The structure of the space of the physical states 物理态空间的结构
1.1 Introduction 1.1 引言
Symmetry principles play a central role in the understanding of natural phenomena. However, it is not always easy to recognize symmetries in physical observations since at a phenomenological level they can manifest as distorted, "rearranged" symmetries. For example, the fundamental symmetry between protons and neutrons, the nucleons, does not manifest as an exact symmetry, but as a "broken symmetry": the charge independence of nuclear interaction is indeed violated by the electromagnetic interaction. In general, various symmetry schemes, which are quite successful, also appear to be in some way approximate symmetry schemes [79, 343, 443, 476, 617], i.e, one has to disregard some phenomenological aspects, eg., mass differences, which violate certain symmetry requirements. A way of looking at this situation is to interpret the observed deviations from the exact symmetry as a phenomenological distortion or rearrangement of the basic symmetry. Examples of rearranged symmetries are easily found in solid state physics: crystals manifest a periodic structure, but do not possess the continuous translational invariance of the Hamiltonian of molecular gas. Ferromagnets present rotational invariance around the magnetization axis, but not the original SU(2) invariance of the Lagrangian. In superconductivity and superfluidity the phase invariance is the one that disappears. 对称性原理在理解自然现象中扮演着核心角色。然而，要在物理观测中识别对称性并非易事，因为在现象学层面上，它们可能表现为扭曲的、"重构"的对称形式。例如，质子与中子（核子）之间的基本对称性并非以精确对称形式呈现，而是表现为"破缺对称性"：核相互作用的电荷独立性确实被电磁相互作用所破坏。一般而言，各种相当成功的对称性方案在某种程度上也表现为近似对称性方案[79, 343, 443, 476, 617]，即必须忽略某些违反特定对称性要求的现象学特征（如质量差异）。理解这种情况的一种方式，是将观测到的对称性偏离现象解释为基本对称性在现象学层面的扭曲或重构。重构对称性的例子在固体物理中随处可见：晶体呈现周期性结构，却不具备分子气体哈密顿量所具有的连续平移不变性。 铁磁体呈现出围绕磁化轴的旋转对称性，但不再保持拉格朗日量原有的 SU(2) 对称性。在超导和超流现象中，消失的则是相位对称性。
The crucial problem one has to face in the recognition of a symmetry is, then, the intrinsic two-level description of Nature: oneaspect of this duality concerns original symmetries ascribed to "basic" entities, the other aspect concerns the corresponding rearranged symmetries of observable phenomena. This two-level description of Nature was soon recognized in Quantum Field Theory (QFT) as the duality between fields and particles. Without
going into the historical developments of this concept, which are outside the purpose of this book, we only recall, as an example, how fundamental this duality is in the renormalization theory, where the distinction is crucial between "bare" and "observed" particles, namely the distinction between basic fields and their physical "manifestation". 识别对称性时面临的关键问题在于自然界本质的双层次描述：这种二元性的一个层面涉及归属于"基本"实体的原始对称性，另一层面则关乎可观测现象对应的重整化对称性。量子场论(QFT)中很早就认识到这种自然界双层次描述体现为场与粒子之间的二元性。本书不拟追溯这一概念的历史发展，仅以重整化理论为例说明这种二元性的根本重要性——其中"裸粒子"与"观测粒子"的区分（即基本场与其物理"显现"的区分）具有决定性意义。
In the following Sections we will focus our attention on somestructural aspects of QFT in order to prepare the tools to be used in the study of the mechanisms through which the dynamics of the basic fields leads to their observable physical manifestation. Thus, the core of our interest will be the structure of the space of the physical fields, which will bring us to study that peculiar nature of QFT consisting in the existence of infinitely many unitarily inequivalent representations of the canonical (anti-)commutation relations, and thus to the analysis of the von Neumann theorem, of the Weyl-Heisenberg algebra, the characterization of the physical fields and the coherent states. Our discussion will include in a unified view, topics such as the squeezing and self-similarity transformations, fractals and quantum deformation of the Weyl-Heisenberg algebra. A glance at the table of contents shows how these subjects are distributed in the various Sections and Appendices. 在后续章节中，我们将重点关注量子场论（QFT）的某些结构特征，以便为研究基本场动力学如何导致其可观测物理表现机制准备工具。因此，我们关注的核心将是物理场空间的结构——这将引导我们研究量子场论的特殊性质，即正则（反）对易关系存在无限多个幺正不等价表示，进而分析冯·诺依曼定理、外尔-海森堡代数、物理场的表征以及相干态。我们的讨论将以统一视角涵盖诸如压缩变换与自相似变换、分形结构以及外尔-海森堡代数的量子形变等主题。通过目录概览可见这些内容在各章节与附录中的分布情况。
1.2 The space of the states of physical particles 1.2 物理粒子的态空间
Let us consider a typical scattering process between two or more particles. By convenient measurements we can identify the kind, the number, the energy, etc., of the particles before they interact (incoming particles); there is then an interaction region which is precluded to observations and finally we can again measurethe kind, the number, the energy, etc., of the particles after the interaction (outgoing particles). The sum of the energies of the incoming particles is observed to be equal to the sum of the energies of the outgoing particles. Incoming particles and outgoing particles are referred to as "physical particles", or else as "observed" or "free" particles, where the word "free" does not exclude the possibility of interaction among them; it means that the interaction among the particles can be considered to be negligible far away, in spaceand time, from the interaction region. Thetotal energy of the system of free particles is given in a good approximation by the sum of the energies of the single particles. We require that the energy of the physical particles is determined as a certain function of their momenta. In solid state physics the physical particles are usually called quasiparticles. 让我们考虑一个典型的两粒子或多粒子散射过程。通过适当的测量，我们可以在粒子相互作用前（入射粒子）确定其种类、数量、能量等参数；随后存在一个无法观测的相互作用区域；最终我们又能测量相互作用后粒子（出射粒子）的种类、数量、能量等参数。实验观测表明，入射粒子总能量等于出射粒子总能量。入射粒子与出射粒子被称为"物理粒子"，或称"可观测粒子"或"自由粒子"——此处"自由"一词并不排除粒子间存在相互作用的可能性，而是指在远离相互作用区域的时空范围内，粒子间的相互作用可以忽略不计。自由粒子系统的总能量在良好近似下等于各单粒子能量之和。我们要求物理粒子的能量必须由其动量确定的某种函数关系给出。 在固态物理学中，物理粒子通常被称为准粒子。
Among the quasiparticles, a special roleis played by those, such as phonons, magnons, etc., called "collective modes", which are responsible for longrange correlation among the elementary components of the system. 在准粒子中，诸如声子、磁振子等被称为"集体模式"的粒子发挥着特殊作用，它们负责系统基本组分之间的长程关联。
The possibility of identifying outgoing and/or incoming particles resides in the possibility of setting our particle detector far away from the interaction region (at a space distance x=±∞ from the interaction region) and to let it be active well before and/ or well after the interaction time (at a time t=±∞ with respect to the interaction time). In other words, we assume that the interaction forces among the particles go to zero at spacetime regions far away from the spacetime interaction region. 1 This is indeed possible in most cases. However, there are important cases in which this "switching off" of the interaction is not possible due to intrinsic properties of the interaction. When this latter situation occurs, we cannot apply the usual methods of perturbation theory. The validity of the perturbative methods relies indeed on the possibility of correctly defining "asymptotic" states for the system under study, namely states properly defined in spacetime regions where the interaction effects are negligible. 识别出射和/或入射粒子的可能性，源于我们能够将粒子探测器置于远离相互作用区域的空间位置（与相互作用区域保持 x=±∞ 的空间距离），并使其在相互作用时刻之前足够早和/或之后足够晚的时间段内保持活跃（相对于相互作用时刻有 t=±∞ 的时间间隔）。换言之，我们假设粒子间的相互作用力在远离时空相互作用区域的时空区域内趋近于零。 1 这在大多数情况下确实可行。然而，也存在因相互作用本身特性而无法实现这种"关闭"的重要情形。当出现后一种情况时，我们便无法应用常规的微扰理论方法。微扰方法的有效性本质上依赖于能否正确定义所研究系统的"渐近"态，即在相互作用效应可忽略的时空区域内明确定义的状态。
We now briefly summarize the main steps in the construction of the Hilbert space for the physical particles. Among several possible strategies [115, 343, 466, 558, 599, 666], we mostly follow [617, 619, 621]. 我们现在简要概述构建物理粒子希尔伯特空间的主要步骤。在多种可能策略中[115, 343, 466, 558, 599, 666]，我们主要遵循[617, 619, 621]的方法。
The state of a single particle is classified by the suffices ( i,s ), where i specifies the spatial distribution of the state, while s specifies other freedoms (e.g., spin, charges, etc.). For simplicity, we assume we are dealing with only one kind of particle (e.g., only electrons, or only protons, etc.). We must use wave packets to specify spatial distributions, because plane waves like exp(ik⋅x) are not normalizable and do not form a countable set. On the other hand, it is well known that an orthonormalized complete set of square-integrablefunctions {fi(x),i=1,2,…} is a countable set. Thus we introduce the creation operators αis† and βis† for particles and their antiparticles, respectively, with spatial distribution fi(x), i.e., in wave-packet states, as 单个粒子的状态由下标( i,s )分类，其中 i 指定状态的空间分布，而 s 指定其他自由度（如自旋、电荷等）。为简化起见，我们假设仅处理一种粒子（如仅电子或仅质子等）。必须使用波包来指定空间分布，因为像 exp(ik⋅x) 这样的平面波不可归一化且不构成可数集。另一方面，众所周知，平方可积函数的正交归一完备集 {fi(x),i=1,2,…} 是可数集。因此我们分别引入粒子及其反粒子的产生算符 αis† 和 βis† ，其空间分布为 fi(x) ，即以波包态形式表示：
In these equations we have also introduced the creation operators αks† and 在这些方程中我们还引入了产生算符 αks† 和
1βks† for particles and their anti-particles of momentum k. Their hermitian conjugates αis,βis and αks,βks, respectively, denote the annihilation operators. In Eqs. (1.1) fi(k) are the Fourier amplitudes of fi(x) 1βks† 表示动量为 k 的粒子及其反粒子的产生算符。它们的厄米共轭 αis,βis 和 αks,βks 则分别表示湮灭算符。在方程(1.1)中， fi(k) 是 fi(x) 的傅里叶振幅。
fi(x)=∫(2π)3d3kfi(k)eik⋅x(1.2)
The possibility of expressing the spatial distribution of the state by a discreteindex follows from the fact that we use the square-integrable functions (wave-packets) fi(x) which form a countable set. The space of states so constructed is a separable Hilbert space, whose states can be expressed as superposition of the countable set of basis vectors fi(k). 通过离散指标表达态的空间分布的可能性源于我们使用了平方可积函数（波包） fi(x) ，这些函数构成可数集。如此构建的态空间是可分希尔伯特空间，其态矢量可表示为可数基矢 fi(k) 的线性叠加。
For the norm we will use the notation ∣fi∣=(fi,fi)1/2=1. In general, any square-integrable normalized function f(x) is expressed as 对于范数，我们将采用符号 ∣fi∣=(fi,fi)1/2=1 。一般而言，任何平方可积的归一化函数 f(x) 可表示为
f(x)=i∑aifi(x),(1.5)
where ai (and f(x) ) may depend on time and ∑i∣ai∣2=1 due to the normalization (f,f)=1 : 其中 ai （及 f(x) ）可能随时间与 ∑i∣ai∣2=1 变化，这源于归一化条件 (f,f)=1 ：
We also introduce, in analogy with Eqs. (1.1), the wave-packet operators associated to the spatial distribution f(x) : 我们类比方程(1.1)，引入与空间分布 f(x) 相关联的波包算子：
where f(k) are the Fourier amplitudes of f(x).
For brevity, we will omit the suffix s when no confusion arises. We next assume the existence of the physical vacuum state ∣0⟩ defined as 此处 f(k) 表示 f(x) 的傅里叶振幅。为简洁起见，若无混淆之虞，我们将省略下标 s 。接着我们假定存在物理真空态 ∣0⟩ ，其定义为
αi∣0⟩=0=βi∣0⟩.(1.8)
The conjugate state ⟨0∣ is such that 共轭态 ⟨0∣ 满足
⟨0∣αi†=0=⟨0∣βi†,(1.9)
and ⟨0∣0⟩=1. From Eqs. (1.1), it is then, for any k , 且 ⟨0∣0⟩=1 。根据方程(1.1)，对于任意 k 有
αk∣0⟩⟨0∣αk†=0=βk∣0⟩=0=⟨0∣βk†(1.10a)(1.10b)
The Hilbert space of physical particle states is cyclically constructed by repeated applications of αi† and βi† on ∣0⟩, e.g., the one-particle state is αi†∣0⟩=∣αi⟩. Let us denote by ni the number of particles with spatial distribution given by fi(x). The n-particle state of the system is specified by the state vector ∣n1,n2,…⟩, where the numbers ni are assigned for each i, and is given by 物理粒子态的希尔伯特空间是通过对 ∣0⟩ 反复施加 αi† 和 βi† 循环构建的，例如单粒子态为 αi†∣0⟩=∣αi⟩ 。我们用 ni 表示空间分布由 fi(x) 给定的粒子数。系统的 n 粒子态由态矢量 ∣n1,n2,…⟩ 确定，其中为每个 i 分配数值 ni ，其表达式为
∣n1,n2…⟩=i∏ni!1(αi†)ni∣0⟩,(1.11)
in the case of boson particles and by 在玻色子情形下，以及
∣n1,n2…⟩=i∏(αi†)ni∣0⟩,(1.12)
in the case of fermion particles. In Eq. (1.12) ∏i denotes product with ni restricted to the values 0 or 1 for any i. The conjugate vectors are introduced in the usual way, e.g., for bosons 对于费米子粒子情形。在方程(1.12)中， ∏i 表示乘积运算，其中 ni 对于任意 i 仅限于取值 0 或 1。共轭向量按常规方式引入，例如对于玻色子
⟨n1,n2…∣=⟨0∣i∏ni!1(αi)ni.(1.13)
The action of αi and αi† on the states of the Hilbert space is given by αi 和 αi† 在希尔伯特空间状态上的作用由下式给出
In a standard fashion [619] one can show that the set {∣n1,n2,…⟩} is an orthonormalized set of vectors in the Hilbert space: 按照标准方法[619]可以证明，集合 {∣n1,n2,…⟩} 构成希尔伯特空间中的一组正交归一化向量：
⟨n1′,n2′,…∣n1,n2,…⟩=i∏δni′ni(1.17)
We are using the simplified notation ∣n1,n2,…⟩≡∣n1⟩⊗∣n2⟩⊗…, where ni can be any non-negative integer for bosons; it is 0 or 1 for fermions. In this notation, the vacuum ∣0⟩ is ∣0,0,…⟩≡∣0⟩⊗∣0⟩⊗…. Eqs. (1.14) show that 我们采用简化记号 ∣n1,n2,…⟩≡∣n1⟩⊗∣n2⟩⊗… ，其中 ni 对玻色子可为任意非负整数；对费米子则为 0 或 1。在此记号下，真空态 ∣0⟩ 表示为 ∣0,0,…⟩≡∣0⟩⊗∣0⟩⊗… 。方程(1.14)表明
Ni∣n1,…,ni,…⟩=ni∣n1,…,ni,…⟩,(1.18)
where Ni=αi†αi is the number operator, or, by restoring the suffix s, 此处 Ni=αi†αi 为粒子数算符，若恢复下标 s 则可表示为
Nis=αis†αis.(1.19)
The total number is given by N=∑i,sNis. Let us consider particles which are bosons. Eqs. (1.14) and (1.17) give for any ni 's
𝟙⟨n1,n2,…∣[αis,αjr†]∣n1,n2,…⟩=1δijδrs,
⟨n1,n2,…∣[αis,αjr]∣n1,n2,…⟩=0=⟨n1,n2,…∣[αis†,αjr†]∣n1,n2,…⟩,
𝟙𝟙⟨n1,n2,…∣[αis,1]∣n1,n2,…⟩=0=⟨n1,n2,…∣[αis†,1]∣n1,n2,…⟩,
and similarly for βis and βis†. Here 𝟙1 denotes the identity operator. Consistency with Eqs. (1.1) then requires that 总粒子数由 N=∑i,sNis 给出。现考虑玻色子情形，方程(1.14)和(1.17)对任意 ni 满足 𝟙⟨n1,n2,…∣[αis,αjr†]∣n1,n2,…⟩=1δijδrs 、 ⟨n1,n2,…∣[αis,αjr]∣n1,n2,…⟩=0=⟨n1,n2,…∣[αis†,αjr†]∣n1,n2,…⟩ 、 𝟙𝟙⟨n1,n2,…∣[αis,1]∣n1,n2,…⟩=0=⟨n1,n2,…∣[αis†,1]∣n1,n2,…⟩ ，对于 βis 和 βis† 亦有类似关系。其中 𝟙1 表示单位算符。与方程(1.1)的自洽性要求
and similarly for βks and βks†. Dueto the delta function δ(k−l) appearing in Eqs. (1.21), these equations areto be understood in the sense of distribution theory, namely 同样适用于 βks 和 βks† 。由于方程(1.21)中出现的δ函数 δ(k−l) ，这些方程需在分布理论的意义上理解，即
In the case of fermions we obtain relations similar to (1.20)-(1.23) with anticommutators replacing the commutators. The vectors of the Hilbert space can also be proven to be, under particle permutations, fully symmetrical states for bosons and fully antisymmetrical states for fermions. 在费米子情形下，我们得到与(1.20)-(1.23)类似的关系式，但需用反对易子替代对易子。可以证明，在粒子置换下，希尔伯特空间中的矢量对玻色子为全对称态，对费米子则为全反对称态。
By repeated applications of creation and annihilation operators one can move from one member to another member in the set {∣n1,n2,…⟩}; however, the operators αks (and βks ) are not bounded operators: due to 通过反复应用产生和湮灭算子，可以在集合 {∣n1,n2,…⟩} 中从一个成员转移到另一个成员；然而，算子 αks （及 βks ）并非有界算子：由于
Eqs. (1.21), they do not map normalizable vectors on normalizable ones; indeed, ⟨0∣αksαks†∣0⟩=δ(0), which is not finite. However, operators associated with spatial distribution fi(x) (or f(x) ) give αis†0⟩∣2=1 (or αfs†0⟩∣2=1 ) dueto Eqs. (1.1) and Eq. (1.4) (or, for αfs†, dueto Eqs. (1.7) and Eq. (1.6)). 方程(1.21)并未将可归一化向量映射为可归一化向量；事实上， ⟨0∣αksαks†∣0⟩=δ(0) 并非有限值。然而，由于方程(1.1)和(1.4)（或对于 αfs† 而言，由于方程(1.7)和(1.6)），与空间分布 fi(x) （或 f(x) ）相关联的算子会产生 αis†0⟩∣2=1 （或 αfs†0⟩∣2=1 ）。
We consider in more detail the mathematical nature of the Hilbert space in the following Section. 我们将在下节更详细地探讨希尔伯特空间的数学性质。
1.3 The Weyl-Heisenberg algebra and the Fock space 1.3 韦尔-海森堡代数与福克空间
We require that it must be possible to express any vector in the Hilbert space, i.e, any physical state of the system, as a superposition of the vectors of the basis. This implies that the Hilbert space must be separable, i.e., it must contain a countable basis of vectors, say {ξn}. In such a case, for any vector ξ of the space and any arbitrary ϵ>0 there exist a sequence {cn} such that ∣ξ−∑ncnξn∣<ϵ, which means that ξ can be approximated by the linear superposition ∑ncnξn to any accuracy. 我们要求必须能够将希尔伯特空间中的任意向量（即系统的任何物理态）表示为基向量的线性叠加。这意味着希尔伯特空间必须是可分的，即必须包含一组可数的基向量，记为 {ξn} 。在这种情况下，对于空间中的任意向量 ξ 和任意 ϵ>0 ，都存在一个序列 {cn} 使得 ∣ξ−∑ncnξn∣<ϵ 成立，这意味着 ξ 可以通过线性叠加 ∑ncnξn 以任意精度逼近。
If we thus require that our space must be a separable Hilbert space, it is not correct to use the set {∣n1,n2,…⟩} as a basis, because this is not a countable set. To provethis, let us consider for simplicity a fermion system where ni can assume only the values 0 or 1 . Then, we consider the set of positive numbers 因此，若要求空间必须是可分离的希尔伯特空间，则使用集合 {∣n1,n2,…⟩} 作为基是不正确的，因为该集合不可数。为证明这一点，我们简化为一个费米子系统考虑，其中 ni 仅能取值 0 或 1。于是，我们考察由正数构成的集合
{0.n1n2…},(1.24)
with ni assuming only the values 0 or 1 . Using the binary system, we see that the set (1.24) covers all the real values in the interval [0,1], i.e, it is a non-countable set. On the other hand, there is a one-to-one correspondence between the set (1.24) and the set {∣n1,n2,…⟩}, and thus we conclude that the latter is a non-countable one. Since the set {∣n1,n2,…⟩} for bosons is larger than the onefor fermions, also in the boson case it is a non-countable set. To remedy this situation, we observe that physical states do not really contain an infinite number of particles. The number of particles can be as large as we want, but does not need to be infinite. We then select a countable subset S from the set {∣n1,n2,…⟩} as follows: 其中 ni 仅取值 0 或 1。通过二进制表示可见，集合(1.24)覆盖了区间 [0,1] 内的所有实数值，即构成不可数集。另一方面，集合(1.24)与集合 {∣n1,n2,…⟩} 存在一一对应关系，故可判定后者亦为不可数集。鉴于玻色子情形下的集合 {∣n1,n2,…⟩} 较费米子情形更大，其同样构成不可数集。为解决此问题，我们注意到物理态实际上并不包含无限多粒子——粒子数可任意大但无需无限。于是我们从集合 {∣n1,n2,…⟩} 中按如下方式选取可数子集 S ：
S={∣n1,n2,…⟩,i∑ni= finite }(1.25)
This set contains the vacuum ∣0,0,…⟩ but does not contain states like ∣1,1,1,…⟩ where ni=1 for all i. Actually, one can extract infinitely many
countable subsets from the set {∣n1,n2,…⟩}, each of them representing a different possible representation of the (anti-)commutation relations of the operators αi,αi†,i=1,2,…. Two of these representations will be said to be unitarily inequivalent representations when one arbitrary vector of one of them cannot be expressed as a superposition of base vectors of the other representations. We will discuss this point below and we will see that the existence of unitarily inequivalent representations is a characterizing feature of QFT, not present in Quantum Mechanics [123, 569, 621, 648, 649]. 该集合包含真空态 ∣0,0,…⟩ ，但不包含诸如 ∣1,1,1,…⟩ 的态（其中对所有 i 均有 ni=1 ）。实际上，可以从集合 {∣n1,n2,…⟩} 中提取出无限多个可数子集，每个子集都代表着算符 αi,αi†,i=1,2,… （反）对易关系的不同可能表示。当任意一个表示的向量无法表示为其他表示基向量的叠加时，这两个表示被称为酉不等价表示。我们将在下文讨论这一点，并将看到酉不等价表示的存在是量子场论区别于量子力学的特征性现象[123, 569, 621, 648, 649]。
We now prove that the set S is a countable set.
