对称性破缺

来自集智百科 - 复杂系统|人工智能|复杂科学|复杂网络|自组织
Jxzhou（讨论 | 贡献）2021年7月26日 (一) 17:53的版本 （→‎Chiral symmetry）
跳到导航 跳到搜索



[[图一：一个小球位于中央山丘的山峰处（C）。这是一种不稳定平衡：一个很小的扰动会使它落到左边(L)或右边(R)稳定点。尽管山丘是对称的，没有理由让球落在哪一侧，但观察到的最终状态仍然是不对称的，它总会落到某一侧]]。

在物理学中，一个作用于系统的（无限）小扰动使系统跨过临界点，通过决定去向分叉的哪个分支来决定系统的命运，这种现象叫做对称性破缺。对于一个观测不到扰动(或“噪声”)的外部观察者来说，这个选择看起来是任意的。这个过程被称为对称性破缺，因为这种转变通常使系统从一个对称但无序的状态进入一个或多个确定的状态。在斑图生成中对称性破缺起着重要作用。

1972年，诺贝尔奖得主P·W·安德森(P.W.Anderson)在《科学》(Science)杂志上发表了一篇名为《多即不同》的论文[1]，文中使用对称性破缺的思想表明，即使还原论是正确的，但它的逆命题建构主义是错误的。建构主义认为，在给出描述各组成部分的理论的情况下科学家可以轻易地预测复杂现象。

对称性破缺可以分为显性对称性破缺和自发对称性破缺两种类型，二者的区别是，在破缺对称性下系统的运动方程是否不变或者基态是否保持不变。

显性对称性破缺

在显性对称性破缺中，描述系统的运动方程在破缺对称下是不同的。在哈密顿力学或拉格朗日力学中，假若系统的哈密顿量(或拉格朗日量)中至少一项显性地打破了给定的对称性，就发生了显性对称性破缺。

自发对称性破缺

在自发对称性破缺中，系统的运动方程是不变的，但系统发生了变化。这是因为系统的背景(时空)——真空态——是非恒定的。这种对称性破缺可以用一个序参量进行参数化。这类对称破缺的一个特殊情况是动力学对称性破缺。

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state ends up in an asymmetric state.[1][2][3] In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.

自发对称破缺是一个自发的对称破缺过程，它使处于对称状态的物理系统最终处于非对称状态。特别地，它可以描述运动方程或拉格朗日方程服从某种对称性，但最低能量真空解不具有该对称性的系统。当系统进入其中一个真空解时，真空解周围的扰动会破坏系统对称性，尽管整个拉格朗日方程仍然保持了对称性。

Overview

In explicit symmetry breaking, if two outcomes are considered, the probability of a pair of outcomes can be different. By definition, spontaneous symmetry breaking requires the existence of a symmetric probability distribution—any pair of outcomes has the same probability. In other words, the underlying laws模板:Clarify are invariant under a symmetry transformation.

The system, as a whole模板:Clarify, changes under such transformations.

Phases of matter, such as crystals, magnets, and conventional superconductors, as well as simple phase transitions can be described by spontaneous symmetry breaking. Notable exceptions include topological phases of matter like the fractional quantum Hall effect.

Examples 例子

Sombrero potential

Consider a symmetric upward dome with a trough circling the bottom. If a ball is put at the very peak of the dome, the system is symmetric with respect to a rotation around the center axis. But the ball may spontaneously break this symmetry by rolling down the dome into the trough, a point of lowest energy. Afterward, the ball has come to a rest at some fixed point on the perimeter. The dome and the ball retain their individual symmetry, but the system does not.[2]

考虑一个对称向上的圆顶，底部环绕着一个槽。如果把一个球放在圆顶的最顶端，这个系统是围绕中心轴旋转对称的。但球体可能会自发地打破这种对称性，因为它会沿着穹顶滚动到能量最低的槽中。然后，球在圆周上某个固定的点上停下来。圆顶和球保持了各自的对称，但系统却没有保持对称性。

文件:Mexican hat potential polar.svg
Graph of Goldstone's "sombrero" potential function [math]\displaystyle{ V(\phi) }[/math].

