Symmetry in Atomic and Quantum Physics 11 Symmetry in Quantum and Atomic PhysicsFrom the interpretive symmetries found in the creativity and subjectivity of human nature, to the sheerfactual objectivity found all around us, we can now shift our focus of symmetry to the largest and mostamazing canvas of them all, the universe.1.1 A Perfectly Symmetrical UniverseIt is widely accepted that 13.7 billion years ago the universe began with an event known as the BigBang, whilst the name holds a tone of ironic simplicity; details of the events at the birth of the universestill prove elusive to us now. It is within these ﬁrst milliseconds of the universe that our journey intosymmetry begins. A number of decades ago a type of theory called Supersymmetric Grand Uniﬁed Theories began toemerge from the scientiﬁc community [Ode87], these theories suggested that in these ﬁrst few momentsthe universe may have been in a state of perfect symmetry, broken only by time. The consequence ofthis would mean that an unlimited number of transformations, reﬂections and rotations would result inno change in the universe. It was even suggested by Vilenkin [Vil83] that the universe was created outof ”nothingness”, he proposes that through quantum tunnelling the universe reaches ”de Sitter” space.Quantum tunnelling is the concept that dictates the probability that particle can penetrate a barrierwhich it does not have the energy to do so [Col01] . De Sitter space is a theoretical concept that wasﬁrst created in order to satisfy Einstein’s theory of general relativity, its important characteristics tous are that it has no matter [Jor06a, Jor06b]. Whilst the fundamentals of his theory are important,this report requires focus more on the symmetry interlaced into these events. With regards to the statethat Vilenkin refers to as ”nothingness”, would there be an inﬁnite symmetry or no symmetry? HeinzPagels describes the geometry of this period as ”The nothingness ’before’ the creation of the universe isthe most complete void we can imagine. No space, time or matter existed. It is a world without place,without duration or eternity...” [Ode87]. Can our laws of symmetry exist outside of our space-time (as itis in our dimension, it is primarily governed by this. Further, is there any possibility for CPT: ’Charge,Parity and Time’ to exist if there is no ”classical” time?)? Conversely, could this be considered, infact, as a ﬂawless state of symmetry? I believe it would be highly informative to investigate whethertime can be considered to be inﬁnite during the period of ”no time”. This would at least allow for anapplication of at least one aspect of CPT, and to consider the parameter of time as an inﬂuential factor;although whether or not it would reﬂect symmetry would be cause for another investigation. In termsof symmetry within our given concept and realm of ’time’, exactly what part ’time’ actually plays onthe symmetry of the universe will be discussed later on in Subsection 1.6 . As we know, the entropy of the universe can only increase in the long run (according to the secondlaw of thermodynamics [Woo99]), hence chaos is created from order, therefore if at a point when timewas initialised it can be assumed that the entropy of the universe was at its lowest and hence it had thepotential for perfect symmetry. If there was any matter added to this state of ”nothingness”, then itwould result in the breaking of the perfect symmetry as the distribution of matter would not be equal.It is our belief that this does reﬂect perfect symmetry, as where there is a ”nothingness”, there can benothing to break symmetry, however we do not, by any means, deem this to be a conclusive statement,as the mathematics involved is beyond the limits of a report such as this.1.2 Part of a Bigger JigsawSince the Big Bang, the universe has continued to expand and many moons, planets, stars and galaxieshave been formed among a host of other celestial bodies. This has comprehensively broken the ”perfectsymmetry” mentioned above as the distribution of matter became unequal. The scale to which thesymmetry has been decreased is daunting indeed, with the universe containing over one hundred billion
Symmetry in Atomic and Quantum Physics 2galaxies, and each of these galaxies encompassing hundreds of billions of stars. Despite this, the universecaries a property that we consider to be astoundingly beautiful, the cosmological principle [HL07] statesthat if the universe is considered over a suitably large area and compared to any another area, thedistribution of matter will be approximately the same. This is quite remarkable considering the size andexpanse of the universe (which is approximately equal to the age of the universe, 13.7 billion, multipliedby a light year, 9,500,000,000,000 [note that disregards any theory on inﬂation at the beginnings of theuniverse as it is outside the realms of application of this report]) in relation to its potential to retainsymmetry. A wonderfully simple analogy of this would be to consider running ones ﬁnger over a planeof glass; whilst it may feel smooth to touch, it is highly uneven at microscopic level [HL07]. It wasunanimously agreed within the group that this is an example of ”beautiful” symmetry, and furtherhelps demonstrates the ’natural’ art inherently found within our world.1.3 How Long is this Piece of String?It has been shown that the universe may have statistical transitional symmetry on a large scale. Likewise,it is important to consider whether there will be the symmetry of time (note the use of future tenseas at present the universe is still expanding). On a small scale, it is impossible to reverse the arrowof time; i.e there is no force that will reverse