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--- advanced_notions:symmetry_breaking [2018/12/17 13:52]
jakobadmin [Abstract]
+++ advanced_notions:symmetry_breaking [2019/01/18 01:15]
68.112.101.198 Typos 2
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 <tabbox History>
 <blockquote>
-Nambu appears to have been the first to recognize that gauge invariancedoes hold true in the BCS Theory but has become hidden. He had iden-tified a profound truth: When the temperature gets cold enough, the fun-damental patterns of electromagnetism—gauge invariance—may be hid-den, as a result of which strange things happen, such as the appearanceof the bosonlike Cooper Pairs. .... In a superconductor, the ground state contains Cooper Pairs. It costsenergy to break up any pair, liberating individual electrons. Once liberated,the electrons have higher energy, the difference from their original bond-ing in pairs being called the “energy gap.” The freed electrons receive thisenergy, which via E=mc2 makes them appear to have gained mass. Thisgave Nambu an idea: If the universe itself was like a superconductor, couldthe masses of particles arise by some analogous mechanism? ... The way that Nambu investigated this possibility was to suppose thatthe mass of a proton or neutron is fundamentally zero and that they ac-quire their masses through the spontaneous violation of some symmetry.To implement this he focused on “chiral symmetry.” Chiral comes fromthe Greek for “hand,” and chirality is a word that refers to the distinctionbetween left- and right-handedness. This is how chiral symmetry relatesto mass.
+Nambu appears to have been the first to recognize that gauge invariance does hold true in the BCS Theory but has become hidden. He had identified a profound truth: When the temperature gets cold enough, the fundamental patterns of electromagnetism—gauge invariance—may be hidden, as a result of which strange things happen, such as the appearance of the boson-like Cooper Pairs. .... In a superconductor, the ground state contains Cooper Pairs. It costs energy to break up any pair, liberating individual electrons. Once liberated,the electrons have higher energy, the difference from their original bonding in pairs being called the “energy gap.” The freed electrons receive this energy, which via E=mc2 makes them appear to have gained mass. This gave Nambu an idea: If the universe itself was like a superconductor, could the masses of particles arise by some analogous mechanism? ... The way that Nambu investigated this possibility was to suppose that the mass of a proton or neutron is fundamentally zero and that they ac-quire their masses through the spontaneous violation of some symmetry.To implement this he focused on “chiral symmetry.” Chiral comes fromthe Greek for “hand,” and chirality is a word that refers to the distinction between left- and right-handedness. This is how chiral symmetry relatesto mass.
-A proton can spin clockwise or anticlockwise, which we may think ofas right- or left-handed, like the two possibilities for a corkscrew. Nowimagine how that spin appears as you catch up and then overtake it. If itwas clockwise as you approached, it will appear to be spinning backwardor anticlockwise when you look back after having passed: Its chirality willchange. This is fine for a massive proton, but for a massless particle, thereis a profound difference. Any massless particle always travels at the speedof light, which is nature’s speed limit. Nothing can move faster than this,so if you see a massless particle spinning left- or right-handed, you cannotovertake it and look back: Its chirality stays fixed.
+A proton can spin clockwise or anticlockwise, which we may think ofas right- or left-handed, like the two possibilities for a corkscrew. Now imagine how that spin appears as you catch up and then overtake it. If it was clockwise as you approached, it will appear to be spinning backward or anticlockwise when you look back after having passed: Its chirality will change. This is fine for a massive proton, but for a massless particle, there is a profound difference. Any massless particle always travels at the speed of light, which is nature’s speed limit. Nothing can move faster than this,so if you see a massless particle spinning left- or right-handed, you cannot overtake it and look back: Its chirality stays fixed.
-The fundamental rule is that chirality can be conserved for masslessparticles but not massive ones.are chirally symmetric, and then he investigated what happens if this sym-metry is spontaneously broken.
+The fundamental rule is that chirality can be conserved for massless particles but not massive ones.are chirally symmetric, and then he investigated what happens if this symmetry is spontaneously broken.
-The result was similar to what he had found in the case of supercon-ductivity: Massive nucleons emerge from the equations. If a nucleon isleft-handed, say, its antimatter counterpart is right-handed. In sponta-neously broken chiral symmetry, nucleon and antinucleon “condense”—act cooperatively—forming analogs of superconductivity’s “Cooper Pairs,”with no chirality.
