# Combining general relativity and quantum theory

Quantum theory wishes to simplify micro behavioral movements in any of earth and cosmos gravitational matter. From meter to highest level of energy all that can be linearly described, are levels of critical masses and the definition of forces binding them?

Electromagnetic, magnetic field and electrical current are bind in presence, there for in meter. Gravitational forces, the truth is that the amount of matter is relatively small and the most of earth space, or our cosmic creation, is hollow.

General relativity theory - implies to behavioral issues in a vertical approach means a two dimensional axis model. There for it cannot fully describe three simultaneous forces; electromagnetic, strong nuclear, weak nuclear forces.

Quantum theory general relativity in motion - Usually described as radial sourced forces binding critical amounts of mass, expanding in space along time.

- Numerical treatment

The most successful numerical approach to quantum field theory begins

with a formulation of quantum mechanics developed by Feynman in which a quantum amplitude is described as a weighted integral over all possible paths (not necessarily obeying the classical equations) which start at the system's initial state and end at the final state. For single particle quantum mechanics the quantum amplitude <qf(tf)| qi(ti)> for a transition from position qi at time ti to position qf at time tf

is written as:

<qf(tf)| qi(ti)> = òdq[t]eiA[q]

where A[q(t)] is the classical action for the path q(t) given by

A = òtitf {½ m q'2 - V(q(t))}dt

This is a sophisticated Wiener integration over function space and is typically an awkward formalism for analytic calculation.

However, it is nicely suited for numerical work since it replaces the normal operator/Hilbert space formalism of quantum mechanics with an explicit integral.

The path integral appropriate for quantum field theory is similar to the equation above except that the integration must be performed over

all possible time evolutions of field configurations rather than particle trajectories.

In our physical problem, a field configuration specifies both the quark and gluon fields as particular functions of space.

A particular time evolution then specifies these fields as functions of space and time. This problem is easily put in a numerically tractable from by replacing the space-time continuum by a grid or lattice of points, conventionally a uniform, four-dimension mesh.

The field theory analogue of the single-particle action given in the equation above is a similar polynomial in the field variables

and their derivatives, integrated over space-time.

Thus, the corresponding discrete field theory action will be a four-dimensional sum of a local density which depends on the lattice

field variables at a specific lattice site and its nearest neighbors.

The actual integration appropriate for the lattice QCD evaluation of an observable O is typically performed as a Monte Carlo average,

<0| O |0> = 1/N Sn=1N O({U}n),

over an ensemble of configurations for the gluon fields {U}n, N ³ n ³ 1.

Each configuration assign a specific 3x3 complex matrix U to each link connecting neighboring sites in the lattice. The ensemble used above is generated by a Metropolis or molecular dynamics algorithm so as to be distributed according to the positive definite statistic weight:

e{b/6SP tr UP} det(D+m)

The sum is over all elementary squares or plaquettes, P, that can be constructed out of four lattice links and UP is the ordered product of the

corresponding U matrices associated with those links.

The quark fields correspond to anti-commuting classical variable and cannot be treated numerically as an integral but instead are represented

by the determinant above. Here D is a nearest neighbor difference operator and 'm', the quark mass matrix. Typically, the force generated by the determinant is computed using a noisy estimator which can be done using, for example, a conjugate gradient method for computing the inverse of the sparse matrix D+m. In addition to the quark mass matrix 'm', the coupling strength, related to the parameter b, is the only other free parameter in the calculation.

This is an ideal formulation for massively parallel computing.

A typical large-scale lattice calculation might work with 324 hypercubic lattice. If each processor in a parallel machine is assigned a 44 subvolume, 4K processors would be required. The most computationally demanding part of a conjugate-gradient iteration requires about 500 floating point operations per lattice site or 128K flops/processor. A 3-component complex vector must be transferred both in and out of the processor for each link that joins the 44 subcube to its neighbors, or 43 · 8 · 6 · 2 or 6K words total. This suggests a reasonably favorable 20:1 computation to communications load

for each processor.