Transcript PROJECTIVE AND CONFORMAL STRUCTURES IN GENERAL RELATIVITY

PROJECTIVE AND
CONFORMAL STRUCTURES
IN GENERAL RELATIVITY
John Stachel
Loops ’07, Morelia
June 25-30, 2007
Work being done in
collaboration with
Mihaela Iftime
Quantum Gravity– The Great
Challenge
• How can we combine the backgroundindependent, dynamical approach to
space-time of General Relativity with
• Quantum theory, which is based on fixed,
absolute background space-time
structures?
• That is the fundamental problem of
quantum gravity
Goal
Our goal is to contribute to the
development of a backgroundindependent, non-perturbative
approach to quantization of the
gravitational field based on the
conformal and projective structures
of space-time.
Outline of the Talk
1)
2)
3)
4)
5)
6)
7)
What quantization is and is not
Measurement analysis and quantization
Processes are primary
Space-time structures
Conformal and projective structures
Variational principle based on them
Hopes for the future
Outline of the Talk
1)
2)
3)
4)
5)
6)
7)
What quantization is and is not
Measurement analysis and quantization
Processes are primary
Space-time structures
Conformal and projective structures
Variational principle based on them
Hopes for the future
What is NOT Being Claimed
Quantization only makes sense
when applied to “fundamental”
structures or entities.
The Mystique Surrounding
Quantum Mechanics
“Anything touched by this formalism
thereby seems to be elevated– or should it
be lowered?– to a fundamental
ontological status. The very words
‘quantum mechanics’ conjure up visions
of electrons, photons, baryons, mesons,
neutrinos, quarks and other exotic
building blocks of the universe.”
The Mystique Surrounding
Quantum Mechanics (cont’d)
“But the scope of the quantum mechanical
formalism is by no means limited to such
(presumed) fundamental particles. There
is no restriction of principle on its
application to any physical system. One
could apply the formalism to sewing
machines if there were any reason to do
so!” (Stachel 1986)
What IS Quantization?
Quantization is just a
way accounting for
the effects of h, the
quantum of action,
on any process
undergone by some
system:
“fundamental”
or
“composite”
What IS Quantization?
Quantization is just a way accounting for
the effects of h, the quantum of action, on
any process involving some system,–
or rather on theoretical models of such a
system-- “fundamental” or “composite”,
in which the collective behavior of a set of
more fundamental entities is quantized
The Quantum of Action
"Anyone who is not
dizzy after his first
acquaintance with
the quantum of
action has not
understood a
word."
Niels Bohr
“Atoms and Human Knowledge”
--Niels Bohr 1957
“..an element of wholeness, so to speak, in the physical
processes, a feature going far beyond the old doctrine
of the restricted divisibility of matter. This element is
called the universal quantum of action. It was
discovered by Max Planck in the first year of this
century and came to inaugurate a whole new epoch in
physics and natural philosophy. We came to
understand that the ordinary laws of physics, i.e.,
classical mechanics and electrodynamics, are
idealizations that can only be applied in the analysis of
phenomena in which the action involved at every stage
is so large compared to the quantum that the latter can
be completely disregarded.
Some Non-fundamental
Quanta
1) quasi-particles: particle-like entities
arising in certain systems of interacting
particles, such as phonons and rotons in
hydrodynamics (Landau 1941)
2) phenomenological photons: quantized
electromagnetic waves in a homogeneous,
isotropic dielectric (Ginzburg 1940)
Solid-state physics: A polariton
laser
Leonid V. Butov (Nature 447, 31 May 2007)
At the foundation of modern quantum physics,
waves in nature were divided into
electromagnetic waves, such as the photon, and
matter waves, such as the electron. Both can
form a coherent state in which individual waves
synchronize and combine. A coherent state of
electromagnetic waves is known as a laser; a
coherent state of matter waves is termed a
Bose–Einstein condensate. But what if a
particle is a mixture of an electromagnetic wave
and matter? Can such particles form a
coherent state? What does it look like?
Solid-state physics: A polariton
laser (cont’d)
If the microcavity is the right width, the
energies of the cavity photon and the exciton
can be made to match up. When this happens
the two mix, forming a new particle. This is a
combination of matter and electromagnetic
waves — an exciton-polariton, or simply
'polariton'. These polaritons inherit some of the
lightness of the cavity photons, and have
masses much smaller than me.
