Preconjugate variables X have commutation relations with the energy-momentum P of the respective system which are of a more general form than just the Hamiltonian one. Since they have been proven useful in their own right for finding new spacetimes we present here a study of them. Interesting examples can be found via geometry: motions on the mass-shell for massive and massless systems, and via group theory: invariance under special conformal transformations of mass-shell, resp. light-cone -both find representations on Fock space. We work mainly in ordinary fourdimensional Minkowski space and spin zero. The limit process from non-zero to vanishing mass turns out to be non-trivial and leads naturally to wedge variables. We point out some applications and extension to more general spacetimes. In a companion paper we discuss the transition to conjugate pairs. 1
In a previous paper we presented the renormalization of Einstein-Hilbert gravity under inclusion of higher derivative terms and proposed a projection down to the physical state space of Einstein-Hilbert. In the present paper we describe this procedure in more detail via decomposing the original double-pole field h µν in the bilinear field sector into a massless and a massive spin two field. Those are associated with the poles at zero mass resp. at non-zero mass of h in the tree approximation. We show that the massive fields have no poles in higher orders hence do not correspond to particles. The Smatrix becomes thus unitary. On the way to these results we derive finiteness properties which are valid in the Landau gauge. Those simplify the renormalization group analysis of the model considerably. We also establish a rigid Weyl identity which represents a proper substitute for a Callan-Symanzik equation in flat spacetime. Contents 1. Introduction 2. Preliminaries 3. Part I: Theory formulated in terms of h 3.1. Rigid Weyl Invariance 3.2. Finiteness in Landau gauge: γ RG c = γ RG h = β 3 = 0 3.3. The RG equation 3.4. No massive higher order zeros 4. Part II: Theory formulated in terms of φ and Σ 4.1. Lagrange multiplier form of the bilinear action 4.2. Propagators 4.3. s-cohomology for φ and Σ sectors 4.4. Projection to Einstein-Hilbert 5. Discussion and Conclusions Appendix A. Appendix A.1. Proof of antighost equation A.2. Optical Theorem References
Recent developments for BPHZ renormalization performed in configuration space are reviewed and applied to the model of a scalar quantum field with quartic self-interaction. An extension of the results regarding the short-distance expansion and the Zimmermann identity is shown for a normal product, which is quadratic in the field operator. The realization of the equation of motion is computed for the interacting field and the relation to parametric differential equations is indicated.
For the case of spin zero we construct conjugate pairs of operators on Fock space. On states multiplied by polarization vectors coordinate operators Q conjugate to the momentum operators P exist. The massive case is derived from a geometrical quantity, the massless case is realized by taking the limit m 2 → 0 on the one hand, on the other from conformal transformations. Crucial is the norm problem of the states on which the Q's act: they determine eventually how many independent conjugate pairs exist. It is intriguing that (light-) wedge variables and hence the wedge-local case seems to be preferred. 1
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