Field Theory on Kappa-spacetime

A general formalism is developed, that allows the construction of field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable -product. Fields are elements of this function algebra. As an example, the Klein-Gordon equation is defined and derived from an action.     

Spacetime path formalism for massive particles of any spin

Earlier work presented spacetime path formalism for relativistic quantum mechanics arising naturally from the fundamental principles of the Born probability rule, superposition, and spacetime translation invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches to relativistic quantum mechanics in the literature. Because time is treated similarly to the three-space coordinates, rather than as an evolution parameter, such approaches have proved particularly useful in the study of quantum gravity and cosmology. The present paper extends the foundational spacetime path formalism to include massive, non-scalar particles of any (integer or half-integer) spin. This is done by generalizing the principle of translational invariance used in the scalar case to the principle of full Poincaré invariance, leading to a formulation for the non-scalar propagator in terms of a path integral over the Poincaré group. Once the difficulty of the non-compactness of the component Lorentz group is dealt with, the subsequent development is remarkably parallel to the scalar case. This allows the formalism to retain a clear probabilistic interpretation throughout, with a natural reduction to non-relativistic quantum mechanics closely related to the well-known generalized Foldy-Wouthuysen transformation.     

Generalized Uncertainty Principle and Deformed Dispersion Relation Induced by Nonconformal Metric Fluctuations

Considering the existence of nonconformal stochastic fluctuations in the metric tensor a generalized uncertainty principle and a deformed dispersion relation (associated to the propagation of photons) are deduced. Matching our model with the so called quantum -Poincaré group will allow us to deduce that the fluctuation-dissipation theorem could be fulfilled without needing a restoring mechanism associated with the intrinsic fluctuations of spacetime. In other words, the loss of quantum information is related to the fact that the spacetime symmetries are described by the quantum -Poincaré group, and not by the usual Poincaré symmetries. An upper bound for the free parameters of this model will also be obtained.     

Configuration ray representations in non-Abelian quantum kinematics and dynamics

Ray extensions of the regular representation of noncompact non-Abelian Lie groups are examined as generalizations of the Cartesian coordinate representation of ordinary quantum mechanics to the case of generalized non-Cartesian coordinates and generalized noncommuting momenta. (The momenta are in fact the generators of the representation, and so they satisfy the Lie algebra of the group.) The concept of configuration ray representation is introduced within this new kinematic formalism as subrepresentations of the regular representation which are embossed with the relativity theory of a given system. The main features of the mathematical formalism leading to these representations in configuration spacetime are discussed, and their importance for non-Abelian quantum kinematics and dynamics is emphasized. Two miscellaneous examples on the calculus of phase functions for configuration ray representations are given.     

On an infinite-dimensional group

We present here an infinite-dimensional Lie algebra, semi-direct product of the Poincaré Lie algebra by an infinite-dimensional abelian Lie algebra. It gives rise to Schur-irreducible subgroups of the unitary group of the (separable) Hilbert space, with a discrete mass-spectrum (real positive isolated mass-eigenvalues). Some related mathematical problems are also examined.     

Unbounded functionals on a group algebra and applications to quantum theory

A certain class of positive functionals on a group algebra is examined that is pertinent to the induced representations of Frobenius and Mackey. Though these functionals are not bounded in theL ¹ norm, continuity still persists to an extent that secures the existence of a continuous group representation obtained from Gelfand's construction. The theory thus developed provides a new aspect of both the improper states in quantum theory and the induced representations of groups. The method is applied to the Poincaré group and it is shown that the representations, in which particles can be accommodated, are determined up to unitary equivalence by unbounded functionals of a simple structure. It is stressed that representations describing an infinitely degenerate vacuum emerge from mass nonzero representations as the mass tends to zero.     

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.     

Matter as Spectrum of Spacetime Representations

Bound and scattering state Schrödinger functions of nonrelativistic quantum mechanics as representation matrix elements of space and time are embedded into residual representations of spacetime as generalizations of Feynman propagators. The representation invariants arise as singularities of rational representation functions in the complex energy and complex momentum plane. The homogeneous space GL(²)U(2) with rank 2, the orientation manifold of the unitary hypercharge-isospin group, is taken as model of nonlinear spacetime. Its representations are characterized by two continuous invariants whose ratio will be related to gauge field coupling constants as residues of the related representation functions. Invariants of product representations define unitary Poincaré group representations with masses for free particles in tangent Minkowski spacetime.     

