Symmetries induced by conserved vector currents in the theory of a scalar field

Let us consider a quantum theory of one scalar, real, local, Poincaré covariant fieldA(x) with the restricted spectrum condition (massive one particle states and a unique vacuum). The asymptotic fieldsA _{in out} (x) are assumed to be irreducible. Our conjecture is that under some technical assumptions the charge of every real, hermitean, locally conserved, Poincaré covariant quantum (pseudo) vector fieldj (x) relatively local toA(x), appearing in this theory-vanishes. This means that in a theory of one scalar, real field with a massive particle one can not expect to get symmetry groups induced by conserved (pseudo) vector currents, only by global, selfadjoint, Poincaré invariant generators.Our arguments can be easily extended to a theory of one complex scalar field, in this case the only symmetry transformation induced by a current can be the gauge transformation.We prove also that under very weak assumptions two fields related to each other by a unitary (or similarity) transformation are equal barring some patological cases.     

Operator-valued connections,Lie connections,and Gauge field theory

Noether's first theorem tells us that the global symmetry groupG _r of an action integral is a Lie group of point transformations that acts on the Cartesian product of the space-time manifold with the space of states and their derivatives. Gauge theory constructs are thus required for symmetry groups that act indiscriminately on the independent and dependent variables where the group structure can not necessarily be realized as a subgroup of the general linear group. Noting that the Lie algebra of a general symmetry groupG _r can be realized as a Lie algebrag _r of Lie derivatives on an appropriately structured manifold,G _r -covariant derivatives are introduced through study of connection 1-forms that take their values in the Lie algebrag _r of Lie derivatives (operator-valued connections). This leads to a general theory of operator-valued curvature 2-forms and to the important special class of Lie connections. The latter are naturally associated with the minimal replacement and minimal coupling constructs of gauge theory when the symmetry groupG _r is allowed to act locally. Lie connections give rise to the gauge fields that compensate for the local action ofG _r in a natural way. All governing field equations and their integrability conditions are derived for an arbitrary finite dimensional Lie group of symmetries. The case whereG _r contains the ten-parameter Poincaré group on a flat space-timeM ₄ is considered. The Lorentz structure ofM ₄ is shown to give a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of the Poincaré group. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of the Poincaré group.     

Additive conservation laws in local relativistic quantum field theory

We consider generatorsQ of symmetry transformations acting additively on asymptotic particle states according to (1.1). [This equation can be derived forQ defined as integral over a conserved local current!]. For simplicity, we consider only the case that all asymptotic fields are scalar. Assuming that elastic scattering occurs at least in an open subset of the scattering manifold we show thatQ is at most alinear combination of generators of the Poincaré group and internal symmetries.     

Direct gauging of the poincaré group. III. Interactions with internal symmetries

The problem of gauging matter fields with a Poincaré invariant action functional that admits anr parameter, semisimple groupG(r) of internal symmetries is considered. A minimal replacement operator for the total groupP ₁₀×G(r) is obtained, together with the requisite compensating 1-forms for both the Poincaré and theG(r) sectors. A basis forP ₁₀×G(r)-invariant Lagrangian densities for the free fields is obtained. Restriction to Lagrangian densities that are at most quadratic in the associated curvature and torsion fields eliminates active coupling between theP ₁₀ free field Lagrangian and theG(r) free field Lagrangian, although there is passive coupling that arises through the requirement of tensorial covariance under general coordinate transformations generated by the local action of the translation group. A superposition principle, modulo passive coupling, thus holds for quadratic free field Lagrangian for the total group:L _TOT=L _P +L _G(r) . Field equations for the matter fields and the compensating fields of theG(r) sector are obtained. Both share the passive coupling toP ₁₀ that is required in order to achieve tensorial covariance, but only the matter fields couple directly to the Poincaré fields and only to the Lorentz sector. For weak Poincaré fields, the field equations for the matter fields and the compensating fields of the internal symmetries go over into the standard field equations of gauge theory for an internal symmetry group.     

Bosonic symmetries of the Dirac equation

We have found on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass the new physically meaningful features of this equation. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found, among which (together with the 32-dimensional pure matrix algebra of invariance) the new spin s=(1,0) multiplet Poincaré symmetry is proved. In order to carry out the corresponding proofs a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.     

