theorems:elitzur_s_theorem
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theorems:elitzur_s_theorem [2017/09/28 10:07] jakobadmin [Researcher] |
theorems:elitzur_s_theorem [2018/05/05 12:51] (current) jakobadmin |
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| ====== Elitzur's Theorem ====== | ====== Elitzur's Theorem ====== | ||
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| - | <tabbox Why is it interesting?> | ||
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| - | <blockquote> | ||
| - | Elitzur demonstrated that a spontaneous breaking of a local symmetry is not possible. | ||
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| - | <cite>https://journals.aps.org/prb/pdf/10.1103/PhysRevB.72.045137</cite> | ||
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| - | </blockquote> | ||
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| - | <tabbox Layman> | + | <tabbox Intuitive> |
| <note tip> | <note tip> | ||
| Line 20: | Line 8: | ||
| </note> | </note> | ||
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| - | <tabbox Student> | + | <tabbox Concrete> |
| To quote [[https://www.youtube.com/watch?v=XM4rsPnlZyg&feature=youtu.be&t=18m38s|Seiberg]]: | To quote [[https://www.youtube.com/watch?v=XM4rsPnlZyg&feature=youtu.be&t=18m38s|Seiberg]]: | ||
| <blockquote>"a gauge symmetry is so big, it's too big to fail."</blockquote> | <blockquote>"a gauge symmetry is so big, it's too big to fail."</blockquote> | ||
| - | A [[:gauge_theory|local symmetry]] means that we have an independent copy of the symmetry (of the group $G$) at each spacetime point. A spacetime point is zero-dimensional. Hence we are dealing with symmetries of zero-dimensional subsystems and | + | A [[advanced_tools:gauge_symmetry|local symmetry]] means that we have an independent copy of the symmetry (of the group $G$) at each spacetime point. A spacetime point is zero-dimensional. Hence we are dealing with symmetries of zero-dimensional subsystems and |
| - | <blockquote>A theorem due to Mermin and Wagner states that a continuous symmetry can only be spontaneously broken in a dimension larger than two. For a discrete symmetry this lower critical dimensionality is one. This is, in fact, well known since in quantum mechanics with finitely many degrees of freedom (corresponding to one-dimensional field theory) tunneling between degenerate classical minima allows for a unique symmetric ground state. [...] The Mermin-Wagner theorem has been restated by Coleman in the framework of field theory. </blockquote> | + | <blockquote>A [[advanced_notions:symmetry_breaking:mermin-wagner_theorem|theorem due to Mermin and Wagner]] states that a continuous symmetry can only be spontaneously broken in a dimension larger than two. For a discrete symmetry this lower critical dimensionality is one. This is, in fact, well known since in quantum mechanics with finitely many degrees of freedom (corresponding to one-dimensional field theory) tunneling between degenerate classical minima allows for a unique symmetric ground state. [...] The Mermin-Wagner theorem has been restated by Coleman in the framework of field theory. <cite>page 525 in Quantum Field Theory by Claude Itzykson, Jean-Bernard Zuber </cite> </blockquote> |
| Therefore, combining the informations that | Therefore, combining the informations that | ||
| Line 77: | Line 65: | ||
| Ref [23] is J. Fröhlich, G. Morchio, F. Strocchi, Higgs phenomenon without symmetry breaking order | Ref [23] is J. Fröhlich, G. Morchio, F. Strocchi, Higgs phenomenon without symmetry breaking order | ||
| parameter. Nucl. Phys. B 190, 553 (1981) | parameter. Nucl. Phys. B 190, 553 (1981) | ||
| - | <tabbox Researcher> | + | <tabbox Abstract> |
| See: | See: | ||
| Line 107: | Line 95: | ||
| S. Elitzur, Phys. Rev. D 12, 3978 (1975); G.F. De Angelis, D. De Falco and F. Guerra, Phys. Rev. D 17, 1624 (1978). The crux of the argument is that gauge invariance decouples the transformations of the fields inside a volume V (in a euclidean functional integral approach) from the transformation of the boundary, so that the boundary conditions are ineffective and cannot trigger non symmetric correlation functions. For a simple account of the argument, see e.g. F. Strocchi, Elements of Quantum Mechanics of Infinite Systems, World Scientific 1985, Part C, Sect. 2.5. | S. Elitzur, Phys. Rev. D 12, 3978 (1975); G.F. De Angelis, D. De Falco and F. Guerra, Phys. Rev. D 17, 1624 (1978). The crux of the argument is that gauge invariance decouples the transformations of the fields inside a volume V (in a euclidean functional integral approach) from the transformation of the boundary, so that the boundary conditions are ineffective and cannot trigger non symmetric correlation functions. For a simple account of the argument, see e.g. F. Strocchi, Elements of Quantum Mechanics of Infinite Systems, World Scientific 1985, Part C, Sect. 2.5. | ||
| - | <cite>P. 195 in Symmetry Breaking by Strocchi<> | + | In order to avoid the vanishing of a symmetry breaking order parameter, one must reconsider the problem by adding to the Lagrangean (19.1) a gauge fixing LGF which breaks local gauge invariance. Then, the discussion of the Higgs mechanism necessarily becomes gauge fixing dependent; this should not appear strange, since the vacuum expectation of φ is a gauge dependent quantity.194 |
| - | </blockquote> | + | [...] |
| - | --> Common Question 1# | + | Since the gauge fixing breaks local gauge invariance, but not the invariance under the global group transformations, the EDDG theorem does not apply and one may consider the possibility of a symmetry breaking order parameter < φ ≯= 0. |
| + | Now, another conceptual problem arises: the starting Lagrangean L is invariant under the U(1) global group and its breaking with a mass gap seems incompatible with the Goldstone theorem. As an explanation of such an apparent conflict, one finds in the literature the statement that the Gold- stone theorem does not apply if the two point function < j0(x)φ(y) > is not Lorentz covariant as it happens in the physical gauges, like the Coulomb gauge. As a matter of fact, the Goldstone-Salam-Weinberg proof of the Gold- stone theorem crucially uses Lorentz covariance; however, the more general proof discussed in Chapter 17 does not assume it, so that the quest of a better explanation remains. | ||
| - | + | <cite>P. 195 in Symmetry Breaking by Strocchi<cite> | |
| - | <-- | + | |
| - | --> Common Question 2# | + | </blockquote> |
| - | + | <tabbox Why is it interesting?> | |
| - | <-- | + | |
| - | + | ||
| - | <tabbox Examples> | + | |
| - | --> Example1# | + | <blockquote> |
| + | Elitzur demonstrated that a spontaneous breaking of a local symmetry is not possible. | ||
| - | + | <cite>https://journals.aps.org/prb/pdf/10.1103/PhysRevB.72.045137</cite> | |
| - | <-- | + | |
| - | --> Example2:# | + | </blockquote> |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| - | + | ||
| - | <tabbox History> | + | |
| </tabbox> | </tabbox> | ||
theorems/elitzur_s_theorem.1506586067.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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