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Another example of a scale-invariant classical field theory is the massless [[scalar field theory|scalar field]] (note that the name [[scalar (physics)|scalar]] is unrelated to scale invariance). The scalar field, {{math|''φ''('''''x''''', ''t'')}} is a function of a set of spatial variables, '''''x''''', and a time variable, {{mvar|t}}.

Another example of a scale-invariant classical field theory is the massless [[scalar field theory|scalar field]] (note that the name [[scalar (physics)|scalar]] is unrelated to scale invariance). The scalar field, {{math|''φ''('''''x''''', ''t'')}} is a function of a set of spatial variables, '''''x''''', and a time variable, {{mvar|t}}.

Another example of a scale-invariant classical field theory is the massless scalar field (note that the name scalar is unrelated to scale invariance). The scalar field,  is a function of a set of spatial variables, x, and a time variable, .

标度不变经典场论的另一个例子是无质量标量场(注意名称“标量”与标度不变性无关)。标量场{{math|''φ''('''''x''''', ''t'')}}是一组空间变量 '''''x''''' 和一个时间变量 {{mvar|t}} 的函数。

 

标度不变的经典场论的另一个例子是无质量标量场(注意标量的名称与尺度不变性无关)。标量场，是一组空间变量 x 和一个时间变量 x 的函数。

Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,

Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,

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

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

:\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi = 0,

 

<math>\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi = 0,</math>，

:x\rightarrow\lambda x,

 

 

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

首先考虑线性理论。像上面的电磁场方程一样，这个理论的运动方程也是一个波动方程: frac {1}{ c ^ 2} frac { partial ^ 2 varphi }{ partial t ^ 2}-nabla ^ 2 varphi = 0，并且在变换下是不变的: x tarrow lambda x，: t right tarrow da t。

<math>t\rightarrow\lambda t.</math>。

The name massless refers to the absence of a term <math>\propto m^2\varphi</math> in the field equation. Such a term is often referred to as a `mass' term, and would break the invariance under the above transformation. In [[relativistic field theory|relativistic field theories]], a mass-scale, {{mvar|m}}  is physically equivalent to a fixed length scale through

The name massless refers to the absence of a term <math>\propto m^2\varphi</math> in the field equation. Such a term is often referred to as a `mass' term, and would break the invariance under the above transformation. In [[relativistic field theory|relativistic field theories]], a mass-scale, {{mvar|m}}  is physically equivalent to a fixed length scale through

and so it should not be surprising that massive scalar field theory is ''not'' scale-invariant.

and so it should not be surprising that massive scalar field theory is ''not'' scale-invariant.

The name massless refers to the absence of a term \propto m^2\varphi in the field equation. Such a term is often referred to as a `mass' term, and would break the invariance under the above transformation. In relativistic field theories, a mass-scale,  is physically equivalent to a fixed length scale through

无质量是指在场方程中没有<math>\propto m^2\varphi</math>项。这一项通常称为“质量”项，它会破坏上述变换下的不变性。在'''Relativistic Field Theories 相对论场理论'''中，质量标度{{mvar|m}}在物理上等同于一个固定的长度标度：

:L=\frac{\hbar}{mc},

 

<math>L=\frac{\hbar}{mc},</math>，

====φ⁴ theory φ4理论====

====φ⁴ theory φ⁴  理论====

The field equations in the examples above are all [[linear]] in the fields, which has meant that the [[scaling dimension]], {{mvar|Δ}}, has not been so important. However, one usually requires that the scalar field [[action (physics)|action]] is dimensionless, and this fixes the [[scaling dimension]] of {{mvar|φ}}. In particular,

The field equations in the examples above are all [[linear]] in the fields, which has meant that the [[scaling dimension]], {{mvar|Δ}}, has not been so important. However, one usually requires that the scalar field [[action (physics)|action]] is dimensionless, and this fixes the [[scaling dimension]] of {{mvar|φ}}. In particular,

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

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