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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}}. |
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− | 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。 |
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− | 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。
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| 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, |
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| :<math>t\rightarrow\lambda t.</math> | | :<math>t\rightarrow\lambda t.</math> |
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− | Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,
| + | 首先考虑线性理论。像上述的电磁场方程一样,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,
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− | and is invariant under the transformation
| + | <math>\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi = 0,</math>, |
− | :x\rightarrow\lambda x,
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− | :t\rightarrow\lambda t.
| + | 并且在进行如下变换时是不变的: |
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| + | <math>x\rightarrow\lambda x,</math> |
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− | 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>。 |
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| 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 |
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| 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. |
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− | 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},
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− | and so it should not be surprising that massive scalar field theory is not scale-invariant.
| + | <math>L=\frac{\hbar}{mc},</math>, |
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− | 名称 massless 是指在字段方程中没有一个 propto m ^ 2 varphi 项。这样的术语通常被称为“质量”术语,它打破了上述变换下的不变性。在相对论场理论中,质量尺度在物理上等同于一个固定长度的尺度: l = frac { hbar }{ mc } ,因此大质量纯量场理论不具有尺度不变性也就不足为奇了。
| + | 因此质量标量场理论不具有标度不变性也就不足为奇了。 |
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− | ====φ<sup>4</sup> theory φ4理论==== | + | ====φ<sup>4</sup> theory φ<sup>4</sup> 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> |