共查询到20条相似文献,搜索用时 15 毫秒
1.
Denis Perrot 《Communications in Mathematical Physics》2001,218(2):373-391
We consider a smooth groupoid of the form Σ⋊Γ, where Σ is a Riemann surface and Γ a discrete pseudogroup acting on Σ by local
conformal diffeomorphisms. After defining a K-cycle on the crossed product C
0(Σ)⋊Γ generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem
of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism
group of the von Neumann algebra L
∞(Σ)⋊Γ.
Received: 1 February 2000 / Accepted: 3 December 2000 相似文献
2.
We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group
associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms
a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants
is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine
their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their
classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem
of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical
case with orthogonal quantum group symmetry. 相似文献
3.
Robert Oeckl 《Czechoslovak Journal of Physics》2001,51(12):1401-1406
The natural generalization of the notion of bundle in quantum geometry is that of bimodule. If the base space has quantum
group symmetries, one is particularly interested in bimodules covariant (equivariant) under these symmetries. Most attention
has so far been focused on the case with maximal symmetry — where the base space is a quantum group and the bimodules are
bicovariant. The structure of bicovariant bimodules is well understood through their correspondence with crossed modules.
We investigate the “next best” case — where the base space is a quantum homogeneous space and the bimodules are covariant.
We present a structure theorem that resembles the one for bicovariant bimodules. Thus, there is a correspondence between covariant
bimodules and a new kind of “crossed” modules which we define. The latter are attached to the pair of quantum groups which
defines the quantum homogeneous space.
We apply our structure theorem to differential calculi on quantum homogeneous spaces and discuss a related notion of induced
differential calculus.
Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June
2001.
This work was supported by a NATO fellowship grant. 相似文献
4.
A Quantum Version of Sanov's Theorem 总被引:3,自引:3,他引:0
We present a quantum version of Sanov's theorem focussing on a hypothesis testing aspect of the theorem: There exists a sequence
of typical subspaces for a given set Ψ of stationary quantum product states asymptotically separating them from another fixed
stationary product state. Analogously to the classical case, the separating rate on a logarithmic scale is equal to the infimum
of the quantum relative entropy with respect to the quantum reference state over the set Ψ. While in the classical case the
separating subsets can be chosen universally, in the sense that they depend only on the chosen set of i.i.d. processes, in
the quantum case the choice of the separating subspaces depends additionally on the reference state. 相似文献
5.
Steven Kenneth Kauffmann 《Foundations of Physics》2011,41(5):805-819
Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent
overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence
between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s
canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables,
but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the
sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase
space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets
for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also
unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan
quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner.
Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features
of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which
can manifest negative densities in phase space. 相似文献
6.
Many argued (Accardi and Fedullo, Pitowsky) that Kolmogorov's axioms of classical probability theory are incompatible with
quantum probabilities, and that this is the reason for the violation of Bell's inequalities. Szabó showed that, in fact, these
inequalities are not violated by the experimentally observed frequencies if we consider the real, “effective” frequencies.
We prove in this work a theorem which generalizes this results: “effective” frequencies associated to quantum events always
admit a Kolmogorovian representation, when these events are collected through different experimental setups, the choice of
which obeys a classical distribution. 相似文献
7.
We take a boundary-value approach to quantum amplitudes arising in gravitational collapse to a black hole. Pose boundary data
on initial and final space-like hypersurfaces Σ
F,I
, separated at spatial infinity by a Lorentzian proper-time interval T. Quantum amplitudes are calculated following Feynman's approach; rotate: T→|T|exp (−iθ) into the complex, where 0< θ≤π/2, and solve the corresponding well-posed complex classical boundary-value problem. We compute the classical Lorentzian action S
class and corresponding semi-classical quantum amplitude, proportional to exp (iS
class). To recover the Lorentzian amplitude, take the limit θ→ 0+ of the semi-classical amplitude. For the classical boundary-value problem with given perturbative boundary data, we compute
an effective spherically-symmetric energy-momentum tensor 〉 T
μν〈
EFF
, averaged over several wavelengths of the radiation, describing the averaged extra energy-momentum contribution in the Einstein
field equations, due to the perturbations. This takes the form of a null fluid, describing the radiation (of quantum origin)
streaming radially outwards. The classical space-time metric, in this region of the space time, is of Vaidya form, justifying
the adiabatic radial mode equations, for spins s = 0 and s = 2. 相似文献
8.
Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of product ruled surfaces
Σ
g
×S
2, where Σ
g
is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabó for g=1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space
of rank two stable bundles over a Riemann surface. (2) We derive the eigenvalue spectrum of the Fukaya–Floer cohomology of
Σ
g
×S
1.
Received: 22 July 1999 / Accepted: 12 June 2000 相似文献
9.
