电子偶素基态的三维相对论方程和规范场中WARD-TAKAHASHI恒等式的研究
摘要
基于强子物理的研究和观察,人们期望夸克禁闭和动力学手征对称破缺等非微扰现象应该成为量子色动力学的两个关键特征。这些非微扰现象的根本原因迄今所知甚不完全。围绕着这一问题,进一步的理论探索就必需建立可靠的非微扰方法。
目前,研究量子色动力学的这两个特征和强子物理的一个自然方法是采用Dyson-Schwinger方法,对于QCD中夸克和胶子的传播子而言,这些方程形成耦合的方程组体系。由于方程呈现明显的相对论协变性,因而在许多领域中有着广泛的应用。尤其是通过构建场论模型如Bethe-Sepeter方程,可以期望很好地理解QCD低能现象。然而,在研究Dyson-Schwinger方程的过程中,人们遇到了很大的阻碍,起因于该方程是无穷迭代的耦合方程组。在最简单情况下,两点传播子耦合到三点的顶角函数。因此,在实际应用中人们只得引入假设来截断方程使其成为封闭体系。针对这种情况,倘若我们能用两点函数把顶角函数完整地表示出来,那么,Dyson-Schwinger就形成可解得封闭体系。无疑,如何准确地表述完整的顶角函数便成为一个关键的课题。
迄今为止,人们在这一课题上做了大量的尝试,但都存在一个症结;那就是在量子场论的考量下,求解Dyson-Schwinger方程的过程中对顶角函数的选择不得不引入不是来自体系对称性的约束条件。这使得顶角函数成为一个附有假设条件的函数,由此导致的结果是DSE的物理解释失去了一定的可靠性。直到不久前[He01],对于Abel规范场理论-QED模型,采用场论中的算符形式探讨了Dyson-Schwinger方程的特殊形式-WT恒等式。文中得到了一些有意义的结论。从这些研究工作中又一次看到了正如所熟知的情况,通常的WT恒等式只表达Green函数的纵向分量,而Green函数的横向分两量从未能正确地被表达出来。从得到的一些顶角函数的形式上看,WT恒等式的反常性质未能很好的考虑进去。这样,一定程度上讲,这些函数的形式是不完善的。为了能更广泛深入研究的这一既是基本也是前沿的课题,我们沿着不同的两条路线研究量子体系的相互作用核形式。一方面,我们采用Green函数的方法,建立了电子偶素基态的三维相对论方程,与通常描写二体费米子体系的BS方程不同,新的方程不仅是一阶微分方程,而且其等效相互作用核具有封闭形式。在Pauli自旋空间中可以得到约化的正能解的Schrodinger方程。在一阶近似下,封闭相互作用核给出正确的单光子交换势,这表明建立的二体相对论方程是正确的。另一方面,规范对称性对规范理论中的基本顶角函数给予强有力的约束,导致了Ward-Takahashi恒等式。本工作提供了由对称性关系决定完全的费米子-玻色子顶角的一种途径。对Abelian规范场论,给出了一系列各种顶角相互耦合的WT恒等式的完整表达式。同时也细致地研究了流守恒的反常现象。这些结果为进一步研究完全Green函数性质及DSE的应用奠定了必要的基础。
本文包括四个方面内容:
第一部分:用Green函数的方法严格地推导了电子偶素基态的严格洛伦茨协变束缚态方程。目前,三维等时相对论方程是在相对论量子场论框架下研究强子束缚态的一种有效工具。其形式是一个严格的洛伦茨协变的束缚态方程,它的有效相互作用核包含了粒子间的全部相互作用。三维基态方程可以等价地约化为泡利-薛定谔方程。并且,在最低阶近似下,等效相互作用势能够正确地被求解出来。
第二部分:在这里我们注意到一个基本事实;量子场论的对称性导致各种Green函数之间的关系式。确切地讲,量子体系的生成泛函可以产生各种粒子传播子的Dyson-Schwinger方程,其中包括熟知的Ward-Takahashi恒等式。在Abel规范场-QED的模型中,通过引入合理的规范群叁数的假设,成功地建立了一般的定域规范变换,使得QED的生成泛函在此变换下具有不变性(这里也称规范不变性)。这是我们工作的根本出发点,它提供了推导各种Dyson-Schwinger方程的理论根据。
由于在量子场论中,以场算符乘积形式出现的各种费米流可能发散,因而量子体系的对称性可能受到破坏而引起反常现象。为了解决这个困难,我们借助于引进的规范群叁数假设,有效地推广了Fujikawa提出的泛函积分方法,建立了适用于各种费米流的计算公式,正确的得到了由积分测度引起的一系列反常项,其结果不仅包含了文献中已经存在的结果,并且又发现了高阶张量流的反常项。同时,根据手征流的反常与规范场的拓扑性质相联系的指数定理,也找到了一种有效办法来判断费米场与规范场相互作用的各种耦合流的奇异性。这些结果为完整的推出反常的WT恒等式提供了存在的可能性。
进一步利用定域规范变换,在放宽拉氏量的物理要求下,导出了各种费米流的WT恒等式。经过观察,我们发现在这些等式中包含了各种顶角函数(矢量,轴矢量,张量)的横向分量。通过求解这些相互耦合的WT恒等式,各种顶角函数的精确表达式可以用两Green函数被完整地推导出来。这样导出的顶角函数是微扰和非微扰都满足的,因为它完全由微扰和非微扰都成立的对称性关系确定。这些工作为进一步的研究打下了坚实的基础。
第三部分:对于标准模型GWS,我们也做了初步的研究。仿照前面的方式,首次正确的找到了一般的规范变换(类似于通常的BRST对称性),使有效拉式量具有规范不变性。在此基础上,推导出了WT恒等式。
在此基础上,关于在标准模型GWS中的反常现象在最简单情形也做了初步的研究。研究表明,由于Higg场的出现,处理反常问题增加了难度。同时指出:Fujikawa提出的两种泛函积分计算反常方法有不等价性。关于理论的WT恒等式需要进一步研究。
第四部分:在附录B中,利用代数恒等式,依照重整化群方程的方法,研究了QCD理论中Green函数在外线动量类空时其值变化的渐进行为。对渐近行为给出了一个可能的表述,很好的说明了非Abel规范场论的渐进自由行为和强耦合在大距离处变强的形式和性质。
