Speculation on NonZero Rest Mass Momentum as Source of Quantum Action-Reaction

  In a previous note (1) we argued that   two flux/flow equations d/dx partial A(x,t)=p ((1a)) and ddt partial A(x,t) = -E ((1b)) lead to the Lorentz invariant solution A(x,t)= -Et+px. These have two stationary solutions (dA=0), namely  dE/dv= dp/dv with x/t=v and p=E(v)v (rest mass nonzero) and  dx/dt = E/p (rest mass i.e. a classical wave). The first solution leads to relativistic/nonrelativistic quantum mechanics if ((1a)) and ((1b)) are written as eigenvalue equations. The second represents classical waves and their eigenfunction equation is the classical wave equation d/dt d/dt cos(A) = 1/vv d/dx docs(A)/dx.

    A periodic solution, we argue, implies action-reaction suggesting force or tension in the system. This seems to be the case for a classical wave on a string with fixed ends or for a sound wave in a gas with pressure. Light treated as a classical wave, however, is more complicated because there is no medium with tension, but rather one has periodically varying electric and magnetic fields. An electric field, however, is considered a force field and so there is an analogy with a periodically changing pressure. A classical wave requires a real periodic solution.

    A quantum mechanical free particle solution (nonzero rest mass) does not require a real periodic solution because there is no medium or obvious real entity changing periodically, but one may establish a complex solution using eigenvalue equations:id/dt partial exp(i A) = E exp(iA) and -id/dx partial exp(iA) = p exp(iA). At first a complex solution seems unphysical, but one may note that for a fixed t (for example), exp(ipx) is 1 at x=0 and  again at pL=2*3.14. Thus there is a special length in the problem which yields the classical mechanical result d/dx partial A = p namely the wavelength 2.*3.14/p.  In other words the quantum solution describes the classical solution at each x= n * wavelength. Within each wavelength region one has an exp(ipx) solution which consists of two out of phase periodic functions such that the modulus is 1. 

   We try to argue that like in a classical wave there should be some impedance to a disturbance (like tension of a string or pressure of a gas). This impedance, however, is not represented by one physical entity (e.g. pressure in a sound wave) as one has a complex solution.

   A complex solution exp(ipx) is not alarming because light treated as a classical wave has two perpendicular entities, namely the electric and magnetic fields, that are both periodic. One may write a classical wave equation for each, but one may write a combined quantum-like wave equation for a photon: i d/dt partial Wi + eijk (-id/dxj) Wk = 0 (where eikj is the Levi-Civita symbol)  which has a vector wavefunction Wi= Electric field (i) (x,t) + i Magnetic field (i) (x,t). 

   The only impedance in the zero rest mass problem seems to be the rest mass (inertia) which resists motion. To be more general, it seems that momentum overall should be an impedance to motion. The entire momentum (E(v)v) is associated with an impact and a change in the momentum of a quantum particle (with rest mass) may be resisted by the entire momentum. In classical physics constant velocity motion seems to be a simple translation from one x point to another. The exp(ipx) solution to ((1a)) and ((1b)) shows that this interpretation only holds for n *wavelength points. Within each wavelength region out of phase two dimensional periodic motion occurs which suggests a kind of action-reaction. This behaviour occurs for various objects such as an electron, nucleon, nucleus, atom or even fairly large molecules etc. It has been argued that this quantum motion may apply for even objects in the macroscopic world e.g. mirrors. Impedance for nonzero rest mass quantum objects is velocity dependent as one may many different velocities associated with a nonzero rest mass momentum while classical waves have one velocity (for a given tension scenario). E.g. light always as a velocity of c because it has only one tension scenario i.e. electric and magnetic fields. Changing wavelengths and frequencies are linked to changing energy and momentum, but not velocity.    

   We try to investigate some of these ideas.