✎ Quantum Field Theory [The Physics Travel Guide]

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

<tabbox {{icon>home}} Formulations> 
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.

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**Recommended Resources**

* [[https://arxiv.org/abs/hep-th/9702027|What is Quantum Field Theory, and What Did We Think It Is?]] by S. Weinberg
  * [[https://arxiv.org/abs/hep-th/9803075|Quantum Field Theory]] by Frank Wilczek

-->Serious QFT by Tom Banks#
 
<blockquote>The standard modern
textbook is An Introduction to Quantum Field Theory, by Peskin and Schroeder [33].
I recommend especially their wonderful Chapter 5, and all of the calculational sections
between 16.5 and 18.5, as well as Chapters 20 and 21. Every serious student of QFT
should work out the final project on the Coleman–Weinberg potential, which can be
found on page 469. Another standard is Weinberg’s three-volume opus [131]. Here
I recommend the marvelous sections on symmetries and anomalies in Volume II. The
technical discussions of perturbative effective field theory are invaluable. The section on
the Batalin–Vilkovisky treatment of general gauge equivalences is also useful. Volume I
should probably be read after completing a first course on the subject. It presents an
interesting but idiosyncratic approach to the logical structure of the field. Volume III on
supersymmetry is full of gems. In my opinion, it is flawed by an idiosyncratic notation
and a tendency to obscure relatively simple ideas in an attempt to give absolutely general
discussions. Finally, let me mention a relatively new book by M. Srednicki [168]. I have
not gone through it thoroughly, and I do not agree with the author’s ordering of
topics, but the pedagogical style of the sections I have read is wonderful. It is clear that
everyone in the field will turn to this book for all those nasty little details about minus
signs and spinor conventions. I think there is also a chance that it will replace Peskin
and Schroeder as a standard textbook.

I have also enjoyed using the books by Bailin and Love [169] and Ramond [170] in
my many years of teaching the subject. The books by Itzykson and Zuber [171] and
Zinn-Justin [111] are more monographs/encyclopediae than textbooks, but they contain a wealth of detail on specific subjects in field theory that can be found nowhere
else.

I mention especially Zinn-Justin’s discussions of the large-order behavior of perturbation theory, of the use of field theory for calculating critical exponents, and of
instantons. Other treatments of the field theory/statistical physics interface can be found
in the book by Drouffe and Itzykson [172], the marvelous book by Parisi [173], and the
book by Ma [174]. Schwinger’s source theory books [175] also belong in the category
of non-textbooks, which contain scads of invaluable information about field theory.
Among older field-theory books, the second volume of Bjorken and Drell [176]
contains lots of useful information, like explicit forms for the spectral representations
for higher spin. The books of Nishijima [177] (Chapters 7 and 8), Bogoliubov and
Shirkov [178], and even Schweber [179] will reward the really serious student of the
subject.

Finally, I want to mention various shorter documents that I think are essential
reading for students of quantum field theory. The most important is Kogut and Wilson [126], still the best introduction I know of to Wilson’s profound ideas about
renormalization. Next is the 1975 Les Houches lecture-note volume Methods in Field
Theory [180], every chapter of which is a gem. Coleman’s book [138] is a collection of
lectures on a variety of topics in field theory. I’ve drawn on it heavily for the material
about instantons and solitons, but the other lectures are also worth reading.

The contributions of Adler, Weinberg, Zimmermann, and Zumino to the 1970 Brandeis Summer
School Lectures, 1 and of Weinberg to the 1964 Brandeis volume, are also worth read-
ing [181]. Much of Weinberg’s material reappears in his textbook [131]. Finally, let me
mention the reprint volume Selected Papers on Quantum Electrodynamics [182] edited
by Schwinger. The contributions of Feynman and Schwinger in particular should be
read by every student of field theory.<cite>Modern Quantum Field Theory by Banks</cite></blockquote>
<--

-->How to get started learning QFT as an undergraduate by Flip Tanedo
see https://fliptomato.wordpress.com/2006/12/30/from-griffiths-to-peskin-a-lit-review-for-beginners/
<--

