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Algebras of currents and diffeomorphisms in arbitrary dimension have extensions which generalize the affine and Virasoro algebras on the circle. A large class of off-shell representations was discovered in Comm. Math. Phys. 214 (2000) 469--491. That paper is not so accessible due to a slightly non-standard normal ordering formalism and cumbersome $p$-jet calculations. The purpose of the present paper is to simplify the derivation using standard OPE methods and a more field-like formalism.

We postulate that the fundamental principles of Quantum Gravity are diffeomorphism symmetry, unitarity, and locality. Local observables are compatible with diffeomorphism symmetry in the presence of diff anomalies, which modify the symmetry algebra upon quantization. We describe the generalization of the Virasoro extension to the diffeomorphism algebra in several dimensions, and its off-shell representations. These anomalies can not arise in QFT, because the Virasoro-like cocycles are functionals of the observer's spacetime trajectory, which is not present in QFT. Possible implications for physics are discussed., Comment: Added reference

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 corrections

We review some key features of Quantum Jet Theory: observer dependence, multi-dimensional Virasoro algebra, and the prediction that spacetime has four dimensions., Comment: To appear in the Invertis Journal of Science and Technology. 10pp

QJT is considered as a regularization of QFT, where the fields are replaced by finite $p$-jets. The regularized phase space is infinite-dimensional, because not all histories are determined by initial conditions. Gauge symmetries are not fully preserved by the regularization, and gauge anomalies arise. These anomalies are of a new type, not present in QFT. They generically diverge when the regulator is removed, but can be made finite with a particular choice of field content, provided that spacetime has at most four dimensions. The field content appears to include unphysical fields that violate the spin-statistics theorem., Comment: 30 pages

In a companion paper we studied field theory in the presence of a physical observer with quantum dynamics. Here we describe the most striking consequence of this assumption: new gauge and diff anomalies arise. The relevant cocycles depend on the observer's spacetime trajectory and can hence not appear in QFT, where this quantity is never introduced. Diff anomalies necessarily arise in every locally nontrivial, non-holographic theory of quantum gravity. Cancellation of the divergent parts of the anomalies only works if spacetime has four dimensions., Comment: 28 pp

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 pp

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 pages

The standard BV complex is never acyclic provided that the equations of motion have solutions and the admissible class of functions is general enough, unless one introduces second-order antifields. This phenomenon is explicitly illustrated for the harmonic oscillator and the free electromagnetic field., Comment: 8 pages

I review the multi-dimensional generalizations of the Virasoro algebra, i.e. the non-central Lie algebra extensions of the algebra vect(N) of general vector fields in N dimensions, and its Fock representations. Being the Noether symmetry of background independent theories such as N-dimensional general relativity, this algebra is expected to be relevant to the quantization of gravity. To this end, more complicated modules which depend on dynamics in the form of Euler-Lagrange equations are described. These modules can apparently only be interpreted as quantum fields if spacetime has four dimensions and both bosons and fermions are present., Comment: In: Mathematical physics research at the leading edge, ed: Charles V. Benton, pp 91-111, 2004 Nova Science Publishers Inc. ISBN 1-59033-905-3

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 1-59454-660-6

QJT (Quantum Jet Theory) is the quantum theory of jets, which can be canonically identified with truncated Taylor series. Ultralocality requires a novel quantization scheme, where dynamics is treated as a constraint in the history phase space. QJT differs from QFT since it involves a new datum: the expansion point. This difference is substantial because it leads to new gauge and diff anomalies, which are necessary to combine background independence with locality. Physically, the new ingredient is that the observer's trajectory is explicitly introduced and quantized together with the fields. In this paper the harmonic oscillator and free fields are treated within QJT, correcting previous flaws. The standard Hilbert space is recovered for the harmonic oscillator, but there are interesting modifications already for the free scalar field, due to quantization of the observer's trajectory. Only free fields are treated in detail, but the complications when interactions are introduced are briefly discussed. We also explain why QJT is necessary to resolve the conceptual problems of quantum gravity., Comment: 115 pp

