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有关能动张量和多极场
用户登陆 | 刷新 | 本版嘉宾: sage yinhow |
yinhow 发表文章数: 727 |
有关能动张量和多极场 能动张量的一种定义是拉氏量对度规的变分, 对于普通物质得到的是能动张量, 对于引力作用量, 得到的是EINSTEIN张量. 从这个定义就可以看出引力的能动张量就不一般了. 如果物质的能动张量守恒, 需要场满足经典的EL方程, 有时为了凑对称性, 可以加入修正项, 这个修正项不需要EL方程的解就可以满足"守恒". 一种考虑把物质的能动张量看做算符, 计算它们的关联函数, 考查它的无穷大项和有限项是否分别满足WARD恒等式. 一种就是CFT中的, 能动张量的LAURENT级数展开系数就是VIRASARO代数, 这可以有很多种模型来实现它. 引力的多极场是出现在闽氏平直时空为背景下的微绕EISNSTEIN方程的的度规中(后NEWTON近似), 度规可以用多极场来表示. 其中要用到谐和坐标条件, 这也是引力波十个极化张量的一个"规范"条件(四个约束).
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可见光 发表文章数: 421 |
Re: 有关能动张量和多极场 这个我大致看懂了 生活充满七彩阳光,是为可见光
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yinhow 发表文章数: 727 |
Re: 有关能动张量和多极场 多谢可可MM的褒奖. 仿照以上的想法, 我们可以在平直时空背景下把引力作用量(RIEMANN曲率标量的积分)展开到度规微扰的二阶项, 从中可以读出很多的弱引力自相互作用模式, 如加速效应, 旋转效应. 可以推广到和物质相互作用的拉氏量来. 这个课题的难点就在于算, 我以前看到的HAWKING一篇文献也是这样算的,不同之处是他的背景时空是ADS空间, 他要算的是两点度规微扰的关联函数. 因为当时ADS/CFT很热门, 这个关联函数同样可以由CFT推出来, 对比一下就可以验证ADS/CFT是否成立. 这篇文献的计算麻烦程度远远超出以上的课题. 不得不佩服HAWKING的算功.
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HPC 发表文章数: 244 |
Re: 有关能动张量和多极场 注释及疑问 1严格来说,是对Action进行度规变分。当然,另外一种便是通过Neother守恒流经过Belinfante程序化而得到。当然,可以证明这两个是等价的。祥见math-ph/0412064。 2我还没有反应过来,为什么守恒可以不用Euler Equation?能否举一个具体的系统? 3我对于经典的一块,还是比较清楚,但是涉及到量子一块,大约是变得复杂而又模糊的缘故而不是很懂,诸如能动张量的重整化,conformal anomaly等。特别的你提到关联函数这个事情,我想知道大家为什么要考察关联函数。或者说关联函数的物理意义在什么地方?顺便提一下。如果严格的来说,当我们把场量子化了,那么能动张量就自动是量子算符了。 Faith, Fashion and Fancy. Welcome to 我的域名:http://hongbaozhang.blog.edu.cn
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卢昌海 发表文章数: 1617 |
Re: 有关能动张量和多极场 :: 我还没有反应过来,为什么守恒可以不用Euler Equation? :: 能否举一个具体的系统? 我想 yinhow 兄指的大概是用散度恒等于零的表达式消去由 Noether 定理给出的 T^{μν} 中的非对称部分。这种表达式的例子是 ∂_λ f^{λμν} (其中 f^{λμν} 是对 λμν 全反对称的函数),它的散度恒为零,即 yinhow 兄所说的 “修正项不需要EL方程的解就可以满足'守恒'”。 宠辱不惊,看庭前花开花落 去留无意,望天空云卷云舒
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HPC 发表文章数: 244 |
Re: 有关能动张量和多极场 但关键的问题,如果就从Belinfante的程序来看,还是用到了Euler Equation。 当然你可以说我可以构造一个T_{ab}=T_{ab}(Belinfante)+Lambda_{ab} if Lamda_{ab} is conserved too。 但我认为这样得到的能动张量并不是物理的能动张量。或者换句话说,物理的能动张量就是Belinfante Tensor. PS 我认为昌海兄的评论就是Belinfante程序吧? Faith, Fashion and Fancy.
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yinhow 发表文章数: 727 |
Re: 有关能动张量和多极场 昌海兄说的就是我想表达的意思, 关联函数的`一些性质, 譬如说极点等等, 和"粒子"的质量, 散射振幅联系起来. 重整化计算过程中的圈圈也和它相关. Greencool Center for Theoretical Physics 这个理论物理中心在哪里? 大致看了一下你的文章, 是否是对标架场变分, 得到LORENTZ指标的能动张量, 然后再转化到通常指标的能动张量表达式, 因为带LORENTZ指标的旋量好处理.
