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---------- (10) |
_{f} S*_{fi}S_{fi} = 1, which guarantees that probability is conserved in the process. Such relationship indicates that the matrix S_{fi} has an inverse, which in turn implies that it is possible to return to the initial state from the final state at least in principle although the probability is almost zero in practice so that the second law of thermodynamics is "almost" never violated. This property is also related to the conservation of information, which caused so much trouble for Stephen Hawking. | |
Figure 01g S-Matrix |
---------- (12) ---------- (13) |
---------- (14) |
---------- (15) |
The s-channel corresponds to two incoming particles with 4-momenta p_{1} and p_{2}, which combine into an intermediate (virtual) particle with real mass and finally emerge as two outgoing particles with 4-momenta p_{3} and p_{4} (Figure 01h,a). Mathematically, s = (p_{1} + p_{2})^{2} = (p_{3} + p_{4})^{2} = 4E^{2}, where we assume they all have energy E to simplify the formula. In t-channel, particle with p_{1} emits virtual particle with imaginary mass, which is absorbed by p_{2}. They depart with p_{3} and p_{4} respectively (Figure 01h,b). Thus, t = (p_{1} - p_{3})^{2} = (p_{4} - p_{2})^{2} = 2E^{2}(1 - cos), where is the angle between | ||
Figure 01h Scattering Channels |
Figure 01i Regge Trajectory [view large image] |
p_{1} and p_{3}. There is an additional u-channel with the role of particles 3, 4 interchanged. Together they form an identity : s + t + u = _{i=1,4} m_{i}^{2}. |
For scalar particles at high energy the scattering amplitude can be generalized to : A(s,t) = -_{J} g_{J}^{2}(s)^{J}/(t - m_{J}^{2}) for the t-channel, and similarly A(t,s) = -_{J} g_{J}^{2}(t)^{J}/(s - m_{J}^{2}) for the s-channel. The finite summation in these formulas yields values way above the experimental data (Figure 01j). Then it was suggested that an infinite sum may offer a more acceptable result. At about the same time, there were hints that the scattering amplitudes from the two different channels may be equal to each other, i.e., A(s,t) ~ A(t,s), which is now known as the "duality" hypothesis. | |
Figure 01j Hadron Scattering [view large image] |
Then a scattering amplitude (known as the Veneziano Amplitude) that satisfies all the stringent requirements of the S-matrix Theory (except the unitarity but including the duality) was discovered in 1968 : | |
Figure 01k Gamma Function [view large image] |
Some comments on the Veneziano amplitude : |
| |
Figure 01l Euler Beta Function [view large image] |
Figure 01m N-point Amplitude [view large image] |
The necessity of having to devise the N-point amplitude is related to the behavior of hadron collision at very high energies. At about 50 times the rest mass energy of the proton, the dominant feature of most events is the sheer number of particles produced. This number is called the multiplicity of the reaction as shown by Figure 01n, in which the number of the product n for the various species is plotted against the squared energy. BTW, the intermediate particles in the S-matrix theory now include all those in the Regge trajectories. This is referred to as Reggeon exchange, which represents the exchange of all the resonances with different masses and spins. It is similar to the mediation by photon in the electromagnetic interaction. The resonances are referred to as hadron excited states in QCD. | |
Figure 01n Multiplicity |