+The result was similar to what he had found in the case of superconductivity: Massive nucleons emerge from the equations. If a nucleon is left-handed, say, its antimatter counterpart is right-handed. In spontaneously broken chiral symmetry, nucleon and antinucleon “condense”—act cooperatively—forming analogs of superconductivity’s “Cooper Pairs,”with no chirality.
 From Nambu’s math emerged a massless particle.12 In this particularcase of chiral symmetry, the massless particle is known as a “pseudo-scalar” boson, as its quantum wave changes sign if viewed in a mirror13—and this change in sign reflects the chiral symmetry that has disappeared.
-His paper, published in 1960, was a remarkable success, as a boson withthese very properties was already known, namely, the pion, the carrier ofthe strong force gripping protons and neutrons in atomic nuclei. It is notexactly massless, but is far lighter than any other strongly interacting par-ticle.14 Thus, Nambu had shown how the breaking of chiral symmetrycould give rise to all the basic players needed to understand the atomicnucleus and the strong force that holds it together. Everything lookedgood.15
+His paper, published in 1960, was a remarkable success, as a boson with these very properties was already known, namely, the pion, the carrier of the strong force gripping protons and neutrons in atomic nuclei. It is not exactly massless, but is far lighter than any other strongly interacting par-ticle.14 Thus, Nambu had shown how the breaking of chiral symmetry could give rise to all the basic players needed to understand the atomicnucleus and the strong force that holds it together. Everything looked good.15
-Any individual snowflake, such as that in Figure 8.2, looks the same ifyou rotate it though a multiple of 60 degrees, exhibiting “sixfold symme-try” under rotation.
+Any individual snowflake, such as that in Figure 8.2, looks the same if you rotate it though a multiple of 60 degrees, exhibiting “sixfold symmetry” under rotation.
 At room temperature, however, the melted snowflake is a drop of water,which appears the same from any orientation. In this situation, water ex-hibits complete rotational symmetry, which is also the property of thebasic laws controlling the behavior of its molecules. In the snowflake,however, where only six discrete rotations survive, the evidence of thisfundamental symmetry has disappeared. Somehow the original symmetryhas been lost. In the jargon of physics, we say that it is hidden or “spon-taneously broken.”
-“Hidden” describes the fact that if our experiences were limited to sub-zero temperatures, we would see the discrete sixfold symmetry of the snowflake while the fundamental complete rotational symmetry of itsbasic molecules would be hidden from us. “Spontaneously broken” refersto the enigmatic change from this complete symmetry, manifested in theliquid phase, to the lesser symmetry of the solid." .... A more realistic example, which builds on the donkey’s dilemma, is thatof a formal dinner where the guests are uniformly positioned around a cir-cular table.4 Midway between each guest is a table napkin. You are there-fore analogous to Buridan’s donkey, in that there is a napkin to your leftand to your right, and the meal cannot begin until you decide which oneto choose. One of the guests, more aggressive than the rest, chooses theirnapkin. This breaks the symmetry and forces everyone to choose the cor-responding napkin around the table. The meal can at last commence. The napkin example contains one further phenomenon of relevance tothis discussion of symmetry breaking. If the guests are very near-sighted,only the pair immediately adjacent to the aggressive guest will see whichnapkin has been chosen. They make their selection, which in turn forcestheir neighbors to do likewise. The end result is that a wave of napkinpickups moves around the entire table. This wave is the analog of what isknown as a Goldstone Boson, which appears when symmetry becomes hidden. In this analogy the phenomenon occurs because the guests are near-sighted. In the jargon: They experience a short-range interaction. For a long-range, far-sighted, situation, the Goldstone Boson disappears; see Chapter 9 ... . This massless Gold-stone Boson is the consequence of the original symmetry being broken.The memory of that symmetry is the collection of individual broken ver-sions that froze out from the original more symmetric state. The Gold-stone Boson is the link connecting them.....
