On the gauge invariance of the non-Abelian Chern-Simons action for D-branes

a r X i v :h e p -l a t /9709149v 1 29 S e p 19971RU-97-78KUNS-1465HE(TH)97/14Gauge Freedom in Chiral Gauge Theory with Vacuum Overlap ∗Yoshio Kikukawa a †aDepartment of Physics and Astronomy,Rutgers University,Piscataway,NJ 08855-0849,USADynamical nature of the gauge degrees of freedom and its effect to fermion spectrum are studied at β=∞for two-and four-dimensional nonabelian chiral gauge theories in the vacuum overlap formalism.It is argued that the disordered gauge degrees of freedom does not contradict to the chiral spectrum of lattice fermion.1.Introduction —Pure gauge limit In the vacuum overlap formalism[1]of a generic chiral gauge theory,gauge symmetry is explic-itly broken by the complex phase of fermion de-terminant.In order to restore the gauge invari-ance,gauge average —the integration along gauge orbit—is invoked.Then,what is required for the dynamical nature of the gauge freedom at β=∞(pure gauge limit)is that the global gauge symmetry is not broken spontaneously and the bosonic fields of the gauge freedom have large mass compared to a typical mass scale of the the-ory so as to decouple from physical spectrum[2,1].However,through the analysis of the waveg-uide model[3,4],it has been claimed that this required disordered nature of the gauge freedom causes the vector-like spectrum of fermion[5].In this argument,the fermion correlation functions at the waveguide boundaries were examined.One may think of the counter parts of these correla-tion functions in the overlap formalism by putting creation and annihilation operators in the overlap of vacua with the same signature of mass.Let us refer this kind of correlation function as bound-ary correlation function and the correlation func-tion in the original definition as overlap correla-tion function .We should note that the boundary correlation functions are no more the observables in the sense defined in the overlap formalism;they cannot be expressed by the overlap of two vacua with their phases fixed by the Wigner-Brillouin| +|v + |v +|v −v −|−2Then the model describes the gauge degrees of freedom coupled to fermion through the gauge non-invariant piece of the complex phase of chiral determinants.Z= [dg] rep. +|ˆG|+ | −|ˆG†|− | ≡ dµ[g].(3)ˆG is the operator of the gauge transformation given by:ˆG=exp ˆa†i n{log g}i jˆa nj .(4) In this limit,one of the possible definitions of the boundary correlation functions is given as fol-lows for the case of the negative mass and in the representation r:φniφ†m j −r≡12δnmδj i |− rλ2 .(8)In four-dimensions,we consider the covariantgaugefixing term and the Faddeev-Popov deter-minant[7],following Hata[9](cf.[10–12]):−12i U nµ−U†nµ −12i U nµ−U†nµ ,with the pure gauge link variable Eq.(2).ˆM ab nm isthe lattice Faddeev-Popov operator.We can show by the perturbation theory inλ(backgroundfield method)that both puregauge models share the novel features of the two-dimensional nonlinear sigma model even in thepresense of the imaginary action.The model isrenormalizable(at one-loop in four-dimensions)andλis asymptotically free.Severe infrared di-vergence occurs and it prevents local order pa-rameters from emerging.Based on these dynamical features,we assumethat the global gauge symmetry does not breakspontaneously for the entire region ofλand thegauge freedom acquires mass M g dynamicallythrough the dimensional transmutation.Withthis assumption,the decoupling of the gauge free-dom could occur as M gր133.Boundary correlation functionThe asymptotic freedom allows us to tamethe gaugefluctuation by approaching the criticalpoint of the gauge freedom.There we can invokethe spin wave approximation for the calculationof the invariant boundary correlation functionφniφ†m i −r,(10)which is free from the IR divergence.At one-looporder,we obtain1(2π)D e ip(n−m)1µsin2pµ+(B(p)−m0)2,(14) andπnπm ′= d D p µ4sin2pµ2(1−m0),(m0=0.5).