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In quantum field theory and statistical mechanics, the Mermin–Wagner theorem (also known as Mermin–Wagner–Hohenberg theorem or Coleman theorem) states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions . Intuitively, this means that long-range fluctuations can be created with little energy cost and since they increase the entropy they are favored. This is because if such a spontaneous symmetry breaking occurred, then the corresponding Goldstone bosons, being massless, would have an infrared divergent correlation function. The absence of spontaneous symmetry breaking in dimensional systems was rigorously proved by in quantum field theory and by David Mermin, Herbert Wagner and Pierre Hohenberg in statistical physics. That the theorem does not apply to discrete symmetries can be seen in the two-dimensional Ising model. == Introduction == Consider the free scalar field of mass in two Euclidean dimensions. Its propagator is: : For small is a solution to Laplace's equation with a point source: : This is because the propagator is the reciprocal of in space. To use Gauss's law, define the electric field analog to be . The divergence of the electric field is zero. In two dimensions, using a large Gaussian ring: : So that the function ''G'' has a logarithmic divergence both at small and large ''r''. : The interpretation of the divergence is that the field fluctuations cannot stay centered around a mean. If you start at a point where the field has the value 1, the divergence tells you that as you travel far away, the field is arbitrarily far from the starting value. This makes a two dimensional massless scalar field slightly tricky to define mathematically. If you define the field by a Monte-Carlo simulation, it doesn't stay put, it slides to infinitely large values with time. This happens in one dimension too, when the field is a one dimensional scalar field, a random walk in time. A random walk also moves arbitrarily far from its starting point, so that a one-dimensional or two-dimensional scalar does not have a well defined average value. If the field is an angle, , as it is in the Mexican hat model where the complex field has an expectation value but is free to slide in the direction, the angle will be random at large distances. This is the Mermin–Wagner theorem: there is no spontaneous breaking of a continuous symmetry in two dimensions. 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「Mermin–Wagner theorem」の詳細全文を読む スポンサード リンク
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