Indivisibility: a brief History of the Fairly Small



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Schwennesen

Fundamental Particles and the Physics of the Miniscule

Ben Schwennesen

Duke University

Professor Hubert Bray

Fundamental Particles and the Physics of the Miniscule



Indivisibility: A Brief History of the Fairly Small

Since the time of the ancient Greeks (and perhaps even earlier), humanity has pondered what components coalesce to form the world around us [1]. Not only have people questioned what matter is formed from, further still, some of history’s greatest minds have sought to identify the smallest parts that give form to everything [6, p. 47]. In other words, the concept of indivisibility and to what objects it might apply is a perplexing one. To Plato, indivisibility was a question of geometry: the four objects essential to all matter (which Greeks, including Plato, considered categorically as incarnations of the four elements—fire, earth, air, and water) corresponded to the tetrahedron, cube, octahedron, and icosahedron [2]. Indeed, the word “atom” was derived from the Greek philosophy of atomism, since the atom was considered fundamental and indivisible upon its discovery (though the atom is now known to be devastatingly divisible, as demonstrated by the Manhattan Project) [1]. The work of scientists like John Dalton, J.J. Thomson, Ernest Rutherford, Niels Bohr, Erwin Schrödinger, and Werner Heisenberg throughout the 19th and 20th centuries brought about a conversion of the atomist philosophy into a full-fledged atomic theory [7]. Experiments in this field led to the discovery of previously unknown particles—particles far smaller than the proton and neutron—giving birth to the Standard Model of particle physics.



The “Spin” of Fundamental Particles

In order to understand the particles of the Standard Model, one must first understand the characteristics used to describe these particles. The angular momentum of a particle is defined in the same way one would define the angular momentum of a macroscopic object: it is the product of its inertia to rotation (a function of the object’s mass distribution relative to the axis of rotation) and its spin velocity (which is proportional to the number of rotations per unit of time) [6, p. 29]. Since angular momentum is a vector quantity, magnitude is not the only information it carries: the direction of angular momentum is defined as being orthogonal to the plane of rotation; thus, if an observer saw a particle rotating clockwise, the particle’s angular momentum would be directed away from this observer [6, p. 30].

Angular momentum came to be identified as an intrinsic aspect of fundamental particles through experiments testing the Zeeman effect, in which a magnetic field affects the energy of a photon released when electrons jump from one orbit around the nucleus to another (an aspect of Bohr’s atomic model, in which angular momentum is quantized) [6, p. 31]. After many years of experiments, the Zeeman effect was found to result in changes in energy that contradicted Bohr’s integer value orbital model; instead, as the two graduate students George Uhlenbeck and Samuel Goudsmit proposed in 1925, the quantum number describing an electron’s angular momentum is a half-integer multiplied by ħ (“h-bar”, equal to Planck’s constant divided by 2π). Along any given axis, therefore, an electron’s rotation about itself could only appear as spin up, + ħ/2, or spin down, - ħ/2 (most physicists will set ħ equal to unity, so that the spins appear as ±1/2) [6, p. 31]. As explained by Wolfenstein and Silva, however, it is vital that one recognizes that “this internal rotation [spin] is just a mental picture, physical rotation does not make sense for fundamental point-like particles” [p. 31]. Crucially, when combined with the orbital angular momentum, the spin gives the total angular momentum, which is a conserved quantity in any particle interaction [6, p. 32].

The sign of a particle’s spin is determined by a measure known as helicity, defined as the projection of the particle’s spin along its direction of motion. A particle with spin pointing along the direction of motion has positive helicity, while spin against the direction of motion indicates negative helicity; by convention, physicists tend to call particles with positive helicity right-handed and those with negative helicity left-handed [6, p. 32]. The helicity of a particle is closely related to another measure, called chirality, which was developed in the context of massive particles [6, p. 81].

For massless particles, most famously photons, chirality is identical to helicity, as the helicity of the particle is, by necessity, a relativistic invariant: the helicity is the same in all reference frames because no frame may exceed the speed of the photon [6, p. 33]. By contrast, massive particles may travel slower than the reference frame of some observer, such that a particle with positive helicity will appear to have negative helicity to the observer in the fast moving frame; hence, helicity is frame-dependent [6, p. 33]. For an object, macroscopic or otherwise, to have chirality, its mirror image must not be identical to itself [3]; hence, in the context of particle physics, massive particles fit this description since they appear to have different helicity when viewed from different frames of reference. The result of this phenomena was striking to physicists: only left-handed chiral fermions and right-handed chiral antifermions participate in the weak interaction; despite the invariance of physical processes under transformations, one interaction occurs and its parity transformation does not [6, p. 81], implying that the Universe does not have parity (positive-negative) symmetry [3].

