Symmetry and the Laws of Nature

N.B: I’m going to make a pdf version of this article!! If the wordpress format is bothersome (especially considering the font isn’t great for the special characters) then consider clicking the link (to be put here) to read.

Full bibliography and credit to all sources listed below.

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““Let proportions be found not only in numbers and measures, but also in sounds, weights, times and positions, and whatever force there is” -Leonardo da Vinci”

If you drive down the motorway at 70 kilometres per hour in a roofless car, you will inevitably “sense” that you are in a state of motion. On the contrary, ride in a sealed compartment of a train equally going at 70 km/h along a straight line of track, and you’ll find that you’re able to walk down the compartment with as much comfort and ease as if you were indoors. The occupants of a sealed car too, without any visual cues, would feel as if they were at rest, even at the greatest speed.  Yet even the Earth, containing us and everything else we know, is not in the state of “rest” that we intuitively perceive – it is travelling around the Sun with an orbital speed of 67,000 miles per hour! This “illusion” of rest is firmly rooted within the the most fundamental, yet arguably elusive, principle found content within nature – symmetry. Besides the Vitruvian Man, the Taj Mahal, and honeycomb – physical emblems and structures displaying immaculate proportion and artistry, besides its lingering presence in the unlikeliest places imaginable – constants of strange attractors (amongst the chaos of fractals and Feigenbaum mapping¹), symmetry resonates at the most rudimentary level of our conceptual understanding of the universal laws. The ubiquity of the most basic truths of our universe depends on the notion of symmetry in order to govern nature.

Permeating and completing some of the most elementary theories of our time, in a physical sense, symmetry almost always relates to invariance upon transformation. A physical system is, therefore, said to possess symmetry if it remains unchanged by a certain kind of operation². Before Albert Einstein’s papers on general and special relativity, symmetry rarely ventured into scientific fields, manifesting largely in objects of architectural practicality and mere appreciation; symmetry as a principle was not regarded to play a significant role in the laws of nature³. Yet when we talk of the invariance of the laws of physics in an inertial system (a non-accelerating reference frame⁴), such as that of our car or train, we are touching on core symmetry principles.

A symmetrical principle which was realised in Einstein’s theory of special relativity was the large-scale homogeneity of the universe (Fig 1) – uniformity with respect to direction⁵ to infer that only the relative motion of an observer travelling (or not) through space can be determined.

Prior to Special Relativity, Galilean invariance, based heavily on the idea of “absolute time” (a conclusive value for universal time which is true for all reference frames⁶), played a key role in Newtonian Relativity. However, Galilean transformations were found to only be valid for non-relativistic systems, and while they hold at scales of our breadth of experimentation, they break down when approaching small distances or the speed of light. Special Relativity explicitly showed that the presence of motion causes space and time to dilate in such a way that the speed of light, c, remains a constant variable (the other cornerstone of the theory) – a symmetry within the transformation⁷ (Fig 2) . It presented the Lorentz invariant, denoted 𝛄 (gamma), a symmetry adopted only in relativistic situations concerning uniform linear motion to extrapolate appropriate dilation.

Conservation Laws

The symmetries found within special relativity are continuous. Continuous symmetries relate invariance with motions, and they are global, manifesting in laws possessing a certain generality for all particles in nature, and exist uniformly throughout the universe. These laws, known as conservation laws, revolutionised how we thought about the physical world, because they provided a limit to the amount of conceivable theories in the universe. From our observations, the basic laws of physics haven’t changed for billions of years, and are constant to the outer reaches of the observable universe⁸. If any of the symmetries were broken, then the laws of physics would have changed according to some parameter, either space or time, and would no longer constitute as reliable⁹. For instance, what use would it be if we were to obtain results for an experiment in London, only to repeat it in Singapore and see that our results are completely different? Or, to initiate an investigation into the value of the gravitational field strength on the surface of the Earth in 1980, only to see it’s completely different by 2017? The asymmetric discrepancy in results would suggest a shift in the laws of physics across space or over time.

