M. N. Stoilov

Duality between constraints and gauge conditions

There are two important sets of seemingly absolutely different objects in any gauge theory: the set of constraints, which generate the local symmetry and the set of gauge conditions, which fix this symmetry; the first one is determined by the Lagrangean of the model, the second is a matter of choice. However, in the transition amplitude constraints and gauge conditions participate in exactly the same way. This suggests the possibility for existence of a model with the same transition amplitude and in which gauge conditions and constraints are interchanged. We investigate the conditions that gauge fixing terms should satisfy so that this dual picture is allowed. En route, we propose to add new terms in the constraints which would generate the gauge transformation of the Lagrange multipliers and construct two BRST charges – one, as usual, for the constraints, and one for the gauge conditions.

A model of p-branes with closed-constraint algebra

A new model of p-branes is introduced and studied. The general solution of this model is equivalent to a particle solution of the covariant p-brane problem which the authors have obtained earlier. The constraint algebra obtained is closed in the sense that its structure functions are field independent.

Kruskal-Penrose Formalism for Lightlike Thin-Shell Wormholes

The original formulation of the “Einstein–Rosen bridge” in the classic paper of Einstein and Rosen (1935) is historically the first example of a static spherically-symmetric wormhole solution. It is not equivalent to the concept of the dynamical and non-traversable Schwarzschild wormhole, also called “Einstein–Rosen bridge” in modern textbooks on general relativity. In previous papers of ours we have provided a mathematically correct treatment of the original “Einstein–Rosen bridge” as a traversable wormhole by showing that it requires the presence of a special kind of “exotic matter” located on the wormhole throat – a lightlike brane (the latter was overlooked in the original 1935 paper). In the present note we continue our thorough study of the original “Einstein–Rosen bridge” as a simplest example of a lightlike thin-shell wormhole by explicitly deriving its description in terms of the Kruskal–Penrose formalism for maximal analytic extension of the underlying wormhole spacetime manifold. Further, we generalize the Kruskal–Penrose description to the case of more complicated lightlike thin-shell wormholes with two throats exhibiting a remarkable property of QCD-like charge confinement.

Critical dimension for a restricted bosonic string

Abstract The BFV-BRST quantization is carried out for a bosonic string model with second class constraints. The critical dimension is found to be d =25. The connection between this string model and point particle with additional local symmetry is discussed.

On the connection between Pauli‐Villars and higher derivative regularizations

We show that in some cases the gauge-invariant Pauli-Villars and higher (covariant) derivative regularizations are equivalent.

Critical dimension for a p-brane with second class constraints

Abstract Further investigation on a bosonic p -brane with second class constraints is carried out. Using the possibility for local abelianization of any constraint algebra, we find that the critical dimension of the model is D = p +24.

Higher equations of motion in N=2 superconformal Liouville field theory

Abstract We present an infinite set of higher equations of motion in N = 2 supersymmetric Liouville field theory. They are in one to one correspondence with the degenerate representations and are enumerated in addition to the U ( 1 ) charge ω by the positive integers m or ( m , n ) respectively. We check that in the classical limit these equations hold as relations among the classical fields.

ON THE CONSTRAINT ALGEBRA STRUCTURE FOR SYSTEMS WITH GAUGE TRANSFORMATIONS DEPENDING ON HIGHER ORDER TIME DERIVATIVES OF THE GAUGE PARAMETERS

In the Hamiltonian approach to the gauge models, the constraints on the one hand and the Hamiltonian and constraints on the other hand have to form closed algebras with respect to the Poisson brackets. We investigate the consequences of this requirement when the dynamical system is invariant under gauge transformations with higher order time derivatives of the gauge parameter. It is demonstrated that the required algebraic structure leads to rigid relations in the constraint algebra.

Fermions as U(1) instantons

Anomalous quantization of the electromagnetic field allows non-trivial (anti) self-dual configurations to exist in four-dimensional Euclidian space-time. These instanton-like objects are described as massless spinor particles.

