Roadmap for Quantum Field Theory
Quantum Field Theory is, quite surprisingly, a very large paradigm shift from quantum mechanics, and so in my opinion is the subject which bears the greatest risk of being misunderstood if visited prematurely.
A small introduction about QFT itself: it is currently the best theoretical framework we have for developing realistic models of fundamental processes, including our very own Standard Model. It carries forward the basic axioms of ordinary quantum mechanics, but is much better suited to handle relativistic effects and variable particle systems - I have made an introductory video highlighting these perspectives over here:
Despite its scary reputation, you could technically start learning QFT barely with knowledge of quantum mechanics up to the quantum harmonic oscillator, and - wait for it - a system of infinite blocks connected by springs in a lattice. I can’t guarantee you’ll appreciate the depth and complexity of QFT, but you will probably get a reasonable understanding of how fields work.
Disclaimer: all of the prerequisites listed below each have their own prerequisites, and so on, I have deliberately suppressed these to avoid a stack overflow.
Definite Prerequisites:
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Quantum Mechanics, at an intermediate level - say, a mastery of all the topics mentioned in Griffiths’ at the level of Sakurai, and especially the harmonic oscillator via ladder operators, angular momentum, scattering and time-dependent perturbation theory (much of QFT is working with perturbative series; even virtual particles can be seen at the level of ordinary quantum mechanics!). Don’t worry about knowing about the Dirac equation or the problems QM faces when extrapolating to the relativistic regime: all good sources provide background on them.
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Special Relativity: As I mentioned, QFT operates in Minkowski space, on a relativistic footing. Knowing about Lorentz transformations, covariance and electromagnetism on a relativistic footing is essential. Being able to solve every convoluted SR problem under the Sun is not a must, but is nonetheless a good measure of one’s SR background.
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Tensor Calculus: This has the added benefit of being a key prerequisite to general relativity too. It’s not terribly difficult, and you’ll see what initially daunting objects like $i\bar\psi_a\gamma^\mu_{ab}\partial_\mu\psi_b$ are even supposed to represent. No need to initially tread into any differential geometry beyond vectors, covectors, two-forms and metrics. The notions of “tensor products” and “direct sums” can be learned from linear algebra (which, unquestionably, you need to have mastered)
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Lagrangian and Hamiltonian mechanics: Despite all the brouhaha about the Hamiltonian formalism in ordinary quantum mechanics, it becomes more economical in QFT to pivot back to the Lagrangian formalism at the opportune moment, mimicking classical field theory. You can pick up the necessary goods from any classical mechanics textbook worth its weight if you’re feeling bold, but I would consider even a basic understanding of the two and their relationship sufficient, since most sources on QFT explicitly delve into this anyway.
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Variational/Functional Calculus: These are concepts that are encountered in classical field theory, including symmetries, conservation laws, Noether’s theorem(s!), actions, etc. Functional calculus plays a large role in the path integral formulation, and functional derivatives are often subjected to rather formal manipulations elsewhere, making it necessary to have a solid background prior to starting.
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Basic Spectral theory $\subset$ Functional Analysis: This is not a conventional recommendation, however I absolutely recommend you learn this first. The basic objects in QFT are not mystical particular vibrations, but rather operator-valued distributions. I estimate that I could have reduced my initial annoyance at the canonical formalism by 50% with a better understanding of distributions, unbounded operators, rigged Hilbert spaces and the like: work merely with the position and momentum operators/spaces if you must.
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Basic Complex Analysis: Tricks like the residue theorem and analytic continuation appear time and time again, especially in integrals that arise while trying to construct manifestly Lorentz-invariant objects.
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Basic Lie Group/Algebra Representation Theory: These become essential for classifying particles, encoding how fields and states transform, and seeing how spin arises as a fundamental property upon the reconciliation of special relativity with quantum mechanics. In particular, the Lorentz group is the main player here, followed by $\mathrm{SU}(n)$. Later on, you’ll employ the latter in constructing Yang-Mills theories. Note that the relevant portion of this topic to physics doesn’t really have much overlap with a corresponding course in finite groups.
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Electrodynamics: The $\mathbf E$ and $\mathbf B$ fields are the earliest real-life examples of (classical, not quantum!) fields, and so a detailed knowledge of their systematics is necessary for understanding their quantized dynamics. Make sure you’re confident in the tensor formulation as well as in working with the gauge fields $\mathbf A$ and $\phi$.
Learn it along the way (or before, for a better understanding):
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Advanced Quantum Mechanics: It’s very helpful to get a feel for certain tools like propagators, the S-matrix, and Grassman algebra within the familiar abode of quantum mechanics before revisiting them in QFT.
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Statistical Field Theory: Several cool ideas like the path integral, correlators, symmetry breaking and Wilsonian effective field theories/RG flows manifest in accessible ways in statistical field theory, despite seeming formal and unintuitive in QFT.
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Differential Geometry: I am a personal fanboy of this topic, so I believe it’s worth learning for its own sake. These ideas become important when looking at gauge theories - interpreting objects like gauge fields in terms of fibre bundles is very elegant, as opposed to common treatments which often seem like a ragtag collection of ad hoc transformations.
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Quantization of Constrained Systems: Especially when you get to a gauge (quantum field) theory like QED, it’s helpful to know what’s going on “close to the metal” as you quantize the familiar classical theory.