In the simplest idealized relativistic model, the spontaneously broken symmetry is summarized through an illustrative scalar field theory. The relevant Lagrangian of a scalar field [math]\displaystyle{ \phi }[/math], which essentially dictates how a system behaves, can be split up into kinetic and potential terms,

在最简单的理想相对论模型中，可以用一个解释性的标量场理论总结自发破对称性。一个标量场 [math]\displaystyle{ \phi }[/math]的拉格朗日量从本质上决定了系统的行为，它可以分解成动能项和势能项：

[math]\displaystyle{ \mathcal{L} = \partial^\mu \phi \partial_\mu \phi - V(\phi). }[/math]

 

 

 

 

(1)

It is in this potential term [math]\displaystyle{ V(\phi) }[/math] that the symmetry breaking is triggered. An example of a potential, due to Jeffrey Goldstone[3] is illustrated in the graph at the left.

正是在势能项 [math]\displaystyle{ V(\phi) }[/math] 中触发了对称性破缺。例如左图所示的 Jeffrey Goldstone 给出的势能函数：

[math]\displaystyle{ V(\phi) = -5|\phi|^2 + |\phi|^4 \, }[/math].

 

 

 

 

(2)

This potential has an infinite number of possible minima (vacuum states) given by

该是函数具有无穷数量的最小值点（真空态）当

[math]\displaystyle{ \phi = \sqrt{5} e^{i\theta} }[/math].

 

 

 

 

(3)

for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to Φ = 0. This state has a U(1) symmetry. However, once the system falls into a specific stable vacuum state (amounting to a choice of θ), this symmetry will appear to be lost, or "spontaneously broken".

对于0到2π之间的任何实数θ。系统也有一个不稳定的真空状态，对应于Φ = 0。这个状态具有U(1)对称。然而，一旦系统落入某个稳定真空状态(相当于选择θ)，这种对称性就会消失，或者说“自发破缺”。

In fact, any other choice of θ would have exactly the same energy, implying the existence of a massless Nambu–Goldstone boson, the mode running around the circle at the minimum of this potential, and indicating there is some memory of the original symmetry in the Lagrangian.

事实上，任何其他θ的选择都将具有完全相同的能量，这意味着无质量的南部-戈德斯通玻色子的存在，这种模式在势能的最小值绕圆运动，并表明存在拉格朗日方程中原始对称性的一些记忆。

