## Quantum Field Theory

### Unitarity Method

In case there are a lot of Feynman graphs to be evaluated in a process, the Unitarity Method can be used to shorten the computing within a manageable time. It is akin to Fluid Dynamics where the motion of individual molecules is collectively aggravated into macroscopic elements governed by a set of equations at another level (known as the Navier-Stokes Equations). It utilizes the fact that the infinities in the loop diagrams are often canceled out leaving a finite result at the end of the calculation. Figure 05n illustrates schematically the idea of one S-matrix corresponding to many Feynman diagrams. It has been shown that together with some conditions imposed on the S matrix, the unitarity method can produce rather accurate result.

#### Figure 05n Unitarity Method [view large image]

The conditions include :
1. Unitarity or conservation of probability.
2. Analyticity in the various energy variables. An analytic function is infinitely differentiable, and can be expressed by Taylor series in some neighborhood of every point.
3. Lorentz invariance.
4. Crossing symmetry, which means the S-matrix of two different processes can be converted to each others, e.g., the Compton effect and pair annihilation, etc.
One of the great successes of this approach in the 1960s was the use of dispersion relations to calculate constraints on the pion-nucleon cross sections. These relations were used extensively in strong interaction physics where a renormalizable field theory eluded physicists for many decades. With the successful development of field theory, this method fell out of favor for many years until the 21st century when it is used to obtain the expected cross sections (from the Standard Model) produced by LHC. The calculated cross section (as a function of colliding energy) is compared with the experimental data, any excess would indicate a new process or new particle is involved. The researchers then re-opened the
case against the infinities produced by supergravity in the 1960s. The argument was rather indirect because it was not feasible to evaluate 1020 terms in the three virtual-graviton loops. The unitarity method has now taken up the challenge and showed that those infinite quantities are indeed finite. The next tests would be the four and five loops computations. Along the way, the technique has also revealed that a graviton can be considered as two copies of gluon (Figure 05o).

.