Indefinite Metric Research Articles

The R operation in the ??4 theory as a consequence of an indefinite metric in the extended space of states (lowest orders)

The R operation in the ??4 theory as a consequence of an indefinite metric in the extended space of states (lowest orders)

Wick polynomials in a space with indefinite metric

Wick polynomials in a space with indefinite metric

The Maslov-Morse index of indefinite metrics

We compute the Maslov-Morse index of geodesics on a manifold with indefinite metric. It is shown that the multiplicity of conjugate points in the sense of Maslov is equal to the signature of a quadratic form obtained by restricting the metric to the space of degeneracy for the projection of the tangent space onto the Lagrangian manifold, if this latter spans the principal axes of the metric.

Soluble Two-Dimensional Models of Generally Covariant Field Theories

Several soluble generally covariant field theories defined on a two-dimensional indefinite metric are discussed. The differential equations for the metrics are nonlinear and hyperbolic. These are solved in closed form.

Continuous Hermitian-indefinite functions

Some classes of Hermitian-indefinite functions and their integral representations are considered, as well as spaces with an indefinite metric that are formed with the aid of Hermitian-indefinite functions.

Piecewise analyticity ofS-matrix elements and nonlocal interactions

In local quantum field theories ultraviolet divergences may be avoided by admitting an indefinite metric in the quantum-mechanical state space. The occurrence of negative-norm and zero-norm states (ghost states) in general leads to a violation of the unitarity of the physicalS-matrix. In order to unitarize the theory in the physical subspace an appropriate procedure has to be employed which leads to the suppression of unphysical channels and as a consequence produces nonanalytic points at the thresholds of these channels which destroy the global analyticity of theS-matrix in thes-plane. If these nonanalytic points occur in the physical region they simulate nonlocal interactions which decrease with a certain power of the distance. The nonlocal effects are strongest if anS-wave 2-particle ghost channel is suppressed and are less aggravating if they involve ghost channels with higher angular momentum and a larger number of particles.

Renormalization in indefinite metric theories

We consider the Lee model,V ⇆ N+θ, and a two neutral scalar meson model with indefinite metrics. It is shown that in general the coupling will cause positive definite bare states to become negative length dressed states through renormalization effects. This can render the theory unphysical depending upon the auxiliary conditions imposed with the indefinite metric.

Gupta-Bleuler formalism with scalar fermions in massless Yang-Mills theory

In the Yang-Mills theory a formalism corresponding to the conventional Gupta-Bleuler formalism in quantum electrodynamics leads to difficulties with regard to the probability interpretation. It is shown that the tree diagrams of theS-matrix are gauge invariant and that nevertheless theS-matrix leads out of the physical subspace Open image in new window with positive semi-definite metric. TheS-matrix projected onto the physical subspace hence is no longer unitary. In order to arrive at a unitary theory the original Gupta-Bleuler formalism has to be extended by introducing a complex scalar isovector field which obeys anticommutation rules. As a consequence, the usual connection between spin and statistics does not hold in the large state space. This does not pose any difficulty because an indefinite metric is used in the state space. The scalar fermions contribute in such a peculiar way as to cancel all bad effects of the Gupta-Bleuler ghosts in the original formulation, yielding, therefore, the correct Feynman diagrams with the usual analyticity properties. It is possible to give a consistent probability interpretation for the asymptotic particles within this formalism without any additional unitarization procedure. In comparison with quantum electrodynamics the ghost structure is much more complicated and the indefinite metric nontrivial. Nevertheless, our notation of good and bad ghosts with some modification remains the most suitable language also in this extended Gupta-Bleuler formalism.

