Relativistic Covariance of Scattering

Abstract Avoiding the assumption that relativistic scattering be describable by interacting fields we find in the Schrödinger picture relativistic scattering closely analogue to the non-relativistic case. On the space of scattering states the invariant mass operator $$M'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> of the interacting time evolution has to be unitarily equivalent to the invariant mass $$M= \sqrt{P^2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:msqrt> </mml:mrow> </mml:math> where P , acting on many-particle states, is the sum of the one-particle four-momenta. For an observer at rest $$P^0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:math> generates the free time evolution. Poincaré symmetry requires the interacting Hamiltonian $$H'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> to Lorentz transform as 0-component of a four-vector and to commute with the four-velocity $$U= P / M $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>=</mml:mo> <mml:mi>P</mml:mi> <mml:mo>/</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> but not with P , else there is no scattering. Even though $$H'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> does not commute with P , the scattering matrix does. The four-velocity U generates translations of states as they are seen by shifted observers. Superpositions of nearly mass degenerate particles such as a $$K_{\text {long}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mtext>long</mml:mtext> </mml:msub> </mml:math> are seen by an inversely shifted observer as a shifted $$K_{\text {long}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mtext>long</mml:mtext> </mml:msub> </mml:math> with an unchanged relative phase. In contrast, the four-momentum P generates oscillated superpositions e.g. a shifted $$K_{\text {short}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mtext>short</mml:mtext> </mml:msub> </mml:math> with a changed relative phase. The probability of scattering of massive particles is shown to be approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and justifies to represent the machinery of actual scattering experiments by the vacuum. In suitable variables the relativistic Hamiltonian of many-particle states is not the sum of a Hamiltonian for the motion of the center and a commuting Hamiltonian for the relative motion but factorizes as their product. They act on different variables of the wave functions.