I'm looking for an article that sets up the framework described by Feynman in Space-Time Approach to Non-Relativistic Quantum Mechanics, but in Special Relativity.
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asked Jan 21, 2011 at 19:16
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5 Answers
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Dear mtrencseni, the special relativistic counterpart of the 1948 Feynman paper you mentioned is the 1949 Feynman paper
http://web.ihep.su/dbserv/compas/src/feynman49b/eng.pdf
called "Spacetime Approach to Quantum Electrodynamics". Note that I used the same Russian server haha. Well, more precisely, Quantum Electrodynamics is one important example of a quantum field theory but the general methods in the Feynman paper were only updated in their "technical details" when people needed to get the similar description for any quantum field theory.
If you were expecting that the special relativistic version of the 1948 paper would still be essentially the same thing, with some $p^2/2m$ replaced by $mc^2/\sqrt{1-v^2/c^2}$, you must feel disappointed. But in fact, it is a reason for a huge happiness.
The fact is that when special relativity is added to the quantum mechanical framework, one immediately encounters many effects that force us to use a fundamentally different classical starting point - field theory instead of quantum mechanics. Why?
Well, if you work with quantum mechanics - whether in an operator framework or in the path-integral approach - it doesn't respect the Lorentz symmetry. To respect the Lorentz symmetry, you need to switch from the non-relativistic one-particle Schrödinger equation to something like the Klein-Gordon equation or the Dirac equation. Both of them have a path-integral description as well although it is a bit subtle.
Take the latter - the Dirac equation - because it's relevant for the same electron that used to be described by the non-relativistic Schrödinger equation. One can show that because of relativity, the equation inevitably predicts solutions with negative energy. While $E=p^2/2m$ only has non-negative values of energy, the condition $E^2-p^2 c^2-m^2 c^4$ has both positive and negative values of $E$ as solutions. You can't avoid it - the squaring of $E$ is a fundamental feature of special relativity.
And indeed, the Dirac equation may be showed to have negative-energy solutions as well. If particles could have arbitrarily low energies, there would be an instability. You could get any energy from an electron, while the electron would be falling to arbitrarily low, negative energy levels.
Nature avoids it because it tries to find the lowest-energy state, dissipate all the excess energy, and call the lowest-energy state "the vacuum" or "the ground state". That's the only way to guarantee that it will be stable. So Nature actually fills all these negative-energy states - it has no choice. There is this "Dirac sea" of negative-energy electrons everywhere. By the Pauli exclusion principle, there can be at most one electron in each state. So the vacuum has 0 electrons in positive energy states and 1 electron in each negative energy state.
If there is a hole missing in this sea of negative-energy electron states, it will look like minus one electron with negative energy, i.e. it will be a positively charged particle, the positron, with a positive energy and positive charge. That's how Dirac predicted antimatter that was abruptly found. He got his Nobel prize for that.
Moreover, in relativity, you will find out that the pair creation of particles is inevitable. The number of particles can't be conserved. In some sense, it's because particles can move back and forth in spacetime - and the particle moving backwards in time is an antiparticle.
For all these reasons, you need to study quantum field theory if you want to combine special relativity with quantum mechanics. By quantizing a classical field, you get a system that looks like a system of particles once again. The number of particles coming from the quantum fields turns out to be integer (in the non-interacting limit) because of the same reason why the energy of a quantum harmonic oscillator is equally spaced. You will be able to recover the mechanics in the non-relativistic limit. But you won't be able to get rid of the phenomena that are implied by the fields - such as the pair creation and pair annihilation. They're real and important.
Relativistic non-field quantum mechanics is kind of inconsistent - or at least, it's not a viable approach to describe the real world. So you will only spend a very limited time with this concept - and you should eventually move to quantum field theory. This is true regardless of the formalism - Schrödinger's equation for the wave function, Heisenberg's equations for the operators, or Feynman's path integral approach. What I say about the fields is a physical insight and physical insights are independent of conventions and the choice of the mathematical machinery.
Best wishes Lubos
answered Jan 21, 2011 at 19:47
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$\begingroup$ I'm not sure the relativistic non-field QM is inconsistent, it just clashes with the Born interpretation (of course it can't be completely accurate since at high energies the number of particles is not constant). QFT has serious problems, so some of this is the pot calling the kettle black, eh? Relativistic particle QM agrees with experiment at low energies and clashes some with the usual framework of QM we all got used to....QFT agrees with experiment better but has foundational problems. I think it is a myth that QFT has solved the problems with Rel. QM so now we can just forget about it. $\endgroup$ Jan 18, 2012 at 16:03
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Feynman published another approach to the path inegral method applied to the single particle Klein-Gordon equation in Appendix A of: Feynman, R. P., "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction", Physical Review 80, 3 (1950), pp. 440--457. In this appendix he replaces the Klein-Gordon equation with a Schrodinger-like equation with a fifth "time" variable. This trick was first utilized by Stueckelberg and also quite early by Fock. It allows for a positive definite probability density, and a localizable relativistic Schrodinger position operator. It puts time and position on the same footing and allows a manifestly covariant description. Many other researchers have studied this type of equation more recently. In some modern variations of this type of theory the mass can go off-shell, even classically.
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What is described above is something called the "world-line approach" to quantum field theory, which is similar in spirit to the non-relativistic "sum over particle paths" picture of the Feynman/Hibbs book. See for instance
Christian Schubert,
"Perturbative quantum field theory in the string inspired formalism",
Phys.Rept.355 (2001) 73-234,
hep-th/0101036.
As Luboš already pointed out, you do need to put in particle creation/annihilation more or less by hand, but the method can nevertheless be a useful computational alternative to standard quantum field theory methods (especially when dealing with non-trivial vacua, e.g. in the background of electromagnetic fields).
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The method introduced here by Feynman is the Path Integral approach to quantum mechanics. When trying to include special relativity, you also need to move from quantum mechanics to quantum field theory. The path integral approach is then also referred to as Functional Integration.
There are many, many books on the subject since it's now considered a standard tool in quantum theory.
Some references are: A book by the man himself: Feynman, Hibbs and Steiber
The tome on path integrals is:
Some books treating Functional Integration and QFT are:
...and pretty much every other modern day QFT book.
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For the full framework, as MisterO explained, any modern textbook on QFT will do (I personally like Tony Zee's book (Quantum Field Theory In A Nutshell) most.
At the end of the paper that you refer to, Feynman remarks "There are other ways of obtaining the Dirac equation which offer some promise of giving a clearer physical interpretation to that important and beautiful equation." I think he refers here to his checkerboard model that he never published (although years later he included it in the book he co-authored with Hibbs. So, as a stepping stone to the full framework, you might want to investigate this Feynman Checkerboard model (there is a lot of material on this toy model on the Internet). And of course the very accessible book QED by Feynman (which, however, is really directed to a lay audience and does not explore the relativistic aspects) can also be helpful to grok the basics of QFT.
