advanced_notions:symmetry_breaking:higgs_mechanism
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advanced_notions:symmetry_breaking:higgs_mechanism [2018/03/29 17:04] jakobadmin [Abstract] |
advanced_notions:symmetry_breaking:higgs_mechanism [2018/12/17 13:56] (current) jakobadmin [Abstract] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | * For an intuitive explanation of the Higgs mechanism, see [[http://jakobschwichtenberg.com/higgs-intuitively/|Understanding the Higgs mechanism intuitively]] by J. Schwichtenberg | + | |
| <blockquote> | <blockquote> | ||
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| ---- | ---- | ||
| + | * For an intuitive explanation of the Higgs mechanism, see [[http://jakobschwichtenberg.com/higgs-intuitively/|Understanding the Higgs mechanism intuitively]] by J. Schwichtenberg | ||
| * See also, the most famous PopSci explanation of the Higgs mechanism: https://www.hep.ucl.ac.uk/~djm/higgsa.html | * See also, the most famous PopSci explanation of the Higgs mechanism: https://www.hep.ucl.ac.uk/~djm/higgsa.html | ||
| * Another great explanation can be found on page 163 in "A Zeptospace Odyssey" by Guidice | * Another great explanation can be found on page 163 in "A Zeptospace Odyssey" by Guidice | ||
| Line 166: | Line 167: | ||
| * Great descriptions can be found in [[https://arxiv.org/pdf/0910.5167v1.pdf|Gravity from a Particle Physicists’ perspective]] by Roberto Percacci and | * Great descriptions can be found in [[https://arxiv.org/pdf/0910.5167v1.pdf|Gravity from a Particle Physicists’ perspective]] by Roberto Percacci and | ||
| * in section 3.3. of [[http://pages.physics.cornell.edu/~ajd268/Notes/QFTIII.pdf|Solitons and Instantons]] by JEFF ASAF DROR | * in section 3.3. of [[http://pages.physics.cornell.edu/~ajd268/Notes/QFTIII.pdf|Solitons and Instantons]] by JEFF ASAF DROR | ||
| - | * Another great summary can be found in https://arxiv.org/pdf/1405.5532.pdf | + | * Another great summary can be found in https://arxiv.org/pdf/1405.5532.pdf |
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | The Higgs mechanism is a crucial ingredient of the [[models:standard_model|standard model of particle physics]]. Without the Higgs mechanism, all particles are not allowed to have a mass, because such terms would violate the [[advanced_tools:gauge_symmetry|gauge symmetry]]. (Breaking of [[advanced_tools:gauge_symmetry|gauge symmetry]] is a bad thing, because the [[advanced_tools:renormalization|renormalizability]], i.e. the removing of the infinities that pop-up in most [[theories:quantum_field_theory|quantum field theory]] calculations, depends on the existence of gauge symmetry.) | + | The Higgs mechanism is a crucial ingredient of the [[models:standard_model|standard model of particle physics]]. Without the Higgs mechanism, all particles are not allowed to have a mass, because such terms would violate the [[advanced_tools:gauge_symmetry|gauge symmetry]]. (Breaking of [[advanced_tools:gauge_symmetry|gauge symmetry]] is a bad thing, because the [[advanced_tools:renormalization|renormalizability]], i.e. the removing of the infinities that pop-up in most [[theories:quantum_field_theory:canonical|quantum field theory]] calculations, depends on the existence of gauge symmetry.) |
| However, we know from experiments that some [[advanced_notions:elementary_particles|elementary particles]], like the electron or also the W-bosons that mediate weak interactions, are massive. These masses can be explained thanks to the Higgs mechanism. The masses then arise as a result of the coupling of the massive particles to the Higgs field and this is possible without breaking gauge symmetry. | However, we know from experiments that some [[advanced_notions:elementary_particles|elementary particles]], like the electron or also the W-bosons that mediate weak interactions, are massive. These masses can be explained thanks to the Higgs mechanism. The masses then arise as a result of the coupling of the massive particles to the Higgs field and this is possible without breaking gauge symmetry. | ||
| Line 207: | Line 208: | ||
| </blockquote> | </blockquote> | ||
| + | <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> | ||
| ---- | ---- | ||
| Line 219: | Line 255: | ||
