results for au:Tsutsui_I in:quant-ph

- Mar 20 2017 quant-ph arXiv:1703.06068v1We propose a general framework of the quantum/quasi-classical transformations by introducing the concept of quasi-joint-spectral distribution (QJSD). Specifically, we show that the QJSDs uniquely yield various pairs of quantum/quasi-classical transformations, including the Wigner-Weyl transform. We also discuss the statistical behaviour of combinations of generally non-commutin quantum observables by introducing the concept of quantum correlations and conditional expectations defined analogously to the classical counterpart. Based on these, Aharonov's weak value is given a statistical interpretation as one realisation of the quantum conditional expectations furnished in our formalism.
- Jul 22 2016 quant-ph arXiv:1607.06406v1We show that the joint behaviour of an arbitrary pair of quantum observables can be described by quasi-probabilities, which are extensions of the standard probabilities used for describing the behaviour of a single observable. The physical situations that require these quasi-probabilities arise when one considers quantum measurement of an observable conditioned by some other variable, with the notable example being the weak measurement employed to obtain Aharonov's weak value. Specifically, we present a general prescription for the construction of quasi-joint-probability (QJP) distributions associated with a given pair of observables. These QJP distributions are introduced in two complementary approaches: one from a bottom-up, strictly operational construction realised by examining the mathematical framework of the conditioned measurement scheme, and the other from a top-down viewpoint realised by applying the results of spectral theorem for normal operators and its Fourier transforms. It is then revealed that, for a pair of simultaneously measurable observables, the QJP distribution reduces to their unique standard joint-probability distribution, whereas for a non-commuting pair there exists an inherent indefiniteness in the choice, admitting a multitude of candidates that may equally be used for describing their joint behaviour. In the course of our argument, we find that the QJP distributions furnish the space of operators with their characteristic geometric structures such that the orthogonal projections and inner products of observables can, respectively, be given statistical interpretations as `conditionings' and `correlations'. The weak value $A_{w}$ for an observable $A$ is then given a geometric/statistical interpretation as either the orthogonal projection of $A$ onto the subspace generated by another observable $B$, or equivalently, as the conditioning of $A$ given $B$.
- Mar 01 2016 quant-ph arXiv:1602.08872v1We clarify the significance of quasiprobability (QP) in quantum mechanics that is relevant in describing physical quantities associated with a transition process. Our basic quantity is Aharonov's weak value, from which the QP can be defined up to a certain ambiguity parameterized by a complex number. Unlike the conventional probability, the QP allows us to treat two noncommuting observables consistently, and this is utilized to embed the QP in Bohmian mechanics such that its equivalence to quantum mechanics becomes more transparent. We also show that, with the help of the QP, Bohmian mechanics can be recognized as an ontological model with a certain type of contextuality.
- Nov 26 2015 quant-ph arXiv:1511.08052v2We present a versatile inequality of uncertainty relations which are useful when one approximates an observable and/or estimates a physical parameter based on the measurement of another observable. It is shown that the optimal choice for proxy functions used for the approximation is given by Aharonov's weak value, which also determines the classical Fisher information in parameter estimation, turning our inequality into the genuine Cramér-Rao inequality. Since the standard form of the uncertainty relation arises as a special case of our inequality, and since the parameter estimation is available as well, our inequality can treat both the position-momentum and the time-energy relations in one framework albeit handled differently.
- Dec 03 2014 quant-ph arXiv:1412.0916v3The notion of trajectory of an individual particle is strictly inhibited in quantum mechanics because of the uncertainty principle. Nonetheless, the weak value, which has been proposed as a novel and measurable quantity definable to any quantum observable, can offer a possible description of trajectory on account of its statistical nature of the value. In this paper, we explore the physical significance provided by this weak trajectory by considering various situations where interference takes place simultaneously with the observation of particles, that is, in prototypical quantum situations for which no classical treatment is available. These include the double slit experiment and Lloyd's mirror, where in the former case it is argued that the real part of the weak trajectory describes an average over the possible classical trajectories involved in the process, and that the imaginary part is related to the variation of interference. It is shown that this average interpretation of the weak trajectory holds universally under the complex probability defined from the given transition process. These features remain essentially unaltered in the case of Lloyd's mirror where interference occurs with a single slit.
