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[[Classical field theory]] is generically described by a field, or set of fields,  ''φ'', that depend on coordinates, ''x''. Valid field configurations are then determined by solving [[differential equations]] for ''φ'', and these equations are known as [[field equation]]s.

[[Classical field theory]] is generically described by a field, or set of fields,  ''φ'', that depend on coordinates, ''x''. Valid field configurations are then determined by solving [[differential equations]] for ''φ'', and these equations are known as [[field equation]]s.

经典场论一般用依赖于坐标''x'' 的场或场集 φ 来描述。然后通过求解 φ 的微分方程来确定有效的场构型，这些方程被称为场方程。

 

经典场论一般用依赖于坐标 x 的场或场集 φ 来描述。然后通过求解 φ 的微分方程来确定有效的场构型，这些方程被称为场方程。

For a theory to be scale-invariant, its field equations should be invariant under a rescaling of the coordinates, combined with some specified rescaling of the fields,

For a theory to be scale-invariant, its field equations should be invariant under a rescaling of the coordinates, combined with some specified rescaling of the fields,

:<math>\varphi\rightarrow\lambda^{-\Delta}\varphi~.</math>

:<math>\varphi\rightarrow\lambda^{-\Delta}\varphi~.</math>

:x\rightarrow\lambda x~,

 

<math>x\rightarrow\lambda x~,</math>，

对于一个理论是标度不变的，它的场方程应该在坐标的重新标度下保持不变，并结合一些特定的重新标度，: x right tarrow lambda x ~ ，: varphi right tarrow lambda ^ {-Delta } varphi ~ 。

<math>\varphi\rightarrow\lambda^{-\Delta}\varphi~.</math>。

The parameter ''Δ'' is known as the [[scaling dimension]] of the field, and its value depends on the theory under consideration.  Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is '''not''' scale-invariant.

The parameter ''Δ'' is known as the [[scaling dimension]] of the field, and its value depends on the theory under consideration.  Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is '''not''' scale-invariant.

The parameter Δ is known as the scaling dimension of the field, and its value depends on the theory under consideration.  Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is not scale-invariant.

The parameter Δ is known as the scaling dimension of the field, and its value depends on the theory under consideration.  Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is not scale-invariant.

参数 δ 称为场的标度维数，其大小取决于所考虑的理论。如果理论中没有固定长度的标尺，尺度不变性通常会成立。相反，一个固定长度尺度的存在表明一个理论不是尺度不变的。

参数 δ 称为场的标度维数，其大小取决于所考虑的理论。如果理论中没有固定长度的标度，标度不变性通常会成立。相反，如果存在固定的长度标度，则表明理论不具有标度不变性。

A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution,  ''φ''(''x''), one always has other solutions of the form

A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution,  ''φ''(''x''), one always has other solutions of the form

A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution,  φ(x), one always has other solutions of the form

A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution,  φ(x), one always has other solutions of the form

尺度不变性的一个结果是，给定一个尺度不变场方程的解，我们可以通过合适地重新标度坐标和场来自动地找到其他解。用专业术语来说，给定一个解 φ (x) ，总会有其它形式的解

标度不变性的一个结果是：给定一个标度不变性场方程的解，我们可以通过适当地缩放坐标和场自动地找到其他解。具体来说，给定一个解φ(x)，总有其他形式的解

:<math>\lambda^{\Delta}\varphi(\lambda x)</math>.

:<math>\lambda^{\Delta}\varphi(\lambda x)</math>.

===Scale invariance of field configurations 场结构中的标度不变性===

===Scale invariance of field configurations 场结构中的标度不变性===