+“Hidden” describes the fact that if our experiences were limited to sub-zero temperatures, we would see the discrete sixfold symmetry of the snowflake while the fundamental complete rotational symmetry of its basic molecules would be hidden from us. “Spontaneously broken” refers to the enigmatic change from this complete symmetry, manifested in the liquid phase, to the lesser symmetry of the solid." .... A more realistic example, which builds on the donkey’s dilemma, is that of a formal dinner where the guests are uniformly positioned around a circular table.4 Midway between each guest is a table napkin. You are there-fore analogous to Buridan’s donkey, in that there is a napkin to your left and to your right, and the meal cannot begin until you decide which one to choose. One of the guests, more aggressive than the rest, chooses their napkin. This breaks the symmetry and forces everyone to choose the cor-responding napkin around the table. The meal can at last commence. The napkin example contains one further phenomenon of relevance to this discussion of symmetry breaking. If the guests are very near-sighted,only the pair immediately adjacent to the aggressive guest will see whichnapkin has been chosen. They make their selection, which in turn forces their neighbors to do likewise. The end result is that a wave of napkin pickups moves around the entire table. This wave is the analog of what is known as a Goldstone Boson, which appears when symmetry becomes hidden. In this analogy the phenomenon occurs because the guests are near-sighted. In the jargon: They experience a short-range interaction. For a long-range, far-sighted, situation, the Goldstone Boson disappears; see Chapter 9 ... . This massless Gold-stone Boson is the consequence of the original symmetry being broken.The memory of that symmetry is the collection of individual broken versions that froze out from the original more symmetric state. The Gold-stone Boson is the link connecting them.....
-Imagine living inside plasma. We would only ever be aware of electro-magnetic radiation oscillating faster than the plasma frequency. Einsteinshowed us that the energy of each photon in an electromagnetic wave isproportional to the frequency of the wave. Thus, a minimum frequencycorresponds to a minimum energy. A lower limit to the amount of energythat a particle can have is a property of a particle with a mass.28 So in sum-mary: The presence of plasma impedes the photon and, in effect, gives itinertia—mass.
+Imagine living inside plasma. We would only ever be aware of electro-magnetic radiation oscillating faster than the plasma frequency. Einstein showed us that the energy of each photon in an electromagnetic wave is proportional to the frequency of the wave. Thus, a minimum frequency corresponds to a minimum energy. A lower limit to the amount of energythat a particle can have is a property of a particle with a mass.28 So in summary: The presence of plasma impedes the photon and, in effect, gives it inertia—mass.
-This is only half the story. The other feature, which Anderson noticed andhas further profound implications for the photon gaining mass, concernsthe way that waves vibrate.
+This is only half the story. The other feature, which Anderson noticed and has further profound implications for the photon gaining mass, concerns the way that waves vibrate.
-Maxwell’s equations, which describe the properties of electric and mag-netic fields, predict that, for light traveling in empty space, these fieldsvary only in the two dimensions that are perpendicular to the direction of travel, not in all three. As a result, electromagnetic waves in free spaceare known as “transverse” waves. This failure to use all of the three availabledimensions is profound; it is intimately connected to the gauge invarianceof Maxwell’s equations and the fact that photons have no mass. Had thephoton been massive, the waves would have vibrated in all directions, bothtransverse and parallel to the direction of travel (see Figure 8.4).29
+Maxwell’s equations, which describe the properties of electric and magnetic fields, predict that, for light traveling in empty space, these fields vary only in the two dimensions that are perpendicular to the direction of travel, not in all three. As a result, electromagnetic waves in free space are known as “transverse” waves. This failure to use all of the three available dimensions is profound; it is intimately connected to the gauge invariance of Maxwell’s equations and the fact that photons have no mass. Had the photon been massive, the waves would have vibrated in all directions, both transverse and parallel to the direction of travel (see Figure 8.4).29
-A wave that oscillates along its path is called a “longitudinal” wave. Thisis the normal state of affairs for sound waves, which are the result of al-ternating regions of high and low pressure in materials such as air; seismicwaves, which propagate through rocks after an earthquake; or waves inthe sea. However, for electromagnetic waves in a vacuum, the longitudinalwave is absent. Anderson now realized that within plasma, an electromagnetic waverecovers this “missing” longitudinal component. Suddenly, all three di-mensions are being used, and the photons have all the characteristics as-sociated with a massive vector boson. Even more remarkable is that thishas happened without spoiling the fundamental gauge invariance of the theory. So if we had lived inside plasma, our experiences of electromag-netic waves would have led us to a gauge-invariant theory where the pho-tons have mass.30
+A wave that oscillates along its path is called a “longitudinal” wave. This is the normal state of affairs for sound waves, which are the result of alternating regions of high and low pressure in materials such as air; seismic waves, which propagate through rocks after an earthquake; or waves in the sea. However, for electromagnetic waves in a vacuum, the longitudinal wave is absent. Anderson now realized that within plasma, an electromagnetic wave recovers this “missing” longitudinal component. Suddenly, all three diomensions are being used, and the photons have all the characteristics associated with a massive vector boson. Even more remarkable is that this has happened without spoiling the fundamental gauge invariance of the theory. So if we had lived inside plasma, our experiences of electromagnetic waves would have led us to a gauge-invariant theory where the photons have mass.30
-Anderson conjectured thatthe two massless entities—the massless photon of QED and the masslessGoldstone Boson of spontaneous symmetry breaking—“seem capable of‘cancelling each other out’ and leaving finite mass bosons only.”
