What background one needs to have to study QFT?

[kostas230]

What background one needs to have to study QFT? I have a good background in calculus, linear algebra, PDEs, and quantum mechanics (at Shankar's level). Are these enough?

 

[ZeroPivot]

you need more math.

 

[Sentin3l]

Generally you need to know physics-wise:

Classical Mechanics (especially Lagrangians)
Electromagnetism
Statistical Mechanics (Can get away without knowing, but it helps on occasion)
Quantum Mechanics (Need to know VERY well)

As well as some others, but you can make due with those four, although it will not be easy. Math-wise however, I would say at minimum (and I stress AT MINIMUM):

Calc
ODE's
Linear Algebra
PDE's (Very important)
Group Theory
Basic topology

 

[WannabeNewton]

(Applied or pure) complex analysis; basic special relativity.

 

[DimReg]

The more prepared you are the better, but I think it would be possible to learn QFT on your background. Keep in mind that QFT builds off of a lot of other fields of physics, so a strong knowledge of electricity and magnetism (if you've never heard of a gauge transformation, your lacking here) and classical mechanics (you should now what the lagrangian and hamiltonian are, and be familiar with noether's theorem) are also good. It also depends on how ready you are to accept new information that you might not understand entirely. You'll have to use some techniques of complex analysis and results of lie algebras, but you won't have time to fully understand those subjects during the class. If that's something you can't handle, then you should learn those subjects first.

If you are self studying, you are probably ok to start. Just make sure you use an accessible book to learn from (I hear good things about Mandl and Shaw). If you are considering taking a class, you should talk to the professor, since only he knows what he expects from his students.

 

[kostas230]

I know pretty well classical mechanics, relativity and electromagnetism to Jackson's level. I am a bit familiar with basic metric topology. We didn't have a statistical mechanics course up 'till now. I don't know a good book in Group Theory, so you could recommend me a book to study from.

 

[WannabeNewton]

You don't need a full fledged book on group theory in the slightest to start QFT. A typical introductory group theory text (e.g. Dummit and Foote) will focus mostly on finite group theory whereas what you want is Lie groups/Lie algebras (perhaps without too much differential topology if you've never seen differential topology before). On that note, check out the following text: https://www.amazon.com/dp/9971966573/?tag=pfamazon01-20

Also, topology is really only important if you want to learn functional analysis and delve into algebraic QFT. It isn't necessary at all for an introduction.

 

[ZombieFeynman]

My 2 cents:

You can certainly get by without pure complex analysis or without any group theory. You definitely don't need to know Lie Algebra. You needn't have even heard of topology.

I think if you understand E&M and Quantum Mechanics at the level you claim (you should be comfortable with the path integral formulation of quantum mechanics too) you can learn a lot of QFT if you choose the right level book. I would look at A. Zee's book or Klauber's book.

 

[kostas230]

I'm learning Real Analysis from Rudin, and Complex Analysis from Lang's book. A colleague of mine said I should open a book on differential geometry.

 

[WannabeNewton]

Seriously you won't need real analysis at the level of papa or baby Rudin or complex analysis at the level of Lang for intro QFT; you won't even need diff geo for intro QFT. Diff geo is much more important if you want to learn GR, not QFT. There's nothing wrong with learning all that math but it won't really help you.

 

[verty]

Kostas, I think just attack QFT now and if you get stuck, then read more on what you are stuck on. Otherwise it can take too long looking at every little thing.

 

[ZeroPivot]

Vector Calculus, Complex Analysis, Optimization. good to know.

 

[radium]

I'm taking QFT this semester. I have taken two semesters of graduate quantum mechanics, graduate E&M, and a graduate math methods course. Math wise, I took a course in topology and group theory as well. I think the most important math topics you should try to get a background in (as a physicist, you don't need to be able to rigorously prove everything) are Lie groups, ODEs, PDEs, Greens functions(!), Fourier analysis, some complex analysis (especially contour integrals), assymptotic methods, and maybe a bit of topology and general group theory. A good book to get an overview of many of these topics is Stone and Goldbart Mathematics for physicists.

