10,829 research outputs found
On the problem of time in two and four dimensions
In general-covariant theories the Hamiltonian is a constraint, and hence
there is no time evolution; this is the problem of time. In the subcritical
free string, the Hamiltonian ceases to be a constraint after quantization due
to conformal anomalies, and time evolution becomes non-trivial and unitary. It
is argued that the problem of time in four dimensions can be resolved by a
similar mechanism. This forces us to challenge some widespread beliefs, such as
the idea that every gauge symmetry is a redundancy of the description.Comment: 9 page
Resolution of the problem of time in quantum gravity
The metric determines the casual structure of spacetime, but in quantum
gravity it is also a dynamical field which must be quantized using this causal
structure; this is the famous problem of time. A radical resolution of this
paradox is proposed: remove the concept of space-like separation entirely. This
can be done by describing all fields in terms on p-jets, living on the
observer's trajectory; all points on the trajectory have time-like separations.
Such a description is necessary to construct well-defined representations the
N-dimensional generalization of the Virasoro algebra Vir(N); this is the
natural quantum extension of vect(N), which is the correct symmetry algebra of
general relativity in N dimensions. The limit p -> oo, necessary for a field
theory interpretation, only exists if N = 4 and there are three fermions for
every two bosons, a relation that is satisfied in the standard model coupled to
gravity.Comment: Name of paper changed since the term "causality paradox" refers to
the problem of time. Material on the generalization of the Virasoro algebra
beyond one dimension expande
Manifestly covariant canonical quantization of gravity and diffeomorphism anomalies in four dimensions
Canonical quantization of gravity requires knowledge about the representation
theory of its constraint algebra, which is physically equivalent to the algebra
of arbitrary 4-diffeomorphisms. All interesting lowest-energy representations
are projective, making the relevant algebra into a four-dimensional
generalization of the Virasoro algebra. Such diffeomorphism anomalies are
invisible in field theory, because the relevant cocycles are functionals of the
observer's trajectory in spacetime. The multi-dimensional Virasoro algebra acts
naturally in the phase space of arbitrary histories, with dynamics playing the
role of first-class constraints. General relativity is regularized by expanding
all fields in Taylor series around the observer's trajectory, and truncating at
some fixed order. This regularized but manifestly general-covariant theory is
quantized in the history phase space, and dynamics is imposed afterwards, in
analogy with BRST quantization. Infinities arise when the regularization is
removed; it is presently unclear how these should be dealt with.Comment: In: Focus on quantum gravity research, ed: David C. Moore, pp
261-310, 2006 Nova Science Publishers Inc. ISBN 1-59454-660-
Extended diffeomorphism algebras and trajectories in jet space
Let the DRO (Diffeomorphism, Reparametrization, Observer) algebra DRO(N) be
the extension of by its four inequivalent Virasoro-like
cocycles. Here is the diffeomorphism algebra in -dimensional
spacetime and describes reparametrizations of trajectories in the
space of tensor-valued -jets. DRO(N) has a Fock module for each and each
representation of . Analogous representations for gauge algebras
(higher-dimensional Kac-Moody algebras) are also given. The reparametrization
symmetry can be eliminated by a gauge fixing procedure, resulting in previously
discovered modules. In this process, two DRO(N) cocycles transmute into
anisotropic cocycles for . Thus the Fock modules of toroidal Lie
algebras and their derivation algebras are geometrically explained.Comment: Expressions for abelian charges corrected. Published versio
Concrete Fock representations of Mickelsson-Faddeev-like algebras
The Mickelsson-Faddeev (MF) algebra can naturally be embedded in a non-Lie
algebra, which suggests that it has no Fock representations. The difficulties
are due to the inhomogeneous term in the connection's transformation law.
Omitting this term yields a ``classical MF algebra'', which has other abelian
extensions that do possess Fock modules. I explicitly construct such modules
and the intertwining action of the higher-dimensional Virasoro algebra.Comment: 9 pages, Late
Symmetries of Everything
I argue that string theory can not be a serious candidate for the Theory of
Everything, not because it lacks experimental support, but because of its
algebraic shallowness. I describe two classes of algebraic structures which are
deeper and more general than anything seen in string theory:
The multi-dimensional Virasoro algebras, i.e. the abelian but non-central
extension of the algebra of vector fields in N dimensions by its module of
closed dual one-forms.
The exceptional simple Lie superalgebra mb(3|8), which is the deepest
possible symmetry (depth 3 in its consistent Weisfeiler grading). The grade
zero subalgebra, which largely governs the representation theory, is the
standard model algebra sl(3)+sl(2)+gl(1). Some general features can be
extracted from an mb(3|8) gauge theory even before its detailed construction:
several generations of fermions, absense of proton decay, no additional gauge
bosons, manifest CP violation, and particle/anti-particle asymmetry.
I discuss classifications supporting the claim that every conceivable
symmetry is known.Comment: Some notes added and flawed definition of mb(3|8) corrected once
again. This became acute after the appearance of a recent paper by D. Frieda
Virasoro 3-algebra from scalar densities
It is shown that the ternary Virasoro-Witt algebra of Curtright, Fairlie and
Zachos can be constructed by applying the Nambu commutator to the vect(1)
realization on scalar densities. This construction is generalized to vect(d),
but the corresponding 3-algebra fails to close.Comment: 5 page
The physical observer I: Absolute and relative fields
Quantum Jet Theory (QJT) is a deformation of QFT where also the quantum
dynamics of the observer is taken into account. This is achieved by introducing
relative fields, labelled by locations measured by rods relative to the
observer's position. In the Hamiltonian formalism, the observer's momentum is
modified: p_i \to p_i - P_i, where P_i is the momentum carried by the field
quanta. The free scalar field, free electromagnetism and gravity are treated as
examples. Standard QFT results are recovered in the limit that the observer's
mass M \to \infty and its charge e \to 0. This limit is well defined except for
gravity, because e = M in that case (heavy mass equals inert mass). In a
companion paper we describe how QJT also leads to new observer-dependent gauge
and diff anomalies, which can not be formulated within QFT proper.Comment: 39 p
Covariant Mickelsson-Faddeev extensions of gauge and diffeomorphism algebras
We construct new extensions of current and diffeomorphism algebras in N>3
dimensions, which are related to the Mickelsson-Faddeev algebra. The result is
compatible with Dzhumadil'daev's classification of diffeomorphism cocycles. We
also construct an extension of the current algebra in N>=5 dimensions which
depends on the fourth Casimir operator.Comment: Minor correction
Why the Mickelsson-Faddeev algebra lacks unitary representations
A simple plausibility argument is given.Comment: Argument sharpened. 4 page
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