advanced_notions:quantum_field_theory:gribov_ambiguities
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advanced_notions:quantum_field_theory:gribov_ambiguities [2017/05/09 07:43] jakobadmin created |
advanced_notions:quantum_field_theory:gribov_ambiguities [2017/11/05 15:34] (current) jakobadmin ↷ Page moved from theories:gauge_theory:gribov_ambiguities to advanced_notions:quantum_field_theory:gribov_ambiguities |
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| ====== Gribov Ambiguities ====== | ====== Gribov Ambiguities ====== | ||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| + | <blockquote> | ||
| + | In electrodynamics, Coulomb gauge is a simple example of a gauge that gives a one-to-one mapping between field strength and potential. This is not the case in the non-Abelian theory, where the Gribov ambiguity shows that there are configurations of field strengths that allow several distinct choices of Coulomb gauge potentials, as well as other configurations that cannot be brought into Coulomb gauge at all [85, 236]. **These issues are often swept under the rug when working in perturbation theory, but cannot be ignored when considering potentials of order 1/g, for which the commutator term in the field strength is comparable in size to the derivative terms**. As suggested by Eq. (10.8), it is precisely such potentials with which we will be concerned. | ||
| + | <cite>page 201 in Classical Solutions in Quantum Field Theory by Erik Weinberg</cite> | ||
| + | </blockquote> | ||
| <tabbox Layman> | <tabbox Layman> | ||
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| <tabbox Student> | <tabbox Student> | ||
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| <blockquote> | <blockquote> | ||
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| <tabbox Researcher> | <tabbox Researcher> | ||
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| + | <blockquote> | ||
| + | There is a difficulty which arises in the cornpactified | ||
| + | case which is not present in the more general one. This | ||
| + | involves gauge fixing and the Gribov ambiguity. ' As | ||
| + | yet the gauge has not been fixed and there is the possibility | ||
| + | that fixing the gauge could affect the topology of | ||
| + | configuration space. The object of gauge fixing is to | ||
| + | choose one member from each class of gauge-equivalent | ||
| + | configurations so that there will be no double counting | ||
| + | of configurations in the path integral. | ||
| + | In the fiber-bundle picture of Dowker, ' | ||
| + | this corresponds | ||
| + | to finding a section in the bundle, reducing each | ||
| + | configuration fiber to a particular configuration in the fiber. The difficulty is that Singer' has proven that | ||
| + | there are no global gauges when space is compactified to | ||
| + | S . That is, there is no global section of the fiber bundle | ||
| + | that Dowker is using. This means that the space of | ||
| + | gauge-fixed configurations is disconnected. The answer is that the Gribov ambiguity allows for | ||
| + | certain zero-action discontinuous evolutions. Because | ||
| + | there are no global sections, the gauge fixing fails to always | ||
| + | specify a unique configuration from each | ||
| + | configuration fiber. When it does fail, there are two (or | ||
| + | more) configurations which are related by a gauge transformation. | ||
| + | Each of these lies on a different section of | ||
| + | the fiber bundle and one's histories may move between | ||
| + | the disconnected sections by way of these configurations. | ||
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| + | <cite>Nontrivial homotopy and tunneling by topological instantons by Arlen Anderson</cite> | ||
| + | </blockquote> | ||
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| + | It was proven by Singer in "Some Remarks on the Gribov Ambiguity", provided that we compactify spacetime to the 4-sphere, that <q>"no continuous choice of exactly one | ||
| + | connection on each orbit can be made. Thus the Gribov ambiguity for the | ||
| + | Coloumb gauge will occur in all other gauges. No gauge fixing is possible. "</q> | ||
| <blockquote> | <blockquote> | ||
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| </blockquote> | </blockquote> | ||
| - | --> Common Question 1# | + | --> What are the physical implications of Gribov ambiguities?# |
| + | <blockquote> | ||
| + | The BRST method used to deal with the gauge symmetry of perturbative Yang-Mills theory does not appear to generalize to the full non-perturbative theory, for a rather fundamental reason. This was first pointed out by **Neuberger** back in 1986 (Phys. Lett. B, 183 (1987), p337-40.), who **argued that, non-perturbatively, the phenomenon of Gribov copies implies that expectation values of gauge-invariant observables will vanish**. | ||
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| + | <cite>http://www.math.columbia.edu/~woit/wordpress/?p=2876</cite> | ||
| + | </blockquote> | ||
| <-- | <-- | ||
| - | --> Common Question 2# | ||
| - | |||
| - | <-- | ||
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| <tabbox Examples> | <tabbox Examples> | ||
advanced_notions/quantum_field_theory/gribov_ambiguities.1494308597.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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