- Oct 06 2014 quant-ph arXiv:1410.0787v1The weak value, introduced by Aharonov et al. to extend the conventional scope of physical observables in quantum mechanics, is an intriguing concept which sheds new light on quantum foundations and is also useful for precision measurement, but it poses serious questions on its physical meaning due to the unconventional features including the complexity of its value. In this paper we point out that the weak value has a direct connection with the wave-particle duality, in the sense that the wave nature manifests itself in the imaginary part while the particle nature in the real part. This is illustrated by the double slit experiment, where we argue, with no conflict with complementarity, that the trajectory of the particle can be inferred based on the weak value without destroying the interference.
- May 14 2013 quant-ph arXiv:1305.2721v1Aharonov's weak value, which is a physical quantity obtainable by weak measurement, admits amplification and hence is deemed to be useful for precision measurement. We examine the significance of the amplification based on the uncertainty of measurement, and show that the trade-offs among the three (systematic, statistical and nonlinear) components of the uncertainty inherent in the weak measurement will set an upper limit on the usable amplification. Apart from the Gaussian state models employed for demonstration, our argument is completely general; it is free from approximation and valid for arbitrary observables $A$ and couplings $g$.
- Feb 15 2013 quant-ph arXiv:1302.3509v1Entanglement is known to be a relative notion, defined with respect to the choice of physical observables to be measured (i.e., the measurement setup used). This implies that, in general, the same state can be both separable and entangled for different measurement setups, but this does not exclude the existence of states which are separable (or entangled) for all possible setups. We show that for systems of bosonic particles there indeed exist such universally separable states: they are i.i.d. pure states. In contrast, there is no such state for fermionic systems with a few exceptional cases. We also find that none of the fermionic and bosonic systems admits universally entangled states.
- Oct 05 2012 quant-ph arXiv:1210.1298v2We present a complex probability measure relevant for double (pairs of) states in quantum mechanics, as an extension of the standard probability measure for single states that underlies Born's statistical rule. When the double states are treated as the initial and final states of a quantum process, we find that Aharonov's weak value, which has acquired a renewed interest as a novel observable quantity inherent in the process, arises as an expectation value associated with the probability measure. Despite being complex, our measure admits the physical interpretation as mixed processes, i.e., an ensemble of processes superposed with classical probabilities.
- Approximated formulae for real quasi-momentum and the associated energy spectrum are presented for one-dimensional Bose gas with weak attractive contact interactions. On the basis of the energy spectrum, we obtain the equation of state in the high temperature region, which is found to be the van der Waals equation without volume correction.
- Sep 22 2010 quant-ph arXiv:1009.4147v3We present a general criterion for entanglement of N indistinguishable particles decomposed into arbitrary s subsystems based on the unambiguous measurability of correlation. Our argument provides a unified viewpoint on the entanglement of indistinguishable particles, which is still unsettled despite various proposals made mainly for the s = 2 case. Even though entanglement is defined only with reference to the measurement setup, we find that the so-called i.i.d. states form a special class of bosonic states which are universally separable.
- Jun 22 2010 quant-ph arXiv:1006.3920v2We show that the various intermediate states appearing in the process of one-way computation at a given step of measurement are all equivalent modulo local unitary transformations. This implies, in particular, that all those intermediate states share the same entanglement irrespective of the measurement outcomes, indicating that the process of one-way computation is essentially unique with respect to local quantum operations.
- Oct 12 2009 quant-ph arXiv:0910.1658v2We present a criterion of separability for arbitrary s partitions of N-particle fermionic pure states. We show that, despite the superficial non-factorizability due to the antisymmetry required by the indistinguishability of the particles, the states which meet our criterion have factorizable correlations for a class of observables which are specified consistently with the states. The separable states and the associated class of observables share an orthogonal structure, whose non-uniqueness is found to be intrinsic to the multi-partite separability and leads to the non-transitivity in the factorizability in general. Our result generalizes the previous result obtained by Ghirardi et. al. [J. Stat. Phys. 108 (2002) 49] for the s = 2 and s = N case.