Theory of Leptons

A unified theory describing four leptons, i.e., muon, electron and two neutrinos together is presented. Leptons are represented by a single fundamental field: Without any interaction the field satisfies a fourth-order differential wave equation. When the influence of the electromagnetic field is taken into account, the appearance of reactive terms in the equation makes it possible to decompose the fundamental field into two Dirac fields (electrons's and muon's) and a residual field. Though the residual field interacts with the electromagnetic field in a way similar to the separated two Dirac fields, the corresponding phenomena are shown to be ineffective owing to the mechanism analogous to Bleuler and Gupta's in quantum electrodynamics, provided the right (or left) handed component of the residual field is assumed not to exist: The proposed nonlocal weak interaction of the fundamental field leads to the coupling, on the mass shell, of the charge exchange current-current type. The current of approximate V - A with small T-mixture type gives the neutrino a small mass, which causes the separation of one neutrino field from the other. From these investigations, the following problems are suggested: i) The existence of two kinds of neutral lepton fields with nearly zero mass would be understood in connection with their appearance through the two-component parts in nature. ii) There would be a possible description of one pair of lepton (for instance, muon and massive neutrino) by means of the indefinite metric in Hilbert space, which is suggested by the existence of two kinds of lepton number conservation. iii) In connection with the occurrence of lepton weak interaction, at least one of the masses of neutrinos, should be finite (non-zero), but less than the electron mass.

Complex masses and acausal propagation in field theory

It is shown that in formulating a field theory with complex masses, keeping much closer to a detailed space-time description than existing treatments, one is naturally led to a modified Yang-Feldman equation. A consistent interpretation emerges, in which the complex masses are not associated with particles, but with a modification of the field propagation law, which becomes acausal. The indefinite metric of the usual treatments plays no role here. An investigation of both the case of interaction with an external potential, and of the fully quantized theory is carried out. No unitarity, or Lorentz invariance troubles arise, but in the case of a fully quantized theory, the contribution of virtual states with arbitrarily high energy, leads to an enhancement of the basic acausality of the propagation law, in a way that may lead to macroscopic deviation from causality.

Regularization of quantum electrodynamics through non-polynomial Lagrangians

A model for regularizing quantum electrodynamics through a nonpolynomial Lagrangian describing the interaction of the electron field with an auxiliary field is given. A gauge-invariant Lagrangian density is obtained. This kind of regularization avoids the difficulties connected with an indefinite metric. Renormalization of quantum electrodynamics is performed for irreducible diagrams. In the hypothesis that this kind of regularization is the physical basis for elimination of divergences in quantum electrodynamics, the physical implications are discussed. In particular, the limits of validity of quantum electrodynamics are discussed in terms of the coupling constant between the electron and the regularizing field. It is found that the present limits of validity of quantum electrodynamics imply for this coupling constant an upper limit larger than the weak-coupling constant. The possibility of regularization through weak interactions is discussed. In this model, for every reasonable coupling constant, the electromagnetic self-mass of the electron is only a fraction of the total mass.

Completeness of states, shadow states, Heisenberg condition, and poles of the S-matrix

A study of the connection between poles of the S-matrix and states of the Hamiltonian of non-relativistic quantum mechanical systems is made with a view to elucidate the concept of shadow states which have been used by one of the authors for the elimination of divergences in quantum field theory with the aid of an indefinite metric. By specific examples we demonstrate that there exist non-relativistic systems for which the S-matrix has poles which correspond to shadow (redunant) states which are not needed in the completeness relation. Systems with such states do not fulfill a condition on S-matrix which was derived by Heisenberg. It is further shown that there exist phase equivalent systems in which these very poles of the S-matrix correspond to genuine bound states which are absolutely necessary to complete the set of states of the system, and these discrete states may have positive or negative norm depending on the choice of the S-matrix.

Is the photon a goldstone particle?

The degenerate vacuum formalism is developed for spin-1 Goldstone particles with an indefinite metric. This formalism eliminates the ambiguities in the early treatments and shows that there is no inconsistency with Poincare invariance. It is argued that if the photon were a Goldstone particle there would be observable consequences, such as apparent angular-momentum nonconservation and violation of the principle of detailed balance.

Quantum electrodynamics in two dimensions

Operator solutions of the Schwinger model are given in several gauges. The correct interpretation for the covariant solution with indefinite metric is discussed. A class of noncovariant gauges, particularly well suited to the analysis of the gauge-invariant algebra is obtained. As a result of this analysis, it is seen that there are no physical electron excitations in the model. Charge sectors can be introduced, but only at the price of violating the spectrum condition. The Coulomb gauge solution is shown to lead to a very pathological indefinite metric. Quantum electrodynamics as limit of a vector meson theory and broken symmetry aspects are discussed in the concluding sections.

Asymptotic scale invariance and the point-source lee model

The renormalized two-point function in the point-source Lee model with indefinite metric is shown to be continuous at short distances. A Ward identity for scale invariance in fixed point-source field models is derived. The dimension of the renormalized field is canonical. The appearance of logarithmic functions in the short-distance behaviour of the model is attributed to the lack of scaling among high-energy eigenstates.

