Allard W., Almgren J.Frederick Geometric Measure Theory and by William K. Allard, Frederick J., Jr. Almgren

By William K. Allard, Frederick J., Jr. Almgren

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Under interchange of the two pions, a ' - {3 and PI - P3, so that AO and Bl are symmetric under interchange of sl and s3 while A l and BO are antisymmetric. GENERALIZATION TO INCLUDE CHARGE AND SPIN 29 It can be verified that the four invariant functions A 0 , 1, EO, 1 satisfy the Mandelstam representation-that is, they have only "dynamical" singularities arising from intermediate states in the various channels-if one accepts the corresponding conjecture for zero spin. , 1 and Y'P1Y'P2, we should have found additional (kinematical) poles in the corresponding invariant functions.

4-3 , which contains P3 P2 Pt FIG. 4-2. The general box diagram. 26 S-MATRIX THEORY OF STRONG INTERACTION only two intermediate particles in Channel III but an arbitrary number (an variety) in Channel I. Cutkosky pointed out 5 that the sum of all such graph can be written as an integral over a product of two absorptive parts Ai' o the type defined above, a circumstance that can be understood in terms of formula (4-3). Later we shall make essential use of this strikingly simpl result. P1 FIG. 4-3.

We have a formula (7-21) in terms of certain single- and double-spectral functions, but how are these functions to be computed? Clearly, the next task is the formulation of a procedure to evaluate V mIl and V III!. t As discussed in reference 21 a correction may be made for the large s3 region of the integral in Eq. (7-20). EVALUATION OF THE LONG-RANGE AND MEDIUM-RANGE FORCES In the previous section we defined a "potential" acting in Channel III, due to Channel II, as the absorptive part for Channel II minus an integral over the elastic double-spectral function p (el)III (s3,s2) for Channel ITI.

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