在附录C中,在Abelian规范场中,结合文中提出的假设与微分几何中Atiyah-Singer指数定理,找到了一种拓扑方法在规范场背景下,判定费米子轴矢量流的奇异性。
值得一提的是,在附录D中,我们研究了QCD理论中各种WT恒等式的形式。由于F-P鬼场的存在,使QCD的情况增加了复杂性。所幸的是,在研究这个课题中,我们注意到了此问题的关键点;在QCD理论中,有效拉式量中的鬼场,不仅是约束场的多余自由度的非物理量,而且也担当着作为规范群叁数的角色。熟知的BRST变换正是如此。
在QED的做法上,根据QCD的特点,我们进一步合理地推广了Fujikawa提出的泛函积分方法,建立了适用于各种费米流的计算公式,正确的得到了由积分测度引起的新的一系列反常项,其明显的特征是这些反常项包含了来自色空间中色荷的贡献。对这些新的反常项的解释,正在研究中.
引入与鬼场相关的且合理的规范群叁数的假设,推导了各种Dyson-Schwinger方程(各种费米流的WT恒等式)。重要的是全新的各种夸克与胶子的顶角函数表达式被完整地推导出来。从这些顶角函数的形式看,其表达式还是相当复杂。然而,通过仔细观察会发现,利用SU(3)群的结构常数,这些结果可以进一步简化。最终的顶角函数形式有了一定的简单性。
这正是Dyson-Schwinger形成可解的封闭体系所需要的。
本文的研究在理论方面给出了一些新的内容和结果,如下所述:
1.研究电子偶素基态过程中(双粒子体系),首次给出正能解的三维相对论方程,虚对的湮灭相互作用势可通过求解三维相对论方程在低级近似下的计算而得到具体的表达式;其形式与Foldy-Wouthuysen技术给出的结果相一致,但表达式中的自旋项系数有些不同。
2.在QED模型中,根据提出的规范群参数假设,找到了一般的定域规范变换,建立了相应的包含各种相互作用的QCD拉式量。在Abel规范场论中,高阶费米张量流的Ward-Takahashi恒等式产生了一新的反常因子。
3.在一般的定域规范变换下,从量子体系的生成泛函导出了各种费米流的Ward-Takahashi恒等式;各种顶角函数的精确表达式可以用两Green函数被完整地推导出;其中自然包含了反常项,完善了原有的结果。
4.在标准模型GWS中,首次正确的找到了一般的定域规范变换,也使得有效拉式量具有规范不变性。然后仿照QCD的量子化方案对理论进行量子化,并推导出与重整化和幺正性紧密相关的Ward-Takahashi恒等式。在这个基础上,研究标准模型GWS的反常性质及WT等式成为可能。
5.在标准模型GWS中,研究指出:Fujikawa提出的两种泛函积分计算反常的方法存在不等价性。
6.在附录B中,利用重整化群方程和严格的代数方法,对QCD理论中顶角Green函数的渐近行为给出了一个可能的新表述。
7.在附录C中,利用微分几何中指数定理,提出了判定费米子轴矢量流奇异性的拓扑方法。
8.在附录D中,探讨了在QCD中表述完全GREEN函数形式的可能性。采用泛函积分方法,通过对各种费米流的定域规范变换,正确地得到了由积分测度引起的一系列新的反常项。结果表明,反常因子包含色空间的贡献。
9.在一般的定域规范变换下,量子体系的生成泛函也导出了一系列新的Ward-Takahashi恒等式;各种顶角函数(包括夸克-胶子顶角函数)相互耦合存在于恒等式中。它们的物理意义及应用需进一步阐释。
10.在附录E中,经过相当繁琐的高阶微扰展开计算,提供了夸克-胶子顶角的WT恒等式的两圈微扰展开的不可约費曼图。
如下所述本文的研究在方法方面给出了一些新的做法:
1.利用Green函数的方法,建立了电子偶素(双费米子系统)的Dirac型相对论方程;其有效相互作用势包含了粒子间所有相互项,对计算非微扰效应提供了有效的一种途径。
2.合理利用了规范群和规范群叁数的性质,成功地建立了一般的定域规范变换。在考虑反常性问题时,我们有效地推广了Fujikawa的做法,建立了使用于各种费米流的计算公式,正确的得到了由积分测度引起的反常项。
3.利用微分几何中指数定理,将各种费米流奇异性与规范场的拓扑性质联系起来。
存在的问题和展望
目前,承于本文的研究内容,在以下两方面正在进行探索:
1.考察强相互作用的过程,期望找到QCD中手征反常的应用.例如,可以通过分析夸克湮灭过程理解反常性质.
2.可推广文中两体系粒子系统的三维相对论方程应用于介子体系,进一步计算介子谱。
3.在Abelian规范场论中,推导完全Green函数形式时,放宽了拉氏量的物理要求,即其Lorentz不变性可能得不到满足,因此,理论需进一步完善。
Based on man observation on hadron physics, the dynamical chiral symmetry breaking (DCSB) and confinement are two crucial features of quantum chromodynamics (QCD). They are expected to occur on QCD. The precise origin of this nonperturbaive phenomenon as well as its relation to quark confinement is still little understood. Further studies of these issues gave to build on reliable nonperturvbative methods.