(16)We also see that the quantum correction due to the gaugefluctuation at one-loop does not affect the leading short-distance nature.There is no symmetry against the spectrum mass gap.There-fore it seems quite reasonable to assume that it holds true asλbecomes large.Since the overlap correlation function does not depend on the gauge freedom and does show the chiral spectrum[1],the above fact means that the entire fermion spec-trum is chiral.The author would like to thank H.Neuberger and R.Narayanan for enlightening discussions. He also would like to thank H.Hata,S.Aoki, H.So and A.Yamada for discussions. REFERENCES1.R.Narayanan and H.Neuberger,Nucl.Phys.B412(1994)574;Phys.Rev.Lett.71(1993) 3251;Nucl.Phys.B(Proc.Suppl.)34(1994) 95,587;Nucl.Phys.B443(1995)305.2. D.Foerster,H.B.Nielsen and M.Ninomiya,Phys.Lett.B94(1980)135;S.Aoki,Phys.Rev.Lett.60(1988)2109;Phys.Rev.D38 (1988)618;K.Funakubo and T.Kashiwa, 60(1988)2113;J.Smit,Nucl.Phys.B(Proc.Suppl.)4(1988)451.3. D.B.Kaplan,Nucl.Phys.B30(Proc.Suppl.)(1993)597.4.M.Golterman,K.Jansen,D.Petcher and J.Vink,Phys.Rev.D49(1994)1606;M.F.L.Golterman and Y.Shamir,Phys.Rev.D51 (1995)3026.5.M.F.L.Golterman and Y.Shamir,Phys.Lett.B353(1995)84;Erratum-ibid.B359(1995) 422.R.Narayanan and H.Neuberger,Phys.Lett.B358(1995)303.6.Y.Kikukawa,KUNS-1445HE(TH)97/08,May1997,hep-lat/9705024.7.Y.Kikukawa,KUNS-1446HE(TH)97/09,July1997,hep-lat/9707010.8.Y.Kikukawa and S.Miyazaki,Prog.Theor.Phys.96(1996)1189.9.H.Hata,Phys.Lett.143B(1984)171.10.A.Borrelli,L.Maiani,G.C.Rossi,R.Sistoand M.Testa,Nucl.Phys.B333(1990)335.11.Y.Shamir,TAUP-2306-95,Dec1995;Y.Shamir,Nucl.Phys.B(Proc.Suppl.)53 (1997)664.12.M.F.L.Golterman and Y.Shamir,Phys.Lett.B399(1997)148.13.E.Witten,Commun.Math.Phys.92(1984)455.。

黑洞四定律黑洞四定律是关于黑洞性质及行为的重要定律，对于理解黑洞的基本性质和其与物理学的关系具有重要意义。

黑洞四定律包括黑洞的质量、面积、角动量和电荷四个方面。

下面将逐一介绍这些定律，并提供一些相关参考内容。

1. 黑洞质量定律：黑洞质量定律，也被称为霍金面积定律，它指出黑洞的表面积和质量之间存在着一种关系，即黑洞的表面积正比于其质量的平方。

这个定律是由物理学家斯蒂芬·霍金于1971年提出的。

霍金面积定律是黑洞热力学理论的基础，它将黑洞与热力学的概念进行了联系，使人们能够通过研究黑洞的热力学性质来更好地理解黑洞。

参考内容：- 斯蒂芬·霍金, "Black Hole Explosions?" Nature, vol. 248, pp. 30-31, 1974.- Andrei Barvinsky, George Kunstatter, "Critical Phenomena in Black Hole Physics," Phys. Lett. B, vol. 389, no. 3-4, pp. 231-236, 1996.- 黄绪淇, "黑洞的热力学与量子论," 物理学报, vol. 52, no. 7, pp. 1359-1366, 2003.2. 黑洞面积定律：根据黑洞的面积定律，也称为麦克斯韦-比克尔定律，黑洞的面积与其事件视界的面积成正比。

黑洞的事件视界是黑洞的边界，当物质趋于接近黑洞时，一旦穿过该边界，就无法再逃脱黑洞的引力。

参考内容：- 麦克斯韦, "On the Dynamical Theory of Gases," Proceedings of the Royal Society of Edinburgh, vol. 2, pp. 1-21, 1867.- J. D. Bekenstein, "Black Holes and Entropy," Physical Review D, vol. 7, no. 8, pp. 2333-2346, 1973.- 斯蒂芬·霍金, "Particle Creation by Black Holes," Communications in Mathematical Physics, vol. 43, no. 3, pp. 199-220, 1975.3. 黑洞角动量定律：黑洞角动量定律是根据黑洞旋转的性质推导出来的定律。

a rXiv:h ep-th/047172v12J ul24IUHET-473Gauge Invariance and the Pauli-Villars Regulator in Lorentz-and CPT-Violating Electrodynamics B.Altschul 1Department of Physics Indiana University Bloomington,IN 47405USA Abstract We examine the nonperturbative structure of the radiatively induced Chern-Simons term in a Lorentz-and CPT-violating modification of QED.Although the coefficient of the induced Chern-Simons term is in general undetermined,the nonperturbative theory appears to generate a definite value.However,the CPT-even radiative corrections in this same formulation of the theory generally break gauge invariance.We show that gauge invariance may yet be preserved through the use of a Pauli-Villars regulator,and,contrary to earlier expectations,this regulator does not necessarily give rise to a vanishing Chern-Simons term.Instead,two possible values of the Chern-Simons coefficient are allowed,one zero and one nonzero.This formulation of the theory therefore allows the coefficientto vanish naturally,in agreement with experimental observations.One of the most interesting terms that arises in the study of Lorentz-and CPT-violating corrections to the standard model action[1,2,3]is the electromagnetic Chern-=1Simons term,with Lagrange density LFµνFµν+¯ψ(i∂−m−e A− bγ5)ψ.