Other examples

  • For ferromagnetic materials, the underlying laws are invariant under spatial rotations. Here, the order parameter is the magnetization, which measures the magnetic dipole density. Above the Curie temperature, the order parameter is zero, which is spatially invariant, and there is no symmetry breaking. Below the Curie temperature, however, the magnetization acquires a constant nonvanishing value, which points in a certain direction (in the idealized situation where we have full equilibrium; otherwise, translational symmetry gets broken as well). The residual rotational symmetries which leave the orientation of this vector invariant remain unbroken, unlike the other rotations which do not and are thus spontaneously broken.
  • 对于铁磁性材料，其基本定律在空间旋转下是不变的。在这里，序参量是衡量磁偶极子密度的磁化强度。在居里温度以上，序参量为零，具有空间不变性，不存在对称性破缺。然而，在居里温度以下，磁化强度变成一个恒定的非零值，指向一个特定的方向(在有充分平衡的理想情况下；否则，平移对称性也会破缺)。使该向量方向不变的旋转对称性仍然保留，而其他旋转对称性自发破缺。
  • The laws describing a solid are invariant under the full Euclidean group, but the solid itself spontaneously breaks this group down to a space group. The displacement and the orientation are the order parameters.
  • 描述固体的定律在完整的欧几里得群下是不变的，但固体本身会自发地将这个群分解为一个空间群。其中位移和方向是序参量。
  • General relativity has a Lorentz symmetry, but in FRW cosmological models, the mean 4-velocity field defined by averaging over the velocities of the galaxies (the galaxies act like gas particles at cosmological scales) acts as an order parameter breaking this symmetry. Similar comments can be made about the cosmic microwave background.
  • 广义相对论具有洛伦兹对称性，但在FRW宇宙模型中，定义为星系速度的平均值(星系在宇宙尺度上的行为就像气体粒子) 的平均 4-速度场，作为序参量会打破这种对称性。对于宇宙微波背景辐射也有类似的评论。
  • For the electroweak model, as explained earlier, a component of the Higgs field provides the order parameter breaking the electroweak gauge symmetry to the electromagnetic gauge symmetry. Like the ferromagnetic example, there is a phase transition at the electroweak temperature. The same comment about us not tending to notice broken symmetries suggests why it took so long for us to discover electroweak unification.
  • 对于电弱模型，如前面所解释的，希格斯场的一个分量提供了将电弱规范对称性破缺到电磁规范对称性的序参量。和铁磁的例子一样，在电弱温度下也有相变。同样的关于我们不倾向于注意破缺对称性的评论，也说明了为什么我们花了这么长时间才发现电弱统一。
  • In superconductors, there is a condensed-matter collective field ψ, which acts as the order parameter breaking the electromagnetic gauge symmetry.
  • 在超导体中有一个凝聚态集体场ψ，它是打破电磁规范对称性的序参量。
  • Take a thin cylindrical plastic rod and push both ends together. Before buckling, the system is symmetric under rotation, and so visibly cylindrically symmetric. But after buckling, it looks different, and asymmetric. Nevertheless, features of the cylindrical symmetry are still there: ignoring friction, it would take no force to freely spin the rod around, displacing the ground state in time, and amounting to an oscillation of vanishing frequency, unlike the radial oscillations in the direction of the buckle. This spinning mode is effectively the requisite Nambu–Goldstone boson.
  • 拿一个细长的圆柱形塑料杆，把两端推到一起。在屈曲之前，系统在旋转下是对称的，因此可见圆柱对称性。但在弯曲之后，它看起来就不同了，而且是不对称的。然而，圆柱对称性的特征仍然存在：忽略摩擦，杆可以不受外力自由地自旋，在时间上取代基态，等于一个频率趋于零的振荡，而不是沿屈曲方向的径向振荡。这种自旋模式实际上是必需的南部-戈德斯通玻色子。
  • Consider a uniform layer of fluid over an infinite horizontal plane. This system has all the symmetries of the Euclidean plane. But now heat the bottom surface uniformly so that it becomes much hotter than the upper surface. When the temperature gradient becomes large enough, convection cells will form, breaking the Euclidean symmetry.
  • 考虑无限水平面上的一层均匀的流体。这个系统具有欧几里得平面的所有对称性。但是现在均匀地加热底部表面，使它变得比上表面热得多。当温度梯度足够大时，就会形成对流单元，打破了欧几里得对称。
  • Consider a bead on a circular hoop that is rotated about a vertical diameter. As the rotational velocity is increased gradually from rest, the bead will initially stay at its initial equilibrium point at the bottom of the hoop (intuitively stable, lowest gravitational potential). At a certain critical rotational velocity, this point will become unstable and the bead will jump to one of two other newly created equilibria, equidistant from the center. Initially, the system is symmetric with respect to the diameter, yet after passing the critical velocity, the bead ends up in one of the two new equilibrium points, thus breaking the symmetry.
  • 考虑一个围绕某个竖直的直径旋转的圆形箍上的珠子。当旋转速度从静止逐渐增加时，珠子最初会停留在环底部的初始平衡点(直观上稳定，重力势最低)。在一定的临界旋转速度下，这一点将变得不稳定，珠子将跳到另外两个新创建的离中心等距离的平衡点中的一个。起初，系统相对直径是对称的，但在通过临界速度后，珠子最终停留在两个新的平衡点中的一个，从而打破了对称性。

Spontaneous symmetry breaking in physics 物理学中的自发对称性破缺

文件:Spontaneous symmetry breaking (explanatory diagram).png
Spontaneous symmetry breaking illustrated: At high energy levels (left), the ball settles in the center, and the result is symmetric. At lower energy levels (right), the overall "rules" remain symmetric, but the symmetric "Sombrero" enforces an asymmetric outcome, since eventually the ball must rest at some random spot on the bottom, "spontaneously", and not all others.

Particle physics 粒子物理

In particle physics, the force carrier particles are normally specified by field equations with gauge symmetry; their equations predict that certain measurements will be the same at any point in the field. For instance, field equations might predict that the mass of two quarks is constant. Solving the equations to find the mass of each quark might give two solutions. In one solution, quark A is heavier than quark B. In the second solution, quark B is heavier than quark A by the same amount. The symmetry of the equations is not reflected by the individual solutions, but it is reflected by the range of solutions.