Consider the state ∣n1,n2,…⟩ belonging to S. Since this state contains only a finite number of particles, there is an integer number, say p, for which np=0 and ni=0 for i>p. Then, to each vector in S we can associate two numbers, i.e., p and N=∑ini. For each product pN there exists only a finite number of vectors in S because for each pN we can distribute a finite number of particles only in a finite number of states. This means that we can label the vectors in S as ξa with a=1,2,… in such a way that pN is not decreasing for a increasing. The set S is thus countable and ξa are orthonormal vectors: (ξa,ξb)=δab. 我们现在证明集合 S 是一个可数集。考虑属于 S 的状态 ∣n1,n2,…⟩ 。由于该状态仅包含有限数量的粒子，存在一个整数 p ，使得当 i>p 时 np=0 且 ni=0 。于是，对于 S 中的每个向量，我们可以关联两个数字，即 p 和 N=∑ini 。对于每个乘积 pN ， S 中只存在有限数量的向量，因为对于每个 pN ，我们只能将有限数量的粒子分配到有限数量的状态中。这意味着我们可以将 S 中的向量标记为 ξa ，其中 a=1,2,… 的取值方式使得 pN 随着 a 的增加而不递减。因此集合 S 是可数的，且 ξa 是正交归一向量： (ξa,ξb)=δab 。
We consider now the linear space HF defined by 我们现在考虑由下式定义的线性空间 HF ：
The vectors in HF thus have finite norm: HF 中的向量具有有限范数：
∣ξ∣2=(ξ,ξ)=a∑∣ca∣2= finite (1.29)
Note that a vector ζ of HF is the null vector, i.e., ζ=0, if and only if all the coefficients ba=(ξa,ζ) (cf. (1.28)) are zero. 注意， HF 中的向量 ζ 是零向量，即 ζ=0 ，当且仅当所有系数 ba=(ξa,ζ) （参见(1.28)）均为零。
The linear space HF is called the Fock space of physical particles. In this space one considers the set D of all the finite summations of the basis vectors of HF : 线性空间 HF 被称为物理粒子的福克空间。在该空间中，人们考虑由 HF 基向量所有有限求和构成的集合 D ：
D={ξN(D)=a=1∑Ncaξa,N finite }.(1.30)
The set D can be proven to be dense in HF, i.e., every vector of D belongs to HF and every vector of HF is either a member of D or the limit of a Cauchy sequence of vectors in D. This last property can be expressed as 可以证明集合 D 在 HF 中稠密，即 D 中的每个向量都属于 HF ，且 HF 中的每个向量要么是 D 的成员，要么是 D 中向量的柯西序列极限。最后这个性质可以表述为
ξ=N→∞limξN(D),(1.31)
where ξ is a vector of HF. Eq. (1.31) has to be understood in the sense of the strong limit (see Appendix A for a definition of strong and weak limit). When the relations (1.20) and (1.21) are computed using vectors belonging to D, then they hold for any vector in HF and we can write the commutation relations in the Hilbert space as 其中 ξ 是 HF 的一个向量。方程(1.31)需在强极限意义下理解（强弱极限的定义参见附录 A）。当使用属于 D 的向量计算关系式(1.20)和(1.21)时，这些关系对 HF 中的任意向量都成立，因此我们可以在希尔伯特空间中将对易关系表示为
and similar relations for βks and βks†. Thealgebra (1.32) generated by αis,αis† and 𝟙1, for any s, and similarly the algebra (1.21), is a Lie algebra and is called the Heisenberg algebra or also the Weyl-Heisenberg (WH) algebra. Instead of Eqs. (1.32) and (1.21) we have anti-commutation relations in the case of fermions. The algebra (1.32) (and (1.21)) is also referred to as the canonical commutation relation (or anti-commutation relation, in the case of fermions) algebra. 对于 βks 和 βks† 也存在类似关系。由 αis,αis† 和 𝟙1 （对于任意 s ）生成的代数(1.32)，以及类似的代数(1.21)，构成李代数，称为海森堡代数或外尔-海森堡(WH)代数。对于费米子情形，方程(1.32)和(1.21)需替换为反对易关系。代数(1.32)（及(1.21)）也被称为正则对易关系（对于费米子情形则为反对易关系）代数。
As a final remark weobservethat assigning the algebra (1.32) (or (1.21)) is not enough to specify the particular countablesubset S onemay select out of the set {∣n1,n2,…⟩}. Since the states are obtained by cyclic operation of αi† on the vacuum state, in order to specify S one needs to assign also the vacuum state annihilated by the αi and on which the αi† operators are defined as shown in Eq. (1.11). 最后需要指出的是，我们观察到仅赋予代数(1.32)（或(1.21)）并不足以从集合 {∣n1,n2,…⟩} 中确定特定的可数子集 S 。由于这些态是通过 αi† 对真空态的循环作用获得的，为了确定 S ，还需要指定被 αi 湮灭的真空态，并在该态上按照方程(1.11)定义 αi† 算符。
1.4 Irreducible representations of the canonical commutation relations 1.4 正则对易关系的不可约表示
We here assume that the particles under consideration are bosons. We use ℏ=c=1. We introduce the dimensionless operators qi and pi defined by 此处我们假设所考虑的粒子是玻色子。采用 ℏ=c=1 ，引入由以下定义的无量纲算符 qi 和 pi ：
This is also called the Weyl-Heisenberg algebra (generated by the operators 𝟙qi,pi,1, for any i ). We note that by a convenient transformation leaving invariant the canonical commutation relations (1.34) the qi and pi operators can be given the dimensions appropriate to the usual phase-space coordinates. Since we are considering a finite number of particles (cf. Eq. (1.25)) the present situation is very similar to that in Quantum Mechanics (QM). In particular, the Hilbert space under consideration is the oscillator realization of the canonical variables qi and pi. It is known to be a complete space and we can use the well-known [660,661] "unitarization" or "extension" procedure, in which one considers the operators 这也被称为 Weyl-Heisenberg 代数（由任意 i 的算符 𝟙qi,pi,1 生成）。我们注意到，通过保持正则对易关系(1.34)不变的一个便利变换，可以使 qi 和 pi 算符具有与常规相空间坐标相称的维度。由于我们考虑的是有限数量的粒子（参见方程(1.25)），当前情形与量子力学(QM)中的情况非常相似。特别地，所考虑的希尔伯特空间是正则变量 qi 和 pi 的振子实现。已知这是一个完备空间，我们可以采用著名的[660,661]"幺正化"或"扩展"程序，其中考虑算符
instead of pi and qi, with σ and τ real parameters. However, it should be stressed that in QM the pi and qi operators are not bounded operators [621, 649,660,661]. In the present QFT case, as noted in the previous Sections, the unbounded operators αk and αk† are smeared out by use of the squareintegrable test functions fi(k) (cf. Eqs. (1.1)), so that in (1.33) we are using wave-packets operators αi and αi†. One can show that the operators Ui(σ) and Vi(σ) are bounded operators and thereforetheir definition can be "extended" on the whole HF (see below). The conclusion is that the Fock space of the physical particles is a representation of the unitary operators Ui(σ) and Vi(τ), with i=1,2,…. 这里使用实数参数 σ 和 τ 来代替 pi 和 qi 。但需要强调的是，在量子力学中 pi 和 qi 算子并非有界算子[621, 649,660,661] ]。在当前量子场论情形下，如前面章节所述，无界算子 αk 和 αk† 通过平方可积测试函数 fi(k) （参见方程(1.1)）进行平滑处理，因此在(1.33)中我们使用的是波包算子 αi 和 αi† 。可以证明算子 Ui(σ) 和 Vi(σ) 是有界算子，因此其定义可以"扩展"到整个 HF 空间（见下文）。结论是：物理粒子的福克空间是酉算子 Ui(σ) 和 Vi(τ) 的一个表示，其中 i=1,2,… 。
We also introduce U(σ) and V(τ) as 我们还引入 U(σ) 和 V(τ) 如下：
where we assume that only a finite number of σi and τi are not zero. The operators U(σ) and V(τ) satisfy the so-called Weyl algebra: 其中我们假设仅有有限个 σi 和 τi 不为零。算子 U(σ) 和 V(τ) 满足所谓的 Weyl 代数：
The relations (1.37) reflect the canonical commutation relations (1.34) (or (1.32)). It is also customary to introduce the so-called Weyl operator W(z) : 关系式(1.37)反映了正则对易关系(1.34)（或(1.32)）。通常还会引入所谓的 Weyl 算子 W(z) ：
W(z)≡exp(iσ⋅τ)U(2σ)V(2τ)(1.38)
with z≡σ+iτ. Eqs. (1.37) then lead to 与 z≡σ+iτ 相关。方程(1.37)由此导出
The knowledge of U(σ) and V(τ) can tell us about piξ and qiξ, respectively, whenever such vectors belong to HF. Indeed, 当这些向量属于 HF 时，对 U(σ) 和 V(τ) 的认知能分别揭示 piξ 和 qiξ 的特性。实际上，
Below we show that any operator which commutes with U(σ) and V(τ) is a multiple of the identity operator, which means that the Fock space is an irreducible representation of the canonical variables qi and pi, i.e, of the annihilation and creation operators of physical particles, or, equivalently, of the Weyl operator introduced above. Sometimes we refer to the Fock space irreducible representation of the Weyl operator as the Weyl system. 下文将证明：任何与 U(σ) 和 V(τ) 对易的算子都是恒等算子的倍数，这意味着福克空间是正则变量 qi 和 pi 的不可约表示——即物理粒子的湮灭与产生算符的表示，或等价地说，是前述外尔算子的表示。我们有时将外尔算子在福克空间中的不可约表示称为外尔系统。
In conclusion, the description of the system in terms of physical (boson) particles naturally leads to canonical variables {qi,pi,i=1,2,…} whose irreducible representation is the Fock space defined above. 综上所述，以物理（玻色子）粒子为出发点的系统描述，自然引出了正则变量 {qi,pi,i=1,2,…} ，其不可约表示正是上述定义的福克空间。
Extension of the Weyl operators on HF 外尔算子在 HF 上的扩展
We introduce 我们引入
UiM(σ)=n=0∑Mn!1(iσpi)n(1.41)
where M is a positive and finite integer. Since the action of any power of αi and αi† on a vector of the basis gives another vector of the basis, the action of any positive power of pi and of qi on vectors ξN in the dense set D creates a superposition of finite number of vectors in D, which is still a vector of D. Then, the sequence of vectors UiM(σ)ξN(D) has a limit for M→∞; thus we define the operation of Ui(σ) on D as 其中 M 是一个正有限整数。由于 αi 和 αi† 的任何幂次作用在基向量上都会产生另一个基向量，因此 pi 和 qi 的任何正幂次作用在稠密集 D 中的向量 ξN 上时，会产生 D 中有限数量向量的叠加，这仍然是 D 中的一个向量。于是，向量序列 UiM(σ)ξN(D) 在 M→∞ 时存在极限；因此我们将 Ui(σ) 在 D 上的运算定义为
Ui(σ)ξN(D)=M→∞limUiM(σ)ξN(D)
Due to the unitarity of Ui(σ), 由于 Ui(σ) 的酉性，
Ui(σ)ξN(D)=ξN(D),(1.42)
from which we conclude that the operator Ui(σ) is a bounded operator and therefore its definition can be "extended" on the whole HF in the following way: let ξ be a vector of HF; if it is a vector of D, the action of Ui(σ) on ξ is well defined. If ξ is not a vector belonging to D, we can find in D a Cauchy sequence {ξN(D)} whose limit is ξ; then we define the action of Ui(σ) on ξ as 由此我们得出结论：算子 Ui(σ) 是有界算子，因此其定义可以按以下方式"延拓"至整个 HF 空间：设 ξ 为 HF 空间中的向量；若其属于 D 空间，则 Ui(σ) 对 ξ 的作用已有明确定义。若 ξ 不属于 D 空间，我们可在 D 中找到一个柯西序列 {ξN(D)} 使其极限为 ξ ；于是将 Ui(σ) 对 ξ 的作用定义为
Ui(σ)ξ=N→∞limUi(σ)ξN(D)(1.43)
In a similar way, we can define the action of Vi(τ) on HF.
