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advanced_notions:quantum_field_theory:ghosts [2018/03/30 10:56] jakobadmin [Concrete] |
advanced_notions:quantum_field_theory:ghosts [2018/03/30 10:57] (current) jakobadmin [Abstract] |
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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> |