No Heading We show that the Dirac-von Neumann formalism for quantum mechanics can be obtained as an approximation of classical statistical
field theory. This approximation is based on the Taylor expansion (up to terms of the second order) of classical physical
variables – maps f : Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures
ρ on Ω having zero mean value and dispersion σ2(ρ) ≈ h. This viewpoint to the conventional quantum formalism gives the possibility to create generalized quantum formalisms based
on expansions of classical physical variables in the Taylor series up to terms of nth order and considering statistical states ρ having dispersion σ2(ρ) = hn (for n = 2 we obtain the conventional quantum formalism). 相似文献
10.
Lajos Molnár 《Communications in Mathematical Physics》2001,217(2):409-421
Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated,
for example, in the following way: Every bijective transformation on the set ℒ of all 1-dimensional subspaces of a Hilbert
space H which preserves the angle between the elements of ℒ is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of H with n∈ℕ fixed.
Received: 28 August 2000 / Accepted: 30 October 2000 相似文献
11.
The Hannay angles were introduced by Hannay as a means of measuring a holonomy effect in classical mechanics closely corresponding to the Berry phase in quantum mechanics. Using parameter-dependent momentum mappings we show that the Hannay angles are the holonomy of a natural connection. We generalize this effect to non-Abelian group actions and discuss non-integrable Hamiltonian systems. We prove an averaging theorem for phase space functions in the case of general multi-frequency dynamical systems which allows us to establish the almost adiabatic invariance of the Hannay angles. We conclude by giving an application to celestial mechanics.Supported by the Deutsche ForschungsgemeinschaftSupported by the Akademie der Wissenschaften zu Berlin 相似文献
12.
Sorkin’s recent proposal for a realist interpretation of quantum theory, the anhomomorphic logic or coevent approach, is based
on the idea of a “quantum measure” on the space of histories. This is a generalisation of the classical measure to one which
admits pair-wise interference and satisfies a modified version of the Kolmogorov probability sum rule. In standard measure
theory the measure on the base set Ω is normalised to one, which encodes the statement that “Ω happens”. Moreover, the Kolmogorov
sum rule implies that the measure of any subset A is strictly positive if and only if A cannot be covered by a countable collection of subsets of zero measure. In quantum measure theory on the other hand, simple
examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation, the quantum cover, which in addition to being a cover of A, satisfies the property that if the quantum measure of A is non-zero then this is also the case for at least one of the elements in the cover. Our work implies a non-triviality result
for the coevent interpretation for Ω of finite cardinality, and allows us to cast the Peres-Kochen-Specker theorem in terms
of quantum covers. 相似文献
13.
We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that lattice structure and classical operations uniquely determine quantum logic underlying Hilbert space. As a consequence of our result, recent proposals for a deduction theorem with quantum operations in an orthomodular lattice as well as a, substitution of quantum operations for the usual standard Hilbert space ones in quantum logic prove to be misleading. Quantum computer quantum logic is also discussed. 相似文献
14.
Andrei Khrennikov 《International Journal of Theoretical Physics》2012,51(8):2488-2502
This paper discusses a possible resolution of the nonobjectivity-nonlocality dilemma in quantum mechanics in the light of experimental tests of the Bell inequality for two entangled photons and a Bell-like inequality for a single neutron. My conclusion is that these experiments show that quantum mechanics is nonobjective: that is, the values of physical observables cannot be assigned to a system before measurement. Bell’s assumption of nonlocality has to be rejected as having no direct experimental confirmation, at least thus far. I also consider the relationships between nonobjectivity and contextuality. Specifically, I analyze the impact of the Kochen-Specker theorem on the problem of contextuality of quantum observables. I argue that, just as von Neumann’s “no-go” theorem, the Kochen-Specker theorem is based on assumptions that do not correspond to the real physical situation. Finally, I present a theory of measurement based on a classical, purely wave model (pre-quantum classical statistical field theory), a model that reproduces quantum probabilities. In this model continuous fields are transformed into discrete clicks of detectors. While this model is classical, it is nonobjective. In this case, nonobjectivity is the result of the dependence of experimental outcomes on the context of measurement, in accordance with Bohr’s view. 相似文献
15.
Weak measurement devices resemble band pass filters: they strengthen average values in the state space or equivalently filter
out some ‘frequencies’ from the conjugate Fourier transformed vector space. We thereby adjust a principle of classical communication
theory for the use in quantum computation. We discuss some of the computational benefits and limitations of such an approach,
including complexity analysis, some simple examples and a realistic not-so-weak approach. 相似文献
16.