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.     

Poincaré partially integrable local representations and mass-spectrum

An example is given of an irreducible representation of a finite-dimensional Lie algebra containing the Poincaré Lie algebra and giving rise to isolated positive masses. In addition the representation is Poincaré partially integrable (which assures the continuous physical spectrum for the energy- momentum vector) and Poincaré-covariant in a weak sense.A connection between this example and some recently published impossibility theorems is shown, and conclusions about a possible future work in this domain are also drawn.     

Maximal acceleration,maximal angular velocity,and causal influence

The upper bounds to acceleration and angular velocity which are suggested by quantum gravitational effects are described, together with the relativistic bound to velocity, by means of a cone ⁺ in the Lie algebra of the Poincaré group. The connection between these bounds and the existence of a minimal measurable length (of the order of Planck's length) is illustrated by means of a simple model. The geometric properties of the cone ⁺ and of other related structures are examined in some detail. The new geometric background requires some modifications of the concepts of causal influence and of spacetime coincidence, which are analyzed and shown to lead to some nonlocal features of the theory. Due to the smallness of Planck's length, these modifications to the causal relations cannot be observed by means of available experimental methods, but they could have some influence on the structure of elementary particles and on the very early cosmology.     

Poincaré-Invariant Markov Processes and Gaussian Random Fields on Relativistic Phase Space

We give a complete characterization, including a Lévy–Itô decomposition, of Poincaré-invariant Markov processes on , the relativistic phase space in 1+1 spacetime dimensions. Then, by means of such processes, we construct Poincaré-invariant Gaussian random fields, and we prove a no-go theorem for the random fields corresponding to Brownian motions on .     

On the integrability of the Lie algebra of the conformal group in quantum field theory

It is shown that the infinitesimal conformal symmetry implies (in any quantum field theory which satisfies the Wightman axioms without invoking locality and global Poincaré symmetry) that there exists a uniquely defined unitary representation of the universal (-sheeted) covering group of the Minkowskian conformal groupSO _e (4,2)/ ₂. Proof was obtained using sufficient conditions for the integrability of a representation of a Lie algebra given by [8].     

Quantum Kinematic Theory of the Poincare Group in Two-Dimensional Spacetime

Non-Abelian quantum kinematics is applied to thePoincare group P ₊ (1, 1),as an example of the quantization-through-the-symmetryapproach to quantum mechanics. Upon quantizing thegroup, generalized Heisenberg commutation relations are obtained, and aclosed Heisenberg–Weyl algebra follows. Then,according to the general theory, the three basicquantum-kinematic invariant operators are calculated;these afford the superselection rules for diagonalizing theincoherent rigged Hilbert space H(P ₊ ) of the regularrepresentation. This paper examines only one of thesediagonalization schemes, while introducing a irreducible spacetime representation carried by isotopicplane-wave eigenvectors of two compatible superselectionoperators (which define a Poincare-invariant linear2-momentum). Thereafter, the principle of microcausality produces massive 2-spinor isotopic states in 1+ 1 Minkowski space. The Dirac equation is thus deducedwithin the quantum kinematic formalism, and the familiarJordan–Pauli propagation kernel in 2-dimensional spacetime is also obtained as a Hurwitzinvariant integral over the group manifold. The maininterest of this approach lies in the adoptedgroup-quantization technique, which is a strictlydeductive method and uses exclusively the assumed Poincaresymmetry.     

Cohomology,central extensions,and (dynamical) groups

We analyze in this paper the process of group contraction which allows the transition from the Einstenian quantum dynamics to the Galilean one in terms of the cohomology of the Poincaré and Galilei groups. It is shown that the cohomological constructions on both groups do not commute with the contraction process. As a result, the extension coboundaries of the Poincaré group which lead to extension cocycles of the Galilei group in the nonrelativistic limit are characterized geometrically. Finally, the above results are applied to a quantization procedure based on a group manifold.     

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.     

States and operators in the spacetime algebra

The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum - and -matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each particle, and it is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed.Supported by a SERC studentship.     

Poincaré covariance and κ-Minkowski spacetime

A fully Poincaré covariant model is constructed as an extension of the κ-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincaré group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincaré covariance is realised à la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of “Poincaré covariance”.