Or not.... I seriously think I may have bitten off more than I can chew, but I am hoping some you guys can baby me through the hard times. So I am working exclusively from Peskin & Shroeder, and let's assume, to be generous to me, that my difficulties are at present largely notational. So to my first question: In the Hamiltonian formulation, they offer the usual formulation for the equation of motion in the real-valued field \(\phi\) as \((\partial^{\mu}\partial_{\mu} +m^2)\phi =0\), which is quite obviously the Klein-Gordon equation. No difficulties here, then. I note that they use, say, \(\text{x}\) for 3-vectors, otherwise they are assumed to be 4-vectors \(x\), as appropriate Then, considering the K-G as an harmonic oscillator, they pass this equation to what they call "Fourier space" a term I am not familiar with, but I assume it is the space of Fourier transforms. This assumption is somewhat strengthened by their statement that "{here} ... we expand the Klein-Gordon equation as \(\phi(\textrm{x},t) = \int\frac{\textrm{d}^3}{(2 \pi)^3}\,e^{i\textrm{p}\cdot\textrm{x}}\phi (\textrm{p}, t)\)". Now this looks a lot like a Fourier integral, but I am accustomed to seeing the constant \(\frac{1}{2\pi}\) pulled outside the integral. Should I care about this? Or have I got the wrong end of the proverbial? My other concern is this: earlier they have defined the momentum density conjugate to \(\phi(\textrm{x})\) as \(\pi(\textrm{x}) = \frac{\partial \mathcal{L}}{\partial \dot{\phi}(\textrm{x})}\), where I assume that \(\dot{\phi}\) is the time derivative and quite obviously \(\mathcal{L}\) is Lagrangian density. But is this not a "different pi" from what appears in the (alleged) Fourier integral? Or has it to do with some subtlety about harmonic oscillators I am unaware of? Or am I being incredibly thick? Please help. As I say, I think I may be being over-ambitious here, but with help I MAY make it to chapter 3!!
You're quite correct to point out that this integral is indeed the Fourier transform of the field \(\phi\). The fact that the factors of \(\pi\) appears inside the integral is not significant and is \(\pi\) the number, not the conjugate momentum. When they mean the conjugate momentum they will always write it \(\pi(x)\) to avoid confusion. Yep, you're right about this. \(\dot{\phi}\) is nothing more than \(\frac{\partial \phi}{\partial t}\) You'll find it all easy once you get used to the physicist idiosyncrasies. Please Register or Log in to view the hidden image!
Are these honestly genuine questions, or did you just feel a desperate need to post something? I mean, come on, none of the language is ambiguous! For instance, they say "Fourier Space" and then write down the Fourier transform of some function - did you really need help joining the dots? Additionally, using a dot to denote a time derivative is completely standard, going back to Newton, and even if this wasn't the case the notation is clearly defined (page 16, line 16)!
The idea is that since the Klein-Gordon equation is linear, it's convenient to solve it in terms of Fourier transforms. They might also call it "transforming to momentum space" because in quantum mechanics, a particle's position and momentum wavefunctions are Fourier transform conjugates. Note it's easy to get confused over whether the Fourier transform is carried out just over the spatial components or whether you also transform the time component. When Peskin & Schroeder start out setting everything up, they're initially just transforming the spatial components of the position wavefunction \(\phi(x)\), and you can also tell both from the way they use different notations for 3-vector and 4-vector dot products, as well as the fact you have \(d^3x\) in the integral rather than \(d^4x\). As to your last questions about \(\pi\) and \(\pi(x)\): In the first case, it obviously doesn't matter whether the constant goes inside or outside the integral, so it's a matter of taste on the part of the authors (as is their convention of writing the integrand after the variables being integrated rather than before). In the second case, yes, \(\pi(x)\) is a function and not the same thing as pi the geometric constant. Personal recommendation: if you want a nice ground-up approach to the subject, with all the math cleanly developed and explained, I would recommend Mark Sredniecki's text- I've worked through sections of the free draft version online and found things made much more sense than what was presented in Peskin & Schroeder. From what I've heard, the P&S book is written for a culture at Stanford where they want students to have a functional knowledge of how to apply quantum field theory before they start worrying about the little details of why things work out certain ways or how people historically figured these things out in the first place.
prom and CptBork: I thank you both for your patient responses. I may well have other questions!! I hope you do not feel, as Guest clearly does, that I have wilfully wasted your time.
Yes, Guest doesn't like you very much does he/she? Different being on the other end of the stick, old chap.
If you are going to work from Peskin & Schroeder, make sure to read the very first few pages (prependix?), as they describe/explain a lot of the conventions and choices of notation used throughout the rest of the book. And again, if that's your textbook of choice, don't spend too much time trying to work backwards and figure out where this stuff came from, that's where Srednicki does a much better job.
I'll be dipping my hands into my pockets in the end of the month for three new textbooks. I want something, akin to what quarkhead really wants - a book which explains their notations as well as presents them well. I hear Zee's book is good. Wanted it a while now. But do you have any other considerations for me?
Zee's book, like other Quantum Field Theory books, has certain prerequisites. You have demonstrated, emphatically, that you lack such prerequisites. A genuine recommendation would be to get hold of some high school mathematics and physics books, then go from there. But hey, if you want to waste your money, then get Zee's book. You'll find it much easier to read if you buy the hardback.
Townsend: A Modern Approach to Quantum Mechanics. That's a good place to start, assuming you already have a decent background in linear algebra and differential equations. I'm not recommending anything more advanced like a quantum field theory book, until I'm convinced you would actually understand it (aside from Srednicki, because it's free).
That actually sounds very good. I've some some linear algebra, so that sounds right down my alley. Thanks.
