Download (Covariant) minimum length and cosmic expansion

Reconciling Spacetime Continuity and
Discreteness using Information Theory
Achim Kempf
Departments of Applied Mathematics and Physics, University of Waterloo
Perimeter Institute for Theoretical Physics
Waterloo, Ontario, Canada
Napoli, October 17, 2008
Summary
Problem:
• Spacetime discreteness versus spacetime continuity
Proposal:
• Unify same way that discrete and continuous information has been unified: Sampling Theory
Applications so far:
• Application to inflationary cosmology
• Possibly measurable effects on cosmic microwave background
(B-Polarization, upcoming Planck satellite)
New results:
• Generalization to arbitrary Riemannian manifolds
• Beginning of generalization to Lorentzian manifolds
Outlook:
• Applications to Quantum Gravity
Spacetime at short distances
A) Quantum theory and general relativity together: indicate existence of some form of minimum length:
Try to resolve position more and more precisely
=> increasing momentum uncertainty
but: momentum gravitates and thus curves space
=> increasing curvature uncertainty
=> increasing distance uncertainties
=> a limit to how precisely distances can be resolved
B) Back of envelope estimate of “minimum resolvable length”: the Planck length, i.e., about 10^(-35)m.
C) Quantum theory: Indeed indicates that spacetime should be discrete:
- “Quantization” literally means “discretization”,
- QFT generally well-defined only on discretized spacetime.
D) General relativity: Spacetime should be a smooth manifold !
- GR makes essential use of differential geometry.
Can spacetime be continuous and discrete?
Situation:
- QT alone indicates space should be discrete
- GR alone indicates space should be continuous
- Should spacetime fields be continuously or discretely defined ?
Proposal (AK, 2001):
Utilize that the problem is analogous to Shannon’s problem to unify description
of discrete and continuous information.
The problem then (in 1940’s) was:
How can a continuous channel (e.g. music) have finite information density?
Information theory in the 1940s:
Problem:
Is there an infinite amount of
information in a continuous
(e.g. music) signal?
How to represent the
information content of
continuous signals through
discrete data ?
Shannon’s solution:
Sampling Theory provides the link between continuous and discrete information.
The basic sampling theorem:
•
Assume f is “bandlimited”, i.e:
f ( x) 
max


•
~
f ( ) e  2i x d
max
Take samples with spacing:
xn 1  xn  (2max ) 1
• Then, exact reconstruction
is possible:
f ( x) 

n
sin[ 2 ( x  xn ) max ]
f ( xn )
 ( x  xn ) max
Further implications:
If the functions, f, are bandlimited, i.e., have finite minimum wavelength,
then also:
• all operators, including the differential operators, can identically also be
written as finite difference operators.
• inner product integrals are identically also series:
(Useful also e.g. in number theory – similarly can be useful in physics even if only as a tool)



