Spinor methods in conformal Killing transport

The equations of conformal Killing transport are discussed using tensor and spinor methods. It is shown that, in Minkowski space-time, the equations for a null conformal Killing vector ξ ^a are completely determined by the corresponding spinor ω ^A and its covariant derivative, which defines a spinor π _A′ . In conformally flat space-time, the covariant derivative of π _A′ is also involved. Some applications to twistor theory are briefly mentioned.     

Spinor theory of general relativity without elementary gravitons

We propose a generally covariant and locally Lorentz invariant theory of a Majorana spinor field ψ_μ^α. Our theory has no elementary spin-2 quanta, but does reproduce Einstein's general relativity as a classical solution. We compare this situation to the possibility of finding classical monopoles in a gauge theory, even though no such elementary object is introduced at the outset.     

Lanczos Potential for the van Stockung Space-Time

The Lanczos Potential is a theoretical useful tool to find the conformal Weyl curvature tensor C _abcd of a given relativistic metric. In this paper we find the Lanczos potential L _abc for the van Stockung vacuum gravitational field. Also, we show how the wave equation can be combined with spinor methods in order to find this important three covariant index tensor.     

Limitations on the existence of massless composite states

Massless particles represented by the fields with mixed spinor indices of SL(2,C) are generally shown to be forbidden in covariant field theory under the assumptions of positivity and covariiance alone. This remains true also in gauge theory (in which a negative metric appears) as far as the particles are gauge invariant. This in particular implies that any dynamical “gauge-type particle” (such as vector A_μ, Rarita-Schwinger ψ_μ etc.) cannot appear unless the system has a corresponding local invariance from the outset.     

Construction of a covariant spinor field theory

A covariant theory is constructed of a spinor field in a space which is represented by the local topological product of a space X_n and a space of values of a geometrical object η. The covariant nonlinear spinor field theory constructed preserves the principles of the theory of the unified field and is compatible with the theory of gauge fields.     

Spinor Wave Equation of Electromagnetic Field

In this paper, we give two spinor wave equations of free electromagnetic field, corresponding to the reducibility and irreducibility representations D ¹⁰+D ⁰¹ and D ¹⁰ of the proper Lorentz group, which are the differential equations of space-time one order. The spinor equations are covariant and are equivalent to Maxwell equations.     

具有广义协变的包含重力场贡献的重力场方程

利用半度规λ^(α)_μ表象的数学工具定义一个对广义坐标具有协变形式的重力场矢势函数ω^(α)_μ≡-cλ^(α)_μ，给出一个具有广义协变的包含重力场贡献的重力场方程R_μν-g_μνR/2+Λg_μν=8πG(T^(Ⅰ)_μν+T^(Ⅱ)_μν) 关键词： 重力场方程 协变形式 能量-动量张量 量子化     

On “Conformal spinor geometry”: An attempt to “Understand” internal symmetry

The natural homomorphism of pure spinors corresponding to a given Clifford algebraC _2n to polarized isotropicn-planes of complex Euclidean spaceE _2n ^c is taken as a starting point for the construction of a geometry called spinor geometry where pure spinors are the only elements out of which all tensors have to be constructed (analytically as bilinear polynomials of the components of a pure spinor).C ₄ andC ₆ spinor geometry are analyzed, but it seems that C₈ spinor geometry is necessary to construct Minkowski spaceM ^3,1.C ₆ spinor field equations give rise in Minkowski space to a pair of Dirac equations (for conformal semispinors) presenting ansu(2) internal symmetry algebra. Mass is generated by breaking spontaneously the originalO(4,2) symmetry of the spinor equation.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

On conformally covariant spinor field equations

A spinor field equation, covariant with respect to the general conformal group (including reflections), should consist in general of not less than eight linear equations and then, in Minkowski space, could be represented by not less than two massless Dirac equations. Their reduction through projectors to only one equation, while not spoiling conformal covariance implies unphysical consequences. It is shown instead that two Dirac equations may be brought unambiguously through a stereographic projection to a manifestly conformal covariant form inE _4,2 space. The physical implications are discussed and it is shown that if the fundamental elementary interactions are expressed in terms of conformal semispinors (which can never appear as free particles), then the corresponding physical Dirac spinors appear in the elementary interactions in terms of their chiral projections. This could indicate both the conformally invariant origin of weak interactions and their fundamental character. The possibility of constructing unified models from conformally invariant Lagrangians is envisaged.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.A preliminary version was issued as Internal Report IC/78/43, ICTP Trieste May 1978, see also Lett. Nuovo Cim.21 (1978), 473.I am indebted to Prof. I. T.Todorov for interesting discussions.     

