Conformal Transformations and Quantum Gravity

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[精彩]量子力学英语词汇

1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-partic le system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性本征矢量eigenvector本征态eigenstate本征值eigenvalue本征值方程eigenvalue equation本征子空间eigensubspace (可以理解为本征矢空间)变分法variatinial method标量scalar算符operator表象representation表象变换transformation of representation表象理论theory of representation波函数wave function波恩近似Born approximation玻色子boson费米子fermion不确定关系uncertainty relation狄拉克方程Dirac equation狄拉克记号Dirac symbol定态stationary state定态微扰法time-independent perturbation定态薛定谔方程time-independent Schro(此处上面有两点)dinger equati on 动量表象momentum representation角动量表象angular mommentum representation占有数表象occupation number representation坐标（位置）表象position representation角动量算符angular mommentum operator角动量耦合coupling of angular mommentum对称性symmetry对易关系commutator厄米算符hermitian operator厄米多项式Hermite polynomial分量component光的发射emission of light光的吸收absorption of light受激发射excited emission自发发射spontaneous emission轨道角动量orbital angular momentum自旋角动量spin angular momentum轨道磁矩orbital magnetic moment归一化normalization哈密顿hamiltonion黑体辐射black body radiation康普顿散射Compton scattering基矢basis vector基态ground state基右矢basis ket ‘右矢’ket基左矢basis bra简并度degenerancy精细结构fine structure径向方程radial equation久期方程secular equation量子化quantization矩阵matrix模module模方square of module内积inner product逆算符inverse operator欧拉角Eular angles泡利矩阵Pauli matrix平均值expectation value （期望值）泡利不相容原理Pauli exclusion principle氢原子hydrogen atom球鞋函数spherical harmonics全同粒子identical partic les塞曼效应Zeeman effect上升下降算符raising and lowering operator 消灭算符destruction operator产生算符creation operator矢量空间vector space守恒定律conservation law守恒量conservation quantity投影projection投影算符projection operator微扰法pertubation method希尔伯特空间Hilbert space线性算符linear operator线性无关linear independence谐振子harmonic oscillator选择定则selection rule幺正变换unitary transformation幺正算符unitary operator宇称parity跃迁transition运动方程equation of motion正交归一性orthonormalization正交性orthogonality转动rotation自旋磁矩spin magnetic monent(以上是量子力学中的主要英语词汇，有些未涉及到的可以自由组合。

Lie algebras, in Infinite-dimensional Lie algebras and groups, Adv. Ser. in Math. Phys. 7,

L1;k (1)
C1;1;k g ' L0;k (0)
L1;k (1)
M1;k (1): Combining Theorem 4.6 and the statement (ii), we obtain the statement (i). Q.E.D.
References
H0 (g(0; 1; 1); L0 ;k (0)
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan. Department of Mathematics, Faculty of Science, Osaka University,
[KK] Kac V.G. and Kazhdan D.A., Structure of representations with highest weight of in nite dimensional Lie algebras, Adv. in Math. 34, (1979), 97{108. [KnPZ] Knizhnik V. G., Polyakov A. M. and Zamolodchikov A.B., Fractal structure in 2D quantum gravity, Mod. Phys. Lett. A 3, (1988), 819{826. [KW1] Kac V.G. and Wakimoto M., Modular invariant representations of in nite dimensional Lie algebras and superalgebras, Proc. Natl. Acad. Soc., 35, (1988), 4956{4960. [KW2] Kac V.G. and Wakimoto M., Classi cation of modular invariant representations of ane Lie algebras, in In nite-dimensional Lie algebras and groups, Adv. Ser. in Math. Phys. 7, (1989), 138{177. [KWn] Kac V. G. and Wang W., Vertex operator superalgebras and their representations, Mathematical aspects of conformal and topological eld theories and quantum groups, 161{191, Contemp. Math. 175, Amer. Math. Soc. Providence, RI, (1994). [Mal] Malikov F.G., Verma modules over Kac-Moody algebras of rank 2, Leningrad Math. J., 2, No. 2, (1991), 269{286. [MFF] Malikov F.G., Feigin B.L. and Fuks D.B., Singular vectors in Verma modules over KacMoody algebras, Funkts. Anal. Prilozhen., 20, No. 2, (1988), 25{37. [MW] Mathieu P. and Walton M.A., Fractional-level Kac-Moody Algebras and Nonunitary Coset Conformal Theories, Prog. Theor. Phys. 102, (1990), 229-254. [MP] Moody R.V. and Pianzola A., Lie algebras with triangular decompositions, Wiley-Interscience Publ., New York, 1995. [P] Polyakov A. M., Quantum Gravity in two dimensions, Mod. Phys. Lett. A 2, (1987), 893-898. [Ray] Ray U., A Characterization of Lie superalgebras for a certain class of graded Lie superalgebras, preprint,to appear in Jour. Alg. [RW] Rocha-Caridi A. and Wallach N.R., Highest weight modules over graded Lie algebras: resolutions, ltrations and character formulas, Trans. Amer. Math. Soc. 20, (1983), 133{162. [TUY] Tsuchiya A., Ueno K. and Yamada Y., Conformal eld theory on moduli family of stable curves with gauge symmetry, in Integrable systems in quantum eld theory and statistical mechanics, Adv. Stud. Pure Math. 19, (1989), 459-566. [V] Verlinde E., Fusion rules and modular transformations in 2D conformal eld theory, Nucl. Phys. B 300, (1998), 360-376. (K. Iohara) (Y. Koga)

