The Dirac Equation and Time Flux
Francesco R. Ruggeri
It was argued in a number of previous notes that for the nonrelativistic case, W(x,t) transforms under x->x+a, t-> t+b (with a and b infinitesimal) as exp(ipx - iEt), except with p replaced with = - i d/dx W(x,t) and E replaced with i d/dt W(x,t). It was suggested this was a driving condition of nonrelativistic quantum mechanics, and if this is the case, one might expect these ideas to carry over into relativistic quantum mechanics. For relativistic quantum mechanics, however, one has the Einstein energy-momentum relationship E*E= p*p + mo*mo (c=1) which leads to -d/dt d/dt W(x,t) and -d/dx d/dx W(x,t) being used. The momentum term is fine because it is related to < p*p conditional >, but the second derivative of time seems to be a little confusing as it represents < E*E conditional > and not <E conditional >. If <E conditional> has physical significance, as it seems to be the case in nonrelativistic quantum mechanics, one might expect an equation with this time flux i.e. i d/dt W(x,t) / W(x,t) = < E conditional > and not < E*E conditional >. Dirac wrote such an equation, but obtaining a single time derivative comes at a price. The Dirac equation, in one dimension, contains matrices, d/dx and d/dt in addition to a two vector wavefunction (U(x,t), V(x,t)) instead of W(x,t). Thus, the Dirac equation in one dimension leads to coupled equations with U(x,t) and V(x,t). The single derivative d/dx is deceptive because U(x,t) can be written in terms of d/dx V(x,t) and vice versa, so ultimately the Dirac equations contain d/dx d/dx which is needed to obtain < p*p conditional >. The question is: How can one obtain < E conditional >? This may be done by using a (d/dt + constant ) -1 operator i,e. almost an inverse time operator, in other words either (E+m)-1 or (E-m) -1 where E can be thought of as id/dt. Thus, the RHS of the Dirac equations is i d/dt U(x,t) or i d/dt V(x,t) which suggests an infinitesimal step forward in time, while the LHS side contains (id/dt +m)-1 or (-id/dt-m) -1 which seems to be representative of undoing such a step. It should be noted that one may take Einstein’s energy-momentum equation E*E = p*p + mo*mo, without any knowledge of quantum mechanics, and cast it as 2x2 matrix equation, linear in E, namely H (U,V) = E (U,V). The matrix H serves as an evolution operator and another matrix serves as a velocity operator. One may derive the Schrodinger zitterbewegung equation for the velocity matrix without any use of wavefunctions and derivatives which are key to quantum mechanics. Nevertheless, the idea of noncommuting matrices, vectors and zitterbewegung is present. There are several choices for the matrix H, but one leads to the one dimensional Dirac coupled equations Thus, the “spin” features may be obtained directly from E*E= p*p+mo*mo without any quantum mechanics. (1), (2). The purpose of this note is to examine some of these ideas.
•Jul 24, 2019•0 citations