We now show that any operator which commutes with U(σ) and V(τ) is a multiple of the identity operator, which implies that the Fock space is an irreducible representation of the canonical variables qi and pi, i.e, of the annihilation and creation operators of physical particles. To see this we note that if ξ is a vector of HF,αiξ=0 for all i, when and only when ξ=c∣0,0,…⟩, due to (1.8), with c an ordinary number. If A is an operator commuting with qi and pi, for all i, i.e, with αi and αi†, for all i, then αiA∣0,0,…⟩=Aαi∣0,0,…⟩=0, i.e., A∣0,0,…⟩=c∣0,0,…⟩, with c an ordinary number. Since any vector of the basis (1.25) is constructed by repeated operations of αi†, e.g., 类似地，我们可以定义 Vi(τ) 对 HF 的作用。现证明任何与 U(σ) 和 V(τ) 对易的算子都是恒等算子的倍数，这意味着福克空间是正则变量 qi 和 pi （即物理粒子的湮灭与产生算子）的不可约表示。为此我们注意到：当且仅当 ξ=c∣0,0,…⟩ 时，由于(1.8)式（其中 c 为普通数）， ξ 才是对所有 i 成立的 HF,αiξ=0 空间向量。若 A 是与 qi 和 pi 对易的算子（即对所有 i 与 αi 和 αi† 对易），则 αiA∣0,0,…⟩=Aαi∣0,0,…⟩=0 ，即 A∣0,0,…⟩=c∣0,0,…⟩ （其中 c 为普通数）。由于基(1.25)中的任何向量都是通过 αi† 的重复作用构造的，例如
with f(n1,n2,…) somefunction of ni (consistent with Eq. (1.14)), we have 其中 f(n1,n2,…) 为 ni 的某函数（与方程(1.14)一致）
A∣n1,n2,…⟩=c∣n1,n2,…⟩,
i.e., using Eq. (1.26) Aξ=cξ for any ξ in HF. This means that 𝟙A=c1, with 𝟙1 the identity operator. 即，对任意 ξ 在 HF 中使用方程(1.26) Aξ=cξ 。这意味着 𝟙A=c1 ，其中 𝟙1 是恒等算子。
Labeling the irreducible representations 标记不可约表示
Since, as already noted in Section 1.3, the choice of the countable basis is not unique, we now consider the problem of labeling the Weyl operators and the Weyl systems. 由于如第 1.3 节所述，可数基的选择并不唯一，我们现在考虑标记 Weyl 算子和 Weyl 系统的问题。
We consider the transformation 我们考虑变换
σ→ρ1σ,τ→ρτ,(1.45)
with ρ a non-zero real c-number, ρ=0. This is a canonical transformation, since ℑm(z1∗⋅z2) in Eq. (1.39) is left invariant under it and the Weyl algebra (1.39) is therefore preserved: 其中 ρ 为非零实常数， ρ=0 。这是一个正则变换，因为方程(1.39)中的 ℑm(z1∗⋅z2) 在此变换下保持不变，因此外尔代数(1.39)得以保持：
where Wρ(z)≡W(ρ1σ+iρτ). We thus see that the transformation parameter ρ acts as a label for the Weyl systems. 此处 Wρ(z)≡W(ρ1σ+iρτ) 。由此可见，变换参数 ρ 充当了外尔系统的标记。
We observe [341] that the transformation (1.45) can be equivalently thought as applied to pi and to qi instead of σi and τi. Let us consider for simplicity one specific value of i (extension to many values of i is straightforward). Therefore, we will omit the index i in the following: 我们注意到[341]，变换(1.45)可以等价地视为作用于 pi 和 qi ，而非 σi 和 τi 。为简化起见，考虑 i 的特定取值（推广到多个 i 值的情况是直接的）。因此下文将省略下标 i ：
p→p(ρ)≡ρ1p,q→q(ρ)=ρq,ρ=0.(1.47)
The action variable J=∫pdq is invariant under (1.47); this clarifies the physical meaning of the invariance of the area ℑm(z1∗⋅z2) under (1.45). Of course, the transformation (1.47) is a canonical transformation: ( [q,p]=i)→([q(ρ),p(ρ)]=i). By inverting Eqs. (1.33) we have 作用量变量 J=∫pdq 在变换(1.47)下保持不变；这阐明了面积 ℑm(z1∗⋅z2) 在变换(1.45)下不变性的物理意义。显然，变换(1.47)是正则变换：( [q,p]=i)→([q(ρ),p(ρ)]=i) 。通过反转方程(1.33)我们得到
so that u2−v2=1. Eqs. (1.48) are then recognized to be nothing but Bogoliubov transformations; specifically, they are the squeezing transformations occurring in solid state physics and in quantum optics [280] and in elementary particle physics, see, e.g., [13, 108]. Let ρ≡e−ζ=ρ(ζ),ζ=∞ and real. Eqs. (1.48) are then put in the form 因此 u2−v2=1 。于是可识别出方程(1.48)正是博戈留波夫变换；具体而言，它们是凝聚态物理、量子光学[280]以及基本粒子物理中出现的压缩变换，参见例如[13, 108]。设 ρ≡e−ζ=ρ(ζ),ζ=∞ 且为实数。则方程(1.48)可表示为
where we have used α(ζ)≡α(ρ(ζ)). The ρ-labeling or parametrization is called the Bogoliubov parametrization of the Weyl algebra or Weyl systems [341]. The generator of the Bogoliubov transformations (1.50) is: 此处我们已使用 α(ζ)≡α(ρ(ζ)) 。这种 ρ 标记或参数化方法称为外尔代数或外尔系统的博戈留波夫参数化[341]。博戈留波夫变换(1.50)的生成元为：
In quantum optics S^(ζ) is called the squeezing operator [279,664], ζ being the squeezing parameter. We notethat ther.h.s. of Eq. (1.51) is an SU(1,1) group element. In fact, by defining K−=21α2,K+=21α†2,Kz=21(α†α+21 ), one easily checks they close the algebra su(1,1) (cf. Appendix C). 在量子光学中， S^(ζ) 被称为压缩算子[279,664]，其中 ζ 为压缩参数。我们注意到方程(1.51)的右侧是一个 SU(1,1) 群元素。事实上，通过定义 K−=21α2,K+=21α†2,Kz=21(α†α+21 )，可轻易验证它们构成了 su(1,1) 代数（参见附录 C）。
In the transition from QM to QFT, namely from finite to infinite number of degrees of freedom, one must operate in the complex linear space EC=E+iE instead of working in CM, where M denotes the (finite) number of degrees of freedom (i=1,2,…,M). Here E denotes a real linear space of square-integrable functions f; we shall denote by F=f+ig,f,g∈E, the elements in EC. The scalar product ⟨F1,F2⟩ in EC is defined through the the scalar product (f,g) in E : 在从量子力学(QM)过渡到量子场论(QFT)时，即从有限自由度到无限自由度的转变过程中，必须在复线性空间 EC=E+iE 中进行操作，而非在 CM 中操作，其中 M 表示(有限的)自由度数量 (i=1,2,…,M) 。此处 E 表示平方可积函数的实线性空间 f ；我们将用 F=f+ig,f,g∈E 表示 EC 中的元素。 EC 中的标量积 ⟨F1,F2⟩ 是通过 E 中的标量积 (f,g) 定义的：
It must be stressed that the use of the complex linear space EC in QFT is required to smear out spatial integrations of field operators by means of test functions f. 必须强调的是，量子场论中使用复线性空间 EC 是为了通过测试函数 f 将场算子的空间积分进行弥散处理。
Our discussion in this Section has been confined to the case of boson operators. We will see that, although fermion operators cannot be traced back to the canonical variables {qi,pi,i=1,2,…}, nevertheless there exist also for them infinitely many unitarily inequivalent Fock spaces which are irreducible representations of the anti-commutation relations. 本节讨论仅限于玻色子算子的情形。我们将看到，尽管费米子算子无法追溯到正则变量 {qi,pi,i=1,2,…} ，但它们同样存在无限多个不可等价幺正化的 Fock 空间，这些空间都是反对易关系的不可约表示。
We now go back to the Weyl-Heisenberg algebra (1.34) (or (1.32), (1.21)). In the following, we will omit for simplicity the suffix i. As customary, we introduce for each i the notation 𝟙e1=ip,e2=iq,e3=i1 [519]. By regarding these as elements of an abstract Lie algebra, we recognize the WH algebra introduced above to be, for each i, a real three-dimensional Lie algebra given by the commutation relations 现在我们回到 Weyl-Heisenberg 代数(1.34)(或(1.32)、(1.21))。为简洁起见，下文将省略下标 i 。按照惯例，我们为每个 i 引入记号 𝟙e1=ip,e2=iq,e3=i1 [519]。将这些视为抽象李代数的元素时，可以认识到上述 WH 代数对于每个 i 而言，都是由对易关系给出的实三维李代数：
[e1,e2]=e3,[e1,e3]=0=[e2,e3].(1.55)
The generic element x of the algebra is written as x=(s;x1,x2)=x1e1+x2e2+se3, with s,x1 and x2 real numbers; or, 该代数的通用元素 x 可表示为 x=(s;x1,x2)=x1e1+x2e2+se3 ，其中 s,x1 和 x2 为实数；或，
𝟙𝟙x=is1+i(τq−σp)=is1+(gα†−g∗α),(1.56)
where we have used Eq. (1.33) and we have put 此处我们使用了方程(1.33)并设...
x1≡−σ,x2≡τ,g=21(σ+iτ),(1.57)
and g∗ is the complex conjugate of g. The commutator of the elements x=(s;x1,x2) and y=(t;y1,y2) is 且 g∗ 是 g 的复共轭。元素 x=(s;x1,x2) 与 y=(t;y1,y2) 的对易子为
[x,y]=B(x,y)e3,B(x,y)=x1y2−x2y1,(1.58)
where B(x,y) is recognized to be the standard symplectic form on the ( x1,x2 ) plane. 其中 B(x,y) 可视为( x1,x2 )平面上的标准辛形式。
It is now possible to construct the Lie group corresponding to the Lie algebra by exponentiation: 现在可以通过指数映射构造对应于该李代数的李群：
𝟙exp(x)=exp(is1)D(g),D(g)=exp(gα†−g∗α).(1.59)
By use of the formula 利用公式
exp(A)exp(B)=exp(21[A,B])exp(A+B)(1.60)
which holds provided [A[A,B]]=0=[B[A,B]], one obtains the multiplication law 在满足 [A[A,B]]=0=[B[A,B]] 的条件下，可得乘法法则：
D(f)D(g)=exp(iℑm(fg∗))D(f+g),(1.61)
from which 由此可得
D(f)D(g)=exp(2iℑm(fg∗))D(g)D(f).(1.62)
This last relation has to be compared with the last one of Eqs. (1.37), of which it provides another realization. Like the operators U and V, the
operators D(g) are bounded operators and defined on the whole HF. In conclusion, the operators 𝟙exp(is1)D(g) form a representation of the group whose elements are specified by three real numbers γ=(s;x1,x2), or by a real number s and a complex number g,γ=(s;g). This group is called the Weyl-Heisenberg group and denoted by W1. The multiplication rule is: 最后这个关系式需与方程(1.37)中的末式相比较，它提供了该方程的另一种实现形式。与算子 U 和 V 类似，算子 D(g) 也是有界算子且定义在整个 HF 空间上。综上所述，算子 𝟙exp(is1)D(g) 构成了一个群的表示，该群的元素由三个实数 γ=(s;x1,x2) 或一个实数 s 与一个复数 g,γ=(s;g) 确定。此群称为 Weyl-Heisenberg 群，记作 W1 。其乘法规则为：
The center of the group W1, i.e., the set of all the elements commuting with every element of W1, is given by the elements (s;0). Let us denote by T(γ) any unitary irreducible representation of W1.T(γ) is also said to be a unitary irreducible representation of the canonical commutation relations ((1.32) or (1.21), or (1.34)). Then the operators T(s;0) form a unitary representation of the subgroup {(s;0)}. They are specified by a real number λ : 群 W1 的中心（即与群 W1 中所有元素可交换的元素集合）由元素 (s;0) 给出。我们用 T(γ) 表示 W1.T(γ) 的任意酉不可约表示——这也称为正则对易关系（(1.32)或(1.21)，或(1.34)）的酉不可约表示。那么算子 T(s;0) 就构成了子群 {(s;0)} 的酉表示，它们由一个实数 λ 参数化确定。
𝟙Tλ(s;0)=exp(iλs)1(1.64)
Furthermore, there are representations for which λ=0 and arespecified by a pair of real numbers, say μ and ν : 𝟙Tμν(γ)=exp{i(μx1+νx2)}1. 此外，存在一些表示方式，其中 λ=0 和由一对实数（例如 μ 和 ν ）指定： 𝟙Tμν(γ)=exp{i(μx1+νx2)}1 。
By generalizing D(g) in Eq. (1.59), we can also introduce the operator 通过推广方程(1.59)中的 D(g) ，我们还可以引入算符
G(g)=exp(∫d3k[gkαk†−gk∗αk])(1.65)
which, acting on αk, generates the transformation 该算符作用于 αk 时，会产生变换
G−1(g)αkG(g)=αk+gk≡αk(g)(1.66)
The commutation rules for the αk(g) and αk†(g) operators are the same as the ones in Eq. (1.21). The transformation (1.66) is thus a canonical transformation since it preserves the commutation rules. Relations similar to Eqs. (1.61) and (1.62) hold for the operator G(g). αk(g) 和 αk†(g) 算符的对易规则与方程(1.21)中的相同。因此，变换(1.66)是一个正则变换，因为它保持了对易规则。对于算符 G(g) ，也存在类似于方程(1.61)和(1.62)的关系。
Clearly, αk(g) acts as the annihilation operator on the state 显然， αk(g) 作为湮灭算符作用于该态
∣0(g)⟩≡G−1(g)∣0⟩(1.67)
but it does not annihilate ∣0⟩. Note that 但它并不湮灭 ∣0⟩ 。需要注意的是
⟨O(g)∣O(g)⟩=1(1.68)
Then we can construct the "new" Fock space HF(g) by using ∣0(g)⟩ as the vacuum and by repeating the construction followed for obtaining HF. In this way we get another representation of the canonical commutation relations (1.21). They are unitarily equivalent representations provided G−1(g) is a unitary operator. In the next Section we show that in QFT there exist infinitely many unitarily inequivalent representations, and we discuss the conditions under which this happens and the related physical meaning. 于是我们可以通过将 ∣0(g)⟩ 作为真空态，并重复用于构建 HF 的构造方法，来建立"新"的福克空间 HF(g) 。这样我们就得到了正则对易关系(1.21)的另一种表示。只要 G−1(g) 是幺正算符，这些表示就是幺正等价的。下一节我们将证明在量子场论中存在无限多个幺正不等价的表示，并讨论这种情况发生的条件及其相关物理意义。
1.6 The Stone-von Neumann theorem 1.6 斯通-冯·诺伊曼定理
By use of Eq. (1.60) we can write Eq. (1.65) as 利用方程(1.60)，我们可以将方程(1.65)表示为
which shows that the inner product of the vacuum state ∣0(g)⟩ for the "new" operators αk(g) with the "old" vacuum ∣0⟩ is 这表明"新"算符 αk(g) 的真空态 ∣0(g)⟩ 与"旧"真空 ∣0⟩ 的内积为
⟨0∣0(g)⟩=exp(−21∫d3k∣gk∣2),(1.71)
which is zero provided 当且仅当
21∫d3k∣gk∣2=∞(1.72)
For instance, this happens when gk=cδ(k), with c a real constant. In such a case, by using the delta function representation 例如，当 gk=cδ(k) （其中 c 为实常数）时就会出现这种情况。此时，利用δ函数的表示法
δ(k)=(2π)31∫d3xeik⋅x(1.73)
and denoting the volume by V=∫d3x, we can formally write 并将体积记为 V=∫d3x ，我们可以形式化地写出
exp(−21∫d3kd3qgkgq∗δ(k−q))=exp(−2(2π)3c2∫d3x∫d3kd3qei(k−q)⋅xδ(k)δ(q))=exp(−21(2π)3Vc2)→0, for V→∞(1.74)
Similarly, for gk=cδ(k) and gk′=c′δ(k),c=c′, we obtain 类似地，对于 gk=cδ(k) 和 gk′=c′δ(k),c=c′ ，我们得到
⟨0(g′)∣0(g)⟩→0, for g′=g and V→∞.(1.75)
Thus Eqs. (1.71) and (1.75) arezero in the infinite volume limit, and in that limit the representations HF(g) and HF(g′) are unitarily inequivalent for each set of c-numbers g={gk=cδ(k),∀k} and g′={gk=c′δ(k),∀k}, with {gk}={gk′},∀k. In other words, there is no unitary generator G−1(g) which maps HF onto itself in the infinite volume limit. If, on
the contrary, the volume is finite, i.e., the number of degrees of freedom is finite, Eqs. (1.71) and (1.75) are not zero and the representations HF(g) and HF(g′) are unitarily equivalent (and therefore physically equivalent): they are related by a unitary transformation. This is what happens in Quantum Mechanics where only systems with a finite number of degrees of freedom are considered. In the case of infinite volume, Eq. (1.70) is instead only a formal relation: the vacuum ∣0(g)⟩ cannot be expressed as a superposition of states belonging to HF (or to HF(g′),g′=g ) in the infinite volume limit. Since the c-numbers g={gk=cδ(k),∀k} span a continuous domain, in the infinite volume limit we have infinitely many unitarily inequivalent representations {HF(g),∀g={gk=cδ(k),∀k}} labeled by g. Due to Eq. (1.66), g is called the shift parameters. In concrete cases one needs to operate at finite volume and the infinite volume limit has to be performed only at the end of the computations. 因此，在无限体积极限下，方程(1.71)和(1.75)为零，此时对于每组 c 数[2]和[3]（其中[4]），表示[0]和[1]是幺正不等价的。换言之，在无限体积极限下不存在将[6]映射到自身的幺正生成元[5]。反之，若体积有限（即自由度数目有限），则方程(1.71)和(1.75)不为零，表示[7]和[8]是幺正等价的（因而物理等价）：它们通过幺正变换相关联。这正是量子力学中仅考虑有限自由度系统时出现的情况。对于无限体积情形，方程(1.70)仅是一个形式关系：真空态[9]在无限体积极限下不能表示为属于[10]（或[11]）的态叠加。由于 c 数[12]跨越连续域，在无限体积极限下我们存在无限多个由[14]标记的幺正不等价表示[13]。根据方程(1.66)，[15]被称为位移参数。 在实际应用中，需要在有限体积下进行计算，无限体积极限仅需在计算完成后取定。