Diederik Aerts Bart D’Hooghe Emmanuel Haven 《International Journal of Theoretical Physics》2010,49(12):2971-2990
We prove a theorem which shows that a collection of experimental data of probabilistic weights related to decisions with respect
to situations and their disjunction cannot be modeled within a classical probabilistic weight structure in case the experimental
data contain the effect referred to as the ‘disjunction effect’ in psychology. We identify different experimental situations
in psychology, more specifically in concept theory and in decision theory, and in economics (namely situations where Savage’s
Sure-Thing Principle is violated) where the disjunction effect appears and we point out the common nature of the effect. We
analyze how our theorem constitutes a no-go theorem for classical probabilistic weight structures for common experimental
data when the disjunction effect is affecting the values of these data. We put forward a simple geometric criterion that reveals
the non classicality of the considered probabilistic weights and we illustrate our geometrical criterion by means of experimentally
measured membership weights of items with respect to pairs of concepts and their disjunctions. The violation of the classical
probabilistic weight structure is very analogous to the violation of the well-known Bell inequalities studied in quantum mechanics.
The no-go theorem we prove in the present article with respect to the collection of experimental data we consider has a status
analogous to the well known no-go theorems for hidden variable theories in quantum mechanics with respect to experimental
data obtained in quantum laboratories. Our analysis puts forward a strong argument in favor of the validity of using the quantum
formalism for modeling the considered psychological experimental data as considered in this paper. 相似文献
17.
Bart D’Hooghe 《International Journal of Theoretical Physics》2010,49(12):3069-3084
We adopt an operational approach to quantum mechanics in which a physical system is defined by the mathematical structure
of its set of states and properties. We present a model in which the maximal change of state of the system due to interaction
with the measurement context is controlled by a parameter which corresponds with the number N of possible outcomes in an experiment. In the case N=2 the system reduces to a model for the spin measurements on a quantum spin-1/2 particle. In the limit N→∞ the system is classical, i.e. the experiments are deterministic and its set of properties is a Boolean lattice. For intermediate
situations the change of state due to measurement is neither ‘maximal’ (i.e. quantum) nor ‘zero’ (i.e. classical). We show
that two of the axioms used in Piron’s representation theorem for quantum mechanics are violated, namely the covering law
and weak modularity. Next, we discuss a modified version of the model for which it is even impossible to define an orthocomplementation
on the set of properties. Another interesting feature for the intermediate situations of this model is that the probability
of a state transition in general not only depends on the two states involved, but also on the measurement context which induces
the state transition. 相似文献
18.
Motivated by a recent use of Glauber dynamics for Monte Carlo simulations of path integral representation of quantum spin
models (Krzakala et al. in Phys. Rev. B 78(13):134428, 2008), we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph G. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality
of Peres and Winkler to the quantum setting. Then we consider the case when G is a regular b-ary tree and prove the same fast mixing results established in Martinelli et al. (Commun. Math. Phys. 250(2):301–334, 2004) for the classical Ising model. Our main tool is an inductive relation between conditional marginals (known as the “cavity
equation”) together with sharp bounds on the operator norm of the derivative at the stable fixed point. It is here that the
main difference between the quantum and the classical case appear, as the cavity equation is formulated here in an infinite
dimensional vector space, whereas in the classical case marginals belong to a one-dimensional space. 相似文献
19.
Alexandru Ivanov Dan Caragheorgheopol 《International Journal of Theoretical Physics》2010,49(12):3146-3152
In quantum mechanics, the Hilbert space formalism might be physically justified in terms of some axioms based on the orthomodular
lattice (OML) mathematical structure (Piron in Foundations of Quantum Physics, Benjamin, Reading, 1976). We intend to investigate the extent to which some fundamental physical facts can be described in the more general framework
of OMLs, without the support of Hilbert space-specific tools. We consider the study of lattice automorphisms properties as
a “substitute” for Hilbert space techniques in investigating the spectral properties of observables. This is why we introduce
the notion of spectral automorphism of an OML. Properties of spectral automorphisms and of their spectra are studied. We prove
that the presence of nontrivial spectral automorphisms allow us to distinguish between classical and nonclassical theories.
We also prove, for finite dimensional OMLs, that for every spectral automorphism there is a basis of invariant atoms. This
is an analogue of the spectral theorem for unitary operators having purely point spectrum. 相似文献
20.
In the classical theory of electromagnetism, the permittivity ε
0 and the permeability μ
0 of free space are constants whose magnitudes do not seem to possess any deeper physical meaning. By replacing the free space
of classical physics with the quantum notion of the vacuum, we speculate that the values of the aforementioned constants could
arise from the polarization and magnetization of virtual pairs in vacuum. A classical dispersion model with parameters determined
by quantum and particle physics is employed to estimate their values. We find the correct orders of magnitude. Additionally,
our simple assumptions yield an independent estimate for the number of charged elementary particles based on the known values
of ε
0 and μ
0 and for the volume of a virtual pair. Such an interpretation would provide an intriguing connection between the celebrated
theory of classical electromagnetism and the quantum theory in the weak-field limit. 相似文献