(1)4This theory has the potential to induce afinite radiatively-generated Chern-Simons term, with∆k CS proportional to b.However,the coefficient of proportionality depends upon the regularization[10,11,12].If the regulator used enforces the gauge invariance of the induced Lagrange density,then∆k CS must necessarily vanish[13];however,this is not particularly interesting,because it excludes the existence of a Chern-Simons term a priori.Other regulators lead to different values of∆k CS,and through a suitable choice, any coefficient of proportionality between the two may be found.This ambiguity has been extensively studied,and several potentially interesting values of∆k CS have been identified[11,14,15,16,17,18].Most notably,if the theory is defined nonperturbatively in b,then the ambiguity is lessened,and a single value appears to be preferred[10,11,14]. However,since this value is nonzero,there is a potential for conflict with the observed vanishing of k CS.We shall continue the analysis of the nonperturbatively-defined theory.While the in-duced Chern-Simons term itself has been extensively studied,the higher-order,CPT-even corrections to this theory have largely been neglected.In a nonperturbatively defined the-ory,the terms of all orders in b are tied together,and so information about the CPT-even terms may help clarify the structure of the CPT-odd Chern-Simons term.We have pre-viously demonstrated[19]that the O(b2)terms in the photon self-energy may violate the Ward identity that enforces the transversality of the vacuum polarization—pµΠµν(p)=0; however,our calculation was limited to the case of m=0.We shall demonstrate here that the same result holds in the opposite limit,when|b2|≪m2(which represents the physi-cal regime for all electrically charged elementary particles).This failure of transversality could represent a significant problem for the nonperturbative formulation of the theory. However,this difficulty may be overcome through the use of a Pauli-Villars regulator. Introducing such a regulator changes the nature of the ambiguity in the Chern-Simons term.When the Ward-identity-violating terms are eliminated,we are left with a theory1in which the coefficient∆k CS of the induced Chern Simons term is not in any way forced to vanish,yet it still may vanish quite naturally.This may represent a resolution of the physical paradox described above.We shall calculate the O(b2)contribution to the zero-momentum photon self-energy,Πµν(p=0),under the assumption thatΠµν(p)may be expanded as a power series in b. However,we do not expect this assumption to be generically valid.The exact fermion propagator,iS(l)=.(3)(l2−m2−b2)2+4[l2b2−(l·b)2]At l=0,the denominator of the rationalized propagator becomes(m2+b2)2.The square root|m2+b2|of this expression arises in the calculation ofΠµν(p=0),and the absolute value leads to behavior that is nonanalytic in b.The b-odd portion of the self-energy(which is just the Chern-Simons term)has different forms for−b2≤m2=0,−b2<m2=0,and m=0.For the b-even terms,we also expect the power-series representation of the self-energy to break down at b2=−m2;moreover,there may exist other thresholds for nonanalytic behavior as well.We shall therefore restrict our attention to the regime in which|b2|≪m2.Since large values of b2are excluded for most physical particles,this is a reasonable restriction.To control the high-energy behavior of our theory,we shall use a Pauli-Villars regula-tor.By utilizing this method of regulation,we may deal with the usual divergences that arise in the photon self-energy[at O(b0)],as well as the Ward-identity-violating terms that appear at O(b2).The use of a single regulator at all orders in b is