In conclusion, we thus have arrived at the so-called Stone-von Neumann theorem [580, 648, 649] (or, simply, von Neumann theorem), which states that for systems with a finite number of degrees of freedom, which is always the case with Quantum Mechanics, the representations of the canonical commutation relations are all unitarily equivalent to each other. In QFT, the number of degrees of freedom is infinite and the von Neumann theorem does not hold: infinitely many unitarily inequivalent representations of the canonical (anti-)commutation relations exist. 综上所述，我们由此得到了所谓的斯通-冯·诺伊曼定理[580, 648, 649]（或简称为冯·诺伊曼定理），该定理指出：对于具有有限自由度的系统（量子力学始终满足此条件），正则对易关系的所有表示都是彼此酉等价的。而在量子场论中，自由度数量是无限的，冯·诺伊曼定理不再成立——存在无限多个正则（反）对易关系的酉不等价表示。
Our discussion is not confined to the relativistic domain. QFT applies also to non-relativistic many-body systems in condensed matter physics. In this last case, one considers the so-called thermodynamic limit in which the infinite volume limit is understood in such a way that the density N/V is kept constant, with N denoting the particle number. One way to visualize this is to consider that at the boundary surfaces of the system the potential barrier is not infinite. Thus wave-packets can spread outside and a continuous distribution of momentum is allowed. 我们的讨论不仅限于相对论领域。量子场论同样适用于凝聚态物理中的非相对论多体系统。在后一种情况下，人们考虑所谓的"热力学极限"，即无限体积极限的理解方式需保持密度 N/V 恒定，其中 N 表示粒子数。一种形象化的理解方式是：假设系统边界表面的势垒并非无限高，因此波包可以向外扩散，从而允许动量的连续分布。
We will comment in the following Section on the physical meaning of the existence of infinitely many unitarily inequivalent representations in QFT. 我们将在下一节中讨论量子场论中存在无限多个幺正不等价表示的物理意义。
It is finally necessary to comment on the case of fermions. We have discussed the von Neumann theorem for the bosonic case where the creation and annihilation operators may be introduced through the operators {qi,pi} as in (1.33). This cannot be the case for the fermion creation and annihilation operators, which need to be directly introduced as in (1.1), without reference to the operators {qi,pi}. However, the result of the von Neumann theorem also holds true for the fermionic case, namely infinitely
many unitarily inequivalent representations of the anti-commutation relations also exist in the infinite volume limit. We will give an explicit example of this in Chapter 2, Example 2b, which may be adopted as an explicit proof of the von Neumann theorem for the fermionic case. 最后有必要对费米子的情况加以说明。我们已讨论了玻色子情形的冯·诺伊曼定理，其中产生湮灭算符可通过(1.33)式中的 {qi,pi} 算符引入。但费米子的产生湮灭算符不能如此处理，它们需要如(1.1)式所示直接引入，而不涉及 {qi,pi} 算符。然而冯·诺伊曼定理的结论同样适用于费米子情形，即在无限体积极限下，反对易关系同样存在无限多个幺不等价的表示。我们将在第 2 章示例 2b 中给出具体实例，该实例可作为费米子情形下冯·诺伊曼定理的显式证明。
A final remark is that the operator D(g) (or G(g) ) introduced above is the generator of coherent states related with the Weyl-Heisenberg group [519]. Essential notions on single mode coherent states (Glauber coherent states) are presented in Appendix B (for their functional integral representation see Appendix N). In Appendix C we discuss how to extract a complete set of coherent states from an over-complete set. 需要补充说明的是，前文引入的 D(g) （或 G(g) ）算符是与 Weyl-Heisenberg 群[519]相关的相干态生成元。关于单模相干态（Glauber 相干态）的基本概念见附录 B（其泛函积分表示参见附录 N）。附录 C 将讨论如何从过完备集中提取完备的相干态集。
In Section 1.3 we have seen that the set {∣n1,n2,…⟩} is not a countable set and thus it cannot be used as a basis for the space of states if werequiresuch a space to be a separable one. Then, we have extracted from {∣n1,n2,…⟩} the subset S as in Eq. (1.25): 在 1.3 节中我们已经看到，集合 {∣n1,n2,…⟩} 不是可数集，因此如果我们要求状态空间是可分的，它就不能用作状态空间的基。于是，我们从 {∣n1,n2,…⟩} 中提取了子集 S ，如方程(1.25)所示：
S={∣n1,n2,…⟩,i∑ni= finite },(1.76)
and shown that this is a countable set. The root of the existence of the infinitely many unitarily inequivalent representations in QFT is in the fact that there are infinitely many ways of choosing a separable subspace out of the original non-separable one. To different countable subsets there correspond different, i.e, unitarily inequivalent, representations of the commutation relations. The meaning of this is that a state vector of a given representation cannot be expressed as a superposition of vectors belonging to another inequivalent representation. We therefore must be careful in selecting the representation describing the physical states of our system under given boundary conditions. Consider, for instance, a ferromagnetic system at a temperature below the Curie temperature. The fact that the ferromagnetic state cannot be expressed as a superposition of non-ferromagnetic (paramagnetic) states means that there is no unitary operator connecting the ferromagnetic phase with the non-ferromagnetic one. Indeed, if such an operator existed, its unitarity would imply that characterizing observables would be left unaltered under its action connecting the ferromagnetic phase to the non-ferromagnetic one. However, for example, the observablemagnetization does change from non-zero to zero in the process of transition from the ferromagnetic to the non-ferromagnetic phase. These phases are thus
physically different in their observable properties and they are therefore to be described by unitarily inequivalent representations. 并已证明这是一个可数集。量子场论中存在无限多个幺正不等价表示的根本原因在于，从原始不可分空间中选择可分子空间存在无限多种方式。不同的可数子集对应着对易关系的不同表示，即幺正不等价表示。这意味着给定表示的状态向量无法表示为另一个不等价表示中向量的叠加。因此，在给定边界条件下选择描述系统物理状态的表示时必须格外谨慎。以低于居里温度的铁磁系统为例，铁磁态无法表示为非铁磁（顺磁）态的叠加，这表明不存在将铁磁相与非铁磁相联系的幺正算符。 确实，若存在这样的算符，其幺正性将意味着可观测量在其连接铁磁相与非铁磁相的作用过程中保持不变。然而，例如磁化强度这一可观测量在从铁磁相过渡到非铁磁相的过程中确实会从非零值变为零值。因此，这些相在可观测性质上具有物理差异，必须通过幺正不等价的表示来描述。
In conclusion, the existence of many unitarily inequivalent representations allows the description of systems which may be in physically different phases under different boundary conditions. Such a situation is excluded in Quantum Mechanics (QM) since there, as we have seen, the von Neumann theorem [648] guarantees that all the representations are unitarily, and therefore physically, equivalent. In this sense, QM can only describe systems in a single specified physical phase. From such a perspective we may say that QFT is drastically different from QM and it provides a much richer framework than Quantum Mechanics. In the course of this book we will see in more detail how the description of physical phases is carried on and what its relation is with the mechanism of the spontaneous breakdown of symmetry. 综上所述，存在许多幺正不等价表示使我们能够描述在不同边界条件下可能处于不同物理相的系统。这种情况在量子力学（QM）中被排除，因为正如我们所见，冯·诺依曼定理[648]保证了所有表示都是幺正（因而物理）等价的。从这个意义上说，量子力学只能描述处于单一特定物理相的系统。由此视角观之，我们可以说量子场论（QFT）与量子力学存在根本性差异，它提供了比量子力学更为丰富的理论框架。在本书后续章节中，我们将更详细地探讨物理相的描述方式及其与对称性自发破缺机制的关系。
We have seen that in the case of the shift transformation, αk→αk(g)=αk+gk, with gk=cδ(k), the vacua ∣0⟩ and ∣0(g)⟩ turn out to be orthogonal (the corresponding representations are unitarily inequivalent). Let us now see the physical meaning of this. 我们已经看到，在平移变换情况下， αk→αk(g)=αk+gk （其中 gk=cδ(k) ）时，真空态 ∣0⟩ 与 ∣0(g)⟩ 是正交的（相应的表示是幺正不等价的）。现在让我们探讨其物理意义。
The number Nk=αk†(g)αk(g) of particles αk(g) in the vacuum ∣0⟩ is given by 真空态 ∣0⟩ 中 αk(g) 粒子的数量 Nk=αk†(g)αk(g) 由以下表达式给出：
⟨0∣αk†(g)αk(g)∣0⟩=∣gk∣2.(1.77)
We then say that there are ∣gk∣2 bosons of momentum k condensed in the state ∣0⟩. The total number of condensed bosons is 我们于是说在状态 ∣0⟩ 中有 ∣gk∣2 个动量为 k 的玻色子发生凝聚。总凝聚玻色子数为
i.e, it is proportional to the system volume V. Thus, an infinite number of bosons is condensed in the vacuum ∣0⟩ in the infinite volume limit. However, the density of these condensed bosons is everywhere finite, even in the infinite volume limit: 即正比于系统体积 V 。因此在无限体积极限下，真空态 ∣0⟩ 中有无限多个玻色子发生凝聚。然而这些凝聚玻色子的密度处处有限，即使在无限体积极限下：
ρ=V1∫d3k∣gk∣2=(2π)31c2.(1.79)
The meaning of gk=cδ(k) is therefore that the density of the boson condensation in the vacuum state ∣0⟩ is spatially homogeneous, i.e., everywhere the same, and finite. This also means that when gk=cδ(k), the transformation (1.66) does not violate the translational invariance of the vacuum state. On the other hand, since ⟨0(g)∣αk†(g)αk(g)∣0(g)⟩=0 everywhere, there are no αk(g) bosons condensed in ∣0(g)⟩. We thus see that the two
vacua ∣0⟩ and ∣0(g)⟩ are different because of their different content in the condensation of αk(g) bosons, being this infinite in one of them in the infinite volume limit. This depicts the physical meaning of the unitary inequivalence between the representations associated to the two vacua. Physical, local observables may thus turn out to be different in the two vacua since the boson condensation density is different in each of them. 因此 gk=cδ(k) 的物理意义在于：真空态 ∣0⟩ 中的玻色子凝聚密度在空间上是均匀分布的，即处处相同且有限。这也意味着当 gk=cδ(k) 时，变换(1.66)不会破坏真空态的平移不变性。另一方面，由于 ⟨0(g)∣αk†(g)αk(g)∣0(g)⟩=0 处处为零，故在 ∣0(g)⟩ 中没有 αk(g) 玻色子发生凝聚。由此可见，两个真空态 ∣0⟩ 与 ∣0(g)⟩ 的差异源于其中 αk(g) 玻色子凝聚含量的不同——在无限体积极限下，其中一个真空态含有无限多凝聚玻色子。这描绘了与两个真空态相关联的表示之间幺正不等价的物理意义。由于两个真空态中玻色子凝聚密度不同，物理上的局域可观测量在两者间可能表现出差异。
In each representation HF(g), for any set g={gk=cδ(k);∀k} (including g=0={gk=0;∀k},⟨0∣αk†αk∣0⟩=0 ), we have a set of creation and annihilation operators {αk†(g),αk(g);∀k}. For each g, i.e., for each representation, we may assume that the associated set {αk†(g),αk(g);∀k} forms an irreducible set of operators. This means that there are, depending on the physical phase in which the system sits, different sets of physical particles appropriate to the system description in that phase. On the other hand, one may always define the action of one set of operators for a given set g on the representation labeled by a different g′(g′=g). For example, the action of αk(g) on HF(g=0) is well defined, as shown in the discussion above, through the mapping αk→αk(g)≡αk+gk,gk=cδ(k). 在每个表示 HF(g) 中，对于任意集合 g={gk=cδ(k);∀k} （包含 g=0={gk=0;∀k},⟨0∣αk†αk∣0⟩=0 ），我们都有一组产生和湮灭算符 {αk†(g),αk(g);∀k} 。对于每个 g ，即对于每个表示，我们可以假设相关联的集合 {αk†(g),αk(g);∀k} 构成了一组不可约算符。这意味着根据系统所处的物理相，存在适用于该相系统描述的不同物理粒子集合。另一方面，人们总可以定义给定集合 g 的算符作用于由不同 g′(g′=g) 标记的表示上的操作。例如，如上文讨论所示，通过映射 αk→αk(g)≡αk+gk,gk=cδ(k) ，可以明确定义 αk(g) 作用于 HF(g=0) 的操作。
Finally, a comment on the operation of normal ordering, by which a given product of a number of operator factors is rearranged in such a way that all the annihilation operators are on the right and all the creation operators on the left. Such a normal ordering is usually denoted by : ⋯:, where dots between the colon denote the operator factors. Expectation value in the vacuum state of normal ordered products is thus zero. However, the discussion above implies that normal ordering is representation dependent, since the annihilation operator in one representation is not such in another unitarily inequivalent representation, and thus normal ordered products have non-zero expectation value in the vacuum of the last representation. A better notation for normal ordering could be: ⋯:g, the label g specifying the representation [90,438] (see also Section 5.3). 最后，关于正规序操作的说明：该操作将多个算符因子的给定乘积重新排列，使所有湮灭算符位于右侧，所有产生算符位于左侧。这种正规序通常表示为: ⋯: ，其中冒号间的点号表示算符因子。因此，正规序乘积在真空态中的期望值为零。然而，上述讨论表明正规序具有表示依赖性，因为某一表示中的湮灭算符在另一幺正不等价表示中并非如此，因此正规序乘积在后一表示的真空态中具有非零期望值。更优的正规序表示法可能是: ⋯:g ，其中标签 g 用于指定表示[90,438]（另见第 5.3 节）。
1.8 The deformation of Weyl-Heisenberg algebra 1.8 外尔-海森堡代数的形变
As we shall see in the present Section and in Chapter 5, the quantumdeformed Hopf algebra is a characterizing structural feature of QFT, intimately related with the existence of the unitarily inequivalent representations of the canonical commutation relations CCR [141-143, 152, 337, 338, 341, 342, 632, 633]. Quantum deformed algebras, usually denoted as q-algebras, are deformations in the enveloping algebras of Lie algebras
whose structure appears to be an essential tool for the description of composed systems. The general properties of q-algebras are better known than those of q-groups. The interest in q-groups arose almost simultaneously in statistical mechanics, in conformal theories, in solid state physics as well as in the study of topologically non-trivial solutions to non-linear equations [207, 359, 437]. The WH algebra admits two inequivalent deformations: one which is properly a q-algebra [144, 145], the other (on which we shall focus our attention here), denoted as q-WH and often referred to as ospq(2∣1), was originated by the seminal work of Biedenharn [75] and MacFarlane [433]. The q-WH algebra is characterized by the property that its intrinsic nature of superalgebra, proper also to the WH algebra itself, plays a non-trivial role, in view of the form of the coproduct. It can therefore be referred to as a Hopf superalgebra [150, 393]. 正如我们将在本节及第五章所见，量子形变的 Hopf 代数是量子场论（QFT）的特征性结构要素，与正则对易关系 CCR[141-143, 152, 337, 338, 341, 342, 632, 633]的幺正不等价表示存在深刻关联。通常标记为 q 代数的量子形变代数，是李代数包络代数的一种形变，其结构已成为描述复合系统的重要工具。相较于 q 群的性质，人们对 q 代数的普遍性质认知更为深入。 q 群的研究兴趣几乎同时在统计力学、共形场论、固体物理以及非线性方程拓扑非平庸解研究中涌现[207, 359, 437]。WH 代数允许两种不等价形变：一种是严格意义上的 q 代数[144, 145]；另一种（本文重点讨论对象）记为 q -WH，常被称为 ospq(2∣1) ，其理论渊源可追溯至 Biedenharn[75]与 MacFarlane[433]的开创性工作。 q -WH 代数的特征在于其超代数的内在性质（这一性质同样适用于 WH 代数本身）在余积形式下发挥着非平凡作用，因此可将其称为 Hopf 超代数[150, 393]。
