从宏观量子电动力学分析色散力毕业论文外文翻译

1、附录a：英文原文dispersion forces within the framework of macroscopic qedchristian raabe and dirk-gunnar welschabstract. dispersion forces, which material objects in the ground state are subject to, originate from the lorentz force with which the fluctuating, object-assisted electromagnetic vacuum acts on t

2、he fluctuating charge and currentdensities associated with the objects. we calculate them within the frame-work of macroscopic qed, considering magnetodielectric objects described in terms of spatially varying permittivities and permeabilities which are complex functions of frequency the result enab

3、les us to give a unified approach to dispersion forces on both macroscopic and microscopic levels. keywords: dispersion forces, lorentz-force approach, qed in linear causal media1. introductionas known, electromagnetic fields can exert forces on electrically neutral, unpolar-ized and unmagnetized ma

4、terial objects, provided that these are polarizable and/or magnetizable. classically, it is the lack of precise knowledge of the state of the sources of a field what lets one resort to a probabilistic description of the field, so that, as a matter of principle, a classical field can be non-fluctuati

5、ng. in practice, this would be the case when the sources, and thus the field, were under strict de-terministic control. in quantum mechanics, the situation is quite different, as field fluctuations are present even if complete knowledge of the quantum state would be achieved; a strictly non-probabil

6、istic regime simply does not exist. similarly, polarization and magnetization of any material object are fluctuating quantities in quantum mechanics. as a result, the interaction of the fluctuating electromagnetic vacuum with the fluctuating polarization and magnetization of material objects in the

7、ground state can give rise to non-vanishing lorentz forces; these are commonly referred to as dispersion forces.in the following we will refer to dispersion forces acting between atoms, between atoms and bodies, and between bodies as van der waals (vdw) forces, casimir-polder (cp) forces and casimir

8、 forces, respectively. this terminology also reflects the fact that, although the three types of forces have the same physical origin, different methods to calculate them have been developed. the cp force that acts on an atom (hamiltonian ra) in an energy eigenstate la) (rala) = nwala) at position r

9、ain the presence of (linearly responding) macroscopic bodies is commonly regarded as being the negative gradient of the position-dependent part of the shift of the energy of the overall system,ea, with the atom being in the state la) and the body-assisted electromagnetic field being in the groundsta

10、te. the interaction of the atom with the field, which is responsible for the energy shift, is typically treated in the electric-dipole approximation, le. hint = -d.e(ra) in the multipolar coupling scheme, and the energy shift is calculated in leading-order perturbation theory. in this way, one finds

11、 1,22=一仏工叮d69 "% img(g，g，q血(1)(p, principal value; wba=wb-wa), where g(r,rw) is the classical (retarded) green tensor (in the frequency domain) for the electric field, which takes the presence of the macroscopic bodies into accoun匸 it can then be argued that, in order to obtain the cp potential

12、 ua(ra) as the position-dependent part of the energy shift, one may replace g(ra,ra,w) in eq. (1) with g(s)(ra,ra,w), where g(s)(r,rw) is the scattering part of the green tensor. hence,叽）"比骼仃g%，m）厶(g) = - a工 0(%)此心 re g(g,几,) dha £()° bwhere ua(ra) has been decomposed into an off-reso

13、nant part uf(ra) and a resonant part u(ra), by taking into account the analytic properties of the green tensor as a function of complex w, and considering explicitly the singularities excluded by the principal-val ne integration in eq. (1).let us restrict our attention to ground-state atol 1 is(forc

14、es on excited atoms lead to dynamical problems in general 2). in this case, there are of course no resonant contributions, as only upward transitions are possible wab <0 in eq. (4). thus, on identifying the (isotropic) ground-state polarizability of an atom as we may write the cp potential of a g

15、round-state atom in the form of (see, e.g. refs. 1-6)讹）=处说牙谒认皿|2,(5)卩(5)= 2c2 jq dg/a(疋)tijg(s)g也,迈)from which the force acting on the ato 111 follows asnow consider, instead of the force on a single ground-state atom, the force on a collection of ground-state atollls distributed with a (coarse-grai

