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Even though the quantized massless ''φ''⁴ is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the '''Wilson-Fisher fixed point''', below.

Even though the quantized massless ''φ''⁴ is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the '''Wilson-Fisher fixed point''', below.

虽然量子化无质量''φ''⁴不是标度不变的，但除了高斯定点外，确实存在标度不变的量子化标量场理论。例如：'''Wilson-Fisher Fixed Point 威尔逊-费雪定点'''

虽然量子化无质量''φ''⁴不是标度不变的，但除了高斯定点外，确实存在标度不变的量子化标量场理论。例如：'''Wilson-Fisher Fixed Point 威尔逊-费雪定点'''。

===Conformal field theory 共形场论===

===Conformal field theory 共形场论===

:<math>G(r)\propto\frac{1}{r^{D-2+\eta}},</math>

:<math>G(r)\propto\frac{1}{r^{D-2+\eta}},</math>

for some particular value of <math>\eta</math>, which is an example of a critical exponent.

for some particular value of <math>\eta</math>, which is an example of a critical exponent.

在这个临界温度下，人们想要计算的物理量之一是存在距离的自旋之间的相互关系。此处通式为：

在这个临界温度下，人们想要计算的物理量之一是存在距离的自旋之间的相互关系。此处通式为：

对于某个特定的<math>\eta</math>值，这是一个临界指数的例子。

对于某个特定的<math>\eta</math>值，这是一个临界指数的例子。

====CFT description CFT描述====

====CFT description 共形场论描述====

The fluctuations at temperature {{math|''T_c''}} are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the '''Wilson-Fisher fixed point''', a particular scale-invariant [[scalar field (quantum field theory)|scalar field theory]].

The fluctuations at temperature {{math|''T_c''}} are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the '''Wilson-Fisher fixed point''', a particular scale-invariant [[scalar field (quantum field theory)|scalar field theory]].

The fluctuations at temperature  are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the Wilson-Fisher fixed point, a particular scale-invariant scalar field theory.

在临界温度处的波动是标度不变的，因此相变时的伊辛模型可以用标度不变的统计场论来描述。事实上，这个理论就是威尔逊-费雪定点，一个特殊的标度不变标量场理论。

 

In this context, {{math|''G''(''r'')}} is understood as a [[correlation function]] of scalar fields,

In this context, {{math|''G''(''r'')}} is understood as a [[correlation function]] of scalar fields,

Now we can fit together a number of the ideas seen already.

Now we can fit together a number of the ideas seen already.

:\langle\phi(0)\phi(r)\rangle\propto\frac{1}{r^{D-2+\eta}}.

 

<math>\langle\phi(0)\phi(r)\rangle\propto\frac{1}{r^{D-2+\eta}}.</math>。

From the above, one  sees that the critical exponent, {{mvar|η}}, for this phase transition, is also an '''anomalous dimension'''. This is because the classical dimension of the scalar field,

From the above, one  sees that the critical exponent, {{mvar|η}}, for this phase transition, is also an '''anomalous dimension'''. This is because the classical dimension of the scalar field,

where {{mvar|D}} is the number of dimensions of the Ising model lattice.

where {{mvar|D}} is the number of dimensions of the Ising model lattice.

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

 

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

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

 

 

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

从上面，我们可以看到，对于这种相变，临界指数也是一个反常的维度。这是因为标量场的经典维数: Delta = frac { D-2}{2}被修改为: Delta = frac { D-2 + eta }{2} ，其中是 Ising 模型格子的维数。

其中{{mvar|D}} 是伊辛模型格子的维数。

So this '''anomalous dimension''' in the conformal field theory is the ''same'' as a particular critical exponent of the Ising model phase transition.

So this '''anomalous dimension''' in the conformal field theory is the ''same'' as a particular critical exponent of the Ising model phase transition.

So this anomalous dimension in the conformal field theory is the same as a particular critical exponent of the Ising model phase transition.

So this anomalous dimension in the conformal field theory is the same as a particular critical exponent of the Ising model phase transition.

Note that for dimension {{math|''D'' ≡ 4−''ε''}}, {{mvar|η}} can be calculated approximately, using the '''epsilon expansion''', and one finds that

Note that for dimension {{math|''D'' ≡ 4−''ε''}}, {{mvar|η}} can be calculated approximately, using the '''epsilon expansion''', and one finds that

:<math>\eta=\frac{\epsilon^2}{54}+O(\epsilon^3)</math>.

:<math>\eta=\frac{\epsilon^2}{54}+O(\epsilon^3)</math>.

对于维度{{math|''D'' ≡ 4−''ε''}}，可以使用epsilon展开式近似地计算{{mvar|η}}，并且可以发现：

:\eta=\frac{\epsilon^2}{54}+O(\epsilon^3).

注意，对于维度，可以使用 epsilon 展开式近似地计算，并且可以发现: eta = frac { epsilon ^ 2}{54} + o (epsilon ^ 3)。

<math>\eta=\frac{\epsilon^2}{54}+O(\epsilon^3)</math>。

In the physically interesting case of three spatial dimensions, we have {{mvar|ε}}=1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that {{mvar|η}} is numerically small in three dimensions.

In the physically interesting case of three spatial dimensions, we have {{mvar|ε}}=1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that {{mvar|η}} is numerically small in three dimensions.

In the physically interesting case of three spatial dimensions, we have =1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that  is numerically small in three dimensions.

在物理上很有趣的三维空间情况下，我们有{{mvar|ε}}=1，因此这种膨胀并不严格可靠。然而，半定量的预测是η在三维上的数值很小。

 

On the other hand, in the two-dimensional case the Ising model is exactly soluble. In particular, it is equivalent to one of the [[Minimal model (physics)|minimal models]], a family of well-understood CFTs, and it is possible to compute {{mvar|η}} (and the other critical exponents) exactly,

On the other hand, in the two-dimensional case the Ising model is exactly soluble. In particular, it is equivalent to one of the [[Minimal model (physics)|minimal models]], a family of well-understood CFTs, and it is possible to compute {{mvar|η}} (and the other critical exponents) exactly,

:<math>\eta_{_{D=2}}=\frac{1}{4}</math>.

:<math>\eta_{_{D=2}}=\frac{1}{4}</math>.

On the other hand, in the two-dimensional case the Ising model is exactly soluble. In particular, it is equivalent to one of the minimal models, a family of well-understood CFTs, and it is possible to compute  (and the other critical exponents) exactly,

另一方面，在二维情况下，伊辛模型是完全可解的。特别地，它等价于'''Minimal Model 最小模型'''之一，即一组很好理解的共形场论，并且可以精确地计算η(和其他临界指数)，

另一方面，在二维情况下，伊辛模型是完全可解的。特别地，它等价于一个极小模型，一个很好理解的 CFTs 族，并且可以精确地计算(和其他临界指数) : eta _ { _ { d = 2} = frac {1}{4}。

<math>\eta_{_{D=2}}=\frac{1}{4}</math>

===Schramm–Loewner evolution===

===Schramm–Loewner evolution===