The q-WH algebra has been shown [142, 143] to be related to coherent states, to squeezed coherent states, to the Bloch functions in periodic potentials, to lattice QM and in general to the physics of discretized (periodic) systems. For completeness, we briefly discuss in this Section the q-deformed WH algebra and its relation with the Fock-Bargmann representation (FBR) in QM. For a more detailed account see [142] and [143]. The relation with coherent states and the theta functions is briefly presented in A ppendix D. 已有研究[142,143]表明， q -WH 代数与相干态、压缩相干态、周期势中的布洛赫函数、晶格量子力学以及离散化（周期性）系统的物理普遍相关。为完备起见，本节将简要讨论 q -变形 WH 代数及其与量子力学中 Fock-Bargmann 表示（FBR）的关联，更详尽的论述参见[142]和[143]。附录 D 简要介绍了该代数与相干态及θ函数的关系。
We consider for simplicity the operators for one single mode. The WH algebra is generated by the operators 𝟙{a,a†,1} with commutation relations 为简化起见，我们考虑单模算子。WH 代数由满足下列对易关系的 𝟙{a,a†,1} 算子生成：
𝟙[a,a†]=1,[N,a]=−a,[N,a†]=a†,(1.80)
and the other commutators vanishing. Here N≡a†a. The representation of (1.80), which here we denote by K, is the Fock space generated by the eigenkets of N with integer (positive and zero) eigenvalues. Any state vector ∣ψ⟩ in K is thus described by the set {cn;cn∈C} defined by ∣ψ⟩=∑n=0∞cn∣n⟩, i.e., by its expansion in the complete orthonormal set of eigenkets {∣n⟩} of N. 其他对易子为零。此处 N≡a†a 。式(1.80)的表示（在此记作 K ）是由 N 的本征态（具有整数（正及零）本征值）生成的福克空间。因此， K 中的任意态矢量 ∣ψ⟩ 可由集合 {cn;cn∈C} 描述，其定义为 ∣ψ⟩=∑n=0∞cn∣n⟩ ，即通过 N 的完备正交本征态集 {∣n⟩} 展开表示。
Upon defining H≡N+21, the three operators {a,a†,H} close on K the relations 定义 H≡N+21 后，三个算符 {a,a†,H} 在 K 上满足闭合关系
{a,a†}=2H,[H,a]=−a,[H,a†]=a†,(1.81)
and the other (anti-)commutators vanishing. These relations are equivalent to (1.80) on K and show the intrinsic nature of superalgebra of such a scheme 其余（反）对易子为零。这些关系等价于 K 上的(1.80)式，揭示了该方案固有的超代数本质
In terms of the operators {aq,aˉq,H;q∈C} the q-deformed version of (1.81), the q-WH algebra, is [150, 393]: 用算符 {aq,aˉq,H;q∈C} 表示时，(1.81)式的 q 变形版本—— q -WH 代数可表述为[150, 393]：
is a quantum superalgebra (graded Hopf algebra) and, consequently, all relations (1.82) are preserved under the coproduct map. 构成一个量子超代数（分次 Hopf 代数），因此所有关系式(1.82)在余积映射下都保持不变。
In the space K (i.e, in the space spanned by the vectors {∣n⟩;n∈N} ), Eqs. (1.82) can be rewritten in the equivalent form [75, 150, 433], which makes them more explicitly analogous to the un-deformed case: 在 K 空间（即由向量 {∣n⟩;n∈N} 张成的空间中），方程(1.82)可以改写为等效形式[75,150,433]，这使得它们与非变形情况的类比关系更为显见：
Eqs. (1.85) and (1.86) are deformations only at the algebra level of (1.80). Thus(1.82)-(1.84) is the relevant mathematical structure. However, we prefer to resort henceforth to (1.86), even though the whole discussion could be based on (1.82), since it is perfectly correct as far as we remain in K and it is the most similar to the usual form (1.80) of the WH algebra. 方程（1.85）和（1.86）仅是对（1.80）在代数层面上的变形。因此（1.82）-（1.84）构成了相关的数学结构。不过，我们更倾向于后续采用（1.86），尽管整个讨论完全可以基于（1.82）展开——只要保持在 K 范围内，它就是完全正确的，并且最接近 WH 代数的常规形式（1.80）。
The q-WH algebra and the Fock-Bargmann representation q -WH 代数与 Fock-Bargmann 表示
In the following, let q be any complex number. The notion of hermiticity for the generators of q-WH algebra associated with complex q is non-trivial and has been studied in [142] and [151] in connection with the squeezing of the generalized coherent states (GCS)q over K. 下文设 q 为任意复数。与复数 q 相关联的 q -WH 代数生成元的厄米性概念具有非平凡性，文献[142]和[151]结合 K 上广义相干态 (GCS)q 的压缩现象对此进行了研究。
We now discuss the functional realization of Eqs. (1.86) by means of finite difference operators in the complex plane, in the Fock-Bargmann representation (FBR) of QM [142, 143, 151, 152]. 我们现在讨论通过复平面上的有限差分算子在 Fock-Bargmann 表示(FBR)中实现方程(1.86)的函数形式[142, 143, 151, 152]。
In the FBR, state vectors are described by entire analytic functions, i.e., uniformly converging in any compact domain of the complex z-plane (see also Appendix D), contrary to the usual coordinate or momentum representation where no condition of analyticity is imposed. 在 FBR 中，态矢量由全纯解析函数描述，即在复 z 平面的任何紧致域内一致收敛（另见附录 D），这与通常的坐标或动量表示不同，后者不施加解析性条件。
The FBR of the operators with commutation relations (1.80) is [519]: 具有对易关系(1.80)的算子的 FBR 表示为[519]：
N→zdzd,a†→z,a→dzd.(1.87)
The corresponding eigenkets of N (orthonormal under the Gaussian measure dμ(z)=π1e−∣z∣2dzdzˉ) are: N 对应的本征态（在高斯测度 dμ(z)=π1e−∣z∣2dzdzˉ) 下正交归一化）为：
un(z)=n!zn,u0(z)=1(n∈N+)(1.88)
The FBR is the Hilbert space generated by the un(z), i.e, the whole space F of entire analytic functions. Each state vector ∣ψ⟩ is associated, in a oneto-one way, with a function ψ(z)∈F by: FBR 是由 un(z) 生成的希尔伯特空间，即整个解析函数空间 F 。每个态矢量 ∣ψ⟩ 通过以下方式与函数 ψ(z)∈F 一一对应：
∣ψ⟩=n=0∑∞cn∣n⟩→ψ(z)=n=0∑∞cnun(z)(1.89)
Note that, as expected in view of the correspondence K→F (induced by ∣n⟩→un(z)), 注意到，正如由 K→F （通过 ∣n⟩→un(z)) 诱导）的对应关系所预期的那样，
Eqs. (1.90) and (1.91) establish the mutual conjugation of a and a† in the FBR, with respect to the measure dμ(z). 方程(1.90)和(1.91)确立了 FBR 中 a 和 a† 关于测度 dμ(z) 的相互共轭关系。
We now consider the finite difference operator Dq defined by: 我们现在考虑由下式定义的有限差分算子 Dq ：
Dqf(z)=(q−1)zf(qz)−f(z),(1.92)
with f(z)∈F,q=eζ,ζ∈C. Dq is the so-called q-derivative operator [76], which, for q→1 ( ζ→0 ), reduces to the standard derivative. By using Eqs. (1.88) and (1.90), it may be written on F as 其中 f(z)∈F,q=eζ,ζ∈C 。 Dq 即所谓的 q 导数算子[76]，当 q→1 （ ζ→0 ）时退化为标准导数。利用式(1.88)和(1.90)，可将其在 F 上表示为
Consistency between (1.92) and (1.93) can be proven by first "normal ordering" the operator (zdzd)n in the form: 式(1.92)与(1.93)的一致性可通过先将算子 (zdzd)n 进行"正规序"排列来证明：
(zdzd)n=m=1∑nSn(m)zmdzmdm(1.94)
where Sn(m) denotes the Stirling numbers of the second kind, defined by the recursion relations [3] 此处 Sn(m) 表示第二类斯特林数，其递归关系定义为[3]
Sn+1(m)=mSn(m)+Sn(m−1)(1.95)
and then expanding in formal power series the exponential (qzdzd−1), and considering the identity: 然后将指数函数 (qzdzd−1) 展开为形式幂级数，并考虑恒等式：
m!1(eθ−1)m=n=m∑∞Sn(m)n!θn(1.96)
Dq satisfies, together with z and zdzd, the commutation relations: Dq 与 z 和 zdzd 共同满足对易关系：
which can be recognized as a realization of relations (1.86) in the space F, with the identification 这可以被识别为关系式(1.86)在空间 F 中的实现，其对应关系为
N→zdzd,a^q→z,aq→Dq(1.98)
where a^q=a^q=1=a† and limq→1aq=a on F. Westress that, while (1.97) are restricted to F, the operators (1.98) are related to the true algebraic structure (1.82)-(1.84). 其中 a^q=a^q=1=a† 和 limq→1aq=a 作用于 F 。我们强调，虽然(1.97)式仅限于 F ，但算子(1.98)与真实的代数结构(1.82)-(1.84)相关。
The relations analogous to (1.90) for the q-deformed case are q 变形情形下与(1.90)式类似的关系式为
The q-commutator [ aq,a^q ] is thus defined on the whole F and acts as 因此， q -对易子[ aq,a^q ]被定义在整个 F 上，其作用表现为
[aq,a^q]f(z)=qNf(z)=f(qz).(1.100)
Eq. (1.100) provides a remarkable result since it shows that the action of the q-WH algebra commutator [aq,a^q], which is a linear form in aq and a^q, may be represented in the FBR as the action of the operator qN which is non-linear in the FBR operators a and a†. 方程(1.100)给出了一个显著结果，因为它表明 q -WH 代数对易子 [aq,a^q] 的作用（这是 aq 和 a^q 的线性形式）在 FBR 中可以表示为算子 qN 的作用，而该算子在 FBR 算子 a 和 a† 中是非线性的。
Finally, we show that the q−WH algebra is related with the squeezing generator. In the Hilbert space of states identified with the space F of entire analytic functions ψ(z), the identity 最后，我们证明 q−WH 代数与压缩生成元相关。在态希尔伯特空间中（该空间与整解析函数 ψ(z) 的空间 F 等同），恒等式
namely, in terms of the FBR operators a and a†, 即用 FBR 算符 a 和 a† 表示时，
z=21(α+α†)→a,dzd=21(α−α†)→a†.(1.103)
In F,α† is indeed the conjugate of α, as discussed in [150] and [519]. In the limit y→0,α and α† turn into the conventional annihilation and creator operators a and a† associated with x and px in the canonical configuration representation, respectively. We then realize that the operator 正如文献[150]和[519]所讨论的， F,α† 确实是 α 的共轭算符。在极限情况下， y→0,α 和 α† 分别转化为与正则位形表示中 x 和 px 相关联的传统湮灭算符 a 和产生算符 a† 。由此我们认识到该算符
where, for simplicity, q=eζ is assumed to be real, is, in the limit y→ 0 , the squeezing operator [142,143] in F, well known in quantum optics [664]. A detailed analysis of the relation between the q-WH algebra and the generator of squeezed coherent states is presented in [142, 143, 151, 152]. As shown in Appendix D, the q-WH algebra is related also to the theta functions, which providean essential tool in the treatment of coherent states [519]. 为简化起见，假设 q=eζ 为实数，在极限 y→ →0 时，就是量子光学[664]中著名的 F 空间压缩算符 [142,143] 。文献[142, 143, 151, 152]详细分析了 q -WH 代数与压缩相干态生成元之间的关系。如附录 D 所示， q -WH 代数还与 theta 函数相关，后者为处理相干态提供了重要工具[519]。
Because the q-algebra has been essentially obtained by replacing the customary derivative with the finite difference operator, the above discussion suggests [142, 143] that whenever one deals with some finite scale (e.g., with some discrete structure, lattice or periodic system, lattice QM) which cannot be reduced to the continuum by a limiting procedure, then a de formation of the operator algebra acting in F should arise. Deformation of the operator algebra is also expected whenever the system under study involves periodic (analytic) functions, since periodicity is nothing but a special invariance under finite difference operators. The special case of the Bloch functions for periodic potentials in QM is studied in [142, 143]. See [142,143] for applications to several cases of physical interest. In Chapter 5 we will discuss the q-deformation of the Hopf algebra in connection with thermal field theory and the general algebraic structure of QFT. 由于 q 代数本质上是通过用有限差分算子替代常规导数而获得的，上述讨论表明[142,143]，当处理无法通过极限过程归约为连续统的有限尺度（例如离散结构、晶格或周期系统、晶格量子力学）时，作用于 F 的算子代数必然会出现形变。当研究系统涉及周期（解析）函数时，同样预期会出现算子代数的形变，因为周期性正是有限差分算子作用下的一种特殊不变性。文献[142,143]研究了量子力学中周期势场布洛赫函数的特例。关于若干物理实际应用案例，可参阅[142,143]。第五章我们将讨论与热场理论相关的 Hopf 代数 q 形变，以及量子场论的一般代数结构。
1.8.1 Self-similarity, fractals and the Fock-Bargmann representation 1.8.1 自相似性、分形与福克-伯格曼表示
It is interesting to consider the FBR and the q-deformation of the WH algebra discussed in the previous Section in connection with self-similarity. We follow closely [634] and [635] in the following. In fractal studies, the selfsimilarity property is referred to as the most important property of fractals (p. 150 in [516]). In fact, a connection will emerge between fractals and q-deformed coherent states. 有趣的是，我们可以将上一节讨论的 FBR 和 WH 代数的 q 变形与自相似性联系起来。下文我们严格遵循文献[634]和[635]的论述。在分形研究中，自相似性被视为分形最重要的特性（参见文献[516]第 150 页）。实际上，分形与 q 变形相干态之间将显现出某种关联。
Let us consider indeed the fractal example provided by the Koch curve (Fig. 1.1). One starts with the step, or stage, of order n=0 : the one
dimensional ( d=1 ) segment u0 of unit length L0, called the initiator [127], is divided by the reducing factor s=3, and the unit length L1=31L0 is adopted to construct the new "deformed segment" u1, called the generator [127], made of α=4 units L1 (step of order n=1 ). The "deformation" of the u0 segment is only possible provided the one-dimensional constraint d=1 is relaxed. The u1 segment "shape" lives in some d=1 dimensions and thus we write u1,q(α)≡qαu0,q=3d1,d=1 to be determined. The index q is introduced in the notation of the deformed segment u1. 让我们具体考察科赫曲线提供的分形实例（图 1.1）。从阶数为 n=0 的步骤（或阶段）开始：将被称为"初始元"[127]的单位长度 L0 的一维( d=1 )线段 u0 ，用缩减因子 s=3 进行分割，并采用单位长度 L1=31L0 来构造新的"变形线段" u1 ，该线段由 α=4 个 L1 单元组成（阶数为 n=1 的步骤），被称为"生成元"[127]。只有当一维约束 d=1 被放宽时， u0 线段的这种"变形"才可能实现。 u1 线段的"形态"存在于某个 d=1 维空间中，因此我们记待确定的 u1,q(α)≡qαu0,q=3d1,d=1 。在变形线段 u1 的表示中引入了下标 q 。
In general, denoting by H(L0) lengths, surfaces or volumes, one has 一般而言，用 H(L0) 表示长度、面积或体积时，可得
H(λL0)=λdH(L0),(1.105)
under the scale transformation: L0→λL0. A square S of side L0 scales to 221S when L0→λL0 with λ=21. A cube V of sameside with samerescaling of L0 scales to 231V. Thus d=2 and d=3 for surfaces and volumes, respectively. Note that S(L0)S(21L0)=p=41 and V(L0)V(21L0)=p=81, respectively, so that in both cases p=λd. For the length L0 it is p=21;2d1=λd and p=λd gives d=1. 在尺度变换下： L0→λL0 。边长为 L0 的正方形 S 在 L0→λL0 时缩放为 221S ，其中 λ=21 。具有相同边长且相同缩放比例 L0 的立方体 V 缩放为 231V 。因此，对于表面积和体积分别有 d=2 和 d=3 。注意 S(L0)S(21L0)=p=41 和 V(L0)V(21L0)=p=81 分别成立，因此在两种情况下都有 p=λd 。对于长度 L0 ，有 p=21;2d1=λd ，而 p=λd 给出 d=1 。
In the case of any other "hypervolume" H one considers the ratio 对于其他任何“超体积” H ，考虑比值
H(L0)H(λL0)=p,(1.106)
and Eq. (1.105) is assumed to be still valid. So, 并假设方程(1.105)仍然成立。因此，
pH(L0)=λdH(L0),(1.107)
i.e, p=λd. For the Koch curve, setting α=p1=4 and q=λd=3d1, the relation p=λd gives 即 p=λd 。对于科赫曲线，设 α=p1=4 和 q=λd=3d1 ，关系式 p=λd 给出
qα=1, where α=4,q=3d1,(1.108)
i.e, 即，
d=ln3ln4≈1.2619.(1.109)
The non-integer d is called the fractal dimension, or the self-similarity dimension [516]. The meaning of Eq. (1.108) is that the measure of the deformed segment u1,q, with respect to the undeformed segment u0, is 1 : u0u1,q=1, i.e, αq=3d4=1. In the following we will set u0=1. 非整数 d 被称为分形维数或自相似维数[516]。方程(1.108)的含义是：变形线段 u1,q 相对于未变形线段 u0 的测度比为 1: u0u1,q=1 ，即 αq=3d4=1 。下文我们将设定 u0=1 。
Steps of higher order n,n=2,3,4,…,∞, can be obtained by iteration of the deformation process. In the step n=2,u2,q(α)≡qαu1,q(α)=(qα)2u0, and so on. For the nth order deformation: 更高阶的步骤 n,n=2,3,4,…,∞ 可通过迭代变形过程获得。在第 n=2,u2,q(α)≡qαu1,q(α)= 步 (qα)2u0 ，依此类推。对于第 n 阶变形：
un,q(α)≡(qα)un−1,q(α),n=1,2,3,…(1.110)
Fig. 1.1 The first five stages of K och curve.
