✎ Symmetry Breaking [The Physics Travel Guide]

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====== Symmetry Breaking ======

<tabbox Why is it interesting?> 
<blockquote>What an imperfect world it would be if every
symmetry was perfect

<cite> B. G. WYBOURNE</cite>
</blockquote>

**Related:**

* [[advanced_notions:symmetry_breaking:goldstones_theorem]]
  * [[advanced_notions:symmetry_breaking:chiral_symmetry_breaking]]
  * [[advanced_notions:symmetry_breaking:higgs_mechanism]]
<tabbox Layman>

<blockquote>
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 [[advanced_notions:symmetry_breaking:higgs_mechanism|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. 
<cite>page 279 in Topological Investigations of Quantized Gauge Theories by R. Jackiw</cite>
</blockquote>

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Student>

* A great introduction is http://www.lassp.cornell.edu/sethna/pubPDF/OrderParameters.pdf
  * For the distinction between symmetry breaking in classical mechanics, quantum mechanics and qft, see https://physics.stackexchange.com/questions/69289/spontaneous-symmetry-breaking-in-classical-mechanics-quantum-mechanics-and-quan
<tabbox Researcher> 
  * A great summary can be found in https://arxiv.org/pdf/1405.5532.pdf

<blockquote>
It is well known that gauge symmetries do not actually get broken, but rather become concealed in what is called—in an abuse of language—the “broken phase” of a theory [28–31].

<cite>https://arxiv.org/pdf/1703.02964.pdf</cite>
</blockquote>

<blockquote>Symmetry breaking in QFT results from a mismatch between
variational symmetries of the Lagrangian and symmetries that can be
implemented as unitary transformations on the Hilbert space of states.
(The inapt adjective ‘spontaneous’ differentiates symmetry breaking that
arises due to the noninvariance of the vacuum state from that due to
explicitly adding asymmetric terms to the Lagrangian.) [...]
Fabri and Picasso
(1966) showed that if the vacuum state $|0\rangle$ is translationally invariant,
then the vacuum is either invariant under the internal symmetry, $Q |0\rangle = 0$
, or there is no state corresponding to $Q |0\rangle$ in the Hilbert space. 
The second case corresponds to SSB. The symmetry is hidden in that
there is no unitary operator to map a physical state to its symmetric
counterparts; instead, the symmetry is (roughly speaking) a map from
one Hilbert space of states to an entirely distinct space. This is usually
described as ‘vacuum degeneracy’, although each distinct Hilbert space
has a unique vacuum state. [...]
Parenti, Strocchi, and Velo (1977) study the features of SSB in classical,
nonlinear field theories; in these theories, solutions to the equations of
motion fall into distinct “sectors,” corresponding to **global field configurations
that cannot be transformed into each other via local perturbations**. The variational symmetries of the Lagrangian then fall into the unbroken
symmetries, for which $Q_V$ converges in the limit, and broken symmetries,
for which $Q_V$ fails to converge. The broken symmetries map between the
“physically disjoint worlds” represented by the distinct global field configurations.
Similarly, in QFT, the degenerate vacua correspond to distinct
global field configurations with minimum energy, with Hilbert spaces built
up from a particular vacuum state.

<cite>http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf</cite></blockquote>

* [[https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwjV6rONpffWAhVOb1AKHcr0Dt0QFggnMAA&url=http%3A%2F%2Fscipp.ucsc.edu%2F~haber%2Fwebpage%2Fsun_son.ps&usg=AOvVaw10NnZLRIGhQEG11Aivzgqy|Notes on the spontaneous breaking of SU(N). and SO(N) via a second-rank tensor multiplet.]] Howard E. Haber. 
  
<tabbox Examples>

--> Example1#

<--

--> Example2:#

<--

-->Why is symmetry breaking only possible in systems with infinite spatial extent#?

<blockquote>
"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)."<cite>Section 8.5 in Gauge Field Theories by Guidry</cite></blockquote>

<blockquote>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. ...<cite>An Invitation to QFT by Alvarez-Gaume et. al.</cite></blockquote>
<--

--> Can we describe symmetry breaking in perturbation theory?#

No!

<blockquote>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.<cite>page 262 in Gauge Field Theories: An Introduction with Applications by Mike Guidry </cite></blockquote>

<--
<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.

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.

<cite>from Infinity Puzzle by Frank Close</cite>
</blockquote>

<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. 
<cite>from End of Physics by Lindley</cite>
</blockquote>
</tabbox>

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