How to get the conditions required to analyse the results?

Consider the quantum field theory (QFT) operator (an operator for each space-time point) that the field amplitude becomes when making the transition from classical field quantities to QFT operators. We will call this the field-amplitude operator. The type of field considered is one in which the classical field amplitude evaluated at a given space-time point is a complex number instead of a real number. In the QFT description, the field amplitude is not an observable and the field-amplitude operator is not Hermitian. Can we still say that an eigenstate of this operator has a definite value of field amplitude (equal to the eigenvalue) even when the field amplitude is not an observable and the eigenvalue is not real number?

As a core idea of gradient boosting problems is related with functional gradient descent algorithm and the solution implies iterative addition of a new model that learns on previous pseudo residuals, then probably it makes sense to transfer somehow the problem into the ordinary equation form (or even, in PDE if it possible formally). The are 2 questions? Does it make sense? Were there any efforts to test this approach? Thank in advance!

Any idea why the solution of the attached equation is always zero at r=0? It seems simple at first look, however, when you start solving, you will see a black hole-like sink which makes the solution zero at r=0 (should not be). I used the variable separation method, I will be happy if you suggest another method or discuss the reasons.

I also attached the graph of the solution, showing the black hole-like sink.

Thanks

The topic considered here is the Klein-Gordon equation governing some scalar field amplitude, with the field amplitude defined by the property of being a solution of this equation. The original Klein-Gordan equation does not contain any gauge potentials, but a modified version of the equation (also called the Klein-Gordon equation in some books for reasons that I do not understand) does contain a gauge potential. This gauge potential is often represented in the literature by the symbol A_i (a four-component vector). Textbooks show that if a suitable transformation is applied to the field amplitude to produce a transformed field amplitude, and another suitable transformation is applied to the gauge potential to produce a transformed gauge potential, the Lagrangian is the same function of the transformed quantities as it is of the original quantities. With these transformations collectively called a gauge transformation we say that the Lagrangian is invariant under a gauge transformation. This statement has the appearance of being justification for the use of Noether’s theorem to derive a conservation law. However, it seems to me that this appearance is an illusion. If the field amplitude and gauge potential are both transformed, then they are both treated the same way as each other in Noether’s theorem. In particular, the theorem requires both to be solutions of their respective Lagrange equations. The Lagrange equation for the field amplitude is the Klein-Gordon equation (the version that includes the gauge potential). The textbook that I am studying does not discuss this but I worked out the Lagrange equations for the gauge potential and determined that the solution is not in general zero (zero is needed to make the Klein-Gordon equation with gauge potential reduce to the original equation). The field amplitude is required in textbooks to be a solution to its Lagrange equation (the Klein-Gordon equation). However, the textbook that I am studying has not explained to me that the gauge potential is required to be a solution of its Lagrange equations. If this requirement is not imposed, I don’t see how any conclusions can be reached via Noether’s theorem. Is there a way to justify the use of Noether’s theorem without requiring the gauge potential to satisfy its Lagrange equation? Or, is the gauge potential required to satisfy that equation without my textbook telling me about that?

The Gamma function,

G(n)= Integral from 0 to infinity [Exp(e^-x^n)]dx

is of the great mathematical and physical importance.

It can be calculated without numerical integration (for practical purposes) via its mathematical and physical properties:

i-minimum of Gamma occurs at x = 1.4616321 and the corresponding value of Gamma(x) is 0.8856032.

ii-Gamma(1.)=Gamma(2.)=1.

iii-Gamma(x)=(x-1.) !

A simple preliminary approach that gives the value of Gamma(x) with an error less than 0.001 is the second-order polynomial expression for the factorial x,

(1.-0.46163*x+0.46163*x*x),x element of [0,1].

For example, this gives:

G(10.5)=11877478.

vs the value of 11899423.084) given by numerical integration.

and Gamma(1.4616)= 0.88527 vs 0.8856032.

Start with a purely classical case to define vocabulary. A charged marble (marble instead of a point particle to avoid some singularities) is exposed to an external electromagnetic (E&M) field. "External" means that the field is created by all charges and currents in the universe except the marble. The marble is small enough for the external field to be regarded as uniform within the marble's interior. The external field causes the marble to accelerate and that acceleration causes the marble to create its own E&M field. The recoil of the marble from the momentum carried by its own field is the self force. (One piece of the charged marble exerts an E&M force on another piece and, contrary to Newton's assumption of equal but opposite reactions, these forces do not cancel with each other if the emitted radiation carries away energy and momentum.) The self force can be neglected if the energy carried by the marble's field is negligible compared to the work done by the external field on the marble. Stated another way, the self force can be neglected if and only if the energy carried by the marble's field is negligible compared to the change in the marble's energy. Also, an analysis that neglects self force is one in which the total force on the marble is taken to be the force produced by external fields alone. The key points from this paragraph are the last two sentences repeated below:

(A) An analysis that neglects self force is one in which the total force on the marble is taken to be the force produced by external fields alone.

(B) The self force can be neglected if and only if the energy carried by the marble's field is negligible compared to the change in the marble's energy.

Now consider the semi-classical quantum mechanical (QM) treatment. The marble is now a particle and is treated by QM (Schrodinger's equation) but its environment is an E&M field treated as a classical field (Maxwell's equations). Schrodinger's equation is the QM analog for the equation of force on the particle and, at least in the textbooks I studied from, the E&M field is taken to be the external field. Therefore, from Item (A) above, I do not expect this analysis to predict a self force. However, my expectation is inconsistent with a conclusion from this analysis. The conclusion, regarding induced emission, is that the energy of a photon emitted by the particle is equal to all of the energy lost by the particle. We conclude from Item (B) above that the self force is profoundly significant.

My problem is that the analysis starts with assumptions (the field is entirely external in Schrodinger's equation) that should exclude a self force, and then reaches a conclusion (change in particle energy is carried by its own emitted photon) that implies a self force. Is there a way to reconcile this apparent contradiction?

There are a few point to consider in this issue

Points pro current emphasis

1. Math is the backbone of a physical theory. Good representation, good quantities of a theory, phenomena but bad math makes for bad theory

2. There is a general skepticism for reconsidering role of mathematized approach in physics Masters syllabi/upgrading role of literature/essay

2. Humans communicate, learn, think & develop construct via language

Arguments Con

1. Math is the elements in theory and "physics product" that is responsible for precision& prediction. Indespensible though, it exists in the mind of some individuals & function as well, in parallel with conception, physical arguments

2. Not all models in physics are mathematical. Some are conceptual

3. Formulations of solutions to physics problems via math techniques and methods is def of mathematical physics. However, this is a certain % of domain of skills.

But syllabus focuses 100% on this