Dimensionalities of Space-Time

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Dimensionalities of Space-Time Yu,
Hoi-Lai 16th Oct. 2008 NTHU
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Basic Questions in searching the Truths of
Nature What is Space? What is Time? What is
the meaning of being somewhere? What is meaning
of moving? Is motion to be defined with
respect to objects or with respect to space? Can
we formulate Physics without referring to time or
space? What is matter? What is causality? What
is the role of Observer in Physics?
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Active Diffeormophism invariant
What disappears in GR is precisely the
background space time that Newton believed to
have been able to detect now the existence
of Gravity is a matter of ones coordinate
choices. Reality is not made by particles and
fields on Spacetime. Reality is made by
particles and fields (incldue gravity), that can
only be localized with respect to each other.
Spacetime emergence of solution of theory
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• Einstein field equation A theory on space, time
  and Matter

Spacetime is curved due to the gravitational
action of matter
Geometry of space time
Matter
Quantization
Quantization
Quantum Gravity?
QM,QED,QCD
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Puzzles in general relativity

• Black holes entropy
• (2)Cosmology Big Bang
• Singularity

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Causal Dynamical Triangularization (Lorentzian
signature)
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• Important consequences of CDT
• Baby Universe eliminated
• Global Times with respect to foliation
• No topology change
• New universality class then Euclidean

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Four-dimensional Picasso. Connecting many
triangular bits of flat space-time to make curvy
space-times demonstrates how our four-dimensional
world can arise from the fuzzy "quantum foam"
that may exist at the tiniest scales. (Image from
researchers who used "non-causal" triangles.)
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Causality implies four-dimensionality! Hausdorff
dimension V(r) rdH Where the gluing for
each N4is chosen such that every single building
Block share a given vertex. Classical spacetime
no matter how curved with finer resolution, it
will always eventually looks like a piece of flat
spacetime Not the case in QG, never resemble a
flat space, no matter how fine. Need a path
integral allows for short-scale fluctuation in
curvature, but in such a way the resulting large
scale geometry does not degenerate
completely Causality require that each spacetime
appearing in the sum over geometry should have
a specific form, namely object that can obtain
by evolving a purely spatial geometry in time
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Volume-Volume Correlator
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Another evidence for spacetime dimension 4
Spectral dimension
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Effective mini-superspace and Wave Function of
Universe (Bounce Solution) Tunneling of from
nothing(zero scale factor a)
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From general scaling of the above action it is
clear that the only chance to obtain the observed
scaling law, expressed in terms of the variable
t/V1/4 4, is by setting a 1/3.
In Euclidean signature and proper time
coordinates, themetrics are of the form
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Nonperturbative Quantum Einstein Gravity
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The phase portrait of Einstein Hilbert trunction
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Now, NGFP predicts 4 dimension at low energies
2 at high energies
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Spectral Dimension
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• Conclusions
• Space-time dimensiona changes from
• 2 to 4 when energy resolutions drops
• Causality requires a 4 dimensional
• Space-time at low energies
• Spherical Symmetric Spacetime is
• perfered at high energies.
• (4) Quantum gravity can be test by gramma
• ray bursts of neutron stars.