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(詳細はphysics, the Landau pole or the Moscow zero is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the physicist Lev Landau and his colleagues.〔Lev Landau, in 〕 The fact that couplings depend on the momentum (or length) scale is the central idea behind the renormalization group. Landau poles appear in theories that are not asymptotically free, such as quantum electrodynamics (QED) or theory—a scalar field with a quartic interaction—such as may describe the Higgs boson. In these theories, the renormalized coupling constant grows with energy. A Landau pole appears when the coupling becomes infinite at a finite energy scale. In a theory purporting to be complete, this could be considered a mathematical inconsistency. A possible solution is that the renormalized charge could go to zero as the cut-off is removed, meaning that the charge is completely screened by quantum fluctuations (vacuum polarization). This is a case of quantum triviality,〔 〕 which means that quantum corrections completely suppress the interactions in the absence of a cut-off. Since the Landau pole is normally identified through perturbative one-loop or two-loop calculations, it is possible that the pole is merely a sign that the perturbative approximation breaks down at strong coupling. Lattice field theory provides a means to address questions in quantum field theory beyond the realm of perturbation theory, and thus has been used to attempt to resolve this question. Numerical computations performed in this framework seems to confirm Landau's conclusion that QED charge is completely screened for an infinite cutoff. ==Brief history== According to Landau, Abrikosov, Khalatnikov,〔L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov, Dokl. Akad. Nauk SSSR 95, 497, 773, 1177 (1954).〕 the relation of the observable charge with the “bare” charge for renormalizable field theories when is given by expression : where is the mass of the particle, and is the momentum cut-off. If and then and the theory looks trivial. In fact, inverting Eq.1, so that (related to the length scale reveals an accurate value of , : As grows, the bare charge increases, to finally diverge at the renormalization point : This singularity is the Landau pole with a ''negative residue'', . In fact, however, the growth of invalidates Eqs.1,2 in the region , since these were obtained for , so that the exact reality of the Landau pole becomes doubtful. The actual behavior of the charge as a function of the momentum scale is determined by the Gell-Mann–Low equation : which gives Eqs.1,2 if it is integrated under conditions for and for , when only the term with is retained in the right hand side. The general behavior of depends on the appearance of the function . According to the standard classification,〔N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, 3rd ed. (Nauka, Moscow, 1976; Wiley, New York, 1980).〕 there are three qualitatively different cases: *(a) if has a zero at the finite value , then growth of is saturated, i.e. for ; *(b) if is non-alternating and behaves as with for large , then the growth of continues to infinity; *(c) if with for large , then is divergent at finite value and the real Landau pole arises: the theory is internally inconsistent due to indeterminacy of for . Landau and Pomeranchuk 〔L.D.Landau, I.Ya.Pomeranchuk, Dokl. Akad. Nauk SSSR 102, 489 (1955); I.Ya.Pomeranchuk, Dokl. Akad. Nauk SSSR 103, 1005 (1955).〕 tried to justify the possibility (c) in the case of QED and theory. They have noted that the growth of in Eq.1 drives the observable charge to the constant limit, which does not depend on . The same behavior can be obtained from the functional integrals, omitting the quadratic terms in the action. If neglecting the quadratic terms is valid already for , it is all the more valid for of the order or greater than unity: it gives a reason to consider Eq.1 to be valid for arbitrary . Validity of these considerations at the quantitative level is excluded by the non-quadratic form of the -function. Nevertheless, they can be correct qualitatively. Indeed, the result can be obtained from the functional integrals only for , while its validity for , based on Eq.1, may be related to other reasons; for this result is probably violated but coincidence of two constant values in the order of magnitude can be expected from the matching condition. The Monte Carlo results seems to confirm the qualitative validity of the Landau–Pomeranchuk arguments, although a different interpretation is also possible. The case (c) in the Bogoliubov and Shirkov classification corresponds to the quantum triviality in full theory (beyond its perturbation context), as can be seen by a reductio ad absurdum. Indeed, if , the theory is internally inconsistent. The only way to avoid it, is for , which is possible only for . It is a widespread belief that both QED and theory are trivial in the continuum limit. In fact, available information confirms only “Wilson triviality”, which just amounts to positivity of for and can be considered as firmly established. Indications of “true” quantum triviality are not abundant and allow different interpretations. 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「Landau pole」の詳細全文を読む スポンサード リンク
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