A natural method for studying both DCSB and confinement in QCD is the complex of Dyson-Schwinger equations (DSE’s) the manifestly relativistically covariant approach has provided the foundation for useful and successful understanding of the phenomena of low–energy QCD by facilitating the construction of realistic field–theoretic models.
The knowledge of the three-point vertices iscrucial in the studies of gauge theories through the use of the Dyson-Schwinger equations (DSE’S), where the transverse part of the vertex plays the crucial role in ensuring multiplicative renormalizahility and in determining the propagator. Therefore, in the past years much effort has been devoted to constructing the transverse part of the vertex based on perturbative constraints on the transverse vertex through the one-loop evaluation of the vertex. Such perturbative constraints are useful because the physically meaningful solutions of the DSE’S must agree with perturbative results in the weak-coupling regime. However, such a constructed vertex is not unique since it is not fixed by the symmetry of the system. The latter provides the key point in determining the transverse part of the vertex; like the longitudinal part of the vertex , the transverse part of the vertex should be determined also by the WT-type relations.
How to solve exactly the transverse part of the vertex and the full vertex function (Green functions) the becomes a curial problem. This is of the nature of our study.
The work in this thesis is composed of four major parts:
1. A rigorous three-dimensional relativistic equation satisfied by positronium bound states is derived from QED generating functional without making use of the four-dimensional Bethe-Salpeter equation and its reduction. The effective interaction kernel in the equation is given by a closed expression that includes all the interactions taking place in the positronium bound states, with the aid of the equations of motion satisfied by some kinds of Green’s functions. Furthermore, it is shown that the equation of bound states can be equivalently reduced to a Pauli-Schrodinger equation. In order to show the applicability of the closed expressions derived, the effective potentials of the corresponding t-channel and s-channel one photon kernels are evaluated , which not only is relativistic but also takes account of the retardation effect properly, no ambiguity being involved. As an illustration, explicit expression for the effective potential is given for the lowest order approximation, which gives the annihilation interaction potential in the process of virtual pair annihilation and the common Breit interaction.