A local gauge symmetry can not possibly be a mere redundancy of the description, provided that 1. charge is nonzero. 2. we also consider divergent gauge transformations, whose brackets with local transformations contain global charge operators. If these conditions hold, unitarity requires the existence of gauge anomalies. We describe the relevant multi-dimensional generalization of affine Kac-Moody algebras, explain why it only arises if the observer's trajectory is explicitly introduced, and contrast it to the Mickelsson-Faddeev algebra which pertains to chiral-fermion type anomalies., Comment: Example of everywhere smooth, invertible gauge generators which vanish at infinity included. Expanded sketch of the origin of observer-dependent anomalies. 10 pages

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 pages

The recently introduced manifestly covariant canonical quantization scheme is applied to gravity. New diffeomorphism anomalies generating a multi-dimensional generalization of the Virasoro algebra arise. This does not contradict theorems about the non-existence of gravitational anomalies in four dimensions, because the relevant cocycles depend on the observer's spacetime trajectory, which is ignored in conventional field theory. Rather than being inconsistent, these anomalies are necessary to obtain a {\em local} theory of quantum gravity.

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

In hep-th/0411028 a new manifestly covariant canonical quantization method was developed. The idea is to quantize in the phase space of arbitrary histories first, and impose dynamics as first-class constraints afterwards. The Hamiltonian is defined covariantly as the generator of rigid translations of the fields relative to the observer. This formalism is now applied to theories with gauge symmetries, in particular electromagnetism and Yang-Mills theory. The gauge algebra acquires an abelian extension proportional to the quadratic Casimir operator. Unlike conventional gauge anomalies proportional to the third Casimir, this is not inconsistent. On the contrary, a well-defined and non-zero charge operator is only compatible with unitarity in the presence of such anomalies. This anomaly is invisible in field theory because it is a functional of the observer's trajectory, which is conventionally ignored., Comment: Added two references. 48 pages

Classical physics is reformulated as a constrained Hamiltonian system in the history phase space. Dynamics, i.e. the Euler-Lagrange equations, play the role of first-class constraints. This allows us to apply standard methods from the theory of constrained Hamiltonian systems, e.g. Dirac brackets and cohomological methods. In analogy with BRST quantization, we quantize in the history phase space first and impose dynamics afterwards. To obtain a truly covariant formulation, all fields must be expanded in a Taylor series around the observer's trajectory, which acquires the status of a quantized physical field. The formalism is applied to the harmonic oscillator and to the free scalar field. Standard results are recovered, but only in the approximation that the observer's trajectory is treated as a classical curve.

Field-theoretic pure gravitational anomalies only exist in $4k+2$ dimensions. However, canonical quantization of non-field-theoretic systems may give rise to diffeomorphism anomalies in any number of dimensions. I present a simple example, where a higher-dimensional generalization of the Virasoro algebra arises upon quantization., Comment: Described Hamiltonian commuting with anomalous diffeomorphism algebra

For sp(n+2) and each exceptional Lie algebra a realization of depth 2 preserving the spaces spanned by a contact one-form and a bilinear form is given. For e_7 and e_6 a realization of depth 1 preserving a lightcone and the space spanned by a bilinear form is also presented. This makes the origin of the exceptions clear.

Fock modules for multi-dimensional Virasoro algebras (non-central extensions of the diffeomorphism algebra vect(N)) have recently been reported. Using ideas from the antifield formalism, I construct new classes of lowest-energy modules, as cohomology groups of a certain Fock complex. The Fock construction involves a passage to p-jets prior to normal ordering, but the abelian charges usually diverge in the limit p --> oo. The requirement of a finite limit imposes severe restrictions on the number of spacetime dimensions and on the order of the Euler-Lagrange (EL) equations. Under some natural assuptions (the EL equations are first order for fermions and second order for bosons, and no reducible gauge symmetries appear), finiteness is only possible when the number of spacetime dimensions N = 4.