Quantum entanglement

Quantum entanglementMaciej LewensteinMaciej Lewenstein has obtained his degree in Physics from Warsaw University. From 1980 he worked at the Center for Theoretical Physics of the Polish Academy of Sciences. He received his doctoral degree in 1983 at the University of Essen and habilitation in 1986 in Warsaw. He became a full Professor in Poland in 1993. In 1995 he joined “Service de Photones, Atomes et Molecules” of CEA in Saclay. In 1998 he became a full professor and a head of the quantum optics theory group at the University of Hannover. In 2005 he started a new theory group at the “Insitut de Ciencias Fotoniques” in Barcelona. His research interests include: quantum optics, quantum information and statistical physics.Chiara MacchiavelloChiara Macchiavello finished her degree in Physics in 1991 and her PhD in 1995 at the University of Pavia. She held a post-doctoral for two years at the University of Oxford. Since 1998 she has been an Assistant Professor at the University of Pavia.Her research interests include quantum information processing and quantum optics.Dagmar BrussSince 2003 Dagmar Bruss is a professor at the Institute of Theoretical Physics at the University of Duesseldorf, Germany. Her research interests include the foundations of quantum information theory, classification of entanglement and quantum optical implementations of quantum computation.AbstractEntanglement is a fundamental resource in quantum information theory. It allows performing new kinds of communication, such as quantum teleportation and quantum dense coding. It is an essential ingredient in some quantum cryptographic protocols and in quantum algorithms. We give a brief overview of the concept of entanglement in quantum mechanics, and discuss the major results and open problems related to the recent scientific progress in this field.IntroductionEntanglement is a property of the states of quantum systems that are composed of many parties, nowadays frequently called Alice, Bob, Charles etc. Entanglement expresses particularly strong correlations between these parties, persistent even in the case of large separations among the parties, and going beyond simple intuition.Historically, the concept of entanglement goes back to the famous Einstein-Podolski-Rosen (EPR) “paradox”. Einstein, who discovered relativity theory and the modern meaning of causality, was never really happy with quantum mechanics. In his opinion every reasonable physical theory should exhibit a so called local realism.Suppose that we consider two particles, one of which is sent to Alice and one to Bob, and we perform independent local measurements of “reasonable” physical observables on these particles. Of course, the results might be correlated, because the particles come from the same source. But Einstein wanted really to restrict the correlations for “reasonable” physical observables to the ones that result from statistical distributions of some hidden (i.e. unknown to us and not controlled by us) variables that characterize the source of the particles. Since quantum mechanics did not seem to produce correlations consistent with a local hidden variable (LHV) model, Einstein concluded that quantum mechanics is not a complete theory. Erwin Schrödinger, in answer to Einstein’s doubts, introduced in 1935 the term “Verschränkung” (in English “entanglement”) in order to describe these particularly strong quantum mechanical correlations.Entanglement was since then a subject of intense discussions among experts in the foundations of quantum mechanics and philosophers of science (and not only science). It took, however, nearly 30 years until John Bell was able to set the framework for experimental investigations on the question of local realism. Bell formulated his famous inequalities, which have to be fulfilled in any multiparty system described by a LHV model. Alain Aspect and coworkers in Paris have demonstrated in their seminal experiment in 1981 that quantum mechanical states violate these inequalities. Recent very precise experiments of Anton Zeilinger’s group in Vienna confirmed fully Aspect’s demonstrations. All these experiments indicate the correctness of quantum mechanics, and despite various loopholes, they exclude the possibility of LHV models describing properly the physics of the considered systems.Entanglement has become again the subject of cover pages news in the 90’s, when quantum information was born. It was very quickly realized that entanglement is one of the most important resources for quantum information processing. Entanglement is a necessary ingredient for quantum cryptography, quantum teleportation, quantum densecoding, and if not necessary, then at least a much desired ingredient for quantum computing.At the same time the theory of entanglement is related to some of the open questions of mathematics, or more precisely linear algebra and functional analysis. A solution of the entanglement problem could help to characterize the so called positive linear maps, i.e. linear transformations of positive definite operators (or physically speaking quantum mechanical density matrices, see below) into positive definite operators.Entanglement of pure statesIn quantum mechanics (QM) a state of a quantum system corresponds to a vector |Psi> in some vector space, called Hilbert space. Such states are called pure states. One of the most important properties of QM is that linear superpositions of state-vectors are also legitimate state-vectors. This superposition principle lies at the heart of the matter-wave dualism and of quantum interference phenomena.Entanglement is also a result of superposition, but in the composite space of the involved parties. Let us for the moment focus on two parties, Alice and Bob. It is then easy to define states which are not entangled. Such states are product states of the form |Φ>= |a>|b>, i.e. Alice has at her disposal |a>, while Bob has |b>. Product states obviously carry no correlations between Alice and Bob. Entangled pure states may be now defined as those which are superpositions of at least two product states, such as|Φ> = α1|a1>|b1> + α2|a2>|b2> + etc.but cannot be written as a single product state in any other basis. All entangled pure states contain strong quantum mechanical correlations, and do not admit LHV models.Entanglement of mixed states and the separability problemVerify whether a given state-vector is a product state or not is a relatively easy task. In practice, however, we often either do not have full information about the system, or are not able to prepare a desired state perfectly. In effect in everyday situations we deal practically always with statistical mixtures of pure states. There exists a very convenient way to represent such mixtures as so called density operators, or matrices. A density matrix rho corresponding to a pure state-vector |Φ> is a projector onto this state. More general density matrices can be represented as sums of projectors onto pure state-vectors weighted by the corresponding probabilities.The definition of entangled mixed states for composite systems has been formulated by Reinhard Werner from Braunschweig in 1989. In fact, this definition determines which states are not entangled. Non-entangled states, called separable states, are mixtures of pure product states, i.e. convex sums of projectors onto product vectors:ρ = Σι pi|ai>|bi><ai|<bi|, (*)where 0 ≤ pi ≤ 1 are probabilities, i.e. Σιpi= 1. The physical interpretation of thisdefinition is simple: a separable state can be prepared by Alice and Bob by using local operations and classical communication. Checking whether a given state is separable or not is a notoriously difficult task, since one has to check whether the decomposition (*) exists or not. This difficult problem is known under the name of “separability or entanglement problem”, and has been a subject of intensive studies in the recent years.Simple entanglement criteriaThe difficulty of the separability problem comes from the fact that rho admits in general an infinite number of decompositions into a mixture of some states, and one has to check whether among them there exists at least one of the form (*). One of the most powerful necessary conditions for separability has been found by one of the fathers of quantum information, the late Asher Peres. Peres (Technion, Haifa) observed that since Alice and Bob may prepare separable states using local operations, Alice may safely reverse the time arrow in her system, which will change the state, but will not produce something unphysical. In general, such a partial time reversal is not a physical operation, and can transform a density operator (which is positive definite) into an operator that is no more positive definite. In fact this is what happens with all pure entangled states. Mathematically speaking partial time reversal corresponds to partial transposition of the density matrix (only on Alice's side). We arrive in this way at the Peres criterion: If a stateρis separable then its partial transposition has to be positive definite.This criterion is usually called positive partial transpose condition, or shortly PPT condition. Amazingly, the PPT condition is not only necessary for separability, but it is also a sufficient condition for low dimensional systems such as two qubits (dimension 2x2)and a system composed of one qubit and one qutrit (dimension 2x3). In higher dimensions, starting from 2x4 and 3x3, this is no longer true: there exist entangled states with positive partial transpose, which are called PPT entangled states.There exist several other necessary or sufficient separability criteria which have been established and frequently discussed in recent years. For example, states that are close to the completely chaotic state (whose density operator is equal to the normalized identity) are necessarily separable. There exist also other criteria that employ entropic inequalities, uncertainty relations, or an appropriate reordering of the density matrix (so called realignment criterion) etc. There exists, however, no general simple operational criterion of separability that would work in systems of arbitrary dimension.Entanglement witnessesThe set of all states P is obviously compact and convex. If ρ1 and ρ2are legitimate states,so is their convex mixture. The set of separable states S is also compact and convex (seeFigure 1). From the theory of convex sets and Hahn-Banach theorem we conclude that for any entangled state there exists a hyperplane in the space of operators separating rhofrom S. Such a hyperplane defines uniquely a Hermitian operator W (observable) which has the following properties: The expectation value of W on all separable states, <W> ≥ 0, whereas its expectation value on ρ is negative, i.e. <W>ρ< 0.Figure 1Such an observable is for obvious reasons called entanglement witness, since it “detects” the entanglement of ρ. Every entangled state has its witnesses; the problem obviously is to find appropriate witnesses for a given state. To find out whether a given state is separable one should check whether its expectation value is non-negative for all witnesses. Obviously this is a necessary and sufficient separability criterion, but unfortunately it is not operational, in the sense that there is no simple procedure to test for all witnesses.Nevertheless, witnesses provide a very useful tool to study entanglement, especially if one has some knowledge about the state in question. They provide a sufficient entanglement condition, and may be obviously optimized (see Figure 2) by shifting the hyperplane in a parallel way towards S.Figure 2Bell inequalitiesAfter introducing the concept of separability and entanglement for mixed states, it is legitimate to ask what is the relation of mixed state entanglement and the existence of a LHV model, which requires that the state cannot violate any of the Bell-like inequalities. Let us discuss an example of such inequalities, the so called Clauser-Horne-Shimony- Holt inequality for two qubits. Let us assume that Alice and Bob measure two binary observables each, namely A 1, A 2, and B 1, B 2. The observables are random variables taking the values +1 or − 1, correlated possibly through some dependence on local hidden variables. It is easy to see that in the classical world, if B 1 + B 2 is zero, then B 1 − B 2 is either +2 or −2, and vice versa. Therefore if we define s = A 1(B 1 + B 2 ) + A 2 (B 1 − B 2 ) , we obtain that 2 ≥ s ≥ −2. This inequality holds also after averaging over various realizations. On the other hand, it can be shown that by taking suitable sets of observables for Alice and Bob we can find pure and even mixed quantum states that violate this inequality.Are Bell-like inequalities similar in this respect to witnesses, i.e. for a given entangled state can one always find a Bell-like inequality that “detects” it? The answer to this question is no, and has been already given by R. Werner in 1989. Even for two qubits there exist entangled states that admit an LHV model, i.e. cannot violate any Bell-like inequality.This observation indicates already that there is more structure in the “eggs” of Figure 1 and Figure 2. Separable states are evidently inside the PPT egg, according to the Peres condition. They admit an LHV model, i.e. they are also inside the LHV egg. But what about PPT entangled states? Do they violate some Bell-like inequality? Peres has formulated a conjecture that this not the case, and there is a lot of evidence that this conjecture is correct, although a rigorous proof is still missing.The distillability problem and bound entanglement Above we have classified quantum states according to the property of being either separable or entangled. An alternative classification approach is based on the possibility of distilling the entanglement of a given state. In a distillation protocol the entanglement of a given state is increased by performing local operations and classical communication on a set of identically prepared copies. In this way one obtains fewer, but “more entangled”, copies. This kind of technique was originally proposed in 1996 by Bennett and coworkers in the context of quantum teleportation, in order to achieve faithful transmission of quantum states over noisy channels. It also has applications in quantum cryptography as a method for quantum privacy amplification in entanglement based protocols in the presence of noise, as pointed out by David Deutsch and coworkers from Oxford.The distillability problem poses the question whether a given quantum state can be distilled or not. A separable state can never be distilled because the average entanglement of a set of states cannot be increased by local operations. Furthermore, the positivity of the partial transpose ensures that no distillation is possible. Thus, a given PPT entangled state is not distillable, and is therefore called bound entangled. There mayeven exist undistillable entangled states which do not have the PPT property. However, this conjecture is not proved at the moment.The first example of a PPT entangled state has been found by Pawel Horodecki from Gdansk in 1997. These states are so called edge states, which means that they cannot be written as a mixture of a separable state and a PPT entangled state. Particularly simple families of states have been suggested by Charles Bennett and coworkers at IBM, New York. They have found the so called unextendible product bases (UPB), i.e. sets of orthogonal product state-vectors, with the property that the space orthogonal to this set does not contain any product vector. It turns out that the projector onto this space is a PPT state, which obviously has to be entangled since it does not contain any product vector in its range (note that all state-vectors in the decomposition of a separable state ρinto a mixture of product states belong automatically to the range of ρ).The existence of bound entanglement is a mysterious invention of Nature. It is an interesting question to ask whether bound entanglement is a useful resource to perform quantum information processing tasks. It was shown so far that this is not the case for communication protocols such as quantum teleportation and quantum dense coding (i.e.a protocol that allows to enhance the transmission of classical information, using entanglement). However, surprisingly, it is possible to distill a secret key in quantum cryptography, starting from certain bound entangled states.Entanglement detectionAs discussed above, entanglement is a precious resource in quantum information processing. Typically in a real world experiment noise is always present and it leads to a decrease of entanglement in general. Thus, it is of fundamental interest for experimental applications to be able to test the entanglement properties of the generated states. A traditional method to this aim is represented by the Bell inequalities, a violation of which indicates the presence of entanglement. However, as mentioned above, not every entangled state violates a Bell inequality. So, not all entangled states can be detected by using this method.Another possibility is to perform complete state tomography, which allows determining all the elements of the density matrix. This is a useful method to get a complete knowledge of the density operator of a quantum system, but to detect entanglement it is an expensive process as it requires an unnecessary large number of measurements. If one has certain knowledge about the state the most appropriate technique is the measurement of the witness observable, which can be achieved by few local measurements. A negative expectation value clearly indicates the presence of entanglement.All these methods have been successfully implemented in various experiments. Recently another method for the detection of entanglement was suggested based on the physical approximation of the partial transpose. It remains a challenge to implement this idea in the laboratory because it requires the implementation of non local measurements.Entanglement measuresWhen classifying a quantum state as being entangled, a natural question is to quantify the amount of entanglement it contains. For pure quantum states there exists a well defined entanglement measure, namely the von Neumann entropy of the density operator of a subsystem of the composite state. For mixed states the situation is more complicated. There are several different possibilities to define an entanglement measure. The so called entanglement cost describes the amount of entanglement one needs in order to generate a given state. An alternative measure is the entanglement of formation, which is a more abstract definition. A further possibility to quantify entanglement is given by the minimum distance to separable states. Finally, motivated by physical applications, one can introduce the distillable entanglement which quantifies the extractable amount of entanglement.Unfortunately all of these quantities are very difficult to compute in general. For example, in order to determine the entanglement of formation one has to find the decomposition of the state that leads to the minimum average von Neumann entropy of a subsystem and this is a very challenging task. So far a complete analytical formula for the entanglement of formation only exists for composite systems of two qubits.Entanglement in multipartite systemsSo far, we have restricted ourselves to the case of composite systems with two subsystems, so called bipartite systems. When considering more than two parties, i.e multipartite systems, the situation becomes much more complex. For example, for the most simple tripartite case of three qubits, a pure state can be either completely separable, or biseparable (i.e. one of the three parties is not entangled with the other two), or genuinely entangled among all three parties. The latter class again consists of inequivalent subclasses, the so called GHZ and W states. This concept can be generalized to mixed states. For more than three parties it is easy to imagine that the number of subclasses grows fast.In recent years there has been much progress in the creation of multipartite entangled states in the laboratory. The existence of genuine multipartite entanglement has also been demonstrated experimentally by using the concept of witness operators.Even if the full classification of multipartite entanglement is a formidable task, certain classes of states, the so called graph states, have been completely characterized and shown to be useful both for quantum computational and quantum error correction protocols. Moreover, a deeper understanding of entanglement has proved to be very fruitful in connection with statistical properties of physical systems. All of these problems are discussed in more details in other sections of this publication.References[1] Einstein, P. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935)[2] J.S. Bell, Physics 1, 195 (1964)[3] P. Horodecki, Phys. Lett. A 232, 333 (1997)[4] M. Lewenstein et al., J. Mod. Opt. 47, 2481 (2000)[5] A. Peres, Phys. Rev. Lett. 77, 1413 (1996)[6] E. Schrödinger, Naturwissenschaften 23, 807 (1935)[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989) Contact information of the author of this article Maciej LewensteinInstitut de Ciènces Fotòniques (ICFO)C/Jordi Girona 29, Nexus 2908034 BarcelonaSpainEmail: maciej.lewenstein@icfo.esChiara MacchiavelloIstituto Nazionale di Fisicadella Materia, Unita' di Pavia Dipartimento di Fisica "A. Volta"via Bassi 6I-27100 PaviaItalyEmail: chiara@unipv.itProf. Dr. Dagmar BrussInst. fuer Theoretische Physik IIIHeinrich-Heine-Universitaet Duesseldorf Universitaetsstr. 1, Geb. 25.32D-40225 Duesseldorf,GermanyEmail: bruss@thphy.uni-duesseldorf.de。

