Relativistic internal time operator

We construct a self-adjoint time operator for massless relativistic systems in terms of the generators of the Poincaré group. The Lie algebra generated by the time operator and the generators of the Poincaré group turns out to be an infinitedimensional extension of the Poincaré algebra. The internal time operator generates two new entities, namely the velocity operator and the internal position operator. The transformation properties of the internal time and position operator under Lorentz boosts are different from what one would expect from relativity theory. This difference reflects the fact that the time concept associated with the internal time operator is radically different from the time coordinate of Minkowski space, due to the nonlocality of the time operator. The spectral projections of the time operator allow us to construct incoming subspaces for the wave equation without invoking Huygens' principle, as in two and one spatial dimensions where Huygens' principle does not hold.     

The Hamiltonian in relativistic systems of interacting particles

The dynamics of a system of relativistically interacting particles is determined by a set of constraints, some combination of which has been frequently identified with the Hamiltonian. These constraints differ from the generators of the Poincaré transformations, among whichp ₀ generates translations along the time axis and hence is to be considered as the energy of the system. There are thus grounds for consideringP ₀ as the appropriate Hamiltonian. In this paper we establish a close relationship between transformations generated by the constraints and those generated by the Poincaré generators. In particular we find that the true Hamiltonian is a rather complicated but well-defined function ofp ₀ and all the constraints. We show that the generators of the entire algebra of the Poincaré group can be realized in such a fashion that the Hamiltonian is correctly included among them, and such that particle world lines in Minkowski space-time generated by this Hamiltonian transform correctly under the Poincaré group.This work was partially supported by the National Science Foundation Grant No. PHY 79-0887 to Syracuse University and by Grant No. PHY 79-09405 to Yeshiva University.     

A discussion of causality and the Lorentz group

Zeeman (1964) has shown that the group of automorphisms for the relation of causality on Minkowski space is that generated by the orthochronous Poincaré (Halpern, 1968) group and dilatations. Here we prove that the group of automorphisms that preserve the time-like vectors of Minkowski space normwise is the complete Poincaré group. We prove that the timelike structure within the null cone of a single event does define the whole structure of Minkowski space. Further, it is shown that only inertial observers can use Minkowski space to describe space-time.     

Poincaré canonical momenta and nambu mechanics

We show that if certain Poincaré-like integrals are conserved, then to each configuration coordinate of a system an entity can be associated that is an acceptable generalization of the notion of canonical momentum: In the particular case of standard mechanics, the canonical momenta are retrieved. Under certain general restrictions, the Poincaré momenta make sense for either mechanical or general systems for which we do not have (or are not aware of) entities (like the Lagrangian) that are generally used to define the momentum. The Poincaré momentum may also make sense for systems whose characteristics are difficult, or impossible, to reconcile with the notion of the usual canonical momentum. It is also relevant for certain cases where a Lagrangian exists, but it leads to a mixture of physical and unphysical entities. In particular, we show that while physical canonical momenta do not generally exist in the new Nambu mechanics (because of the dimensionality of state vector space), the Poincaré momenta exist, they are physical, and have the properties we could have expected for the mechanics.     

The foundations of the poincaré group and the validity of general relativity

After discussions about accepted ideas concerning the nonlocalisability of the photon, the interpretation of the Minkowski space-time, the wave-corpuscle duality ideas of Niels Bohr and the concept of elementary particle by Eugene Wigner, the validity of the Poincaré group is brought into question and some other ideas are developed. Lukierski, Nowicki and Ruegg showed that the successes of the Poincaré group are preserved if we deform the group by introducing a constant κ. Such deformation replaces the Poincaré Hopf algebra by another one. We call such a deformation a mathematical deformation. The main inconvenience of this mathematical deformation is that the coproduct is not commutative. The consequence is that a two-particle state is defined in an ambiguous way because we must say which is the first particle and which is the second one. The only mathematical deformation of the Poincaré group which preserves the commutativity of the coproduct is the trivial one, that is the Poincaré Hopf algebra itself. That is why we reject the mathematical deformation of Lukierski, Nowicki and Ruegg. That is also why we propose what we call a physical deformation of the Poincaré group, which means that we reinterpret the Poincaré Hopf algebra, with the same constant κ. Our proposal has four advantages:

1.

1. The constant x has the dimensions of a mass. When this constant becomes infinite, we are left with the Poincaré group with its main successes.

2.

2. The two-particle states are unambiguously defined.

3.

3. The constant κ may be chosen in such a way that the search for a missing mass in the universe is useless.

4.

4. It consists in the disappearing of unphysical irreducible representations of the Poincaré group.

With the constant κ, we arrive at a reformulation of special relativity where the energy is no longer additive. This would imply a change in general relativity where the density of matter must be different from the density of energy. Unfortunately, we are not able to propose a substitute for the general relativity theory. Obviously, when the constant κ goes to infinity, the new general relativity would become the standard general relativity.     

Relativistic Nonlocal Quantum Field Theory

A theory is defined to be relativistic if its Hamiltonian, total momenta, and boost's generators satisfy commutation relations of the Poincaré group. Field theories with usual local interactions are known to be relativistic. A simple example of a relativistic nonlocal theory is found. However, it has divergences. Some conditions are obtained which are necessary in order that a nonlocal theory be relativistic and divergenceless.     