Fermions: Leptons

The Standard Model electron is of a type of particle called a fermion (after Enrico Fermi), all of which have half integer spin; all particles that make up matter are fermions [6, p. 31]. Specifically, electrons are part of the class of fermions known as leptons, distinguished from other the other class of fermions, known as quarks, in that they do not participate in the strong interaction (more on this later) [0, pg. 70]. Three generations of leptons exist, with two particles in each generation: the first in each pair is charged while the second is not. The first generation leptons, the electron and electron-neutrino (e & νe), are found commonly in nature [0, p. 70]. The muon and the muon-neutrino (μ & νμ) make up the second generation, and are also fairly natural observations, at least in certain environments. The third and final generation, consisting of the tau and tau-neutrino (τ & ντ), have only been observed in laboratories [0, p. 71]. Neutrinos are exceptionally elusive particles due to their lack of charge and extremely low mass: detection of a particle that will not respond outright to electromagnetism and the strong force (and effectively not to gravitation, either) is incredibly difficult. As such, in order to significantly impede the approximate 1012 electron neutrinos that pass through any given human’s body every second, one would need a block of lead “ninety thousand million million meters thick” [0, pg. 74].

There is an advantage to the elusive nature of the neutrinos in that they allow particle physicists to study the weak interaction in an uncluttered manner. Though the probability of their reactions are miniscule, they are plentiful enough that reactions are not excessively difficult to study [0, p. 74]. A canonical example of a reaction involving neutrinos is given by:

,

where n is a neutron and p is a proton. An example of this reaction where the neutron is contained in an atom is [0, p. 74]:



To understand how the neutron in this reaction is changed into a proton, one must first understand another type of fermion—quarks.



Fermions: Quarks

Quarks were first identified by experiments involving pions (π+, π0, or π-), which were only understood at the time as particles that mediated the conversion of protons to neutrons and vice versa [6, p. 53]. In 1964, Murray Gell-Mann and George Zweig independently published proposals which argued that protons and neutrons were in fact composite particles made up of some as-yet unidentified fundamental particles; Gell-Mann was responsible for the name “quark,” derived from James Joyce’s Finnegans Wake [6, p. 58]. Soon, it became clear that protons and neutrons consisted of three quarks each (any particle of this composition is known as a baryon [6, p. 60]), inexorably bound together by the strong interaction. The two “flavors” (types) of quarks found in protons and neutrons are the up quark (u), with a charge of +2/3, and the down quark (d), with a charge of -1/3. Hence, since there must be a +1 charge on the proton and no charge on the neutron, the proton consists of two up and one down quark (p = uud) while the neutron consists of one up and two down quarks (n = udd) [6, p. 58]. Pions, furthermore, came to be seen as the combination of a quark with an antiquark—i.e. the antimatter counterpart of a quark (π+ = u, ) [6, p. 58]. (To be consistent with relativity, the production of a corresponding antiparticle for each known particle must be possible in the Universe, though they need not exist simultaneously. Indeed, when particles and antiparticles are close enough to interact, they undergo annihilation reactions which release massive amounts of energy and sometimes produce exotic particles [0, p. 99].) Any particle composed of a quark and an antiquark, like the pion, is called a meson; baryons and mesons are considered types of hadrons, any composite particle of quarks bound together by the strong interaction [6, p. 62].

The formulation of the pion as a meson allows for explanation of the neutrino reaction described earlier: to convert a neutron to a proton, a d quark must be changed into a u [0, p. 75]. Pions most commonly decay into muon/muon antineutrino pairs or antimuon/muon neutrino pairs, due to the favorable helicity of these particles (i.e., due to the Universe’s lack of parity symmetry on the scale of fundamental particles [6, p. 81]). These or corresponding first generation lepton pairs, which are less likely but still possible decay products, lead to the conversion of the d quark into a u (or vice-versa) via the weak interaction [6, p. 59; 0, p. 76]:

& .

These reactions demonstrate a crucial rule in particle physics, the conservation of electrical charge: any reaction must preserve the total charge of all particles [0, p. 76]. In the above examples, the left hand sides of the equations have total charge of -1/3, all from the down quark, while the right hand sides have -1/3 total charge, with +2/3 from the up quark and -1 from the electron, i.e., (0) + (-1/3) = (+2/3) + (-1) [0, p. 76].

There are four other flavors of quarks aside from those found in baryonic matter, which form generations similar to those of the leptons (with u and d being the first generation): the charm (c) and strange (s) quarks form the second generation, while the top (t) and bottom (b) quarks constitute generation three [0, p. 107]. These generations are distinct from those of the leptons, however, in that they are identified by flavor numbers (U = 1 & D = -1, C = 1 & S = -1, T = 1 & B = -1). (Note that the flavor numbers of antiquarks are the same but with opposite parity [0, p. 107].) These other flavors lead to other composite particles, though they are rarely observed in nature [0, p. 111].

