, these symmetries are treated identically. However from a physical standpoint, the first (geometric in origin) symmetry is regarded as accidental, whereas the second is a genuine physical symmetry. Importantly, as we will expand upon below, not all accidental physical symmetries are geometric in origin. However what philosophically concerns us is the status of the genuine physical symmetries. It is these genuine physical symmetries that have been regarded as meta-laws, or laws which all dynamical laws must obey.

How faithful, then, are ‘genuine’ physical symmetries to the notion of Symmetry itself?

Well, as I’ll spend the remainder of this post arguing, not as much as the corresponding conservation laws. Not only are conservation laws more physical than symmetry groups, they also stand up better to the onset of ordering (spontaneous symmetry-breaking), and age better with scientific progress. This does not mean that conservation laws should be used to explain symmetries, nor should they be viewed as entirely disconnected from symmetries. Rather, I believe conservation laws should be regarded as more central to the physical impression of Symmetry than the so-called symmetries of physics.

What really is a conservation law?

Whereas symmetry groups are abstract mathematical objects that we indirectly use to express invariances in dynamical equations, conservation laws more intimately connect to physical quantities, and more directly express the physical relationships between those quantities.

Conservation laws are expressed in terms of continuity equations, which dictate the dynamical relationships between the amount of a physical quantity in a particular region and the flow of that quantity in or out of the region. While this might sound simple, it is far from trivial. Stricter than conservation, these continuity equations really describe local conservation. This means that the total amount of a quantity cannot change over time, and additionally that such quantities cannot be destroyed in one place and created in another. Instead, the conserved quantity must flow through space.

Simply put, a conservation law is a cosmic constraint on what is possible. No matter what you do, however hard you try, you cannot break these laws. The Universe forbids it.

## Order and Spontaneous Symmetry-Breaking

In stark contrast with their relative ascribed explanatory power, these ‘meta-laws’ of conservation turn out to be more robust than the meta-laws of symmetry. If Symmetry is indeed related to regularity and pattern, then it must be associated with Order. Yet in the physics community, symmetry is treated (most generally) as antithetical to order.

Working in the regime of Classical Mechanics (macroscopic, low velocity, and small patches of spacetime), under the assumptions of homogeneity and isotropy, any closed system should obey translation invariance. Yet we find ourselves surrounded by solid objects (like tables and chairs), most of which microscopically consist of atoms arranged in lattice-based structures. These crystalline patterns are typically quite regular, and can be simply characterized in terms of a unit cell and the ‘lattice vectors’ that separate cells. Such systems still exhibit symmetry, albeit of a reduced nature: translating all units cell by any integer number of lattice vectors preserves the crystalline structure. The continuous translation symmetry ‘guaranteed’ by the symmetry meta-law is broken.

To explain scenarios of this nature, physicists appeal to the Landau-Ginzburg theory of phase transitions. On this view, the dynamical laws – which describe evolution in time –  remain symmetric. The ‘initial’ state, however, is not beholden to the same symmetry meta-laws. To the contrary, the ground state of a system is defined as the state which minimizes the (free) energy of that system. When there are multiple macroscopic options for the ground state which all have the same energy, the system spontaneously chooses one, thereby breaking the underlying symmetry. This phenomenon, known as spontaneous symmetry-breaking (SSB), is crucial to most order in the physical world and, by extension, to the complexity of phenomena in our Universe.

Philosophically, this puts us in a tough spot. Meta-laws should not need caveats, and symmetry should not be in opposition to order. The conclusion that symmetries must be broken in order to incite regularity in the physical world runs counter to the essence of Symmetry itself.

Conservation laws provide a promising alternative that does not bend or buckle with the onset of ordering. Even in the presence of SSB, taking the table or the chair on its own as a closed system, momentum is still conserved – albeit vacuously so. Additionally, another detail tends to be forgotten or misconstrued: It is often said that when continuous translation invariance is broken down to discrete translation invariance, momentum is no longer a good, or well-defined quantity. This conveys entirely the wrong message. Within the crystal, it is true that momentum in the strict sense cannot be defined, and must be replaced with quasimomentum, a quantity that can only take on discrete values. Nonetheless, quasi-momentum is not the poor man’s momentum. It’s discretization reflects the allowed modes of collective oscillation of the lattice. The discrete values correspond to the commensurate wavelengths, physically manifesting harmony.

## Paradigm-Shift and the Breakdown of Symmetry

Another variant of ‘symmetry-breaking’ in physics is associated not with a particular paradigm, but with the process of paradigm-shift itself. According to historian of science Thomas Kuhn in his book The Structure of Scientific Revolutions, there are two kinds of Science: normal science, which works to fit observations and measurements within a paradigm, and paradigm-shift science, which seeks to find a new framework that better explains a broader set of observations. Historical paradigms in physics include the ancient Greek atomism, Aristarchus’s heliocentrism, Newton’s mechanics, and Einstein’s relativity, and paradigm shift usually occurs when a large body of evidence is at odds with the predominant view of the world.

