I should add maybe that Lévy-Leblond has made great contributions to analyzing the representation structure of the Galilei group in QM, and in another paper has done an analysis with the inhomogeneous Galilei group similar to what what Wigner did with the Poincare group years before. Therefore, what he states in his papers, at first glance, bears a certain weight, at least to me. This is why I am unsure

Schrodinger is non relativistic, only Galilean invariant.

However do not off hand interprate what you mean by linearize. Both Schrodinger and ,Dirac are linear equations supporting superposition.

Dirac using 4 spinors is Lorenz invariant.

Maybe I was a little bit confusing at first. I will try to clarify here: "linearization" in this case means: turning the second-order-in-space differential equation into a first-order-in-space differential equation. It is a well-known way to get the DIrac equation out of the Klein-Gordon equation.

Now it becomes interesting: when you try to do the same with the Schrödinger equation which Lévy-Leblond has done in his1967 paper linked above, he ended up with a 4-component spinor equation which -- as is argued in the same paper and in many other references -- is Galilei covariant. And I am saying: it is not.

Because when you try to do the linearization, in order to introduce dimensionless matrices that span a Clifford algebra, you need to introduce a velocity scale in the resulting lienarization approach, like "c". As a consequence, the equation you end up with is a 4-spinor equation, which at a closer look is nothing else than the non-relativistic limit of the DIrac equation, which still has a "c" in it.

So what's my issue? My issue is: an equation including a velocity scale like "c" simply cannot be Galilei-covariant, no way. And using the quite common "c=1" unit convention does not make this velocity scale go away. And of course it isn't a relativistic equation either, because it represents the "nearly"-non-relativistic limit.

I am wondering why -- where this linearization approach is discussed at all anyway -- Galieli covariance is stressed. Not only in the paper above, but also in some textbook references (not many anyway). So maybe there is some error in my reasoning?

I should add maybe that Lévy-Leblond has made great contributions to analyzing the representation structure of the Galilei group in QM, and in another paper has done an analysis with the inhomogeneous Galilei group similar to what what Wigner did with the Poincare group years before. Therefore, what he states in his papers, at first glance, bears a certain weight, at least to me. This is why I am unsure

Dear Aleksei,

I think I understand now a little more of the intricacies involved in the overall "linearization" approach taken by Lévy-Leblond (and others, later).

Upfront, I admit I still have to find the flaw in the claim on Galilei covariance of the "linearized Schrödinger equation" (25) in the above-mentioned paper.

But what I know 100 % is that for "linearizing" of the Schrödinger equation and turning it into (25), you definitely have to break velocity invariance, i.e. introduce a velocity scale. I have done the detailed calculation with all units in place, and I am struggling how to see a Galilei covariance in this equation. Moreover, this equation (25) is absolutely well-known, and later in the paper it is recognized anyway that it is nothing else than the non-relativistic limit of the Dirac equation (IIIf. p.298).

That means: all the math is there and absolutely clear (bearing in mind that c=1)! No issue with any calculation whatsoever, only the interpretation is what I am struggling with!

Here is my interpretation:

1.) If you do the math right, in the approach taken for turning the "scalar" second-order-in-space differential equation into a "4-spinor" first-order-in-space differential equation, you have to introduce a velocity scale. This velocity scale means: you explicitly break Galilei covariance. The equation you end therefore still contains that scale, and of course: because it is nothing else but the non-relativistic limit of the Dirac equation, involving the well-known Dirac matrices). This is a physics textbook knowledge kind-of equation, well-known to (well, nearly) every physics student.

Summary: you break Galilei covariance explicitly, and of course never reach Lorentz covariance (which is never claimed anyway of course).

2.) Because the equation you end up with is a 4-spinor equation, physics intuition should at least signal to you that the whole framework has left non-relativistic physics anyway. Why? Because a 4-spinor is actually a 2x2-spinor or a bi-spinor, and is directly linked with the existence of an internal degree of freedom ("internal charge"), which eventually is linked to the existence of antiparticles. A 4-spinor in non-relativistic physics has no fundamental relevance or meaning whatsoever! In fact, as seen in the proceeding calculation following (25) in the paper above, a 4-spinor \Phi is found to have a "small" and a "large" component \phi and \xi, respectively. What other interpretation should be given to those degrees of freedom if not the one given by the Dirac equation (here: in the non-relativistic limit)? Again: this is standard textbook knowledge.

3.) "Linearization" by itself is ambiguous in meaning. the approach taken by Lévy-Leblond is turning a (d/dt, d^2/dx^2) equation into a (d/dt, d/dx) equation, thereby putting space and time derivative on an equal footing. This also smells of leaving the non-relativistic arena. You have to introduce a velocity scale "c" for doing so. That's why we end up with a "nearly"-relativistic equation, and *not* with a Galilei-covariant equation! This again is the reason why we end up with a 4-spinor equation, and not a 2-spinor equation that has been the goal in the first place.

3.) The Pauli equation with g=2 is "only" a side-effect of the whole approach, the reason being that you have a spinor equation in front of you. But you only get it via eliminating 2 degrees of freedom, either the upper (small) or the lower (large) component of (25), and at the same time: getting rid of "c"! This is again textbook knowledge because the equation you are dealing with the whole time is the well-known non-relativistic Dirac equation anyway! So where's the news?

The point is: you get the Pauli equation also by heuristically substituting p --> sigma. p in all equations, with a correct g=2 factor. The whole "linearization" approach is total overkill and absolutely misleading.

Interestingly enough (not covered by the paper): if you only analyse the time-independent Schrödinger equation and "linearize" it, you will successfully be able to do _without_ introducing a velocity scale.

A totally different topic is the nature of spin itself, as you correctly point out. Here Lévy-Leblond has shown in another paper in 1963 that the analysis of the representation theory of the Galilei group leads to two space-time characteristics of a point particle: mass and spin. Just as Wigner did for the Poincaré group way back earlier. This is fundamental insight that tells you both are of non-relativistic nature, whereas the paper above leads you along a wrong path of thinking.

My bottom line is: the whole linearization approach is at least partially a red herring. It does lead to a spinor equation, but as time and space are put on an equal footing, a velocity scale needs to be introduced, thereby explicitly breaking Galilei invariance (and covariance of the resulting equations).

If you want to understand the nature of spin: analyse the representation theory of the Galilei group and find out that SU(2) is the universal covering group of SO(3), and that 1-particle states also allow for a non-trivial representation of SU(2) when at rest. This is the same result obtained by Lévy-Leblond as Wigner achieved in 1939 for the Poincare group.

We use a hybrid Hamiltonian to map these operations. The question asked regarding C is that it is not a constant velocity when framing the references between Planck embedded instantaneous functions (Landscape) and the electromagnetic operations which are known for C as a velocity related action. Thus the speed of C, as it were, is a misnomer and will not serve as the cosmological constant. I have moved my focus to a less tangible/cognitive, but more promising, method of satisfying the modelling of relativistic space dimensional operations by developing the precise Planck landscape operations and path integrals. I will be publishing soon for open forums. The main reason I tell you this is it will end up in a cyclical paradox/ambiguous if pursued in the fashion we are accustomed to using the language and concepts presently applied. Non-metric space operations has special challenges, and communicating those concepts also requires a specialized set of principles and terminology, which has made my efforts take a bit longer than expected. This has been my focus for the last 3 years, and I am now satisfied I can proceed. The models you find in our archives all fit together neatly now, and have a high value once they are all connected together.

that should read " a less tangible/cognitive logical argument", but provide a more promising method " (apologies)