弦论揭秘：自学向导网上预览

作者简介
戴维德 • 麦克马洪是当代著名的美国物理学家。作为物理研究人员，他曾在圣迪亚（SaБайду номын сангаасdia）国家实验室工作过七年。他是《线性代数揭秘》、《量子力学揭秘》、《相对论揭秘》、《MATLAB®揭秘》和《复变函数揭秘》等自学导读书籍的作者。
译者简介
郑中，生于 1979 年，四川隆昌人，理学硕士、工程师。本科就读于成都理工大学资源经济系，后考入中国科学院地球化学研究所，毕业后在矿业公司和资源规划院工作过，现仍主要从事地质矿产工作。译者博客：/u/warlong /cqzg
II
目录
译者序言作者前言第一章导论（Introduction）广义相对论快览（A Quick Overview of General Relativity）量子论快速入门（A Quick Primer on Quantum Theory）标准模型（The Standard Model）量子引力场（Quantizing the Gravitational Field）弦论中一些基本分析（Some Basic Analysis in String Theory）统一与基本常数（Uniﬁcation and Fundamental Constants）弦论概览（String Theory Overview）弦论类型（Types of String Theories） M论 D膜高维小结小测验第二章经典弦I：运动方程（The Classical String I: Equations of Motion）相对论性点粒子（The Relativistic Point Particle）时空中的弦（Strings in Space-Time）弦的运动方程（Equations of Motion for the String）波里雅科夫作用量（The Polyakov Action）数学旁白：欧拉示性数（Mathematical Aside: The Euler Characteristic）光锥坐标（Light-Cone Coordinates）波动方程解（Solutions of the Wave Equation）具有自由端点的开弦（Open Strings with Free Endpoints）闭弦（Closed String）具有固定端点的开弦（Open Strings with Fixed Endpoints）泊松括号公式（Poisson Brackets）小测验第三章经典弦II：对称性与世界面流（The Classical String II: Symmetries and Worldsheet Currents）能动张量（The Energy-Momentum Tensor）波里雅科夫作用量的对称性（Symmetries of the Polyakov Action）庞加莱变换（Poincaré transformations）再参数化（reparameterization）外尔变换（Weyl transformation）转换为平直世界面度规（Transforming to a Flat Worldsheet Metric）源于庞加莱不变性的守恒流（Conserved Currents from Poincaré Invariance）哈密顿公式（The Hamiltonian）小结小测验

施瓦茨_克里斯托弗反变换的快速收敛算法及其应用

T
把( Y ) 在 ( T L ) 附近作泰勒展开: ( Y ) U ( Y ) + [ YcT L ] ( T L - TL ) 所以 (T L ) k+ 1 U ( T L ) k + [ YcTL ]
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Abstract
T his paper present s a numerical met hod for Inverse Schw arz - Christof fel
( ISC) transformat ions. By combining t he relax at ion met hod w ith t he recursive secant technique, t he nonlinear equat ion set occurred in ISC mapping is solved rapidly. No special init ial value is needed and the convergence is guaranteed by adjust ing a conver gent criterion. Inserting some spurious vert ices and removing singularit ies are recom mended to avoid t he t rouble in int eg rals and make t he com put at ion rapid and accurat e. Based on above schemes a sing le prog ram is developed to deal w it h any polygon problems quickly . As an example, the square coax ial line is analyzed in det ail. From the results, w e can conclude t hat this algorithm can be used in the sim ilar problems including curve boundaries. Key words inverse Schw arz - Christoff el t ransformat ions, relaxation, recurrent secant approach, algorit hm , conformal mapping 静磁场、弹性场以及其他场类问题中。它能把上半复平面映射为多角形所围成的区域, 其反变换 ( ISC) 则能把多角形所包围的区域映射为上半复平面。在多数情况下 ISC 比 S - C 更复杂, 很难求到解

中英文地理信息系统(GIS)英语词汇表

accreditation 委派accuracy 准确度acquisition 获取activity patterns 活动模式added value 附加值adjacency邻接Aeolian 伊奥利亚人的, 风的, 风蚀的Age of Discovery 发现的年代aggregation聚合algorithm, definition算法,定义ambiguity 不明确analytical cartography 分析制图application programming interfaces(APIs) 应用编程接口ARCGis 美国ESRI公司开发的世界先进的地理信息系统软件ArcIMS 它是个强大的，基于标准的工具，让你快速设计和管理Internet地图服务ArcInfo 在ArcGIS软件家族中，ArcInfo是GIS软件中功能最全面的。