From a perspective that privileges symmetry, such paradigm shifts can be devastating. Historically, the bodies of evidence that have precipitated such shifts have often done so by undermining symmetries that we previously believed the Universe to obey. As the mathematical structures used in our theories grow increasingly complex, so too do the symmetry groups of the mathematical objects. The progress of science distorts symmetries. Conservation laws on the other hand benefit from theoretical advancements. To illustrate this point, we will trace spacetime symmetries and conservation laws along the historical trajectory from Classical Mechanics to General Relativity.

In the context of Classical Mechanics, the symmetric essence of genuine spacetime symmetries is almost self-evident. The regularity of space and time appear with striking elegance and universality, reflected in the simplicity of conserved quantities: energy, momentum, and angular momentum. These physical quantities are mathematically unadorned, and depend cleanly on direct physical observables. If symmetry concerns the relation of parts to the whole, then these physical symmetries paint space (time) as the whole, and prescribe the relations between different points in space (time). Impressionistically, the proper spacetime symmetries of CM are about as physically symmetric as a circle is geometrically symmetric – which is to say absolutely.

Reality is not so simple. With Special Relativity (SR) disbanding any notions of absolute time, this picture of physical symmetry begins to deteriorate. The definitions of momentum, angular momentum, and energy (among many others) are largely modified by the relativistic term $\gamma = 1/\sqrt\left\{1 – v2/c2\right\}$, where $c$ is the speed of light, and $v$ is the velocity. On the one hand, this represents a divergence of the quantities of fundamental physical interest from naturally observable quantities. Furthermore, space on its own cannot be seen as a whole, and points in space cannot be seen as parts. On the other hand, SR uncovers connections between space and time: the only sensible metric of distance in SR involves both space and time. Additionally, the re-defined energy contains a new term, the ‘rest mass’, which encodes the inter-convertibility of mass and energy. The relationships between parts are undoubtedly more complex, but one could argue that the whole, space and time, matter and energy, exhibits greater unity.

General Relativity (GR) complicates this picture even further, departing from assumptions of homogeneity and isotropy in favor of a more ~general~ theoretical framework. In GR, spacetime itself is curved, which leads to challenges in generalizing the idea of conservation of energy. While energy, defined in the SR sense is no longer conserved, physicists have had some success in recovering conservation by further modifying the definition of ‘energy’. One approach, pioneered by Landau and Lifshitz, introduces a mathematical quantity called a ‘pseudo-tensor’, which incorporates gravitational energy and momentum due to the curvature of spacetime.

Historically, it would seem that the simple physical symmetries – all of the niceties of our picture of the world – get more and more distorted. SR stretches the circle of CM along one axis, making it into an oval; GR stretches the oval along the other axis until it becomes oblong; Who knows how unrecognizable the circle will be when all is said and done (if we are fortunate enough to find the Grand Unified Theory). On this view, the symmetries of our physical theories inevitably drift further and further away from anything resembling the concept of Symmetry.

### A Modeling-Based Perspective

Such an understanding is both pessimistic and problematic. To understand why, let’s take a quick detour and venture far from physics.

Suppose there were a real-estate agent who lived in the rural countryside, surrounded by sweeping pastures in every direction. The agent, tasked with pricing vacant plots of land, could likely build a simple and accurate model only taking into account a few factors – the size of the plot, and the arability of the land. Their simple model would obey various “symmetries”. For instance, the location (i.e. position) likely would not matter. Moreover, this model would hold throughout the entire countryside giving them a false sense of general applicability.

Now if that real-estate agent were to move to the city, they would quickly find that their model fails – location is paramount in urban environments. Any accurate model accounting for both rural and urban real-estate pricing would necessarily be more complex and less “symmetric”.

Conversely, an agent dealing only in the city could devise an accurate model based primarily on location and size. Such a model would invariably disregard the arability of the land.

To belabor the point, broader sets of observations necessitate more complicated models. If we privilege the mathematical symmetries of our physical models, we are standing symmetries up just to knock them down.

Newton’s mechanics was developed based on phenomena like apples falling from trees, and balls rolling down ramps, not particles (quantum) tunneling across barriers, or black holes colliding. These experiments only gave physicists access to phenomena at low velocities (relative to the speed of light), and small patches of space (relative to the size of the Universe), and so these theories were inherently anthropocentric.

If instead of human-sized, we were galaxy-sized, our base assumptions would be entirely different. Almost all phenomena we would encounter would be close to the speed of light, and we would observe that, as $\gamma \approx \infty$ for $v \approx c$, momentum is infinite, and ~independent~ of velocity (in this regime). If we were as massive and compact as a black hole, we would undoubtedly notice the curvature of spacetime and develop a simple high-curvature physical theory.