Generalised Chern–Simons actions for 3d gravity and κ-Poincaré symmetry

We consider Chern–Simons theories for the Poincaré, de Sitter and anti-de Sitter groups in three dimensions which generalise the Chern–Simons formulation of 3d gravity. We determine conditions under which κ-Poincaré symmetry and its de Sitter and anti-de Sitter analogues can be associated to these theories as quantised symmetries. Assuming the usual form of those symmetries, with a timelike vector as deformation parameter, we find that such an association is possible only in the de Sitter case, and that the associated Chern–Simons action is not the gravitational one. Although the resulting theory and 3d gravity have the same equations of motion for the gauge field, they are not equivalent, even classically, since they differ in their symplectic structure and the coupling to matter. We deduce that κ-Poincaré symmetry is not associated to either classical or quantum gravity in three dimensions. Starting from the (non-gravitational) Chern–Simons action we explain how to construct a multi-particle model which is invariant under the classical analogue of κ-de Sitter symmetry, and carry out the first steps in that construction.     

Direct gauging of the Poincaré group V. Group scaling,classical gauge theory,and gravitational corrections

Homogeneous scaling of the group space of the Poincaré group,P ₁₀, is shown to induce scalings of all geometric quantities associated with the local action ofP ₁₀. The field equations for both the translation and the Lorentz rotation compensating fields reduce toO(1) equations if the scaling parameter is set equal to the general relativistic gravitational coupling constant 8Gc ^–4. Standard expansions of all field variables in power series in the scaling parameter give the following results. The zeroth-order field equations are exactly the classical field equations for matter fields on Minkowski space subject to local action of an internal symmetry group (classical gauge theory). The expansion process is shown to breakP ₁₀-gauge covariance of the theory, and hence solving the zeroth-order field equations imposes an implicit system ofP ₁₀-gauge conditions. Explicit systems of field equations are obtained for the first- and higher-order approximations. The first-order translation field equations are driven by the momentum-energy tensor of the matter and internal compensating fields in the zeroth order (classical gauge theory), while the first-order Lorentz rotation field equations are driven by the spin currents of the same classical gauge theory. Field equations for the first-order gravitational corrections to the matter fields and the gauge fields for the internal symmetry group are obtained. Direct Poincaré gauge theory is thus shown to satisfy the first two of the three-part acid test of any unified field theory. Satisfaction of the third part of the test, at least for finite neighborhoods, seems probable.     

Constraint algebra from local Poincaré symmetry

A rigorous derivation of the constraint algebra between lapse, shift and Lorentz Hamiltonians is presented assuming that only local Poincaré symmetry constraints are present in the theory. It is also shown that the Dirac-Arnowitt-Deser-Misner form of the Hamiltonian is merely a consequence of the local Poincaré symmetry identities.     

Space-time theories and symmetry groups

This paper addresses the significance of the general class of diffeomorphisms in the theory of general relativity as opposed to the Poincaré group in a special relativistic theory. Using Anderson's concept of an absolute object for a theory, with suitable revisions, it is shown that the general group of local diffeomorphisms is associated with the theory of general relativity as its local dynamical symmetry group, while the Poincaré group is associated with a special relativistic theory as both its global dynamical symmetry group and its geometrical symmetry group. It is argued that the two groups are of equal significance as symmetry groups of their associated theories.     

The preferred null symmetries of a null electromagnetic field interacting with a null free gravitational field

The real null vector 1 ^a of the Newman-Penrose formalism is preferred to correspond to a geometrical symmetry as well as a dynamical symmetry. The 16 types of geometrical symmetries expressed through the vanishing of the Lie derivatives of certain tensor fields with respect to 1 ^a are examined separately. Two types of dynamical symmetries are imposed simultaneously on 1 ^a : A null electromagnetic field and a null gravitational field are both chosen to have the same propagation vector 1 ^a . By adopting freedom conditions on 1 ^a , it is shown that the symmetries of the null electromagnetic field are shared neither by the free gravitational field nor by the gravitational potentials. In fact the following five preferred null symmmetries are found to be proper: motion, affine collineation, special curvature collineation, curvature collineation, and Ricci collineation. The scalars characterizing the coupled fields are found to be constant with respect to 1 ^a .     

Construction of field algebras with quantum symmetry from local observables

It has been discussed earlier that (weak quasi-) quantum groups allow for a conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses the reconstruction of quantum symmetries and algebras of field operators. For every algebraA of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey a local braid relation.     

