Quantum gravity: unification of principles and interactions G. Esposito, INFN Seminar, January 11, 2007
1.1 Unification of guiding principles • Space and time are unified into the spacetime manifold (M,g), where • Light-cone structure of spacetime (g(X,X)<0: timelike; g(X,X)=0: null; g(X,X)>0: spacelike). • The classical laws of nature are written in tensor language.
Gravity in GR • Gravity is the curvature of spacetime. • Gravity couples to the energy-momentum tensor of matter through the Einstein equations
Gravity as a non-Abelian gauge theory • The action functional for gravity is diffeomorphism-invariant. • With hindsight, this was the first example of non-Abelian gauge theory (about 38 years before Yang-Mills theory).
Quantum mechanics (outline) • The world on atomic or sub-atomic scale is described by a Hilbert-space structure, or suitable generalizations. • Hydrogen-atom: infinite-dimensional Hilbert space; Spin-1/2 particle: space of spin-states is finite-dimensional.
Schrodinger picture • Schrodinger equation
Heisenberg picture • Operators evolve in time according to
Weyl quantization • Group-theoretical reformulation of CCR:
Feynman (Lagrangian approach) • Sum over all space-time histories for the Green kernel, i.e.
Tools of QM • QM regards wave functions only as a tool to study bound states (discrete spectrum), scattering states (continuous spectrum), and to evaluate probabilities (of finding the values taken by the observables of the theory).
QM in the laboratory • It is meaningless to talk about an elementary phenomenon on atomic (or sub-atomic) scale unless it is registered. • QM in the laboratory needs also an external observer and reduction of the wave packet.
1.2 Spacetime singularities • In geometry, geodesics are curves whose tangent vector X moves by parallel transport, so that eventually
The physics of geodesics: • Timelike geodesics: trajectories of freely-falling observers. • Null geodesics: trajectories of photons.
Singularity-free spacetimes • Criterion/definition: spacetime (M,g) is singularity-free if all timelike and null geodesics can be extended to arbitrary values of their affine parameter. • At a spacetime singularity in GR, all laws of classical physics would break down.
History of singularity theorems • Penrose: singularities in gravitational collapse • Hawking, Geroch, Hawking—Ellis, HAWKING-PENROSE, Kriele: spacetime singularities are generic properties of GR, provided that energy conditions hold. Very little use of the Einstein equations is made. Key role of topological and global methods in GR (plus Morse theory for Lorentzian manifolds).
1.3 Unification of all fundamental interactions • Fully established unifications: • Maxwell: electricity and magnetism (electromagnetism). • Einstein: space and time (spacetime manifold); inertial and gravitational mass. • Standard model: electromagnetic and weak forces (electroweak forces) jointly with strong.
Current status of the 4 fundamental interactions • 1. Electromagnetic • 2. Weak • 3. Strong • 4. Gravitational The large-scale structure of the universe is ruled by gravity only. All unifications beyond Maxwell involve non-Abelian gauge groups (YM or Diff(M)).
Three extreme views • 1. Gravity first in the very early universe, then all other fundamental interactions. • 2. Gravity might result from Quantum Field Theory (Sakharov). • 3. The vacuum of particle physics as a cold quantum liquid in equilibrium. Photons, gravitons and gluons as collective excitations of this liquid.
2.1 Old unification • S= action functional • There exist vector fields on the space of histories, such that
Type-I gauge theories • These equations define type-I gauge theories (e.g. Maxwell, Yang—Mills, Einstein). • All these theories, being gauge theories, need supplementary conditions, since the second functional derivative of S is not an invertible operator. After imposing such conditions, the theories are ruled by a differential operator of D’Alembert (or Laplace) type, or a non-minimal operator at the very worst.
Type-II gauge theories • Lie bracket of vector fields as for type-I theories, but the structure constants are promoted to structure functions. • Example: simple supergravity (a SUSY gauge theory of gravity, with a symmetry relating bosonic and fermionic fields) in 4 spacetime dimensions, with auxiliary fields.
Example of type-III • Theories with gravitons and gravitinos such as Bose-Fermi supermultiplets of both simple and extended supergravity in any number of spacetime dimensions, without auxiliary fields.
From SUGRA to GR • GR is naturally related to SUSY, since gauge-invariant Rarita-Schwinger Eqs. imply Ricci-flatness in 4 dimensions, which is then equivalent to vacuum Einstein equations!
Dirac is more fundamental! • The Dirac operator is here more fundamental, since the spacetime metric is entirely re-constructed from the gamma-matrices, in that
2.2 New unification • The emphasis is no longer on fields (sections of vector bundles in classical FT, operator-valued distributions in QFT) but rather on extended objects such as strings. • The various string theories are different aspects of a more fundamental theory, called M-theory.
Functional integrals for brane cosmology • In the braneworld picture, branes are timelike surfaces embedded into bulk space-time. Their normal vector N is therefore spacelike with respect to the bulk metric, i.e.
Towards the effective action • Action functional and effective action:
Vector fields on the space of histories • In general, there exist vector fields on the space of histories such that
Bulk and brane ghost operators • The bulk and brane ghost operators are therefore, in general,
Full bulk integration • The full bulk integration means
Wave function of the bulk • Thus, one first evaluates the cosmological wave function of the bulk space-time, i.e.
Five-dimensional action where
Full effective action • Eventually, the effective action results from
Bulk BRST transformations • This scheme is invariant under infinitesimal BRST transformations with anticommuting T fields
Brane BRST transformations as well as
4 key issues (among the many) • Impact of Planck-scale physics on cosmological observations (A)(B)(C). • Will GR retain its fundamental role (D)(E) ? • Renormalization group methods a viable way to do non-perturbative quantum gravity? • Singularity avoidance in quantum cosmology or string theory?