International Journal of Modern Physics A

It is pointed out that in the Dirac formalism ψ^†γ⁰ψ is not a correct spinorial expression and should be replaced by where C₀ is a Lorentz-invariant quantity with different spinorial indices than γ₀. Although has the same matrix value as γ⁰ in flat space-time, the two quantities are different in curved space-time. However, γ₀ retains its role as the time component of a 4-vector, and as a factor in the parity and time reversal states.

We show that the algebraic quantization method of Heisenberg and the analytical method of Schrödinger are not necessarily equivalent when applied to systems of identical particles. Heisenberg quantization is a natural approach, but inherently more ambiguous and difficult than Schrödinger quantization. We apply the Heisenberg method to the examples of two identical particles in one and two dimensions, and relate the results to the so-called fractional statistics known from Schrödinger quantization. For two particles in d dimensions we look for linear, Hermitian representations of the symplectic Lie algebra sp(d, R). The boson and fermion representations are special cases, but there exist other representations. In one dimension there is a continuous interpolation between boson and fermion systems, different from the interpolation found in Schrödinger quantization. In two dimensions we find representations that can be realized in terms of multicomponent wave functions on a three-dimensional space, but we have no clear physical interpretation of these representations, which include extra degrees of freedom compared to the classical system.

We present a method whereby the equations of motion yielding the explicit two-component spinor form of the complete Maxwell-Dirac theory in complex Minkowski space may be directly derived from two variational principles. One of these dynamical statements gives rise to the first half of the electromagnetic theory which particularly appears as the equations of motion involving a slightly modified version of the free part of the conventional Maxwell Lagrangian density. The other principle actually involves a holomorphic two-spinor expression for the full Maxwell-Dirac Lagrangian density which leads to the second half along with the Dirac equations that carry the relevant covariant derivative operator. It is shown explicitly how the Bianchi identities of the complete theory can be established in a manifestly covariant way. A system of essentially equivalent wave equations for positive-frequency photons and electrons is then exhibited. In particular, we make use of certain skew-symmetric operators to obtain an extended form of the Feynman-Gell-Mann equations.

Approximate formulae are given for the effective Hamiltonian H_||(t) governing the time evolution in a subspace ℋ_|| of the state space ℋ. The properties of matrix elements of H_||(t) and the eigenvalue problem for H_||(t) are discussed in the case of two-dimensional ℋ_||. The eigenvectors of H_||(t) for the short time region are found to be different from those for the long time region. The decay law of particles described by the eigenvectors of H_||(t) is shown to have the form of the exponential function multiplied by some time-independent factor, equal to 1 only in the case of the -invariant theory. Some general properties of the matrix elements of H_||(t) are tested in the Lee model.

The problem of defining Poisson brackets for a degenerate Lagrangian without introduction of canonical variables is discussed. More generally, we introduce and give a complete geometrical description of a class of Poisson brackets on a presymplectic manifold. The construction is illustrated by both finite-dimensional and field-theoretic examples.

We propose and investigate a new conformal invariant integrable field theory called bosonic superconformal affine Toda theory. This theory can be viewed either as the affine generalization of the so-called bosonic superconformal Toda theory studied by the authors sometime earlier, or as the generalization to the case of half-integer conformal weights of the conformal affine Toda theory, and can also be obtained from the Hamiltonian reduction of WZNW theory (with an affine WZNW group). The fundamental Poisson stracture is established in terms of the classical r matrix. Then the exchange algebra for the chiral vectors is obtained as well as the reconstruction formula for the classical solutions. The dressing transformations of the fundamental fields are found explicitly, and the Poisson-Lie structure of the dressing group is also constructed with the aid of classical exchange algebras, which turns out to be the semiclassical limit of the quantum affine group. The conformal breaking orbit of the model is also studied, which is called bosonic super loop Toda theory in the context. In addition, the quantum exchange relation and quantum group symmetry are discussed briefly.

We investigate through the dilute instanton gas approximation the long-distance properties of the (1+1)-dimensional electrodynamics with massless fermions and Higgs-like bosons. Generally fermions have a charge different from that of bosons and are confined. We find the “mesonic” and “baryonic” spectrum of the model. The fermions become free when their charge becomes equal to that of bosons. This first-order phase transition is investigated as well. The order parameter is found to be the fermion-antifermion condensate, which is zero in confining phase and nonzero in nonconfining phase. The effective long-distance Lagrangians in both phases are found.

We describe a real-time classical solution of c=1 string field theory written in terms of the phase space density, u(p, q, t), of the equivalent fermion theory. The solution corresponds to tunnelling of a single fermion above the filled Fermi sea and leads to amplitudes that go as exp(−C/g_str). We discuss how one can use this technique to describe nonperturbative effects in the Marinari-Parisi model. We also discuss implications of this type of solution for the two-dimensional black hole.