Method for Studying Nonrenormalizable Field Theories

A method is presented for studying nonrenormalizable quantum field theories in terms of renormalizable theories. The theory of a Hermitian scalar field in quintic self-interaction is modified by including in the Lagrangian a second-order derivative of the field variable. This additional term introduces a heavy particle with indefinite metric that acts as a cutoff on the momentum-space integrations. The most divergent part of this theory is shown to be analytically equivalent to the conventional ${\ensuremath{\varphi}}^{4}$ theory. It is then possible to use unitarity to relate the next most divergent part to the renormalizable most divergent part.

Operators, absolutely indefinitely bounded below, in spaces with indefinite metric

Let r be the spectral radius of an operatorU, absolutely indefinitely bounded below. It is proved that r = c1/α, where c is the exact lower bound ofU and α is a number occurring in the definition of the I-metric. A bound is obtained for the dimensionality of the direct sum of root lineals ofU (c ≥ 1), corresponding to eigenvalues whose absolute values are smaller than unity.

Finite model of four-fermion interactions with indefinite metric

The scattering operator theory for fermions is developed in the framework of an indefinite metric Hilbert space starting with the second power Dirac equation. It is based on Lorentz invariance, strong pseudounitarity and strong Bogoliubov causality. These assumptions, together with operator derivatives and generalized convolution operators lead to a strong scattering operator equation in the indefinite metric Hilbert space. There is enough latitude in the homogeneous solutions of the scattering operator equation for the specification of interaction. The second-order two- and four-point functions are explicitly computed for the four-fermion interactions. The scattering operator theory does not involve renormalization in the Feynman-Dyson sense. There are no ultraviolet divergences encountered and hence no momentum cut-off needed. The vacuum diagrams are divergent in perturbation theory and call for a space-time cut-off. Arguments are presented to show that all the higher order functions lead to convergent integrals. The present approach, though mathematically satisfactory, is at least so far not fully amenable to physical interpretation because of the indefinite metric.

One-Parameter Subgroups of Unitary Groups with Indefinite Metric and in Particular of the Conformal Group

Motivated by a problem concerning two-variable expansions of covariant scattering amplitudes, and by recent theories involving indefinite metrics and the conformal group, we study subgroups of unitary groups with indefinite metric. The one-parameter subgroup case reduces to finding canonical matrices for pseudo-Hermitian operators with respect to orthonormal bases. By decomposing the space on which such operators act as far as possible as an orthogonal direct sum of invariant subspaces, one obtains invariant subspaces having indecomposable primary components. The general results, summarized in tables of canonical forms valid for any finite dimension, are supplemented by more detailed tables for low dimensions, including the case of the conformal group of space-time.

An infinite-component free field carrying compounds lying on a Regge trajectory

One herein constructs an infinite-component generalized free field ϕ(x) which carries an infinite tower of unstable (excepting the lowest-mass member) self-compounds which is defined by a spin trajectory. ϕ(x) transforms locally under the Poincare group. The norm or propagator of ϕ(x) can be written as an infinite partial (in spinj) series of contributions of positive-definite metric, which, with the suggested constraint on the mass2 (squared) functionmj2 as |j|→∞, permits summation by the Sommerfeld-Watson transformation. In close analogy to the Gupta-Bleuler form of quantum electrodynamics, the consequent Reggeized propagator then involves contributions of indefinite metric completely equivalent to the positive-metric partial series. The indefinite metric is here directly associated with the instability of the higher Regge poles. The constraint onmj2 together with the effect of invariance underTP for the ϕ propagator leads to a particular dispersion relation for our mass2 functionmΛZ2. The latter involves Λ=(j+1/2)2,Z=P2 the fourmomentum squared, on a completely reciprocal basis. It also, necessarily, introduces two mass2 continua (r>4m02,r<0). The latter, tachyon component suggests that ϕ has to be interpreted as a fundamental field which is not directly observed. The phenomenological field having no tachyon component and which is directly observed will be discussed on the basis of unitarity in a subsequent paper. The propagator of the free field ϕ clearly obeys unitarity in a trivial way in terms of the free states defined by itself.