在粒子物理学中，载力子通常由规范对称的场方程表示；它们的方程预测到某些测量值在场的任何点上都是相同的。例如，场方程可以预测两个夸克的质量是常数。通过求解方程来求每个夸克的质量可能会得到两个解。在一个解中，夸克A比夸克B重；在第二个解中，夸克B比夸克A重相同的量。方程的对称性不是由单个解来反映的，而是由解的范围来反映的。

An actual measurement reflects only one solution, representing a breakdown in the symmetry of the underlying theory. "Hidden" is a better term than "broken", because the symmetry is always there in these equations. This phenomenon is called spontaneous symmetry breaking (SSB) because nothing (that we know of) breaks the symmetry in the equations.[4]:194–195

一个实际的测量只反映了一个解，代表了其潜在理论的对称性的破缺。在这里“隐藏”是比“破坏”更好的术语，因为对称性总是存在于这些方程中。这种现象被称为自发对称破缺(SSB)，因为(我们所知道的)没有任何东西会打破方程中的对称性。

Chiral symmetry 手性对称性

模板:Main article Chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of the strong interactions in particle physics. It is a property of quantum chromodynamics, the quantum field theory describing these interactions, and is responsible for the bulk of the mass (over 99%) of the nucleons, and thus of all common matter, as it converts very light bound quarks into 100 times heavier constituents of baryons. The approximate Nambu–Goldstone bosons in this spontaneous symmetry breaking process are the pions, whose mass is an order of magnitude lighter than the mass of the nucleons. It served as the prototype and significant ingredient of the Higgs mechanism underlying the electroweak symmetry breaking.

手性对称破缺是粒子物理中影响强相互作用手性对称的自发对称破缺的一个例子。手性对称性破缺是量子色动力学(描述这些相互作用的量子场理论)的一种特性，它是核子的大部分质量(超过99%)的成因，因此也是所有普通物质的主要成因，因为它将非常轻的束缚夸克转化为100倍重量的重子的成分。在这个自发对称破缺过程中，近似的南部-戈德斯通玻色子是介子，其质量比核子的质量轻一个数量级。它是电弱对称破缺的希格斯机制的原型和重要组成部分。

Higgs mechanism 希格斯机制

模板:Main article

The strong, weak, and electromagnetic forces can all be understood as arising from gauge symmetries. The Higgs mechanism, the spontaneous symmetry breaking of gauge symmetries, is an important component in understanding the superconductivity of metals and the origin of particle masses in the standard model of particle physics. One important consequence of the distinction between true symmetries and gauge symmetries, is that the spontaneous breaking of a gauge symmetry does not give rise to characteristic massless Nambu–Goldstone physical modes, but only massive modes, like the plasma mode in a superconductor, or the Higgs mode observed in particle physics.

强、弱和电磁力都可以理解为来自规范对称。希格斯机制，即规范对称的自发对称破缺机制，是理解金属超导性和粒子物理标准模型中粒子质量起源的重要组成部分。区分真正的对称性和规范对称性的一个重要的结果，是规范对称性的自发破缺不产生典型的无质量Nambu-Goldstone物理模式，而是只产生有质量的模式，像超导体中的等离子体模式,或者粒子物理学中观察到的希格斯模式。

In the standard model of particle physics, spontaneous symmetry breaking of the SU(2) × U(1) gauge symmetry associated with the electro-weak force generates masses for several particles, and separates the electromagnetic and weak forces. The W and Z bosons are the elementary particles that mediate the weak interaction, while the photon mediates the electromagnetic interaction. At energies much greater than 100 GeV, all these particles behave in a similar manner. The Weinberg–Salam theory predicts that, at lower energies, this symmetry is broken so that the photon and the massive W and Z bosons emerge.[5] In addition, fermions develop mass consistently.

Without spontaneous symmetry breaking, the Standard Model of elementary particle interactions requires the existence of a number of particles. However, some particles (the W and Z bosons) would then be predicted to be massless, when, in reality, they are observed to have mass. To overcome this, spontaneous symmetry breaking is augmented by the Higgs mechanism to give these particles mass. It also suggests the presence of a new particle, the Higgs boson, detected in 2012.

Superconductivity of metals is a condensed-matter analog of the Higgs phenomena, in which a condensate of Cooper pairs of electrons spontaneously breaks the U(1) gauge symmetry associated with light and electromagnetism.

Condensed matter physics

Most phases of matter can be understood through the lens of spontaneous symmetry breaking. For example, crystals are periodic arrays of atoms that are not invariant under all translations (only under a small subset of translations by a lattice vector). Magnets have north and south poles that are oriented in a specific direction, breaking rotational symmetry. In addition to these examples, there are a whole host of other symmetry-breaking phases of matter — including nematic phases of liquid crystals, charge- and spin-density waves, superfluids, and many others.