 

[aworrystudent]

You may need contour integral, Lagrange/Hamiltonian formulation of mechanics, classical electrodynamics especially specially relativity and the covariant formulation of Maxwell eqns, and a bit Lie group/algebra

Unless you are really interested in mathematician's QFT, e.g. Quantum Field Theory by Gerald B. Folland, real analysis should not be necessary...

Have a look at Mark Srednicki's quantum field theory, check what kind of background is needed

 

[Arsenic&Lace]

Notice how neither graduate nor undergraduate physics departments require students to study real analysis, differential geometry, abstract algebra, or topology (from what I know). Taking any of these courses is not a horrible idea, but it is important to recognize that the math department has a very different culture and purpose for the study of these subjects than physics, and you can probably get by simply learning what you need on your own rather than going through any of these courses (none of which are even remotely trivial).

The only major missing piece in your first post list was Lagrangian mechanics, but the basic formalism can be understood quickly by a good student.

 

[ZombieFeynman]

Notice how neither graduate nor undergraduate physics departments require students to study real analysis, differential geometry, abstract algebra, or topology (from what I know). Taking any of these courses is not a horrible idea, but it is important to recognize that the math department has a very different culture and purpose for the study of these subjects than physics, and you can probably get by simply learning what you need on your own rather than going through any of these courses (none of which are even remotely trivial).

As a physics graduate student doing fairly mathematical Condensed Matter Theory, I WHOLLY agree with the above statements.

I double majored in mathematics and physics as an undergraduate. I took plenty of pure math as well as applied. I am certain nearly all of my time doing pure mathematics would have been better spent learning additional computer science and taking more courses in experimental methods of physics.

 

[Arsenic&Lace]

I was doing the exact same thing, and have all but stopped taking pure math courses.

I often suspect the math department is a bit of a rogue department. Much of what they learn in the purest of pure math courses never seems to percolate down to other fields; even the most esoteric physics (save for string theory) is at least useful to experimenters. L'Hospitals rule was invented long before it was proven to work, which raises doubts about whether the machinery of say, analysis, is merely a titanic linguistic rigamarole or truly a useful invention. Indeed, I heard from one theorist that he found the ways of thinking about series he learned in analysis to be damaging to his understanding of field theory, because you simply cannot take the mathematician's point of view in advanced physics. If you want to find out if a person is a mathematician at heart, show him how renormalization works, and see if he or she has a seizure. It is fascinating how much useful mathematics was invented by applied mathematicians and physicists; the fear, loathing, and disdain for application in the "pure" math department is somewhat reprehensible.

/rant

Also, in my undergraduate research I was dragged kicking and screaming into the world of programming, and discovered that I loved it; I think certain CS courses, especially practicums and low level "how to program" courses could be incredibly handy.

 

[aworrystudent]

I heard from professors that for most interacting field theories (except phi^4 in 2+1 dimension), it has not been shown that they are consistent with the Wightman axioms. And proven realistic 3+1 dimensional interacting field theories will not lead to new physics (SUSY, string?...) anyway

 

[kostas230]

Notice how neither graduate nor undergraduate physics departments require students to study real analysis, differential geometry, abstract algebra, or topology (from what I know). Taking any of these courses is not a horrible idea, but it is important to recognize that the math department has a very different culture and purpose for the study of these subjects than physics, and you can probably get by simply learning what you need on your own rather than going through any of these courses (none of which are even remotely trivial).

The only major missing piece in your first post list was Lagrangian mechanics, but the basic formalism can be understood quickly by a good student.

Well, I am self-studying the material; I'm not taking any courses...

Also, I didn't mention Lagrangian and Hamiltonian mechanics because I thought it was obvious since one cannot go to a good book in quantum mechanics without knowing the Hamiltonian and Lagrangian formalisms.

 

[Arsenic&Lace]

Ah ok, I encountered field theory in research before I'd finished the full quantum mechanics sequence, and so I did not realize that the Lagrangian formalism was already covered.