2. In the second part of this thesis,The symmetries of gauge theories lead to a variety of exact relations among the Green’s functions including the transverse WT relation. This provides an approach to determine the transverse part of the vertex. We are interested in deriving various full Green functions through general Ward-Takahashi identities (WTIs) for quantized field theories. With the help of a postulate of gauge group parameter, the general local gauge transformation laws preserving the gauge-invariance of the generating functional itself of QED model have been established successfully. By using path-integral technique, the various
WTIs with resulting anomaly terms are derived under the gauge transformations. The arising of Jacobian factor from the integration measure gives a viable possibility to express full Green function. As a consequence, the complete expressions of the full vector, the full axial-vector, the full tensor vertex functions and so on are presented respectively by solving the complete set of the WTIs in the momentum space without considering the constraint imposing any Ansatz. In addition, anomaly function also provides an effective means to judge the divergence of variant coupling currents on fields.
3. The third part is mainly concerned with the quantum theory of the quantized Glashow-Weinverg-Salam model.. With the help of a postulate of gauge group parameter, the infinitesimal gauge transformation laws preserving the gauge–invariance of the generating functional itself of the quantized Glashow-Weinberg–Salam model have been established precisely. As viewed from the origin of the symmetry, the new transformations are the same as the standard Becchi-Rouet-Stora ones which also stem from the invariance of the full quantum Lagrangian itself. These new transformations are indispensable for both calculation of the anomaly in GWS model and derivation of the generalized WT identity. A straightforward application of the infinitesimal transformation generally leads the fermionic integral measure relating with the transformation to the path–integral derivation of the anomaly associated with WT identity.
In the simplest case, by making use of a general path-integral prescription, a series of the expressions of the anomaly factor including the contribution for left–handed and right–handed fields have been carried out in coordinate space.
Herein we explore the difference between the two integral- approaches. For purpose of comparison, by means of an alternative path–integral regulation scheme proposed in Ref.[Fu04], the possible effect of the Higgs coupling on the anomaly in the WS model, is precisely estimated. Thus we first present that, in general, the Higgs field gives rise to the influence on the anomaly with respect to tensor currents. As a consequence, it is shown that the two path–integral prescriptions are not completely equivalent to each other in the case of GWS model.
4. In addition, a new exposition of proper asymptotic behavior of Green Functions for large and small external momenta in QCD is presented in appendix B. Herein we try to demonstrate the confinement property of QCD.
In appendix C, A topologcal way to distinguish the divergence of operator product of fermion current in quantum field theory is presented. By virtue of the topologically non-trivial Jacobian factor of the integration measure in the path-integral formulation of the theory, the singularity of operator product of fermion current linked with topological property of gauge field can be examined in a gauge background correctly.
5.In appendix D, We make use of the symmetry of the generating functional to generate the identities for various full Green functions in quantized non- Abelian gauge field theory-QCD model. With the help of a postulate of gauge group parameter, such identities with the anomaly terms are exactly derived under a general local gauge transformation law maintaining the symmetry. Hence the usual BRS transformation is regarded as a special case in the new symmetry. A key point is that ghost field possesses a duel purpose. Thus the path-integral derivation of the anomaly factors associated with Ward-Takahashi identities for variant fermionic currents gives a viable possibility to express the complete vertex function. As a consequence, the complete expressions of the full tensor, the full axial-tensor vertex function, and so on are well expressed respectively in terms of the fermion propagators by solving the complete set of the WTIs in the momentum space in terms of these WTIs. The resulting expression contains only a few types of Green’s functions which not only are easily calculated by the perturbation method, but also suitable for investigation by a certain nonperturbative approach. However, there are several problematic aspects to need further study in this paper;
1) The application of the three-dimensional relativistic two-body wave equations presented in this thesis to calculating the meson spectra will be studied later.
2) The derived complete expression of the transverse Ward-Takahashi(WT) relations for the fermion–boson vertex (vector vertex) in momentum space in four-dimensional Abelian gauge theory should lead to the same results one calculated in perturbation theory.