Successful Quantization
Successful quantization of some classical
formalism does not mean that one has
achieved a deeper understanding of
reality– or better, an understanding of a
deeper level of reality. It means that one
has successfully understood the effects of
the quantum of action on the phenomena
(processes) described by the formalism
Peaceful Coexistence in QG
Having passed beyond the quantum
mystique, one is free to explore how to
apply quantization techniques to various
formulations of a theory without the need
to single one out as the unique “right”
one. One might say: “Let a hundred
flowers blossom, let a hundred schools
contend”
Three Morals of This Tale
(1)
If two such quantizations at different
levels are carried out, one may then
investigate the relation between them
Example: Crenshaw demonstrates: “A
limited equivalence between microscopic
and macroscopic quantizations of the
electromagntic field in a dielectric”
[Phys. Rev. A 67 033805 (2003)]
Three Morals of This Tale
(1 cont’d)
If two such quantizations at the same
level are carried out, one may also
investigate the relation between them
Example: the relation between loop
quantization and usual field quantization
of the electromagnetic field: If you
“thicken” the loops, they are equivalent
(Ashtekar and Rovelli 1992)
Three Morals of This Tale
(2)
The search for a method of
quantizing space-time structures
associated with the Einstein
equations is distinct from:
The search for an underlying theory
of all “fundamental” interactions
Carlo Rovelli, 2004
I see no reason why a quantum theory of
gravity should not be sought within a
standard interpretation of quantum
mechanics (whatever one prefers). …A
common [argument] is that in the Copenhagen interpretation the observer must
be external, but it is not possible to be
external from the gravitational field. I
think that this argument is wrong; if it
was correct it would apply to the Maxwell field as well (Quantum Gravity, 370).
Carlo Rovelli, 2004
We can consistently use the Copenhagen
interpretation to describe the interaction
between a macroscopic classical apparatus and
a quantum-gravitational phenomenon
happening, say, in a small region of
(macroscopic) spacetime. The fact that the
notion of spacetime breaks down at short scale
within this region does not prevent us from
having the region interacting with an external
Copenhagen observer (ibid.)
Three Morals of This Tale
(3)
An attempt to quantize the conformal
and projective structures does not
negate, and need not replace,
attempts to quantize other spacetime structures. Everything depends
on the utility of the results in
explaining some physical processes
Outline of the Talk
1)
2)
3)
4)
5)
6)
7)
What quantization is and is not
Measureability analysis and quantization
Processes are primary
Space-time structures
Conformal and projective structures
Variational principle based on them
Hopes for the future
Commutation Relations
One central method of taking into
account the quantum of action is by
means of introducing commutation
relations between various particle (nonrel QM) or field (SR QFT) quantities into
the formalism.
But these commutation relations have
more than a purely formal significance
Peter G. Bergmann
Collaborator of
Einstein
Pioneer in study of
quantization of
“generally
covariant” theories,
including GR
Bergmann and Smith 1982
Measurability Analysis for the Linearized
Gravitational Field
“Measurability analysis identifies those
dynamic field variables that are
susceptible to observation and
measurement (“observables”), and
investigates to what extent limitations
inherent in their experimental
determination are consistent with the
uncertainties predicted by the formal
theory.”
Bergmann & Smith 1982 (cont’d)
Measurability analysis of linearized GR,
treated as massless spin 2 field, showed
limits of co-measurability of “electric”
and “magnetic” components of the
linearized Riemann tensor coincide with
limits of co-definability imposed by the
covariant commutation relations.
Bergmann & Smith 1982 (cont’d)
As in Bohr-Rosenfeld analysis of the comeasurability of electric and magnetic
field components, rather than a test point
particle, they had to use averages over
test bodies occupying space-time regions,
and with a similar treatment of the
commutation relations.
Louis Crane
”Categorical
Geometry and
the Mathematical Foundations of
Quantum
Gravity” (2006)
Louis Crane:
”Categorical Geometry and the Mathematical Foundations
of Quantum Gravity” (2006)
“The ideal foundation for a quantum theory of
gravity would begin with a description of a
quantum mechanical measurement of some
part of the geometry of some region; proceed to
an analysis of the commutation relations
between different observations, and then
hypothesize a mathematical structure for
space-time which would contain these relations
and give general relativity in a classical limit.
We do not know how to do this at present.”
Stachel
”Prolegomena to any future Quantum
Gravity” (2007)
“ ‘measurability analysis’… is based on ‘the relation
between formalism and observation’; its aim is to shed
light on the physical implications of any formalism: the
possibility of formally defining any physically
significant quantity should coincide with the possibility
of measuring it in principle; i.e., by means of some
idealized measurement procedure that is consistent
with that formalism.
Non-relativistic QM and special relativistic quantum
electrodynamics, have both passed this test ; and its use
in QG is discussed in Section 4.
Amelino-Camelia and Stachel
”Measurement of the space-time interval between two events …” (2007)
We share the point of view emphasized
by Heisenberg and Bohr and Rosenfeld,
that the limits of definability of a
quantity within any formalism should
coincide with the limits of measurability
of that quantity for all conceivable (ideal)
measurement procedures. For wellestablished theories, this criterion can be
tested. For example, in spite of a serious
challenge, source-free quantum electrodynamics was shown to pass this test.
