附录
附录A：英文原文
Dispersion Forces within the Framework of Macroscopic QED
Christian Raabe and Dirk-Gunnar Welsch
Abstract. 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 the 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 enables 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 media
1. Introduction
As known, electromagnetic fields can exert forces on electrically neutral, unpolar-ized and unmagnetized material 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-fluctuating. 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-probabilistic 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 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 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 rAin the presence of (linearly responding) macroscopic bodies is cornmonly 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 ground
state. The interaction of the atorn 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 [1,2]
()ba b A A ab ba a d r r G d d p E ∑?∞??+-=?020
,,Im ωωωωωπμ (1)
(P, principal value; Wba==Wb-Wa), where G(r,r',w) is the classical (retarded) Green tensor (in the frequency domain) for the electric field, which takes the presence of the macroscopic bodies into account. It can then be argued that, in order to obtain the CP potential 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,r',w) is the scattering part of the Green tensor. Hence,
()(),A r a
A a a r U r U E =? (2) ()()()∑?∞??+-=b
ba A A S ab ab ab A or a
d i r r G d d c r U 022220,,1ξξωξωξπε (3) ()()()()∑??Θ-=b ba ab A A S ab ab ab A r a d r r G d c r U ωωωε,,R
e 1220 (4)
where Ua(rA) has been decomposed into an off-resonant part U~f(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 at0111S. (Forces 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 ground-state atom in the form of (see, e.g. Refs. [1-6])
from which the force acting on the at0111 follows as
()()A A F r U r =-? (7)
Now consider, instead of the force on a single ground-state atom, the force on a collection of ground-state at0111S distributed with a (coarse-grained) nUInber density ''7(r) inside a space region of volume Vr-iI. 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 the volume Vm, Le.
Since the collection of atoms can be regarded as constituting a weakly dielectric body of susceptibility XNI(r, i~),
Eq. (8) gives the Casimir force acting on such a body. Note that 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 can be given by means of the Lorentz-force approach to dispersion forces, as developed in Refs. [8,9].
2. Lorentz Force
Let us consider macroscopic QED in a linearly, locally and causally responding medium with given (complex) permittivity c( r, w) and perrneability p( r, w). Then, if the current density that enters the macroscopic Maxwell equations is
the source-quantity representations of the electric and induction fields
read as
where the retarded Green tensor G(r,r',w) corresponds to the prescribed medium. In Eqs.
(12) and (13), it is assumed that the medium covers the entire space so that solutions of the homogeneous Maxwell equations do not appear. 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 magnetization currents attributed to the medium, the total charge and current densities are given by
where
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(r,w)l~(r''w')), where the angle brackets denote classical and/or quan-turn averaging, is sufficient to C0111pute the correlation functions (e(r,w)Et(r',w')),
(t(r,w),E(r',w')), (l(r,w)Bt(r',w')) and (t(r,w)B(r',w')), from which the
(slowly varying part of the) Lorentz force density follows as
Where the limit r'~ r 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 volurne Vf\,l is then given by the volume integral
which can be rewritten as the surface integral
where T(r) is (the expectation value of) Maxwell's 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 resulting 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 radiation force that acts on the material inside the chosen space region of volume Vm (see also Ref. [10]).
3. Dispersion Force
As 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 material. 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 [11,12] it can be shown that the relevant current correlation function reads as
(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 that
and
Let us consider, for instance, an isolated dielectric body of volume Vl\rI and susceptibility XrvI(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:
where GrvI(r, r',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 supplernented with a surface integral,
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 with
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 S0111e part of a larger body. In the latter case, Minkowski's tensor is 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 ground-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. (Note that, as a consequence of these simplifying assumptions, only
the static value e == c(w ---+ 0) of the permittivity of the cavity medium appears in the formula.] In contrast, the corresponding result on the basis of Minkowski's stress tensor is
which can indeed noticeably differ from Eq. (26).
附录B ：中文译文
从宏观量子电动力学分析色散力
Christian Raabe and Dirk-Gunnar 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。我们通过宏观量子电动力学计算他们，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来描述。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；宏观量子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非零的洛伦兹力的作用；这些通常被称为色散力。
我们将把原子之间的色散力作用下，原子和身体之间，和体之间的范德瓦尔斯（VDW ）力，卡西米尔垸（CP ）力和Casimir 力，分别。这个术语也反映了一个事实，虽然军队三类型具有相同的物理起源，计算不同的方法已被开发。在一个原子行为的CP 力（Hamilton RA ）在一个能量本征态La ）（RALA ）= = nwala ））在位置雨的存在（线性响应）宏观体cornmonly 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= = -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[ 1,2 ]
()ba b A A ab ba a d r r G d d p E ∑?∞??+-=?020
,,Im ωωωωωπμ (1)
（P ，主值；W ba == W b -W a ），其中G （R ，R ，W ）是经典的（弱智）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 （A R ，A R ，W ）在公式（1）和G （s ）（A R ，A R ，W ），其中G （s ）（R ，R ，W ）的格林张量的散射部分。因此，
()(),A r a
A a a r U r U E =? (2) ()()()∑?