a parachute jumper’s descent. If there was, then whywould the Earth continue to rotate about the Sun when gravity was reversed? [Pit99] Therefore, theaforementioned hypothesis can be proven to be true. However, how does this apply to the universe ona larger scale? This question primarily concerns if the universe will begin to contract naturally, thuswill the inﬂuence of gravity eventually overcome the kinetic force driving the universe’s expansion. Weknow that the universe is currently expanding, due to the redshift caused by the Doppler Eﬀect [AC95],however there is much debate as to whether it will continue to do so. There are three possibilities: ﬁrst, if the density of the universe is below what is known as a criticallevel, the universe will continue to expand for an inﬁnite period of time, eventually tearing apart all formsof matter, even subatomic matter [Par11a], into a universe tending towards de Sitter space. Secondly,the forces of gravity and expansion could reach a point of equilibrium, where there will be a state of noexpansion or contraction. Finally, the universe could eventually collapse under the force of gravity inan event known as ”The Big Crunch”. It is known that the universe continues to expand in accordance with the Hubble Constant [oTdfpa94], however whether it will continue to do so is still very debatable. A collaborative paper [Kal02] concludesthat it is possible for all three possibilities to occur, all be it in the extremely distant future. Evidently,the fate of the universe whose outcome provides the best symmetry, in regards to time, is the one inwhich the universe contracts back into the singularity from which all was born so many billions of yearsago, the Big Crunch. If this was the case, it would certainly provide an astonishing level of symmetryover the universe. However, the extent to which the expansion would be reﬂected in the contraction isvery much undiscovered; leading to the question of ”Would the inﬂuence of gravity provide a perfectinverse for the expansion?” It is not yet known whether the events which followed nanoseconds after theBig Bang, such as the separation and creation of the electromagnetic and weak forces, would reverseand, for example, collapse back into the primal ’electroweak’ force which preceded today’s current forces[Sat96]. Furthermore, would the factor of time-scale be exactly symmetrical, given that reversing eﬀectsdid occur? For example, would the collapse of forces into the electroweak force exist for exactly the10 picoseconds [Bak07] that it initially existed after the Big Bang, or would the current asymmetricalnature of the universe aﬀect how quickly or slowly the reverse would happen? It is apparent from thecurrent uncertainty that, over the coming years, this subject will be one of much debate and interest.
Symmetry in Atomic and Quantum Physics 31.4 Quantum Entanglement and Quantum SpinThe phenomenon known as Quantum Entanglement is one such area of physics that demonstratesan innate presence of symmetry in its nature. First oﬀ, Quantum Entanglement is the theory that,under certain circumstances, two particles can become ’entangled’, meaning their properties, such asspeed, momentum, energy, and notably spin, become linked [VV10]. What’s signiﬁcant about thisprocess is that these properties remain linked regardless of spatial (and temporal) positioning, relativeto each other. For example, if a property of one entangled particle is measured, the same property ofthe entangled pair, a theoretically inﬁnite distance away, would instantaneously be deﬁned; this can bedescribed as a ”faster-than-light communication” [WCY05] between two particles, known as non-locality. This is where symmetry plays an interesting part. One property that can become entangled isQuantum Spin, which has exactly two spin orientations (or Quantum States), known as ’Spin Up’ and’Spin Down’ [ea04]. Due to the nature of an entangled pair, one particle must always have a Spin Up,and the other a Spin Down, thus must always be opposites. However, their spin properties are notsimply the reverse of each other, but an exact reﬂectional symmetry of each other’s properties. It isyet unknown exactly why entangled particles reﬂect their properties in this manner, but this occurrencedoes demonstrate the power and practicality of symmetry; the natural application of symmetry allowstwo particles to remain linked, transcending, essentially, space and time. The power of this ’timetravelling’ connection, thus, is demonstrated by it innately counteracting Albert Einstein’s theory ofgeneral relativity; the statement that ’nothing can travel faster than light’. [Got02] This is demonstrated,for example, by taking two unknown entangled particles, keeping one on Earth and sending the otherto Alpha Centauri, the closest star to Earth at approximately 4.27 light years. Both particles haveunmeasured properties, and therefore are undeﬁned. If these particles were not entangled, then to knowthe Spin of both, it would take 4.27 years (at the very least, given light transmission as the fastest meansof travel) to receive the data from Alpha Centauri, but by measuring the single entangled particle onEarth, it instantaneously ﬁxes and deﬁnes, by symmetry, the state of the entangled partner, that largedistance away [Par11c].1.5 Symmetric Wave FunctionsAnother case of symmetry comes from light, itself; or more speciﬁcally, the particles which make up light,called photons. Photons are part of a subatomic group called bosons, which have speciﬁc properties.Essentially, bosons are particles with absolutely symmetrical wave functions, and therefore have moreconsistent behaviours [MF01], implying far more predictable actions, in