Hmmm… apparently making QM play nice with Special Relativity isn’t quite as simple as using the Dirac equation instead of the Schrodinger equation, because the Dirac equation has negative energy solutions, and making it impossible for electrons to “decay” into these negative energy states requires kludges.
Quantized wave function solves the negative energy problem, at the expense of introducing a bunch of infinities, some of which are easier to work around (renormalize) than others. For example, there is no way to usefully quantize gravitational field.
This isn’t quite true. What solves the problem isn’t quantizing wave functions, its insisting that positive energy propagate forward in time- i.e. picking the Feynman propagator (instead of the retarded or advanced propagator, etc) that solves the problem. You still have to make a division between the positive energy and negative energy pole in the propagator (unfortunately, all observers can’t agree on which states have positive and what states have negative energy, which is the basis of the Unruh effect- two observers accelerating relative two each other cannot agree on particle number).
Also, its a misconception that you can’t simply quantize the gravitational field. If you treat GR as an effective theory you can make calculations of arbitrary accuracy with a finite number of measured parameters, with just canonical quantization. The standard model is ALSO not a renormalizable field theory (not since the addition of neutrino masses). Weinberg has recently tried to make the argument that maybe GR + canonical quantization (i.e. gravity is asymptotically safe)
Thanks for the corrections, my area is mostly classical GR, not Standard Model physics. And a good point on the Unruh effect. As for quantizing GR, note the “useful” disclaimer. I am deeply suspicious of any technique that treats GR as an effective field theory on some background spacetime, as it throws away the whole reason why GR is unlike any other field theory. Weinberg is especially prone to to doing that, so, while I respect anything he does in HEP, I don’t put much stock into his GR-related efforts. If anything, I expect the progress to come from the entropic gravity crowd, with nothing to quantize.
When I worked in physics I did perturbative QCD stuff in graduate school and then effective theories for medium energy scattering, and finally axiomatic quantum field theories as a postdoc before I left physics for a field with actual employment opportunities (statistics/big data stuff).
But why shouldn’t GR be treated as just another field theory? It certainly has the structure of a field theory. Feynman and then Weinberg managed to show that GR is THE self-consistent, massless spin-2 field theory- so to that extent it IS just another field theory.
Treating GR as an effective theory works. I doubt that the theory is asymptotically safe, but for an effective theory, who cares? Why should we treat the matter part of the action any differently than we treat the spacetime piece of the action?
But why shouldn’t GR be treated as just another field theory? It certainly has the structure of a field theory.
That’s a separate discussion, but let me just note that the action would have to be summed not just over all paths (in which spacetime?), but also over all possible (and maybe impossible) topologies, as well.
::does some reading on Wikipedia::
Hmmm… apparently making QM play nice with Special Relativity isn’t quite as simple as using the Dirac equation instead of the Schrodinger equation, because the Dirac equation has negative energy solutions, and making it impossible for electrons to “decay” into these negative energy states requires kludges.
(Why is it that, the more I learn about QM, the more it seems like one kludge after another?)
Quantized wave function solves the negative energy problem, at the expense of introducing a bunch of infinities, some of which are easier to work around (renormalize) than others. For example, there is no way to usefully quantize gravitational field.
This isn’t quite true. What solves the problem isn’t quantizing wave functions, its insisting that positive energy propagate forward in time- i.e. picking the Feynman propagator (instead of the retarded or advanced propagator, etc) that solves the problem. You still have to make a division between the positive energy and negative energy pole in the propagator (unfortunately, all observers can’t agree on which states have positive and what states have negative energy, which is the basis of the Unruh effect- two observers accelerating relative two each other cannot agree on particle number).
Also, its a misconception that you can’t simply quantize the gravitational field. If you treat GR as an effective theory you can make calculations of arbitrary accuracy with a finite number of measured parameters, with just canonical quantization. The standard model is ALSO not a renormalizable field theory (not since the addition of neutrino masses). Weinberg has recently tried to make the argument that maybe GR + canonical quantization (i.e. gravity is asymptotically safe)
Thanks for the corrections, my area is mostly classical GR, not Standard Model physics. And a good point on the Unruh effect. As for quantizing GR, note the “useful” disclaimer. I am deeply suspicious of any technique that treats GR as an effective field theory on some background spacetime, as it throws away the whole reason why GR is unlike any other field theory. Weinberg is especially prone to to doing that, so, while I respect anything he does in HEP, I don’t put much stock into his GR-related efforts. If anything, I expect the progress to come from the entropic gravity crowd, with nothing to quantize.
When I worked in physics I did perturbative QCD stuff in graduate school and then effective theories for medium energy scattering, and finally axiomatic quantum field theories as a postdoc before I left physics for a field with actual employment opportunities (statistics/big data stuff).
But why shouldn’t GR be treated as just another field theory? It certainly has the structure of a field theory. Feynman and then Weinberg managed to show that GR is THE self-consistent, massless spin-2 field theory- so to that extent it IS just another field theory.
Treating GR as an effective theory works. I doubt that the theory is asymptotically safe, but for an effective theory, who cares? Why should we treat the matter part of the action any differently than we treat the spacetime piece of the action?
That’s a separate discussion, but let me just note that the action would have to be summed not just over all paths (in which spacetime?), but also over all possible (and maybe impossible) topologies, as well.