f ( x)* g ( x) dx 
1
2max


n  
f ( xn ) * g ( xn )
Relevance to Quantum Gravity?
Proposal: Natural UV cutoff could be a generalized bandwidth cutoff:
•
Minimum length = minimum wavelength = bandwidth cutoff
•
Covariant version on curved space?
• Cutoff on the spectrum of the Laplacian
• i.e., function space of fields spanned by eigenfunctions only up to a maximal eigenvalue
•
Spacetime then still continuous manifold.
•
All fields then fully determined if known only on sufficiently densely spaced
sample points.
•
Derivatives are also finite difference operators.
•
Integrals are also series.
Bandwidth as a natural UV cutoff
•
Can use continuous rep => display e.g. symmetries (Killing fields).
•
Can use any one of the equivalent representations on sufficiently dense lattices => display
UV finiteness. => Also, no lattice needs to grow when spacetime is expanding.
•
Spacetime then possesses a finite density of degrees of freedom in information theoretic
sense. Channel capacity depends on:
- bandwidth, i.e., here: UV cutoff
- and noise, i.e., here: quantum fluctuations
•
Sampling methods should be useful in all quantum gravity theories with discreteness:
–
–
–
stabilize the dimension of a lattice description (e.g., in spin foam models)
avoid having to find or define a continuum limit
(because then the theory is equivalently continuous already.)
Or even only as a technical tool as in number theory, turning sums into integrals and vice versa.
Indications for information theoretic
cutoff from Quantum Gravity?
•
Quantum gravity arguments (Garay et al), also strings (Witten):
–
there should exist a minimum length uncertainty
xmin ( x) , i.e.:
(noncovariant !)
•
Result (AK 2002):
Functions (wave functions or fields etc) in the domain of an operator X
with lower uncertainty bound always possess the sampling property.
How does sampling theory work?
•
•
Consider N-dimensional function space, spanned by functions
Its functions are:
f ( x) 
N
 b ( x) 
i 1
•
i
We use: a j 
•
Generically, we can invert the matrix
•
Thus:
f ( x) 
N
i
a j  f (x j )
Assume N samples of f known:
=> Can reconstruct f at all x!
N
 b (x )
i 1
Bij  bi ( x j )
1
b
(
x
)
B
 i ij a j 
i , j 1
bi (x) .
N
i
j
i
to obtain the
i .
1
b
(
x
)
B
 i ij f ( x j )
i , j 1
How to recover Shannon sampling?
•
On an interval of length L, consider the space of bandlimited functions (in Fourier
series sense).
•
This function space has finite dimension n(L),
•
=> N(L) = the number of samples necessary for reconstruction.
•
Fourier frequencies: n/L. Thus: N(L) is proportional L for large L.
=> For large L, the density of necessary samples per unit length,
D = n(L)/L
converges to a constant.
=> When taking limit L  infinity: recover the Shannon sampling theorem.
Shannon sampling in n dimensions ?
Generalization to functions on R^n by Landau, with different methods:
The minimum sample density, D, is proportional to the volume, B, of the
allowed region in frequency space:
D  2 B
n
Notice:
• The allowed frequency range need not be centred about 0.
Example:
To capture a HiFi FM radio station’s signal at 98.2 MHz
with 20KHz bandwidth it suffices to take 40K samples/sec.
• This is called the case of “bandpass” (will occur later in relativistic case).
Remark: Sampling theory is a rich field with dedicated journals and conferences.
Sampling on curved manifolds
AK, R.T.Martin, Phys.Rev.Lett., 100, 021304 (2008)
Consider nested bounded submanifolds Mi of M.
•
•
Assume minimum wavelength cutoff, i.e., cutoff on spectrum of Laplacian:
–
Fields are in space spanned by eigenfunctions only up to a maximum eigenvalue of Laplacian.
–
Since have IR and UV cutoff: this space is finite-dimensional, say N(Mi).
–
Thus, N(Mi) generic sample points suffice.
Now use a result of spectral geometry, Weyl’s asymptotic formula:
–
•
N(Mi) is approx proportional to the covariant volume V(Mi) of the submanifold, for large UV cutoff.
Let sequence of submanifolds, Mi, approach the unbounded M:
N(Mi) / V(Mi) stays finite as one considers larger and larger submanifolds Mi
 the sampling property is maintained as Mi  M.
Sampling on curved manifolds
• Note (a fact also used in noncommutative geometry):
There are correction terms to the Weyl formula for N(Mi):
• Lowest order term (Weyl):
Volume term = integral of “cosmological constant”.
• Next terms:
proportional to curvature R and integrals over higher powers of R’s.
• This suggests:
– Curvature may be encodable entirely in terms of sampling theory, i.e. in terms
of the equivalence classes of minimum density lattices.
– I.e., the curvature is then encoded in the distribution of the density of degrees
of freedom.
Use of curved space sampling?
• Sampling theory on curved space should be a useful mathematical tool in
approaches to quantum gravity with discrete space:
– Stabilizes dimension of the lattices
– Problem of continuum “limit” replaced by an equivalence to the continuum
theory.
– Preserve external symmetries and associated conservation laws
– Avoid the phenomenon (see Jacobson) that the discrete creation of degrees of
freedom due to cosmic expansion tends to be highly nonadiabatic, thus creating
far more particles than consistent with observations.
– Even only as a mathematical tool, these methods can be useful to turn series
into more manageable integrals (as in number theory).
Generalization to Lorentzian manifolds
Key problem:
Lorentz contraction => No covariant notion of minimum distance or (wave-) length !
Motivation:
–
Applications to quantum gravity approaches with discrete events rather than discrete points in space.
–
Calculate covariant predictions for possible effects of quantum gravity on the cosmic microwave background.
–
Re-address mode creation in expanding spacetimes (where each wavelength gets stretched along with the expansion).
–
Re-investigate, now fully covariantly, the associated problem of the vacuum energy of these expanding modes.
Ansatz: Cutoff on the spectrum of the d’Alembertian or Dirac operator.
–
Effective action could contain a series of terms in powers of d’Alembertian which are each small but such that the
series has a finite radius of convergence => get cutoff on spectrum of d’Alembertian.
Findings: Time dilatation and Lorentz contraction conspire to yield, [AK, Phys.Rev.Lett., 92, 221301 (2004)]:
–
Wavelengths smaller than Planck length exists but virtually freeze in their dynamics - this is a covariant feature!
–
Spatial modes obey temporal sampling theorem and vice versa, with covariantly transforming bandwidths.
Sampling theory of the manifold itself,
i.e., sampling of the metric tensor?
Sampling theorems for the metric seem impossible:
– Sampling theorems require taking samples at a certain minimum density
– But before one has the metric everywhere, one cannot say how densely spaced
sample points are.
Latest result (out in a few days):
Measure spacetime shape using quantum distance measurements:
– Measure propagator matrix elements, i.e. essentially < xa |  | xb >
– Find lowest N eigenvalues
– Use spectral geometry to reconstruct manifold, up to UV cutoff.
Outlook
• Fully spacetime covariant sampling of Lorentzian manifolds themselves.
• Application to Quantum gravity models
Any discrete theory of gravity, spin foam models.
Sampling in presence of black hole horizon.
Use for description, e.g., of holography.
• Applications to inflationary cosmology:
Could a fully spacetime covariant UV cutoff measurably affect the CMB?
Each spatial mode has temporal sampling theorem
Hard cutoff of the spectrum of d’Alembertian, e.g., in Minkowski space:
2
2
| p  p |  M cutoff
2
0
Every spatial mode  p (t ) has a temporal sampling theorem:
Case A)
If spatial mode has sub-Planckian spatial wavelength, i.e. small spatial momenta
=> the temporal bandwidth is approx [-M,M].
=> roughly Planck density of temporal sample times t_n suffices.
(1+d) dimensional covariant sampling theory
Case B)
If spatial mode  p (t ) has trans-Planckian wavelengths, i.e. large spatial momenta :

