1,948 research outputs found
Complex Moduli of Physical Quanta
Classical mechanics can be formulated using a symplectic structure on
classical phase space, while quantum mechanics requires a
complex-differentiable structure on that same space. Complex-differentiable
structures on a given real manifold are often not unique. This letter is
devoted to analysing the dependence of the notion of a quantum on the
complex-differentiable structure chosen on classical phase space.Comment: Some minor cosmetic changes made and new refs. adde
Noncommutative Manifolds from the Higgs Sector of Coincident D-Branes
The Higgs sector of the low-energy physics of n of coincident D-branes
contains the necessary elements for constructing noncommutative manifolds. The
coordinates orthogonal to the coincident branes, as well as their conjugate
momenta, take values in the Lie algebra of the gauge group living inside the
brane stack. In the limit when n=\infty (and in the absence of orientifolds),
this is the unitary Lie algebra u(\infty). Placing a smooth manifold K
orthogonally to the stack of coincident D-branes one can construct a
noncommutative C*-algebra that provides a natural definition of a
noncommutative partner for the manifold K.Comment: 10 page
Quantum-Mechanical Dualities from Classical Phase Space
The geometry of the classical phase space C of a finite number of degrees of
freedom determines the possible duality symmetries of the corresponding quantum
mechanics. Under duality we understand the relativity of the notion of a
quantum with respect to an observer on C. We illustrate this property
explicitly in the case when classical phase space is complex n-dimensional
projective space. We also provide some examples of classical dynamics that
exhibit these properties at the quantum level.Comment: 8 pages, Late
Gerbes and Heisenberg's Uncertainty Principle
We prove that a gerbe with a connection can be defined on classical phase
space, taking the U(1)-valued phase of certain Feynman path integrals as Cech
2-cocycles. A quantisation condition on the corresponding 3-form field strength
is proved to be equivalent to Heisenberg's uncertainty principle.Comment: 12 pages, 1 figure available upon reques
Quantum-Mechanical Dualities on the Torus
On classical phase spaces admitting just one complex-differentiable
structure, there is no indeterminacy in the choice of the creation operators
that create quanta out of a given vacuum. In these cases the notion of a
quantum is universal, i.e., independent of the observer on classical phase
space. Such is the case in all standard applications of quantum mechanics.
However, recent developments suggest that the notion of a quantum may not be
universal. Transformations between observers that do not agree on the notion of
an elementary quantum are called dualities. Classical phase spaces admitting
more than one complex-differentiable structure thus provide a natural framework
to study dualities in quantum mechanics. As an example we quantise a classical
mechanics whose phase space is a torus and prove explicitly that it exhibits
dualities.Comment: New examples added, some precisions mad
Quantum Dynamics on the Worldvolume from Classical su(n) Cohomology
A key symmetry of classical -branes is invariance under worldvolume
diffeomorphisms. Under the assumption that the worldvolume, at fixed values of
the time, is a compact, quantisable K\"ahler manifold, we prove that the Lie
algebra of volume-preserving diffeomorphisms of the worldvolume can be
approximated by , for . We also prove, under the same
assumptions regarding the worldvolume at fixed time, that classical Nambu
brackets on the worldvolume are quantised by the multibrackets corresponding to
cocycles in the cohomology of the Lie algebra .Comment: This is a contribution to the Special Issue on Deformation
Quantization, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Semiclassical Expansions, the Strong Quantum Limit, and Duality
We show how to complement Feynman's exponential of the action so that it
exhibits a Z_2 duality symmetry. The latter illustrates a relativity principle
for the notion of quantum versus classical.Comment: 5 pages, references adde
Quantum States from Tangent Vectors
We argue that tangent vectors to classical phase space give rise to quantum
states of the corresponding quantum mechanics. This is established for the case
of complex, finite-dimensional, compact, classical phase spaces C, by
explicitly constructing Hilbert-space vector bundles over C. We find that these
vector bundles split as the direct sum of two holomorphic vector bundles: the
holomorphic tangent bundle T(C), plus a complex line bundle N(C). Quantum
states (except the vacuum) appear as tangent vectors to C. The vacuum state
appears as the fibrewise generator of N(C). Holomorphic line bundles N(C) are
classified by the elements of Pic(C), the Picard group of C. In this way Pic(C)
appears as the parameter space for nonequivalent vacua. Our analysis is
modelled on, but not limited to, the case when C is complex projective space.Comment: Refs. update
A Quantum is a Complex Structure on Classical Phase Space
Duality transformations within the quantum mechanics of a finite number of
degrees of freedom can be regarded as the dependence of the notion of a
quantum, i.e., an elementary excitation of the vacuum, on the observer on
classical phase space. Under an observer we understand, as in general
relativity, a local coordinate chart. While classical mechanics can be
formulated using a symplectic structure on classical phase space, quantum
mechanics requires a complex-differentiable structure on that same space.
Complex-differentiable structures on a given real manifold are often not
unique. This article is devoted to analysing the dependence of the notion of a
quantum on the complex-differentiable structure chosen on classical phase
space
A Quantum-Gravity Perspective on Semiclassical vs. Strong-Quantum Duality
It has been argued that, underlying M-theoretic dualities, there should exist
a symmetry relating the semiclassical and the strong-quantum regimes of a given
action integral. On the other hand, a field-theoretic exchange between long and
short distances (similar in nature to the T-duality of strings) has been shown
to provide a starting point for quantum gravity, in that this exchange enforces
the existence of a fundamental length scale on spacetime. In this letter we
prove that the above semiclassical vs. strong-quantum symmetry is equivalent to
the exchange of long and short distances. Hence the former symmetry, as much as
the latter, also enforces the existence of a length scale. We apply these facts
in order to classify all possible duality groups of a given action integral on
spacetime, regardless of its specific nature and of its degrees of freedom.Comment: 10 page
- …