i.e, for any n 图 1.1 科赫曲线的前五个阶段。即，对于任意 n
un,q(α)=(qα)nu0.(1.111)
By requiring that u0un,q(α) be 1 for any n, this gives (qα)n=1 and Eq. (1.109) is again obtained. We stress that the fractal is mathematically defined in the limit of infinite iterations of the deformation process, n→∞. The fractal is the limit of the deformation process for n→∞. The definition of fractal dimension is indeed more rigorously given starting from (qα)n=1 in the n→∞ limit [47, 127]. Self-similarity is defined only in the n→∞ limit. Since Ln→0 for n→∞, the Koch fractal is a curve which is non-differentiable everywhere [516]. 通过要求对于任意 n ， u0un,q(α) 为 1，由此可得 (qα)n=1 ，并再次得到方程(1.109)。我们强调，分形在数学上定义为变形过程无限迭代的极限 n→∞ 。分形是变形过程在 n→∞ 时的极限。分形维数的定义实际上是从 (qα)n=1 出发，在 n→∞ 极限下更严格给出的[47,127]。自相似性仅在 n→∞ 极限下定义。由于在 n→∞ 时 Ln→0 ，科赫分形是一条处处不可微的曲线[516]。
Eqs. (1.110) and (1.111) express, in the n→∞ limit, the self-similarity property of a large class of fractals (the Sierpinski gasket and carpet, the Cantor set, etc.) [47, 127]. Our discussion can be extended to self-affine fractals (invariance under anisotropic magnification is called self-affinity). 方程(1.110)和(1.111)在 n→∞ 极限下表达了一大类分形(谢尔宾斯基垫片、谢尔宾斯基地毯、康托尔集等)的自相似特性[47,127]。我们的讨论可以推广到自仿射分形(各向异性放大下的不变性称为自仿射性)。
Summarizing the discussion of [634], we consider the complex α-plane and note that applying Eq. (1.100) to the basis provided by the functions un(α) (cf. Eq. (1.88) where now we have changed z into α ) we have: 总结[634]的讨论，我们考虑复 α 平面，并注意到将方程(1.100)应用于由函数 un(α) 提供的基(参见方程(1.88)，其中现在我们将 z 改为 α )，可得：
qNun(α)=n!(qα)n,u0(α)=1,(n∈N+)(1.112)
We recall that the FBR is the Hilbert space generated by the un(α), i.e., the space F of entire analytic functions. Eq. (1.112) applied to the coherent state functional (D.2) (cf. Appendix D), gives 我们回顾 FBR 是由 un(α) 生成的希尔伯特空间，即全纯解析函数的空间 F 。将方程(1.112)应用于相干态泛函(D.2)(参见附录 D)，可得
Eq. (1.111), with u0 set equal to 1 , is then obtained by projecting out the nth component of ∣qα⟩ and restricting to real qα,qα→ℜe(qα) : 令 u0 等于 1 时，方程(1.111)可通过投影出 ∣qα⟩ 的第 n 个分量并限制于实数 qα,qα→ℜe(qα) 而获得：
The operator (a)n thus acts as a "magnifying" lens [127]. The nth iteration can be "seen" by applying (a)n to ∣qα⟩ and restricting to real qα : 算子 (a)n 因此充当了一个"放大"透镜[127]。通过将 (a)n 应用于 ∣qα⟩ 并限制为实数 qα ，可以"看到"第 n 次迭代：
⟨qα∣(a)n∣qα⟩=(qα)n=un,q(α),qα→ℜe(qα).(1.117)
In conclusion, the nth fractal stage of iteration, with n=0,1,2,…,∞, is represented, in a one-to-one correspondence, by the nth term in the coherent state series Eq. (1.113). The operator qN applied to ∣α⟩ (Eq. (1.113)) "produces" the fractal in the functional form of the coherent state ∣qα⟩.qN is also called the fractal operator [634]. 综上所述，第 n 次分形迭代阶段（当 n=0,1,2,…,∞ 时）与相干态级数 Eq.(1.113)中的第 n 项形成一一对应关系。将算子 qN 应用于 ∣α⟩ （Eq.(1.113)）会在相干态 ∣qα⟩.qN 的函数形式中"生成"分形，该算子也被称为分形算子[634]。
The study of the fractal properties may thus be carried on in the space F of the entire analytic functions, by restricting, at theend, the conclusions to real α,α→ℜe(α). Since in Eq. (1.111) it is q=1(q<1), actually one needs to consider the " q-deformed" algebraic structure of which the space F provides a representation. 因此，分形性质的研究可以在全解析函数空间 F 中进行，最终将结论限制到实数 α,α→ℜe(α) 。由于在 Eq.(1.111)中 q=1(q<1) ，实际上需要考虑" q 变形"的代数结构，而空间 F 提供了该结构的一个表示。
Eq. (1.114) expresses the invariance of the coherent state under the action of the operator qα1a and allows us to consider the coherent functional
ψ(qα) as an "attractor" in F. This reminds us of the fixed point equation W(A)=A, where W is the Hutchinson operator [127], characterizing the iteration process for the fractal A in the n→∞ limit. Eq.(1.114)表达了相干态在算子 qα1a 作用下的不变性，并允许我们将相干泛函 ψ(qα) 视为 F 中的"吸引子"。这让我们联想到不动点方程 W(A)=A ，其中 W 是 Hutchinson 算子[127]，表征了分形 A 在 n→∞ 极限下的迭代过程。
The connection between fractals and the ( q-deformed) algebra of the coherent states is formally established by Eqs. (1.115), (1.116) and (1.117). 分形与相干态的（ q 变形）代数之间的关联通过方程(1.115)、(1.116)和(1.117)得以形式化建立。
Moreover, the fractal operator qN is associated with the squeezing transformation (cf. the previous Section). This establishes the relation between the fractal generator process and squeezed coherent states (see also [634]). 此外，分形算子 qN 与压缩变换相关联（参见前节）。这确立了分形生成过程与压缩相干态之间的关系（另见文献[634]）。
In conclusion, for the case of fractals generated iteratively according to a prescribed recipe (deterministic fractals), the functional realization of fractal self-similarity has been obtained in terms of the q-deformed algebra of coherent states. Fractal study can thus beincorporated into the theory of entire analytical functions. From the discussion it appears that the reverse is also true: under a convenient choice of the q-deformation parameter and by a suitable restriction to real α, coherent states exhibit fractal properties in the q-deformed space of the entire analytical functions. 综上所述，对于按照既定规则迭代生成的分形（确定性分形），其自相似性的函数实现已通过相干态的 q 变形代数获得。因此，分形研究可被纳入整解析函数理论体系。讨论表明反之亦然：通过适当选择 q 变形参数并对实 α 进行恰当限制时，相干态在整解析函数的 q 变形空间中展现出分形特性。
Therelation between fractals and coherent states, originally conjectured in [636], introduces dynamical considerations in the study of fractals and of their origin, as well as geometrical insight into coherent state properties. Fractals appear to be global systems arising from local deformation processes. 分形与相干态之间的关系最初在[636]中提出猜想，为分形及其起源的研究引入了动力学考量，同时也为相干态特性提供了几何视角。分形似乎是局部形变过程产生的全局系统。
1.9 The physical particle energy and momentum operator 1.9 物理粒子的能量与动量算符
Let us now make a more specific statement about the energy and the momentum operator of physical particles in the language of the Fock space. We consider a one particle (boson or fermion) wave-packet state αi†⟩=αi†∣0⟩, 现在让我们用福克空间的术语更具体地阐述物理粒子的能量和动量算符。我们考虑一个单粒子（玻色子或费米子）波包态 αi†⟩=αi†∣0⟩ ，
αi†∣0⟩=∫(2π)3/2d3kfi(k)αk†∣0⟩(1.118)
The energy operator H0 is introduced by requiring 能量算符 H0 的引入要求
H0αi†∣0⟩=∫(2π)3/2d3kEkfi(k)αk†∣0⟩(1.119)
with real Ek. Since this should be true for any square-integrable function fi(k), we have 对于任意平方可积函数 fi(k) ，此关系式应成立，因此我们有
H0αk†∣0⟩=Ekαk†∣0⟩(1.120)
Such a relation must be understood in the sense of distributions, i.e, in the sense of (1.119), since the states involved in (1.120) are not elements of the Fock space due to the unboundness of the αk† operators. 这种关系必须在分布的意义上理解，即按照(1.119)式的含义，因为由于 αk† 算符的无界性，(1.120)式中涉及的态并非 Fock 空间的元素。
We now consider the scattering of many particles. Before the collision, the system energy is the sum of the energies of the particles entering in the collision region (the incoming particles), say the particles Ain,Bin,Cin, etc. After the collision, the total energy is the sum of the energies of the particles outgoing from the collision region, say the particles Aout ,Bout , Cout , etc. If the experimental setup is such that any exchange of energy between our particle system and the environment is negligible, then as a result of our measurement we find that the total energy after the collision equals the total energy before the collision (the principle of conservation of energy). Thus, for many particles we require that 现在我们考虑多粒子散射问题。碰撞前系统能量等于进入碰撞区域（入射粒子）的各粒子能量之和，即粒子 Ain,Bin,Cin 等的能量总和。碰撞后总能量则为离开碰撞区域的各粒子（如粒子 Aout ,Bout 、 Cout 等）能量之和。若实验装置确保粒子系统与环境间的能量交换可忽略不计，则通过测量可发现碰撞前后总能量守恒（能量守恒原理）。因此对于多粒子系统，我们要求满足
Since Ek is real, we require that H0=H0†. From (1.121) we can derive 由于 Ek 是实数，我们要求 H0=H0† 。根据(1.121)式可以推导出
[H0,αk†]=Ekαk†,[H0,αk]=−Ekαk(1.123)
These commutators imply that the form of H0 has to be 这些对易关系意味着 H0 的形式必须为
H0=s∑∫d3kEksαks†αks+H1,(1.124)
where the suffix s (cf. Section 1.2) has been restored and H1 commutes with αks† and αks. Since the Fock space is an irreducible representation of αks† and αks,H1 must be a c-number multiple of the identity operator. On the other hand, its value is determined by the vacuum expectation value of (1.124) (cf. Eq. (1.122)): 其中下标 s （参见第 1.2 节）已恢复，且 H1 与 αks† 和 αks 对易。由于 Fock 空间是 αks† 的不可约表示， αks,H1 必须是单位算子的 c 数倍。另一方面，其值由(1.124)式的真空期望值决定（参见方程(1.122)）：
H1=⟨0∣H0∣0⟩=0.(1.125)
Thus, by considering also the operators βks and βks†, we have 因此，通过同时考虑算子 βks 和 βks† ，我们得到
H0=s∑∫d3kEks(αks†αks+βks†βks)(1.126)
In a similar way we can introduce the momentum operator P0 as 同样地，我们可以引入动量算符 P0 为
P0=s∑∫d3kk(αks†αks+βks†βks),(1.127)
with 其中
[P0,αks†]=kαks†,[P0,αks]=−kαks,(1.128)
and 而
P0∣0⟩=0.(1.129)
Although the operators H0 and P0 are well defined only on the dense set D, the operators eiH0t and eiP0⋅x, with real t and x, are well defined on the whole HF. In terms of these last operators, the above commutation relations should be replaced by 尽管算符 H0 和 P0 仅在稠密集 D 上有明确定义，但对于实数 t 和 x ，算符 eiH0t 和 eiP0⋅x 在整个 HF 上都有良好定义。用这些最终算符表示时，上述对易关系应替换为
1.10 The physical Fock space and the physical fields 1.10 物理福克空间与物理场
Eqs. (1.121), (1.126) and (1.127) give an exact meaning to the statement that the total energy and the total momentum of a system of free particles is given by the sum of the energies and of the momenta, respectively, of each particle (cf. Eq. (1.19) for the definition of the number operator). In particular, we take this to be the definition of the free or physical particle state: it is the state where the total energy and the total momentum are given by the sum of the energy and of the momentum, respectively, of each constituent particle. The Fock space of the free particle states is thus the one where the Hamiltonian operator and the momentum operator assume the form (1.126) and (1.127), respectively. We will call it the physical Fock space. 方程(1.121)、(1.126)和(1.127)精确表述了以下陈述：自由粒子系统的总能量和总动量分别由各粒子的能量和动量之和给出（参见数算符定义方程(1.19)）。特别地，我们将此作为自由或物理粒子态的定义：即总能量和总动量分别由组成粒子的能量和动量之和给出的状态。因此，自由粒子态的福克空间就是哈密顿算符和动量算符分别取(1.126)和(1.127)形式的空间，我们称之为物理福克空间。
We define the free or physical field ϕ(x), with x denoting x,t, by 我们定义自由或物理场 ϕ(x) ，其中 x 表示 x,t ，通过
In general, ϕ(x) is a onecolumn matrix. The fact that the energy Ek of a physical particle is a certain function of its momentum means that the physical field ϕ(x) must solve a linear homogeneous equation: 通常， ϕ(x) 是一个单列矩阵。物理粒子的能量 Ek 是其动量的特定函数，这意味着物理场 ϕ(x) 必须满足一个线性齐次方程：
∧(∂)ϕ(x)=0.(1.132)
The differential operator Λ(∂) is in general a square matrix. Its operation is defined on the Fourier transform as Λ(∂)e−ik⋅x=Λ(ik)e−ik⋅x,k⋅x≡kμxμ=Ekt−k⋅x. The "wave functions" u(k) and v(k) are solutions of 微分算子 Λ(∂) 通常是一个方阵。其运算在傅里叶变换上定义为 Λ(∂)e−ik⋅x=Λ(ik)e−ik⋅x,k⋅x≡kμxμ=Ekt−k⋅x 。所谓的"波函数" u(k) 和 v(k) 是该算子的解。
∧(ik)u(k)=0, and ∧(−ik)v(k)=0(1.133)
respectively. 分别。
Physical particles are thus ingoing and outgoing particles far from the region of interaction. In solid state physics, as mentioned in Section 1.2, the physical particles arecalled quasiparticles. We will call in-fields or out-fields the fields referring to ingoing or outgoing physical particles, respectively, and denote them by ϕin and/or ϕout . In the following Chapters, whenever no misunderstanding arises we will drop the 'in' and/ or 'out' indexes. Infields and out-fields will also be generically called asymptotic fields since they describe particles in spacetime regions where interactions are not felt. The free field equations of type (1.132), in fact, do not contain any information about the interactions. Although the physical particles undergo interaction processes, the language we have set up till now cannot describe such dynamical processes; thus we need another source of information to describe the dynamics of a physical system. The concept of free field is pertinent to one of the aspects of the two-level description of Nature. We thus assume the existence of basic entities, the Heisenberg or interacting fields, in order to account for interactions, the other aspect of this duality. Heisenberg fields satisfy basic relations characterizing the dynamics, the Heisenberg equations. We will come back to this point in the following Chapter. 因此，物理粒子即指远离相互作用区域的入射与出射粒子。如 1.2 节所述，在固态物理学中，这些物理粒子被称为准粒子。我们将分别称描述入射与出射物理粒子的场为入场或出场，并记作 ϕin 和/或 ϕout 。在后续章节中，只要不引起歧义，我们将省略"入"和/或"出"的下标。入场与出场也统称为渐近场，因为它们描述的是时空中感受不到相互作用的粒子。事实上，(1.132)型自由场方程并不包含任何相互作用信息。尽管物理粒子会经历相互作用过程，但我们迄今建立的语言体系尚无法描述这类动力学过程；因此需要引入新的信息源来描述物理系统的动力学行为。自由场概念与自然界双重描述的一个层面密切相关。 因此，我们假设存在基本实体——海森堡场或相互作用场，以解释相互作用的这一对偶性另一面。海森堡场满足表征动力学的基本关系，即海森堡方程。我们将在下一章回到这一点。
A ppendix A 附录 A
Strong limit and weak limit 强极限与弱极限
Consider in full generality a linear metric vector space F (namely a vector space endowed with addition of its elements, multiplication by a scalar and inner product). Let ξ be an element of F and ∣ξ∣≡(ξ,ξ)1/2 denote the norm of ξ. A sequence of elements {ξn} of F is said to be a Cauchy sequence if for every ϵ>0 one can find an N>0 such that 完全一般性地考虑一个线性度量向量空间 F （即配备元素加法、标量乘法和内积的向量空间）。设 ξ 是 F 的一个元素， ∣ξ∣≡(ξ,ξ)1/2 表示 ξ 的范数。若对于任意 ϵ>0 都能找到 N>0 使得 F 中的元素序列 {ξn} 满足条件，则称该序列为柯西序列。
∣ξn−ξm∣<ϵ,(A.1)
whenever n,m>N. A sequence {ξn} converges to an element {ξ} of F if for every ϵ>0 there exist an N>0 such that 当 n,m>N 时。若对于每个 ϵ>0 ，存在一个 N>0 使得序列 {ξn} 收敛于 F 中的元素 {ξ} ，则称该序列收敛
∣ξn−ξ∣<ϵ, for n>N,(A.2)
and we write 记作
n→∞lim∣ξn−ξ∣=0(A.3)
A space F with the property that all Cauchy sequences of elements of F have a limit that also belongs to F is called complete. A Hilbert space is a linear metric vector space that is also complete 若空间 F 满足其所有柯西序列的极限仍属于 F ，则称该空间完备。希尔伯特空间是兼具完备性的线性度量向量空间
A subset D of elements ξ of F is said to be dense in F if, for any element ζ of F, one can construct a sequence of elements of D that has ζ as its limit. If this dense set has a countable basis, the space F is called separable. In other words, this means that one can find a countable set of orthogonal elements {ξi}(i=1,2,…) such that any vector ζ can be written as 若对于 F 中任意元素 ζ ，都能构造出 D 中元素的序列使其极限为 ζ ，则称 F 的子集 D 在 F 中稠密。若该稠密集具有可数基，则称空间 F 可分。换言之，即存在一组可数的正交元素 {ξi}(i=1,2,…) ，使得任意向量 ζ 均可表示为
ζ=i=0∑∞ciξi(A.4)
where ci are complex constants. The equality in (A.4) is understood in the sense of 其中 ci 为复常数。(A.4)式中的等式应理解为
N→∞limζ−i=0∑Nciξi=0(A.5)
In a complete space F, the vector ξ is said to be the strong limit [7,351,537] of the sequence {ξn;ξn∈F} if (A.3) is satisfied. 在完备空间 F 中，若满足(A.3)式，则称向量 ξ 为序列 {ξn;ξn∈F} 的强极限[7,351,537]
On the other hand, the sequence ξn is said to be weakly convergent to ξ if for any arbitrary vector η in F it is 另一方面，若对于 F 中任意向量 η 都满足，则称序列 ξn 弱收敛于 ξ
n→∞lim(η,ξn)=(η,ξ).(A.6)
Moreover, if the sequence ξn is a bounded operator sequence, i.e., ∣ξn∣≤M for any n and for some constant M independent of n, as wegenerally assume for the states of the Fock space, ξn is weakly convergent to ξ if 此外，若序列 ξn 为有界算子序列（即对任意 n 及与 n 无关的某常数 M ，满足 ∣ξn∣≤M ——我们通常对福克空间态作此假设），则当 ξn 弱收敛于 ξ 时
n→∞lim(ηi,ξn)=(ηi,ξ),(A.7)
for all the elements ηi of a dense set in F (the Fock space). Sometimes one writes w−limn→∞ξn=ξ to denote the weak limit convergence. 对于 F （福克空间）中稠密集的所有元素 ηi 。有时人们写作 w−limn→∞ξn=ξ 来表示弱极限收敛。
Appendix B 附录 B
Glauber coherent states 格劳伯相干态
We shall briefly consider here some essentials of single-mode coherent states (CS) commonly called canonical CS, or Glauber coherent states, or FockBargmann coherent states [285, 519]. The functional integrals based on Glauber coherent states will be discussed in Appendix N.1. Group related generalized CS will be considered in Appendices C and N.2. 我们在此简要考虑单模相干态（CS）的一些基本要素，这些态通常被称为正则 CS、格劳伯相干态或福克-巴格曼相干态[285, 519]。基于格劳伯相干态的函数积分将在附录 N.1 中讨论。与群相关的广义 CS 将在附录 C 和 N.2 中予以考虑。
The un-normalized coherent state has the form 未归一化的相干态具有如下形式
one can alternatively rewrite (B.1) as 可将式(B.1)改写为
∣z⟩=n∑n!zn∣n⟩.(B.3)
It is easy to see that ∣z⟩ is an eigenstate of a with the eigenvalue z. This is a straightforward implication of the operator formula 显然 ∣z⟩ 是 a 的本征态，对应的本征值为 z 。这是算子公式的直接推论
The corresponding completeness relation (or resolution of unity) for single-mode CS can be easily derived with the aid of (B.3). Indeed, 单模相干态对应的完备性关系（或称单位分解）可借助(B.3)式轻松导出。事实上，
wherethe polar decomposition z=reiθ has been used. There is yet another frequently used form of the completeness relation that is particularly useful in the path integral formalism. To obtain it we use 其中使用了极坐标分解 z=reiθ 。完备性关系还有另一种常用形式，在路径积分形式中特别有用。为得到它，我们使用
x^=2ωmℏ(a+a†),p^=i2ℏωm(a†−a).(B.10)
Clearly 𝟙[x^,p^]=iℏ1. Equations (B.10) imply that 显然 𝟙[x^,p^]=iℏ1 。方程(B.10)表明
a=2ℏmωx^+i2ℏmω1p^(B.11)
In a similar fashion we decompose z according to the rule 类似地，我们按照规则分解 z
z=2ℏmωx+i2ℏmω1p,x,p∈R(B.12)
Direct consequences of (B.10) and (B.12) are (B.10)和(B.12)的直接推论是
Relation (B.13) provides an interpretation for the labels x and p, while (B.14) indicates that states ∣z⟩ saturate the Heisenberg uncertainty relation. It is customary to utilize an alternative notation for ∣z⟩, namly ∣x,p⟩≡∣z⟩.