- Jun 16 2009 quant-ph arXiv:0906.2626v2Boundary effects in quantum mechanics are examined by considering a partition wall inserted at the centre of a harmonic oscillator system. We put an equal number of particles on both sides of the impenetrable wall keeping the system under finite temperatures. When the wall admits distinct boundary conditions on the two sides, then a net force is induced on the wall. We study the temperature behaviour of the induced force both analytically and numerically under the combination of the Dirichlet and the Neumann conditions, and determine its scaling property for two statistical cases of the particles: fermions and bosons. We find that the force has a nonvanishing limit at zero temperature T = 0 and exhibits scalings characteristic to the statistics of the particles. We also see that for higher temperatures the force decreases according to 1/sqrtT, in sharp contrast to the case of the infinite potential well where it diverges according to sqrtT. The results suggest that, if such a nontrivial partition wall can be realized, it may be used as a probe to examine the profile of the potentials and the statistics of the particles involved.
- Sep 12 2008 quant-ph arXiv:0809.1932v5We present a family of entanglement measures R_m which act as indicators for separability of n-qubit quantum states into m subsystems for arbitrary 2 ≤m ≤n. The measure R_m vanishes if the state is separable into m subsystems, and for m = n it gives the Meyer-Wallach measure while for m = 2 it reduces, in effect, to the one introduced recently by Love et al. The measures R_m are evaluated explicitly for the GHZ state and the W state (and its modifications, the W_k states) to show that these globally entangled states exhibit rather distinct behaviors under the measures, indicating the utility of the measures R_m for characterizing globally entangled states as well.
- May 26 2008 quant-ph arXiv:0805.3625v3Entanglement of multipartite systems is studied based on exchange symmetry under the permutation group S_N. With the observation that symmetric property under the exchange of two constituent states and their separability are intimately linked, we show that anti-symmetric (fermionic) states are necessarily globally entangled, while symmetric (bosonic) states are either globally entangled or fully separable and possess essentially identical states in all the constituent systems. It is also shown that there cannot exist a fully separable state which is orthogonal to all symmetric states, and that full separability of states does not survive under total symmetrization unless the states are originally symmetric. Besides, anyonic states permitted under the braid group B_N should also be globally entangled. Our results reveal that exchange symmetry is actually sufficient for pure states to become globally entangled or fully separable.
- May 26 2008 quant-ph arXiv:0805.3632v2We study the possibility of testing local realistic theory (LRT), envisioned implicitly by Einstein, Podolsky and Rosen in 1935, based on the Bell inequality for the correlations in the decay modes of entangled K or B-mesons. It is shown that such a test is possible for a restricted class of LRT, despite the passive nature of decay events and/or the non-unitary treatment of the correlations which invalidate the test for general LRT. Unfortunately, the present setup of the KEKB (Belle) experiment, where the coherence of entangled B-mesons has been confirmed recently, does not admit such a test due to the inability of determining the decay times of the entangled pairs separately. The indeterminacy also poses a problem for ensuring the locality of the test, indicating that improvement to resolve the indeterminacy is crucial for the test of LRT.
- Feb 19 2007 quant-ph arXiv:quant-ph/0702167v2We present a novel formulation of quantum game theory based on the Schmidt decomposition, which has the merit that the entanglement of quantum strategies is manifestly quantified. We apply this formulation to 2-player, 2-strategy symmetric games and obtain a complete set of quantum Nash equilibria. Apart from those available with the maximal entanglement, these quantum Nash equilibria are extensions of the Nash equilibria in classical game theory. The phase structure of the equilibria is determined for all values of entanglement, and thereby the possibility of resolving the dilemmas by entanglement in the game of Chicken, the Battle of the Sexes, the Prisoners' Dilemma, and the Stag Hunt, is examined. We find that entanglement transforms these dilemmas with each other but cannot resolve them, except in the Stag Hunt game where the dilemma can be alleviated to a certain degree.
- Dec 04 2006 quant-ph arXiv:quant-ph/0612011v3The paper wishes to demonstrate that, in quantum systems with boundaries, different boundary conditions can lead to remarkably different physical behaviour. Our seemingly innocent setting is a one dimensional potential well that is divided into two halves by a thin separating wall. The two half wells are populated by the same type and number of particles and are kept at the same temperature. The only difference is in the boundary condition imposed at the two sides of the separating wall, which is the Dirichlet condition from the left and the Neumann condition from the right. The resulting different energy spectra cause a difference in the quantum statistically emerging pressure on the two sides. The net force acting on the separating wall proves to be nonzero at any temperature and, after a weak decrease in the low temperature domain, to increase and diverge with a square-root-of-temperature asymptotics for high temperatures. These observations hold for both bosonic and fermionic type particles, but with quantitative differences. We work out several analytic approximations to explain these differences and the various aspects of the found unexpectedly complex picture.
