Spinorial Space-time Research Articles

Ultra-high energy physics and standard basic principles

It has not yet been elucidated whether the observed flux suppression for ultra- high energy cosmic rays (UHECR) at energies above � 4x1 0 19 eV is a signature of the Greisen-Zatsepin-Kuzmin (GZK) cutoff or a consequence of other phenomena. In both cases, violations of the standard fundamental principles of Physics can be present and play a significant role. They can in particular modify cosmic-ray interactions, prop- agation or acceleration at very high energy. Thus, in a long-term program, UHECR data can hopefully be used to test relativity, quantum mechanics, energy and momentum con- servation, properties... as well as the elementariness of standard particles. Data on cosmic rays at energies � 10 20 eV may also be sensitive to new physics generated well beyond Planck scale. A typical example is provided by the search for possible sig- natures of a Lorentz symmetry violation (LSV) associated to a privileged local reference frame (the vacuum rest frame, VRF). If a VRF exists, the internal structure of standard particles at ultra-high energy can undergo substantial modifications. Similarly, the con- ventional particle symmetries may cease to be valid at such energies instead of heading to a grand unification and the structure of may no longer be governed by standard quantum field theory. Then, the question whether the notion of Planck scale still makes sense clearly becomes relevant and the very grounds of Cosmology can undergo essential modifications. UHECR studies naturally interact with the interpretation of WMAP and Planck observations. Recent Planck data analyses tend to confirm the possible existence of a privileged space direction. If the observed phenomenon turns out to be a signature of the spinorial space-time (SST) we suggested in 1996-97, then conventional Particle Physics may correspond to the local properties of standard matter at low enough energy and large enough distances. This would clearly strengthen the cosmological relevance of UHECR phenomenology and weaken the status of the Planck scale hypothesis. Another crucial observation is that, already before incorporating standard matter and relativity, the SST geometry naturally yields a Ht = 1 law where t is the age of the Universe and H the ratio between relative speeds and distances at cosmic scale. As standard cosmology is not required to get such a fundamental result, the need for a conventional Planck scale is far from obvious and the study of UHECR can potentially yield evidence for an alter- native approach including new physics and new ultimate constituents of matter. UHECR may in particular allow to explore the possible indications of the existence of a transition scale at very high energy where the standard laws would start becoming less and less dominant and new physics would replace the conventional fundamental principles. We discuss prospects of searches for potential signatures of such a phenomenon.

Pre-Big Bang, fundamental Physics and noncyclic cosmologies

Detailed analyses of WMAP and Planck data can have significant implications for noncyclic pre-Big Bang approaches incorporating a new fundamental scale beyond the Planck scale and, potentially, new ultimate constituents of matter with unconventional basic properties as compared to standard particles. Cosmic-ray experiments at the highest energies can also yield relevant information. Hopefully, future studies will be able to deal with alternatives: i) to standard physics for the structure of the physical vacuum, the nature of space-time, the validity of quantum field theory and conventional symmetries, the interpretation of string-like theories...; ii) to standard cosmology concerning the origin and evolution of our Universe, unconventional solutions to the cosmological constant problem, the validity of inflationary scenarios, the need for dark matter and dark energy... Lorentz-like symmetries for the properties of matter can then be naturally stable space-time configurations resulting from more general primordial scenarios that incorporate physics beyond the Planck scale and describe the formation and evolution of the physical vacuum. A possible answer to the question of the origin of half-integer spins can be provided by a primordial spinorial space-time with two complex coordinates instead of the conventional four real ones, leading to a really new cosmology. We discuss basic questions and phenomenological topics concerning noncyclic pre-Big Bang cosmologies and potentially related physics.

EMERGENCE OF THE LORENTZIAN STRUCTURE IN CLASSICAL FIELD THEORY

The Lorentzian metric structure allows one to implement the relativistic notion of causality in any field theory and to define a notion of time dimension. We propose that at the microscopic level the metric is Riemannian and that the Lorentzian structure, usually thought as fundamental, is in fact an effective property, that emerges in some regions of a 4-dimensional space with a positive definite metric. We argue that a decent classical field theory for scalars, vectors and spinors in flat spacetime can be constructed, and that gravity can be included under the form of a covariant Galileon theory instead of general relativity.