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sage 发表文章数: 1125 |
Re: 有关能动张量和多极场 但我认为这样得到的能动张量并不是物理的能动张量 it depends on what you mean by physical. Strictly speaking, none of them is physical since they are not gauge invariant. One can go to another by a gauge transformation. The real physical quantity are gauge invariant green's functions (or correlation functions). Roughly speaking, knowing a complete set of Green's functions of a quantum theory, we can obtain all the physical observables.
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sage 发表文章数: 1125 |
Re: 有关能动张量和多极场 conformal anomaly ---------------------------- on of my earlier articles have something about it. one of the famous consequnce is that we have to have 26(10) space-time dimensions in bosonic(super) string theories.
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HPC 发表文章数: 244 |
Re: 有关能动张量和多极场 1 GCTP is located in Beijing. Generally speaking, I am fairly free here. 2 I am not sure what you mean by gauge invariance. For example, I think energy momentum tensor can be fixed by Einstein equation. Specifically speaking, energy momentum can be observed by its gravitational effect, although it has some unambiguities in non-gravitational physics. Faith, Fashion and Fancy.
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sage 发表文章数: 1125 |
Re: 有关能动张量和多极场 1 GCTP is located in Beijing. Generally speaking, I am fairly free here. 2 I am not sure what you mean by gauge invariance. For example, I think energy momentum tensor can be fixed by Einstein equation. Specifically speaking, energy momentum can be observed by its gravitational effect, although it has some unambiguities in non-gravitational physics. roughly speaking, my impression is that in GR, there is no gauge invariant distinction between 'matter' and 'gauge field'. T_{mu, nu}, i think, transforms under diffeomorphism like a tensor. something can be fixed EOM, but it does not have to be gauge invariant (just like the vector portential in E & M).
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yinhow 发表文章数: 727 |
Re: 有关能动张量和多极场 conformal anomaly 如果经典的作用量在度规的表达变换下不变, 那么能动张量的迹为零, g^{ab}T_{ab}=0, 但如果能动张量取为相应算符的平均值, 就不为零, 出现反常,g^{ab}
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HPC 发表文章数: 244 |
Re: 有关能动张量和多极场 1.gauge invariance the metric variational energy momentum tensor and the Belinfante tensor is equivalent and gauge invariant. (cf. math-ph/0412064 and references therein) 2. conformal anomaly I am aware of this result about trace anomaly. But I am not familiar some principles techniques to derive this result. Faith, Fashion and Fancy.
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sage 发表文章数: 1125 |
Re: 有关能动张量和多极场 1.gauge invariance the metric variational energy momentum tensor and the Belinfante tensor is equivalent and gauge invariant. (cf. math-ph/0412064 and references therein) ====================================================== Energy momentum tensor contains, by definition, energy and momentum, which are not gauge invariant quantities. It does not have to be gauge invariant to be meaningful either since by itself, it is not a physical quantity.
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yinhow 发表文章数: 727 |
Re: 有关能动张量和多极场 2. conformal anomaly I am aware of this result about trace anomaly. But I am not familiar some principles techniques to derive this result 参考文献: S. Deser, M. J. Duff and C. J. Isham, Non-local Conformal Anomalies, Nucl. Phys. B111 (1976) 45. 另一种计算是用Schwinger-DeWitt技巧, 这种优美的计算方法一直吸引着我, 同时也一直困扰着我, 到现在也没有弄懂. 等手头的工作告一段落的话, 我想好好学学. 还有一种是热核系数法, 主要是有Gilkey搞出来的.
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可见光 发表文章数: 421 |
Re: 有关能动张量和多极场 Energy momentum tensor contains, by definition, energy and momentum, 准确的说,Energy momentum tensor 包含能量和动量密度分量,而不是能量和动量本身。作为能量和动量密度是一个二阶张量的不同分量;而作为能量和动量本身,它们对应一个一阶张量(时空四矢)的不同分量。在规范不变性方面,我不知道能量和动量密度、能量和动量本身它们是否有区别。 生活充满七彩阳光,是为可见光
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sage 发表文章数: 1125 |
Re: 有关能动张量和多极场 准确的说,Energy momentum tensor 包含能量和动量密度分量,而不是能量和动量本身。作为能量和动量密度是一个二阶张量的不同分量;而作为能量和动量本身,它们对应一个一阶张量(时空四矢)的不同分量。在规范不变性方面,我不知道能量和动量密度、能量和动量本身它们是否有区别。 Just like you said, we are talking about energy and momentun 'densities ' (which is almost always true in field theory). In general relativity, in a general background, strictly speaking, the notion energy and momentum (again, i mean 'density' through this abuse of language) is only local, in the sense we could choose a local intertia frame, blah, blah. However, there is no globally preferred frame. Our usual intuitive notion of energy and momentun 4-vector will apply to a special case when we have a localized distribution of matter and the space-time far away from it is Minkowskian. In this case, from an observer far away like us, we have a preferred frame, our minkowskian frame!. We coud them define energy as a spatial integral of T_00, etc. We do also have the notion of a 4-vector (E, P1, P2, P3) which transform covariantly under lorentz transformation in our frame.