它包含ArcView和ArcEditor 所有功能，并加上高级空间处理和数据转换ArcNews 美国ESRI向用户终生免费赠送的ArcNews报刊ArcSDE ArcSDE在ESRI GIS软件和DBMS之间提供通道，是一个空间数据引擎ArcUser Magazine 为ESRI用户创建的报刊ArcView 桌面GIS和制图软件，提供数据可视化，查询，分析和集成功能，以及创建和编辑地理数据的能力ARPANET ARPA 计算机网（美国国防部高级研究计划局建立的计算机网）aspatial data 非空间数据？Association of Geographic Information (AGI) 地理信息协会attribute data 属性数据attributes, types 属性，类型attributive geographic data 属性地理数据autocorrelation 自相关Autodesk MapGuide 美国Autodesk公司生产的Web GIS软件Automated mapping/facility management(AM/FM) systems 自动绘图/设备管理系统facilities 设备avatars 化身A VIRIS 机载可见光/红外成像光谱仪azimuthal projections 方位投影batch vectorization 批量矢量化beer consumption 啤酒消费benchmarking 基准Berry, Brianbest fit line 最优线binary counting system 二进制计算系统binomial distribution 二项式分布bivariate Gaussian distribution 二元高斯分布block encoding 块编码Bosnia, repartitioning 波斯尼亚，再分离成两个国家buffering 缓冲区分析Borrough, PeterBusiness and service planning(retailing) application in petroleum and convenience shopping 石油和便利购物的业务和服务规划（零售）应用business drivers 业务驱动business, GIS as 业务，地理信息系统作为Buttenfield, Barbaracadasters 土地清册Callingham, Martincannibalizing 调拨Cartesian coordinate system笛卡尔坐标系Cartograms 统计地图cartographic generalization 制图综合cartographic modeling 地图建模cartometric transformations 量图变换catalog view of database 数据库目录视图census data人口普查数据Census of Population 人口普查central Place Theory 中心区位论central point rule 中点规则central tendency 中心倾向centroid 质心choropleth mapping分区制图choosing a GIS 选择一个地理信息系统class 类别classification generalization 分类综合client 客户端client-server C/S结构客户端-服务器cluster analysis 聚类分析clutter 混乱coastline weave 海岸线codified knowledge 编码知识COGO data 坐标几何数据COGO editing tools 坐标几何编辑工具Collaboration 协作Local level 地方级National level 国家级Collection-level metadata 获取级元数据Commercial-off-the-shelf (COTS) systems 成熟的商业化系统chemas-microsoft-comfficeffice" />>> Commom object request broker architecture (CORBA) 公共对象请求代理体系结构Community, GIS 社区，地理信息系统Competition 竞争Component GIS 组件地理信息系统Component object model (COM) 组件对象模型Computer assisted mass appraisal (CAMA) 辅助大量估价，>>Computer-aided design (CAD)-based GIS 基于计算机辅助制图的地理信息系统Models 数据模型Computer-aided software engineering (CASE) tool 计算机辅助软件工程工具Concatenation 串联Confidence limits 置信界限Conflation 异文合并Conformal property 等角特性Confusion matrix 混淆矩阵Conic projections 圆锥投影Connectivity 连接性Consolidation 巩固Constant term 常数项Contagious diffusion 传染扩散Continuing professional development (CPD) 持续专业发展Coordinates 坐标Copyright 版权Corridor 走廊Cost-benefit analysis 成本效益分析Cost-effectiveness evaluation 成本效率评估Counting method 计算方法Cresswell, PaulCustomer support 客户支持Cylindrical Equidistant Projection 圆柱等距投影Cylindrical projections 圆柱投影> >Dangermond, Jack 美国ESRI总裁>> dasymetric mapping 分区密度制图>>data 数据>>automation 自动化>>capture costs 获取代价>>capture project 获取工程>>collection workflow 采集工作流>> compression 压缩>>conversion 转换>>definition 定义>>geographic, nature of 地理数据，数据的性质>> GIS 地理信息系统>>industry 产业>>integration 集成>>mining 挖掘>>transfer 迁移>>translation 转化>>data model 数据模型>> definition 定义>>levels of abstraction 提取等级>> in practice 实际上>>types 类型>>database 数据库>>definition 定义>>design 设计>>generalization 综合>>global 全球的>>index 索引>>multi-user editing 多用户编辑>> structuring 结构>>database management system (DBMS) 数据库管理系统>>capabilities 能力>>data storage 数据存储>>geographic extensions 地理扩展>>types 类型>>Dayton Accord 达顿协定，1995年12月达顿协定(DAYTON ACCORD)签订,巴尔干和平已经实现,波斯尼亚(包括黑塞哥维那)再被分解成两个国家>>decision support 决策支持>>deductive reasoning 演绎推理>>definitions of GIS 地理信息系统的各种定义>>degrees of freedom 自由度>>density estimation 密度估算>>dependence in space 空间依赖>>desktop GIS 桌面地理信息系统>>desktop paradigms 桌面范例>>Digital Chart of the World (DCW) 世界数字化图>>digital divide 数字鸿沟>>Digital Earth 数字地球>>Digital elevation models (DEMs) 数字高程模型>>Digital line graph (DLG) 数字线划图>>Digital raster graphic (DRG) 数字影像图>>Digital representation 数字表现>>Digital terrain models 数字地形模型>>Digitizing 数字化>>DIME (Dual Independent Map Encoding) program 美国人口调查局建立的双重独立地图编码系统>> Dine CARE >>Discrete objects 离散对象>>Douglas-Poiker algorithm 道格拉斯-普克算法，一种矢量数据抽稀算法>>Dublin Core metadata standard 都柏林核心元数据标准>>Dynamic segmentation 动态分割>>Dynamic simulation models 动态仿真模型>>> >Easting 朝东方>>Ecological fallacy 生态谬误>>e-commerce 电子商业>>editing 编辑>>education 教育>>electromagnetic spectrum 电磁光谱>>ellipsoids 偏振光椭圆率测量仪>>of rotation 旋转的>>emergency evacuation 应急撤退>>encapsulation 封装>>environmental applications 环境应用>>environmental impact 环境影响>>epidemiology 流行病学>>equal area property 等面积特性>>Equator 赤道>>ERDAS ERDAS公司是世界上最大的专业遥感图像处理软件公司，用户遍布100多个国家，软件套数超过17000套。

群论英文

LIE GROUPS IN PHYSICS1
version 25/06/07
Institute for Theoretical Physics Utrecht University Beta Faculty
2007
English version by G. ’t Hooft Original text by M.J.G. Veltman B.Q.P.J. de Wit and G. ’t Hooft
pions mπ+ ≈ 140 MeV/c2 mπ0 ≈ 135 MeV/c2 mπ− ≈ 140 MeV/c2
∆ particles m∆++ ≈ 1231 MeV/c2 m∆+ ≈ 1232 MeV/c2 m∆0 ≈ 1233 MeV/c2 m∆− ≈ 1235 MeV/c2
Table 1: Masses of nucleons, pions and ∆ particles, expressed in MeV/c2 . two possible decay modes: ∆+ → n π + and ∆+ → p π on 2 Quantum mechanics and rotation invariance 3 The group of rotations in three dimensions 4 More about representations 5 Ladder operators 6 The group SU (2) 7 Spin and angular distributions 8 Isospin 9 The Hydrogen Atom 10 The group SU(3) 11 Representations of SU(N); Young tableaus 12 Beyond these notes 1 7 14 22 26 31 39 45 48 55 60 61

