Download A path towards quantum gravity

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
Dirac canonical quantization
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 freelyfalling 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, HAWKINGPENROSE, 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 nonAbelian 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.
Lorenz gauge for Maxwell theory
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.
Type-III gauge theories
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 gammamatrices, 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:
Gauge-fixed 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?