Masao Ninomiya

Absence of neutrinos on a lattice: (I). Proof by homotopy theory

Abstract It is shown, by a homotopy theory argument, that for a general class of fermion theories on a Kogut-Susskind lattice an equal number of species (types) of left- and right-handed Weyl particles (neutrinos) necessarily appears in the continuum limit. We thus present a no-go theorem for putting theories of the weak interaction on a lattice. One of the most important consequences of our no-go theorem is that is not possible, in strong interaction models, to solve the notorious species doubling problem of Dirac fermions on a lattice in a chirally invariant way.

ADLER-BELL-JACKIW ANOMALY AND WEYL FERMIONS IN CRYSTAL

The Adler-Bell-Jackiw (ABJ) axial anomaly is derived from the physical point of view as the production of Weyl particles and it is used to show the absence of the net production of particles for lattice regularized chirally invariant theories with locality. An analogy or a simulation is pointed out between the Weyl fermion theory and gapless semiconductors where two energy bands have pointlike degeneracies. For such materials, in the presence of parallel electric and strong magnetic fields, there exists an effect similar to the ABJ anomaly that is the movement of the electrons in the energy-momentum space from the neighborhood of one degeneracy point to another one. The longitudinal magneto-conduction becomes extremely strong.

Absence of neutrinos on a lattice: (II). Intuitive topological proof

Abstract An intuitive topological proof is given of the no-go theorem for putting Weyl fermions in weak interaction on a lattice, or for constructing chiral invariant lattice QCD, which was proved by a homotopy theory argument in our preceding paper (Absence I). This theorem hangs on the existence of the charge (e.g. fermion number), and thus on the complex-field formulation and on locality. If we relax the assumptions for the no-go theorem, for instance the existence of the charge, and thus use the real-field formulation, we can construct a model that has only one two-component field. We can assign this model an only approximately conserved charge.

A no-go theorem for regularizing chiral fermions

Abstract We present a no-go theorem for regularizing chiral fermions in a general and abstract form, together with a review of our lattice no-go theorem for chiral fermions.

Renormalization group and quantum gravity

Abstract We study non-perturbative effects of the quantum fluctuation of gravity by means of the ϵ-expansion around two dimensions and the renormalization group method. By introducing the fractal dimension as an order parameter we show that the space-time possesses a self-similar structure. We also point out that an operator such as the cosmological term, the scalar and fermion mass terms ceases to exist as a local one as the number of matter fields increases. This fact may be taken as a suggestion as to how to answer the naturalness problem, in particular the vanishing of the cosmological term. It is argued that there exists potential pathology in the ϵ-expansion in the 2+ϵ dimensional gravity and we give the solution that it is a double expansion in ϵ and 1 −c where c is the number of matter fields (c=N S + 1 2 N F ) .

Axial Anomaly and Index Theorem for Manifolds With Boundary

Abstract The use of the APS theorem for detecting the axial anomaly in manifolds with a boundary is discussed. The necessity of imposing non-local boundary conditions for the euclidean Dirac operator is explained and a physical interpretation is provided. The notion of a “handed” non-local boundary condition is introduced. The index for the Dirac operator for general gauge fields is found for a cylinder and for a disk. A similar analysis is also carried out for general separable gauge fields in four dimensions.

Lattice gravity and strings

Abstract We are concerned with applications of the simplicial discretization method (Regge calculus) to two-dimensional quantum gravity with emphasis on the physically relevant string model. Beginning with the discretization of gravity and matter we exhibit a discrete version of the conformal trace anomaly. Proceeding to the string problem we show how the direct approach of (finite difference) discretization based on Nambu action corresponds to unsatisfactory treatment of gravitational degrees. Based on the Regge approach we then propose a discretization corresponding to the Polyakov string. In this context we are led to a natural geometric version of the associated Liouville model and two-dimensional gravity.

Diffeomorphism symmetry in simplicial quantum gravity

Abstract A construction of simplicial quantum gravity with an extremely strong form of diffeomorphism symmetry of the functional integral measure ( Π d l e −S pg ) is proposed using the method of mathematical induction. By means of the formalism of introducing a continuous four-space approximating a lattice model in our previous paper this symmetry results in a “free gas” behavior of the vertices of the lattice on a C ∞ manifold. Furthermore, we argue that this kind of a simplicial quantum gravity model leads to a gravitational theory described by a local action in the long wavelength (continuum) limit.

Nonlocal SU(3) current algebra

Abstract We construct the nonlocal (parafermionic) SU(3) current algebra with Z N -symmetry in two dimensions, following the Z-algebra a la Lepowsky and Wilson. By computing the characters of the representation, we show that the simplest N =2 algebra (central charge of the conformal sector c = 6 5 ) is equivalent to a combined system of the tri-critical ( c = 7 10 ) and the critical ( c = 1 2 ) Z 2 -Ising models.

PREGEOMETRIC QUANTUM LATTICE: A GENERAL DISCUSSION

Abstract We put forward an idea that the fundamental, i.e. pregeometric, structure of spacetime is given by an abstract set, so called abstract simplicial complex ASC. Thus, at the pregeometric level there is no (smooth) spacetime manifold. However, we argue that the structure described by an abstract simplicial complex is dynamical. This dynamics is then assumed to ensure that ASC can be realized as a lattice on a four-dimensional manifold with the simplest topologies dominating. We rewrite the pregeometric model, which is quantized using euclidean path-integral formalism, in an exact way so that as a four-dimensional manifold with the simples topologies dominating. is done by definition. The first step in bringing the continuum into the arena is to build up a lattice on a four-dimensional manifold from a given ASC. In fact, we choose a specific lattice: The Regge calculus lattice, i.e. a piecewise linear (flat) metric spacetime manifold. Secondly, we introduce a smooth (C ∞ ) manifold (described by a metric tensor g μν ) to approximate the Regge calculus manifold (described by a metric tensor g μν RC ). It turns out that after integrating (and summing) out all other degrees of freedom than the metric tensor field g μν , the resulting continuum theory is nonlocal (as would be expected). However, it is our main point to show that the nonlocality is not very severe since it is only of finite range. We argue that the points in the introduced continuum which represent lattice points have so great quantum fluctuations that they are in a high temperature phase with no long-range correlations. In other words, although the effective action for the continuum formulation is not totally local, it is effectively so because it has only finite range nonlocalities. We can prove this kind of weak locality of the effective action by means of a general high-temperature theorem. Then we claim that the resulting local (or rather almost local) model with reparametrization invariance and g μν as a field gives essentially the ordinary Einsteins gravity theory in the long wavelength limit.