There are several known examples of matter that cannot be described by spontaneous symmetry breaking, including: topologically ordered phases of matter, such as fractional quantum Hall liquids, and spin-liquids. These states do not break any symmetry, but are distinct phases of matter. Unlike the case of spontaneous symmetry breaking, there is not a general framework for describing such states.[6]

Continuous symmetry

The ferromagnet is the canonical system that spontaneously breaks the continuous symmetry of the spins below the Curie temperature and at h = 0, where h is the external magnetic field. Below the Curie temperature, the energy of the system is invariant under inversion of the magnetization m(x) such that m(x) = −m(−x). The symmetry is spontaneously broken as h → 0 when the Hamiltonian becomes invariant under the inversion transformation, but the expectation value is not invariant.

Spontaneously-symmetry-broken phases of matter are characterized by an order parameter that describes the quantity which breaks the symmetry under consideration. For example, in a magnet, the order parameter is the local magnetization.

Spontaneous breaking of a continuous symmetry is inevitably accompanied by gapless (meaning that these modes do not cost any energy to excite) Nambu–Goldstone modes associated with slow, long-wavelength fluctuations of the order parameter. For example, vibrational modes in a crystal, known as phonons, are associated with slow density fluctuations of the crystal's atoms. The associated Goldstone mode for magnets are oscillating waves of spin known as spin-waves. For symmetry-breaking states, whose order parameter is not a conserved quantity, Nambu–Goldstone modes are typically massless and propagate at a constant velocity.

An important theorem, due to Mermin and Wagner, states that, at finite temperature, thermally activated fluctuations of Nambu–Goldstone modes destroy the long-range order, and prevent spontaneous symmetry breaking in one- and two-dimensional systems. Similarly, quantum fluctuations of the order parameter prevent most types of continuous symmetry breaking in one-dimensional systems even at zero temperature. (An important exception is ferromagnets, whose order parameter, magnetization, is an exactly conserved quantity and does not have any quantum fluctuations.)

Other long-range interacting systems, such as cylindrical curved surfaces interacting via the Coulomb potential or Yukawa potential, have been shown to break translational and rotational symmetries.[7] It was shown, in the presence of a symmetric Hamiltonian, and in the limit of infinite volume, the system spontaneously adopts a chiral configuration — i.e., breaks mirror plane symmetry.

Dynamical symmetry breaking

Dynamical symmetry breaking (DSB) is a special form of spontaneous symmetry breaking in which the ground state of the system has reduced symmetry properties compared to its theoretical description (i.e., Lagrangian).

Dynamical breaking of a global symmetry is a spontaneous symmetry breaking, which happens not at the (classical) tree level (i.e., at the level of the bare action), but due to quantum corrections (i.e., at the level of the effective action).

Dynamical breaking of a gauge symmetry 模板:Ref is subtler. In the conventional spontaneous gauge symmetry breaking, there exists an unstable Higgs particle in the theory, which drives the vacuum to a symmetry-broken phase. (See, for example, electroweak interaction.) In dynamical gauge symmetry breaking, however, no unstable Higgs particle operates in the theory, but the bound states of the system itself provide the unstable fields that render the phase transition. For example, Bardeen, Hill, and Lindner published a paper that attempts to replace the conventional Higgs mechanism in the standard model by a DSB that is driven by a bound state of top-antitop quarks. (Such models, in which a composite particle plays the role of the Higgs boson, are often referred to as "Composite Higgs models".)[8] Dynamical breaking of gauge symmetries is often due to creation of a fermionic condensate — e.g., the quark condensate, which is connected to the dynamical breaking of chiral symmetry in quantum chromodynamics. Conventional superconductivity is the paradigmatic example from the condensed matter side, where phonon-mediated attractions lead electrons to become bound in pairs and then condense, thereby breaking the electromagnetic gauge symmetry.

Generalisation and technical usage

For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes. However, if the system is sampled (i.e. if the system is actually used or interacted with in any way), a specific outcome must occur. Though the system as a whole is symmetric, it is never encountered with this symmetry, but only in one specific asymmetric state. Hence, the symmetry is said to be spontaneously broken in that theory. Nevertheless, the fact that each outcome is equally likely is a reflection of the underlying symmetry, which is thus often dubbed "hidden symmetry", and has crucial formal consequences. (See the article on the Goldstone boson.)

When a theory is symmetric with respect to a symmetry group, but requires that one element of the group be distinct, then spontaneous symmetry breaking has occurred. The theory must not dictate which member is distinct, only that one is. From this point on, the theory can be treated as if this element actually is distinct, with the proviso that any results found in this way must be resymmetrized, by taking the average of each of the elements of the group being the distinct one.