Amelino-Camelia and Stachel
(cont’d)
In the case of quantum gravity, our situation is
rather the opposite. In the absence of a fully
accepted, rigorous theory, exploration of the
limits of measurability of various quantities can
serve as a tool to provide clues in the search for
such a theory: If we are fairly certain of the
results of our measurability analysis, the
proposed theory must be fully consistent with
these results.”
Outline of the Talk
1)
2)
3)
4)
5)
6)
7)
What quantization is and is not
Measurement analysis and quantization
Processes are primary
Space-time structures
Conformal and projective structures
Variational principle based on them
Hopes for the future
Lee Smolin
“[R]elativity theory and quantum theory each ...
tell us-- no, better, they scream at us- that our
world is a history of processes. Motion and
change are primary. Nothing is, except in a
very approximate and temporary sense. How
something is, or what its state is, is an illusion.
It may be a useful illusion for some purposes,
but if we want to think fundamentally we must
not lose sight of the essential fact that 'is' is an
illusion. So to speak the language of the new
physics we must learn a vocabulary in which
process is more important than, and prior to,
stasis.“ (p. 53).
Carlo Rovelli, 2004
“The data from a local experiment
(measurements, preparation, or just
assumptions) must in fact refer to the
state of the system on the entire boundary of a finite spacetime region. The field
theoretical space ... is therefore the space
of surfaces Σ [where Σ is a 3d surface
bounding a finite spacetime region] and
field configurations φ on Σ . Quantum
dynamics can be expressed in terms of an
[probability] amplitude W[Σ , φ].
Rovelli, 2004 (cont’d)
Following Feynman’s intuition, we can
formally define W[Σ , φ] in terms of a
sum over bulk field configurations that
take the value φ on Σ. … Notice that the
dependence of W[Σ, φ] on the geometry
of Σ codes the spacetime position of the
measuring apparatus. In fact, the relative
position of the components of the
apparatus is determined by their physical
distance and the physical time elapsed
between measurements, and these data
are contained in the metric of Σ.” (p. 23).
Rovelli, 2004 (cont’d)
Consider now a background independent
theory. Diffeomorphism invariance implies
immediately that W[Σ , φ] is independent of Σ
... Therefore in gravity W depends only on the
boundary value of the fields. However, the
fields include the gravitational field, and the
gravitational field determines the spacetime
geometry. Therefore the dependence of W on
the fields is still sufficient to code the relative
distance and time separation of the components
of the measuring apparatus! (p. 23)
Rovelli, 2004 (cont’d)
What is happening is that in backgrounddependent QFT we have two kinds of
measurements: those that determine the
distances of the parts of the apparatus and the
time elapsed between measurements, and the
actual measurements of the fields’ dynamical
variables. In quantum gravity, instead,
distances and time separations are on an equal
footing with the dynamical fields. This is the
core of the general relativistic revolution, and
the key for back-ground- independent QFT
Outline of the Talk
1) What quantization is and is not
2) Measurement analysis and quantization
3) Processes are primary:
Loops– but what kind?
1) Space-time structures
2) Conformal and projective structures
3) Variational principle based on them
4) Hopes for the future
Loops and Holonomies
Loop integrals and holonomies may
well be the most suitable set of
observables for quantization. But:
Loop integrals of a one-form are
independent of all other space-time
structures. Why confine the loops to
space-like hypersurfaces?
Example: The Two AharonovBohm Effects
There are two Aharonov-Bohm
effects:
1) Magnetic: involving space-like
loops
2) Electric: involving time-like loops
Aharonov-Bohm Effect (cont’d)
In the terms of modern differential
geometry, the Aharonov-Bohm effect can
be understood to be the holonomy of the
complex-valued line bundle representing
the electromagnetic field. The connection
on the line bundle is given by the
electromagnetic potential A, and thus the
electromagnetic field strength is the
curvature of the line bundle F=dA. The
integral of A around a closed loop is the
holonomy…
Electric Aharonov-Bohm effect
Just as the phase of the wave function deends upon the magnetic vector potential,
it also depends upon the scalar electric
potential. By constructing a situation in
which the electrostatic potential varies
for two paths of a particle, through regions of zero electric field, an observable
Aharonov-Bohm interference phenomenon from the phase shift has been predicted; again, the absence of an electric field
means that, classically, there would be no
effect.
Outline of the Talk
1)
2)
3)
4)
5)
6)
7)
What quantization is and is not
Measurement analysis and quantization
Processes are primary
Space-time structures
Conformal and projective structures
Variational principle based on them
Hopes for the future
Space-time Structures
There are a number of
space-time structures that
play an important role in
the general theory of
relativity.
Pseudo-Metric tensor field
The chronogeometry is
represented
mathematically
by a pseudoRiemannian
metric tensor
field on a fourdimensional
manifold
Bernhard Riemann- From Globally to
Locally Euclidean
Assume that the
spatial metric is
locally Euclidean,
but globally non
flat: there is a
curvature that
varies from point to
point. What kind of
curvature?
