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Most treatments of quantization assume

• Linear unitary evolution of field

• Possibly non-free fields, so plane waves cannot be used

If the non-relativistic approximation limit of the KG field is considered, these assumptions will be automatically obtained.

Then, the time component of the static gauge field acts as the electric potential. Both the harmonic oscillator potential (uniform Charge) (probably should not be interpreted this way) and the hydrogen atom potential (point charge) originate from simple charge densities.

Thus, the motivation problem becomes:

• Why use KG field?

• Why and how does it correspond to point particles?

• Why does the potential of a gauge field become the potential of a particle when the field is quantized into particles?

Classical correspondence refers to the expectation value version of the point particle Lagrange-equation (ref-15, p.116) (wiki:Ehrenfest_theorem), e.g.

there also new operator in the speed of expectation

Where the non-commutativity or is controlled by the very small Planck constant

Note the difference between classical energy and quantum energy, similar to the difference between mean and variance. For example, . We can consider the standard deviation . There is the uncertainty principle . The equality holds <==> harmonic-oscillator-ground-state

The Schrodinger eq is the non-relativistic limit of the KG eq, and Newton’s equation is the non-relativistic limit of relativistic point particles. So, can it be proven that the KG eq also has a point particle limit? At this point, should the definition of “expectation” use the charge density of the KG, . However, the charge of the KG eq is not positive definite, making it even further from the meaning of classical particles. However, the energy of the KG eq is positive definite (even with electromagnetic potential). For the Dirac eq, the charge is positive definite but the energy is not positive definite.

How to make the expected value version of the point particle Lagrange-equation correspond to the Lagrange-equation of the field?

Other signs that the Schrodinger wave function corresponds to point particles:

• Feynman path integrals use the path statistics weighted by the point particle Lagrangian to calculate the propagator of the Schrodinger eq. Question proof that it satisfies Ehrenfest theorem and then satisfies Schrodinger eq

The derivation of the behavior of the Schrodinger wave function under Galileo transformation, in brief, assumes that in addition to the non-relativistic space-time coordinate transformation , it is also assumed that the wave function changes phase , the result will be (ref-16)

Question Simplified calculations and better motivated explanations for the phase transformation? Perhaps it can be related to the use of static energy phase when KG eq is approximated to Schrodinger eq. In short, neither Schrodinger eq nor electromagnetic field are purely Galileo transformation invariant.

Galileo boost gives the transformation of the wave function as , the δ action of this action is the operator

Time translation -> δ action , Hamiltonian

Spatial translation -> δ action , momentum operator

Rotation -> δ action , angular momentum operator

Phase -> δ action

For QM in space, the Lie bracket of boost and spatial translation , or equivalently

For the following discussion, why is it an infinite-dimensional generalization of a finite-dimensional quadratic form? Because the source of the Lagrangian is the finite-dimensional quadratic form (and then integral)

Mimicking the finite-dimensional case, (ref-3, p.143–144. ref-23, p.218–222) diagonalize one quadratic form with an orthonormal basis in a (positive definite) quadratic form space.

Using differentiation to find the extreme values or first-order stable values of the Hermitian operator

on , let (orthogonal in the sense of )

Another quadratic form on a finite-dimensional standard quadratic form corresponds to an infinite-dimensional case where the wave function is in some or Hilbert space, acting as a quadratic form space. The energy is another quadratic form.

Its first derivative forall ==> ==> exists

States with different eigenvalues are orthogonal

There exists an orthogonal basis for a finite-dimensional standard quadratic form, such that , and is in a form close to a standard quadratic form, i.e., diagonalizing . can be written as or . The corresponding infinite-dimensional case is that there exists an orthogonal basis of the Hilbert space such that the energy is diagonalized. However, in the case of Hilbert space, diagonalization of the energy expectation might be in the form of a countable sum.

However, sometimes a countable sum is not enough, and an integral form is required, and the orthogonal basis is not the space where the wave function resides, even if the coefficients of the wave function’s orthogonal basis expansion might be in the space of the eigenvalues. (Question What is the specific construction method for the structure of the eigenvalue space?) An example is the existence of unbound states in the electrostatic hydrogen atom model.

Schrodinger eq evolution preserves the eigenvalue space

The energy is a quadratic form, and its second-order differential polynomial is

Divide the space into the eigenspace and its orthogonal complement . In the direction, the second-order differential is positive definite or negative definite, and the energy will increase or decrease. According to Lagrangian variational theory, the energy of the Schrodinger eq is a time-conserved quantity

are separated in the sense of limit, unless at . , so the evolution will be confined to the eigenspace

Specific solution for the Schrodinger eq evolution in the eigenvalue space. is a constant-coefficient linear ODE from for each spatial point , solution i.e., essentially static except for the phase factor which oscillates in time according to .