In terms of the phase-space variables/ operators we can directly write 关系式(B.13)为标签 x 和 p 提供了诠释，同时(B.14)表明态 ∣z⟩ 达到了海森堡不确定性关系的饱和极限。通常我们会采用另一种符号表示 ∣z⟩ ，即 ∣x,p⟩≡∣z⟩ 。利用相空间变量/算符，我们可以直接写出
An important signature of CS is their over-completeness. In fact, one should note that relation (B.17) (resp. (B.8)) appears exactly like a resolution of unity used for self-adjoint operators. There is, however, a difference in that the one-dimensional projection operators ∣x,p)(x,p∣ are not mutually orthogonal. Indeed, 相干态的一个重要特征是其过完备性。事实上，需要注意关系式(B.17)（相应地(B.8)）形式上完全类似于自伴算子所用的单位分解。然而不同之处在于，一维投影算子 ∣x,p)(x,p∣ 并非相互正交。实际上，
For this reason it is usually said that the set of CS is over-complete. In fact, the previous result shows that CS are not orthogonal for any ∣z⟩ and ∣z′⟩. If, however, the numerical distance ∣z−z′∣ is large the states are almost orthogonal. This is because the angle θ(zz′) between the states can be calculated through the relation 因此，通常称相干态集合是过完备的。事实上，前文结果表明对于任意 ∣z⟩ 和 ∣z′⟩ ，相干态并不正交。然而，当数值距离 ∣z−z′∣ 较大时，这些态几乎正交。这是因为态间夹角 θ(zz′) 可通过以下关系计算得出：
Inasmuch as ∣z−z′∣≫1 then vectors ∣z⟩ and ∣z′⟩ are close to being orthogonal. In Appendix D we consider the problem of extracting a complete set
of CS from an over-complete set. We will see that in order to do that one needs to introduce a regular lattice called the von Neumann lattice. 由于 ∣z−z′∣≫1 ，矢量 ∣z⟩ 和 ∣z′⟩ 近乎正交。附录 D 中我们将讨论如何从过完备集中提取完备的相干态集合。我们将看到，为此需要引入称为冯·诺伊曼格点的规则晶格。
If we make use of the resolution of the unity (B.8) we can write for a general state ∣ψ⟩ 若利用单位分解(B.8)式，对于一般态 ∣ψ⟩ 可表示为
∣ψ⟩=∫2πidzdz∗e−∣z∣2⟨z∣ψ⟩∣z⟩.(B.21)
Here 这里
⟨z∣ψ⟩=n=0∑∞n!(z∗)n⟨n∣ψ⟩≡fψ(z∗).(B.22)
From the fact that ∑n∣⟨n∣ψ⟩∣2=1 it is clear that the series (B.22) converges for all z∗, and thus it represents a complex function that is holomorphic on the whole complex plane C. Such functions are said to be entire. Decomposition (B.21) indicates that the function fψ(z∗) is itself a representation of ∣ψ⟩ (in {∣z⟩} basis), and can be regarded as the element of the Hilbert space. The representation fψ(z∗) is called a holomorphic representation and the corresponding Hilbert space is known as the Fock-Bargmann, or Segal-Fock-Bargmann space of entire analytical functions [55, 242, 564]. 由 ∑n∣⟨n∣ψ⟩∣2=1 可知，级数(B.22)对所有 z∗ 都收敛，因此它表示一个在整个复平面 C 上全纯的复函数。这类函数被称为整函数。分解式(B.21)表明函数 fψ(z∗) 本身就是 ∣ψ⟩ 的一种表示（基于 {∣z⟩} 基），可视为希尔伯特空间的元素。这种 fψ(z∗) 表示称为全纯表示，对应的希尔伯特空间称为福克-巴格曼空间或西格尔-福克-巴格曼整解析函数空间[55, 242, 564]。
Appendix C 附录 C
Generalized coherent states 广义相干态
The Glauber coherent states considered in Appendix B, have the three following properties: they are eigenstates of lowering operators [60], they are minimum uncertainty states [500], and they may be generated via translation (or displacement) operators [519]. 附录 B 中讨论的 Glauber 相干态具有以下三个特性：它们是降算符的本征态[60]，是最小不确定态[500]，且可通过平移（或位移）算符生成[519]。
Various generalizations of the above coherent states have been proposed [379, 519], which maintain only some of the above conditions. Here we consider the generalized coherent states generated via displacement operators [518, 519], related to a Lie group G. Such states have been used in many applications in atomic and nuclear physics and in statistical mechanics [33, 172, 173, 533]. 学界已提出上述相干态的各种广义形式[379,519]，这些形式仅保留部分上述条件。本文研究通过位移算符生成的广义相干态[518,519]，其与李群 G 相关。此类态在原子核物理与统计力学领域已有诸多应用[33,172,173,533]。
The generalized coherent states related to a Lie group G are constructed in the following way: let D(g),g∈G be an irreducible unitary representation of G acting in some Hilbert space H. We choose a normalized fiducial state vector in H and denote it as ∣0⟩ (the reason for this notation will be clear shortly). The generalized coherent states corresponding to G are then defined as 与李群 G 相关的广义相干态构建方式如下：设 D(g),g∈G 为作用于某希尔伯特空间 H 的 G 不可约酉表示。我们选取 H 中的归一化基准态矢量，记作 ∣0⟩ （此记号的缘由将很快阐明）。对应于 G 的广义相干态则定义为
∣0(g)⟩=D(g)∣0⟩ for ∀g∈G.(C.1)
We say that two coherent states ∣0(g1)⟩ and ∣0(g2)⟩ represent the same state (or are physically equivalent) in H if 若两个相干态 ∣0(g1)⟩ 与 ∣0(g2)⟩ 在 H 中表征相同状态（或物理等价），则称：
Here the phase factor α∈R may depend both on g1 and g2. As g runs in ∣0(g)⟩ through G, states ∣0(g)⟩ travel through the Hilbert space H. In general, however, physically equivalent states will be visited many times during this procedure. Defining the stability group H∣0⟩ the group of transformations leaving ∣0⟩ invariant (modulo a phase factor), i.e, 此处相位因子 α∈R 可能同时依赖于 g1 和 g2 。当 g 在 ∣0(g)⟩ 中经由 G 变化时，态 ∣0(g)⟩ 将在希尔伯特空间 H 中移动。然而一般而言，在此过程中物理上等价的状态会被多次遍历。定义稳定子群 H∣0⟩ 为保持 ∣0⟩ 不变的变换群（模去相位因子），即
H∣0⟩={h∈G:D(h)∣0⟩=eiβ(h)∣0⟩,β(h)∈R},(С.З)
we see from (C.2) that g2−1g1∈H∣0⟩. Note that H∣0⟩ is indeed a subgroup of G, because if h1 and h2 belong to H∣0⟩ then also h1−1h2 does. 由(C.2)式可见 g2−1g1∈H∣0⟩ 。注意到 H∣0⟩ 确实是 G 的子群，因为若 h1 和 h2 属于 H∣0⟩ ，则 h1−1h2 亦然。
In this connection and for future reference, we recall that given a subgroup H of a group G, the (left) coset of H with respect to g∈G, written as gH, is defined as the set of all elements {gh;h∈H}. An elementary theorem from group theory asserts that two cosets g1H and g2H for g1=g2 are either identical or completely disjoint. In this way the group G can be partitioned into disjoint cosets. The collection of cosets of the subgroup H in the group G is usually denoted as G/H and called the coset (or quotient) space of G modulo H. Despite the fact that both G and H are groups, the coset space G/H is generally not a group. Only in cases when H is a normal subgroup of G (i.e, when gH=Hg for all g∈G ) then one can formulate group operations in G/H. For instance, the product law for two cosets g1H and g2H can be simply defined as the coset (g1g2)H : (g1H)(g2H)=(g1g2)H. With this the associativity is obvious, the identity element can be taken as E≡eH=H, and the inverse of the coset gH is g−1H. In such cases the coset space G/H is called the factor group. 在此背景下并为将来参考，我们回顾：给定群 G 的子群 H ，关于 g∈G 的（左）陪集记为 gH ，其定义为所有元素 {gh;h∈H} 的集合。群论中的一个基本定理断言，对于 g1=g2 的两个陪集 g1H 和 g2H 要么完全相同，要么完全不相交。通过这种方式，群 G 可以被划分为不相交的陪集。子群 H 在群 G 中的陪集集合通常记为 G/H ，称为 G 模 H 的陪集（或商）空间。尽管 G 和 H 都是群，但陪集空间 G/H 通常不构成群。仅当 H 是 G 的正规子群时（即对所有 g∈G 满足 gH=Hg ），才能在 G/H 中定义群运算。例如，两个陪集 g1H 和 g2H 的乘积法则可简单定义为陪集 (g1g2)H ： (g1H)(g2H)=(g1g2)H 。由此结合律显然成立，单位元可取为 E≡eH=H ，而陪集 gH 的逆元为 g−1H 。在此情形下，陪集空间 G/H 称为商群。
It is also convenient to recall that the algebra of a d-dimensional Lie algebra is given by the commutators 同样方便回顾的是， d 维李代数的代数结构由对易子给出
[Ta,Tb]=iCabcTc.(C.4)
Cabc are the structure constants and Ta are hermitian matrices - the group generators ( a=1,…,d ). The adjoint or regular representation of the Lie algebra is then defined so that Cabc 是结构常数， Ta 是厄米矩阵——即群生成元（ a=1,…,d ）。李代数的伴随表示或正则表示由此定义为：
(Ta)bc=iCbac or equivalently (Ta)bc=−iCabc.(C.5)
Thus Ta is a d×d matrix. For instance, the adjoint representation for SU(2)≅SO(3) has d=3 and hence its representation space is three dimensional vector space with matrix elements (Ta)bc given explicitly by 因此 Ta 是一个 d×d 矩阵。例如， SU(2)≅SO(3) 的伴随表示具有 d=3 ，故其表示空间为三维向量空间，其矩阵元 (Ta)bc 可显式表示为：
(Ta)bc=−iϵabc(С.6)
The fundamental or defining representation of the Lie algebra corresponds to the defining matrix representation. For instance, the fundamental representation of SO (3) corresponds to 3×3 orthogonal matrices of determinant 1. The corresponding representation space is thus three-dimensional. The fundamental representation of SU(2) has two-dimensional representation space. 李代数的基本表示或定义表示对应于定义性的矩阵表示。例如， SO (3)的基本表示对应于行列式为 1 的 3×3 正交矩阵，其表示空间因此是三维的。而 SU(2) 的基本表示则具有二维表示空间。
In future considerations we shall simply denote H∣0⟩ as H with the implicit knowledge that H is associated with a fiducial state. We note that both g1 and g2 in (C.2) are part of the same stability group H∣0⟩ in G. 在后续讨论中，我们将简单地将 H∣0⟩ 记作 H ，并默认 H 与一个基准态相关联。需注意的是，(C.2)式中的 g1 和 g2 同属于 G 中的同一稳定子群 H∣0⟩ 。
Let dg be the left-invariant group measure (Haar measure), i.e., for any fixed g0∈G,d(g0⋅g)=dg. Consider now the operator 设 dg 为左不变群测度（哈尔测度），即对于任意固定的 g0∈G,d(g0⋅g)=dg 。现考虑算子
B=∫Gdg∣O(g)⟩⟨O(g)∣(C.7)
Due to the invariance of the measure we have for any g′∈G 由于测度不变性，对任意 g′∈G 有
D(g′)BD†(g′)=∫Gdg∣0(g′⋅g)⟩⟨0(g′⋅g)∣=B(C.8)
So B commutes with all D(g), and hence it must be proportional to the unit operator. This is a result of the fundamental lemma of group theory which asserts that any linear operator commuting with all the operators of an irreducible representation of some group G must be a multiple of the unit operator, i.e, 因此 B 与所有 D(g) 对易，故必为单位算子的比例常数。此结果源于群论基本引理：与某群 G 不可约表示中所有算子对易的线性算子必为单位算子的倍数，即
𝟙D(g)B=BD(g),∀g∈G⇒B=c−11(С.9)
This lemma is known as the first Schur lemma.
Having measure dg on G, the measure on the coset space G/H is naturally induced by dg. We shall denote this induced measure as dx. With the help of (C.9) the resolution of the unity can be written as 该引理称为第一舒尔引理。在 G 上给定测度 dg 后，商空间 G/H 上的测度自然由 dg 诱导产生。我们将此诱导测度记为 dx 。借助(C.9)式，单位分解可表示为
the condition (C.11) cannot be fulfilled with c=0. It is thus meaningful to confine only to representations D(g) for which the integral (C.12) is finite, i.e, square integrable representations. 条件(C.11)无法通过 c=0 实现时。因此，仅限定于使积分(C.12)有限的表示 D(g) 才有意义，即平方可积表示。
SU(2) coherent states SU(2)相干态
The SU(2) group has three generators J1,J2,J3. The SU(2) algebra is SU(2) 群有三个生成元 J1,J2,J3 。 SU(2) 代数为
[J+,J−]=2J3[J3,J±]=±J±.(С.13)
Here the ladder operators are defined as J±=J1±iJ2. The unitary irreducible representations of the SU(2) algebra are finite-dimensional and are spanned by the states ∣j,m⟩, such that 此处阶梯算子定义为 J±=J1±iJ2 。 SU(2) 代数的酉不可约表示是有限维的，由态 ∣j,m⟩ 张成，满足
As the fiducial vector we choose the state ∣j,−j⟩, i.e, the state which is annihilated by the lowering operator: J−∣j,−j⟩=0. In this way each representation has its unique fiducial state - "vacuum state" ∣0⟩≡∣j,−j⟩. The stability group for this state is the subgroup of rotations around the z-axis, thus H=U(1). According to Eq. (C.10), distinct coherent states are labeled by x∈M=G/H. By noting that M=SU(2)/U(1)≅S2 we can identify x with the spherical angular variables θ and φ. The associated state can be written as ∣0(θ,φ)⟩ : 我们选择基准向量为态 ∣j,−j⟩ ，即被降算符湮灭的态 J−∣j,−j⟩=0 。这样每个表示都有其独特的基准态——"真空态" ∣0⟩≡∣j,−j⟩ 。该态的稳定子群是绕 z 轴旋转的子群，因此 H=U(1) 。根据式(C.10)，不同的相干态由 x∈M=G/H 标记。注意到 M=SU(2)/U(1)≅S2 ，我们可以将 x 与球面角变量 θ 和 φ 对应起来。相关态可表示为 ∣0(θ,φ)⟩ ：
∣0(θ,φ)⟩=D(θ,φ)∣0⟩=exp[iθ(J⋅n)]∣0⟩,(C.17)
with the unit vector n=(sinφ,cosφ,0). Using the Gauss decomposition formula 其中 n=(sinφ,cosφ,0) 为单位矢量。利用高斯分解公式
one can alternatively use the more economical form 也可以采用更简洁的形式
∣0(θ,φ)⟩=(1+∣ξ∣2)−jeξJ+∣0⟩≡∣0(ξ)⟩.(С.19)
Relation (C.19) is an analogueof the canonical coherent staterelation (B.1). The scalar product of two coherent states ∣0(ξ)⟩ can be written in the form 关系式(C.19)是典型相干态关系式(B.1)的类比。两个相干态 ∣0(ξ)⟩ 的标量积可表示为
Here m=(sinθcosφ,sinθsinφ,cosθ) is the unit vector parametrizing M=S2 (similarly for m′ ). For this reason it is sometimes convenient to use notation ∣0(θ,φ)⟩=∣0(ξ)⟩=∣0(m)⟩. 此处 m=(sinθcosφ,sinθsinφ,cosθ) 是参数化 M=S2 的单位向量（ m′ 同理）。因此有时使用 ∣0(θ,φ)⟩=∣0(ξ)⟩=∣0(m)⟩ 表示法更为便利。
According to Eq. (C.10) the resolution of the unity reads 根据式(C.10)，单位分解可表示为
where we have used the usual convention 此处我们采用了常规约定
dξdξ∗≡dℜe(ξ)dℑm(ξ)
where ℜe and ℑm denote the real and the imaginary parts, respectively. This relation will be useful in the construction of the SU(2) coherent state functional integral. 其中 ℜe 与 ℑm 分别表示实部和虚部。该关系式在构建 SU(2) 相干态泛函积分时将发挥重要作用。
SU (1,1) coherent states SU (1,1) 相干态
The group SU(1,1) is the group of unitary unimodular matrices of the form 该群 SU(2)是由形如...的酉幺模矩阵构成的群
g=(αβ∗βα∗),detg=1,(C.26)
i.e, matrices that preserve the quadratic form ∣α∣2−∣β∣2. The algebra of the SU(1,1) group is 即保持二次型...不变的矩阵。SU(2)群的李代数为
[J+,J−]=−2J3[J3,J±]=±J±(C.27)
with the ladder operators J±=J1±iJ2.