目前,研究量子色动力学的这两个特征和强子物理的一个自然方法是采用Dyson-Schwinger方法,对于QCD中夸克和胶子的传播子而言,这些方程形成耦合的方程组体系。由于方程呈现明显的相对论协变性,因而在许多领域中有着广泛的应用。尤其是通过构建场论模型如Bethe-Sepeter方程,可以期望很好地理解QCD低能现象。然而,在研究Dyson-Schwinger方程的过程中,人们遇到了很大的阻碍,起因于该方程是无穷迭代的耦合方程组。在最简单情况下,两点传播子耦合到三点的顶角函数。因此,在实际应用中人们只得引入假设来截断方程使其成为封闭体系。针对这种情况,倘若我们能用两点函数把顶角函数完整地表示出来,那么,Dyson-Schwinger就形成可解得封闭体系。无疑,如何准确地表述完整的顶角函数便成为一个关键的课题。
迄今为止,人们在这一课题上做了大量的尝试,但都存在一个症结;那就是在量子场论的考量下,求解Dyson-Schwinger方程的过程中对顶角函数的选择不得不引入不是来自体系对称性的约束条件。这使得顶角函数成为一个附有假设条件的函数,由此导致的结果是DSE的物理解释失去了一定的可靠性。直到不久前[He01],对于Abel规范场理论-QED模型,采用场论中的算符形式探讨了Dyson-Schwinger方程的特殊形式-WT恒等式。文中得到了一些有意义的结论。从这些研究工作中又一次看到了正如所熟知的情况,通常的WT恒等式只表达Green函数的纵向分量,而Green函数的横向分两量从未能正确地被表达出来。从得到的一些顶角函数的形式上看,WT恒等式的反常性质未能很好的考虑进去。这样,一定程度上讲,这些函数的形式是不完善的。为了能更广泛深入研究的这一既是基本也是前沿的课题,我们沿着不同的两条路线研究量子体系的相互作用核形式。一方面,我们采用Green函数的方法,建立了电子偶素基态的三维相对论方程,与通常描写二体费米子体系的BS方程不同,新的方程不仅是一阶微分方程,而且其等效相互作用核具有封闭形式。在Pauli自旋空间中可以得到约化的正能解的Schrodinger方程。在一阶近似下,封闭相互作用核给出正确的单光子交换势,这表明建立的二体相对论方程是正确的。另一方面,规范对称性对规范理论中的基本顶角函数给予强有力的约束,导致了Ward-Takahashi恒等式。本工作提供了由对称性关系决定完全的费米子-玻色子顶角的一种途径。对Abelian规范场论,给出了一系列各种顶角相互耦合的WT恒等式的完整表达式。同时也细致地研究了流守恒的反常现象。这些结果为进一步研究完全Green函数性质及DSE的应用奠定了必要的基础。
本文包括四个方面内容:
第一部分:用Green函数的方法严格地推导了电子偶素基态的严格洛伦茨协变束缚态方程。目前,三维等时相对论方程是在相对论量子场论框架下研究强子束缚态的一种有效工具。其形式是一个严格的洛伦茨协变的束缚态方程,它的有效相互作用核包含了粒子间的全部相互作用。三维基态方程可以等价地约化为泡利-薛定谔方程。并且,在最低阶近似下,等效相互作用势能够正确地被求解出来。
第二部分:在这里我们注意到一个基本事实;量子场论的对称性导致各种Green函数之间的关系式。确切地讲,量子体系的生成泛函可以产生各种粒子传播子的Dyson-Schwinger方程,其中包括熟知的Ward-Takahashi恒等式。在Abel规范场-QED的模型中,通过引入合理的规范群叁数的假设,成功地建立了一般的定域规范变换,使得QED的生成泛函在此变换下具有不变性(这里也称规范不变性)。这是我们工作的根本出发点,它提供了推导各种Dyson-Schwinger方程的理论根据。
由于在量子场论中,以场算符乘积形式出现的各种费米流可能发散,因而量子体系的对称性可能受到破坏而引起反常现象。为了解决这个困难,我们借助于引进的规范群叁数假设,有效地推广了Fujikawa提出的泛函积分方法,建立了适用于各种费米流的计算公式,正确的得到了由积分测度引起的一系列反常项,其结果不仅包含了文献中已经存在的结果,并且又发现了高阶张量流的反常项。同时,根据手征流的反常与规范场的拓扑性质相联系的指数定理,也找到了一种有效办法来判断费米场与规范场相互作用的各种耦合流的奇异性。这些结果为完整的推出反常的WT恒等式提供了存在的可能性。
进一步利用定域规范变换,在放宽拉氏量的物理要求下,导出了各种费米流的WT恒等式。经过观察,我们发现在这些等式中包含了各种顶角函数(矢量,轴矢量,张量)的横向分量。通过求解这些相互耦合的WT恒等式,各种顶角函数的精确表达式可以用两Green函数被完整地推导出来。这样导出的顶角函数是微扰和非微扰都满足的,因为它完全由微扰和非微扰都成立的对称性关系确定。这些工作为进一步的研究打下了坚实的基础。
第三部分:对于标准模型GWS,我们也做了初步的研究。仿照前面的方式,首次正确的找到了一般的规范变换(类似于通常的BRST对称性),使有效拉式量具有规范不变性。在此基础上,推导出了WT恒等式。
在此基础上,关于在标准模型GWS中的反常现象在最简单情形也做了初步的研究。研究表明,由于Higg场的出现,处理反常问题增加了难度。同时指出:Fujikawa提出的两种泛函积分计算反常方法有不等价性。关于理论的WT恒等式需要进一步研究。
第四部分:在附录B中,利用代数恒等式,依照重整化群方程的方法,研究了QCD理论中Green函数在外线动量类空时其值变化的渐进行为。对渐近行为给出了一个可能的表述,很好的说明了非Abel规范场论的渐进自由行为和强耦合在大距离处变强的形式和性质。
在附录C中,在Abelian规范场中,结合文中提出的假设与微分几何中Atiyah-Singer指数定理,找到了一种拓扑方法在规范场背景下,判定费米子轴矢量流的奇异性。