-->Classical Solutions in Quantum Field Theory#

<diagram>
| AA|~@2| General overview, why should we care about classical solutions? |AA=[[advanced_notions:quantum_field_theory:solitons]]
| |!@4| | | | | |
| AA|~@2| Kink solution, stability of the kink, boundary conditions, topological charge |AA=[[models:scalar_1plus1]]
| |!@4| | | | | |
| BB|~@2| Duality |BB=[[models:sine_gordon|Sine-Gordon]] and [[models:thirring|Thirring Model]]
| |!@4| | | | | |
| BB|~@2| Vortex solution, solitons in gauge theories |BB=[[models:abelian_higgs_2plus1|Abelian 2+1 Higgs Model]]
| |!@4| | | | | |
| BB||   |BB=[[advanced_notions:quantum_field_theory:sphalerons]]
| |!@4| | | | | |
| BB||   |BB=[[advanced_notions:quantum_field_theory:instantons]]
| |!@4| | | | | |
| BB||   |BB=[[advanced_notions:quantum_field_theory:qcd_vacuum]]
</diagram>

<--
-->Lifecycle of a Quantum Field Theorist#

- Write down a **[[formalisms:lagrangian_formalism|Lagrangian density]]** $L$. This is a polynomial in **[[theories:quantum_field_theory:canonical|fields]]** $\psi$ and their derivatives. For example $$L[\psi] = \partial_\mu\psi\partial^\mu\psi - m^2\psi^2 + \lambda\psi^4$$
  -  Write down the **[[theories:quantum_mechanics:path_integral|Feynman path integral]]**. Roughly speaking this is $$\int e^{i\int L[\psi]} D\psi$$ The value of this integral can be used to compute "[[advanced_notions:cross_section|cross sections]]" for various processes.
  -  Calculate the Feynman path integral by expanding as a formal power series in the "coupling constant" $\lambda$. $$a_0 + a_1\lambda + a_2\lambda + \cdots$$ The $a_i$ are finite sums over [[advanced_tools:feynman_diagrams|Feynman diagrams]]. Feynman diagrams are a graphical shorthand for finite dimensional integrals.
  - Work out the integrals and add everything up.
  - Realise that the finite dimensional integrals do not converge.
  - Regularise the integrals by introducing a ``cutoff'' $\epsilon$ (there is usually an infinite dimensional space of possible regularisations). For example $$\int_\mathcal{R} {1\over x^2} dx \longrightarrow \int_{|x|>\epsilon} {1\over x^2} dx$$
  - Now we have the series $$a_0(\epsilon) + a_1(\epsilon)\lambda + \cdots$$ **Amazing Idea:** Make $\lambda$, $m$ and other parameters of the Lagrangian depend on $\epsilon$ in such a way that terms of the series are independent of $\epsilon$.
  -  Realise that the new sum still diverges even though we have made all the individual $a_i$'s finite. **No good way of fixing this is known**. It appears that the resulting series is in some sense an asymptotic expansion.
  -  Ignore step 8, take only the first few terms and compare with experiment.
  -  Depending on the results to step 9: Collect a Nobel prize or return to step 1.

There are many problems that arise in the above steps

* [Problem 1] The Feynman integral is an integral over an infinite dimensional space and there is **no** analogue of Lebesgue measure.
  * [Solution] Take what the physicists do to evaluate the integral as its definition.
  * [Problem 2] There are many possible cutoffs.  This means the value of the integral depends not only on the Lagrangian but also on the choice of cutoff.
  * [Solution] There is a group $G$ called the **group of finite [[advanced_tools:renormalization|renormalizations]]** which acts on both Lagrangians and cutoffs.
QFT is unchanged by the action of $G$ and $G$ acts transitively on the space of cutoffs.  So, we only have to worry about the space of Lagrangians.
  * [Problem 3] The resulting formal power series (even after renormalization) does not converge.
  * [Solution] Work in a formal power series ring.

<cite>[[https://arxiv.org/abs/math-ph/0204014|Lectures on Quantum Field Theory]] by R. E. Borcherds, A. Barnard</cite>
</blockquote>

<--

</tabbox>

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