From configuration to dynamics: Emergence of Lorentz signature in classical field theory

The Lorentzian metric structure used in any field theory allows one to implement the relativistic notion of causality and to define a notion of time dimension. This article investigates the possibility that at the microscopic level the metric is Riemannian, i.e. locally Euclidean, and that the Lorentzian structure, that we usually consider as fundamental, is in fact an effective property that emerges in some regions of a 4-dimensional space with a positive definite metric. In such a model, there is no dynamics nor signature flip across some hypersurface; instead, all the fields develop a Lorentzian dynamics in these regions because they propagate in an effective metric. It is shown that one can construct a decent classical field theory for scalars, vectors and (Dirac) spinors in flat spacetime. It is then shown that gravity can be included but that the theory for the effective Lorentzian metric is not general relativity but of the covariant Galileon type. The constraints arising from stability, the equivalence principle and the constancy of fundamental constants are detailed and a phenomenological picture of the emergence of the Lorentzian metric is also given. The construction, while restricted to classical fields in this article, offers a new view on the notion of time.

Large spin behavior of anomalous dimensions and short–long strings duality

We consider the semi-classical string soliton solution of Gubser, Klebanov and Polyakov which represents highly excited states on the leading Regge trajectory, with large spin in AdS5. A prescription relates this soliton solution with the corresponding field theory operators with many covariant derivatives, whose anomalous scaling dimension grows logarithmically with the spacetime spin. We explicitly derive the coefficients in the large spin expansion of the anomalous dimension in the leading ln nS/Sn and next-to-leading ln nS/Sn + 1 orders. We develop an iteration procedure which, in principle, allows us to derive all terms in the large spin expansion of the anomalous scaling dimension of twist two operators. Our string theory results are consistent with the conjectured ‘reciprocity’ relation, which has been verified to hold in perturbation theory up to five loops in N = 4 SYM. We also derive a duality relation between long and short strings.

Cosmic rays and tests of fundamental principles

It is now widely acknowledged that cosmic rays experiments can test possible new physics directly generated at the Planck scale or at some other fundamental scale. By studying particle properties at energies far beyond the reach of any man-made accelerator, they can yield unique checks of basic principles. A well-known example is provided by possible tests of special relativity at the highest cosmic-ray energies. But other essential ingredients of standard theories can in principle be tested: quantum mechanics, uncertainty principle, energy and momentum conservation, effective space-time dimensions, hamiltonian and lagrangian formalisms, postulates of cosmology, vacuum dynamics and particle propagation, quark and gluon confinement, elementariness of particles... Standard particle physics or string-like patterns may have a composite origin able to manifest itself through specific cosmic-ray signatures. Ultra-high energy cosmic rays, but also cosmic rays at lower energies, are probes of both "conventional" and new Physics. Status, prospects, new ideas, and open questions in the field are discussed. The Post Scriptum shows that several basic features of modern cosmology naturally appear in a SU(2) spinorial description of space-time without any need for matter, relativity or standard gravitation. New possible effects related to the spinorial space-time structure can also be foreseen. Similarly, the existence of spin-1/2 particles can be naturally related to physics beyond Planck scale and to a possible pre-Big Bang era.

Equivalence of three-dimensional spacetimes

A solution to the equivalence problem in three-dimensional gravity is given and a practically useful method to obtain a coordinate invariant description of local geometry is presented. The method is a nontrivial adaptation of the Karlhede invariant classification of spacetimes of general relativity. The local geometry is completely determined by the curvature tensor and a finite number of its covariant derivatives in a frame where the components of the metric are constants. The results are presented in the framework of real two-component spinors in three-dimensional spacetimes, where the algebraic classifications of the Ricci and Cotton–York spinors are given and their isotropy groups and canonical forms are determined. As an application we discuss Gödel-type spacetimes in three-dimensional general relativity. The conditions for local space and time homogeneity are derived and the equivalence of three-dimensional Gödel-type spacetimes is studied and the results compared with previous work on four-dimensional Gödel-type spacetimes.