We would expect the outcomes of the experiments in London and Singapore, performed with the same equipment and methodology, to be identical. The physical laws governing the experiments should be symmetric under space translation (in this case, the differing locations of the two cities at any given moment)¹⁰. This is known as translational invariance, and may be approximated using Euclidean space in non-relativistic situations. Since the universe is homogeneous¹¹ a set of, therefore arguably redundant, coordinates in the space can be varied continuously without changes in the laws. In 1915 mathematician Emmy Noether proved that such continuous symmetries, such as translational invariance, had a deeper impact on the laws of physics. Through Noether’s theorem, published in 1918, she showed that for every such continuous global symmetry there exists a global conservation law. All space-time symmetries, including not just the aforementioned spatial translation, but also translations in time and rotational symmetry are considered to be global and continuous, thus all requiring their respective accompanying conservation laws.

Concerning spatial invariance, provided any two locations can be infinitesimally close to one another on a worldline¹² (a section of curved spacetime), one can see that the symmetry of the Lagrangian is equal to the equation for conservation of momentum. Assuming that the worldline is two-dimensional in this instance, an object moving between the two locations with constant velocity, v, (Δd/Δt) and kinetic energy ½mv², would have an “action”, S, along the straight segment from point x₁ to point x₂ in the Lagrangian (expressed as shown in Equation 1). Here, varying the distance points x₁ and x₂  by some fixed displacement β is meaningless, and cancels out such as in x₂ + β – (x₁ + β) = x₂–x₁ ,  showing the identical outcome in such a process¹³.

(Source of Drawing: https://smashtheparadox.wordpress.com)

One can then use the principle of least action, the kinetic energy (KE) of the particle minus the potential energy (PE) of the particle integrated over time¹⁴. Taking three points,  A, B and C on the worldline and translating just the middle coordinate, B (Fig 3) in order to minimise the overall action gives the derivative of the action: dS/dx² = 0. Therefore the actions of the two segments, 1 and 2, connecting the three points would be added together and simplified to produce the equation for conservation of momentum (Eq 2a and 2b), provided we take direction into account. Of course, we could add infinitely more points representing events on the particle world line between x₁ and x₂ and hence introduce new segments, but momentum will always be conserved for the spatial translation of this particle.

(Source of Drawing: http://www.smashtheparadox.wordpress.com)

(Source of Drawing: http://www.smashtheparadox.wordpress.com)

Equally, rotational invariance (rotating a system about a set angle) is a continuous symmetry which corresponds to the conservation of angular momentum, the product of angular velocity and an object’s moment of inertia¹⁵. This is due to the uniformity of space in all directions, or isotropy¹⁶. In a similar vein, time translational invariance in an isolated system too equates to a conservation law – the conservation of energy. These laws demonstrate an irrevocable symmetry within nature, overriding the asymmetry and irregularity of everyday situations¹⁷. For instance, the Earth, with its asymmetrical orbit, travels around the sun in a state of inertia due to no exterior forces (Fig 4), thus conserving its angular and linear momentum and complying with Newton’s first law¹⁸. The laws that govern its motion can be derived from the same symmetry principles which govern the motion  of our car.

The Early Universe

Heating up an isolated magnetic material beyond the Curie Temperature (the critical point at which the ferromagnetic property disappears¹⁹) would result in it undergoing a phase change and becoming demagnetised²⁰. Its domains, which were all previously arranged facing a randomly chosen direction upon alignment, would accordingly lose their collective direction of orientation and transition into a disordered state, which juxtaposingly also exhibits rotational symmetry. Cooling the magnet again would would result in spontaneously breaking this rotational symmetry of domain disorder (hence inducing magnetism). This is because the symmetry found at high temperature has an energy configuration which is unstable, and is easily broken when transitioning to low energy states. Some crystalline states of matter, such as ice, exhibit a broken symmetry which is similar, but smaller, than the continuous symmetry of space. Thus, phase changes and states of matter in our universe correspond to energy levels which affect possible symmetrical states.