theory. Because of this symmetry, more than one boson can occupy the exact same Quantum State at anyone period of time. Therefore, unlike other particles (known as fermions, which include protons andelectrons), whose behaviour is erratic and individual; bosons demonstrate uniﬁed or ’group’ behaviour,where each particle acts in a far more coordinated manner [Par11a]. In the aforementioned case ofthe photons, this group behaviour is demonstrated in an experiment known as the ”Quantum DoubleSlits” experiment, initially created by Thomas Young and developed vastly later on, which harboursinteresting results connected to symmetry. In short, the Quantum Double Slits experiment initiallyattempted to demonstrate whether photons act as waves or particles, and was tested by shining a beamof light through two minuscule slits in front of a solid screen. This demonstrated light as a wave,as the second screen had what is known as ’interference fringes’, or bands of dark and lighter shadescreated by a variation in distance travelled to diﬀerent areas of the screen, with light from each slitinterfering with each other [Bli04]. Given that the light is coherent and monochromatic (most notablyfound in laser light), which essentially produces a pure light; the interference fringes will be an exactreﬂectional symmetry with respect to the midpoint of the two slits. This demonstrates the regularityand symmetrical properties of boson wave functions when grouped together (into one beam of light)[RAS06]. However, what is even more startling is that manipulating this experiment slightly results in
Symmetry in Atomic and Quantum Physics 4the demonstration of a photon as a particle, as well as a wave, as well as another case of symmetry.Instead of shining a beam of light at the two slits, it was made so only one boson, one photon, wasshone at the slits any one time. As this was only one particle or wave, it had to traverse througheither one or the other slit, and as it had no other form of light to interfere with, logically in theory, itdidn’t have any means of interference; therefore interference fringes could not be generated. Conversely,however, when a plethora of single photons of light were ﬁred in practice, one by one, with the resultantposition marked after each attempt, eventually, the original interference pattern appeared, without anyinterference present [Par11b]. This creates an identity symmetry with the previous example, but moreinterestingly, demonstrates that the symmetric wave function of a photon is identical to its probabilitydistribution of where to most likely ﬁnd the particle, the interference pattern; therefore demonstratingthat the interference pattern has a symmetrical probability distribution. This is theorised to be due tothe single photon interfering with itself through the midpoint of the two slits and, in practice, somehowpassing through both slits, thereby symmetrically splitting itself into two when traversing to the screen,resulting in the two sides of the probability distribution.1.6 Synoptic Symmetry BreakingA ﬁnal case of symmetry within the realms of Physics and the universe takes into account a few ideasalready discussed previously in Subsections 1.3 and 1.5 . As mentioned before, bosons are particles withsymmetric wave functions, but they are also particles associated with the fundamental forces, and in thefollowing case, the electromagnetic and the strong and weak forces. Because of this, bosons are typicallyknown as ”force carriers” [Wax10]. Under conditions we now consider as ’normal’, i.e. conditions reﬂective of the universe as it iscurrently, three of the four fundamental forces have an abundance of symmetrical properties, notablythe reﬂective symmetry and charge exchange of the strong force. This allows these forces to act ina predictable and coherent manner, akin to the symmetrical probability waves of photons. However,one force lacks this array of properties: the weak force. This goes a partial way in explaining why theweak force (with relative strength of 1025 ) is substantially weaker than the strength of its comparativeother, the strong force (of strength 1038 ); the lack of symmetrical properties implies that its inherentbehaviours do not possess the same focus that the strong force does. This is most evidential by the factthe weak force can act mainly upon left-handed systems (thus an asymmetrical application), such asquarks and leptons, and the large mass of its bosons inferring a substantially short aﬀect ﬁeld (ie. itsrange). [Cro08] What is crucial about the fact that the W and Z bosons of the weak force are heavy is thefactor of energy levels. When compared to the electromagnetic force, the two are exceptionally diﬀerentat low energy levels. However, as the energy of the two systems rise beyond perceivable levels by thestandards of today’s universe, the two forces become more and more symmetrical, tending towards oneidentical set of properties; properties of the primal electroweak force. [RAS05] These levels of energyare parallel to those seen from the outset of the Big Bang, and thus go some way to explaining howthe single electroweak force gave way to two such diﬀerent forces. In this sense, with the Big Bang asthe starting point of time, time is what creates the diminishing energy levels of the electroweak force,therefore ’time’ can be considered as the ”symmetry breaker” of the electromagnetic and weak forcesymmetry. [McC07] This, therefore, goes some small way in demonstrating how much less symmetricalthe great, big universe is now, compared to the inﬁnite symmetry that it potentially had at the veryfoundation of time, or, in fact, before time.
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