Karl Friedrich Gauss & Surface
Curvature
What are the radii R1,
R2, .. of the circles
that best fit the
normal cross sections
of the surface?
The product of the
inverses 1/(RmaxRmin)
is the Gaussian
curvature
Higher Dimensions
Riemann extended Gauss’ ideas
about curvature of a 2-D surface to
higher dimensions, and related this
curvature to the (metric) Riemann
tensor:
R[μν][κλ]
Another Kind of Curvature
But there is another kind of
curvature, more important
for geometrizing gravitation
To understand it, we must
turn back to Hermann
Grassmann and affine spaces
Affine Space –
Parallelism is all!
Forget about
distance, keep
concept of parallel
lines;
Only the ratio of
parallel intervals
has meaning;
Affine Curvature
Tullio Levi Civita did
for affine space what
Riemann had done
for Euclidean space:
He went from global
to local, and this
enabled a new
interpretation of
curvature
Affine Curvature
Geodesic Deviation
Geodesic Deviation (cont’d)
Is there any relative acceleration between
two nearby freely-falling test bodies (i.e.,
each following a geodesic)
The amount of such relative accelerations
in various directions is a measure of the
components of the affine curvature tensor
Hermann Weyl and Affine
Connections
The inertiogravitational field is
represented by a
symmetric affine
connection Γκμν on
the space-time
manifold
But Geodesic Deviation
Physically = Tidal Forces !!
Tidal Force
“The tidal force is a secondary
effect of the force of gravity and
is responsible for the tides. It
arises because the gravitational
field is not constant across a
body's diameter” (Wikipedia).
Parallel Displacement: A Post-Mature
Concept
Riemann went from global Euclidean to local
Euclidean with Gaussian curvature (1854)
Grassmann developed global affine geometry
(1844, 1862)
Someone should have gone from global affine
to locally affine with Riemannian curvature by
1880 at the latest– but no one did (poor AE!)
Compatibility Conditions
In GR there are compatibility conditions
between the chrono-geometry and the
inertio-gravitational field.
Mathematically:
The covariant derivative of the metric with
respect to the connection must vanish:
g;k = 0.
Physical Interpretations
Pseudo-metric tensor field g:
Chrono-geometry governs behavior of
rods and clocks, proper spatial, temporal
and null intervals between events;
Affine connection field Γκμν :
Inertio-gravitational field governs
behavior of (structureless) particles,
geodesic curves (paths) together with
preferred parameterization =proper
tiime);
Physical Interpret’ns (cont’d)
Compatibility conditions Dg = 0 :
1) proper spatial and temporal intervals
are preferred parameters along affine
parallel (geodesic) paths
2) rods and clocks still read proper spatial
and temporal intervals as they move in
inertio-gravitational field
Original 2nd Order Formalism
Start with just the metric field g and derive
from it the unique symmetric connection (the
Christoffel symbols {...}) that identically satisfies
the compatibility conditions.
Variation of a second order Lagrangian
(integrand = the densitized curvature scalar
√-g R )
leads to the field equations of second order in
the metric. This is the route first taken by
Hilbert, and still followed by most textbooks.
First Order Formalism
It is also possible to treat metric and
connection initially as independent
structures, and then derive the
compatibility conditions between them,
together with the first order field
equations (written initially in terms of the
connection), from a so-called Palatini
variational principle (not really what
Palatini did, but AE called it so).
Metric-Connection Duality
In this approach, metric and connection are in a
sense dual, like coordinates and momenta in
mechanics.
Major technical advances in the canonical
quantization program came when:
rather than taking the three-metric as
coordinates (geometrodynamics)
A three-connection was taken as coordinates
(loop quantum gravity)
Tetrads, Forms
Either the second or first order method
may be combined with a tetrad
formalism for the metric, introducing
tetrad vectors eμA (& dual eμA) and tetrad
metric ηAB , with corresponding local
symmetry group SO(3,1) or. if spinor
formalism is used, SU(2), together with
Tetrads, Forms (cont’d)
a corresponding mathematical
representation of the connection,
e.g., matrix of connection one-forms
ΓμAB , or tetrad components of the
connection ΓCAB , or corresponding
spin connections with local
symmetry group GL(4, R).
Outline of the Talk
1)
2)
3)
4)
5)
6)
7)
What quantization is and is not
Measurement analysis and quantization
Processes are primary
Space-time structures
Conformal and projective structures
Variational principle based on them
Hopes for the future
Further Decomposition
But neither metric nor connection is irreducible.
One can go a step further and decompose both
metric and affine connection themselves.
In homogeneous (flat) geometries:
metric geometry is a special case of
conformal geometry;
affine geometry is a special case of
projective geometry
.
Felix Klein’s Erlangen
Program
"The remarkable
power of this
program is revealed
in its applicability to
situations which
Klein himself had not
yet envisaged."
H. S. M. Coxeter
Erlangen Program (cont’d)
Different geometries can
co-exist, because they dealt
with different types of
propositions and
invariances related to
different types of
symmetry transform’ns.