∞??+-=b ba A A S ab ab ab A or a d i r r G d d c r U 022220,,1ξξ
ωξωξπε (3)
()()()()∑??Θ-=b ba ab A A S ab ab ab
A r a d r r G d c r U ωωωε,,Re 1220 (4)
在()A A U R 已被分解成了谐振部分U ~ F （A R ）和一个谐振部分U ~（A R ），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（1）。
让我们限制我们关注的基态。（对激发态原子导致一般[ 2 ]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）]。因此，在确定的（各向同性）基态的原子极化率
我们可以写在一个基态原子的电位（见，例如参考文献 [ 6 ]）。
从原子的受力情况如下
()()A A F r U r =-? (7)
现在考虑，而不是力在一个单一的基态原子，一系列的基态at0111s 分布有力（粗粒度）nuinber 密度''7年（R ）的内部体积VR II 空间区域。当原子的相互作用可以忽略不计，它允许简单地增加CP 力对单个原子得到的力作用在原子由于他们与外界的体积V M ，i ，e
由于原子的集合可以被视为构成的敏感性弱介电体。
方程（8）给出了Casimir 力作用于一身。注：本公式的特例都已经使用利弗席兹[ 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xrvi （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。[ 8,9 ]。
2 洛伦兹力
让我们在一个线性考虑宏观量子电动力学，本地和因果的响应介质与给定的（复杂）的介电常数C （R ，W ）和通透性P （R ，W ）。然后，然后，如果电流密度进入宏观麦斯威尔方程是
源的数量表示的电场和感应域
看成
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间，解麦斯威尔方程不出现。自由空间区域可以通过限制E ~ 1和J-1 ~ 1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
，
和
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子电动力学有效的，在任何情况下，它是明确的，相关函数的知识（在（R，W）我~（R ''w’）），其中括号表示古典或量子平均，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
的限制~ R R必须以这样一种方式，不同的自我力量的理解，这将是在一个统一的正式提出（散装）中，省略。力的物质在一个体积V由下式给出的体积积分
可改写为表面积分
T（R）是（期望值）麦斯威尔的应力张量（反对闵可夫斯基的应力张量），这是（正式）与微观的电动力学应力张量相同的。请注意，从式（18）式（19），导致从一个学期（缓慢变化的坡印廷矢量）部分被省略，而固定的条件下是有效的。如果在（R）可以被视为是典型的电流密度产生经典的辐射，在（R）~ J class（R，T），然后洛伦兹力以这种方式计算给出了传统的辐射力，在选择的空间区域的体积vl'vl材料的行为（参见参考文献[ 10 ]）。
3 色散力
在SEC已经提到。1，色散力是如果在得到（R）的识别与噪声电流密度归因于材料的极化和磁化。让我们限制我们的注意力在零温度限制，乐。让我们假设，整个系统是在地面状态。（泛化的热状态是简单的。）从宏观量子电动力学中的散射和吸收的线性媒体[11,12]可以表明有关电流的相关函数读取为
（I，单位张量）。组合式。（12），（13），（15），（16）和（20），和使用的格林张量的标准属性，可以显示
和
[注：()()()()????,,,,0r E r j r B r ρωωωω++''''==在地面状态]。插入式。（21）和（22）
式（17）最终产生的色散力的密度，根据式（18），作用于所选择的空间区域内的inagnetoclielectric 材料色散力可以计算。
让我们考虑，例如，一个孤立的介电体体积/V V L R 和磁化率(,)M x R M 在任意磁介电体的存在，这是从介电体分离。在这种情况下，公式进一步评估（18）导致下列公式对介电体的分散力：
在Gm （R ，R ’，我~）的系统，包括介电体的格林张量。当介电体不是一个孤立的方程，但一些较大的方程的一部分，方程（23）必须辅以表面积分，
这可能被视为反映屏蔽效应由于身体的剩余部分。在这一点上，应该说，如果闵可夫斯基的应力张量被用来计算的介电体的力，方程（24）可以换成
虽然这两个方程（24）和（25）适当降低方程（23）当介电体是孤立的，它们的差异在体是一个更大的方程的曲面积分。在后者的情况下，闵可夫斯基张量，因此预计将导致错误的甚至自相矛盾的结果[ 9.13 ]。应该指出的是，洛伦兹力的方法来分散力和基于闵可夫斯基张量或相关的数量的方法之间的差异不一定是小。例如，基态的洛伦兹力（每单位面积），作用于一个几乎完美的反射平面板在平面介质腔由几乎完全反射壁
提供读取距离。博士和板的左、右腔壁，分别是足够大的。[注意，当这些简化的假设的结果，只有静态值E = = C （W —+ 0）型腔的介质的介电常数，出现在公式。] 相比之下，对闵可夫斯基应力张量的基础上相应的结果[ 14 ]，
确实可以明显不同于公式（26）。