Symmetry groups, density-matrix equations and covariant Wigner functions

A representation theory for Lie groups is developed taking the Hilbert space, say , of the w^*-algebra standard representation as the representation space. In this context the states describing physical systems are amplitude wave functions but closely connected with the notion of the density matrix. Then, based on symmetry properties, a general physical interpretation for the dual variables of thermal theories, in particular the thermofield dynamics (TFD) formalism, is introduced. The kinematic symmetries, Galilei and Poincaré, are studied and (density) amplitude matrix equations are derived for both of these cases. In the same context of group theory, the notion of phase space in quantum theory is analysed. Thus, in the non-relativistic situation, the concept of density amplitude is introduced, and as an example, a spin-half system is algebraically studied; Wigner function representations for the amplitude density matrices are derived and the connection of TFD and the usual Wigner-function methods are analysed. For the Poincaré symmetries the relativistic density matrix equations are studied for the scalar and spinorial fields. The relativistic phase space is built following the lines of the non-relativistic case. So, for the scalar field, the kinetic theory is introduced via the Klein–Gordon density-matrix equation, and a derivation of the Jüttiner distribution is presented as an example, thus making it possible to compare with the standard approaches. The analysis of the phase space for the Dirac field is carried out in connection with the dual spinor structure induced by the Dirac-field density-matrix equation, with the physical content relying on the symmetry groups. Gauge invariance is considered and, as a basic result, it is shown that the Heinz density operator (which has been used to develope a gauge covariant kinetic theory) is a particular solution for the (Klein–Gordon and Dirac) density-matrix equation.     

Supergravity in (2 + 1) dimensionsfrom (3 + 1)-dimensional supergravity

In the context of the formalism proposed by Stelle-West and Grignani-Nardelli, it is shown that Chern-Simons supergravity can be consistently obtained as a dimensional reduction of (3 + 1)-dimensional supergravity, when written as a gauge theory of the Poincaré group. The dimensional reductions are consistent with the gauge symmetries, mapping (3 + 1)-dimensional Poincaré supergroup gauge transformations onto (2 + 1)-dimensional Poincaré supergroup ones.     

The Poincaré-Lyapounov-Nekhoroshev Theorem

We give a detailed and mainly geometric proof of a theorem by N. N. Nekhoroshev for hamiltonian systems in n degrees of freedom with k constants of motion in involution, where 1≤k≤n. This state's persistence of k-dimensional invariant tori, and local existence of partial action-angle coordinates, under suitable nondegeneracy conditions. Thus it admits as special cases the Poincaré-Lyapounov theorem (corresponding to k=1) and the Liouville-Arnold one (corresponding to k=n) and interpolates between them. The crucial tool for the proof is a generalization of the Poincaré map, also introduced by Nekhoroshev.     

Relativistic Chaplygin Gas with Field-Dependent Poincaré Symmetry

The relativistic generalization of the Chaplygin gas, put forward by Jackiw and Polychronakos, is derived in Duval's Kaluza–Klein framework, using a universal quadratic Lagrangian. Our framework yields a simplified proof of the field-dependent Poincaré symmetry. Our action is related to the usual Nambu–Goto action [world volume] of d-branes in the same way as the Polyakov and the Nambu action are in string theory.     

Hopf-Algebra Description of Noncommutative-Spacetime Symmetries

A brief summary is given of the results reported in [hep-th/0306013], in collaboration with G. Amelino-Camelia and F. D'Andrea. It is focused on the analysis of the symmetries of -Minkowski noncommutative spacetime, described in terms of a Weyl map. The commutative-spacetime notion of Lie-algebra symmetries must be replaced by the one of Hopf-algebra symmetries. However, in the Hopf-algebra sense, it is possible to construct an action in -Minkowski, which is invariant under a 10-generators Poincaré-like symmetry algebra.     

The cauchy problem for non-linear Klein-Gordon equations

We consider in ⁿ⁺¹,n2, the non-linear Klein-Gordon equation. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincaré covariant then the non-linear representation of the Poincaré Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincaré group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincaré group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra.     

Asymptotic locality and the structure of local internal symmetries

Symmetries are investigated from the local viewpoint. Using the Haag-Ruelle construction, the action of a local internal symmetry on the asymptotic states is determined. A condition of asymptotic locality is derived and used to show that the symmetry acts linearly and locally on the asymptotic fields. Within a field theoretical framework it is shown that the internal symmetry must commute with the Poincaré group. The general structure of an internal symmetry is determined. The uniqueness of the representation of the Poincaré group is discussed, and a simple example of an infinite component field is given to indicate what occurs when there are infinitely degenerate particle multiplets.