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sage 发表文章数: 1125 |
Re: 有关能动张量和多极场 The fact that 'source' is not gauge invariant could be illustrated in gauge theory as well. In abelian gauge theory such as QED, we have a well defined notion of current and charge. suppose we have a charged fermion in the theory, current which satisfy \partial_mu F^{mu nu}=j^nu j^mu = \bar{\psi} gamma^mu \psi is gauge invariant and conserved in the ordinary sense, \partial_mu j^mu=0. However, this is only case in gauge theory where this is true. The main reason, as we will see, is that U(1) gauge field is 'neutral', in the sense that they could not be the source of themselves. Now, suppose we have a non-abelian gauge theory generated by a set of generators (t^a). For SU(2), t^a s will be Pauli matrices. Then, the current will be j^mu = \bar{\psi} gamma^mu t^a \psi which is obviously not invariant under gauge transformation (transforms instead as t^a). Actually, we have (might contain a factor of 2, i, and indices errors) D^mu F_{mu nu} = \partial^mu F_{mu nu} + g [A, F] = j_nu We see that D^{nu} j_nu =0. However, this is not conservation of charge and current in the usual sense. Actually, what is conserved in the usual sense seems to be j_nu - g[A,F]. This agree with intuition that there is no good gauge invariant way of defining source. We can shift between matter (j) and gauge field [A,F] by doing gauge transformations. Notice again, this intimately related to the fact that gauge field could the source of themselves. In gravity, the situation is similar (although more complicated since \partial_mu only have local meaning). 'Soure current' would be T_{mu nu}. Gauge field will be graviton. it is an interesting exercise to see the structure similar to that of the gauge theory by doing a weak field expansion.
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yinhow 发表文章数: 727 |
Re: 有关能动张量和多极场 说的很清楚, 赞!!
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星空浩淼 发表文章数: 1743 |
Re: 有关能动张量和多极场 当年Maxwell得到电磁场的波动方程(即达朗贝尔方程)之后,就因此而预言了电磁波的存在,并且预言光也是一种电磁波。我曾经推得的Yang-Mills场满足的波动方程异常复杂(可以用类Dirac方程推得),我当时甚至怀疑,是否由于非线性,使得胶子不是类光的,而是存在一个等效质量(例如,从Yang-Mills场满足的波动方程中,分离出质量不为零的Klein-Gordon 方程对应的项)。但是,从胶子的自由传播子中好象又看不出这一点。 如果有一天为以太平反,我相信,以太相对于任何惯性参照系的宏观经典速度恒为零。 唯有与时间赛跑,方可维持一息尚存
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sage 发表文章数: 1125 |
Re: 有关能动张量和多极场 当年Maxwell得到电磁场的波动方程(即达朗贝尔方程)之后,就因此而预言了电磁波的存在,并且预言光也是一种电磁波。我曾经推得的Yang-Mills场满足的波动方程异常复杂(可以用类Dirac方程推得),我当时甚至怀疑,是否由于非线性,使得胶子不是类光的,而是存在一个等效质量(例如,从Yang-Mills场满足的波动方程中,分离出质量不为零的Klein-Gordon 方程对应的项)。但是,从胶子的自由传播子中好象又看不出这一点。 Normally when we talk about mass of a particle (light-like or not), we are talk about almost free state (asymptotic state when the interactions beyond quadratic order could be turned off). Normally, this can be achieved by seperating particles. gluon is different because of asymptotic freedom. However, they are 'free' when they are very close to each other. in this case, all the non-linear terms should drop out. 如果有一天为以太平反,我相信,以太相对于任何惯性参照系的宏观经典速度恒为零。 why? because of dragging?
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卢昌海 发表文章数: 1617 |
Re: 有关能动张量和多极场 :: 如果有一天为以太平反,我相信,以太相对于任何惯性参照系的宏观经典速度恒为零。 星空兄需要定义一下什么是以太?什么是“以太相对于惯性参照系的宏观经典速度”? 早年的以太之所以被摒弃, 是因为人们试图为它构筑“机械”模型,而那些模型无法与相对性原理(从而无法与观测)相一致(哪个“机械”模型能在一切参照系中都具有完全相同的性质呢?)。从现代场论的角度看,真空是最接近以太的概念,只不过现代场论中的真空是(局域)Lorentz不变的,因此不具有当年人们赋予以太的优越参照系地位。 星空兄所说的“为以太平反”,如果是把现代场论中的真空作为以太的话,那只是引进alias,算不上平反。如果指的不是现代场论中的真空,则需要单独定义一下。不过这与原主题相距甚远,不妨另开一个主题。 宠辱不惊,看庭前花开花落 去留无意,望天空云卷云舒
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