Timeline for When is the "minus sign problem" in quantum simulations an obstacle?
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| Jun 29, 2011 at 14:46 | comment | added | wsc | In fact, an idea that might help clarify: Nikolay Prokofiev has this technique he calls "Diagrammatic Monte Carlo," which I saw him give a talk on once, but I was young and didn't understand. The gist of it as I took it though, is that he uses a Metropolis algorithm to walk through the space of all possible diagrams (relevant to the given problem) at all orders. The diagrams themselves are easily evaluated -- but there's still a sign problem! There is no way to sample the space of diagrams probabilistically with positive definite weights! | |
| Jun 29, 2011 at 14:37 | comment | added | wsc | @Bjorn -- that's right, we can write down diagrams and they'll always represent some complex amplitude. There are no grassmann numbers in the integrals or anything of the sort. The purpose of quantum monte carlo techniques is not to evaluate these integrals (although even at relatively low order, I suppose it's a fine technique - there is no sign problem then, but if the integrals oscillate rapidly it will still be difficult to get an accurate numerical estimate), it is to evaluate the path integral, to all orders, without any reference to diagrams. | |
| Jun 29, 2011 at 9:05 | comment | added | BjornW | @wsc: thanks, I do have that I'll have a look :) but the fermionic feynman diagrams do return a "simple" complex probability don't they? I guess I have some trouble relating the path integral to the feynman diagrams. perhaps that is for another question... | |
| Jun 29, 2011 at 1:35 | comment | added | wsc | @Bjorn -- functions of grassmann numbers return grassmann numbers, or explicitly, any function $f(\psi)=c_1+c_2\psi$ with $c_1,c_2$ complex scalars (so $e^\psi=1+\psi$ exactly). If you have a copy of Shankar's QM handy, the last chapter has perhaps the best introduction to these ridiculous beasts that I know of. | |
| Jun 28, 2011 at 21:08 | comment | added | BjornW | i too would like some comment on the fermionic sum.. doesn't the exp() function in this context always return a complex number (or a real number in case of the wick rotation) even for a fermionic lagrangian density integral? or is it required to return a higher-dimensional object when handling fermionic fields in the path integral? | |
| Jun 27, 2011 at 15:55 | comment | added | lurscher | why you can't represent grassmanian numbers effectively? | |
| Jun 27, 2011 at 10:50 | history | answered | Kostya | CC BY-SA 3.0 |