The crucial concept in physics theories is the order parameter. If there is a field (often a background field) which acquires an expectation value (not necessarily a vacuum expectation value) which is not invariant under the symmetry in question, we say that the system is in the ordered phase, and the symmetry is spontaneously broken. This is because other subsystems interact with the order parameter, which specifies a "frame of reference" to be measured against. In that case, the vacuum state does not obey the initial symmetry (which would keep it invariant, in the linearly realized Wigner mode in which it would be a singlet), and, instead changes under the (hidden) symmetry, now implemented in the (nonlinear) Nambu–Goldstone mode. Normally, in the absence of the Higgs mechanism, massless Goldstone bosons arise.

The symmetry group can be discrete, such as the space group of a crystal, or continuous (e.g., a Lie group), such as the rotational symmetry of space. However, if the system contains only a single spatial dimension, then only discrete symmetries may be broken in a vacuum state of the full quantum theory, although a classical solution may break a continuous symmetry.

Nobel Prize

On October 7, 2008, the Royal Swedish Academy of Sciences awarded the 2008 Nobel Prize in Physics to three scientists for their work in subatomic physics symmetry breaking. Yoichiro Nambu, of the University of Chicago, won half of the prize for the discovery of the mechanism of spontaneous broken symmetry in the context of the strong interactions, specifically chiral symmetry breaking. Physicists Makoto Kobayashi and Toshihide Maskawa, of Kyoto University, shared the other half of the prize for discovering the origin of the explicit breaking of CP symmetry in the weak interactions.[9] This origin is ultimately reliant on the Higgs mechanism, but, so far understood as a "just so" feature of Higgs couplings, not a spontaneously broken symmetry phenomenon.

See also

Notes

  • 模板:Note Note that (as in fundamental Higgs driven spontaneous gauge symmetry breaking) the term "symmetry breaking" is a misnomer when applied to gauge symmetries.

References

  1. ↑ Anderson, P.W. (1972). "More is Different" (PDF). Science. 177 (4047): 393–396. Bibcode:1972Sci...177..393A. doi:10.1126/science.177.4047.393. PMID 17796623.
  2. ↑ Edelman, Gerald M. (1992). Bright Air, Brilliant Fire: On the Matter of the Mind. New York: BasicBooks. p. 203. https://archive.org/details/brightairbrillia00gera. 
  3. ↑ Goldstone, J. (1961). "Field theories with " Superconductor " solutions". Il Nuovo Cimento. 19 (1): 154–164. Bibcode:1961NCim...19..154G. doi:10.1007/BF02812722. S2CID 120409034.
  4. ↑ Steven Weinberg (20 April 2011). Dreams of a Final Theory: The Scientist's Search for the Ultimate Laws of Nature. Knopf Doubleday Publishing Group. ISBN 978-0-307-78786-6. https://books.google.com/books?id=Rsg3PE_9_ccC. 
  5. ↑ A Brief History of Time, Stephen Hawking, Bantam; 10th anniversary edition (1998). pp. 73–74.模板:ISBN?
  6. ↑ Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang (2010). "Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order". Phys. Rev. B. 82 (15): 155138. arXiv:1004.3835. Bibcode:2010PhRvB..82o5138C. doi:10.1103/physrevb.82.155138. S2CID 14593420.
  7. ↑ Kohlstedt, K.L.; Vernizzi, G.; Solis, F.J.; Olvera de la Cruz, M. (2007). "Spontaneous Chirality via Long-range Electrostatic Forces". Physical Review Letters. 99 (3): 030602. arXiv:0704.3435. Bibcode:2007PhRvL..99c0602K. doi:10.1103/PhysRevLett.99.030602. PMID 17678276. S2CID 37983980.
  8. ↑ William A. Bardeen; Christopher T. Hill; Manfred Lindner (1990). "Minimal dynamical symmetry breaking of the standard model". Physical Review D. 41 (5): 1647–1660. Bibcode:1990PhRvD..41.1647B. doi:10.1103/PhysRevD.41.1647. PMID 10012522.
  9. ↑ The Nobel Foundation. "The Nobel Prize in Physics 2008". nobelprize.org. Retrieved January 15, 2008.

External links

模板:Wikiquote

取自“https://wiki.swarma.org/index.php?title=对称性破缺&oldid=24763”