advanced_notions:quantum_field_theory:instantons
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advanced_notions:quantum_field_theory:instantons [2018/03/24 14:48] 63.143.42.250 ↷ Links adapted because of a move operation |
advanced_notions:quantum_field_theory:instantons [2018/05/05 11:52] jakobadmin ↷ Links adapted because of a move operation |
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| ====== Instantons ====== | ====== Instantons ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | <blockquote> | + | <tabbox Intuitive> |
| - | The vacuum which we inhabit is filled with such instantons at a density of the order of one instanton | + | |
| - | per femtometer in every direction. (The precise quantitative theoretical predictions of this [ScSh98] suffer | + | |
| - | from an infrared regularization ambiguity, but numerical simulations demonstrate the phenomenon [Gru13].) | + | |
| - | This “instanton sea” that fills spacetime governs the mass of the η′-particle [Wit79, Ven79] as well as | + | |
| - | other non-perturbative chromodynamical phenomena, such as the quark-gluon plasma seen in experiment | + | |
| - | [Shu01]. It is also at the heart of the standard hypothesis for the mechanism of primordial baryogenesis | + | |
| - | [Sak67, ’tHo76, RiTr99], the fundamental explanation of a universe filled with matter.<cite>https://arxiv.org/abs/1601.05956</cite></blockquote> | + | |
| - | + | ||
| - | <blockquote> | + | |
| - | Of all the solutions, the instantons have interested mathematicians most; for physicists they give a semi-classical understanding of some of the [[advanced_tools:topology|topological]] effects that are present in Yang-Mills theory. | + | |
| - | + | ||
| - | <cite>Topological Investigations of Quantized Gauge Theories, by R. Jackiw (1983)</cite> | + | |
| - | </blockquote> | + | |
| - | + | ||
| - | <tabbox Layman> | + | |
| In quantum mechanics, potential barriers are no longer as tough as they are in classical mechanics. Instead, any physical system can tunnel through a potential barrier with some probability. | In quantum mechanics, potential barriers are no longer as tough as they are in classical mechanics. Instead, any physical system can tunnel through a potential barrier with some probability. | ||
| Line 25: | Line 9: | ||
| Instantons are sequences of field configurations that describe how a field tunnels through a potential barrier. | Instantons are sequences of field configurations that describe how a field tunnels through a potential barrier. | ||
| Such potential barriers exist, for example, when there is not only one state with minimum energy but many. Between these possible ground states, we usually have a potential barrier. However, in a quantum theory the field can transform itself from one ground state configuration into another ground state configuration and such a process is called an instanton. | Such potential barriers exist, for example, when there is not only one state with minimum energy but many. Between these possible ground states, we usually have a potential barrier. However, in a quantum theory the field can transform itself from one ground state configuration into another ground state configuration and such a process is called an instanton. | ||
| - | <tabbox Student> | + | |
| + | <tabbox Concrete> | ||
| In contrast to other solitons like, for example, [[advanced_notions:topological_defects:magnetic_monopoles|monopoles]], instantons can not be interpreted as "particle-like". Instead instantons is a continuous set of field configurations that describe how the field tunnels from one vacuum configuration into another. Nevertheless, we call instantons also topological solitons, because they describe field configurations with finite (Euclidean) energy. | In contrast to other solitons like, for example, [[advanced_notions:topological_defects:magnetic_monopoles|monopoles]], instantons can not be interpreted as "particle-like". Instead instantons is a continuous set of field configurations that describe how the field tunnels from one vacuum configuration into another. Nevertheless, we call instantons also topological solitons, because they describe field configurations with finite (Euclidean) energy. | ||