Unsurprisingly, these phase changes are adopted on the grander scale of the whole universe. One of the greatest contemporary challenges in physics stems from one of the Universe’s most mysterious asymmetries²¹: the fundamental interactions of the Universe. The unification of these forces: electromagnetism, the strong and weak forces, and gravity, would call for an absolute theory of everything – a symmetrical alignment which has many contenders, including String Theory which incorporates mirror symmetries of its own²². However impossible it is today to incorporate gravity into quantum theory, at the Planck time after the Big Bang, the symmetries of nature were all presumably restored at the extremely high temperatures into a highly symmetric configuration. Just like the rotational symmetry of the domains of the magnet, all the forces of nature had equal strength, and the elementary particles comprising the Grand Unified Theory (GUT) were the same²³. However the highly symmetrical unification of the forces during the Planck Era, like a pencil standing on its tip, was too unstable to last and the consequent expansion and cooling of the universe resulted in the fragmentation of fundamental interactions. The era of supergravity (the combined result of general relativity and supersymmetry) came to an end and GUT matter, a superposition of all the matter today, split off into respective elementary particles.  

Symmetry Violation

In nature there exists another group of like symmetries – discrete symmetries which do not yield conservation laws²⁴.  In 1928, Paul Dirac predicted that some physical laws, such as Maxwell’s equations, are invariant under such discrete symmetries. These symmetries, namely charge conjugation (C), Parity (P), and Time reversal (T) were originally thought to apply invariantly to all elementary particle interactions²⁵. However this was proven not to be the case when James Cronin and Val Fitch were awarded the Nobel prize in 1980 “for the discovery of violations of fundamental symmetry principles in the decays of neutral kaons” . They showed that CP symmetry, a theoretical symmetry between matter and antimatter, had to be violated for certain elementary particles. CP symmetry suggests that if you conjugate the charge of an elementary particle (say, change an electron into its antiparticle – the positron) and then apply parity (the reflection of spatial coordinates) the particle is exactly interchangeable with its original state²⁶. The violation of this symmetry would hence imply distinct and non-transferable differences between matter and antimatter.

During the grand unification epoch²⁷, equal amounts of matter and antimatter were created, and when they came in contact they immediately annihilated one another, producing photons (the echo of this ancient event is Cosmic Microwave Background radiation, discovered by Robert Wilson and Arno Penzias in May 1964²⁸). Contradictorily, our universe today consists of matter which was not annihilated in this primordial event, about one in every billion quarks, implying a violation of symmetry which is proven by the existence of our world (composed of matter).

CP violation was proposed by physicist Andrei Sakharov as a hypothetical solution to the excess of matter at the beginning of the universe. He inferred that the diverse initial conditions of matter and antimatter (known as the theory of baryogenesis²⁹) would have lead to an imperfect annihilation with cosmological implications! In 2001, CP violation was also discovered present in the decay of B-meson particles and anti B-mesons in an experiment at the BABAR detector at the Stanford Linear Accelerator Centre³⁰. The B factories (specific asymmetrical accelerators required for obtaining such experimental data) announced measurements for the decay time of B0 Mesons (in red) and anti B0 mesons (in blue) to be different (Fig 5). The inherent CP violation in weak interactions of the  B0 →J/ψ K0 decay (where J/ψ consists of one charm and one anti-charm quark and K0 is a neutral kaon) shows the discrepancy between matter and antimatter particles³¹.

CPT symmetry, a combination of the aforementioned CP symmetry and time reversal, T, is a theoretical symmetry that applies to all relativistic quantum field theories. However time reversal as a standalone symmetry is found to be violated by all but a handful of natural systems. For example, a film showing two colliding billiard balls can be watched backwards, and the “backwards” event will look plausible, because it will still appear as two particles colliding and separating (Fig 6). However this scenario is one of a restricted context. In the case of, say, a brick wall being demolished by dynamite, a “backwards” film of this event will not result in a physically feasible situation³² due to entropy, the measure of disorder in a system³³. The Second Law of Thermodynamics is not time reversal invariant because a disordered state cannot be reordered without significant energy transfer (hence still increasing entropy)³⁴. Therefore the initial conditions of certain outcomes, such as an explosion or demolition, are too improbable to be time reversal invariant.

