advanced_notions:quantum_field_theory:ghosts
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
advanced_notions:quantum_field_theory:ghosts [2018/03/30 10:56] jakobadmin |
advanced_notions:quantum_field_theory:ghosts [2018/03/30 10:57] (current) jakobadmin [Abstract] |
||
|---|---|---|---|
| Line 16: | Line 16: | ||
| <blockquote>The physical reason why ghosts may show up, is the non-local nature of the gauge- | <blockquote>The physical reason why ghosts may show up, is the non-local nature of the gauge- | ||
| fixing procedure. If we demand, for instance, | fixing procedure. If we demand, for instance, | ||
| - | $$∂µAµ = 0 , \tag{(3.1)}$$ | + | $$∂_µA^µ = 0 , \tag{(3.1)}$$ |
| then the transition from some other gauge choice to this one requires knowledge of the | then the transition from some other gauge choice to this one requires knowledge of the | ||
| field values of a given configuration over all of space-time. Since gauge transformations | field values of a given configuration over all of space-time. Since gauge transformations | ||
| Line 26: | Line 26: | ||
| needed.<cite>[[https://arxiv.org/pdf/hep-th/9812204.pdf|TOPOLOGICAL ASPECTS OF QUANTUM CHROMODYNAMICS]] by Gerard ’t Hooft</cite></blockquote> | needed.<cite>[[https://arxiv.org/pdf/hep-th/9812204.pdf|TOPOLOGICAL ASPECTS OF QUANTUM CHROMODYNAMICS]] by Gerard ’t Hooft</cite></blockquote> | ||
| - | "One of the most insightful treatment of ghosts in quantum field theory appears in lecture notes for the Basko Polje Summer School (1976) by [[http://www.nbi.dk/~lautrup/papers/ghoulies.pdf|Benny Lautrup entitled Of Ghoulies and Ghosties]]." http://scipp.ucsc.edu/~haber/ph218/ | + | ---- |
| + | |||
| + | * "One of the most insightful treatment of ghosts in quantum field theory appears in lecture notes for the Basko Polje Summer School (1976) by [[http://www.nbi.dk/~lautrup/papers/ghoulies.pdf|Benny Lautrup entitled Of Ghoulies and Ghosties]]." http://scipp.ucsc.edu/~haber/ph218/ | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
| + | The group of [[advanced_tools:gauge_symmetry|gauge transformations]] $G$ means the bundles automorphisms which preserve the Lagrangian. ([[http://www.mathunion.org/ICM/ICM1978.2/Main/icm1978.2.0881.0886.ocr.pdf|Source]]) | ||
| - | + | The gauge group is simply one fiber of the bundle, i.e. for example, $SU(2)$. | |
| - | + | ||
| - | The group of gauge transformations $G$ means the bundles automorphisms which preserve the Lagrangian. ([[http://www.mathunion.org/ICM/ICM1978.2/Main/icm1978.2.0881.0886.ocr.pdf|Source]]) | + | |
| - | + | ||
| - | The gauge group is simply one fibre of the bundle, i.e. for example, $SU(2)$. | + | |
| We denote the space of all connections by $A$. Now, to get physically sensible results we must be careful with these different notions: | We denote the space of all connections by $A$. Now, to get physically sensible results we must be careful with these different notions: | ||
| <blockquote> | <blockquote> | ||
| - | Integration should therefore be carried out on the quotient space $\mathcal{G}=A/G$. Now $A$ is a linear space but $\mathcal{G}$ is only a manifold and has to be treated with more respect. Thus for integration purposes a Jacobian term arises which, in perturbation theory, gives rise to the well-known Faddeev-Popov "ghost" particles. Nonperturbatively it seems reasonable that global topological features of $\mathcal{G}$ will be relevant. | + | Integration should, therefore, be carried out on the quotient space $\mathcal{G}=A/G$. Now $A$ is a linear space but $\mathcal{G}$ is only a manifold and has to be treated with more respect. Thus for integration purposes a Jacobian term arises which, in perturbation theory, gives rise to the well-known Faddeev-Popov "ghost" particles. Nonperturbatively it seems reasonable that global topological features of $\mathcal{G}$ will be relevant. |
| <cite>Geometrical Aspects of Gauge Theories by M. F. Atiyah</cite> | <cite>Geometrical Aspects of Gauge Theories by M. F. Atiyah</cite> | ||
advanced_notions/quantum_field_theory/ghosts.1522400168.txt.gz · Last modified: 2018/03/30 08:56 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
