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Note: some chapters of Robert Klauber's excellent book are downloadable for free.
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In my intermittent progress through Robert Klauber's excellent "Student Friendly Quantum Field Theory" I'm currently working through the relativistic versions of the Schrödinger equation (prior to hitting QFT-proper).
These are indexed by spin. Schrödinger's non-relativistic equation which is the staple of introductory quantum mechanics courses, works for spin-1/2 fermions like the electron to which it is usually applied.
The most direct relativistic counterpart is Klein-Gordon, which Schrödinger considered first but couldn't make work for the hydrogen atom. This is because it actually applies to spin-0 particles (scalar bosons such as the Higgs particle).
The correct relativistic equation for the electron and other spin-1/2 fermions is named after Dirac, while vector bosons such as the photons (spin 1) have their own Proca equation.
So I was wondering if the spin-2 graviton has its own equation .. but then I recalled that quantum theory can't do gravity yet - see this.
Here's a convenient, semi-impenetrable table.
Needless to say, some comments are called for...
ReplyDelete1. Interestingly all of these relativistic equations which include mass are really equations about the expression (mc/h) which has dimensions of inverse length (it is the Compton wavelength (^-1), discovered before relativistic QM). Of course it contains all three key constants: mass, c (light), h (bar quantum) - so it is really all about that wavelength and its relativistic transformations (at different spins).
2. These equations can become more standardised in a clearer notation, such as the pure spinor form.
3. There is an intermediate form of QM which is non-relativistic, but where Psi is a (Pauli) spinor introduced to include spin in the Schrodinger Wave Equn.
4. The usual form of the spin 2 equation is a linearised form of GR equations, which looks like the others with a higher spin phi.
Thanks for the clarifications. I have some more reading to do before I can comment more constructively!
DeleteCannot wait for the book review - it is held in high regard by its Amazon reviewers. I cannot remember whether you have Zee (QFT in Nutshell). I don't know how equation centric this book is, but no doubt all will be explained soon.
ReplyDeleteI am still wondering whether to get the Least Action book.
Robert Klauber has a useful book-website, http://www.quantumfieldtheory.info/, where you may download some chapters and reassure yourself that there are *plenty* of equations. This is a hand-holding *textbook* which reminds me in style of Open University courseware (the level is far higher).
DeleteI have the Zee book but that makes too many presumptions about the background formation of the student - too big a jump from a first QM course in a self-study context.
If one is comfortable intuitively with Lagrangian Mechanics, Jennifer Coopersmith's book is a comfortable read, of interest mainly for its historical treatment. I suspect a nice-to-have rather than a must-have.
Thanks I have now downloaded and read Chp1,2,3a. I see what you mean about the Course format nature of this style.
DeleteThere is more to be said about the difference between QFT and NRQM in terms of the state vector. In that context I am intrigued by the reference to Sections 7.4.3 and 7.4.4 mentioned in Ch1, but which I cannot access.
Section 7.4 is a detailed introductory mathematical treatment of the S Operator and S Matrix. Perhaps it's time to invest in the book, if only as a refresher. The author is quite opinionated!
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