| Such processes cannot be described by [[advanced_notions:quantum_field_theory:perturbation_theory|perturbation theory]], but instead only with the help of [[advanced_tools:non-perturbative_qft|non-perturbative methods]]. This follows since the wave function of tunnel processes is proportional to $e^{1/x}$ or $e^{1/x^2}$ and the Taylor expansion of such functions vanishes. Hence such effects do not appear in a perturbative expansion also, of course, these effects exist. | Such processes cannot be described by [[advanced_notions:quantum_field_theory:perturbation_theory|perturbation theory]], but instead only with the help of [[advanced_tools:non-perturbative_qft|non-perturbative methods]]. This follows since the wave function of tunnel processes is proportional to $e^{1/x}$ or $e^{1/x^2}$ and the Taylor expansion of such functions vanishes. Hence such effects do not appear in a perturbative expansion also, of course, these effects exist. | ||
| - | The [[advanced_notions:quantum_field_theory:qcd_vacuum|ground state]] of, for example, [[models:qcd|QCD]] consists of an infinite number of degenerate states that are separated by a finite energy barrier. An instanton is a description how the field tunnels (not meant in a spatial sense) through one of these barriers into another vacuum. During the tunnel process the field, also in the ground state at the beginning and end of the process, goes continuously through a set of field configurations that do not correspond to a ground state, i.e. non-zero field energy. This is meant when we say that an instanton "has" finite field energy. | + | The [[advanced_notions:quantum_field_theory:qcd_vacuum|ground state]] of, for example, [[models:standard_model:qcd|QCD]] consists of an infinite number of degenerate states that are separated by a finite energy barrier. An instanton is a description how the field tunnels (not meant in a spatial sense) through one of these barriers into another vacuum. During the tunnel process the field, also in the ground state at the beginning and end of the process, goes continuously through a set of field configurations that do not correspond to a ground state, i.e. non-zero field energy. This is meant when we say that an instanton "has" finite field energy. |
| A detailed discussion of instantons written with the needs of students in mind can be found [[http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-1/|here]]. | A detailed discussion of instantons written with the needs of students in mind can be found [[http://jakobschwichtenberg.com/demystifying-the-qcd-vacuum-part-1/|here]]. | ||
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| * Aspects of Symmetry by S. Coleman | * Aspects of Symmetry by S. Coleman | ||
| * and ABC of Instantons by M. Shifman et al. | * and ABC of Instantons by M. Shifman et al. | ||
| - | <tabbox Researcher> | + | |
| + | |||
| + | <tabbox Abstract> | ||
| One way to write the instanton potential is | One way to write the instanton potential is | ||
| Line 171: | Line 158: | ||
| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| - | --> Example1# | ||
| - | + | <blockquote> | |
| - | <-- | + | The vacuum which we inhabit is filled with such instantons at a density of the order of one instanton |
| + | per femtometer in every direction. (The precise quantitative theoretical predictions of this [ScSh98] suffer | ||
| + | from an infrared regularization ambiguity, but numerical simulations demonstrate the phenomenon [Gru13].) | ||
| + | This “instanton sea” that fills spacetime governs the mass of the η′-particle [Wit79, Ven79] as well as | ||
| + | other non-perturbative chromodynamical phenomena, such as the quark-gluon plasma seen in experiment | ||
| + | [Shu01]. It is also at the heart of the standard hypothesis for the mechanism of primordial baryogenesis | ||
| + | [Sak67, ’tHo76, RiTr99], the fundamental explanation of a universe filled with matter.<cite>https://arxiv.org/abs/1601.05956</cite></blockquote> | ||
| - | --> Example2:# | + | <blockquote> |
| + | Of all the solutions, the instantons have interested mathematicians most; for physicists they give a semi-classical understanding of some of the [[advanced_tools:topology|topological]] effects that are present in Yang-Mills theory. | ||
| - | + | <cite>Topological Investigations of Quantized Gauge Theories, by R. Jackiw (1983)</cite> | |
| - | <-- | + | </blockquote> |
| - | <tabbox FAQ> | + | <tabbox FAQ> |
| - | + | ||
| - | <tabbox History> | + | --> How do instantons cause vacuum decay?# |
| + | see https://physics.stackexchange.com/questions/127879/how-do-instantons-cause-vacuum-decay | ||
| + | <-- | ||
| </tabbox> | </tabbox> | ||
advanced_notions/quantum_field_theory/instantons.txt · Last modified: 2021/10/01 11:46 by 79.35.127.248
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