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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fermion doubling ： ウィキペディア英語版 Fermion doubling The fermion doubling problem is a problem that is encountered when naively trying to put fermionic fields on a lattice. It consists in the appearance of spurious states, such that one ends up having 2''d'' fermionic particles (with ''d'' the number of discretized dimensions) for each original fermion. In order to solve this problem, several strategies are in use, such as Wilson fermions and staggered fermions. ==Mathematical overview== The action of a free Dirac fermion in ''d'' dimensions,〔As the lattice discretization is always defined in Euclidean spacetime, we will suppose the suitable Wick rotation has been performed. Herefore, no difference will be made between covariant and cotravariant indices.〕 of mass ''m'', and in the continuum (i.e. without discretization) is commonly given as :$S\; =\; \backslash int\; d^dx\; \backslash ;\; \backslash bar\backslash psi\; (\backslash partial\backslash !\backslash !\backslash !/\; +\; m)\; \backslash psi\; \backslash ;.$ Here, the Feynman slash notation was used to write :$\backslash gamma\_\backslash mu\; a^\; =\; a\backslash !\backslash !\backslash !/$ where γμ are the gamma matrices. When this action is discretized on a cubic lattice, the fermion field ψ(''x'') is replaced with a discretized version ψ''x'', where ''x'' now denotes the lattice site. The derivative is replaced by the finite difference. The resulting action is now: :$S\; =\; a^d\; \backslash sum\_\; \backslash frac1\; (\backslash bar\backslash psi\_x\backslash gamma\_\backslash mu\backslash psi\_\; -\; \backslash bar\backslash psi\_\backslash gamma\_\backslash mu\backslash psi\_x)\; +\; a^d\; \backslash sum\_x\; m\backslash bar\backslash psi\_x\backslash psi\_x\; \backslash ;,$ where ''a'' is the lattice spacing and $\backslash hat\backslash mu$ is the vector of length ''a'' in the μ direction. When one computes the inverse fermion propagator in momentum space, one readily finds:〔 :$S^(p)\; =\; m\; +\; \backslash frac\; ia\; \backslash sum\_\backslash mu\; \backslash gamma\_\backslash mu\; \backslash sin(p\_\backslash mu\; a)\; \backslash ;.$ Due to the finite lattice spacing the momenta ''p''μ have to be inside the Brillouin zone, which is typically taken to be the interval (). When simply taking the limit ''a'' → 0 in the above inverse propagator, one recovers the correct continuum result. However, when instead expanding this expression around a value of ''p''μ where one or more of the components are at the corners of the Brillouin zone (i.e. equal to /''a''), one finds the same continuum form again, although the sign in front of the gamma matrix can change.〔Due to these changes in sign, the chiral anomaly cancels exactly, which is not in agreement with phenomenology.〕 This means that, when one of the components of the momentum is near /''a'', the discretized fermion field will again behave like a continuum fermion. This can happen with all ''d'' components of the momentum, leading to —with the original fermion with momentum near the origin included— 2''d'' different "tastes" (in analogy to flavor).〔As the action of scalars contains second derivatives, a similar procedure in this case would lead to a quadratic inverse propagator, which does not have these doublers.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fermion doubling」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.