16、ned) nuinber density h7(r) inside a space region of volume vrdl when the mutual interaction of the atoms can be disregarded, it is permissible to simply add up the cp forces on the individual atoms to obtain the force acting on the collection of atoms due to their interaction with the bodies outside

17、 the volume vm, le.f = / d3r?/(r)f(r) = d3r d<27?(r)a(i<)vtrg(s)(r,. (8)jvm2亦0& jvm josince the collection of atoms can be regarded as constituting a weakly dielectric body of susceptibility xni(r, i)，.“(r)a(迈)i eoxm(r,诋),(9)eq. (8) gives the casimir force acting on such a body. note that

18、special cases of this formula were already used by lifshitz 7 in the study of casimir forces between dielectric plates. the question is how eq. (8) can be generalized to an arbitrary ground-state body whose susceptibility xrvi (r，i)is not necessarily small. an answer to this and related questions ca

19、n be given by means of the lorentz-force approach to dispersion forces, as developed in refs. 8,9.2. lorentz forcelet us consider macroscopic qed in a linearly, locally and causally responding medium with given (complex) permittivity c( r, w) and peitneability p( r, w). then, if the current density

20、that enters the macroscopic maxwell equations isjn(f) = dw+ h.c，(10)丿0the source-quantity representations of the electric and induction fields/*ooaroge(r) = / du; e(r,w) + h.c. t b(r) = /+ h. c. (11)jqjo read aswhere the retarded green tensor g(r,w) corresponds to the prescribed medium. in eqs. (12)

21、 and (13), it is assumed that the medium covers the entire space so that solutions of the homogeneous maxwell equations do not appea匚 free-space regions can be introduced by performing the limits e 1 and j-l 1, but not before the end of the actual calculations.because of the polarization and/or magn

22、etization currents attributed to the medium, the total charge and current densities are given byroo,roo a(14)d(r) = / d<vp(r,(j) + h. c., j(r) = / dwj(ro) + h.c., jo jowhere2(r,3)= -£() k(2)一 le(r,u;)| +pn(r>u?)(15)pn(r, a) =3)j(r,uz) = 3£o£(r,3)- le(r,cv)-l-v x ll - “-i(r,3)p(r

23、,3) +jn(r,uz) x xdvg,?,) jn(rz,uz).(16)as we have not yet specified the current density in(r) in any way, the above formu-las are generally valid so far, and they are valid both in classical and in quantum electrodynamics, in any case, it is clear that knowledge of the correlation func-tion (in(i;w)

24、l(rnwr), where the angle brackets denote classical and/or quan-turn averaging, is sufficient to colllpute the correlation functions (e(r,w)et(rw,), (gwj'etw'), (l(r,w)bt(r,w,) and (t(r,w)b(rw'), from which the (slowly varying part of the) lorentz force density follows asfl(r)+ t(r,3)e(r&

25、#39;,)+(r,3)x b(r>z)rr，(17)where the limit fr must be understood in such a way that divergent self-forces,which would be formally present even in a uniform (bulk) medium, are omitted the force on the matter in a volume vf,l is then given by the volume integralfl= d3rfl(r),(18)which can be rewritt

26、en as the surface integral(19)fl= / da-t(r)t/皿where t(r) is (the expectation value of) maxwelfs stress tensor (as opposed to minkowski's stress tensor), which is (formally) identical with the stress tensor in microscopic electrodynamics. note that in going from eq. (18) to eq. (19), a term resul

27、ting from the (slowly varying part of the) poynting vector has been omitted, which is valid under stationary conditions. if in(r) can be regarded as being a classical current density producing classical radiation, in(r)jclass (r, t), then the lorentz force computed in this way gives the classical ra

28、diation force that acts on the material inside the chosen space region of volume vm (see also ref. flo).3. dispersion forceas already mentioned in sec. 1，the dispersion force is obtained if in(r) is identified with the noise current density attributed to the polarization and magnetization of the mat

29、erial. let us restrict our attention to the zero-temperature limit, le. let us assume that the overall system is in its ground state. (the generalization to thermal states is straightforward.) from macroscopic qed in dispersing and absorbing linear media 11j2| it can be shown that the relevant curre

