共查询到20条相似文献,搜索用时 46 毫秒
1.
A test space is a collection of non-empty sets, usually construed as the catalogue of (discrete) outcome sets associated with a family
of experiments. Subject to a simple combinatorial condition called algebraicity, a test space gives rise to a “quantum logic”—that is, an orthoalgebra. Conversely, all orthoalgebras arise naturally from
algebraic test spaces. In non-relativistic quantum mechanics, the relevant test space is the set ℱ F(H) of frames (unordered orthonormal bases) of a Hilbert space H. The corresponding logic is the usual one, i.e., the projection lattice L(H) of H. The test space ℱ F(H) has a strong symmetry property with respect to the unitary group of H, namely, that any bijection between two frames lifts to a unitary operator. In this paper, we consider test spaces enjoying
the same symmetry property relative to an action by a compact topological group. We show that such a test space, if algebraic,
gives rise to a compact, atomistic topological orthoalgebra. We also present a construction that generates such a test space
from purely group-theoretic data, and obtain a simple criterion for this test space to be algebraic.
PACS: 02.10.Ab; 02.20.Bb; 03.65.Ta. 相似文献
2.
Ashoke Sen 《General Relativity and Gravitation》2012,44(5):1207-1266
Logarithmic corrections to the extremal black hole entropy can be computed purely in terms of the low energy data—the spectrum
of massless fields and their interaction. The demand of reproducing these corrections provides a strong constraint on any
microscopic theory of quantum gravity that attempts to explain the black hole entropy. Using quantum entropy function formalism
we compute logarithmic corrections to the entropy of half BPS black holes in N=2{{\mathcal N}=2} supersymmetric string theories. Our results allow us to test various proposals for the measure in the OSV formula, and we
find agreement with the measure proposed by Denef and Moore if we assume their result to be valid at weak topological string
coupling. Our analysis also gives the logarithmic corrections to the entropy of extremal Reissner–Nordstrom black holes in
ordinary Einstein–Maxwell theory. 相似文献
3.
C. J. Isham 《International Journal of Theoretical Physics》1997,36(4):785-814
A major problem in the consistent-histories approach to quantum theory is contending with the potentially large number of
consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way.
However, in this paper the alternative approach is considered in which all consistent sets are kept, leading to a type of
‘many-world-views’ picture of the quantum theory. It is shown that a natural way of handling this situation is to employ the
theory of varying sets (presheafs) on the spaceB of all nontrivial Boolean subalgebras of the orthoalgebraUP of history propositions. This approach automatically includes the feature whereby probabilistic predictions are meaningful
only in the context of a consistent set of history propositions. More strikingly, it leads to a picture in which the ‘truth
values’ or ‘semantic values’ of such contextual predictions are not just two-valued (i.e., true and false) but instead lie
in a larger logical algebra—a Heyting algebra—whose structure is determined by the spaceB of Boolean subalgebras ofUP. This topos-theoretic structure thereby gives a coherent mathematical framework in which to understand the internal logic
of the many-world-views picture that arises naturally in the approach to quantum theory based on the ideas of consistent histories. 相似文献
4.
Considering the fundamental role symmetry plays throughout physics, it is remarkable how little attention has been paid to
it in the quantum-logical literature. In this paper, we discuss G-test spaces—that is, test spaces hosting an action by a group G—and their logics. The focus is on G-test spaces having strong homogeneity properties. After establishing some general results and exhibiting various specimens
(some of them exotic), we show that a sufficiently symmetric G-test space having an invariant, separating set of states with affine dimension n, is always representable in terms of a real Hilbert space of dimension n+1, in such a way that orthogonal outcomes are represented by orthogonal unit vectors. 相似文献
5.
William Demopoulos 《Foundations of Physics》2010,40(4):368-389
The quantum logical and quantum information-theoretic traditions have exerted an especially powerful influence on Bub’s thinking
about the conceptual foundations of quantum mechanics. This paper discusses both the quantum logical and information-theoretic
traditions from the point of view of their representational frameworks. I argue that it is at this level—at the level of its
framework—that the quantum logical tradition has retained its centrality to Bub’s thought. It is further argued that there
is implicit in the quantum information-theoretic tradition a set of ideas that mark a genuinely new alternative to the framework
of quantum logic. These ideas are of considerable interest for the philosophy of quantum mechanics, a claim which I defend
with an extended discussion of their application to our understanding of the philosophical significance of the no hidden variable
theorem of Kochen and Specker. 相似文献
6.
