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Sharpen the statement about P vs NP |
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:<math>Z = \int D \sigma \; \rho[\sigma]</math>
where <math>D \sigma</math> represents the measure for the sum over all configurations <math>\sigma(x)</math> of the bosonic fields, weighted by
:<math>\rho[\sigma]=\det(M(\mu,\sigma))\exp(-S[\sigma])</math>
where <math>S</math> is now the action of the bosonic fields, and <math>M(\mu,\sigma)</math> is a matrix that encodes how the fermions were coupled to the bosons. The expectation value of an observable <math>A[\sigma]</math> is therefore an average over all configurations weighted by <math>\rho[\sigma]</math>
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* Meron-cluster algorithms. These achieve an exponential speed-up by decomposing the fermion world lines into clusters that contribute independently. Cluster algorithms have been developed for certain theories,<ref name='Wiese-cluster'>S. Chandrasekharan and U.-J. Wiese, "Meron-Cluster Solution of Fermion Sign Problems", [http://prl.aps.org/abstract/PRL/v83/i16/p3116_1 Phys. Rev. Lett. 83, 3116–3119 (1999)] [http://arxiv.org/abs/cond-mat/9902128 arXiv:cond-mat/9902128]</ref> but not for the Hubbard model of electrons, nor for [[Quantum chromodynamics|QCD]], the theory of quarks.
* Stochastic quantization. The sum over configurations is obtained as the equilibrium distribution of states explored by a complex [[Langevin equation]]. So far, the algorithm has been found to evade the sign problem in test models that have a sign problem but do not involve fermions.<ref>G. Aarts, "Can stochastic quantization evade the sign problem? The relativistic Bose gas at finite chemical potential", [http://prl.aps.org/abstract/PRL/v102/i13/e131601 Phys. Rev. Lett. 102, 131601 (2009)], [http://arxiv.org/abs/0810.2089 arXiv:0810.2089]</ref>
* Fixed node method. One fixes the ___location of nodes (zeros) of the multiparticle wavefunction, and uses Monte-Carlo methods to obtain an estimate of the energy of the ground state, subject to that constraint.<ref>H. J. M. van Bemmel et al, "Fixed-node quantum Monte Carlo method for lattice fermions", [http://prl.aps.org/abstract/PRL/v72/i15/p2442_1 Phys. Rev. Lett. 72, 2442–2445 (1994)]</ref>
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