The Behavior of Particles and Quantum Field Theory

One of the key developments of science in the 20th century was the recognition that electrons and photons (eventually encompassing all particles, as well) have both wave-like and particle-like properties; as such, they are now regarded as quanta [6, p. 37]. Mathematically, quanta may be described using the framework of quantum field theory (QFT), which Wolfenstein and Silva laud as “the consistent theory joining quantum mechanics and special relativity” [6, p. 37]. Each quanta is described by its own quantum field; the consistent QFT of electron and photon fields, quantum electrodynamics (QED), was put forth by Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga in the mid-20th century. QED describes the nature of interactions between electrons and photons, and though precise details about the theory would be enough to fill entire books, the introduction of Feynman diagrams allowed its ideas to be communicated in terms far simpler than the actual mathematics behind them (see Figure 1 for an example) [6, p. 38]. Similar QFTs which emerged later arguably owe their credibility to the elegance of the model laid out by Feynman and the other pioneers of QED [0, p. 202].




Figure 1: Feynman diagram representing a second-order interaction, specifically electron-electron scattering

Retrieved from: http://chemphys.armstrong.edu/secrest/Astro/astro_1.html


Feynman’s contributions to particle physics are not limited to his contributions to QED. In his work on the fairly well-known double slot experiment for electrons, Feynman could not relate to the contemporary methods of doing quantum mechanical calculations, so he developed a “beautifully elegant” method for describing probabilities in interactions on the smallest scale [0, p. 50]. The essence of the experiment is that quantum probabilities are not additive: when opening one of two slots to allow electrons to pass through, then opening only the other in another trial, and finally opening both simultaneously, the distribution of electrons in the final trial does not correspond to a summation of the first two, as would be expected macroscopically [0, p. 49]. Feynman proposed that the probability of an event on the quantum scale is related to a quantity in the complex field called the amplitude. To calculate the probability of travelling from a to b, one must add the amplitude of each possible path (no matter how unlikely) and take the absolute square [0, p. 51]:



probability(a,b) = |Amp(a,b)path 1 + Amp(a,b)path 2 + … + Amp(a,b)path n |2

Bosons and the Fundamental Forces

Bosons (after Satyendra Bose), the second top-level classification in the Standard Model, are particles with integer spin (including zero); bosons are the force carriers for the fundamental interactions [6, p. 31]. Wolfenstein and Silva assert that one of the greatest achievements of particle physics to date was the narrowing of unthinkable numbers of phenomena down to their root in the four fundamental forces, all of which may be described mathematically [6, p. 109]. Each fundamental force is relevant at different scales. Gravity is significant only on the macroscopic level, as it is so very weak: the strength of the electromagnetic force—the best understood of the fundamental interactions—divided by the force of gravity is a massive number, 1036 (if this number were smaller, the cosmos would be far less vast and very short lived) [4, p. 2]. As described earlier, the theory of QED is a highly successful model of the electromagnetic force, which acts from the planetary scale all the way down to quarks and leptons [6, p. 110].

Models similar to QED have emerged to describe the strong and weak interactions, which are relevant only on the subatomic scale [6, p. 111]. Quantum chromodynamics (QCD) is the theory that emerged to describe the strong force between quarks; though QCD is similar to QED in outline, the mathematics of QCD is more difficult to utilize in practice, as the theory is not ‘well-behaved’ and models actually become more important with increasing complexity (whereas in QED, increasingly complex Feynman diagrams are less important to total amplitude) [0, p. 203]. QCD earned its name by explaining quarks as having three distinct ‘strong charges,’ deemed ‘colors,’ in analogy with the three primary colors [6, p. 115]. A fascinating property of QCD explains why isolation of an individual quark in nature is thought impossible: as the distance between quarks increases, so does the force between them [0, p. 203]. Thus, as one tries to pull out a single quark (or gluon, to be discussed below), the energy in the quantum field between them becomes so great that a particle-antiparticle pair will appear instead and form new hadrons [6, p. 116].

Finally, the weak interaction model was developed to explain certain decays which could not be explained by any of the other three forces [6, p. 111]. In attempting to develop a formal theory for the weak interaction, its properties were found to be so similar to QED that it was possible to develop a single mathematical theory describing both forces, which came to be known as electroweak theory [0, p. 204]. The unification of these two forces is counterintuitive, as they act over entirely different ranges: electromagnetism essentially has an unbounded range, while the weak interaction is constrained to scales less than 10-18 meters [0, p. 204]. Since the weak field suggested by electroweak theory acted identically to the electromagnetic field, an explanation for this short range was necessarily to come from some new aspect of particle physics; this explanation turned out to be the only scalar field in the Standard Model, the Higgs field, which was found to interact with each of the other fields (all of which are either gauge or tensor fields) [0, p. 205].