Michael H. Freedman Alexei Kitaev Zhenghan Wang 《Communications in Mathematical Physics》2002,227(3):587-603
Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates,
which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models”
having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the
state is discrete, H≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently
simulated on a quantum computer. The conclusion is two-fold:
1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”.
2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might
suggest a new quantum algorithm.
Received: 4 May 2001 / Accepted: 16 January 2002 相似文献
7.
M. Combescot O. Betbeder-Matibet 《The European Physical Journal B - Condensed Matter and Complex Systems》2007,55(1):63-76
The purpose of this paper is to show how the diagrammatic expansion
in fermion exchanges of scalar products of N-composite-boson
(“coboson”) states can be obtained in a practical way. The hard
algebra on which this expansion is based, will be given in an independent publication.
Due to the composite nature of the particles, the scalar products
of N-coboson states do not reduce to a set of Kronecker symbols, as
for elementary bosons, but contain subtle exchange terms between two or
more cobosons. These terms originate from Pauli exclusion between the
fermionic components of the particles. While our many-body
theory for composite bosons leads to write these scalar products as
complicated sums of products of “Pauli scatterings” between
two cobosons, they in fact correspond to fermion exchanges
between any number P of quantum particles, with
2 ≤P≤N. These P-body exchanges are nicely represented by the
so-called “Shiva diagrams”, which are topologically different from
Feynman diagrams, due to the intrinsic many-body nature of the Pauli
exclusion from which they originate. These Shiva diagrams in fact
constitute the novel part of our composite-exciton many-body theory
which was up to now missing to get its full
diagrammatic representation. Using them, we can now “see” through
diagrams the physics of any quantity in which enters N interacting
excitons — or more generally N composite bosons —, with fermion
exchanges included in an
exact — and transparent — way. 相似文献
8.
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
9.
10.
We study the ground state phase diagram of the two dimensional t — t′ — U Hubbard model concentrating on the competition between antiferro-, ferro-, and paramagnetism. It is known that unrestricted
Hartree–Fock- and quantum Monte Carlo calculations for this model predict inhomogeneous states in large regions of the parameter
space. Standard mean field theory, i.e., Hartree–Fock theory restricted to homogeneous states, fails to produce such inhomogeneous
phases. We show that a generalization of the mean field method to the grand canonical ensemble circumvents this problem and
predicts inhomogeneous states, represented by mixtures of homogeneous states, in large regions of the parameter space. We
present phase diagrams which differ considerably from previous mean field results but are consistent with, and extend, results
obtained with more sophisticated methods.
PACS: 71.10.Fd, 05.70.Fh, 75.50.Ee 相似文献
11.
Any unitary operation in quantum information processing can be implemented via a sequence of simpler steps — quantum gates.
However, actual implementation of a quantum gate is always imperfect and takes a finite time. Therefore, searching for a short
sequence of gates — efficient quantum circuit for a given operation, is an important task. We contribute to this issue by
proposing optimization of the well-known universal procedure proposed by Barenco et al. [Phys. Rev. A 52, 3457 (1995)]. We also created a computer program which realizes both Barenco’s decomposition and the proposed
optimization. Furthermore, our optimization can be applied to any quantum circuit containing generalized Toffoli gates, including
basic quantum gate circuits.
相似文献
12.
The Cohen—Glashow Very Special Relativity (VSR) algebra is defined as the part of the Lorentz algebra which upon addition
of CP or T invariance enhances to the full Lorentz group, plus the space—time translations. We show that noncommutative space—time,
in particular noncommutative Moyal plane, with light- like noncommutativity provides a robust mathematical setting for quantum field theories which are VSR invariant and hence set the stage for building
VSR invariant particle physics models. In our setting the VSR invariant theories are specified with a single deformation parameter,
the noncommutativity scale ╕NC. Preliminary analysis with the available data leads to ╕NC ≳ 1–10 TeV. 相似文献
13.
S. N. Mayburov 《International Journal of Theoretical Physics》2010,49(12):3192-3198
The quantum space-time with Dodson-Zeeman topological structure is studied. In its framework, the states of massive particle
m correspond to the elements of fuzzy ordered set (Foset), i.e. the fuzzy points. Due to their partial ordering, m space coordinate x acquires principal uncertainty σ
x
. Schroedinger formalism of Quantum Mechanics is derived from consideration of m evolution in fuzzy phase space with minimal number of additional axioms. The possible particle’s interactions on fuzzy manifold
are studied and shown to be gauge invariant. 相似文献
14.