- Feb 23 2006 quant-ph arXiv:quant-ph/0602178v2Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners' Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.
- Apr 01 2005 quant-ph cond-mat.other math-ph math.MP nlin.CD nlin.SI physics.bio-ph arXiv:quant-ph/0503233v2A simple and general formulation of the quantum game theory is presented, accommodating all possible strategies in the Hilbert space for the first time. The theory is solvable for the two strategy quantum game, which is shown to be equivalent to a family of classical games supplemented by quantum interference. Our formulation gives a clear perspective to understand why and how quantum strategies outmaneuver classical strategies. It also reveals novel aspects of quantum games such as the stone-scissor-paper phase sub-game and the fluctuation-induced moderation.
- We quantize the 1-dimensional 3-body problem with harmonic and inverse square pair potential by separating the Schrödinger equation following the classic work of Calogero, but allowing all possible self-adjoint boundary conditions for the angular and radial Hamiltonians. The inverse square coupling constant is taken to be $g=2\nu (\nu-1)$ with ${1/2} <\nu< {3/2}$ and then the angular Hamiltonian is shown to admit a 2-parameter family of inequivalent quantizations compatible with the dihedral $D_6$ symmetry of its potential term $9 \nu (\nu -1)/\sin^2 3\phi$. These are parametrized by a matrix $U\in U(2)$ satisfying $\sigma_1 U \sigma_1 = U$, and in all cases we describe the qualitative features of the angular eigenvalues and classify the eigenstates under the $D_6$ symmetry and its $S_3$ subgroup generated by the particle exchanges. The angular eigenvalue $\lambda$ enters the radial Hamiltonian through the potential $(\lambda -{1/4})/r^2$ allowing a 1-parameter family of self-adjoint boundary conditions at $r=0$ if $\lambda <1$. For $0<\lambda<1$ our analysis of the radial Schrödinger equation is consistent with previous results on the possible energy spectra, while for $\lambda <0$ it shows that the energy is not bounded from below rejecting those $U$'s admitting such eigenvalues as physically impermissible. The permissible self-adjoint angular Hamiltonians include, for example, the cases $U=\pm {\bf 1}_2, \pm \sigma_1$, which are explicitly solvable and are presented in detail. The choice $U=-{\bf 1}_2$ reproduces Calogero's quantization, while for the choice $U=\sigma_1$ the system is smoothly connected to the harmonic oscillator in the limit $\nu \to 1$.
- We show that the U(2) family of point interactions on a line can be utilized to provide the U(2) family of qubit operations for quantum information processing. Qubits are realized as localized states in either side of the point interaction which represents a controllable gate. The manipulation of qubits proceeds in a manner analogous to the operation of an abacus. Keywords: quantum computation, quantum contact interaction, quantum wire
- Dec 04 2003 quant-ph arXiv:quant-ph/0312028v1Defects or junctions in materials serve as a source of interactions for particles, and in idealized limits they may be treated as singular points yielding contact interactions. In quantum mechanics, these singularities accommodate an unexpectedly rich structure and thereby provide a variety of physical phenomena, especially if their properties are controlled properly. Based on our recent studies, we present a brief review on the physical aspects of such quantum singularities in one dimension. Among the intriguing phenomena that the singularities admit, we mention strong vs weak duality, supersymmetry, quantum anholonomy (Berry phase), and a copying process by anomalous caustics. We also show that a partition wall as a singularity in a potential well can give rise to a quantum force which exhibits an interesting temperature behavior characteristic to the particle statistics.
- Sep 12 2003 quant-ph arXiv:quant-ph/0309095v1We calculate the quantum statistical force acting on a partition wall that divides a one dimensional box into two halves. The two half boxes contain the same (fixed) number of noninteracting bosons, are kept at the same temperature, and admit the same boundary conditions at the outer walls; the only difference is the distinct boundary conditions imposed at the two sides of the partition wall. The net force acting on the partition wall is nonzero at zero temperature and remains almost constant for low temperatures. As the temperature increases, the force starts to decrease considerably, but after reaching a minimum it starts to increase, and tends to infinity with a square-root-of-temperature asympotics. This example demonstrates clearly that distinct boundary conditions cause remarkable physical effects for quantum systems.