necessary,because in the nonperturbative formalism,there is actually only a single Feynman diagram that contributes to the photon self-energy at O(e2).This diagram is the usual QED vacuum polarization,but with the usual fermion propagator replaced by the b-exact S(l).We shall use a symmetric integration prescription in our calculation,because this is appropriate at O(b0),and it preserves the expected transformation properties of the loop integral. (Moreover,without symmetric integration,additional ambiguities in∆k CS would exist.) The Pauli-Villars regularization of the photon self-energy involves the introduction of afictitious species of heavy fermions,whose contribution to the self-energy is subtractive. This subtraction renders the Lorentz-invariant part of the self-energyfinite and preserves its gauge invariance.Previously,it has been believed that the same subtraction will cause the O(b)Chern-Simons term to vanish.However,we shall show that this is not necessarily the case.Thefictitious Pauli-Villars particles have the exact propagatoriS M(l)=where M is the particles’large mass,and b M is the Lorentz-violating coefficient appropri-ate to thesefictitious particles.Previous analyses have assumed that b M=b,and indeed this must be the case in a perturbatively defined theory,in which the b interaction is treated as a vertex.However,in the nonperturbative formulation,this restriction is ab-sent,and b and b M exist as distinct objects.Depending upon what properties we demand for the self-energy,b M may take on different values.Even if we place quite strong(yet quite natural)conditions on the form ofΠµν(p),we are still left with a discrete freedom in our choice of b M;either b M=b or b M=−b will suffice.In fact,the weakest condition that we may place upon b M is no condition at all.If we choose not to worry about the failure of the Ward identity at O(b2),then the only restriction on the Pauli-Villars propagator is that it must cancel the divergences at O(b0). In that case,any value of b M is acceptable.We might be tempted to choose a vanishing b M,which would produce a formulation of the theory equivalent to that used in[10]. However,since Lorentz violation is already present in the theory,there is no reason to prefer this value.Similarly,the choice b M=b,while aesthetically pleasing,has nothing special to recommend it.We are left with an arbitrary b M,which will generate an arbitrary Chern-Simons term.Moreover,since b and b M need not point in the same direction,the coefficient∆k CS of the induced Chern-Simons term need not even be parallel to b;this is an even greater degree or arbitrariness in∆k CS than has been found previously.The above scenario illustrates an interesting point.In the theory as we have described it,there can exist a radiatively-induced Chern-Simons term even if b vanishes,because the regulator involves the Lorentz-violating parameter b M.In a general Lorentz-violating quantumfield theory,the Lagrange density need not be the only source of the viola-tion.The regularization prescription can also break Lorentz invariance,and the Lorentz violation of the regulator need not be determined by the behavior of any terms in the unrenormalized action.It may seem somewhat unnatural for the regulator to become an additional source of Lorentz violation.However,it is well established thatfictitious Pauli-Villars fermions may generate radiative corrections that are qualitatively different from those generated by a theory’s real fermions.For example,in massless QED,with a Pauli-Villars regulator,it is thefictitious heavy particles which are solely responsible for the anomalous nonconservation of the axial vector and dilation currents.However,the specific scenario presented in the previous paragraph is probably unrealistic,because we had to abandon the Ward identity in order to avoid placing any restrictions on b M.The key to developing a more sophisticated theory—one in which gauge invariance holds at all orders—is the determination of the O(b2)part of the self-energy.Since we are only interested in the situation in which we may expandΠµν(p)as a power series in b, we may use the approximation scheme developed in[10]to expand the exact propagator to second order in b.WefindS(l)≈il−m(−i bγ5)il−m(−i bγ5)il−m.