15.
We report on an “anti-Gleason” phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical
observables can carry a non-linear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commuting
functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies
in the intersection of two seemingly remote mathematical theories—symplectic topology and the theory of topological quasi-states.
We use this quasi-state to estimate the error of the simultaneous measurement of non-commuting Hamiltonians.
M. Entov partially supported by E. and J. Bishop Research Fund and by the Israel Science Foundation grant # 881/06.
L. Polterovich partially supported by the Israel Science Foundation grant # 11/03. 相似文献
16.
In anomaly-free quantum field theories the integrand in the bosonic functional integral—the exponential of the effective action
after integrating out fermions—is often defined only up to a phase without an additional choice. We term this choice ``setting
the quantum integrand'. In the low-energy approximation to M-theory the E8-model for the C-field allows us to set the quantum integrand using geometric index theory. We derive mathematical results of independent
interest about pfaffians of Dirac operators in 8k+3 dimensions, both on closed manifolds and manifolds with boundary. These theorems are used to set the quantum integrand
of M-theory for closed manifolds and for compact manifolds with either temporal (global) or spatial (local) boundary conditions.
In particular, we show that M-theory makes sense on arbitrary 11-manifolds with spatial boundary, generalizing the construction
of heterotic M-theory on cylinders.
The work of D.F. is supported in part by NSF grant DMS-0305505. The work of G.M. is supported in part by DOE grant DE-FG02-96ER40949 相似文献
17.
Fabio Gavarini 《Czechoslovak Journal of Physics》2001,51(12):1330-1335
The “quantum duality principle” states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal
Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as
well. We extend this to a much more general result: namely, for any principal ideal domainR and for each primepεR we establish an “inner” Galois’ correspondence on the categoryHA of torsionless Hopf algebras overR, using two functors (fromHA to itself) such that the image of the first and the second is the full subcategory of those Hopf algebras which are commutative
and cocommutative, modulop, respectively (i.e., they are“quantum function algebras” (=QFA) and“quantum universal enveloping algebras” (=QUEA), atp, respectively). In particular we provide a machine to get two quantum groups — a QFA and a QUEA — out of any Hopf algebraH over a fieldk: apply the functors tok[ν] ⊗k H forp=ν.
A relevant example occurring in quantum electro-dynamics is studied in some detail.
Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June
2001 相似文献
18.
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
19.
David Finkelstein 《International Journal of Theoretical Physics》1982,21(6-7):489-503
The mathematical language presently used for quantum physics is a high-level language. As a lowest-level or basic language
I construct a quantum set theory in three stages: (1) Classical set theory, formulated as a Clifford algebra of “S numbers” generated by a single monadic operation, “bracing,” Br = {…}. (2) Indefinite set theory, a modification of set theory
dealing with the modal logical concept of possibility. (3) Quantum set theory. The quantum set is constructed from the null
set by the familiar quantum techniques of tensor product and antisymmetrization. There are both a Clifford and a Grassmann
algebra with sets as basis elements. Rank and cardinality operators are analogous to Schroedinger coordinates of the theory,
in that they are multiplication or “Q-type” operators. “P-type” operators analogous to Schroedinger momenta, in that they transform theQ-type quantities, are bracing (Br), Clifford multiplication by a setX, and the creator ofX, represented by Grassmann multiplicationc(X) by the setX. Br and its adjoint Br* form a Bose-Einstein canonical pair, andc(X) and its adjointc(X)* form a Fermi-Dirac or anticanonical pair. Many coefficient number systems can be employed in this quantization. I use the
integers for a discrete quantum theory, with the usual complex quantum theory as limit. Quantum set theory may be applied
to a quantum time space and a quantum automaton.
This material is based upon work supported in part by NSF Grant No. PHY8007921. 相似文献
20.
D. J. Foulis R. J. Greechie G. T. Rüttimann 《International Journal of Theoretical Physics》1993,32(10):1675-1690
Test spaces are mathematical structures that underlie quantum logics in much the same way that Hilbert space underlies standard quantum logic. In this paper, we give a coherent account of the basic theory of test spaces and show how they provide an infrastructure for the study of quantum logics. IfL is the quantum logic for a physical systemL, then a support inL may be interpreted as the set of all propositions that are possible whenL is in a certain state. We present an analog for test spaces of the notion of a quantum-logical support and launch a study of the classification of supports. 相似文献