量子力学索引英汉对照

21-centimeter line, 21厘米线AAbsorption, 吸收Addition of angular momenta, 角动量叠加Adiabatic approximation, 绝热近似Adiabatic process, 绝热过程Adjoint, 自伴的Agnostic position, 不可知论立场Aharonov-Bohm effect, 阿哈罗诺夫—玻姆效应Airy equation, 艾里方程；Airy function, 艾里函数Allowed energy, 允许能量Allowed transition, 允许跃迁Alpha decay, α衰变；Alpha particle, α粒子Angular equation, 角向方程Angular momentum, 角动量Anomalous magnetic moment, 反常磁矩Antibonding, 反键Anti-hermitian operator, 反厄米算符Associated Laguerre polynomial, 连带拉盖尔多项式Associated Legendre function, 连带勒让德多项式Atoms, 原子Average value, 平均值Azimuthal angle, 方位角Azimuthal quantum number, 角量子数BBalmer series, 巴尔末线系Band structure, 能带结构Baryon, 重子Berry's phase, 贝利相位Bessel functions, 贝塞尔函数Binding energy, 束缚能Binomial coefficient, 二项式系数Biot-Savart law, 毕奥—沙法尔定律Blackbody spectrum, 黑体谱Bloch's theorem, 布洛赫定理Bohr energies, 玻尔能量；Bohr magneton, 玻尔磁子；Bohr radius, 玻尔半径Boltzmann constant, 玻尔兹曼常数Bond, 化学键Born approximation, 玻恩近似Born's statistical interpretation, 玻恩统计诠释Bose condensation, 玻色凝聚Bose-Einstein distribution, 玻色—爱因斯坦分布Boson, 玻色子Bound state, 束缚态Boundary conditions, 边界条件Bra, 左矢Bulk modulus, 体积模量CCanonical commutation relations, 正则对易关系Canonical momentum, 正则动量Cauchy's integral formula, 柯西积分公式Centrifugal term, 离心项Chandrasekhar limit, 钱德拉赛卡极限Chemical potential, 化学势Classical electron radius, 经典电子半径Clebsch-Gordan coefficients, 克—高系数Coherent States, 相干态Collapse of wave function, 波函数塌缩Commutator, 对易子Compatible observables, 对易的可观测量Complete inner product space, 完备内积空间Completeness, 完备性Conductor, 导体Configuration, 位形Connection formulas, 连接公式Conservation, 守恒Conservative systems, 保守系Continuity equation, 连续性方程Continuous spectrum, 连续谱Continuous variables, 连续变量Contour integral, 围道积分Copenhagen interpretation, 哥本哈根诠释Coulomb barrier, 库仑势垒Coulomb potential, 库仑势Covalent bond, 共价键Critical temperature, 临界温度Cross-section, 截面Crystal, 晶体Cubic symmetry, 立方对称性Cyclotron motion, 螺旋运动DDarwin term, 达尔文项de Broglie formula, 德布罗意公式de Broglie wavelength, 德布罗意波长Decay mode, 衰变模式Degeneracy, 简并度Degeneracy pressure, 简并压Degenerate perturbation theory, 简并微扰论Degenerate states, 简并态Degrees of freedom, 自由度Delta-function barrier, δ势垒Delta-function well, δ势阱Derivative operator, 求导算符Determinant, 行列式Determinate state, 确定的态Deuterium, 氘Deuteron, 氘核Diagonal matrix, 对角矩阵Diagonalizable matrix, 对角化Differential cross-section, 微分截面Dipole moment, 偶极矩Dirac delta function, 狄拉克δ函数Dirac equation, 狄拉克方程Dirac notation, 狄拉克记号Dirac orthonormality, 狄拉克正交归一性Direct integral, 直接积分Discrete spectrum, 分立谱Discrete variable, 离散变量Dispersion relation, 色散关系Displacement operator, 位移算符Distinguishable particles, 可分辨粒子Distribution, 分布Doping, 掺杂Double well, 双势阱Dual space, 对偶空间Dynamic phase, 动力学相位EEffective nuclear charge, 有效核电荷Effective potential, 有效势Ehrenfest's theorem, 厄伦费斯特定理Eigenfunction, 本征函数Eigenvalue, 本征值Eigenvector, 本征矢Einstein's A and B coefficients, 爱因斯坦A,B系数；Einstein's mass-energy formula, 爱因斯坦质能公式Electric dipole, 电偶极Electric dipole moment, 电偶极矩Electric dipole radiation, 电偶极辐射Electric dipole transition, 电偶极跃迁Electric quadrupole transition, 电四极跃迁Electric field, 电场Electromagnetic wave, 电磁波Electron, 电子Emission, 发射Energy, 能量Energy-time uncertainty principle, 能量—时间不确定性关系Ensemble, 系综Equilibrium, 平衡Equipartition theorem, 配分函数Euler's formula, 欧拉公式Even function, 偶函数Exchange force, 交换力Exchange integral, 交换积分Exchange operator, 交换算符Excited state, 激发态Exclusion principle, 不相容原理Expectation value, 期待值FFermi-Dirac distribution, 费米—狄拉克分布Fermi energy, 费米能Fermi surface, 费米面Fermi temperature, 费米温度Fermi's golden rule, 费米黄金规则Fermion, 费米子Feynman diagram, 费曼图Feynman-Hellman theorem, 费曼—海尔曼定理Fine structure, 精细结构Fine structure constant, 精细结构常数Finite square well, 有限深方势阱First-order correction, 一级修正Flux quantization, 磁通量子化Forbidden transition, 禁戒跃迁Foucault pendulum, 傅科摆Fourier series, 傅里叶级数Fourier transform, 傅里叶变换Free electron, 自由电子Free electron density, 自由电子密度Free electron gas, 自由电子气Free particle, 自由粒子Function space, 函数空间Fusion, 聚变Gg-factor, g—因子Gamma function, Γ函数Gap, 能隙Gauge invariance, 规范不变性Gauge transformation, 规范变换Gaussian wave packet, 高斯波包Generalized function, 广义函数Generating function, 生成函数Generator, 生成元Geometric phase, 几何相位Geometric series, 几何级数Golden rule, 黄金规则"Good" quantum number, “好”量子数"Good" states, “好”的态Gradient, 梯度Gram-Schmidt orthogonalization, 格莱姆—施密特正交化法Graphical solution, 图解法Green's function, 格林函数Ground state, 基态Group theory, 群论Group velocity, 群速Gyromagnetic railo, 回转磁比值HHalf-integer angular momentum, 半整数角动量Half-life, 半衰期Hamiltonian, 哈密顿量Hankel functions, 汉克尔函数Hannay's angle, 哈内角Hard-sphere scattering, 硬球散射Harmonic oscillator, 谐振子Heisenberg picture, 海森堡绘景Heisenberg uncertainty principle, 海森堡不确定性关系Helium, 氦Helmholtz equation, 亥姆霍兹方程Hermite polynomials, 厄米多项式Hermitian conjugate, 厄米共轭Hermitian matrix, 厄米矩阵Hidden variables, 隐变量Hilbert space, 希尔伯特空间Hole, 空穴Hooke's law, 胡克定律Hund's rules, 洪特规则Hydrogen atom, 氢原子Hydrogen ion, 氢离子Hydrogen molecule, 氢分子Hydrogen molecule ion, 氢分子离子Hydrogenic atom, 类氢原子Hyperfine splitting, 超精细分裂IIdea gas, 理想气体Idempotent operaror, 幂等算符Identical particles, 全同粒子Identity operator, 恒等算符Impact parameter, 碰撞参数Impulse approximation, 脉冲近似Incident wave, 入射波Incoherent perturbation, 非相干微扰Incompatible observables, 不对易的可观测量Incompleteness, 不完备性Indeterminacy, 非确定性Indistinguishable particles, 不可分辨粒子Infinite spherical well, 无限深球势阱Infinite square well, 无限深方势阱Inner product, 内积Insulator, 绝缘体Integration by parts, 分部积分Intrinsic angular momentum, 内禀角动量Inverse