(5)Naively,this might appear equivalent to treating the b term as an interaction vertex.3However,there are subtle differences that characterize our nonperturbative approach; since there is only a single diagram,there is a complex interplay between terms at different orders in b.The one-loop photon self-energy isΠµν(p)=−ie2 d4k(2π)4tr γµi k−mγνi k−mγµi k−m(−i bγ5)i k−m+(µ↔ν) .(7)=ie2 d4k(k2−m2)4tr{γµ(k+m)b(k−m)γν(k−m)b(k+m) +[γµ(k+m)γν(k+m)b(k−m)b(k+m)+(µ↔ν)]}(8)The only Lorentz structures that may be present inΠµνb2(0)are bµbνand gµνb2,so itis simplest to evaluate the self-energy by calculating gµνΠµνb2(0)and bµbν(2π)4tr{[2(k b k−m2b)+4(k−2m)(k+m)b](k−m)b(k+m)}(2π)416m2[2(b·k)2−b2k2]−10m4b24π2.(11)4Similarly,we also havebµbνb2 d4k(k2−m2)4(12)=−4ie2 d4k(k2−m2)4(13)e2b2=−24π2 2bµbν+gµνb2 .(15) This is the same result that we found previously in the m=0case[19].Since pµ(2bµbν+ gµνb2)=0,there is the potential for a violation of gauge invariance.However,since we are using a Pauli-Villars regularization,there is also a contribution fromΠµνM(p)that is second order in b M.This contribution is precisely e2bµ;(16)8π2this is twice the value found in[10,11,14],since there are equal contributions from b and b M.This is a novel situation.Unambiguouslyfinite yet undetermined radiative corrections generally involve continuously variable parameters that describe the arbitrariness of the results[20],and indeed,in more general formulations of the theory we are discussing,there is a continuous ambiguity in the induced Chern-Simons term.However,in this instance, only a discrete set of values for∆k CS is allowed.Since zero is one of these allowed values, this formulation allows for the existence of a gauge-invariant Lagrange density,but it does not force the density to be so invariant.The Pauli-Villars formulation therefore allows the theory to possess a naturally vanishing induced Chern-Simons term without externally enforcing the gauge invariance of the Lagrange density.Moreover,since the Pauli-Villars regulator is covariant,it should be possible to generalize our calculations to occur in a weakly curved spacetime[21].As we remarked in[19],the O(b)and O(b2)terms in the photon self-energy both violate gauge invariance in the strongest fashions that are allowed by their tensor structures.5The Chern-Simons term violates the gauge invariance of the Lagrange density;however, because the induced density necessarily involves the antisymmetricǫ-tensor,the Wardidentity is preserved and the integrated action remains gauge invariant.Πµνb2(p)possessesno such special structure,and so it violates the Ward identity explicitly.In order to ensure gauge invariance,we have therefore been forced to place a very strong condition on the tensor structure of the second-order correction to the self-energy;we have insisted that it must vanish at p=0.Since there is more than one way to enforce this condition, we have found more than one possible value for∆k CS.This is the origin of the discrete ambiguity in this formulation of the theory.We may now ask what may happen if the mass scale m2is not large compared with the Lorentz-violation scale|b2|.When the real fermions are massless,their O(b2)contributiontoΠµν(p)is exactly theΠµνb2(0)of equation(15).In this instance,as in the|b2|≪m2case,the use of a power series expansion to determine the second-order portion of the photon self-energy is justified,so the similarity of the results is somewhat unsurprising. If a Pauli-Villars regulator is used in