QM & Relativity: The Hardest Problems to Solve

[gonadas91]

Hi everyone,

according to some texts I ve read, there is not a clear definition for a quantum mechanical theory involving relativity. The most similar approach is that of Klein Gordon and Dirac equations, but there is not an analogy Schrödinger equation when we use relativity in QM. Can anyone tell me what are the hardest problems found to deal with both theories to coexist?

Thanks!

 

[DEvens]

Special relativity is fine with quantum mechanics. Relativistic quantum field (RQF) theory does just fine. You can write down your theory in a relativistic covariant form, and so keep all the symmetries that are required under special relativity.

General relativity is a lot harder. At a quantum field theory level the problem is one of infinities. In RQF there are infinities when you get to loops. You get rid of them using a process called renormalization.

https://en.wikipedia.org/wiki/Renormalization

The key here is, the symmetries of the system allow you to prove that there are a finite number of independent infinities. You can get rid of all of them by adjusting a finite number of parameters. In quantum electrodynamics you need to adjust the mass and charge of the electron, and set the photon mass to zero, and nothing else. That gets rid of all infinities. So two little parameters and you have the entire theory. And it produces truly stunning agreement between theory and experiment. For example:

https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment

But in general relativity this program fails. Each order of loop introduces a new infinity. That means there are arbitrarily many different parameters that must be adjusted to get rid of the infinities. A theory with arbitrarily many parameters has no predictive power and so is a problem.

There are several approaches people have chosen to taken to this. One is the "string" approach. Another is the "loop" approach. There are others. People get very excited about these approaches. So far, none has emerged as victorious with tested predictions that distinguish it from other theories.

 

[Nugatory]

Can anyone tell me what are the hardest problems found to deal with both theories to coexist?

Mark Srednicki's QFT textbook works through the hard spots pretty well, and has the added advantage that a prepublication draft is available online for free: http://web.physics.ucsb.edu/~mark/qft.html

 

[atyy]

Special relativity is fine with quantum mechanics. Relativistic quantum field (RQF) theory does just fine. You can write down your theory in a relativistic covariant form, and so keep all the symmetries that are required under special relativity.

General relativity is a lot harder. At a quantum field theory level the problem is one of infinities. In RQF there are infinities when you get to loops. You get rid of them using a process called renormalization.

https://en.wikipedia.org/wiki/Renormalization

The key here is, the symmetries of the system allow you to prove that there are a finite number of independent infinities. You can get rid of all of them by adjusting a finite number of parameters. In quantum electrodynamics you need to adjust the mass and charge of the electron, and set the photon mass to zero, and nothing else. That gets rid of all infinities. So two little parameters and you have the entire theory. And it produces truly stunning agreement between theory and experiment. For example:

https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment

But in general relativity this program fails. Each order of loop introduces a new infinity. That means there are arbitrarily many different parameters that must be adjusted to get rid of the infinities. A theory with arbitrarily many parameters has no predictive power and so is a problem.

There are several approaches people have chosen to taken to this. One is the "string" approach. Another is the "loop" approach. There are others. People get very excited about these approaches. So far, none has emerged as victorious with tested predictions that distinguish it from other theories.

This view is out of date. General relativity is as fine as QED nowadays in relativistic QFT. This was the great breakthrough of Kenneth Wilson. Basically, both are not fine, but we are fine with things that are not fine :)

 

[bhobba]

This view is out of date. General relativity is as fine as QED nowadays in relativistic QFT. This was the great breakthrough of Kenneth Wilson. Basically, both are not fine, but we are fine with things that are not fine :)

Here is a link that gives the detail:
http://arxiv.org/abs/1209.3511

To the OP - don't be turned off by the math. You will likely still get the gist.

Thanks
Bill

 

[gonadas91]

Thanks to all for the answers!