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-

Why the Mickelsson-Faddeev algebra lacks unitary representations

A simple plausibility argument is given.Comment: Argument sharpened. 4 page

Extended diffeomorphism algebras and trajectories in jet space

Let the DRO (Diffeomorphism, Reparametrization, Observer) algebra DRO(N) be the extension of $diff(N)\oplus diff(1)$ by its four inequivalent Virasoro-like cocycles. Here $diff(N)$ is the diffeomorphism algebra in $N$-dimensional spacetime and $diff(1)$ describes reparametrizations of trajectories in the space of tensor-valued $p$-jets. DRO(N) has a Fock module for each $p$ and each representation of $gl(N)$. 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 $diff(N)$. 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