The conformal factor and DiffS1/S1

Two-dimensional gravity coupled with an open string is considered to geometrically construct a string field theory following Bowick and Rajeev. The phase space of this system can be thought of as an extended loop space, and is in fact a Kähler manifold. Each loop carries a conformal factor. Bowick and Rajeev's proposal is modified so that the ghost fields stand on the same footing as the x^μ-fields. The closed string field is modified to be the Kähler potential of the extended loop space. The canonical line bundle is not needed for the equation of motion for a closed string field. A covariant (free) open string field theory is constructed as a by-product in this approach.     

The nonlinear group representation method in the nonlinear spinor theory

Nonlinear spinor equations are derived in the paper by the nonlinear symmetry-group representation method. The basic field transforms according to the linear spinor representation of the orthochronous Lorentz group Lt. The internal symmetry group G^r is realized as a group of nonlinear transformations of the field . The invariant nonlinear spinor equations constituting the group G^r are found in terms of the covariant derivative. The group SU(2) is considered as an example.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 49–55, December, 1972.The author thanks D. D. Ivanenko and D. F. Kurdgelaidze for stimulating discussions and support.     

Protecting the Conformal Symmetry via Bulk Renormalization on Anti deSitter Space

The problem of perturbative breakdown of conformal symmetry can be avoided, if a conformally covariant quantum field j{\varphi} on d-dimensional Minkowski spacetime is viewed as the boundary limit of a quantum field f{\phi} on d + 1-dimensional Anti-deSitter spacetime (AdS). We study the boundary limit in renormalized perturbation theory with polynomial interactions in AdS, and point out the differences as compared to renormalization directly on the boundary. In particular, provided the limit exists, there is no conformal anomaly. We compute explicitly the one-loop “fish diagram” on AdS₄ by differential renormalization, and calculate the anomalous dimension of the composite boundary field j²{\varphi^2} with bulk interaction kf⁴{\kappa \phi^4}.     

Conformal anomaly in 4D gravity-matter theories non-minimally coupled with a dilaton

The conformal anomaly for 4D gravity-matter theories, which are non-minimally coupled with the dilaton, is systematically studied. Special care is taken of rescaling of fields, treatment of total derivatives, hermiticity of the system operator and the choice of measure. Scalar, spinor and vector fields are taken as the matter quantum fields and their explicit conformal anomalies in the gravity-dilaton background are found. The cohomology analysis is carried out and some new conformal invariants and trivial terms, involving the dilaton, are obtained. The symmetry of the constant shift of the dilaton field plays an important role. The general structure of the conformal anomaly is examined. It is shown that the dilaton affects the conformal anomaly characteristically for each case: (1) [Scalar] The dilaton changes the conformal anomaly only by a new conformal invariant, I₄; (2) [Spinor] The dilaton does not change the conformal anomaly; (3) [Vector] The dilaton changes the conformal anomaly by three new (generalized) conformal invariants, I₄, I₂, I₁. We present some new anomaly formulae which are useful for practical calculations. Finally, the anomaly induced action is calculated for the dilatonic Wess-Zumino model. We point out that the coefficient of the total derivative term in the conformal anomaly for the 2D scalar coupled to a dilaton is ambiguous. This resolves the disagreement between earlier calculations and the result of Hawking and Bousso.     