Hence it excludes proton decay and supersymmetry. The main predictions of a gauge model based on the exceptional simple Lie superalgebra mb(3|8) (a localized version of su(3)+su(2)+u(1)) are reviewed.

In p-form lattice gauge theory, the fluctuating variables live on p-dimensional cells and interact around (p+1)-dimensional cells. It has been argued that the continuum version of this model should be described by (p-1)-gerbes. However, only connections and curvatures for gerbes are understood, not covariant derivatives. Using the lattice analogy, an alternative definition of gerbes is proposed: sections are functions phi(x,s), were x is the base point and s is the surface element. In this purely local formalism, there is a natural covariant derivative. The Yang-Baxter equation, and more generally the simplex equations, arise as zero-curvature conditions. The action of algebras of vector fields and gerbe gauge transformations, and their abelian extensions, are described., Comment: Added four references

Degenerate modules of the exceptional infinite-dimensional simple Lie superalgebras vle(3|6), ksle(5|10) and mb(3|8) have recently been constructed by Kac and Rudakov, and by Grozman, Leites and Shchepochkina. I rederive their results using a formalism which is contragredient to theirs; instead of finding singular vectors in induced modules, I build reducible tensor modules ("forms") from elementary differentials. There is a discrepancy between my result for ksle(5|10) and Kac' and Rudakov's one. Since the grade zero subalgebra of vle(3|6) and mb(3|8) is sl(3)+sl(2)+gl(1), gauge theories based on these algebras can be viewed as a "second-gauged" version of the standard model, where the rigid sl(3)+sl(2)+gl(1), symmetry is made local not only in spacetime ("first gauging"), but in the internal directions as well. An attempt to construct such a second-gauged theory is presented. Some predictions regarding the fermion spectrum, absense of new gauge bosons, and CP violation follow immediately.

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 expanded

Dual Pfaff equations (of the form \tilde D^a = 0, \tilde D^a some vector fields of degree -1) preserved by the exceptional infinite-dimensional simple Lie superalgebras ksle(5|10), vle(3|6) and mb(3|8) are constructed, yielding an intrinsic geometric definition of these algebras. This leads to conditions on the vector fields, which are solved explicitly. Expressions for preserved differential form equations (Pfaff equations), brackets (similar to contact brackets) and tensor modules are written down. The analogous construction for the contact superalgebra k(1|m) (a.k.a. the centerless N=m superconformal algebra) is reviewed., Comment: An error in the definition of mb(3|8) has been corrected, and the paper has been greatly revised and expanded. In addition to Pfaff equations (preserved form equations), I give dual Pfaff equations (preserved vector field equations), solve the resulting conditions for the vector fields, and write down the brackets and tensor modules explicitly

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. Friedan

The recently constructed Fock representations of $N$-dimensional diffeomorphism and current algebras are reformulated in terms of one-dimensional currents, satisfying Virasoro and affine Kac-Moody algebras., Comment: Theorem 3 corrected

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, Latex

Dzhumadil'daev has classified all tensor module extensions of $diff(N)$, the diffeomorphism algebra in $N$ dimensions, and its subalgebras of divergence free, Hamiltonian, and contact vector fields. I review his results using explicit tensor notation. All of his generic cocycles are limits of trivial cocycles, and many arise from the Mickelsson-Faddeev algebra for $gl(N)$. Then his results are extended to some non-tensor modules, including the higher-dimensional Virasoro algebras found by Eswara Rao/Moody and myself. Extensions of current algebras with $d$-dimensional representations are obtained by restriction from $diff(N+d)$. This gives a connection between higher-dimensional Virasoro and Kac-Moody cocycles, and between Mickelsson-Faddeev cocycles for diffeomorphism and current algebras., Comment: Corrected treatment of special 2-dim cocycles. Added section on div-free, Hamiltonian and contact subalgebras. 34 pages, Latex2e

