Dimensionalities of Space-Time Yu, Hoi-Lai 16 th Oct NTHU

Basic Questions in searching the Truths of Nature: 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?

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 one’s 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 Active Diffeormophism invariant

Einstein field equation ： A theory on space, time and Matter Geometry of space time Matter Quantization Quantum Gravity ？ Spacetime is curved due to the gravitational action of matter QM,QED,QCD… Quantization

(1)Black holes entropy (2)Cosmology: Big Bang Singularity Singularity Puzzles in general relativity

Causal Dynamical Triangularization (Lorentzian signature)

Important consequences of CDT: (1)Baby Universe eliminated (2)Global Times with respect to foliation (3) No topology change (4) New universality class then Euclidean

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.)

Causality implies four-dimensionality! V(r) ~ r d H Hausdorff dimension: V(r) ~ r d H Where the gluing for each N 4 is 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

Volume-Volume Correlator:

Another evidence for spacetime dimension 4: Spectral dimension

Effective mini-superspace and Wave Function of Universe (Bounce Solution): Tunneling of from nothing(zero scale factor a)

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/V 1/4 4, is by setting α = 1/3. In Euclidean signature and proper time coordinates, themetrics are of the form

Nonperturbative Quantum Einstein Gravity

The phase portrait of Einstein Hilbert trunction

Now, NGFP predicts: 4 dimension at low energies 2 at high energies

Spectral Dimension

Conclusions: (1)Space-time dimensiona changes from 2 to 4 when energy resolutions drops (2)Causality requires a 4 dimensional Space-time at low energies (3)Spherical Symmetric Spacetime is perfered at high energies. (4) Quantum gravity can be test by gramma ray bursts of neutron stars.