QF vs. QM

[AntonioLao]

For free particle radiation quantum field (QF) relativistic equations, their amplitude solutions have the form exp(-ip×x) and exp(+ip×x) where i is the imaginary unity and the linear momentum function p is a tensor in 4D, a vector in 3D, 2D, and a spinor in 1D, but becomes a scalar in 0D (zero dimension). Likewise, the position function x is a tensor in 4D, position vector in 3D, 2D, and a spinor in 1D. It becomes the null vector in 0D. These 2 amplitude solutions practically replaced the separated eigenvector wave functions f and y of nonrelativistic Schrödinger’s quantum wave mechanics (QM). Moreover, the scalar inner dot product p×x always satisfies the uncertainty principle of QM: p×x³ħ where ħ is Dirac’s constant or equivalently Planck’s constant of action divided by 2p.

The QF solutions exp(-ip×x) and exp(+ip×x) become exp(-iET) and exp(+iET) where and when p×x is replaced by its alternative uncertainty expression for total energy E and total time interval parameter T between the initial and final state of a particular interaction. For the case of the quantum electromagnetic field of radiation of photons, the amplitude solution exp(-iET) corresponds to incoming photons of absorption while exp(+iET) corresponds to outgoing photons of emission. As unitary operators, exp(-iET) is equivalent to the y-loop Hadamard operator and exp(+iET) is equivalent to the Y-loop Hadamard operator of squares of energy as real conjugate product of the real Lagrangian and the real Hamiltonian energy functions.

[Profpat]

So does the quantum particle create the quantum field? Or does the quantum field create the quantum particle? Or are both created simultaneously?

[AntonioLao]

both created simultaneously

For all practical purposes, the creation and annihlation operators seem to work simultaneously but once electrons and protons are created they practically live forever. Together with the photons, these 3 groups of elementary particles live for beyond eternity.