Each fundamental interaction is mediated by a force-carrying particle (though for gravitation this particle remains hypothetical). Exchange particles, described precisely, are actually disturbances of the given force’s field caused by interactions between particles subject to the force [0, p. 208]. The most well-known of these force carriers is the photon, the mediator of the electromagnetic force, which has zero mass and travels at the speed of light [6, p. 114]. The exchange particles for the strong interaction are a set of eight massless particles known as gluons. Since gluons are also subjects to the strong force they carry, individual gluons cannot be isolated from the strong field [6, p. 115]. For the weak field, it turns out that three types of disturbances exist, all of which have mass: the W+ and W- bosons, which form a particle-antiparticle pair (as they have opposite electric charges and equal mass), and the neutral Z boson. The mass of the Z boson is about 13% greater than the W bosons, which are, in turn, about 85 times the mass of a proton [6, p. 115]. In principle, all the exchange particles are massless, but the interactions of the W and Z bosons with the Higgs field grant them large masses [0, p. 208]. Though gravitation in Einstein’s theory of general relativity is suspected to be quantized, as with the other forces, which would imply a massless force-carrier due to gravity’s infinite range (deemed the graviton), no predictive theory of quantum gravity has emerged [6, p. 117].



Conclusion: The Large Hadron Collider and the Power of Experimentation

Many of the particles described by the Standard Model are only known to exist through experiments that subject ordinary particles to extremely high energies, usually by means of high-speed collisions. The Large Hadron Collider (LHC) is a circular facility, 27 km long, constructed at CERN (Conseil Européen pour la Recherche Nucléaire) in Geneva, Switzerland [6, p. 215; 0, p. 191]. The LHC replaced another project at CERN, the LEP (Large Electron-Positron) collider; according to Wolfenstein and Silva, the LHC can “accelerate protons to 99.9999991% of the speed of light” [p. 215]. Collisions between these rapidly moving protons occur about 600 million times per second at four sites around the facility, each with their own detectors: ATLAS, CMS, LHCb, and ALICE; the former two are devoted to finding the elusive Higgs boson. Curving the paths of the protons requires special superconducting magnets operating at around -271 degrees Celsius [6, p. 215]. To store the data from all the collisions (estimated to be about 15 million gigabytes per year), CERN had to establish an international network of computers to share the load, which came to be called ‘the Grid’ [6, p. 216].

The amounts of technology crammed into the LHC is almost incomprehensible, all for the sake of knowledge of a world so small we can never truly understand it. After decades of construction, planning, and anticipation, CERN announced that it had finally identified the Higgs boson at a mass of about 125 giga-electron volts (GeV); in 2013, Peter Higgs and François Englert were awarded the Nobel Prize in physics for their original proposals of the Higgs mechanism [5]. The mass of 125 GeV, as it turned out, left physicists with many questions: since “no evidence for supersymmetry or for any competing ideas—such as “technicolor” and “warped extra dimensions”—turned up during the first run of the LHC from 2010 to 2013,” new theories of particle physics were mandated by the Higgs [5]. The hope of many theorists is that further experiments at higher energies will reveal evidence for some theory of quantum gravity, as opposed to the virtual dead end supplied by the multiverse theory [5]. For now, however, particle physicists can rest peacefully, knowing that the mass of the Higgs boson at 125 GeV does not complete the Standard Model, and therefore new physics is still out there waiting to be discovered.

References

[0] Allday, J. (1998). Quarks, Leptons, and the Big Bang. Bristol: Institute of Physics Pub.

[1] Giunta, C. J., & Jensen, W. B. (2010). Atoms in Chemistry: From Dalton's Predecessors to Complex Atoms and Beyond, pp. 7-19. Washington, DC: American Chemical Society.

[2] Gregory, A. (2000). Plato's Philosophy of Science. London: Duckworth.

[3] Poppitz, E. (n.d.). Chirality, particle physics, and “theory space”. Retrieved February 12, 2016, from http://www.physics.utoronto.ca/~poppitz/poppitz/Talks_files/chiralityaspen.pdf

[4] Rees, M. J. (2000). Just Six Numbers: The Deep Forces that Shape the Universe. New York: Basic Books.

[5] Wolchover, N. (2015, June 6). A New Theory to Explain the Higgs Mass. Retrieved February 13, 2016, from http://www.wired.com/2015/06/new-theory-explain-higgs-mass/



[6] Wolfenstein, L., & Silva, J. P. (2011). Exploring Fundamental Particles. Boca Raton, FL: CRC Press.

[7] Zheng, A. (2012, July-December). The Evolution of Atomic Theory. Young Scientists Journal, 5(12), 74.


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