beta decay, 逆β衰变Inverse Fourier transform, 傅里叶逆变换KKet, 右矢Kinetic energy, 动能Kramers' relation, 克莱默斯关系Kronecker delta, 克劳尼克δLLCAO technique, 原子轨道线性组合法Ladder operators, 阶梯算符Lagrange multiplier, 拉格朗日乘子Laguerre polynomial, 拉盖尔多项式Lamb shift, 兰姆移动Lande g-factor, 朗德g—因子Laplacian, 拉普拉斯的Larmor formula, 拉摩公式Larmor frequency, 拉摩频率Larmor precession, 拉摩进动Laser, 激光Legendre polynomial, 勒让德多项式Levi-Civita symbol, 列维—西维塔符号Lifetime, 寿命Linear algebra, 线性代数Linear combination, 线性组合Linear combination of atomic orbitals, 原子轨道的线性组合Linear operator, 线性算符Linear transformation, 线性变换Lorentz force law, 洛伦兹力定律Lowering operator, 下降算符Luminoscity, 照度Lyman series, 赖曼线系MMagnetic dipole, 磁偶极Magnetic dipole moment, 磁偶极矩Magnetic dipole transition, 磁偶极跃迁Magnetic field, 磁场Magnetic flux, 磁通量Magnetic quantum number, 磁量子数Magnetic resonance, 磁共振Many worlds interpretation, 多世界诠释Matrix, 矩阵；Matrix element, 矩阵元Maxwell-Boltzmann distribution, 麦克斯韦—玻尔兹曼分布Maxwell’s equations, 麦克斯韦方程Mean value, 平均值Measurement, 测量Median value, 中位值Meson, 介子Metastable state, 亚稳态Minimum-uncertainty wave packet, 最小不确定度波包Molecule, 分子Momentum, 动量Momentum operator, 动量算符Momentum space wave function, 动量空间波函数Momentum transfer, 动量转移Most probable value, 最可几值Muon, μ子Muon-catalysed fusion, μ子催化的聚变Muonic hydrogen, μ原子Muonium, μ子素NNeumann function, 纽曼函数Neutrino oscillations, 中微子振荡Neutron star, 中子星Node, 节点Nomenclature, 术语Nondegenerate perturbationtheory, 非简并微扰论Non-normalizable function, 不可归一化的函数Normalization, 归一化Nuclear lifetime, 核寿命Nuclear magnetic resonance, 核磁共振Null vector, 零矢量OObservable, 可观测量Observer, 观测者Occupation number, 占有数Odd function, 奇函数Operator, 算符Optical theorem, 光学定理Orbital, 轨道的Orbital angular momentum, 轨道角动量Orthodox position, 正统立场Orthogonality, 正交性Orthogonalization, 正交化Orthohelium, 正氦Orthonormality, 正交归一性Orthorhombic symmetry, 斜方对称Overlap integral, 交叠积分PParahelium, 仲氦Partial wave amplitude, 分波幅Partial wave analysis, 分波法Paschen series, 帕邢线系Pauli exclusion principle, 泡利不相容原理Pauli spin matrices, 泡利自旋矩阵Periodic table, 周期表Perturbation theory, 微扰论Phase, 相位Phase shift, 相移Phase velocity, 相速Photon, 光子Planck's blackbody formula, 普朗克黑体辐射公式Planck's constant, 普朗克常数Polar angle, 极角Polarization, 极化Population inversion, 粒子数反转Position, 位置；Position operator, 位置算符Position-momentum uncertainty principles, 位置—动量不确定性关系Position space wave function, 坐标空间波函数Positronium, 电子偶素Potential energy, 势能Potential well, 势阱Power law potential, 幂律势Power series expansion, 幂级数展开Principal quantum number, 主量子数Probability, 几率Probability current, 几率流Probability density, 几率密度Projection operator, 投影算符Propagator, 传播子Proton, 质子QQuantum dynamics, 量子动力学Quantum electrodynamics, 量子电动力学Quantum number, 量子数Quantum statics, 量子统计Quantum statistical mechanics, 量子统计力学Quark, 夸克RRabi flopping frequency, 拉比翻转频率Radial equation, 径向方程Radial wave function, 径向波函数Radiation, 辐射Radius, 半径Raising operator, 上升算符Rayleigh's formula, 瑞利公式Realist position, 实在论立场Recursion formula, 递推公式Reduced mass, 约化质量Reflected wave, 反射波Reflection coefficient, 反射系数Relativistic correction, 相对论修正Rigid rotor, 刚性转子Rodrigues formula, 罗德里格斯公式Rotating wave approximation, 旋转波近似Rutherford scattering, 卢瑟福散射Rydberg constant, 里德堡常数Rydberg formula, 里德堡公式SScalar potential, 标势Scattering, 散射Scattering amplitude, 散射幅Scattering angle, 散射角Scattering matrix, 散射矩阵Scattering state, 散射态Schrodinger equation, 薛定谔方程Schrodinger picture, 薛定谔绘景Schwarz inequality, 施瓦兹不等式Screening, 屏蔽Second-order correction, 二级修正Selection rules, 选择定则Semiconductor, 半导体Separable solutions, 分离变量解Separation of variables, 变量分离Shell, 壳Simple harmonic oscillator, 简谐振子Simultaneous diagonalization, 同时对角化Singlet state, 单态Slater determinant, 斯拉特行列式Soft-sphere scattering, 软球散射Solenoid, 螺线管Solids, 固体Spectral decomposition, 谱分解Spectrum, 谱Spherical Bessel functions, 球贝塞尔函数Spherical coordinates, 球坐标Spherical Hankel functions, 球汉克尔函数Spherical harmonics, 球谐函数Spherical Neumann functions, 球纽曼函数Spin, 自旋Spin matrices, 自旋矩阵Spin-orbit coupling, 自旋—轨道耦合Spin-orbit interaction, 自旋—轨道相互作用Spinor, 旋量Spin-spin coupling, 自旋—自旋耦合Spontaneous emission, 自发辐射Square-integrable function, 平方可积函数Square well, 方势阱Standard deviation, 标准偏差Stark effect, 斯塔克效应Stationary state, 定态Statistical interpretation, 统计诠释Statistical mechanics, 统计力学Stefan-Boltzmann law, 斯特番—玻尔兹曼定律Step function, 阶跃函数Stem-Gerlach experiment, 斯特恩—盖拉赫实验Stimulated emission, 受激辐射Stirling's approximation, 斯特林近似Superconductor, 超导体Symmetrization, 对称化Symmetry, 对称TTaylor series, 泰勒级数Temperature, 温度Tetragonal symmetry, 正方对称Thermal equilibrium, 热平衡Thomas precession, 托马斯进动Time-dependent perturbation theory, 含时微扰论Time-dependent Schrodinger equation, 含时薛定谔方程Time-independent perturbation theory, 定态微扰论Time-independent Schrodinger equation, 定态薛定谔方程Total cross-section, 总截面Transfer matrix, 转移矩阵Transformation, 变换Transition, 跃迁；Transition probability, 跃迁几率Transition rate, 跃迁速率Translation,平移Transmission coefficient, 透射系数Transmitted wave, 透射波Trial wave function, 试探波函数Triplet state, 三重态Tunneling, 隧穿Turning points, 回转点Two-fold degeneracy , 二重简并Two-level systems, 二能级体系UUncertainty principle, 不确定性关系Unstable particles, 不稳定粒子VValence electron, 价电子Van der Waals interaction, 范德瓦尔斯相互作用Variables, 变量Variance, 方差Variational principle, 变分原理Vector, 矢量Vector potential, 矢势Velocity, 速度Vertex factor, 顶角因子Virial theorem, 维里定理WWave function, 波函数Wavelength, 波长Wave number, 波数Wave packet, 波包Wave vector, 波矢White dwarf, 白矮星Wien's displacement law, 维恩位移定律YYukawa potential, 汤川势ZZeeman effect, 塞曼效应。