Regardless of our starting point, our expectations for physical theories are grounded by our earliest attempts to understand the universe. Naturally, we develop unjustified assumptions about the world based on limited observations, which we project onto our theories. What gave us the right to prize the absoluteness of space and time, and then to treat the turn toward relativity as a departure from this ideal? What justification do we have in telling the universe that it must be flat (no curvature) and look the same in all directions?

## Beyond Continuous Spacetime Symmetries

As alluded to above, there are many different types of symmetry in physics. Without going into depth, I want to give two more examples that illustrate the advantages of a perspective based on conservation laws under scientific progress. The first example looks at the continuous internal symmetries of the Standard Model, while the second extends the discussion to discrete symmetries.

### Continuous Internal Symmetries

The Standard Model of particle physics, which jointly described the forces of electromagnetism, and the strong and weak nuclear forces, is often viewed as one of the crowning achievements of the 20th century. In this formulation, each force has a corresponding internal symmetry group, mathematically referred to as $U\left(1\right)$, $SU\left(3\right)$, and $SU\left(2\right)$ respectively. Their product, $SU\left(3\right) × SU\left(2\right) × U\left(1\right)$, is touted as encapsulating all of our knowledge about these three forces.

While the determination of these symmetry groups – and the identification of the fundamental particles – was undoubtedly a tremendous achievement, in some sense it is disingenuous to say that this product symmetry group is the symmetry of the Standard Model, when there are other groups, like $SU\left(5\right)$, which contain all of the same structure.

Conservation laws historically fare better under adaptations to our particle physics models. For instance, before electromagnetism and the weak nuclear force were united in electro-weak theory, even classical theories for electromagnetism featured a $U\left(1\right)$ symmetry group, related to the conservation of electric charge. In electro-weak theory, the associated with charge conservation is a slanted subgroup of $SU\left(2\right) × U\left(1\right)$. While the particular symmetry group is altered the conservation law is preserved.

### Discrete Symmetries

While Noether’s theorems only apply to continuous symmetry groups, physical theories (and physical systems) can also exhibit discrete symmetries. Our Universe is thought to obey CPT symmetry, in which the mathematical equations governing dynamics are unchanged under simultaneous conjugation of charge (C), inversion of spatial parity (P) or chirality, and reversal of time (T).

Before in-depth experimentation with the weak nuclear force however, it was firmly believed that physics was invariant under parity inversion alone. Electromagnetism, the strong nuclear force, and gravity (when viewed as a force) all remain the same under the interchange of directionality in space. Said another way, if one were to view the physical phenomena of these forces reflected in a mirror, the virtual, reflected objects would obey the same force laws. As a result, parity symmetry was taken by many to be a meta-law, a constraint on possible force laws.

With the weak nuclear force, chiral or left- and right-handed particles were found to break this symmetry, rendering the symmetry accidental, and calling into question why all but one force obey such an accidentalsymmetry.

Fixating on the breaking or preservation of chiral symmetry misses the mark. After all, who are we to say that a Universe that can be predictably mirrored is more Symmetric? The anchoring of our initial expectations to the electromagnetism and strong nuclear forces does not give us reason to yearn for a parity-symmetry Universe.

Although conservation laws in the traditional sense only exist for continuous symmetries, I believe an equally appealing analog exists for discrete symmetries. In this context, conservation laws take the form of ‘cosmic constraints’ on what is possible to know and to communicate.

If parity symmetry were genuine, that would mean that it is fundamentally impossible, physically speaking, to unambiguously distinguish left from right. The breaking of parity symmetry by the weak nuclear force implies that such a distinction is possible. Likewise, CPT symmetry tells us that for all intents and purposes, a particle propagating forward in time and traveling in a particular direction is  the same as its anti-particle going backwards in time and moving in the opposite direction. Phrased this way, symmetry satisfaction and symmetry breaking can be viewed on equal footing, and one is not inherently more physically Symmetric than the other.

## Our Mathematical Folly

Physics has benefited tremendously from mathematics, which has remarkably provided the language for succinctly understanding the inner workings of the cosmos. But translating literally from mathematics to physics can be dangerous. In particular, viewing mathematical symmetries of physical models as physical symmetries pins Symmetry against unity and pattern.

While conservation laws possess the same mathematical content as their corresponding symmetries, they are enhanced by the same forces that distort symmetries. In Classical Mechanics, we view kinetic and potential energy not as two disparate entities, but as two parts that comprise a whole: energy. In much the same way, Special Relativity reveals that mass and energy are two sides of the same coin. General Relativity declares that the curvature of spacetime also contributes to energy. While mathematically, the relationships between quantities become ever more muddled, physically, the relationships between fundamental entities are ever more illuminated. And as the relationships between parts come into focus, so too does the commensurate nature of the whole.

Said another way, focusing on conservation laws rather than the mathematical symmetries of the representational objects, we see a beautiful, harmonious ecosystem rather than a skewed circle. Conservation laws capture that unity and celebrate the extent of our understanding in a way ‘symmetries’ in physics do not  – they align more closely with the concept of Symmetry.

Jacob A. Marks