值得一提的是,在附录D中,我们研究了QCD理论中各种WT恒等式的形式。由于F-P鬼场的存在,使QCD的情况增加了复杂性。所幸的是,在研究这个课题中,我们注意到了此问题的关键点;在QCD理论中,有效拉式量中的鬼场,不仅是约束场的多余自由度的非物理量,而且也担当着作为规范群叁数的角色。熟知的BRST变换正是如此。
在QED的做法上,根据QCD的特点,我们进一步合理地推广了Fujikawa提出的泛函积分方法,建立了适用于各种费米流的计算公式,正确的得到了由积分测度引起的新的一系列反常项,其明显的特征是这些反常项包含了来自色空间中色荷的贡献。对这些新的反常项的解释,正在研究中.
引入与鬼场相关的且合理的规范群叁数的假设,推导了各种Dyson-Schwinger方程(各种费米流的WT恒等式)。重要的是全新的各种夸克与胶子的顶角函数表达式被完整地推导出来。从这些顶角函数的形式看,其表达式还是相当复杂。然而,通过仔细观察会发现,利用SU(3)群的结构常数,这些结果可以进一步简化。最终的顶角函数形式有了一定的简单性。
这正是Dyson-Schwinger形成可解的封闭体系所需要的。
本文的研究在理论方面给出了一些新的内容和结果,如下所述:
1.研究电子偶素基态过程中(双粒子体系),首次给出正能解的三维相对论方程,虚对的湮灭相互作用势可通过求解三维相对论方程在低级近似下的计算而得到具体的表达式;其形式与Foldy-Wouthuysen技术给出的结果相一致,但表达式中的自旋项系数有些不同。
2.在QED模型中,根据提出的规范群参数假设,找到了一般的定域规范变换,建立了相应的包含各种相互作用的QCD拉式量。在Abel规范场论中,高阶费米张量流的Ward-Takahashi恒等式产生了一新的反常因子。
3.在一般的定域规范变换下,从量子体系的生成泛函导出了各种费米流的Ward-Takahashi恒等式;各种顶角函数的精确表达式可以用两Green函数被完整地推导出;其中自然包含了反常项,完善了原有的结果。
4.在标准模型GWS中,首次正确的找到了一般的定域规范变换,也使得有效拉式量具有规范不变性。然后仿照QCD的量子化方案对理论进行量子化,并推导出与重整化和幺正性紧密相关的Ward-Takahashi恒等式。在这个基础上,研究标准模型GWS的反常性质及WT等式成为可能。
5.在标准模型GWS中,研究指出:Fujikawa提出的两种泛函积分计算反常的方法存在不等价性。
6.在附录B中,利用重整化群方程和严格的代数方法,对QCD理论中顶角Green函数的渐近行为给出了一个可能的新表述。
7.在附录C中,利用微分几何中指数定理,提出了判定费米子轴矢量流奇异性的拓扑方法。
8.在附录D中,探讨了在QCD中表述完全GREEN函数形式的可能性。采用泛函积分方法,通过对各种费米流的定域规范变换,正确地得到了由积分测度引起的一系列新的反常项。结果表明,反常因子包含色空间的贡献。
9.在一般的定域规范变换下,量子体系的生成泛函也导出了一系列新的Ward-Takahashi恒等式;各种顶角函数(包括夸克-胶子顶角函数)相互耦合存在于恒等式中。它们的物理意义及应用需进一步阐释。
10.在附录E中,经过相当繁琐的高阶微扰展开计算,提供了夸克-胶子顶角的WT恒等式的两圈微扰展开的不可约費曼图。
如下所述本文的研究在方法方面给出了一些新的做法:
1.利用Green函数的方法,建立了电子偶素(双费米子系统)的Dirac型相对论方程;其有效相互作用势包含了粒子间所有相互项,对计算非微扰效应提供了有效的一种途径。
2.合理利用了规范群和规范群叁数的性质,成功地建立了一般的定域规范变换。在考虑反常性问题时,我们有效地推广了Fujikawa的做法,建立了使用于各种费米流的计算公式,正确的得到了由积分测度引起的反常项。
3.利用微分几何中指数定理,将各种费米流奇异性与规范场的拓扑性质联系起来。
存在的问题和展望
目前,承于本文的研究内容,在以下两方面正在进行探索:
1.考察强相互作用的过程,期望找到QCD中手征反常的应用.例如,可以通过分析夸克湮灭过程理解反常性质.
2.可推广文中两体系粒子系统的三维相对论方程应用于介子体系,进一步计算介子谱。
3.在Abelian规范场论中,推导完全Green函数形式时,放宽了拉氏量的物理要求,即其Lorentz不变性可能得不到满足,因此,理论需进一步完善。
Based on man observation on hadron physics, the dynamical chiral symmetry breaking (DCSB) and confinement are two crucial features of quantum chromodynamics (QCD). They are expected to occur on QCD. The precise origin of this nonperturbaive phenomenon as well as its relation to quark confinement is still little understood. Further studies of these issues gave to build on reliable nonperturvbative methods.
A natural method for studying both DCSB and confinement in QCD is the complex of Dyson-Schwinger equations (DSE’s) the manifestly relativistically covariant approach has provided the foundation for useful and successful understanding of the phenomena of low–energy QCD by facilitating the construction of realistic field–theoretic models.