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a r X i v :g r -q c /9903049v 1 13 M a r 1999CONFORMAL TRANSFORMATIONS ANDQUANTUM GRAVITYFATIMAH SHOJAI ∗,1and ALI SHOJAI †,2,1and MEHDI GOLSHANI 3,11Institute for Studies in Theoretical Physics and Mathematics,P.O.Box 19395–5531,Tehran,IRAN.2Department of Physics,Tarbiat Moddarres University,P.O.Box 14155-4838,Tehran,IRAN.3Department of Physics,Sharif University of Technology,P.O.Box 11365–9161,Tehran,IRAN.February 7,2008AbstractRecently[1],it was shown that quantum eﬀects of matter could be identiﬁed with the conformal degree of freedom of the space–time metric.Accordingly,one canintroduce quantum eﬀects either by making a scale transformation(i.e.changingthe metric),or by making a conformal transformation(i.e.changing all physicalquantities).These two ways are investigated and compared.Also,it is argued that,the ultimate formulation of such a quantum gravity theory should be in the frameworkof the scalar–tensor theories.1Quantum Eﬀects,Scale,and Conformal Transfor-mationsIn a previous paper[1]it was shown that the application of an idea of de-Broglie leads to the fact that the quantum eﬀects of matter are equivalent to a speciﬁc conformal factor of the space–time metric.In the de-Broglie–Bohm relativistic quantum theory[3],the Klein–Gordon equation1−2Ψ+Ψ=0(1)¯h2is replaced with the quantum Hamilton–Jacobi equation:∇µS M2(2) and the countinuty equation:ρ1A”bar”sign over any quantity indicates that it is expressed in terms ofwhere all covariant derivatives are calculated with respect to the space–time metricM=Q)1/2(5)√ρ;α=¯h2m2(6) It muse be noted thatQ.If the above metric is written as conformally transformed of another metric(gµν)whichQ(8) Then the Hamilton–Jacobi equation in terms of this metric is:gµν∇µS∇νS=gµν(the background metric)the quantum eﬀects are included in the variable massIn our previous work[1],the conformal transformation was applied only to the space–time metric.Other quantities like mass,density and so on were assumed to posses no transformation.This is because the above conformal transformation which incorporates the quantum eﬀects of matter into a speciﬁc conformal factor,is in fact a scale transfor-mation guessed from the Hamilton–Jacobi equation(2).As the conformal transformation is more general than scale transformation which is used in[1],it seems preferable to make a conformal transformation,in which all physical quantities are transformed,instead of making only a scale transformation.Now,applying the conformal transformation given by the equation(7),we have:ρ=φ3/2ρ(11) Therefore the equation(2)reads:∇µS∇µS=φ2m2(1+Q(13) This equation determines the conformal degree of freedom of the space–time metric.With the aid of the guiding equation in the physical metric(Pµ=muµ=∇µS)and the equation(12),the equation of motion takes the form:duµ2(∂µgαβ)uαuβ=0(14) which is the geodesic equation for a free particle.But with respect to the background metric(Pµ=uµ=uµτ−1gαβ uβ=1M uµ∇νuµτ−1gαβ uβ=−1uµdφτ−1uαgαβ∂µφτ=∇αandφ=M,a simple calculation shows that the right hand side of the above equation is the quantum force.Here,It must be noted that the conformal factor is obtained only from one of the equations of motion,i.e.the Hamilton–Jacobi equation.The continouty equation in the physical metric is:∇µ(ρ∇µS)=0(17) Soρ∇µφρThe right hand side of this expression is of the order of¯h2and higher[1].The conformal transformation used to show the equivalence between quantum eﬀects and cuavature of space–time has many applications.It is possible to discuss the physical properties of the conformal factor(13)as it was done in our earlier paper[1]about the scale factor(8).Correspondingly,one can start from gravity–matter action and then proceed to eliminate the matter quantum potential and transform metric and physical quantities conformally.Therefore,the quantum gravity action is:A= d4x√g φ2∇µφκ φ3∇µSρQ)−1/2(19) whereΛis a lagrangian multiplier introduced in order toﬁx the conformal factor.Varia-tion of this action with respect toρ,S andΛleads to:1.The equation of motion ofφ:2∇µφφ+Λ+6ρm∇µS−2ρ∇µ ∇µS =0(21)3.The equation of motion of particles:4∇µSκ2(1+Q2−2 (1+√ρ =0(22) 4.The modiﬁed Einstien equations forGµν−1gµν=1gµν−2−∇ν]φ+3∇µφφ2−1∇αφφ2+ρmφ2∇αSκm∇µSκρ2ΛQ)−1/2 4(1+Q4φ∇α√ρQ)−3/24φ∇ν Λφ(1+√ρ +α∇ν√ρQ)−3/2Q(24)Then,the equations of motion can be solved,leading to the background metric and other physical quantities.2Concluding RemarksRecently[1],it was shown that quantum eﬀects could be contained in the conformal factor of the space–time metric.In this paper,the diﬀerence between introducing the quantum eﬀects by conformal transformation or scale transformation are discussed.No essential diﬀerence is found.But,some points must be noted here.Theﬁrst important point is about the geodesic equation(14,15).In the background metric,this equation resembles the geodesic equation in Brans–Dicke theory.Consideration of the matter quantum eﬀects, leads to the physical metric in which a particle moves on the geodesic of Branse–Dicke theory written in Einstein guage.This point supports the suggestion that the discussion of quantum gravity requires a scalar–tensor theory.Previously this was suggested when disscusing Bohmian quantum gravity[4].Secondly,since as the dimensional coupling between matter and gravity is resulted from the breaking of the conformal invariance,this formulation shows that the conformal frame isﬁxed by the distribution of matter at quantum level.Thus,the Bohmian quantum theory singles out the prefered frame.In the present paper and the earlier one[1]we have used lagrangian multiplier,in order toﬁx the conformal factor by quantum potential.We suggested now that it must be possible to write a scalar–tensor theory which automatically leads to the correct equations of motion.This has the advantage that the conformal factor would beﬁxed by the equations of motion,and not by introducing a lagrangian multiplier by hand.We shall elaborate on this point in a forthcoming paper[2].References[1]F.Shojai,and M.Golshani,Int.J.Mod.Phys.A,Vol.13,No.4,677,1998.[2]F.Shojai,A.Shojai,and M.Golshani,”Scalar–Tensor Theories and Quantum Grav-ity”,to appear in Mod.Phys.Lett.A.[3]D.Bohm,Phys.Rev.,85,No.2,166,(1952);D.Bohm,Phys.Rev.,85,No.2,180,(1952);D.Bohm and B.J.Hiley,The undivided universe,Routledge,London,(1993);P.R.Holland,The Quantum Theory of Motion,Cambridge University Press,(1993).[4]F.Shojai,and M.Golshani,Int.J.Mod.Phys.A,Vol.13,No.13,2135,1998.。