Field Equations of Arbitrary Spin in Space-time with Torsion

A two spinor lagrangian formulation of field equations for massive particle of arbitrary spin is proposed in a curved space-time with torsion. The interaction between fields and torsion is expressed by generalizing the situation of the Dirac equation. The resulting field equations are different (except for the spin-1/2 case) from those obtained by promoting the covariant derivatives of the torsion free equations to include torsion. The non linearity of the equations, that is induced by torsion, can be interpreted as a self-interaction of the particle. The spin-1 and spin-3/2 cases are studied with some details by translating into tensor form. There result the Proca and Rarita-Schwinger field equations with torsion, respectively.

Integrability and transcendentality

We derive the two-loop Bethe ansatz for the twist operator sector of gauge theory directly from the field theory. We then analyse a recently proposedperturbative asymptotic all-loop Bethe ansatz in the limit of large spacetime spin at largebut finite twist, and find a novel all-loop scaling function. This function obeys theKotikov–Lipatov transcendentality principle and does not depend on the twist. Under theassumption that one may extrapolate back to leading twist, our result yields an all-loopprediction for the large spin anomalous dimensions of twist 2 operators. Thelatter also appears as an undetermined function in a recent conjecture of Bern,Dixon and Smirnov for the all-loop structure of the maximally helicity violatingn-point gluon amplitudes of gauge theory. This potentially establishes a direct link between the worldsheet and the spacetimeS matrix approach. A further assumption for the validity of our prediction is thatperturbative BMN (Berenstein–Maldacena–Nastase) scaling does not break down atfour-loop level or beyond. We also discuss how the result gets modified if BMN scaling doesbreak down. Finally, we show that our result qualitatively agrees at strong coupling with aprediction of string theory.

The Spinning and Curving of Spacetime: The Electromagnetic and Gravitational Fields in the Evans Field Theory

The unification of the gravitational and electromagnetic fields achieved geometrically in the generally covariant unified field theory of Evans implies that electromagnetism is the spinning of spacetime and gravitation is the curving of spacetime. The homogeneous unified field equation of Evans is a balance of spacetime spin and curvature and governs the influence of electromagnetism on gravitation using the first Bianchi identity of differential geometry. The second Bianchi identity of differential geometry is shown to lead to the conservation law of the Evans unified field, and also to a generalization of the Einstein field equation for the unified field. Rigorous mathematical proofs are given in appendices of the four equations of differential geometry which are the cornerstones of the Evans unified field theory: the first and second Maurer-Cartan structure relations and the first and second Bianchi identities. As an example of the theory, the origin of wavenumber and frequency is traced to elements of the torsion tensor of spinning spacetime.

Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere

Any eigenfunction of the laplacian on the sphere is given in terms of a unique set of directions: these are Maxwell's multipoles, their existence and uniqueness being known as Sylvester's theorem. Here, the theorem is proved by realising the multipoles are pairs of opposite vectors in Majorana's sphere representation of quantum spins. The proof involves the physicist's standard tools of quantum angular momentum algebra, integral kernels, and gaussian integration. Various other proofs are compared, including an alternative using the calculus of spacetime spinors.

Massive Spinors and dS/CFT Correspondence

Using the map between free massless spinors on d+1 dimensional Minkowski spacetime and free massive spinors on $dS_{d+1}$, we obtain the boundary term that should be added to the standard Dirac action for spinors in the dS/CFT correspondence. It is shown that this map can be extended only to theories with vertex $({\bar\p}\p)^2$ but arbitrary $d\ge1$. In the case of scalar field theories such an extension can be made only for $d=2,3,5$ with vertices $\phi^6$, $\phi^4$ and $\phi^3$ respectively.

Closed strings in Ramond-Ramond backgrounds

We write the IIB Green-Schwarz action in certain general classes of curved backgrounds threaded with Ramond-Ramond fluxes. The fixing of the kappa symmetry in the light-cone gauge and the use of supergravity Bianchi identities simplify the task. We find an expression that truncates to quartic order in the spacetime spinors and relays interesting information about the vacuum structure of the worldsheet theory. The results are particularly useful in exploring integrable string dynamics in the context of the holographic duality.

ELEMENTARY PARTICLES AND SPIN REPRESENTATIONS

We emphasize that the group-theoretical considerations leading to SO (10) unification of electroweak and strong matter field components naturally extend to spacetime components, providing a truly unified description of all generation degrees of freedoms in terms of a single chiral spin representation of one of the groups SO (13,1), SO (9,5), SO (7,7) or SO (3,11). The realization of these groups as higher-dimensional spacetime symmetries produces unification of all fundamental fermions is a single spacetime spinor.