By abstracting the
underlying symmetry
groups from the
geometries, the relations
between them can be reestablished at the group
level.
Erlangen Program (example)
The distinction between affine geometry and
projective geometry lies just in the fact that
affine-invariant notions such as parallelism are
the proper subject matter of the first, while not
being principal notions in the second. Since the
group of affine geometry is a (non-empty)
subset of the group of projective geometry, any
notion invariant in projective geometry is a
priori meaningful in affine geometry; but not
the other way round. If you add required
symmetries, you have a more powerful theory
(which is deeper and more general) but with
fewer concepts and theorems .
Elie Cartan takes conformal and
projective from global to local
Sur les variétés à
connexion
projective (1922)
Sur les espaces a
connexion
conforme (1923)
G-Structures
G-structures are the continuation of the
Klein program in spaces that are only
locally flat in the appropriate sense of the
word.
Keep the local group but drop the
translations, replacing them with some
non-flat Cartan connection
Conformal Structure
If one abstracts from the fourvolume-defining property of the
metric, one gets the conformal
structure on the manifold.
Physically, this conformal structure
represents the causal structure of
space-time. One can measure the
conformal structure of space-time by
using massless test entities
The Eikonal Equation for Light
gμνΦ,μΦ,ν = 0 ,
where Φ is the phase of the light
wave,
is invariant under conformal
transformations
Light rays are the bi-characteristics
of this equation
Projective Structure
Similarly, if one abstracts from the
preferred parametrization (proper length
along spacelike, proper time along
timelike) of the affine parallel curves
associated with an affine connection, one
gets a projective structure on the
manifold.
Physically the projective structure picks
out the class of preferred paths in spacetime. One can measure the projective structure
by use of massive test bodies
Projective Transformations
Γ’κμν= Γκμν + δκμ pν+ δκν pμ, ,
where pν is an arbitrary vector, preserve
the affine parallel paths.
Since they affect only the trace of the
affine connection, we can form
projectively invariant quantities by
removing the trace:
Projective parameters:
Πκμν= Γκμν–1/5(δκμΓν+δκνΓμ),
Γμ= Γκμκ
Parallel paths:
d2xκ/dλ2 + Πκμν (dxμ/dλ)(dxν/dλ) = 0
where λ is the projective parameter,
invariant under all projective
transformations
E-P-S: From Conformal and
Projective Strs to Metric Str
Ehlers, J., Pirani, F.,
and Schild, A. (1973).
"The Geometry of
Freefall and Light
Propagation", in L.
O' Raifeartaigh (ed.),
General Relativity:
Papers in Honour of
J.L. Synge. (Oxford:
Clarendon Press), 6384.
The geometry of free fall and light propagation
J. Ehlers, F.A.E. Pirani and A. Schild
The geometry of the space-time has a
conformal Lorentzian structure that
explains the phenomenon of light
propagation.
In order to obtain the Lorentzian metric
structure of the space-time, just as in
General Relativity theory, these authors
carry out the following steps.
E-P-S ( cont’d)
Firstly, they introduce a projective structure
that represents the movement of free test
particles, suffering gravitational effects only.
Then, a compatibility condition is necessary to
link the projective and conformal structures in
order to obtain a symmetric linear connection,
the so called Weyl connection. This
mathematical condition is physically justified
by the experimental fact that a particle with
mass can be made to chase a photon arbitrarily
closely .
E-P-S ( concl’n)
Finally, they introduce the axiom which affirms
that the Ricci tensor of the connection must be
symmetric so that the Weyl connection becomes
the linear connection associated with a
Lorentzian metric.
Once this axiom at hand has been physically
justified, they deduce the existence of a
Lorentzian metric structure, uniquely
determined except for a constant factor.
2 Compatibility Conditions
A compatibility condition between
the causal and projective structures
can be defined, leading to a Weyl
connection and a further condition
that guarantees the existence of a
metric compatible with the affine
connection
First Compatibility Condition
If it is obeyed, the affine connection is
fixed uniquely. So concept of proper time
along a timelike world line emerges from
this condition.
But this proper time subject to a second
clock effect: two initially synchronous
clocks will not run at the same rate when
reunited.
Second Compatibility
Condition
If it is obeyed, the second clock effect
disappears.
Moreover, the conformal factor is fixed,
so a unique four-volume element is
defined
Conformal
Structure
Volume
Structure
Projective
Structure
Weyl
connection
Lorentz
metric
Linear Symmetric
connection
Summary
• Conformal structure: measured by
massless test entities
• Projective structure: measured by
massive test entities
• First compatibility condition: Concept of
proper time emerges
• Second compatibility condition: Concept
of proper four-volume emerges
Outline of the Talk
1)
2)
3)
4)
5)
6)
7)
What quantization is and is not
Measurement analysis and quantization
Processes are primary
Space-time structures
Conformal and projective structures
Variational principle based on them
Hopes for the future
New Palatini-Type Variational
Principle
In a manifold with metric, its
conformal and affine structures
suffice to determine that metric up to
an overall constant factor
(Weyl1921).
