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Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in D spacetime dimensions can be redefined in a lower number of dimensions d, by taking all the fields to be independent of the location in the extra D − d dimensions. For example, consider a periodic compact dimension with period L. Let x be the coordinate along this dimension. Any field can be described as a sum of the following terms:

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  • تخفيض الأبعاد (ar)
  • Dimensional reduction (en)
  • Réduction dimensionnelle (fr)
  • 차원 축소 (물리학) (ko)
  • 維度減化 (zh)
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  • في الإحصاء، والتعلم الآلي، ونظرية المعلومات فإن تخفيض الأبعاد أو تقليل الأبعاد هو عملية تقليل عدد المتغيرات العشوائية تحت ظروف.عن طريق الحصول على مجموعة من المتغيرات الرئيسية. يمكن تقسيم المناهج إلى اختيار الخصائص واستخراج الخصائص (ar)
  • En physique, une réduction dimensionnelle est une procédure par laquelle, étant donnée une théorie formulée sur un espace-temps de dimension , on construit une autre théorie formulée sur un sous-espace de dimension . Dans la suite nous allons décrire brièvement plusieurs procédures de réduction communément utilisées. (fr)
  • 이론물리학에서, 차원 축소(次元縮小, 영어: dimensional reduction)는 고차원에 정의된 장론으로부터, 더 낮은 차원에 존재하는 장론을 구성하는 방법이다. (ko)
  • 维度减化（英語：Dimensional reduction）是紧化理论中紧致化的维度的大小变为零时的临界情况。在物理学中，通过将所有的场独立存在于额外维度D中，时空维数D的理论能够被较少数量的额外维度D重新定义。 例如，考虑一个周期性的紧凑的维度的L时期。让x成为沿着这条维度的坐标。任何场 可以被描述为以下单元的总和： An 作为一个常数。根据量子力学，这一单元具有沿着x轴的动量nh/L，在那里 h 是普朗克常数。因此，当L达到0时，这个动量达到了无限大，能量也一样，除非n = 0。然而n = 0提供了一个关于 x恒定的场。因此在这个场的限制下，并在有限的能量下， 将不依赖于 x。 这种说法进行了概括。紧凑的维度对所有场施加了特定的边界条件，例如在周期性维度的情况下的周期性边界条件，并且在其他情况下通常为诺伊曼边界条件或狄利克雷边界条件。现在假设紧凑的维度的尺度是L；那么沿这个维度的梯度的可能的特征值是1/L的整数或半整数倍（取决于精确的边界条件）。在量子力学中，这个特征值是场的动量，因此与其能量有关。当L → 0时，除零之外的所有特征值都到无穷大，而能量也是如此。因此，在这个极限情况下，在有限能量的情况下，零是唯一可能的沿着紧凑尺寸的梯度下的特征值，这意味着没有任何东西依赖于这个维度。 (zh)
  • Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in D spacetime dimensions can be redefined in a lower number of dimensions d, by taking all the fields to be independent of the location in the extra D − d dimensions. For example, consider a periodic compact dimension with period L. Let x be the coordinate along this dimension. Any field can be described as a sum of the following terms: (en)
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  • في الإحصاء، والتعلم الآلي، ونظرية المعلومات فإن تخفيض الأبعاد أو تقليل الأبعاد هو عملية تقليل عدد المتغيرات العشوائية تحت ظروف.عن طريق الحصول على مجموعة من المتغيرات الرئيسية. يمكن تقسيم المناهج إلى اختيار الخصائص واستخراج الخصائص (ar)
  • Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in D spacetime dimensions can be redefined in a lower number of dimensions d, by taking all the fields to be independent of the location in the extra D − d dimensions. For example, consider a periodic compact dimension with period L. Let x be the coordinate along this dimension. Any field can be described as a sum of the following terms: with An a constant. According to quantum mechanics, such a term has momentum nh/L along x, where h is Planck's constant. Therefore, as L goes to zero, the momentum goes to infinity, and so does the energy, unless n = 0. However n = 0 gives a field which is constant with respect to x. So at this limit, and at finite energy, will not depend on x. This argument generalizes. The compact dimension imposes specific boundary conditions on all fields, for example periodic boundary conditions in the case of a periodic dimension, and typically Neumann or Dirichlet boundary conditions in other cases. Now suppose the size of the compact dimension is L; then the possible eigenvalues under gradient along this dimension are integer or half-integer multiples of 1/L (depending on the precise boundary conditions). In quantum mechanics this eigenvalue is the momentum of the field, and is therefore related to its energy. As L → 0 all eigenvalues except zero go to infinity, and so does the energy. Therefore, at this limit, with finite energy, zero is the only possible eigenvalue under gradient along the compact dimension, meaning that nothing depends on this dimension. Dimensional reduction also refers to a specific cancellation of divergences in Feynman diagrams. It was put forward by Amnon Aharony, Yoseph Imry, and Shang-keng Ma who proved in 1976 that "to all orders in perturbation expansion, the critical exponents in a d-dimensional (4 < d < 6) system with short-range exchange and a random quenched field are the same as those of a (d–2)-dimensional pure system." Their arguments indicated that the "Feynman diagrams which give the leading singular behavior for the random case are identically equal, apart from combinatorial factors, to the corresponding Feynman diagrams for the pure case in two fewer dimensions." This dimensional reduction was investigated further in the context of supersymmetric theory of Langevin stochastic differential equations by Giorgio Parisi and Nicolas Sourlas who "observed that the most infrared divergent diagrams are those with the maximum number of random source insertions, and, if the other diagrams are neglected, one is left with a diagrammatic expansion for a classical field theory in the presence of random sources... Parisi and Sourlas explained this dimensional reduction by a hidden supersymmetry." (en)
  • En physique, une réduction dimensionnelle est une procédure par laquelle, étant donnée une théorie formulée sur un espace-temps de dimension , on construit une autre théorie formulée sur un sous-espace de dimension . Dans la suite nous allons décrire brièvement plusieurs procédures de réduction communément utilisées. (fr)
  • 이론물리학에서, 차원 축소(次元縮小, 영어: dimensional reduction)는 고차원에 정의된 장론으로부터, 더 낮은 차원에 존재하는 장론을 구성하는 방법이다. (ko)
  • 维度减化（英語：Dimensional reduction）是紧化理论中紧致化的维度的大小变为零时的临界情况。在物理学中，通过将所有的场独立存在于额外维度D中，时空维数D的理论能够被较少数量的额外维度D重新定义。 例如，考虑一个周期性的紧凑的维度的L时期。让x成为沿着这条维度的坐标。任何场 可以被描述为以下单元的总和： An 作为一个常数。根据量子力学，这一单元具有沿着x轴的动量nh/L，在那里 h 是普朗克常数。因此，当L达到0时，这个动量达到了无限大，能量也一样，除非n = 0。然而n = 0提供了一个关于 x恒定的场。因此在这个场的限制下，并在有限的能量下， 将不依赖于 x。 这种说法进行了概括。紧凑的维度对所有场施加了特定的边界条件，例如在周期性维度的情况下的周期性边界条件，并且在其他情况下通常为诺伊曼边界条件或狄利克雷边界条件。现在假设紧凑的维度的尺度是L；那么沿这个维度的梯度的可能的特征值是1/L的整数或半整数倍（取决于精确的边界条件）。在量子力学中，这个特征值是场的动量，因此与其能量有关。当L → 0时，除零之外的所有特征值都到无穷大，而能量也是如此。因此，在这个极限情况下，在有限能量的情况下，零是唯一可能的沿着紧凑尺寸的梯度下的特征值，这意味着没有任何东西依赖于这个维度。 (zh)
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