On the invariant regularization of the standard model

We show that the regularization of the Standard Model proposed by Frolov and Slavnov describes a nonlocal theory with quite simple Lagrangian. 22 February 1999 * e-mail: mstoilov@inrne.bas.bg The construction of gauge invariant regularization for the chiral theories (even for the anomaly free ones) was an open problem for a long time. Only recently such a regularization was proposed for the Standard Model [1]; some generalisations and important insights on the issue can be found in Refs.[2]–[4]. The new regularization (called generalised Pauli– Villars regularization) is a version of the standard gauge-invariant Pauli–Villars one where one regularizes entire loops, but not separate propagators. The main difference is that infinitely many regulator fields are used. Therefore, in order to specify the regularization completely, one needs for any divergent diagram a recipe for how to handle the infinite sum of the terms due to regulator fields. In this letter our aim is to show that the contribution of the regulator fields can be calculated on the Lagrangian level, so as to give a nonlocal theory. We begin with a short description of the generalised Pauli–Villars regularization of the Standard Model. In Ref.[1] a construction [5] is used, where all one-generation matter fields are combined into a single chiral SO(10) spinor ψ+ (which is also a chiral Lorentzian spinor) and all gauge fields — into an SO(10) gauge field. The gauge field Lagrangian is regularized by the higher covariant derivative method and is not considered in [1] (and neither is here). In addition to the original fields an infinite set of commuting and anticommuting Pauli–Villars fields (φr and ψr respectively, r ≥ 1) is added. These new fields are simultaneously chiral Lorentzian spinors and non-chiral SO(10) ones. The explicit form of the mass terms for the regulator fields is determined by the requirement that they are nonzero, real, SO(10) and Lorentzian scalars and by the chirality properties of the fields. As a result the Lorentz and SO(10) charge conjugation matrices (CD and C) have to be used. (Basic feature of any charge conjugation matrix C is that Cψ transforms under the conjugate representation to that of ψ.) A list of properties of CD and C used in this work is given in Appendix. 1 The one-generation matter field regularized Lagrangian of the Standard Model reads Lreg = ψ̄+i 6 Dψ+ + ψ̄ri 6 Dψr + 1 2 Mr(ψ T r CDCΓ11ψr + ψ̄rCDCΓ11ψ̄ T r ) + φ̄rΓ11i 6 Dφr − 1 2 Mr(φ T r CDCφr − φ̄rCDCφ̄ T r ). (1) Here 6 D is the covariant derivative with respect to the SO(10) gauge field A, 6 D = γ(∂μ − igA μ σij), σij are the SO(10) generators; Mr = Mr, where M is a (large) mass parameter (Pauli–Villars mass) and a summation over r ≥ 1 is assumed. This form of Mr is crucial for the convergence of the diagrams in the model [2] while the concrete sign of Mr does not matter. Introducing the projectors on the irreducible spinor representations: Π± = 1 2 (1 ± γ) and P± = 1 2 (1 ± Γ11) the chirality properties of the fields read: Π−ψ+ = Π−ψr = Π−φr = 0, P−ψ+ = P±ψr∓ = P±φr∓ = 0, where ψr = ψr+ + ψr− and analogously for φr. Any SO(10) gauge model is anomaly free, and so it is not a big surprise that (1) could be rewritten in a vector-like form. Following [3] we introduce variables Ψr = ψr+ + CCDψ̄ T r−, Φr = φr+ + CCDφ̄ T r−. (2) Both these new fields are SO(10) chiral and Lorentzian non-chiral spinors contrary to the original ones. P−Ψr = P−Φr = 0, Π+Ψr = ψr+; Π−Ψr = CCDψ̄ T r−, Π+Φr = φr+; Π−Φr = CCDφ̄ T r−. Using definitions (2) and the ones following from them Ψ̄r = ψ̄r+ − ψ T r−CCD and Φ̄r = φ̄r+ − φ T r−CCD, eq.(1) takes the form Lreg = ψ̄+i 6 Dψ+ + Ψ̄r(i 6 D −Mr)Ψ + Φ̄r(i 6 D +Mr)Φ. (3) 2 The Berezian corresponding to the change of variables (2) is 1, which guarantees that Lagrangians (1) and (3) describes one and the same theory. Now we want to reformulate (3) as a higher derivative theory. Following [6] our first step is to replace the commuting Pauli–Villars fields by anticommuting ones. The idea is to consider instead of L = Φ̄(i 6 D −M)Φ (4) the following one L = (Φ̄ + χ̄)(i 6 D −M)(Φ + χ), (5) where χ is an additional dynamical field. This Lagrangian has a very large Stuckelbergtype gauge symmetry. Its fixing produces Faddeev–Popov ghosts (η and η̄) which have statistics, opposite to Φ, i.e. they are normal anticommuting spinors. A particular gauge choice brings (5) into (4) plus ghosts terms trivially decoupled from the dynamics; another gauge choice leaves only L = −η̄(i 6 D −M)η (6) (plus decoupled Φ-terms). Thus eqs.(4) and (6) describe the same physics. This is true provided there are no sources for the field Φ (and η) in the model and this is exactly the situation with the Pauli–Villars fields. Applying the procedure described above to all Φr the Lagrangian (3) could be rewritten as: Lreg = ψ̄+i 6 Dψ+ + Ψ̄r(i 6 D +Mr)Ψ − η̄r(i 6 D −Mr)η, (7) where all fields are anticommuting now. Our next step is to combine different terms in (7) into a single higher derivative Lagrangian. It was shown in [6] that the Lagrangian L = gψ̄(i 6 D −m1)(i 6 D −m2)ψ after suitable Legendre transformation could be put into the form L = g |g| ( ψ̄1(i 6 D −m1)ψ1 − ψ̄2(i 6 D −m2)ψ2 )Abstract We show that the gauge invariant regularization of the Standard Model proposed by Frolov and Slavnov describes a nonlocal theory with quite simple Lagrangian.