this case,the Ward identity may be preserved,but the Chern-Simons term cannot be made to vanish.The choice is between∆kµCS=e24π2bµ.So this formulation of the theory does not allow for∆k CS to vanishnaturally if there are massless fermions coupled to the gaugefield.When a power series expansion in b is not justified,the gauge noninvariant portions of the self-energy must be calculated directly from the nonperturbative propagator(3). Only if the result is of the form A(b2)bµbν+B(b2)gµνb2,with A(b2)=2B(b2)≤0,will the Pauli-Villars regulator be capable of restoring the Ward identity.If we do choose to use a Pauli-Villars regulator and also insist upon gauge invariance as a requirement of our theory,then any breakdowns of the Ward identity when|b2|≪m2can be interpreted as giving restrictions on the allowed values of b.Applied to the standard model,this could imply that the scale of b-generated Lorentz violation should be smaller than the smallest nonzero fermionic mass scale present.However,this is all merely conjectural,since we do not know the general form taken by the self-energy when|b2|≪m2,and because of the substantial additional complexity of the standard model.This work has demonstrated that there exist new,unexpected subtleties in the struc-ture of the radiatively-induced Chern-Simons term.We have identified new types of ambiguities in the value of the Chern-Simons coefficient;these ambiguities are derived from the use of a Pauli-Villars regulator and a nonperturbative formulation of the theory. If we do not insist that the Ward identity hold at second order,then the coefficient is completely arbitrary and is not even constrained to be parallel to b.However,if,more reasonably,we do insist on transversality of the vacuum polarization at higher orders, then there is a binary ambiguity in the value of the induced coefficient.If m=0,either of two nonzero values for∆k CS is allowed.However,if|b2|≪m2,then∆k CS=0is allowed, in addition to a particular nonzero value.This may provide a satisfactory explanation for why the electromagnetic Chern-Simons coefficient vanishes in nature.6AcknowledgmentsThe author is grateful to V.A.Kosteleck´y,M.P´e rez-Victoria,and R.Jackiw for many helpful discussions.This work is supported in part by funds provided by the U.S.Depart-ment of Energy(D.O.E.)under cooperative research agreement DE-FG02-91ER40661. References[1]D.Colladay,V.A.Kosteleck´y,Phys.Rev.D55,6760(1997).[2]D.Colladay,V.A.Kosteleck´y,Phys.Rev.D58,116002(1998).[3]V.A.Kosteleck´y,R.Lehnert,Phys.Rev.D63,065008(2001).[4]R.Jackiw,S.Templeton,Phys.Rev.D23,2291(1981).[5]J.Schonfeld,Nucl.Phys.B185,157(1981).[6]S.M.Carroll,G.B.Field,R.Jackiw,Phys.Rev.D41,1231(1990).[7]S.M.Carroll,G.B.Field,Phys.Rev.Lett.79,2394(1997).[8]M.Goldhaber,V.Trimble,J.Astrophys.Astron.17,17(1996).[9]V.A.Kosteleck´y,ne,A.G.M.Pickering,Phys.Rev.D65,056006(2002).[10]R.Jackiw,V.A.Kosteleck´y,Phys.Rev.Lett.82,3572(1999).[11]M.P´e rez-Victoria,Phys.Rev.Lett.83,2518(1999).[12]M.P´e rez-Victoria,JHEP04,032(2001).[13]S.Coleman,S.L.Glashow,Phys.Rev.D59,116008(1999).[14]J.M.Chung,Phys.Lett.B461,138(1999).[15]J.M.Chung,P.Oh,Phys.Rev.D60,067702(1999).[16]W.F.Chen,Phys.Rev.D60,085007(1999).[17]J.M.Chung,Phys.Rev.D60,127901(1999).[18]A.A.Andrianov,P.Giacconi,R.Soldati,JHEP02,030(2002).[19]B.Altschul,Phys.Rev.D69,125009(2004).[20]R.Jackiw,Int.J.Mod.Phys.B14,2011(2000).[21]V.A.Kosteleck´y,Phys.Rev.D6*******(2004).7。