The holonomy of the supercovariant connection and Killing spinors

We show that the holonomy of the supercovariant connection for M-theory backgrounds with $N$ Killing spinors reduces to a subgroup of $SL(32-N,\bR)\st (\oplus^N \bR^{32-N})$. We use this to give the necessary and sufficient conditions for a background to admit $N$ Killing spinors. We show that there is no topological obstruction for the existence of up to 22 Killing spinors in eleven-dimensional spacetime. We investigate the symmetry superalgebras of supersymmetric backgrounds and find that their structure constants are determined by an antisymmetric matrix. The Lie subalgebra of bosonic generators is related to a real form of a symplectic group. We show that there is a one-one correspondence between certain bases of the Cartan subalgebra of $sl(32, \bR)$ and supersymmetric planar probe M-brane configurations. A supersymmetric probe configuration can involve up to 31 linearly independent planar branes and preserves one supersymmetry. The space of supersymmetric planar probe M-brane configurations is preserved by an $SO(32,\bR)$ subgroup of $SL(32, \bR)$.

A semi-classical limit of the gauge/string correspondence

A world-sheet sigma model approach is applied to string theories dual to four-dimensional gauge theories, and semi-classical soliton solutions representing highly excited string states are identified which correspond to gauge theory operators with relatively small anomalous dimensions. The simplest class of such states are strings on the leading Regge trajectory, with large spin in AdS 5. These correspond to operators with many covariant derivatives, whose anomalous dimension grows logarithmically with the space–time spin. In the gauge theory, the logarithmic scaling violations are similar to those found in perturbation theory. Other examples of highly excited string states are also considered.

A New Approach to Spinors and Some Representations of the Lorentz Group on Them

We give a geometric realization of space-time spinors and associated representations, using the Jordan triple structure associated with the Cartan factors of type 4, the so-called spin factors. We construct certain representations of the Lorentz group, which at the same time realize bosonic spin-1 and fermionic spin-\({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}\) wave equations of relativistic field theory, showing some unexpected relations between various low-dimensional Lorentz representations. We include a geometrically and physically motivated introduction to Jordan triples and spin factors.

Transformations of Ramond–Ramond fields and space–time spinors

We explicitly construct the SO(d,d) transformations of Ramond-Ramond field strengths and potentials, along with those of the space-time supersymmetry parameters, the gravitinos and the dilatinos in type-II theories. The results include the case when the SO(d,d) transformation involves the time direction. The derivation is based on the compatibility of SO(d,d) transformations with space-time supersymmetry, which automatically guarantees compatibility with the equations of motion. It involves constructing the spinor representation of a twist that an SO(d,d) action induces between the local Lorentz frames associated with the left- and right-moving sectors of the worldsheet theory. The relation to the transformation of R-R potentials as SO(d,d) spinors is also clarified.

T-duality, space-time spinors and R–R fields in curved backgrounds

We obtain the T-duality transformations of space-time spinors (the supersymmetry transformation parameters, gravitinos and dilatinos) of type-II theories in curved backgrounds with an isometry. The transformation of the spinor index is shown to be a consequence of the twist that T-duality introduces between the left- and right-moving local Lorentz frames. The result is then used to derive the T-duality action on Ramond–Ramond field strengths and potentials in a simple way. We also discuss the massive IIA theory and, using duality, give a short derivation of “mass”-dependent terms in the Wess–Zumino actions on the D-brane world-volumes.

A unifying topological action for heterotic and type II superstring theories

The heterotic and type II superstring actions are identified in different anomaly-free decompositions of a single topological sigma-model action depending on bosonic and fermionic coordinates, X μ and ϱ A respectively, and of their topological ghosts. This model results from gauge-fixing the topological gauge symmetry δX μ = ϵ μ(z, z) ( μ = 1, 2,…, 10) and δϱ α = ϵ α(z, z) ( α = 1, 2,…, 16). From another viewpoint the heterotic and type H superstring actions emerge as two different gauge-fixings of the same closed two-form. Comments are also made concerning the possibility of relating ϱ α to a Majorana-Weyl space-time spinor superpartner of X μ .