Lorentz invariants in particle-wave mechanical systems

Abstract In these notes we detail a number of new results involving the Lorentz invariants associated with the special relativistic extension of Newton’s second law proposed in [10] and not included in that text. We first summarise existing results for the two Lorentz invariants $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> and $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> and the angle $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> which is the angle in which Lorentz invariance appears as a translational invariance. We then determine new integral formulae involving these quantities and a new relationship which connects the applied forces with the same invariances. This latter relation turns out to be equivalent to the partial differential equation arising from the invariance of the velocity equation $$\textrm{d}x/\textrm{d}t = u(x, t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>d</mml:mtext> <mml:mi>x</mml:mi> <mml:mo>/</mml:mo> <mml:mtext>d</mml:mtext> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> under a Lorentz transformation. We then provide some specific examples which confirm the validity of the new relation connecting the applied forces with the invariances.