+Anderson conjectured that the two massless entities—the massless photon of QED and the massless Goldstone Boson of spontaneous symmetry breaking—“seem capable of‘cancelling each other out’ and leaving finite mass bosons only.”
 ...
-Although Anderson had identified the way forward, he had not actuallyidentified any flaws in Goldstone’s argument.34 The complete solution hadto be found.
+Although Anderson had identified the way forward, he had not actually identified any flaws in Goldstone’s argument.34 The complete solution had to be found.
-There is some irony to the fact that the key to the answer was already inone of Nambu’s seminal papers. Foreshadowing even Anderson’s insight,in 1961 Nambu, and his collaborator Giovanni Jona-Lasinio, had remarkedthat in superconductivity there would have been Nambu-GoldstoneBosons “in the absence of Coulomb [electrostatic] interaction.”35 In effect,this recognizes that the Goldstone theorem applies only if there are nolong-range forces, such as electromagnetic forces, present. Conversely, inthe presence of the electromagnetic force, Goldstone’s massless bosonvanishes.
+There is some irony to the fact that the key to the answer was already in one of Nambu’s seminal papers. Foreshadowing even Anderson’s insight,in 1961 Nambu, and his collaborator Giovanni Jona-Lasinio, had remarked that in superconductivity there would have been Nambu-GoldstoneBosons “in the absence of Coulomb [electrostatic] interaction.”35 In effect,this recognizes that the Goldstone theorem applies only if there are no long-range forces, such as electromagnetic forces, present. Conversely, in the presence of the electromagnetic force, Goldstone’s massless boson vanishes.
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 <blockquote>
-In the original conceptionof these theories, supersymmetry was broken "globally''—that is,it happened everywhere at once, as if a giant cosmic switch hadbeen thrown that abruptly and ubiquitously turned off the equiv alence between fermions and bosons. This kind of global symme try breaking is mathematically possible biit is thought ugly bymost physicists, who dislike the idea of some sort of physicalchange happening at the same time everywhere. Global symme try breaking does not strictly contradia Einstein's prohibition ofphysical influences that travel faster than the speed of light,because it is "pre-wired" into the theory. But it strikes physicistsas a violation of the spirit of relativity, and they much prefer theo ries in which symmetry breaking is done "locally"—that is, thereis at every point some quantity that decides whether supersym metry is on or off, and the breaking of supersymmetry can beginat one place and spread out, like ripples on a pond.
+In the original conception of these theories, supersymmetry was broken "globally''—that is,it happened everywhere at once, as if a giant cosmic switch had been thrown that abruptly and ubiquitously turned off the equivalence between fermions and bosons. This kind of global symmetry breaking is mathematically possible but is thought ugly by most physicists, who dislike the idea of some sort of physicalchange happening at the same time everywhere. Global symmetry breaking does not strictly contradict Einstein's prohibition of physical influences that travel faster than the speed of light,because it is "pre-wired" into the theory. But it strikes physicists as a violation of the spirit of relativity, and they much prefer theories in which symmetry breaking is done "locally"—that is, there is at every point some quantity that decides whether supersymmetry is on or off, and the breaking of supersymmetry can begin at one place and spread out, like ripples on a pond.
 <cite>from End of Physics by Lindley</cite>
 </blockquote>

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