The representation theory of non-centrally extended Lie algebras of Noether symmetries, including spacetime diffeomorphisms and reparametrizations of the observer's trajectory, has recently been developped. It naturally solves some long-standing problems in quantum gravity, e.g. the role of diffeomorphisms and the causal structure, but some new questions also arise., Comment: 14 pages, LaTex

I construct lowest-energy representations of non-centrally extended algebras of Noether symmetries, including diffeomorphisms and reparametrizations of the observer's trajectory. This may be viewed as a new scheme for quantization. First classical physics is formulated as the cohomology of a certain Koszul-Tate (KT) complex, using not only fields and antifields but also their conjugate momenta. Then all fields are expanded in a Taylor series around the observer's present position, and terms of order higher than p are truncated. Finally, quantization is carried out by replacing Poisson brackets by commutators and imposing the KT cohomology in Fock space. This procedure is consistent for finite p, but the limit p\to\infty leads to difficulties., Comment: 46 pages, LaTex

The method of constrained Hamiltonian systems can be used to reduce Fock modules. It is applied to the Virasoro algebra, where a possibly new realization is found., Comment: 5 pages, Latex

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 version

A class of Fock representations of non-central extensions of the super-diffeomorphism algebra in (N+1|M) dimensions is constructed, by superization of the paper [physics/9705040]. The representations act on trajectories in (N|M)-dimensional superspace, the extra dimension being the parameter along the trajectory. The restrictions to various subalgebras are considered. In particular, the centrally extended superconformal algebra is obtained by restriction to the contact superalgebra K(1|1). This shows that one of the basic assumptions in superstring theory (the distinguished nature of the superconformal algebra) is incorrect., Comment: 31 pages, LaTeX

We describe a class of non-central extensions of the diffeomorphism algebra in N-dimensional spacetime, and construct lowest-energy modules thereof, thus generalizing work of Eswara-Rao and Moody. There is one representation for each representation of $Vir\ltimes_{k_0}\km{gl(N)}$ (an extension of the semi-direct product). Similar modules are constructed for gauge algebras., Comment: Minor changes in published version. 13 pages, Latex

Manifestly consistent Fock representations of non-central (but ``core-central'') extensions of the $Z^N$-graded algebras of functions and vector fields on the $N$-dimensional torus $T^N$ are constructed by a kind of renormalization procedure. These modules are of lowest-energy type, but the energy is not a linear function of the momentum. Modulo a technical assumption, reducibility conditions are proved for the extension of $vect(T^N)$, analogous to the discrete series of Virasoro representations., Comment: LaTeX, 21 pages

The Feigin-Fuks construction of irreducible lowest-weight Virasoro representations is reviewed using physics terminology. The procedure consists of two steps: constructing invariants and applying them to the Fock vacuum. We attempt to generalize this construction to the diffeomorphism algebra in higher dimensions. The first step is straightforward, but the second is difficult, due to the appearence of infinite Schwinger terms. This might be avoided by imposing constraints on the fields, which should be of the recently discovered conformal type. The resulting representations are reminiscent of quantum gravity., Comment: 36 pages, Plain TeX

Conformal fields are a new class of $Vect(N)$ modules which are more general than tensor fields. The corresponding diffeomorphism group action is constructed. Conformal fields are thus invariantly defined., Comment: LaTex, 9 pages

Conformal fields are a recently discovered class of representations of the algebra of vector fields in $N$ dimensions. Invariant first-order differential operators (exterior derivatives) for conformal fields are constructed., Comment: 17 pages

$Vect(N)$, the algebra of vector fields in $N$ dimensions, is studied. Some aspects of local differential geometry are formulated as $Vect(N)$ representation theory. There is a new class of modules, {\it conformal fields}, whose restrictions to the subalgebra $sl(N+1) \subset Vect(N)$ are finite-dimensional $sl(N+1)$ representations. In this regard they are simpler than tensor fields. Fock modules are also constructed. Infinities, which are unremovable even by normal ordering, arise unless bosonic and fermionic degrees of freedom match., Comment: 16 pages