This circumstance enables us to set
up a new first order, Palatini-type
variational principle for general
relativity.
Further Breakup
Break up the metric into its volumedetermining part (essentially, the
determinant of the metric) and its
conformal, causal structuredetermining part (with unit
determinant);
Further Breakup (cont’d)
Break up the affine connection into
its projective, preferred pathdetermining part (the trace-free part
of the connection) and its preferred
parameter-determining part ( the
trace of the connection).
Each of these four parts may then be
varied independently
Pseudo-Metric
The pseudo-metric gν can be built up out
of two parts gν and γ.
The first part gν
is a tensor density of weight -1/2, with
determinant = -1, invariant under
conformal transformations.
Pseudo-Metric (cont’d)
The second part γ is a scalar density of weight
+1/2 and transforms under conformal
transformations as:
γ → 'γ = φ γ .
Then, we may define
gν= γ gν ,
so it is a pseudo-metric tensor; with det g= -γ4
and inverse
gν = γ-1gν
Pseudo-Metric (cont’d)
Since det gν = -1, the components of its
inverse gν will be cubic functions of the
components of gν.
Thus, the Lagrangian will be cubic in
these variables– important for
quantization?
The Connection
The connection Γκμν can be built up out of
two parts, Pκμν and Γμ . The first part Pκμν
is traceless, Pκμκ = 0, and invariant under
projective transformations.
The second part Γμ transforms under a
projective transformation as:
Γμ→ Γμ+Tμ .
The Connection (cont’d)
The connection is then:
Γκμν = Pκμν +(1/5) (δκμΓν + δκνΓμ) .
Since Pκμν is traceless,
Γμ = Γκμκ ,
Conformal Curvature
Tensor
There is a decomposition of the Riemann tensor in
terms of the Weyl conformal curvature tensor C[μ] [κλ] ,
the traceless part of the Ricci tensor Sν
Sν = Rμ -1/4 Rgν ,
and the Ricci scalar R:
R[ν][κλ] = C[ν][κλ] + E[ν][κλ] + (1/12) R G[ν][κλ] ,
where:
E[ν][κλ] = 1/2(Sκgλ -Sλgκν +Sλgκ -Sκ gλ) ,
G[ν][κλ] = gκgλ -gλgκν
Conformal Curvature
Tensor (cont’d)
The conformal curvature tensor with one
raised index C [κλ] = γ-1gκ C[κ][κλ] is
invariant under conformal transformations, so it cannot depend on γ but only
on g ν.
This means that C[][κλ] must be γ times a
function of gν. and its derivatives.
Projective Curvature Tensor
The affine curvature tensor A [κλ] , formed
from the affine connection and its first
derivatives, can be written in terms of Pκμν and
Γμ , and their derivatives. There is a decomposition of the affine curvature tensor in terms of
the projective curvature tensor Pν [κλ] , which is
also traceless, and the Ricci tensor Aν of the
affine curvature tensor:
A [κλ] = P [κλ]+ 2 (δνμP[κλ])+ 2(δν[κPλ]μ) ,
Projective Curvature
Tensor(cont’d)
where:
Pκλ = (4/15)Aκλ - (1/15)Aλκ .
This definition of Pκλ simplifies the breakup of
the affine curvature tensor, and it has a very
simple transformation law under projective
transformations.
The projective curvature tensor P [κλ] is
invariant under projective transformations, so
it cannot depend on Γ .
Lagrangian
The Lagrangian density is then:
L = γgνAμν (Pκμν, Γμ),
and variations are to be taken with
respect to
γ , gν, Pκμν and Γμ.
Compatibility Conditions
Variations with respect to Pκμν and Γν give
2 compatibility conditions between the
conformal and projective structures:
Dλ{[γgν]-(1/5)(δμλDκ[γgνκ] + δνλDκ[ γgκ]) = 0.
Dν[γgν]= 0 ,
where Dλ stands for the covariant
derivative w.r.t. the affine connection
Weyl Space
Dλ{[γgν]-(1/5)(δμλDκ[γgνκ] + δνλDκ[γgκ]) = 0
implies that the conformal and projective
structures together define a Weyl space.
If we define a tensor density Qλν:
Dλ[γgν] = Qλν,
then the equation above can be written:
Qλν-1/5(δμλQκνκ + δνλ Qκκ) = 0 .
These are 36 equations for 40 unknowns
Weyl Space (cont’d)
Setting:
Qλν = - γgν Qλ ,,
where Qλ is an arbitrary one-form, it is
easily shown it is the general solution.
But the condition:
Dλ[γgν] = - γgνQλ
defines a Weyl space .
