How Can I Understand QFT in My Own Way?

[jordi]

Hello, I am new in this forum. I studied physics up to the equivalent of a MSc. This included mainly QFT and GR. Then, instead of following a PhD, I went to finance. Now, after a few years, I have decided I want to go back and study QFT. Not full time, but part time (in my free time). I already know I am not intelligent enough to make important discoveries. What I want is to understand QFT in my own way.

I recall that what disturbed me in QFT (and in QM) was that I could not relate the concepts to mathematical structures. For me, it is necessary to relate concepts to mathematical concepts. But studying QFT was not done in this way: you study the Klein-Gordon, and Dirac, and QED quantization, and you see many tricks that pave the path to go fast and to the point: you can calculate. But when I studied, I could not reach to the point to relate all those concepts in mathematical form. I am sure that people more intelligent than me can do that, though.

Now I know (or at least, I have read about it), that all the stuff about operators, commutators and anticommutators have something to do with group and algebra representations. And that the Feynman-Kac theorem relates the functional analysis view (values of operators under some Hilbert space states) with the probabilistic view (eg, correlation functions). But almost always, in QFT books the functional analysis view is stressed, as opposed to the probabilistic view. I would like to develop the probabilistic view from scratch: I have my probability space, well defined, then I see the random variables, the martingales, the Radon-Nikodim theorem how is it applied, ... ie, I can study math books and then physics is only to use theorems, with the hypothesis taking some particular values.

Yes, I know that this is not a modern approach. I have read many good physicists saying that mathematizing physics to the extreme does not lead to good insights in Nature. But I do not want to have good insights in Nature. I just want to understand QFT my way.

Please, help!

 

[pellman]

It's tough going. I have been in the same boat basically. The main thing working against you is that there are no really good QFT texts. As one quantum theorist I know put it, "Most QFT texts are just authors showing off: 'See what I know!'"

I recommend

(1) reading through Messiah's Quantum Mechanics to prepare. It is probably the most thorough treatment and is available in one volume from Dover cheap.

(2) read some classical field theory

(3) work carefully through a rotation of a vector field and understand how the components in one frame are related to those in the other frame. Understand fully why the orbital angular momentum operator generates the scalar part of a rotation and why the spin operator generates the rotation of the components. (If you don't know what I mean by "generate" in this context, then you definitely need to do this.) You can't say you understand spin if you don't have a firm grasp of this. Use a three-component vector field because it is easier to visualize. Then you can generalize to spin 1/2.

(4) Get this, because you won't find it spelled out in any QFT text. The objects described by QFT theory are the same quantum states that are handled by quantum mechanics. The major difference is that particle numbers are not fixed. Mathematically, for instance, this means that the state vector describing a positron and electron has non-zero inner product with a state describing two photons. The "quantum fields" describe how these states interact and how the states of differing quantum number are related to each other. If you keep this in mind, it might help you see where the QFT texts are leading you.

Best of luck.
Todd

 

[simic4]

Hey jordi,

i totally understand where you are coming from. I myself am trying to understand qft from the ground up.

weinbergs book volume I is pretty good at this. its pretty damn hard though, i recommend trying to understand the first few chapters of it indirectly..ie: reading weinberg till you are confused then looking elsewhere and coming back to weinberg with your new insights.

for example weinberg doesn't look at the Klen Gordon equation, see it looks like a harmonic oscillator, and then postulate creation/annihilation operators.

i advise you:

1. learn the representation theory of the lorentz group really well.

this will give you insight into what a fundamental particle actually is.

you will find that wave equations (like the dirac eqn, or a massive vector field) are really just projection operators which effectively make a reducible non-unitary reprepsentation reducible + unitary.

if you understand this deeply enough you will be able to come up with your own, correct wave equations for higher spin fields.

you will be able to quantize them too.

hagen and singh worked this out in the 70s. you can find their papers on spires.

however, this is all more advanced. make sure you know peskin ch. 3 really well, and do the problems there. the first step to understanding representation theory is understanding spinors,, and then combing spinors to make higher spin fields..

2. learn and ponder haags thm , , then visit peskin ch. 4. this should give you an idea as to why its so hard to find a mathematically rigorous approach to qft: it doesn't really exist!

3. to better understand renormalization study it in statistical mechanics, theory of critical phenomena,,,this subject is called the renornalization group.. then ponder why the infinities that arise in qft and see if you can convince yourself that they are not a problem (some think it is some dont).reference for haag: Haag's Theorem and Clothed Operators, Phys. Rev. 115, 706–710 (1959)

if you cannot access the reference or any other reference you may need on spires email me, id be more than happy to send it to you! (i can axcess these things online through my university's account!)

 

[simic4]

one more thing, the spin statistics thm is a good foundational thing to know as well. i plan on learning it soon.

 

[reilly]

jordi -- Believe me, if you can handle Black-Scholes, advanced topics like martingales, Feynman as Feyman-Kac, efficient markets and all that stuff, then you have considerable more knowledge and sophistication than you need for mastering basic QFT. I'm a one-time-QFT researcher and teacher who has dabbled in finance -- and I still can get confused about selling short.

Everyone learns differently, so I'll give you my personal take. First, I'm strongly biased in favor of history and in reading the "founding fathers" prior to taking on more modern acounts. For example. if you read Dirac before Weinberg, then Weinberg will not be anywhere as difficult as without Dirac. To me this is a no-brainer.

First, review QM from Dirac -- and he discusses QFT as well. And, in spite of his abstract notation, Dirac's reasoning is often physically based. Note that Dirac pretty much formulated QED in 1927, with the help of his amazing intuition, in a form that has not really changed that much over the past 80 years. Then, in my view, read S. Schweber's QED and the Men Who Made It, which presents QFT up to the Standard Model, that is the book is all about the physics of QED, and the work of Feynman, Schwinger, Tomonaga, Oppenheimer, Weiskopf, Pauli, Dirac, Heisenberg and their colleagues. With this under your belt, Weinberg's books will be a piece of cake -- well almost. Both books I mention are physics books, not books on math applied to physics. (If you can get it, Schwinger's Dover book, Quantum Electrodynamics, is invaluable with many original papers by Dirac, Schwinger, Feynman, ...)

Walk before you run to Weinberg, Gross, Bjorken-Drell, all of which I highly recommend -- Weinberg is highly formal, advanced, Gross is very physically based, and Bjorken-Drell covers all the myriad details of basic QED and QFT.

Hope some of this is helpful.
Regards.
Reilly Atkinson