advanced_notions:quantum_field_theory:ghosts
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advanced_notions:quantum_field_theory:ghosts [2017/11/14 14:49] jakobadmin [Student] |
advanced_notions:quantum_field_theory:ghosts [2018/03/30 10:57] (current) jakobadmin [Abstract] |
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| ====== Ghosts ====== | ====== Ghosts ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | <tabbox Layman> | + | <tabbox Intuitive> |
| - | <note tip> | + | <blockquote>As I have discussed several times introducing these redundancies makes live much easier. But this can turn against you, if you need to make approximations. Which, unfortunately, is usually the case. Still their benefits outweighs the troubles. |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
| - | </note> | + | |
| - | + | ||
| - | <tabbox Student> | + | |
| - | "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 remarkable consequences of these redundancies is that they even affect our description of the most fundamental particles in our theories. Here, I will concentrate on the gluons of the strong interactions (or QCD). On the one hand because they play a very central role in many phenomena. But, more importantly, because they are the simplest particles exhibiting the problem. This follows essentially the old strategy of divide and conquer. Solve it for the simplest problem first, and continue from there. |
| + | Still, even the simplest case is not easy. The reason is that the redundancies introduced auxiliary quantities. These act like some imaginary particles. These phantom particles are called also ghosts, because, just like ghosts, they actually do not really exist, they are only there in our imagination. Actually, they are called Faddeev-Popov ghosts, honoring those two people who have introduced them for the very first time. | ||
| - | + | Thus, whenever we calculate quantities we can actually observe, we do not see any traces of these ghosts. But directly computing an observable quantity is often hard, especially when you want to use eraser-and-pencil-type calculations. So we work stepwise. And in such intermediate steps ghosts do show up. But because they only encode information differently, but not add information, their presence affects also the description of 'real' particles in these intermediate stages. Only at the very end they would drop out. If we could do the calculations exactly.<cite>http://axelmaas.blogspot.de/2016/10/redundant-ghosts.html</cite></blockquote> | |
| - | <tabbox Researcher> | + | |
| - | <note tip> | + | <tabbox Concrete> |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | |
| - | --> Common Question 1# | + | <blockquote>The physical reason why ghosts may show up, is the non-local nature of the gauge- |
| + | fixing procedure. If we demand, for instance, | ||
| + | $$∂_µA^µ = 0 , \tag{(3.1)}$$ | ||
| + | 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 | ||
| + | do not affect physical information, the information transmitted over space-time in order | ||
| + | to realize the gauge (3.1), is unphysical. This is the explanation of the emergence of | ||
| + | ghosts. | ||
| + | We can avoid ghosts, if the gauge fixing at any point x in space-time, is done in | ||
| + | such a way that no knowledge of the field values in points other than the point x is | ||
| + | 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/ | ||
| - | --> Common Question 2# | ||
| - | <-- | + | <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]]) | |
| - | <tabbox Examples> | + | |
| - | --> Example1# | + | The gauge group is simply one fiber 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: | |
| - | <-- | + | |
| + | <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. | ||
| + | |||
| + | <cite>Geometrical Aspects of Gauge Theories by M. F. Atiyah</cite> | ||
| + | </blockquote> | ||
| + | <tabbox Why is it interesting?> | ||
| - | --> Example2:# | ||
| - | |||
| - | <-- | ||
| - | | ||
| - | <tabbox History> | ||
| </tabbox> | </tabbox> | ||
advanced_notions/quantum_field_theory/ghosts.1510667379.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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