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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. |
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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.
| + | 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。 |
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− | 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。 | |
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| 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 |
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− | 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
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| 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 | | 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 |
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| :<math>\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)</math> | | :<math>\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)</math> |
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− | For a particular field configuration, φ(x), to be scale-invariant, we require that
| + | 对于特定的场构型''φ''(''x''),要具有标度不变性,我们就要满足: |
− | :\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)
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− | 对于特定的字段配置 φ (x) ,要使其具有尺度不变性,我们需要: varphi (x) = lambda ^ {-Delta } varphi (lambda x)
| + | <math>\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)</math> |
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| where ''Δ'' is, again, the [[scaling dimension]] of the field. | | where ''Δ'' is, again, the [[scaling dimension]] of the field. |
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− | where Δ is, again, the scaling dimension of the field.
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| 其中 δ 是场的标度维数。 | | 其中 δ 是场的标度维数。 |
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| 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. |
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− | 我们注意到这个条件相当有限制性。一般来说,即使是标度不变场方程的解也不是标度不变的,在这种情况下,对称性被称为自发破缺。
| + | 我们注意到这个条件限制性很强。一般来说,即使是标度不变场方程的解也不是标度不变的,在这种情况下,对称性出现'''Spontaneously Broken 自发破缺'''。 |
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| ===Classical electromagnetism 经典电磁学 === | | ===Classical electromagnetism 经典电磁学 === |
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| An example of a scale-invariant classical field theory is electromagnetism with no charges or currents. The fields are the electric and magnetic fields, E(x,t) and B(x,t), while their field equations are Maxwell's equations. | | An example of a scale-invariant classical field theory is electromagnetism with no charges or currents. The fields are the electric and magnetic fields, E(x,t) and B(x,t), while their field equations are Maxwell's equations. |
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− | 比例不变的经典场论的一个例子是没有电荷和电流的电磁学。电场是电场和磁场,e (x,t)和 b (x,t) ,而它们的场方程是麦克斯韦方程组。
| + | 标度不变的经典场论的一个实例是没有电荷和电流的电磁学。场是电场和磁场,'''E'''('''x''',''t'') 和 '''B'''('''x''',''t''),而它们的场方程是麦克斯韦方程组。 |
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| With no charges or currents, [[electromagnetic field#Light as an electromagnetic disturbance|these field equations]] take the form of [[wave equation]]s | | With no charges or currents, [[electromagnetic field#Light as an electromagnetic disturbance|these field equations]] take the form of [[wave equation]]s |
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| where ''c'' is the speed of light. | | where ''c'' is the speed of light. |
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− | With no charges or currents, these field equations take the form of wave equations
| + | 在没有电荷或电流的情况下,这些场方程采用波动方程的形式: |
− | :\nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}
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− | :\nabla^2\mathbf{B} = \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2}
| + | <math>\nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}</math> |
− | where c is the speed of light.
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| + | <math>\nabla^2\mathbf{B} = \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2}</math> |
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− | 没有电荷和电流,这些场方程采用波动方程的形式: nabla ^ 2 mathbf { e } = frac {1}{ c ^ 2} frac { partial ^ 2 mathbf { e }{ partial t ^ 2} : nabla ^ 2 mathbf { b } = frac {1}{ c ^ 2} frac { b }{ partial t ^ 2}其中 c 是光速。
| + | 其中 c 是光速。 |
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| These field equations are invariant under the transformation | | These field equations are invariant under the transformation |
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| :<math>t\rightarrow\lambda t.</math> | | :<math>t\rightarrow\lambda t.</math> |
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− | These field equations are invariant under the transformation
| + | 这些场方程在进行如下变换下是不变的: |
− | :x\rightarrow\lambda x,
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− | :t\rightarrow\lambda t.
| + | <math>x\rightarrow\lambda x,</math>, |
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− | 这些字段方程在变换下是不变的: x right tarrow lambda x,: t right tarrow lambda。
| + | <math>t\rightarrow\lambda t.</math>。 |
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| Moreover, given solutions of Maxwell's equations, '''E'''('''x''', ''t'') and '''B'''('''x''', ''t''), it holds that | | Moreover, given solutions of Maxwell's equations, '''E'''('''x''', ''t'') and '''B'''('''x''', ''t''), it holds that |
| '''E'''(λ'''x''', λ''t'') and '''B'''(λ'''x''', λ''t'') are also solutions. | | '''E'''(λ'''x''', λ''t'') and '''B'''(λ'''x''', λ''t'') are also solutions. |
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− | Moreover, given solutions of Maxwell's equations, E(x, t) and B(x, t), it holds that
| + | 此外,已知'''E'''('''x''', ''t'')和'''B'''('''x''', ''t'')是麦克斯韦方程组的解,则可以认为'''E'''(λ'''x''', λ''t'')和'''B'''(λ'''x''', λ''t'')也是解。 |
− | E(λx, λt) and B(λx, λt) are also solutions. | |
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− | 给出了麦克斯韦方程组 e (x,t)和 b (x,t)的解,证明了 e (λx,λt)和 b (λx,λt)也是解。
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| ===Massless scalar field theory 无质量标量场理论=== | | ===Massless scalar field theory 无质量标量场理论=== |