p2  M 2

| p0 |


p2  M 2
=> frequency range of  p (t ) is finite and decreasing for small wavelengths (bandpass).
Recall (Landau): volume of frequency range determines minimum sample density.
=> need less and less temporal samples, the dynamics freezes out, becomes trivial.
Intuition?
The freezing out of transplanckian small wavelength’s dynamics is covariant due to
interplay of Lorentz contraction and time dilatation.
Second quantization with the cutoff:
• Use 1st quantization rules
(following Schwinger and
deWitt)…
• …to obtain repr. indep.
form of the action, e.g.:
p̂ j   i x j 
x̂ j   x j 
L   d 3 x  * f ( xi ,i x j )
 ( | f ( x̂ i , p̂ j ) | )
• Adopt new QG-motivated CCRs (and UCRs).
• Find new Hilbert space representations for fields.
• 2nd quantize.
Sketch of calculations:
• Take, e.g., tensor
mode’s action:
a 2 '2
S   d d y (  ( y j ) 2 )
2
• Write in proper distances and
representation independently as:
1
S   d (, A A)  a 2 (, pˆ 2)
2a
• Here: A( )    i
3

a' 3
3a'
ˆ
ˆ
p
x


j
j
a j 1
a
• Now replace [ xˆi , pˆ j ]  i i , j
by new CCRs.
• Find new Hilbert space representation.

Mode equations each have a start time:
 k ' '  r ( )  k '  s( )  k  0
Irregular singular point at the mode’s creation time!
Thus, in particular:
 (kn ) (c ) 
 k (c ) or divergent for  c
=> Nontrivial to even specify any initial conditions to identify the vacuum state!
Notice:
•
The situation is mathematically similar to that described by Bojowald in the context of the
Wheeler deWitt equation.
•
Here, as there, the singularity may lead us to single out the correct initial condition. Maybe
the equations contain more than we can see so far.
Mode creation through expansion
Recall:
Significant effect on CMB possible
• oscillations in power spectrum
• modification of scalar/tensor consistency condition etc.
Literature: experimental signature for non-covariant cutoff worked out
Now:
Let us reconsider the entire ansatz: what if (1+d) dim. covariant?
– no sharp minimum length ?
– no sharp starting time ?
– no sudden onset of initial conditions ?
Sampling theory on expanding spacetimes
Cut off spectrum of d’Alembertian on expanding spacetime, e.g. for 1+1dim (e.g. in QFT path integral):
   a 3 (t ) ( dtd a 3 (t )
d
dt
 a(t ) k 2 )
Here, k is the comoving momentum. Analysis yields:
For each fixed spatial mode
Case A)
 k (t )
again we have a temporal sampling theorem.
Mode has large spatial wavelength
=> it again requires Planckian density of temporal sampling points, as expected.
Case B)
Mode has transplanckian spatial wavelength
=> it requires much less density of temporal sampling points.
New result for de Sitter and power law expansion: (with RT Martin, forthcoming paper)
Number of sample points (or temporal degrees of freedom) for any spatial mode is finite in any
time interval [-infinity, t_1] (and increasing as expected when expanding).
Sampling theory on expanding spacetimes
Conclusion so far:
• With covariant UV cutoff, comoving modes do not get created at
fixed time
• Instead, they only acquire the possibility for significant nontrivial
dynamics when they become larger than the cutoff scale.
• Now need concrete model for how the cutoff arises in QFT from QG
(such as Generalized Uncertainty Principle for noncovariant case.)
• Then, could calculate concretely how the modes slowly unthaw
rather than being created at once.
Hidden places in Hilbert spaces:

Hamiltonian: has the general form:
ˆ k (t )  k (t )ak  *k (t )ak
where: 
The fields are:
Observe:
3
ˆ 
ˆ '  L̂
d
k

k
k

a ( ) 2
k 
e
2
3
iyk ˆ
d
k
e
 k ( )

ˆ ( , y) 

k2


Hˆ ( ) 
a ( ) 2
e
The Hamiltonian and the fields continually acquire new
creation/annihilation operators
Or they lose them during a contraction phase!

Mode creation through expansion
Some parts of the Hilbert space are not being acted upon by H nor the
fields!
 Degrees of freedom disappear and (re)appear in shrinking and expanding
phases:
The state of such a decoupled mode may be nontrivial, e.g. after
contraction.
“Where” does the information of such decoupled modes exist, except
mathematically?
Could it be an interface to transplanckian degrees of freedom beyond
QFT ?
Unlikely: Note that unitarity of time evolution still holds, because H stays
self-adjoint.