Weyl Space (cont’d)
The affine connection so defined is actually
unique. If one carries out a conformal
transformation on the metric gν →φ gν , then
a corresponding transformation:
Qλ → Qλ - ∂λ log φ
leaves the connection invariant. Thus, in a
Weyl space, the connection is fixed, but the
metric is only fixed up to a conformal factor.
From Weyl to PseudoRiemannian Space
Imposing the second equation:
Dν[γgν]= 0
forces Qλ to vanish. We then have a
pseudo-Riemannian metric and
connection
Relation to Causal Sets
This breakup is of special
interest because the
causal set theory
approach to quantum
gravity, developed by
Rafael Sorkin (but cf.
Kronheimer and
Penrose), is based on
taking the conformal
structure and the fourvolume structure as the
primary constituents of
the classical theory, and
then replacing them with
discretized versions
Rel’n to Causal Sets (cont’d)
The causal set approach combines
the two ideas of:
discreteness (four-volume structure)
and order (conformal structure)
to produce a structure, on which a
theory of quantum gravity can be
based.
.
Problems With Causal Sets
However, in causal set theory:
1) quantization of four-volume is simply
inserted by hand;
2) no attention has been paid to the affine
connection, and hence the possibility of
finding quantum analogues of the
traceless (projective) and trace parts of
the connection;
3) no real quantization has been achieved:
where is h?
Outline of the Talk
1)
2)
3)
4)
5)
6)
7)
What quantization is and is not
Measurement analysis and quantization
Processes are primary
Space-time structures
Conformal and projective structures
Variational principle based on them
Hopes for the future
“Zukunftsmusik” (Promissory
Notes)
Program of work:
Program
Extend the classical measurability
analysis of the conformal and projective
structures from point particles to
extended massless test entities and
massive test bodies.
Program
Investigate effects of quantum of action
on the measurability and comeasurability of these structures
See what this implies about possible
commutation relations between them
Program
Investigate the effect of the quantum of
action on the compatibility conditions.
In particular, what are the implications
for possible quantization of the proper
time and proper four-volume?
Program
• Reformulate the theory in terms of
Cartan connections and curvatures for
the conformal and projective structures
• Investigate the conformal and projective
holonomies, and:
• Their relation to LQG based on metric
and affine connection
Program
The conformal structure singles out null
hypersurfaces, which suggest application of
null-hypersurface quantization techniques.
Null hypersurface breaks up (2+1) naturally
into spacelike 2-surfaces plus null congruence,
suggesting use of conformal 2-structure as the
dynamical degrees of freedom of the field.
Program
Carry out the quantization program for
the cylindrical wave models (midisuperspaces) with both degrees of
polarization, and compare with existing
results
Final Summary
Classically, the conformal and projective
structures provide a sort of minimal basis
for constructing Lorentzian space-times
based on the behavior of massless and
massive entities.
Study of their quantization seems a
worthwhile avenue of approach to the
problem of quantum gravity
Decomposition of PseudoMetric
Pseudo-metric Structure: gμν = ηAB eμA eνB
Conformal Structure: {k eμA } , where k is any
non-vanishing scalar field
Volume Structure: {│eμA│} , where the value of
the determinant remains fixed
Decomposition of Affine
Structure
Affine Structure: ΓμAB , matrix-valued one-form
Due to antisymmetry, ΓμAA vanishes, so we have
to construct the trace
The physical components of the connection are:
ΓBAC = ΓμAC eμB
Decomposition of Affine
Structure (cont’d)
Their trace is:
ΓC = ΓAAC = ΓμAC eμA
We convert it into a one-form by contracting it with eνC:
Γν = eνC ΓC
Cartan-Maurer Equations
Insert this decomposition into the CartanMaurer structure equations to determine
the Riemann two-form, etc.
Intersection of G-structures of first or second
order
I. Sanchez-Rodrıguez
Abstract. The concept of G-structure of first or
second order includes the majorityof
geometrical structures that can be defined on a
manifold. We analyse the intersection of Gstructures with different kinds of G. In
particular, we discuss the relations among
conformal, (semi) Riemannian, volume and
projective geometrical structures on a fixed
manifold. Some consequences are pointed out
relative to geometrical aspects of General
Relativity.
Sanchez-Rodriguez
The results, that I will show here, prove
that given a volume structure and a
conformal structure, we obtain a uniquely
determined metric structure.
This suggests that we must investigate the
physical motivation in order to introduce a
volume structure as a component of the
space-time geometry.
Reductions of a Principal
Bundle
Let G be a Lie group and let P(M,G) be a
principal bundle over M with G as
structure group.
If we have a closed subgroup H of G, we
define a reduction Q(M,H) of P, or an Hreduction, as a subbundle Q of P with H
as structure group.
Semi-Riemannian structures
Let η be the standard scalar product on Rn of a
fixed signature.