As our technology advances, we are becoming increasingly able to probe at ever smaller scales and ever higher energies. These insights uncover symmetries within nature undetectable from our macroscopic viewpoint. Certain symmetries of the universe, deemed too unstable to exist, have been discovered to be broken – their broken forms appearing at lower energy levels in our observable, few-dimensional world. In the future it is hoped that new symmetries, such as supersymmetry will be able to intrinsically piece together a new, extended superspace³⁵ with the capability of unifying fermions and bosons into symmetric configurations. These configurations will undoubtedly require the monumental discovery of new super particles (sparticles) uncovered in future high energy collisions.

In conclusion, the power symmetry evinces in our world is clear. Its non-exclusionary applications can be observed from the mundanity of train journeys, to the formation of our entire universe. It is perhaps the most potent principle capable of uniting forces with matter, and the laws of nature with physical happenstance. There is inevitably more to symmetry than we currently know, and it is vital that we look towards it on our quest to civilisational advancement and universal understanding.

Bibliography

1. 1. Field, M. and Golubitsky M. (1992). Symmetry in Chaos: A search for Pattern in Mathematics, Art and Nature. Great Britain: Oxford University Press.
   2. Symmetries and Conservation Laws. Available at: http://www.phy.pku.edu.cn/~qhcao/resources/class/QM/symmetry.pdf
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   7. Weinberg, S. (2011). Symmetry: A ‘Key to Nature’s Secrets’. A New York Review of Books, 27 October. Available at:  http://www.nybooks.com/articles/2011/10/27/symmetry-key-natures-secrets/
   8. Kaku, M. (2008). Physics of the Impossible. London: Penguin Group.
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   17. Quote from Gross, D. (2016). Beyond Beauty: The Predictive Power of Symmetry. World Science Festival. Available at: https://www.youtube.com/watch?v=X6HobTJ2jnk
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   31. Sciolla, G. (2006). The Mystery of CP Violation. MIT Physics Annual. Available at: http://web.mit.edu/physics/news/physicsatmit/physicsatmit_06_sciollafeature.pdf
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Figure References:

Figure 1: Wright, E (2005) Las Campanas Redshift Survey. Available at: http://www.astro.ucla.edu/~wright/lcrs.html

Figure 2: https://en.wikipedia.org/wiki/Minkowski_diagram

Figure 4: Nave, R.  http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

Figure 5: Sourced from: Sciolla, G. (2006). The Mystery of CP Violation. MIT Physics Annual. Available at: http://web.mit.edu/physics/news/physicsatmit/physicsatmit_06_sciollafeature.pdf

Figure 6: Sourced from: https://www.reddit.com/r/askscience/comments/1ciybw/can_someone_explain_how_time_works_and_what_it_is/

Equations 1,2a and 2b, and Figure 3 are illustrated by the author of the essay, cited from: Hank, J. and Tuleja, S. and Hancova, M. (2003). Symmetries and Conservation Laws: Consequences of Noether’s Theorem. Available at: http://www.eftaylor.com/pub/symmetry.html

Additional sources utilised during research:

1. Kaku, M. (2004). Parallel Worlds: A Journey Through Creation, Higher Dimensions, and the Future of the Cosmos. United States: Doubleday.
2. Gleick, J. (1987). Chaos: Making a new Science. United States: Viking Books.
3. Wade, D. (2006) Symmetry: The Ordering Principle. New York: Walker and Company.
4. In Our Time: Symmetry (2007). Podcast from BBC Radio 4 In Our Time. Available at: http://www.bbc.co.uk/programmes/b00776v8