✎ Quantum Field Theory [The Physics Travel Guide]

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====== Quantum Field Theory ======

<tabbox Overview> 
The most popular formulations of quantum field theory are

* [[theories:quantum_field_theory:canonical]] $\leftrightarrow$ Quantum Field Theory in [[basic_tools:hilbert_space|Hilbert space]],
  * [[theories:quantum_field_theory:path_integral|Path Integral Quantum Field Theory]] $\leftrightarrow$ Quantum Field Theory in [[basic_tools:phase_space]]

----

Canonical quantum field theory can be summarized as follows:

<diagram>
|AA|AA=[[theories:classical_field_theory|Classical Field Theory]]
||!|
||!|AA|AA=[[advanced_tools:quantization|canonical quantization]]
||!@4|
|AA|-@2|BB|-@2|CC||AA=[[theories:quantum_field_theory:canonical|Canonical Quantum Field Theory]]|BB=calculation of transition amplitudes|CC=probabilities for scatter processes or decays 
</diagram>

The main components of quantum field theory are:

- **Theory of free fields**. Here we neglect any interaction terms and only calculate what different fields do when they are on their own.
  - **Theory of interacting fields**. Here we derive the correct terms in our equations that describe interactions using [[advanced_tools:gauge_symmetry|gauge symmetry]]. Unfortunately, it turns out that we can't solve these equations in closed form but only approximately. There is a powerful mathematical machinery that allows us to calculate results approximately. At the heart of this machinery are the famous [[advanced_tools:feynman_diagrams|Feynman diagrams]]. In addition, an important trick (the interaction picture) allows us to reuse most of the things we derived for free fields. 
  - **[[advanced_tools:renormalization|Renormalization]]**. The main problem in quantum field theory is that calulcations are technically extremely challenging. One reason for this is that when we calculate things naively we end always end up with the result "infinity". This is physically nonsensical and we need to take greater care. The mathematical machinery that allows us to tame these infinities is known as renormalization.

</tabbox>

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