The knowledge of the three-point vertices iscrucial in the studies of gauge theories through the use of the Dyson-Schwinger equations (DSE’S), where the transverse part of the vertex plays the crucial role in ensuring multiplicative renormalizahility and in determining the propagator. Therefore, in the past years much effort has been devoted to constructing the transverse part of the vertex based on perturbative constraints on the transverse vertex through the one-loop evaluation of the vertex. Such perturbative constraints are useful because the physically meaningful solutions of the DSE’S must agree with perturbative results in the weak-coupling regime. However, such a constructed vertex is not unique since it is not fixed by the symmetry of the system. The latter provides the key point in determining the transverse part of the vertex; like the longitudinal part of the vertex , the transverse part of the vertex should be determined also by the WT-type relations.
How to solve exactly the transverse part of the vertex and the full vertex function (Green functions) the becomes a curial problem. This is of the nature of our study.
The work in this thesis is composed of four major parts:
1. A rigorous three-dimensional relativistic equation satisfied by positronium bound states is derived from QED generating functional without making use of the four-dimensional Bethe-Salpeter equation and its reduction. The effective interaction kernel in the equation is given by a closed expression that includes all the interactions taking place in the positronium bound states, with the aid of the equations of motion satisfied by some kinds of Green’s functions. Furthermore, it is shown that the equation of bound states can be equivalently reduced to a Pauli-Schrodinger equation. In order to show the applicability of the closed expressions derived, the effective potentials of the corresponding t-channel and s-channel one photon kernels are evaluated , which not only is relativistic but also takes account of the retardation effect properly, no ambiguity being involved. As an illustration, explicit expression for the effective potential is given for the lowest order approximation, which gives the annihilation interaction potential in the process of virtual pair annihilation and the common Breit interaction.
