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see also Goldstone's theorem, Chiral Symmetry Breaking,Higgs Mechanism
For aesthetic and practical reasons, we prefer theories with a high degree of symmetry, but because Nature is asymmetric, we must also account for this symmetry breaking. The oldest and most primitive idea for symmetry breaking is that of "approximate" symmetries. One supposes that there are terms in the Lagrangian that violate the symmetry, but they are "small". More refined is the concept of spontaneous symmetry breaking, introduced by W. Heisenberg in condensed-matter physics, and extended by him as well as by J. Goldstone and Y- Nambu to the particle-physics domain. Here the dynamical equations are completely symmetric, but energetic considerations of stability indicate that the ground state is asymmetric. […] Anomalous breaking of symmetries - the third, most subtle mechanism- arises from quantum mechanical effects, in a way whose fundamental origin remains obscure. Certainly there are no energetic or stability considerations as in spontaneous breaking. Our only clue comes from perturbation theory: there does not exist a regularization procedure which respects the anomalously broken symmetries. page 279 in Topological Investigations of Quantized Gauge Theories by R. Jackiw
There are different ways a symmetry can be realized in a theory:
Unbroken The classical symmetry exists at the quantum level. Only invariant quantities are non-zero Hidden By redefining the fields the new fields do no longer possess the original symmetry. However, a, possibly involved and non-linearly realized, version of the original symmetry persists, and the theory exhibits the symmetry as if unbroken Metastable The classical symmetry exists at the quantum level. Only invariant quantities are non-zero. The theory is changing discontinuously when a source is applied which selects a stratum or orbit Spontaneously broken The classical symmetry is explicitly broken by a source which selects a particular stratum. This breaking persists when the source is sent to zero, and the limiting theory does no longer possess the symmetry. Only in this limiting case, the Goldstone theorem applies Explicitly broken An external current breaks the symmetry by singling out a stratum or an orbit. The breaking vanishes when taking the source to zero. Otherwise the symmetry is also spontaneously broken Anomaly A classical symmetry is explicitly broken by quantum effects. Usually, this comes from the non-invariance of the path integral measure [34]Brout-Englert-Higgs physics: From foundations to phenomenology by Axel Maas
In modern physics the most important type of symmetry breaking is spontaneous symmetry breaking.
What an imperfect world it would be if every symmetry was perfect
B. G. WYBOURNE
"Heisenberg Ferromagnet: The infinite degeneracy of the possible ground states, corresponding to an infinite number of possible directions for the aligned ground state spin, depends crucially on the assumption of an unlimited spatial extent for the system. Mathematically, the infinite size of the system means that there is no unitary operator that can connect the different ground states and they lie in different Hilbert spaces. This is because the ferromagnet has an infinite moment of inertia, implying that no finite amount of energy can rotate one ground state into another. If the ferromagnet is of limited spatial extent the different ground states are separated by finite energy barriers and may tunnel into each other; the situation then resembles band structure in the solid state. The degeneracy of the ground state is lifted since one of the linear combinations resulting from the degenerate vacuum will lie lower in energy than all others and will be the (non-degenerate) physical ground state. For a relativistic field theory exhibiting spontaneous symmetry breaking we may expect a related situation. The vacua are degenerate only if the universe is of infinite spatial extent, for only then will the vacua be orthogonal (§13.1.7). However, from the radius of the known universe one estimates that the frequency of rotation from one vacuum to another is negligible (Taylor, 1976, §5.3)."Section 8.5 in Gauge Field Theories by Guidry
A very important point to keep in mind is that once the system described by the theory chooses a vacuum determined by a value of $\langle \phi \rangle$, all other possible vacua of the theory are inaccessible in the infinite volume limit. This means that any two vacuum states $|0_1\rangle$, $|0_2\rangle$ corresponding to different vacuum expectation values of the scalar field are orthogonal $\langle 0_1 |0_2 \rangle =0$ and, moreover, cannot be connected by any local observable $\hat O(x)$, $\langle 0_1 | \hat O(x) |0_2 \rangle =0$. Heuristically, this can be understood by thinking that in the infinite volume limit switching from one vacuum into another requires changing the vacuum expectation value of the field everywhere in space at the same time, something that cannot be done by any local operator of the theory. Notice that this is radically different from our expectations based on the quantum mechanics of a system with a finite number of degrees of freedom where symmetries do not break spontaneously, i.e. the ground state is always symmetrical. Let us make these arguments a bit more explicit since they are very important in understanding how symmetry breaking works. […] It is clear that unless the directions $\hat r , \hat r'$ are parallel, the scalar product vanishes in the infinite volume limit. Hence we can build disconnected Hilbert spaces for each different direction $\hat r$. If the spatial volume is finite, the scalar product is non- vanishing and the ground states associated to different directions will mix, so that the lowest ground state will preserve the symmetry. It is only in the limit V → ∞, when the states are orthogonal, that we obtain spontaneous symmetry breaking. It is clear that if the volume is finite but large, the mixing of the different ground states is very highly suppressed, so that for many practical purposes we can approximate this finite volume theory by the theory with Goldstone bosons. A similar argument can be carried out in field theory. …An Invitation to QFT by Alvarez-Gaume et. al.
When a symmetry is broken spontaneously the true vacuum cannot be reached by perturbative expansion from the normal one. Spontaneous symmetry breaking is a phase transition and is manifestly a non-perturbative effect. (For example, the acquisition of mass by the gauge bosons through the agency of a third state of polarization must be a phase transition-we cannot change the number of degrees of freedom for a particle in infinitesimal steps. An analogous situation occurs in superconductivity: the superconducting solution cannot be obtained by perturbative expansion about free-particle states because a phase transition is involved. For an excellent discussion of this see Mattuck (1976), §5.4 and ch . 17.) As such, it is not easily handled in relativistic quantum field theories and heavy reliance must still be placed on models. We can, however, develop a consistent perturbation theory about one of the degenerate vacua if a potential with such degenerate vacua is introduced by hand, as in the Landau-Ginsberg theory of phase transitions. If done a t the macroscopic level this is equivalent to using phenomenology t o approximate the non-perturbative effects that generate the new vacuum, and perturbation theory within the new vacuum. Obviously this is not a completely satisfactory state of affairs from a microscopic point of view, but this approach allows calculations that would otherwise be impossible.page 262 in Gauge Field Theories: An Introduction with Applications by Mike Guidry
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.
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.
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 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.
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
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.
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…..
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.
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. 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 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
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.”
…
Although Anderson had identified the way forward, he had not actuallyidentified any flaws in Goldstone’s argument.34 The complete solution hadto 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.
from Infinity Puzzle by Frank Close
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. from End of Physics by Lindley
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