a r X i v :q u a n t -p h /0005035v 1 10 M a y 2000Tr.J.of Physics 1(2008),1–6.cT ¨UB ˙ITAK Inertial Force,Equivalence Principle andQuantum MechanicsMinoru OmoteDepartment of Physics ,Keio University ,Hiyoshi ,Yokohama Susumu KamefuchiAtomic Energy Research Institute ,Nihon University ,Kanda-Surugadai ,TokyoReceived February 1,2008AbstractOn the basis of a manifestly covariant formalism of non-relativistic quantum mechanics in general coordinate systems,proposed by us recently,we derive general expressions for inertial forces.The results enable us further to discuss,and to explain the validity of,the equivalence principle in non-relativistic quantum mechanics.1.IntroductionIn the usual formulation of non-relativistic quantum mechanics (NRQM)basic quantities such as fields or probability amplitudes are taken to be projective representations of the 4-dimensional Galilei group G 4,and the manifest covariance of the formalism is thereby impaired.As shown previously [1],this shortcoming can be removed,however,by employing,instead of G 5,a 5-dimensional form G 5for Galilei transformations [2].Here,basic quantities can be given as vector representations of the group G 5,so that the manifest covariance is achieved,just as Lorentz covariance is in relativistic quantum mechanics (RQM).In a recent paper [3]the result of [1]has been extended by considering more general transformations than G 5,i.e.,those which combine inertial with non-inertial coordinate systems.In this way we have obtained generally covariant equations for basic quantities that should hold in arbitrary coordinate systems,inertial or non-inertial.Naturally,the equations thus obtained contain the terms corresponding to inertial forces.And it is the purpose of this paper to discuss such terms,placing emphasis upon the problem of Einstein’s equivalence principle.Our conclusion then is that NRQM is compatible with this principle.1In what follows we shall exclusively be concerned with field equations for a non-relativistic field ψ( x ,t ).However,when ψsatisfies a basic equation of the Schr¨o dinger type such as (9)below,the probability amplitude ϕ( x ,t )for a single particle,arising from the quantized ψ,does also satisfy one and the same equation.Thus,all arguments about covariance of ψapply as well to ϕ.2.5-dimensional Transformations:G 5and G ′5We begin by noticing that the Lagrangian L =(1/2)m ˙ x 2of a free particle of mass m is not invariant under G 4: x → x ′=R x − v t,t ′=t with R being a 3×3orthogonal matrix.However,¯L ≡L −m ˙s remains invariant,provided the new variable s transforms under G 4as s →s ′=s +f with f ( x ,t )≡− v ·(R x )+(1/2) v 2t.Thus,instead of G 4we employ G 5,a central extension of G 4,such thatx ′i =R ijx j −viu (R i j x j)+1u 2x 4,(1)where (x 1,x 2,x 3)≡ x ,x 4≡ut,x 5≡s/u with [u ]=[v ],or x ′µ≡Λµνx ν.Under (1)ηµνx µx ν= x 2−2ts is invariant,where the metric tensor ηµν = ηµν is such that ηij =δij (i,j =1,2,3),η45=η54=−1and others =0.Note that ηµνx µx νwith x ±≡(x 4±x 5)/√u ˙˜A j x j +12u 2x 40˙˜A j (τ)˙˜A j (τ)dτ,(2)where ˜Ai ≡R j i A j .Obviously,this rule for x 5→x ′5is not unique ,and the above is the simplest choice:the arbitrariness arising from this will not affect the final results,however.The coordinate system S obtained from S 0by (2)is non-inertial in general.Thus non-inertial systems will be restricted hereafter to those specified by R (t )and A(t ).In the coordinate system S the metric tensor g ′µνis given byg ′ij =δij ,g ′i 4=0,g ′i 5=−1u 2R ji ¨˜A i x ′j;(3)2and the affine connectionΓ′λµν(x′)byΓ′i4j=1u2R i j¨R k j x′k−1u2R ij¨˜A j,Γ′544=−2u3R ij···˜A j x′i,(4)others=0.3.Generally Covariant Field EquationsFor illustration we consider in the following a scalarfieldφ(x)and a spinorfieldχ(x):the generalization to the case of Bargmann-Wignerfields of higher spins is straightforward.As shown in[3],a Klein-Gordon type equationηµν∂µ∂νφ(x)=0forφ(x)in S0takes, in S,the following form:g′µν∂′µ∂′νφ′(x′)=0.(5) Further,this should be supplemented by a subsidiary condition of invariant form:(i¯h∂′5−mu)φ′(x′)=0,(5′) with which to eliminate the extra degree of freedom.On the other hand,a spinorfieldχ(x)is to behave as a scalar under(2),i.e.,χ′(x′)=χ(x),and a Dirac-type equationγµ∂µχ(x)=0in S0is generalized,in S,toγ′µ(x′) ∂′µ+Γ′µ(x′) χ′(x′)=0.