The unitary irreducible representations for SU(1,1) are labeled by the eigenvalues of Casimir operators. Because the rank of SU(1,1) is 1 , there is only one (quadratic) Casimir operator for the su(1,1) algebra, i.e, 其中包含阶梯算子...。SU(2)群的不可约酉表示由 Casimir 算子的本征值标记。由于 SU(2)的秩为 1，其代数中仅存在一个(二次)Casimir 算子，即...
However, in contrast with SU(2) there is more than one way in which the spectrum {j,m} may be realized. This is because the Casimir operator (C.28) is a semi-definite operator. Owing to this, there are four classes (socalled series) of the unitary irreducible representations of SU(1,1) (see, e.g., [519]). All of these representations are infinite dimensional. It should be emphasized that SU(1,1) has also non-unitary representations (eg., nonunitary principal series [387]), but we shall refrain from considering nonunitary representations. Analogous algebraic considerations as in SU(2) would lead one to four classes of unitary irreducible representations in the SU(1,1) case. These are:
(1) The principal continuous series Cj(q0) : 但与 SU(2) 不同的是， {j,m} 谱的实现方式不止一种。这是由于卡西米尔算子(C.28)是一个半定算子。因此， SU(1,1) 的酉不可约表示存在四类（即所谓级数）（参见文献[519]）。所有这些表示都是无限维的。需要强调的是， SU(1,1) 也存在非酉表示（例如非酉主级数[387]），但我们将不考虑非酉表示。与 SU(2) 情形类似的代数考量会导出 SU(1,1) 情形下的四类酉不可约表示，分别为：(1) 主连续级数 Cj(q0) ：
j=−21+is,m=q0+n(s∈R+,n∈Z,q0∈R,∣q0∣≤21)
(2) The principal discrete series Dj+(q0) : (2) 主离散系列 Dj+(q0) ：
j=−∣q0∣−n~,m=−j+n(n~,n∈N0,q0∈R,∣q0∣≤21)
(3) The principal discrete series Dj−(q0) : (3) 主离散序列 Dj−(q0) ：
j=−∣q0∣−n~,m=j−n(n~,n∈N0,q0∈R,∣q0∣≤21)
(4) The supplementary continuous series Ej(q0) : (4) 补充连续系列 Ej(q0) ：
−21<j<−∣q0∣,m=q0+n(n∈Z,q0∈R,∣q0∣≤21).
The so-called Bargmann index q0 cannot be determined from algebraic considerations alone and the representation must be labeled both by value of j and q0. However, SU(1,1) has the maximal compact subgroup U(1) for whose unitary representations the possible values of Bargmann index is restricted to 0 or 1/2. These representations are known as Bargmann representations. 所谓的巴格曼指数 q0 仅凭代数考量无法确定，该表示必须同时通过 j 和 q0 的值来标记。然而， SU(1,1) 具有最大紧子群 U(1) ，对于其酉表示而言，巴格曼指数的可能取值被限制为 0 或 1/2 。这些表示被称为巴格曼表示。
In the following we confine ourselves to the discussion of Dj+only:
The principal discrete series Dj+:2j=−n~,m=−j+n(n~,n∈N0) i.e., j=0,−21,−1,−23,…,m=∣j∣,∣j∣+1,∣j∣+2,…. 下文我们仅讨论 Dj+ ：主离散系列 Dj+:2j=−n~,m=−j+n(n~,n∈N0) ，即 j=0,−21,−1,−23,…,m=∣j∣,∣j∣+1,∣j∣+2,… 。
Asthefiducial "ground" state ∣0⟩ wechoosethestate ∣j,−j⟩. Similarly as in the SU(2) case, such a state is annihilated by J−: i.e, J−∣j,−j⟩=0. Thus each representation in Dj+has its uniquefiducial vector. Thestability group for ∣j,−j⟩ is the subgroup of rotations around the z-axis. The coherent states ∣0(x)⟩ are then completely determined by points x on the coset space M=SU(1,1)/U(1)≅H+2. The manifold H+2 represents the upper sheet of the two-sheet hyperboloid: H+2={m;m2=m32−m22−m12=1,m3>0}. This two-dimensional surface can be conveniently parametrized by the hyperbolic "angular" variables τ and φ according to prescription 作为基准"基态" ∣0⟩ ，我们选择状态 ∣j,−j⟩ 。与 SU(2) 情形类似，此类状态会被 J− 湮灭：即 J−∣j,−j⟩=0 。因此 Dj+ 中的每个表示都有其唯一的基准向量。 ∣j,−j⟩ 的稳定群是围绕 z 轴旋转的子群。相干态 ∣0(x)⟩ 则由陪集空间 M=SU(1,1)/U(1)≅H+2 上的点 x 完全确定。流形 H+2 表示双叶双曲面的上叶： H+2={m;m2=m32−m22−m12=1,m3>0} 。这个二维曲面可以方便地通过双曲"角"变量 τ 和 φ 按照以下规则进行参数化。
m=(sinhτcosφ,sinhτsinφ,coshτ)(C.30)
Parameter x∈M can be identified with variables τ and φ. The coherent state ∣0(τ,φ)⟩ can thus be written as 参数 x∈M 可与变量 τ 和 φ 等同。因此相干态 ∣0(τ,φ)⟩ 可表示为
with the unit vector n=(sinφ,cosφ,0). The Gauss decomposition allows us to write D(τ,φ) in the ordered form 其中 n=(sinφ,cosφ,0) 为单位向量。高斯分解使我们能将 D(τ,φ) 写成有序形式
Here the pseudo-Euclidean scalar product is defined as m′⋅m=m3′m3−m2′m2−m3′m3. Defining ∣0(τ,φ)⟩=∣0(ζ)⟩≡∣0(m)⟩ we can write the resolution of the unity as 这里伪欧几里得标量积定义为 m′⋅m=m3′m3−m2′m2−m3′m3 。定义 ∣0(τ,φ)⟩=∣0(ζ)⟩≡∣0(m)⟩ 后，我们可以将单位分解写成
Note that the integral is convergent (i.e., the representation is square integrable) only when ∣j∣>1/2. Only such representations will concern us here. In the end the resolution of the unity reads 注意该积分仅在 ∣j∣>1/2 时收敛（即表示是平方可积的）。我们在此仅关注此类表示。最终单位分解表达式为
Here again dζdζ∗≡dℜe(ζ)dℑm(ζ). The resolution of unity (C.38) will serve as a useful starting point in setting up the SU(1,1) coherent state functional integral in Appendix N.2. 这里再次提到 dζdζ∗≡dℜe(ζ)dℑm(ζ) 。单位分解（C.38）将作为建立附录 N.2 中 SU(1,1) 相干态泛函积分的有用起点。
A ppendix D 附录 D
q-W H algebra, coherent states and theta functions q - WH 代数、相干态与θ函数
As an application of the result expressed by Eq. (1.100), we shall show that the action of the commutator [ aq,a^q ] may be related in the Fock-Bargmann representation (FBR) to the action of the coherent states (CS) displacement operator. 作为式(1.100)所表达结果的应用，我们将证明在 Fock-Bargmann 表示(FBR)中，对易子[ aq,a^q ]的作用可以与相干态(CS)位移算子的作用相关联。
The FBR provides a transparent frame to describe the usual CS [379, 380,519]. In this A ppendix we change notation with respect to A ppendix B. Here we replace z by α, so that the CS are now written as: FBR 为描述常规 CS[379,380,519]提供了透明框架。本附录变更了附录 B 的记号体系，将 z 替换为 α ，因此 CS 现表示为：
The relation between the CS and the basis {un(z)} (Eq. (1.88)) of the entire analytic function is here made explicit: un(α)=e21∣α∣2⟨n∣α⟩. The unitary displacement operator D(α) in (D.1) is given by: CS 与解析函数基 {un(z)} （式(1.88)）的关系在此显式表达为 un(α)=e21∣α∣2⟨n∣α⟩ 。式(D.1)中的幺正位移算符 D(α) 由下式给出：
Eq. (D.4) is nothing but the WH group law, also referred to as the Weyl integral representation (cf. Eqs. (1.61) and (1.62)). 式(D.4)正是 WH 群律，亦称为 Weyl 积分表示（参见式(1.61)与(1.62)）。
In order to extract a complete set of CS {∣αn⟩}, from the over-complete set {∣α⟩}, it is necessary to introduce a regular lattice L in the α-complex plane [55, 519]. The points (lattice vectors) αn of L({αn∈C;n=(n1,n2);nj∈Z}) are given by αn=n1Ω1+n2Ω2≡n⋅Ω, with the two lattice periods Ωj,j=1,2 linearly independent, i.e, such that ℑm(Ω1Ωˉ2)=0. 为了从过完备集 {∣α⟩} 中提取完备的 CS {∣αn⟩} 集，需要在 α 复平面[55,519]中引入规则格点 L 。格点 L({αn∈C;n=(n1,n2);nj∈Z}) 的向量 αn 由 αn=n1Ω1+n2Ω2≡n⋅Ω 给出，其中两个格点周期 Ωj,j=1,2 线性无关，即满足 ℑm(Ω1Ωˉ2)=0 。
We recall [519] that the set {∣αn⟩} (with the exclusion of the vacuum state ∣0⟩≡∣α0⟩ ) can be shown to be complete, invoking square integrability along with analyticity [56], if the lattice elementary cell has area ℑm(Ω1Ωˉ2)=π ( L is called, in this case, the von Neumann lattice). 我们回顾[519]可知，若格点元胞面积为 ℑm(Ω1Ωˉ2)=π （此时 L 称为冯·诺伊曼格点），通过结合平方可积性与解析性[56]，可以证明集合 {∣αn⟩} （除真空态 ∣0⟩≡∣α0⟩ 外）是完备的。
The lattice vectors αn describe the discrete translational invariance of L:αn+m=αn+αm, i.e., 晶格矢量 αn 描述了 L:αn+m=αn+αm 的离散平移对称性，即
eαndαd∣α⟩α=αm=∣αn+m⟩.(D.6)
The denumerable set of points αn is now mapped onto the set {zn;zn∈C} with zn≡eαn. Assuming that the two periods Ω1 and Ω2 have imaginary parts incommensurate with π and among themselves, such a map is oneto-one and no point zn lies on the real axis in the z plane (notice that the set {zn} does not constitute a lattice in z, but it has the structure of concentric circles). 可数点集 αn 现被映射到集合 {zn;zn∈C} ，其中 zn≡eαn 。假设两个周期 Ω1 和 Ω2 的虚部与 π 及彼此之间不可公度，则该映射是一一对应的，且 z 平面上没有任何点 zn 位于实轴上（注意集合 {zn} 在 z 中不构成格点，而是具有同心圆结构）。
The function z=eα, which interpolates among these points, is analytical in its domain of definition, and, along with the basis functions {un(α)}, the new functions {u~n(z)≡un(lnz)=un(α)}, with u~n(z)∈F, may be introduced ( F denotes the space of the entire analytical functions). It is then straightforward to check that, in F, 在这些点之间进行插值的函数 z=eα 在其定义域内是解析的，并且与基函数 {un(α)} 一起，可以引入新函数 {u~n(z)≡un(lnz)=un(α)} （其中 u~n(z)∈F ， F 表示全纯函数空间）。随后不难验证，在 F 中，
where we utilized Eq. (D.4), and the notation fm(α)≡exp(−2∣α∣2)um(α). The operator [aq,a^q]∣q=qn, which realizes the mapping f~(zm)→f~(qnzm), can thus be thought of as extendible to the map on the α-plane ∣αm⟩→∣αn+m⟩. 这里我们运用了方程(D.4)及符号 fm(α)≡exp(−2∣α∣2)um(α) 。实现映射 f~(zm)→f~(qnzm) 的算子 [aq,a^q]∣q=qn ，因此可视为可扩展为 α 平面 ∣αm⟩→∣αn+m⟩ 上的映射。
In the FBR we have, for any f∈F 在 FBR 表示中，对于任意 f∈F
D(β)f(α)=exp(−2∣β∣2)exp(αβ)f(α−βˉ),f∈F,(D.9)
so that, by using Eq. (D.7), we can write, for q=eζ and z=eα, 因此，利用方程(D.7)，我们可以对 q=eζ 和 z=eα 写出：
[aq,a^q]f~(z)=e21∣ζ∣2qˉαD(−ζˉ)f(α).(D.10)
Eqs. (D.8) and (D.10) show the relation between the q-WH algebra operator [aq,a^q] and the CS displacement operator in the frame of the theory of entire analytical functions. 方程(D.8)和(D.10)展示了在整解析函数理论框架下， q -WH 代数算子 [aq,a^q] 与 CS 位移算子之间的关系。
We conclude that the existence of a quantum deformed algebra signals the presence of finite lengths in the theory and provides the natural framework for the physics of discretized systems, the q-deformation parameter being related with the lattice constants. 我们得出结论：量子变形代数的存在标志着理论中存在有限长度，并为离散化系统的物理提供了自然框架，其中 q 变形参数与晶格常数相关。
The lattice structure is also of crucial relevance in the relation between theta functions and the complete system of CS. In order to establish such a relation, welook for the common eigenvectors ∣θ⟩ of the CS operators D(αn) associated to the regular lattice L [519]. A common set of eigenvectors exists if and only if all the D(αn) commute, i.e, when the D(Ωj) commute, as indeed happens on the von Neumann lattice (cf. Eq. (D.5)). 晶格结构在 theta 函数与 CS 完备系统之间的关系中也具有关键重要性。为建立这种关联，我们寻找与规则晶格 L [519]相关联的 CS 算子 D(αn) 的共同本征向量 ∣θ⟩ 。当且仅当所有 D(αn) 对易时（即当 D(Ωj) 对易时，正如冯·诺伊曼晶格上发生的情形（参见方程(D.5)）），才会存在一组共同的本征向量。
The eigenstates ∣θ⟩ are characterized by two real numbers ϵ1 and ϵ2, so that we denote them by ∣θϵ⟩, which are eigenvectors of D(Ωi) : 本征态 ∣θ⟩ 由两个实数 ϵ1 和 ϵ2 表征，因此我们将其记作 ∣θϵ⟩ ，它们是 D(Ωi) 的本征向量：
D(Ωj)∣θϵ⟩=eiπϵj∣θϵ⟩,j=1,2,0≤ϵj≤2.(D.11)
We thus see that ∣θϵ⟩, which belongs to a space which is the extension [519] of the Hilbert space where the operators D(α) act, corresponds to a point on the two-dimensional torus. The action of D(α) on ∣θϵ⟩ generates a set of generalized coherent states. Use of Eqs. (D.11) and (D.4) gives 由此可见， ∣θϵ⟩ 属于一个作为希尔伯特空间扩展[519]的空间（其中算子 D(α) 作用其上），对应于二维环面上的一个点。 D(α) 对 ∣θϵ⟩ 的作用生成了一组广义相干态。利用方程(D.11)和(D.4)可得
D(αm)∣θϵ⟩=eiπFϵ(m)∣θϵ⟩,(D.12)
with αm=m⋅Ω, an arbitrary lattice vector, and 其中 αm=m⋅Ω 为任意格矢，且
Fϵ(m)=m1m2+m1ϵ1+m2ϵ2.(D.13)
On the other hand, the system of CS is associated, in the FBR, with a corresponding set of entire analytic functions, say θϵ(α). Eq. (D.9) with αˉ=−αm shows that Eq. (D.12) may be written as 另一方面，在 FBR 表示中，CS 系统与一组对应的全纯解析函数（记作 θϵ(α) ）相关联。结合 αˉ=−αm 的方程(D.9)表明，方程(D.12)可表示为
which is the functional equation for the theta functions [69, 484, 519]. We emphasize that such relation is obtained by considering the CS system corresponding to the von Neumann lattice L. The relation with the q−WH algebra is established by realizing that in F the functional equation (D.14) reads 这是关于θ函数的函数方程[69, 484, 519]。我们强调，这一关系是通过考虑对应于冯·诺伊曼格点 L 的 CS 系统而获得的。与 q−WH 代数的联系是通过认识到在 F 中，函数方程(D.14)表现为：
The commutator [ aq,a^q ] acts as shift operator on the von Neumann lattice whereas it acts as z-dilatation operator ( z→qz ) in the space of entire analytic functions or, else, as the U(1) generator of phase variations in the z-plane, arg(z)→arg(z)+θ, when q=eiθ. 交换子[ aq,a^q ]在冯·诺伊曼格点上作为位移算子作用，而在全解析函数空间中则作为 z -膨胀算子( z→qz )作用，或者当 q=eiθ 时，在 z 平面上作为相位变化的 U(1) 生成元( arg(z)→arg(z)+θ )作用。
It is remarkable that Eqs. (D.7), (D.8) and (D.10) represent the action of the q-WH algebra commutator [aq,a^q], (bi-)linear in aq and a^q, through the action of the CS displacement operator which is non-linear in the FBR operators a and a†. Conversely, the non-linear operator D(α) is represented by the linear form [aq,a^q] in the q-WH algebra. 值得注意的是，方程(D.7)、(D.8)和(D.10)通过 CS 位移算子的作用，展现了 q -WH 代数交换子 [aq,a^q] （在 aq 和 a^q 中为(双)线性）的行为，而该位移算子在 FBR 算子 a 和 a† 中是非线性的。相反，非线性算子 D(α) 在 q -WH 代数中由线性形式 [aq,a^q] 表示。
The reader is referred to [143] for the relation between q-WH algebra and lattice QM, and q-WH algebra and Bloch functions. 关于 q -WH 代数与格点量子力学的关系，以及 q -WH 代数与布洛赫函数的关系，读者可参阅文献[143]。
Footnotes 脚注
1 Self-interaction, which dresses each particle and is responsible for wave function renormalization, is implicitly taken into account in the concept of physical particle. ↩ 1 自相互作用既修饰了每个粒子，又导致了波函数重整化，这一概念已隐含在物理粒子的定义中。↩
