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:
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.
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.
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)
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.
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.
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.
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.
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.
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):
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.
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.
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.
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.
Electrodynamics isn’t appreciated enough! Lots of people find it stuffed with tedious computations and tricks, but the fact is that it remains one of the most practical, wide-ranging subjects and a key cornerstone of one’s physics learning journey.
A big disclaimer that I should add is that my viewpoint is skewed towards high energy physics, so I am not aware of the details of electrodynamical applications to say plasma physics. Sorry.
Also before I commence, I just want to put it out there: Griffiths’ Electrodynamics book is a national treasure.
Definite prerequisites
The mathematical prerequisites are, on the whole, very similar to those for quantum mechanics - with the sole exception of linear algebra. So an understanding of multivariable calculus and differential equations is necessary to commence. Knowledge of basic PDE and Fourier series is also helpful, but can be learned concurrently.
Most electrodynamics sources provide an overview of the requisite vector calculus, which is probably the largest mathematical hurdle. It’s possible to pick it up simultaneously after a brief overview of vector fields, vector operators and Gauss’ and Stokes’ theorems: don’t waste your time solving too many abstract problems beforehand, since you’ll have to apply them anyway while doing E&M problems. It’s good to be able to visualise all of these mathematical notions clearly (they’re really quite intuitive once you get the hang of it)
Wave optics: Essentially just the high school coverage of light and waves, this goes without saying. A qualitative understanding of the key optical phenomena should be coupled with knowledge of non-rigorous derivations of the key formulae. Don’t worry, the motivation behind all the seemingly ad hoc rules that pervade high school physics will become clear in due course.
Newtonian mechanics: Duh.
High school electromagnetism: now this varies hugely from one education system to another, I’ve seen some high school physics syllabi extend right until Maxwell’s equations, while some go up only to electric force, potentials and some basic electromagnetic relations. My advice: make sure you at least know the latter. In fact, you can specialise to electric fields and potentials only, the high school/pop-sci explanations of magnetism are generally not great, and so you can rely on a solid electrodynamics textbook to teach you this.
Learn it along the way (or before, for a better understanding):
Try performing some computational electrodynamics! There are lots of finite element methods, discretisation schemes and numerical methods to be played with here, and these all serve as a robust introduction to more advanced simulations, for example in quantum chemistry
Quantum mechanics: There are a few historical routes that you can use to enter the quantum domain! First is spontaneous emission á la Einstein, and the other (more famous one) is Planck’s radiation curve - see if you can find some more conflicts between the two.
Special relativity: A lot of E&M textbooks are bundled together with a chapter or two on special relativity anyway, but it’s fun to start exploring and thinking about early on!
Classical field theory: You get to put to the test all of your field theoretic knowledge! Sadly, most of the content is not very helpful for practical electrodynamics, but it assumes utmost importance if you want to probe the theoretical features of Maxwell’s equations, and electrodynamics as a gauge theory.
Physics-wise I’ve enjoyed reading about instantons, large N gauge theories, topological string theory (I definitely recomomend the review by Vafa, it delves into toric geometry) and some superstring theory (RNS formalism, GSO projection, black hole microstates). Some other cool stuff is viewing the string B-field as a cohomology class, and probing the geometrical structure of CFTs (the sort of thing they teach mathematicians about QFT and CFT). That said, I feel like my current interests have slightly shifted to be more gravity-oriented. As such, topics like the holographic principle, the information paradox, the S-matrix in a gravitational theory, AdS black holes (and general black hole thermodynamics) and proto-quantum gravity have really interested me. I am also looking at real algebraic topology - stuff like CW approximations and Postnikov towers (I really want to see what the $\mathrm{Fivebrane}$ group is all about), and I also picked up Nakahara again because of it’s nice review of anomalies and $\mathrm{GL(4,\mathrm R)}$ gauge theory.
I’ve also been exploring some interesting deep learning models in depth, some of which echo my interests of 1 whole year (!) ago, including Graph CNNs for drug design, LSTMs and VAEs for music generation, and R-CNNs for object detection and recognition. I solved some Project Euler problems too, here’s my account:
String theory (and indeed AdS/CFT) in itself is a huge vortex that one gets sucked into, so I am cautious about spending too much time there - besides, there is something entirely unsatisfying about learning something at the pinnacle of physics research as AdS/CFT from distilled lecture notes without a sufficiently strong background in strings, and indeed, an appreciation of other fields of HEP. That said, I am very fond of D-brane dynamics.
Gauge theories: both from the point of view of constrained dynamics à la Henneaux and Teitelboim as well as phenomenological aspects. The former is elegant and powerful, the latter is so far-reaching it’s ridiculous. This is supplemented bvy Tong’s TASI Lectures on Solitons which are possibly my favourite lectures of all time - wonderfully and cleanly presented, with applications in string theory. I now love D3 brane stacks and want to explore D-brane bound states (I am of course defaulting to the D-brane bible by Clifford Johnson) as well as D-brane wrapping and (choral sounds) K-theoretic classification of RR charges on D-branes.
Anomalies: lots of modern viewpoints here - the Atiyah-Singer index theorem, the Dai-Freed theorem, descent equations, Green-Schwarz anomaly cancellation. Four references in particular stood out: Dai-Freed Anomalies in Particle Physics (with spectral sequences and homology!), TASI Lectures on Anomalies, a lightning review culminating in M-theory anomalies, Anomalies in Quantum Field Theory by Serone (which had a nice review of SQM and the descent equations) and Lectures on Anomalies by Bilal (the Weinberg of sorts)
AdS3/CFT2: rather than jumping straight to 5 dimensions, I thought I would dip my toes in its (much more concrete) 3-dimensional cousin. Naturally, this led to the conformal bootstrap, minimal models and Verma modules. Something which I would really like to know about is the CS/WZW correspondence, but the only good source is currently nLab, I am yet to picture what a “circle 3-bundle” is. I am also delving into the renormalisation group properly: statements like “a QFT is just an effective field theory on an RG flow between two CFTs in the IR and the UV” are super-intriguing
Like I said - rather than learning string theory proper, I have lately preferred to learn about ideas that are perhaps better grounded in string theory, but have wide implications in ordinary super-Yang Mills, for example: Montonen-Olive duality, as well as compactifications. To anyone learning string theory: I suggest you drop Becker, Becker, Schwarz now. It’s downright impossible to learn any actual string theory properly from it, save for the big picture. It seems like it would probably be the best “summary” book as a reference to people who are already experts, but I would not recommend learning from it for the first time.
In mathematics (if you don’t count most of the previous stuff as mathematics), I’ve really just been reading up on physics-motivated algebraic geometry and algebraic topology (AT II on MIT CourseWare is very well written), index theorems for elliptical operators and the McKay correspondence, for fun.
