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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'''Scaling Dimension 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。

参数 {{mvar|Δ}} 称为场的'''Scaling Dimension 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

where ''Δ'' is, again, the [[scaling dimension]] of the field.

where ''Δ'' is, again, the [[scaling dimension]] of the field.

其中 δ 是场的标度维数。

其中 {{mvar|Δ}} 是场的标度维数。

We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will '''not''' be scale-invariant, and in such cases the symmetry is said to be [[spontaneously broken]].

We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will '''not''' be scale-invariant, and in such cases the symmetry is said to be [[spontaneously broken]].

where {{mvar|D}} is the combined number of spatial and time dimensions.

where {{mvar|D}} is the combined number of spatial and time dimensions.

上面例子中的场方程在场中都是线性的，这意味着标度维数{{mvar|Δ}}并不是那么重要。然而，通常要求标量场的作用是无量纲的，这就固定了φ的标度维数。特别是：

:\Delta=\frac{D-2}{2},

上面例子中的场方程在场中都是线性的，这意味着标度维数，，并没有那么重要。但是，通常需要标量场操作是无量纲的，这修正了。特别是: Delta = frac { D-2}{2} ，其中是空间维度和时间维度的组合数。

<math>\Delta=\frac{D-2}{2},</math>

 

Given this scaling dimension for {{mvar|φ}}, there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless [[Phi to the fourth|φ⁴ theory]] for {{mvar|D}}=4. The field equation is

Given this scaling dimension for {{mvar|φ}}, there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless [[Phi to the fourth|φ⁴ theory]] for {{mvar|D}}=4. The field equation is

:<math>\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0.</math>

:<math>\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0.</math>。

Given this scaling dimension for , there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless φ⁴ theory for =4. The field equation is

已知φ的标度维数，则无质量标量场理论的某些非线性修正也是标度不变的。例如，{{mvar|D}}=4的无质量'''φ⁴theory φ⁴理论'''。场方程是：

给定这个尺度维数，无质量纯量场理论的非线性修正也是尺度不变的。一个例子是 = 4的无质量 φ < sup > 4  理论。场方程是: frac {1}{ c ^ 2} frac { partial ^ 2 varphi }{ partial t ^ 2}-nabla ^ 2 varphi + g varphi ^ 3 = 0。

<math>\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0.</math>。

(Note that the name {{mvar|φ}}⁴ derives from the form of the [[Phi to the fourth#The Lagrangian|Lagrangian]], which contains the fourth power of {{mvar|φ}}.)

(Note that the name {{mvar|φ}}⁴ derives from the form of the [[Phi to the fourth#The Lagrangian|Lagrangian]], which contains the fourth power of {{mvar|φ}}.

(Note that the name 4 derives from the form of the Lagrangian, which contains the fourth power of .)

(注意，{{mvar|φ}}4的名称来自拉格朗日量的形式，它包含{{mvar|φ}}的四次幂)

 

(注意，名称4来源于拉格朗日函数的形式，它包含. 的四次方。)

When {{mvar|D}}=4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is {{mvar|Δ}}=1. The field equation is then invariant under the transformation

When {{mvar|D}}=4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is {{mvar|Δ}}=1. The field equation is then invariant under the transformation

:<math>\varphi (x)\rightarrow\lambda^{-1}\varphi(x).</math>

:<math>\varphi (x)\rightarrow\lambda^{-1}\varphi(x).</math>

When =4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is =1. The field equation is then invariant under the transformation

当{{mvar|D}}=4(如三维空间维数和一维时间维数)时，标量场标度维数为{{mvar|Δ}}=1。场方程在进行如下变换下是不变的：

:\varphi (x)\rightarrow\lambda^{-1}\varphi(x).

当 = 4时。3空间维和1时间维) ，标量场标度维为 = 1。字段方程在变换下是不变的: x right tarrow lambda x，: t right tarrow lambda t，: varphi (x) right tarrow lambda ^ {-1} varphi (x)。

:<math>x\rightarrow\lambda x,</math>

:<math>t\rightarrow\lambda t,</math>

:<math>\varphi (x)\rightarrow\lambda^{-1}\varphi(x).</math>

The key point is that the parameter {{mvar|g}} must be dimensionless, otherwise one introduces a fixed length scale into the theory: For  {{mvar|φ}}⁴  theory, this is only the case in {{mvar|D}}=4.

The key point is that the parameter {{mvar|g}} must be dimensionless, otherwise one introduces a fixed length scale into the theory: For  {{mvar|φ}}⁴  theory, this is only the case in {{mvar|D}}=4.

Note that under these transformations the argument of the function {{mvar|φ}} is unchanged.

Note that under these transformations the argument of the function {{mvar|φ}} is unchanged.

The key point is that the parameter  must be dimensionless, otherwise one introduces a fixed length scale into the theory: For  4 theory, this is only the case in =4.

关键是参数{{mvar|g}}必须是无量纲的，否则就会引入一个固定的长度标度到理论中：对于{{mvar|φ}}⁴理论，只有在{{mvar|D}}=4时才会出现这种情况。注意，在这些变换下，函数{{mvar|φ}}的参数是不变的。

 

==Scale invariance in quantum field theory 量子场论中的标度不变性==

==Scale invariance in quantum field theory 量子场论中的标度不变性==