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===Conformal field theory 共形场论===

===Conformal field theory 共形场论===

Scale-invariant QFTs are almost always invariant under the full [[conformal symmetry]], and the study of such QFTs is [[conformal field theory]] (CFT). [[operator (physics)|Operators]] in a CFT have a well-defined [[scaling dimension]], analogous to the [[scaling dimension]], ''∆'', of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as [[anomalous scaling dimension]]s.

Scale-invariant QFTs are almost always invariant under the full [[conformal symmetry]], and the study of such QFTs is [[conformal field theory]] (CFT). [[operator (physics)|Operators]] in a CFT have a well-defined [[scaling dimension]], analogous to the [[scaling dimension]], ''∆'', of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as [[anomalous scaling dimension]]s.

在完全共形对称条件下，标度不变的量子场论几乎总是不变的，对此类量子场论的研究就是共形场论（CFT）。共形场论中的算子具有定义明确的标度维数，类似于前面所讨论的经典场标度维数 ''∆''。然而，共形场论中算子的标度维数与经典理论中场的标度维数不同。在共形场论中出现的附加贡献称做'''Anomalous Scaling Dimensions 异常标度维数'''。

在完全共形对称条件下，标度不变的量子场论几乎总是不变的，对此类量子场论的研究就是共形场论（CFT）。共形场论中的算子具有定义明确的标度维数，类似于前面所讨论的经典场标度维数 ''∆''。然而，共形场论中算子的标度维数与经典理论中场的标度维数不同。在共形场论中出现的附加贡献称做'''Anomalous Scaling Dimensions 异常标度维数'''。

The φ⁴ theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be [[conformal anomaly|anomalous]]. A classically scale invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called [[Inflation (cosmology)|cosmic inflation]], as long as the theory can be studied through [[perturbation theory]].<ref>{{cite journal|last=Salvio, Strumia|title=Agravity|journal=JHEP  |volume=2014 |issue=6|pages=080|date=2014-03-17|url=http://inspirehep.net/record/1286134|arxiv = 1403.4226|bibcode = 2014JHEP...06..080S|doi=10.1007/JHEP06(2014)080}}</ref>

The φ⁴ theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be [[conformal anomaly|anomalous]]. A classically scale invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called [[Inflation (cosmology)|cosmic inflation]], as long as the theory can be studied through [[perturbation theory]].<ref>{{cite journal|last=Salvio, Strumia|title=Agravity|journal=JHEP  |volume=2014 |issue=6|pages=080|date=2014-03-17|url=http://inspirehep.net/record/1286134|arxiv = 1403.4226|bibcode = 2014JHEP...06..080S|doi=10.1007/JHEP06(2014)080}}</ref>

上面的φ⁴理论例子表明，量子场论的耦合参数可以是标度依赖的，即使相应的经典场论是标度不变(或共形不变)。如果是这种情况，则称经典标度(或共形)不变性为异常。经典的标度不变场论，当量子效应打破其中的标度不变性，可以为接近指数级膨胀的早期宇宙提供了一种解释，即为'''Cosmic Inflation 宇宙膨胀'''，只要该理论可以通过'''Perturbation Theory 微扰理论'''研究。

 

上面的φ⁴理论例子表明，量子场论的耦合参数可以是标度依赖的，即使相应的经典场论是标度不变(或共形不变)。如果是这种情况，则称经典标度(或共形)不变性为异常。经典的标度不变场论，当量子效应打破其中的标度不变性，可以为接近指数级膨胀的早期宇宙提供了一种解释，即为宇宙膨胀，只要该理论可以通过微扰理论研究。

==Phase transitions 相变==

==Phase transitions 相变==

In statistical mechanics, as a system undergoes a phase transition, its fluctuations are described by a scale-invariant statistical field theory. For a system in equilibrium (i.e. time-independent) in  spatial dimensions, the corresponding statistical field theory is formally similar to a -dimensional CFT. The scaling dimensions in such problems are usually referred to as critical exponents, and one can in principle compute these exponents in the appropriate CFT.

In statistical mechanics, as a system undergoes a phase transition, its fluctuations are described by a scale-invariant statistical field theory. For a system in equilibrium (i.e. time-independent) in  spatial dimensions, the corresponding statistical field theory is formally similar to a -dimensional CFT. The scaling dimensions in such problems are usually referred to as critical exponents, and one can in principle compute these exponents in the appropriate CFT.

在21统计力学，当一个系统经历相变时，它的波动用一个尺度不变的统计场理论来描述。对于一个处于平衡状态的系统(即。时间无关)的空间维度，相应的统计场理论形式上类似于一维 CFT。这类问题中的标度维数通常被称为临界指数，原则上可以在适当的 CFT 中计算这些指数。

在统计力学中，当某个系统经历相变时，其波动可以用标度不变的统计场论来描述。对于在{{mvar|D}}空间维度中处于平衡状态(即时间无关)的系统，相应的统计场论形式上类似于{{mvar|D}}维共形场论。这类问题中的标度维数通常称为'''Critical Exponents 临界指数'''，原则上可以在适当的共形场论中计算这些指数。

===The Ising model Ising模型===

===The Ising model Ising 模型===

An example that links together many of the ideas in this article is the phase transition of the [[Ising model]], a simple model of [[ferromagnet]]ic substances. This is a statistical mechanics model, which also has a description in terms of conformal field theory. The system consists of an array of lattice sites, which form a {{mvar|D}}-dimensional periodic lattice. Associated with each lattice site is a [[magnetic moment]], or [[spin (physics)|spin]], and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)

An example that links together many of the ideas in this article is the phase transition of the [[Ising model]], a simple model of [[ferromagnet]]ic substances. This is a statistical mechanics model, which also has a description in terms of conformal field theory. The system consists of an array of lattice sites, which form a {{mvar|D}}-dimensional periodic lattice. Associated with each lattice site is a [[magnetic moment]], or [[spin (physics)|spin]], and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)