- We investigate the spectral and symmetry properties of a quantum particle moving on a circle with a pointlike singularity (or point interaction). We find that, within the U(2) family of the quantum mechanically allowed distinct singularities, a U(1) equivalence (of duality-type) exists, and accordingly the space of distinct spectra is U(1) x [SU(2)/U(1)], topologically a filled torus. We explore the relationship of special subfamilies of the U(2) family to corresponding symmetries, and identify the singularities that admit an N = 2 supersymmetry. Subfamilies that are distinguished in the spectral properties or the WKB exactness are also pointed out. The spectral and symmetry properties are also studied in the context of the circle with two singularities, which provides a useful scheme to discuss the symmetry properties on a general basis.
- We provide a systematic study on the possibility of supersymmetry (SUSY) for one dimensional quantum mechanical systems consisting of a pair of lines $\R$ or intervals [-l, l] each having a point singularity. We consider the most general singularities and walls (boundaries) at $x = \pm l$ admitted quantum mechanically, using a U(2) family of parameters to specify one singularity and similarly a U(1) family of parameters to specify one wall. With these parameter freedoms, we find that for a certain subfamily the line systems acquire an N = 1 SUSY which can be enhanced to N = 4 if the parameters are further tuned, and that these SUSY are generically broken except for a special case. The interval systems, on the other hand, can accommodate N = 2 or N = 4 SUSY, broken or unbroken, and exhibit a rich variety of (degenerate) spectra. Our SUSY systems include the familiar SUSY systems with the Dirac $\delta(x)$-potential, and hence are extensions of the known SUSY quantum mechanics to those with general point singularities and walls. The self-adjointness of the supercharge in relation to the self-adjointness of the Hamiltonian is also discussed.
- We study the possibility of supersymmetry (SUSY) in quantum mechanics in one dimension under the presence of a point singularity. The system considered is the free particle on a line R or on the interval [-l, l] where the point singularity lies at x = 0. In one dimension, the singularity is known to admit a U(2) family of different connection conditions which include as a special case the familiar one that arises under the Dirac delta-potential. Similarly, each of the walls at x = l and x = -l admits a U(1) family of boundary conditions including the Dirichlet and the Neumann boundary conditions. Under these general connection/boundary conditions, the system is shown to possess an N = 1 or N = 2 SUSY for various choices of the singularity and the walls, and the SUSY is found to be `good' or `broken' depending on the choices made. We use the supercharge which allows for a constant shift in the energy, and argue that if the system is supersymmetric then the supercharge is self-adjoint on states that respect the connection/boundary conditions specified by the singularity.
- To describe a quantum system whose potential is divergent at one point, one must provide proper connection conditions for the wave functions at the singularity. Generalizing the scheme used for point interactions in one dimension, we present a set of connection conditions which are well-defined even if the wave functions and/or their derivatives are divergent at the singularity. Our generalized scheme covers the entire U(2) family of quantizations (self-adjoint Hamiltonians) admitted for the singular system. We use this scheme to examine the spectra of the Coulomb potential $V(x) = - e^2 / | x |$ and the harmonic oscillator with square inverse potential $V(x) = (m \omega^2 / 2) x^2 + g/x^2$, and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra. We further show that, for any parity invariant singular potentials $V(-x) = V(x)$, the spectrum is determined solely by the eigenvalues of the characteristic matrix $U \in U(2)$.
- Feb 07 2002 quant-ph arXiv:quant-ph/0202037v4Quantization of a harmonic oscillator with inverse square potential $V(x)=(m{\omega^2}/2){x^2}+g/{x^2}$ on the line $-\infty<x<\infty$ is re-examined. It is shown that, for $0<g<3{\hbar^2}/(8m)$, the system admits a U(2) family of inequivalent quantizations allowing for quantum tunneling through the potenatial barrier at $x=0$. In the family is a distinguished quantization which reduces smoothly to the harmonic oscillator as $g\to 0$, in contrast to the conventional quantization applied to the Calogero model which prohibits the tunneling and has no such limit. The tunneling renders the classical caustics anomalous at the quantum level, leading to the possibility of copying an arbitrary state from one side $x>0$, say, to the other $x<0$.