Half-sided modular inclusion and the construction of the Poincaré group

In this note we will give a construction of the Poincaré group out of the modular groups of the wedge algebras provided the groups act on the algebra of every double cone like the associated Lorentz boosts. This construction will use the concept of half-sided modular inclusions instead of the first and second cohomology of the Poincaré group as used by Brunetti, Guido and Longo. By our method we obtain directly the Poincaré group and not its covering group.     

Relativistic kinetic momentum operators

In quantum theory, in the relativistic configuration r-space, the kinetic momenta, corresponding to the half of the non-Euclidean distance in the Lobachevsky velocity space, are introduced. These operators, coinciding up to the constant factor with the generators of translations of the r-space, are the exterior derivatives of noncommutative differential calculus.     

Finite-dimensional relativistic quantum mechanics

A finite-dimensional relativistic quantum mechanics is developed by first quantizing Minkowski space. Two-dimensional space-time event observables are defined and quantum microscopic causality is studied. Three-dimensional colored even observables are introduced and second quantized on a representation space of the restricted Poincaré group. Creation, annihilation, and field operators are introduced and a finite-dimensional Dirac theory is presented.     

An Algebraic Characterization of Vacuum States in Minkowski Space. III. Reflection Maps

Employing the algebraic framework of local quantum physics, vacuum states in Minkowski space are distinguished by a property of geometric modular action. This property allows one to construct from any locally generated net of observables and corresponding state a continuous unitary representation of the proper Poincaré group which acts covariantly on the net and leaves the state invariant. The present results and methods substantially improve upon previous work. In particular, the continuity properties of the representation are shown to be a consequence of the net structure, and surmised cohomological problems in the construction of the representation are resolved by demonstrating that, for the Poincaré group, continuous reflection maps are restrictions of continuous homomorphisms.     

Derivation of Poincaré covariance from causality requirements in field theory

It is proved that the group of covariance of a non-second quantized theory of scalar fields on Minkowski space is uniquely restricted to the causality group, constituted by the group of dilatations and by the orthochronous Poincaré group, if certain causality requirements in field theory are assumed.Research supported by U.S. Air Force under Grant No. AF-AFOSR-385-67.     

q-deformed Minkowski space based on a q-Lorentz algebra

The Hilbert space representations of a non-commutative -deformed Minkowski space, its momenta and its Lorentz boosts are constructed. The spectrum of the diagonalizable space elements shows a lattice-like structure with accumulation points on the light-cone. Received: 23 January 1997 / Published online: 10 March 1998     

Conformal invariance of relativistic equations for arbitrary spin particles

We show that any Poincaré-invariant equation for particles of zero mass and of discrete spin provide a unitary representation of the conformal group, and find an explicit expression of the conformal group generators in terms of Poincaré group generators.     

On the state space of the dipole ghost

A particular representation of SO(4, 2) is identified with the state space of the free dipole ghost. This representation is then given an explicit realization as the solution space of a 4th-order wave equation on a spacetime locally isomorphic to Minkowski space. A discrete basis for this solution space is given, as well as an explicit expression for its SO(4, 2) invariant inner product. The connection between the modes of dipole field and those of the massless scalar field is clarified, and a recent conjecture concerning the restriction of the dipole representation to the Poincaré subgroup is confirmed. A particular coordinate transformation then reveals the theory of the dipole ghost in Minkowski space. Finally, it is shown that the solution space of the dipole equation is not unitarizable in a Poincaré invariant manner.     

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.     

Static vacuum solution of direct Poincaré gauge theory in ten dimensions with four external

A torsion-free solution of the free gauge field equations of direct Poincaré gauge theory on a ten-dimensional Minkowski space is constructed. This solution exhibits nontrivial curvature two-forms, but shaves the metric structure down to that of a four-dimensional Minkowski space. Universality of this solution with respect to the choice of the free field Lagrangian is established.     

Non-commutative spacetime in very special relativity

The purpose of Very Special Relativity is to show that the ISIM(2) subgroup of the Poincaré group is sufficient to describe the spacetime symmetries of the so far observed physical phenomena. A deformation of such group, called DISIMb(2), was later introduced. In the present work, we present a novel non-commutative spacetime structure, underlying the DISIMb(2), that allows us to construct explicitly the generators of the group. Exploiting the Darboux map technique, we then construct a point particle Lagrangian that lives in the non-commutative phase space proposed by us.     

Geometro-Differential Conception of Extended Particles and the Semigroup of Trajectories in Minkowski Space-Time

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal Poincaré symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup.     

Magnetohydrodynamics and cosmology

It is well known that the Maxwell equations are connected to Minkowski space-time and to the Poincaré group. If we pass to the De Sitter universe with constant curvature, i.e., to the projective relativity, we must generalize the Maxwell equations in such a way to make them invariants for the Fantappié group. We thus obtain more general equations which can be interpreted as equations of magnetohydrodynamics and which reunite in a single theory electromagnetism and relativistic hydrodynamics.     

q-deformed conformal and Poincaré algebras on quantum 4-spinors

We investigate quantum deformation of conformal algebras by constructing the quantum space forsl _q (4). The differential calculus on the quantum space and the action of the quantum generators are studied. We derive deformedsu(2,2) algebra from the deformedsl(4) algebra using the quantum 4-spinor and its conjugate spinor. The quantum 6-vector inso _q (4,2) is constructed as a tensor product of two sets of 4-spinors. We obtain theq-deformed conformal algebra with the suitable assignment of the generators which satisfy the reality condition. The deformed Poincaré algebra is derived through a contraction procedure.Work partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (#030083)