advanced_notions:symmetry_breaking:higgs_mechanism
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advanced_notions:symmetry_breaking:higgs_mechanism [2018/03/29 17:01] jakobadmin [Concrete] |
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 | ||
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| <tabbox Concrete> | <tabbox Concrete> | ||
| - | --> Short Description# | ||
| - | |||
| Spontaneous symmetry breaking means that the vacuum configuration of the Higgs field is no longer invariant under the complete [[advanced_tools:group_theory|group]] $G$, but only under some subgroup $H \subset G$. In mathematical terms, this means the vacuum expectation value (VEV) $\langle \Phi_i \rangle$ is not invariant under the action of some elements of the group $g \in G$ | Spontaneous symmetry breaking means that the vacuum configuration of the Higgs field is no longer invariant under the complete [[advanced_tools:group_theory|group]] $G$, but only under some subgroup $H \subset G$. In mathematical terms, this means the vacuum expectation value (VEV) $\langle \Phi_i \rangle$ is not invariant under the action of some elements of the group $g \in G$ | ||
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| \end{equation} | \end{equation} | ||
| where $1$ denotes the one-dimensional representation of the gauge group. Here we can see that $A R\stackrel{!}{=} R $, because otherwise we would add apples to oranges. Therefore if we want to know which gauge fields become massive we need to compute the product $A\otimes R =R$. Then we can check which generators (= elements of $A$) can be combined with the element of the scalar representation $R$ that gets a VEV and yield again an element of $R$. These generators become massive and all others remain massless. This means, we need to check which generators can act on our VEV $\langle \Phi_i \rangle$ and yield again an element of the scalar representation. If we act with some generator on $\langle \Phi_i \rangle$ and get something that is not an element of the scalar representation, then the corresponding generator annihilates the VEV. (This is analogous to the angular momentum ladder operators acting on the highest state of a representation. The raising operator annihilates the state, because going any higher would be no longer a state inside the original representation. In contrast, the lowering operator simply yields another state of the representation when one acts on the highest state and therefore does not annihilate the state.) This means the generator is unbroken. | where $1$ denotes the one-dimensional representation of the gauge group. Here we can see that $A R\stackrel{!}{=} R $, because otherwise we would add apples to oranges. Therefore if we want to know which gauge fields become massive we need to compute the product $A\otimes R =R$. Then we can check which generators (= elements of $A$) can be combined with the element of the scalar representation $R$ that gets a VEV and yield again an element of $R$. These generators become massive and all others remain massless. This means, we need to check which generators can act on our VEV $\langle \Phi_i \rangle$ and yield again an element of the scalar representation. If we act with some generator on $\langle \Phi_i \rangle$ and get something that is not an element of the scalar representation, then the corresponding generator annihilates the VEV. (This is analogous to the angular momentum ladder operators acting on the highest state of a representation. The raising operator annihilates the state, because going any higher would be no longer a state inside the original representation. In contrast, the lowering operator simply yields another state of the representation when one acts on the highest state and therefore does not annihilate the state.) This means the generator is unbroken. | ||
| - | <-- | + | |
| --> Higgs mechanism in Superconductivity# | --> Higgs mechanism in Superconductivity# | ||
| Line 166: | Line 165: | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
| - | * 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 | ||
| <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. | ||
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| </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> | ||
| ---- | ---- | ||
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| + | -->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 278: | 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: | ||
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| <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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