Abstract
In previous work we discussed the quantization of paths in spacetime. Building on these ideas we have developed a mathematically coherent theory addressing a number of open questions concerning Loop Quantum Gravity. Our approach develops a discrete spacetime and shows that macroscopic spacetime is a renormalization limiting form. Weaving together a number of our previous results we then prove that quantum states invariant under either an external group of local diffeomorphisms of spacetime or by contrast quantum states invariant under the internal action of a compact Lie group are common in a well-defined sense. These form the building blocks of invariant fields and Lagrangians. A form of supersymmetry and noncommutative spacetime naturally emerges, which predicts a massless graviton and its companion gravitino.
Highlights
We have developed a mathematically coherent theory addressing a number of open questions concerning Loop Quantum Gravity
The Poincare group is a locally compact Lie group with 10 generators, and the translational group is an abelian subgroup generated by the energy-momentum 4-vector Pμ
As for local diffeomorphism-invariant quantum states [3,7], that quantum states invariant under the action of compact Lie groups are common in the sense that the weakly closed convex hull of every normal state contains such a state
Summary
A left-handed Weyl 2-spinor is an element of a 2-dimensional vector space F with a basis of Clifford variables denoted ψA (A=1,2). Where I2 × 2 is the 2 × 2 identify matrix æçççè10 10÷÷÷÷öø This Dirac spin operator representation acts on a 4-dimensional vector space. Defining g γ5 to be the composite matrix operator iγ0γ1γ2γ3, we can write this state |ψ > in the form y These correspond to the left handed and right handed 2-spinor components of the state;. Two) four vector ψR and is defined as yR = æçççççççççççècc0012 has only two ö÷÷÷÷÷÷÷÷÷÷÷÷÷ø These correspond to the left handed chiral Weyl 2-spinoræççççècc12ö÷÷÷÷ø and the right handed Wely E = F Å F*. Received September 11, 2018; Accepted October 15, 2018; Published October 22, 2018 Their product is a closed loop which corresponds to a scalar λ times the identity matrix with λ=1 in this case. The advantage of this approach is that spinor calculations become a sequence of potentially simpler topological transformations with connections to knot theory
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