Michio Kaku

Gauge Theory of the Conformal and Superconformal Group

Abstract We show that the locally scale invariant Weyl theory of gravity is the gauge theory of the conformal group. Proper conformal transformations are gauged by a non-propagating gauge field. A gauge theory for the superconformal group is obtained which is locally scale, Lorentz, and chiral invariant but not locally supersymmetric despite remarkable cancellations.

Unified field theories with U(N) internal symmetries: G

Gauge theories for extended SU(N) conformal supergravity are constructed which are invariant under local scale, chiral, proper conformal, supersymmetry and internal SU(N) transformations. The relation between intrinsic parity and symmetry properties of their generators of the internal vector mesons is established. These theories contain no cosmological constants, but technical problems inherent to higher derivative actions are pointed out.

Lattice formulation of general relativity

Abstract Using Wests recent Sp(4) invariant formulation of the Einstein action, we give a gauge invariant lattice formulation of general relativity. The lattice spacing provides an Sp(4) invariant ultraviolet cutoff for the quantum theory.

Gauge Field Theory of Covariant Strings

Abstract We present a gauge covariant second-quantized field theory of strings which is explicitly invariant under the gauge transformations generated by the Virasoro algebra. Unlike the old field theory strings [1] this new formulation is Lorentz covariant as well as gauge covariant under the continuous group Diff(S1) and its central extension. We derive the free action: L=Φ(X) † P[i∂ τ −(L 0 −1)]PΦ(X) , in the same way that Feynman derived the Schrodinger equation from the path integral formalism. The action is manifestly invariant under the gauge transformation δΦ(X)= ∑ n=1 ∞ ϵ −n L −n Φ(X) , where P is a projection operator which annihilates spurious states. We give three distinct formulations of this operator P to all orders, the first based on extracting the operator from the functional formulation of the Nambu-Goto action, and the second and third based on inverting the Shapovalov matrix on a Verma module. This gauge covariant formulation can be easily extended to the Green-Schwarz superstring [2,3]. One element application of these methods is to re-express the old Neveu-Schwarz-Ramond model as a field theory which is manifestly invariant under space-time supersymmetric transformations.

Locality in the gauge-covariant field theory of strings

Abstract Recently, we wrote down the gauge-covariant field theory of the free bosonic, super, and heterotic strings. These second quantized actions were derived from path integrals in the same way as Feynman derived the Schrodinger equation. These actions possess all the local gauge invariance of the super Virasoro algebra. These actions, however, are non-local. It has been conjectured that these actions can be made local by adding auxiliary fields. In this paper, we prove this conjecture to all orders , making our action explicitly local.

Ghost-free formulation of quantum gravity in the light-cone gauge

Abstract By choosing the light-cone gauge, we remove all redundant components of the metric tensor as well as all Faddeev-Popov ghosts from the Einstein Lagrangian, from which Feynman rules for two independent transverse components can be immediately formulated. The 2 + 2 decomposition of the metric tensor proves to be simpler than the usual canonical 1 + 3 decomposition, so that all spurious components can be explicitly eliminated from the spectrum. Because the Lagrangian is now only a function of independent components, it is possible to study the unresolved problem of the functional measure for quantum gravity. We are presently studying the measure problem.

Quantization of conformal gravity: Another approach to the renormalization of gravity

Abstract The non-renormalizability of quantum gravity poses a great problem to the construction of any unified field theory of all known interactions. Normally, we start with a unitary theory of gravity and investigate its renormalization properties. This is the first of a series of papers where we start with the opposite approach, beginning with a renormalizable theory and investigating its unitarity structure. In particular, we study non-perturbative approaches to the quantization of conformal gravity. Using ADM coordinates, we perform the canonical quantization of the Weyl action Cμναβ2, which is renormalizable and is also local scale invariant. Although this theory is certainly not unitary in perturbation theory, we speculate that unitarity may be restored when we approach this theory non-perturbatively, by examining the possibility of different phase transitions.

Why are there two BRST string field theories

Abstract One of the mysteries of string field theory is why there are two distinct BRST field theories. We solve this puzzle by (1) explicitly constructing a geometric vertex that smoothly interpolates between the other theories, (2) proving that this geometric vertex links the other two by a one-dimensional reparametrization, (3) proving that both BRST theories are thus gauge-fixed versions of a higher, geometric field theory . The geometric theory also explains the origin of the baffling four-string interaction, which now emerges as a gauge artifact, the counterpart of the four-fermion instantaneous Coulomb term of QED.

DERIVING THE FOUR-STRING AND OPEN-CLOSED STRING INTERACTIONS FROM GEOMETRIC STRING FIELD THEORY

One of the baffling questions concerning the covariant open string field theory is why there are two distinct BRST theories and why the four-string interaction appears in one version but not the other. We solve this mystery by showing that both theories are gauge-fixed versions of a higher gauge theory, called the geometric string field theory, with a new field, a string vierbein , which allows us to gauge the string length and σ-parametrization. By fixing the gauge, we can derive the “endpoint gauge” (the covariantized light cone gauge), the “midpoint gauge” of Witten, or the “interpolating gauge” with arbitrary string lengths. We show explicitly that the four-string interaction is a gauge artifact of the geometric theory (the counterpart of the four-fermion instantaneous Coulomb term of QED). By choosing the interpolating gauge, we produce a new class of four-string interactions which smoothly interpolate between the endpoint gauge and the midpoint gauge (where it vanishes). Similarly, we can extract the closed string as a bound state of the open string, which appears in the endpoint gauge but vanishes in the midpoint gauge. Thus, the four-string and open-closed string interactions do not have to be added to the action as long as the string vierbein is included.

2D Gravity and Matrix Models

To any finite order in perturbation theory, one does not see any of the interesting nonperturbative properties of gauge theories, such as confinement, tunneling, formation of strings, etc. As a consequence, two approximations have been developed, large N methods and lattice gauge theory, to analyze gauge theories in the nonperturbative regime. However, both approaches are still in their infancy, and neither has given us definitive results.