- Nov 12 2001 quant-ph arXiv:quant-ph/0111057v2We investigate the system of a particle moving on a half line x >= 0 under the general walls at x = 0 that are permitted quantum mechanically. These quantum walls, characterized by a parameter L, are shown to be realized as a limit of regularized potentials. We then study the classical aspects of the quantum walls, by seeking a classical counterpart which admits the same time delay in scattering with the quantum wall, and also by examining the WKB-exactness of the transition kernel based on the regularized potentials. It is shown that no classical counterpart exists for walls with L < 0, and that the WKB-exactness can hold only for L = 0 and L = infinity.
- We study the spectral properties of one-dimensional quantum wire with a single defect. We reveal the existence of the non-trivial topological structures in the spectral space of the system, which are behind the exotic quantum phenomena that have lately been found in the system.
- The Schroedinger operator with point interaction in one dimension has a U(2) family of self-adjoint extensions. We study the spectrum of the operator and show that (i) the spectrum is uniquely determined by the eigenvalues of the matrix U belonging to U(2) that characterizes the extension, and that (ii) the space of distinct spectra is given by the orbifold T^2/Z_2 which is a Moebius strip with boundary. We employ a parametrization of U(2) that admits a direct physical interpretation and furnishes a coherent framework to realize the spectral duality and anholonomy recently found. This allows us to find that (iii) physically distinct point interactions form a three-parameter quotient space of the U(2) family.
- We analyze the spectral structure of the one dimensional quantum mechanical system with point interaction, which is known to be parametrized by the group U(2). Based on the classification of the interactions in terms of symmetries, we show, on a general ground, how the fermion-boson duality and the spectral anholonomy recently discovered can arise. A vital role is played by a hidden su(2) formed by a certain set of discrete transformations, which becomes a symmetry if the point interaction belongs to a distinguished U(1) subfamily in which all states are doubly degenerate. Within the U(1), there is a particular interaction which admits the interpretation of the system as a supersymmetric Witten model.
- We study systems with parity invariant contact interactions in one dimension. The model analyzed is the simplest nontrivial one --- a quantum wire with a point defect --- and yet is shown to exhibit exotic phenomena, such as strong vs weak coupling duality and spiral anholonomy in the spectral flow. The structure underlying these phenomena is SU(2), which arises as accidental symmetry for a particular class of interactions.
- The quantum dynamics of a free particle on a circle with point interaction is described by a U(2) family of self-adjoint Hamiltonians. We provide a classification of the family by introducing a number of subfamilies and thereby analyze the spectral structure in detail. We find that the spectrum depends on a subset of U(2) parameters rather than the entire U(2) needed for the Hamiltonians, and that in particular there exists a subfamily in U(2) where the spectrum becomes parameter-independent. We also show that, in some specific cases, the WKB semiclassical approximation becomes exact (modulo phases) for the system.
- We study classical and quantum caustics for system with quadratic Lagrangian. Gaussian slit experiment is examined and it is pointed out that the focusing around caustics is stabilized against initial momentum fluctuations by quantum effect.
- We study caustics in classical and quantum mechanics for systems with quadratic Lagrangians. We derive a closed form of the transition amplitude on caustics and discuss their physical implications in the Gaussian slit (gedanken-)experiment. Application to the quantum mechanical rotor casts doubt on the validilty of Jevicki's correspondence hypothesis which states that in quantum mechanics, stationary points(instantons) arise as simple poles.
- A path-integral quantization on a homogeneous space G/H is proposed based on the guiding principle `first lift to G and then project to G/H'. It is then shown that this principle gives a simple procedure to obtain the inequivalent quantizations (superselection sectors) along with the holonomy factor (induced gauge field) found earlier by algebraic approaches. We also prove that the resulting matrix-valued path-integral is physically equivalent to the scalar-valued path-integral derived in the Dirac approach, and thereby present a unified viewpoint to discuss the basic features of quantizing on $G/H$ obtained in various approaches so far.
- The path integral on a homogeneous space $ G/H $ is constructed, based on the guiding principle `first lift to $ G $ and then project to $ G/H $'. It is then shown that this principle admits inequivalent quantizations inducing a gauge field (the canonical connection) on the homogeneous space, and thereby reproduces the result obtained earlier by algebraic approaches.