(6) The subsidiary condition forχ′(x′)can be of the same form as(5′).In the above theγ′-matrices and the spin connectionΓ′µ(x′)are given,in terms of the f¨u nfbein hµa(x′)and of4×4γ-matrices satisfyingγaγb+γbγa=2ηab,as follows:γ′µ(x′)=hµa(x′)γa,Γ′λ(x′)=1in accordance with(5′).Substituting this into(5)and using(3),we thenfind forψ′( x′,t′):i¯h ∂2m∇′2+H′inert ψ′( x′,t′)(9)withH′inert≡−mR jk¨˜A k x′j−i¯h˙Rℓj R k j x′k∂′ℓ.(9′) Here we confirm that for¨ A=0and˙R=0the equation(9)reduces to the free-field equation,and H′inert thus provides a general expression for inertial forces.Next,for the case ofχ′(x′)we may write,in terms of2-component spinorsψ′1andψ′2,χ′(x′)=exp −ims′u ˙R ijR k j x′k+1u ˙˜Ai,h54(x′)=12u2˙˜Ai˙˜A i,h44(x′)=h55(x′)=1,h i5(x′)=h4i(x′)=h45(x′)=0.(11)From(4)and(11)it follows,however,that D′λhνb(x′)=0,henceΓ′µ(x′)=0.Equation (6)together with(7),(11)and(10)then leads us toi¯h ∂2m∇′2+H′inert ψ′1( x′,t′)(12)with H′inert being the same as(9′).b)Taking a t′-dependent3×3orthogonal matrix¯R(t′)we now adopt f¨u nfbein h′µa such ash′i j=˜R j k h i k,h′5i=¯R i j h5j,h′µa=hµa for others(11′) with hµa’s given by(11).In this caseΓ′µ=0,and in factΓ′i=0,Γ′5=0andΓ′4= 14˙¯Rℓj˜Rkjǫkℓmσm,(13)which is to be regarded as a kind of inertial force.As remarked earlier,the situation is basically the same for general Bargmann-Wigner fields.Thus,summarizing the above results we can say that when a special hµa(or the gauge)is chosen,the inertial forces are all described by H′inert,irrespective of spins.44.2.ExamplesLet us now rewrite H′inert and H′spin for some special cases.a)H′inert:If we put R=I and A(t)=12σ.(16)Thus,if we choose Ω= Ω′,then the2nd term in H′inert plus H′spin provides− Ω·( L+ S), a result being true only in a special gauge,however.It is to be remarked here that the above results agree with the Newton approximations to the corresponding cases of general relativity.5.Equivalence PrincipleLet us start by considering an inertial system S0,where all quantities concerned are denoted by unprimed symbols.Thefieldφ(x)orχ(x)then satisfies the free-field equation (5)or(6)withΓµ=0.As shown in[3],the interaction with an external gravitational or Newton potentialΦ( x)can be introduced there by making a series of replacements:1)ηµν→hµa hνbηab,γµ→hµaγa;2)h54→h54−1u2Φ( x)∂5∂5 φ(x)=0,γµ∂µ−1∂tΨ( x,t)= −¯h2We next go over to a non-inertial system S via the transformation(2)with R(t)=I and A(t)=12 a t′2).Now,for definitenesslet us consider the case of uniform gravitation:Φ( x)= g· x( g:const.vector).Then, the equation forΨ′( x′,t′)≡ψ′( x′,t′)orψ′1( x′,t′)turns out,after a phase transformation exp(m a· g t3/6i¯h),to bei¯h ∂2m∇′2+m( g− a)· x Ψ′( x′,t′).(20)The above result has an important physical meaning.That is,if we choose a non-inertial system S such as a= g,then the gravitational effect will completely disappear there,being in accordance with Einstein’s equivalence principle.The situation is basically the same for general cases ofΦ( x)and also of Bargmann-Wignerfields.6.Conclusionsa)Fully utilizing the generally covariant formalism we have derived the general ex-pressions for inertial forces in a purely quantum-mechanical manner.b)NRQM is compatible with the equivalence principle.c)In our formalism other related problems can also be discussed within the framework of NRQM,i.e.,without recourse to any other theories like classical mechanics.d)So far as the covariance under space-time transformations is concerned,no differ-ence seems to exist formally between NRQM and RQM.We may therefore say that the former is a self-supporting,independent theory,which is comparable to the latter.References[1]M.Omote,S.Kamefuchi,.Y.Takahashi,Y.Ohnuki,Fortschr.Phys.,37(1989)933.[2]D.E.Soper,Classical Field Theory(John Wiley and Sons,New York,1976).[3]M.Omote,S.Kamefuchi,Proceedings of the International Symposium”QuantumTheory and Symmetries”,18-22July1999,Goslar,Germany,eds.H.D.Doebner et al.(World Scientific,Singapore),to be published.6。