| + | -->Can the Higgs interactions be considered a fifth fundamental force? # | ||
| + | Yes! See https://physics.stackexchange.com/questions/1080/why-isnt-higgs-coupling-considered-a-fifth-fundamental-force and also [[http://inspirehep.net/record/256768/files/Pages_from_C87-01-24_1-18.pdf|THE FIFTH FORCE]] by James D. Bjorken | ||
| + | <-- | ||
| --> Is there an "inverse" Higgs mechanism?# | --> Is there an "inverse" Higgs mechanism?# | ||
| Line 276: | Line 314: | ||
| It is well known that breaking of a local gauge symmetry is impossible (see the quotes below.) | It is well known that breaking of a local gauge symmetry is impossible (see the quotes below.) | ||
| - | However, there is symmetry breaking when the Higgs field acquires a non-zero vev. It is a global part of the gauge group that gets broken. However, there are no [[advanced_notions:symmetry_breaking:goldstones_theorem|Goldstone bosons]], because these correspond to the gauge degrees of freedom and become the longitudinal polarizations of the gauge bosons that become massive. | + | However, there is symmetry breaking when the Higgs field acquires a non-zero vev. It is a global part of the gauge group that gets broken. However, there are no [[theorems:goldstones_theorem|Goldstone bosons]], because these correspond to the gauge degrees of freedom and become the longitudinal polarizations of the gauge bosons that become massive. |
| This is discussed nicely in "Quantum Field Theory - A Modern Perspective" by V. P. Nair at page 268: | This is discussed nicely in "Quantum Field Theory - A Modern Perspective" by V. P. Nair at page 268: | ||
| Line 420: | Line 458: | ||
| <cite>[[https://arxiv.org/abs/cond-mat/0503400|Is electromagnetic gauge invariance spontaneously violated in superconductors?]] by Martin Greiter</cite></blockquote> | <cite>[[https://arxiv.org/abs/cond-mat/0503400|Is electromagnetic gauge invariance spontaneously violated in superconductors?]] by Martin Greiter</cite></blockquote> | ||
| - | The fact that spontaneous breaking of a local symmetry is impossible is the message of [[advanced_notions:elitzur_s_theorem]]. For a gauge symmetry, we have a copy of the symmetry group $G$ at each spacetime point. Thus symmetry breaking would need to happen at each spacetime point individual, i.e. in each zero-dimensional subsystem. A spacetime point is zero-dimensional and there is no symmetry breaking in systems of dimension lower than 2. This is known as the [[advanced_notions:symmetry_breaking:mermin-wagner_theorem|Mermin-Wagner theorem]]. | + | The fact that spontaneous breaking of a local symmetry is impossible is the message of [[theorems:elitzur_s_theorem]]. For a gauge symmetry, we have a copy of the symmetry group $G$ at each spacetime point. Thus symmetry breaking would need to happen at each spacetime point individual, i.e. in each zero-dimensional subsystem. A spacetime point is zero-dimensional and there is no symmetry breaking in systems of dimension lower than 2. This is known as the [[advanced_notions:symmetry_breaking:mermin-wagner_theorem|Mermin-Wagner theorem]]. |
| <blockquote> | <blockquote> | ||
| - | However, strictly speaking a //local gauge// symmetry can //never// be broken because we would want and observable to serve as an order parameter, but all such are necessarily gauge-invariant; this is the content of //[[advanced_notions:elitzur_s_theorem]]//. | + | However, strictly speaking a //local gauge// symmetry can //never// be broken because we would want and observable to serve as an order parameter, but all such are necessarily gauge-invariant; this is the content of //[[theorems:elitzur_s_theorem]]//. |
| <cite>[[https://books.google.de/books?id=LtdqCQAAQBAJ&lpg=PA30&ots=xKAr3CFF-b&dq=banks%20finite%20temperature%20behaviour%20of%20the%20lattice&hl=de&pg=PA9#v=onepage&q&f=false|Classification of topological defects and their relevance to cosmology and elsewhere]] by T.W.B. Kibble</cite> | <cite>[[https://books.google.de/books?id=LtdqCQAAQBAJ&lpg=PA30&ots=xKAr3CFF-b&dq=banks%20finite%20temperature%20behaviour%20of%20the%20lattice&hl=de&pg=PA9#v=onepage&q&f=false|Classification of topological defects and their relevance to cosmology and elsewhere]] by T.W.B. Kibble</cite> | ||
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