30、nt correlation function reads as(20)-v x im “t(r,3)l6(r - ?) x p(i, unit tensor). combining eqs. (12), (13), (15), (16) and (20), and making use of standard properties of the green tensor, one can then show thatandq(r,3)®(f3)= 一一 d(w-u/) x x -inig(r, rz,u) x(22)let us consider, for instance, an

31、 isolated dielectric body of volume vlrl and susceptibility xrvl(r,w) in the presence of arbitrary rnagnetodielectric bodies, which are well separated from the dielectric body. in this case, further evaluation of eq. (18) leads to the following formula for the dispersion force on the dielectric body

32、:f = 一£ l 齐l <icvtr gm(r, r i4)rr, where grvl(r, f,i)is the green tensor of the system that includes the dielectric body. when the dielectric body is not an isolated body but a part of some larger body (again in the presence of arbitrary magnetodielectric bodies), eq. (23) must be supplement

33、ed with a surface integral,daxm(r,策)gm(k 疋)f(24)which may be regarded as reflecting the screening effect due to the residual part of the body.at this point it should be mentioned that if minkowski's stress tensor were used to calculate the force on a dielectric body, eq. (24) would be replaced w

34、ithf(mink)=鼎:曲 l d3rvxm(rj<)trgm(r)r,i<)rr. (25)although both eq. (24) and (25) properly reduce to eq. (23) when the dielectric body is an isolated one, they differ by a surface integral in the case where the body is soi 1 le part of a larger body. in the latter case, minkowski's tensor is

35、 hence expected to lead to incorrect and even self-contradictory results 9,13. it should be pointed out that the differences between the lorentz-force approach to dispersion forces and approaches based on minkowski's tensor or related quantities are not necessarily small, for instance, the groun

36、d-state lorentz force (per unit area) that acts on an almost perfectly reflecting planar plate in a planar dielectric cavity bounded by almost perfectly reflecting walls reads provided the distances di. and dr of the plate to the left and right cavity walls, respectively, are sufficiently large. (no

37、te that, as a consequence of these simplifying assumptions, onlythe static value e = c(w 一+ 0) of the permittivity of the cavity medium appears in the formula. in contrast, the corresponding result on the basis of minkowskifs stress tensor is which can indeed noticeably differ from eq. (26).hmink)=几

38、胪 1 _ "240"(27)附录b：中文译文从宏观量子电动力学分析色散力christian raabe and dirkgimnar welsch摘要：色散力是基础材质的主要课题，是来自于波动的洛伦兹力，是对象变化的电荷 和电流密度与对象相关联的真空电磁行为。我们通过宏观量了电动力学计算他们， 考虑磁性电介质物体在空间上介电常数和磁导率变化的频率的复变函数來描述。结 果使我们能够提供一个统一的方法，从宏观和微观两个层面来研究色散力。 关键词：色散力；洛伦兹力方法；宏观量了电动力学线性因果关系1引言众所周知，电磁领域可以发挥的力量在电中性的非偏振光和非磁化材料的物体, 但这

39、些极化或磁化。传统上，这是一场让人感兴趣的领域，一个精确的描述源状态 概率的知识，作为一个原则问题，一个经典场可以是非脉动的。在实践中，会有这 样的情况，当然，因此现场，严格确定性控制下。在量子力学中，情况是完全不同 的，作为字段波动目前即使量子态的完全的知识将实现；严格的非概率政权根本不 存在。同样，任何物质对象的极化和磁化的量子力学的波动量。作为一个结果，的 波动与波动的极化和在地面状态物质磁化电磁真空能产生菲零的洛伦兹力的作用； 这些通常被称为色散力。我们将把原了z间的色散力作用下，原了和身体z间，和体z间的范德瓦尔斯 (vdw)力，卡西米尔垸(cp)力和casimir力,分别。这个术语

40、也反映了一个事 实，虽然军队三类型具有和同的物理起源，计算不同的方法已被开发。在一个原了 行为的cp力(hamilton ra)在一个能量木征态la) (rala) = = nwala)在位置 雨的存在(线性响应)宏观体commonly看作是对整个系统，能量转移位置相关的部 分的负梯度ea,与原子处于la)和体辅助电磁场在地面状态。同场的原子的相互 作用，这是负责的能量转移，通常是在电偶极近似处理，乐。提示二二有效剂量(ra) 在多极耦合方案，和能量转移是领先的阶摄动理论计算。在这种方式中，人们发现"2】 2(p,主值；wba = wb-wa),其屮g (r, r, w)是经典的(弱

