Claude Bloch. Scientific Works Oeuvre Scientifique by R. Balian (Eds.)

By R. Balian (Eds.)

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Defining then the quantities 6μ = [davT^\ Q = [davj\ (3,12) where dov is the surface element on σ such that da 4 > 0; the equations (3,5) and (3,7) show that Θμ and Q have the same value if σ = σ(1) or if σ = σ (2) . These quantities are thus constants of collision*14*; they represent the total energy-momentum and electric charge of the system. 5 Nr. 8 22 4. Quantization. If all the calculations of the preceding section are carried out in such a way that the order of the factors is always preserved, the results will still hold if the field functions are non-commuting operators.

We shall write y>n for yPfö'), un for ιΐη(χ'ή), y>+ for y>+in(x'n), n and Hn for H\ (x'n , x'j[, χή")· Finally, ζη will stand for the three points x'n, χ'ή, x'n > and άξη for dxn dx„ dx'n'. We shall now try to extend the formulas (5,6) and (5,9) to the case of a non-localized interaction. These formulas are obtained from the field equations by a certain number of algebraic operations: additions, multiplications, integrations. The same operation can be performed as well with a non-localized interaction, and the result should be very similar.

3,5) = 0, "-ΐ(<' + "£-£/*)- du du dx dxv -gfH (3,6) can be identified as the energy-momentum tensor of the system. The same method applied to the infinitesimal Lorentz transformations leads to the conservation equation of angular momentum. The Lagrange function is also invariant under gauge transformations ψ^-βίαψ, ψ+-> e~ίαψ+, a = constant. The corresponding infinitesimal transformation δψ = ίδαψ, δψ+ = —ΐδαψ + , διι = 0 gives 6LQ = 0. From (3,2) we obtain then the conservation equation !

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