2. In the second part of this thesis,The symmetries of gauge theories lead to a variety of exact relations among the Green’s functions including the transverse WT relation. This provides an approach to determine the transverse part of the vertex. We are interested in deriving various full Green functions through general Ward-Takahashi identities (WTIs) for quantized field theories. With the help of a postulate of gauge group parameter, the general local gauge transformation laws preserving the gauge-invariance of the generating functional itself of QED model have been established successfully. By using path-integral technique, the various
WTIs with resulting anomaly terms are derived under the gauge transformations. The arising of Jacobian factor from the integration measure gives a viable possibility to express full Green function. As a consequence, the complete expressions of the full vector, the full axial-vector, the full tensor vertex functions and so on are presented respectively by solving the complete set of the WTIs in the momentum space without considering the constraint imposing any Ansatz. In addition, anomaly function also provides an effective means to judge the divergence of variant coupling currents on fields.
3. The third part is mainly concerned with the quantum theory of the quantized Glashow-Weinverg-Salam model.. With the help of a postulate of gauge group parameter, the infinitesimal gauge transformation laws preserving the gauge–invariance of the generating functional itself of the quantized Glashow-Weinberg–Salam model have been established precisely. As viewed from the origin of the symmetry, the new transformations are the same as the standard Becchi-Rouet-Stora ones which also stem from the invariance of the full quantum Lagrangian itself. These new transformations are indispensable for both calculation of the anomaly in GWS model and derivation of the generalized WT identity. A straightforward application of the infinitesimal transformation generally leads the fermionic integral measure relating with the transformation to the path–integral derivation of the anomaly associated with WT identity.
In the simplest case, by making use of a general path-integral prescription, a series of the expressions of the anomaly factor including the contribution for left–handed and right–handed fields have been carried out in coordinate space.
Herein we explore the difference between the two integral- approaches. For purpose of comparison, by means of an alternative path–integral regulation scheme proposed in Ref.[Fu04], the possible effect of the Higgs coupling on the anomaly in the WS model, is precisely estimated. Thus we first present that, in general, the Higgs field gives rise to the influence on the anomaly with respect to tensor currents. As a consequence, it is shown that the two path–integral prescriptions are not completely equivalent to each other in the case of GWS model.
4. In addition, a new exposition of proper asymptotic behavior of Green Functions for large and small external momenta in QCD is presented in appendix B. Herein we try to demonstrate the confinement property of QCD.
In appendix C, A topologcal way to distinguish the divergence of operator product of fermion current in quantum field theory is presented. By virtue of the topologically non-trivial Jacobian factor of the integration measure in the path-integral formulation of the theory, the singularity of operator product of fermion current linked with topological property of gauge field can be examined in a gauge background correctly.
5.In appendix D, We make use of the symmetry of the generating functional to generate the identities for various full Green functions in quantized non- Abelian gauge field theory-QCD model. With the help of a postulate of gauge group parameter, such identities with the anomaly terms are exactly derived under a general local gauge transformation law maintaining the symmetry. Hence the usual BRS transformation is regarded as a special case in the new symmetry. A key point is that ghost field possesses a duel purpose. Thus the path-integral derivation of the anomaly factors associated with Ward-Takahashi identities for variant fermionic currents gives a viable possibility to express the complete vertex function. As a consequence, the complete expressions of the full tensor, the full axial-tensor vertex function, and so on are well expressed respectively in terms of the fermion propagators by solving the complete set of the WTIs in the momentum space in terms of these WTIs. The resulting expression contains only a few types of Green’s functions which not only are easily calculated by the perturbation method, but also suitable for investigation by a certain nonperturbative approach. However, there are several problematic aspects to need further study in this paper;
1) The application of the three-dimensional relativistic two-body wave equations presented in this thesis to calculating the meson spectra will be studied later.
2) The derived complete expression of the transverse Ward-Takahashi(WT) relations for the fermion–boson vertex (vector vertex) in momentum space in four-dimensional Abelian gauge theory should lead to the same results one calculated in perturbation theory.
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[We01]S.Weinberg, The Quantum Theory of Fields (II), Cambridge University Press P36 (2001)
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[Wa03]H.J.Wang , Ph.D. Thesis, K-N Elastic Scattering and Renormalization of σ-ωModel , (2003)
[Wa50]J. Ward, Phys. Rev, 78, 182 (1950).
[Wa75] Y.Watanable et al., Phys. Rev. Lett. 35, 898(1975).
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[Yo87]B.L. Young, Introduction to quantum field theory, Science Presss,Beijing China (1987).
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