Good to hear, but you need to have some patience. When I was in high school, they simply called the subject "calculus" as if the whole thing could be taught in a year. As I later discovered, we pretty much spent that whole year covering but a small portion of the work Newton mostly discovered on his own (or with help from a small handful of colleagues) at the same age. Calculus has been in development for hundreds of years and is still in development today. After differential calculus you'll need to learn integral calculus, then multivariable calculus, then vector calculus, then complex variable calculus... meanwhile around the time you start the vector calculus, you'll need to begin learning about differential equations and abstract linear algebra (especially the concept of an inner product space), and as a prereq for these last two subjects, you'd already need to have taken a more basic course in linear algebra where concepts like matrix diagonalization and vector spaces are first introduced. Long story short, if you've just started differential calculus, you'll almost certainly need a couple more years of intense study to learn all the necessary pieces, and that's just on the math side. Don't expect quantum physics to make any sense either if you can't solve classical problems like estimating the precession rate of a spinning top or solving the differential equation for a spring. You'll also need to have a decent background in electricity and magnetism- I already knew how to derive Maxwell's equations before I was even finished with my first course in QM. There's a damn good reason they don't start teaching the theory of quantum fields 'til grad school. The prerequisites I've described are just the ones you need for ordinary undergraduate quantum mechanics, never mind when quantum fields start coming into play...
I feel kind of cheated, because we were never taught calculus at school, so in that sense, I feel a bit behind. Thankfully my algebra is ok, and we were taught that. I am on the side, trying to learn complex algebra. It's been an adventure so far. One I am enjoying. I know about many of these concepts, and I've spent some time learning about inner product spaces and the scalar product, cauchy sequences, Hilbert spaces, vector spaces. I am under the impression what I have learned though, only skims the surface. So again, it's a big learning curve. I am doing a lot of work on the electromagnetism side, on my spare time. I am trying to find all types of information I can work with. Sure, I'm not a wizz-kid yet at that side of things, but I find it all very interesting. Also, it yields some beautiful mathematics, as I am sure you will agree Please Register or Log in to view the hidden image!
http://sciforums.com/showpost.php?p=2624704&postcount=28 Cheated here seems to be misused, unless you meant that you cheated yourself. Also, linear algebra is not high school algebra, but the algebra of vectors and matrices, including concepts like determinant, trace, eigenvectors, and eigenvalues. A generalization of trace and matrix multiplication is the contraction of tensors, and thus if you knew linear algebra, and even a gloss of tensor contraction from Wikipedia, you ought to be able to respond to this: http://sciforums.com/showpost.php?p=2623011&postcount=2 http://sciforums.com/showpost.php?p=2623592&postcount=26
It's hopeless. Look, while I obviously have no proprietary rights over a thread of mine own creation, my queries were well-motivated (in spite of what one member here suspects), and I do feel a certain right to be slightly aggrieved at the turn this thread has taken. Likewise virtually all other threads here. Why is that pretty much all threads here descends to slanging matches? I really don't get it. What has this sub-forum become? Are we not supposed to be an "intelligent community", as proudly displayed on our banner? Sorry, but I see very little evidence of so-called intelligent discussion here at present, only bickering
Amen to that. I have brought this up more than once. Is degenerative for threads. I could argue back, but I'd like to respect your thread.
I sort of feel cheated too because in high school they sometimes pretended we were learning something substantial. All the math and physics I learned in high school, you could wake me up at 4 in the morning and shoot me full of heroin and I'd still be able to do it on the fly, heck I'd be able to derive and prove every single piece of it off the top of my head and pretty much write a standard textbook on it from scratch. They rarely gave us proofs, and when we would see a "proof" it often involved grievous errors and mistakes (i.e. you can't prove a general theorem in geometry just by drawing a finite number of cases, even if you could draw and measure those cases with impossibly perfect precision). Our school system was a one-size-fits-all kind of deal. For each grade they had three different levels of advancement, so they didn't have me trying to learn math with people intent on becoming lifelong junkies, but even the most advanced classes would have put Gauss to sleep when he was 8. Most proofs and concepts I had to learn on my own and on my own time (usually I'd do that while other people wasted their time memorizing formulas instead of learning how to deduce and apply those formulas on the fly). I also bought a couple of books and taught myself basic calculus a year ahead of schedule. Even then, with all the extra knowledge I had piled on, I knew math like a 10 year old knows Shakespeare. High schools teach you how to learn, that's their real purpose (aside from cranking out a generic labour force), so you can pick up more advanced knowledge once you get older and specialize. What they call "algebra" in high school is a very different creature from what "algebra" refers to at the university level. Go check out an abstract algebra textbook from a university library or on Google books, and you'll quickly understand what I mean- it would be pretty pointless to fill a book with 1000 pages of quadratic equations, wouldn't it? I think your impression reflects the reality quite accurately. Seeing you speak of "learning about" Hilbert spaces while you're still working on vector spaces and inner product spaces kind of made my eyes bulge out of my head. It would be like teaching a kid about encryption algorithms before he's learned what a prime number is. That's why people tend to get snide with you here- you want to jump into advanced topics without spending the years needed to digest the concepts on which these topics are built. It's not even about the time spent, or paying your dues to a bunch of scientists before they'll listen; it's a simple matter of needing to learn things in a certain order, just like you don't put the roof on a house before you've laid the foundations. Let me be very clear about this, the foundations in this case are enormous, and you won't be learning them in less than a year unless you can read and think as quickly as Superman. It does yield some beautiful mathematics, and I strongly advise that you learn those mathematics first before jumping down the rabbit hole. These aren't the kinds of things you'll find in the appendix of some intro calculus book, these are detailed subjects worthy of having entire courses dedicated solely to them. Electrostatics before electromagnetism. Coulomb's law before Gauss's, etc. etc.
