Unitarity and Symmetry

It is of interest to investigate general constraints of the transition matrix such as unitarity and symmetry. These properties can be established by applying the principle of conservation of energy to nonabsorbing particles ( i 0 and jj,i 0). We begin our analysis by defining the S matrix in terms of the T matrix by the relation 

First we consider the unitarity property. Application of the divergence theorem to the total fields E = Eg + Ee and H = Hg + i e in the domain D bounded by the surface S and a spherical surface Sc situated in the far-field region, yields 

We consider the real part of the above equation and since the boimded domain D is assumed to be lossless Sg 0 and /j,g 0) it follows that  

We next seek to find a series representation for the total electric field. For this purpose, we use the decomposition 

The coefEcients and are determined by the incoming field. Since in the far-field region and become incoming vector spherical waves, 

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Additional properties of the transition matrix for particles with specific symmetries will be discussed in the next chapter. The exact infinite transition matrix satisfies the unitarity and symmetry conditions (1.110) and (1.112), respectively. However, in practical computer calculations, the tnmcated transition matrix may not satisfies these conditions and we can test the unitarity and symmetry conditions to get a rough idea regarding the convergence to be expected in the solution computation. 

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