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Recently in 2009 a novel solutions was proposed to address the problem with quantum gravity. The author is Petr Horava, who is a string theorist co-authoring paper with the like of Ed Witten, e.g., the "Horava-Witten domain wall" in M theory. He should be one of the more creditable sources among the many trying to revise relativity over the years. The main thrust of the theory is that at very short distance the 4 dimensional spacetime breaks down into 3 space + 1 time (Figure 21). It is believed that the Lorentz symmetry would be violated if space is emergent (at short distance), i.e., if space is not merely a scaffold for physical phenomena to play out. At low energies, general relativity is recovered from this underlying framework, and the fabric of spacetime re-stitches. The theory seems to be working so far: | |

## Figure 21 Broken Spacetime [view large image] |
the infinties that plague other theories of quantum gravity have been removed, it produces a well-behaved graviton, and also matches with computer simulations of quantum gravity. |

- The followings provide further details on this theory:
- Lorentz Symmetry - It is observed that electrons move with high speed in graphene (sheet of carbon atoms just one atom thick) near absolute zero temperature, therefore relativistic theory (implying Lorentz symmetry) is required to describe them. However, these electrons move only at a small fraction of the speed of light a normal temperature, there is no need to take relativistic effects into account. It is surmised if the same thing is true for our universe - The cool cosmos today satisfying Lorentz symmetry could be very different near the moment of Big Bang with no link to Lorentz symmetry.
- Anisotropic Scaling - In condensed-matter systems (including solid, liquid, superfluid, superconductor, ...), the degree of anisotropy between space and time is measured by the
*dynamical critical exponent*z in the transformations : - Detailed Balance - The transition from the static action (with z = 1) to dynamical (involving time) is accomplished by invoking the concept of detailed balance, which states that the transition rates between each pair of states
*i*and*j*obey :

P_{ij}_{i}= P_{ji}_{j}

where P_{ij}is the transition probability from*i*to*j*and_{i}is the equilibrium probability in state*i*. In a stochastic process in which the distribution of future states depends only on the present state, detailed balance insures the system is time reversible. - Renormalizability - The divergence of a quantum process comes from the integration of the 4-momentum p (or sometimes denoted as k) for the virtual particles (both fermion and boson) to infinity. The integration "4-volume" element d
^{4}p contributes a p^{4}term, which can be "diluted" by a p^{-1}term from each fermion propagator (internal line), a p^{-2}term from each boson propagator, and in addition the coupling constant in each vertex will contribute a (-n) power of p if its unit has n-dimension of p (in order to keep the total dimension of the graph to be the same). The origin of the other negative power p terms can be traced ultimately to the number of spacetime derivatives (within individual term) in either the Lagrangian or the field equation. While the Dirac equation involves only first order derivatives, the field equations for most boson are written down in second order derivatives. Thus, if we denote the Divergence as D, then the degree of divergence can be written as :

D = 4 - F - 2B - nV ---------- (26)

where F and B are respectively the number of fermion and boson propagators (internal lines), V is the number of vertices in the Feynman diagram, which is divergent if D 0. The coupling constants g for any gauge theory is dimensionless, so n = 0. (see more in Renormalizable Theories)

Since the number of spatial derivatives in the Lifshitz scalar theory depends on z as shown in Eq.(25), the rule for divergence in Eq.(26) now becomes :

D = 4 - F - 2zB - nV ---------- (27)

if the mediating boson is the Lifshitz scalar field. Thus the value of z > 1 has a drastic effect on reducing the divergence on any Feynman graph, and so a non-renormalizable theory such as the one in General Relativity would become viable once again. - Construction of a Theory of Gravity - In analogy to the formulation of the Lifshitz scalar field for z=3, the action for a theory of gravity is constructed with a kinetic term (involving time derivative), and a potential term somewhat similar to the action in static. The scalar field is now replaced by the spatial metric g
_{ij}'s. The transition is done first for z=2, then expanded to z=3. In this theory of quantum gravity, z=1 represents a system with Lorentz symmetry (IR limit); z=2 means a non-relativistic description is sufficient; while z=3 produces massive gravitons interacting at short range (UV distance). - The IR Limit - Perturbation of the z=3 theory enables the flow to the IR limit, where general relativity and the speed of light is recovered. In addition, the cosmological constant emerges from the reduction process naturally.
- Predictions - This theory predicts that the universe did not start with a "Bang", but a "Bounce" from a contraction phase. Fluctuation to short range interaction in certain circumstances may create the illusion of dark matter. As for the dark energy, there is a parameter that can be fine-tuned to produce a value of the vacuum energy in agreement with the calculation from particle physics. The theory also alters the physics of black holes - especially microscopic black holes, which may form at the very highest energies.

x bxt b ^{z}tAs illustrated in Figure 22, z = 1 is for system in a single phase. When the system is at the boundary of transition from one phase to anther as shown by the curves in the diagram, then z = 2. At the triple point or tri-critical point, the critical exponent changes to z = 3. An additional term with time derivative is introduced into the Lagrangian density L for a system off equilibrium as shown below :
| |||

## Figure 22 Phase Diagram [view large image] |
where is the Lifshitz scalar field for the condensed system, i = 1,...D (D is the spatial dimension), a is a dimensionless coupling constant, and is a high-energy scale parametrizing the strength of the higher derivative operator. |

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