Broken Weyl Invariance and the Origin of Mass

A massless Weyl-invariant dynamics of a scalar, a Dirac spinor, and electromagnetic fields is formulated in a Weyl space, W₄, allowing for conformal rescalings of the metric and of all fields with nontrivial Weyl weight together with the associated transformations of the Weyl vector fields , representing the D(1) gauge fields, with D(1) denoting the dilatation group. To study the appearance of nonzero masses in the theory the Weyl symmetry is broken explicitly and the corresponding reduction of the Weyl space W₄ to a pseudo-Riemannian space V₄ is investigated assuming the breaking to be determined by an expression involving the curvature scalar R of the W₄ and the mass of the scalar, selfinteracting field. Thereby also the spinor field acquires a mass proportional to the modulus of the scalar field in a Higgs-type mechanism formulated here in a Weyl-geometric setting with providing a potential for the Weyl vector fields . After the Weyl-symmetry breaking, one obtains generally covariant and U(1) gauge covariant field equations coupled to the metric of the underlying V₄. This metric is determined by Einstein's equations, with a gravitational coupling constant depending on , coupled to the energy momentum tensors of the now massive fields involved together with the (massless) radiation fields.     

CRAVITATION AND SPINOR GAUGE FIELD

Basing on the Lorentz covariance and SO (4, 2) symmetry of Dirac theory, anobvious covariant theory of spinor gauge field is obtained by expanding the Lorentztransformation to general coordinate tranformation and making the SO (4, 2) to belocalized. We have proved that, by the gauge independence, the symmetrygroup is reduced to the localized rotation of Lorentz group in Riemann space automa-tically. So our theory is the natural generalization of Dirac theory in curved space.We have also proved that, the spinor gauge field can not appear in flat space, thenthe existence of spinor gauge field is closely related to the curvature. The differencesbetween our theory and Utiyama and Kibble theories are also discussed, and it is poin-ted out that the so-called scalar property of Dirac wave function in general relativity isa misunderstanding caused by the unobvious covariance of those theories, even inthose theories We can not distinguish what is the genuine gauge. field and what is theeffect of the structure of space. In obvious covariant theory this paradox disappears.     

Some properties of the two-point functions in a dilatationally covariant field theory

The object of our concern are some properties of the two-point functions in a model of dilatationally covariant field theory. We examine the one- and two-dimensional irreducible representations of the dilatation group. For the one-dimensional case we obtain either a massless free field theory or a theory of an interacting field which does not contribute on the mass shell μ=p²=0 and is characterized by a spectral function μ^r₊?1. In the two-dimensional case both fields differ from the free field, their spectral functions ρ_ij(μ) do not vanish identically and are products of two factors, a polynomial of order up to two in ln μ and μ^r₊?-1. The differences between the case of internal symmetry and the case of dilatations are emphasized. The formula for the form factor in the Araki-Haag limit is given.     

Dark viscous fluid coupled with dark matter and future singularity

Motivated by Kerner and Man’s fermions tunneling method of dimension 4 black holes, in this paper, we further improve the analysis to investigate Hawking radiation of charged Dirac particles with spin 1/2 from general non-extremal rotating charged black holes with two parameters and a non-zero cosmological constant in minimal five-dimensional gauged supergravity. For space-times with different horizon topology and different dimensions, constructing a set of appropriate γ ^μ matrices for general covariant Dirac equation is an important technique for the fermion tunneling method. By introducing a set of appropriate matrices γ ^μ and employing the ansatz for the spin-up spinor field, we successfully recovered the tunneling probability of charged Dirac particles and the expected Hawking temperature of the black hole, which is exactly consistent with that obtained by other methods. Moreover, the fermion tunneling method can be directly applied to the other five-dimensional charged black holes, which strengthens the validity and power of the fermion tunneling method.     

On behaviour of the form factor of decay of the 63-1k63-1k63-1kinto the dalitz pair in a region of small invariant masses

Dependence of the decay form factor \(F_{\pi ^0 \to e^ + e^ - \gamma } (s)\) on the invariant mass squared of the Dalitz pairs=(p ₊+p _?)² at smalls is calculated with the use of bound-state wave functions-solutions of a covariant single-time equation for a system of two spinor quarks. A quasipotential of one-gluon exchange and an oscillator potential are chosen as quasipotential.