41、智)格林张量(在 频域中)的电场，以宏观体的存在下，考虑。它可以被认为，为了获得电位ua(ra) 作为能量转移的位置相关的部分，一个可以取代g (r八,心，w)在公式(1)和 g (s)(心，心，w),其屮g (s) (r, r, w)的格林张量的散射部分。因此，比“ 一匕(g ) = /；()，epu； h）=-亠工 0（%）砧心 re g匕,r4, %） dha£（）c b在a（&）已被分解成了谐振部分uf （心）和一个谐振部分u （&）,考虑到 的格林张量的解析性质复杂的w函数，并考虑明确的奇点排除主集成式（do让我们限制我们关注的基态。（对激发态原了导致一般2

42、 动力学问题的力量）。 在这种情况下，当然没有共振的贡献，只有向上转换公式可能wab<0 （4） 0因 此，在确定的（各向同性）基态的原了极化率&（） = liin 半 22 _ idibf，（5）我们可以写在一个基态原子的电位（见，例如参考文献ml）。方 r°°叫）=云应 l dee2a（2e）trg（s）（ra,ra3c）,从原子的受力情况如下f（rj = -v（/（rj（7）现在考虑，而不是力在一个单一的基态原了，一系列的基态atollls分布有力（粗 粒度）nuinber密度“7年（r）的内部体积vr ii空间区域。当原了的相互作用可以 忽略不计，它允

43、许简单地增加cp力对单个原了得到的力作用在原了由于他们与外界 的体积vm，i, ef = l 讯“仗用二一/（r,r&）（8）由于原了的集合可以被视为构成的敏感性弱介电体.“（r）a（迈）i eoxm（r,诋）,（9）方程（8）给出了 casimir力作用于一身。注：本公式的特例都已经便用利弗席兹7 在介质平板间casimir作用力的研究。问题是如何平衡（8）可以推广到任意的基态 体的敏感性xwi （r, i）不一定是小。这个回答相关的问题可以由洛伦兹力的方法 来分散力，发达国家作为参考文献。&9。2洛伦兹力让我们在一个线性考虑宏观量了电动力学，本地和因果的响应介质与给定的（复

44、 杂）的介电常数c （r, w）和通透性p （r, w）o然后，然后，如果电流密度进入 宏观麦斯威尔方程是jn(r) = /丿0(10)dcjjn(r,w) + h.c.,广ooe(r) = / du; e(r,w) + h.c., jq看成/oo九b(r) = /d3b(3)+ h.c. (11)jo源的数量表示的屯场和感应域(12)e(r,w)=仏3 / d 屮 g(r,f,3) jn(ruj),b(r,) = /£0v x /d%g(ri<3) jn(r；3),(13)在格林张量g (r, r, w)对应于规定的培养基。均衡器。(12)和(13),它是假 定介质覆盖整个空

45、间，解麦斯威尔方程不出现。自由空间区域可以通过限制e1和 j-11中引入的，但不是在实际计算中结束。由于极化或磁化电流归因于媒体，总 电荷和电流密度由p(r) = / dcjp(r,(j) + h. c.,j(r) = j d3j(r,3)+hc.,(14)的3)= -£() k(2)一 le(r,u;)| +pn(r>u?)(15)9n(fw)=(血)-p jjra)j(r,3)= 3£ok(r,3)- lle(r,cv) + x ll - /z_1(r,cj)jb(r,u/) + 厲。) x x一初)/d3fg(r,3)以1<3)(16)我们尚未指定的电流密度(r)以任何方式，上面的公式是普遍有效的，到目前 为止，他们都是在经典和量了电动力学有效的，在任何情况下，它是明确的，相关 函数的知识(在(r, w)我 (r”w，),其中括号表示占典或量了平均，足以计算 相关函数(e (r, w)等(r, w), (t (r, w), e (r, w), (l (r, w) bt (r, w)和(t (r, w) b (r, w),其中(缓慢变化的)部分的洛伦兹力密度为oooofl(r)=d