Oη(n) -structures or (semi) Riemannian
structures are the reductions of LM with
subgroup
Oη(n) :={a ε GL(n,R) : a†a = In }
where
In is the identity matrix in GL(n,R)
a† is the unique matrix such that
η(v, a†w) = η(av,w), v,w ε Rn.
Conformal Structures
The COη(n)-structures or conformal structures
are the reductions of LM with subgroup
COη(n) := {a  GL(n,R) : a†a = k In, k > 0}
A COη(n)-structure on M determines and is
determined by a set {g} of metric tensors which
are proportional by a positive factor to a given
metric tensor g on M (i.e., g  {g} if and only if
g = ωg, with ω: M → R+).
Volume Structures
The SL±(n)-structures or volume structures are the
reductions of LM with subgroup
SL± (n) := {a  GL(n,R) : |det(a)| = 1}
An SL± (n)-structure on M can be understood as a
selection, for every point x  M, of a maximal set of
basis of TmM with the same unoriented volume, in the
sense of linear algebra.
It determines a family of local volume forms, which
when applied to a basis belonging to the SL± (n)structure, gives +1 or −1.
The existence of volume structures does not depend on
the orientability of M as in the case of SL(n)-structures,
but if M is an orientable manifold a SL± (n)-structure
on M determines and is determined by a global volume
form, unique except for the sign.
Semi-Riemannian Structures
It follows that:
CO(n) ∩ SL±(n) = O(n)
And
GL(n,R) = CO(n)  SL± (n)
Theorem: The (semi) Riemannian
structures on M are the intersections of
conformal and volume structures on M.
In other words:
Given a volume structure and a
conformal structure, we obtain a
uniquely determined metric
structure.
Application to geometrical
structures of second order
The second order frame bundle of M, F2M(M,G2(n)) is
the bundle of 2-jets at 0  Rn of inverse of charts of M
The group G2(n) is the group of 2-jets at 0  Rn of
(local) diffeomorphisms φ of Rn, with φ(0) = 0
It is described as the semi-direct product of Lie groups
GL(n,R)  S2(n), with the product law
(a, t)•(a’, t’):= (aa’, a−1 t(a’, a’) + t’),
Affine and Projective
structures
We can identify F2M with the first prolongation (LM)1 of
LM.
A G-structure of second order on M is a G-reduction of
F2(M), with G being a subgroup of
G2(n) ≡ GL(n,R)  Sym2(n).
The symmetric linear connections of M are examples of Gstructures of second order, when
G = GL(n,R)  {0}
The projective structures on M are also examples of it,
when G = GL(n,R)x p, where
p := {t ε S2(n) : tijk = δij pk + δikpj }
for some (pi ) ε Rn.
Projective & volume structures
Theorem
The intersection of a projective structure on M and
the first prolongation of a volume structure on M
gives a symmetric linear connection on M.
Theorem
The intersection of the first prolongation of a
conformal structure on M and the first prolongation of a
volume structure on M gives a symmetric linear
connection on M.
Space-Time Structures
Physically the conformal structure is
determined by the local behavior of null
wave-fronts and rays; and the projective
structure by the local behavior of freely
falling massive test particles, respectively.
In general relativity, these structures may
be taken as fundamental, and the pseudoRiemannian metric and affine connection
derived from them.
Two Degrees of Freedom
Various initial value problems in GR
may be reformulated on the basis of the
conformal-projective breakup-- in
particular, null-initial value problems
and the 2+2 decomposition of the field
equations-- with the aim of investigating
how best to isolate the two degrees of
freedom of the gravitational field, a
question of crucial importance for their
quantization.
Measurability Analysis
The quantum of action sets limits on the
co-measurability of various physically
measurable quantities, and thereby
determines their commutation relations.
Hence, the co-measurability of quantities
derived from the conformal and
projective structures, such as the
conformal 2-structure, will be analyzed
as a heuristic guide to their quantization.
Phonons
A phonon is a quantized mode of vibration
occurring in a rigid crystal lattice, such as the
atomic lattice of a solid. The study of phonons
is an important part of solid state physics,
because phonons play an important role in
many of the physical properties of solids, such
as the thermal conductivity and the electrical
conductivity. In particular, the properties of
long-wavelength phonons gives rise to sound in
solids -- hence the name phonon. In insulating
solids, phonons are also the primary mechanism by which heat conduction takes place.
Phonons (cont’d)
Phonons are a quantum mechanical version of a special
type of vibrational motion, known as normal modes in
classical mechanics, in which each part of a lattice
oscillates with the same frequency. These normal modes
are important because, according to a well-known
result in classical mechanics, any arbitrary vibrational
motion of a lattice can be considered as a superposition
of normal modes with various frequencies; in this
sense, the normal modes are the elementary vibrations
of the lattice. Although normal modes are wave-like
phenomena in classical mechanics, they acquire certain
particle-like properties when the lattice is analysed
using quantum mechanics. They are then known as
phonons. Phonons are bosons possessing zero spin.