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This work is a discussion on the energy parallax theory developed in [1] [2] based on the multiplicity of the solutions theorem. This theory is compared with the perturbation theory in mathematical physics. The perturbation theory uses the increment of a solution which can be formalized with a Taylor series development. With the energy parallax theory, the convergence property of the Taylor series of the energy of a system is the key to decide to include additional solutions, defined on the so-called energy spaces [2]. The development is supported using various examples in quantum mechanics (i.e. Rayleigh-Schr ödinger perturbation theory) and wave theory with the Electromagnetic (EM) energy density (i.e. evanescent waves within the skin layer of a dielectric material). Finally, we discuss the Woodward effect [3] and the application of the energy parallax when assuming that the variations of EM energy density can trigger such effect within asymmetric cavities.

Perturbation theory has played an important role in the development of mathematics and physics from the end of the 19th century. With the pioneering work of H. Poincaré in the theory of dynamical systems, the perturbation theory found a major application in the emergence of quantum mechanics with the preliminary works of M. Bore and W. Heisenberg [

In quantum mechanics, perturbation theory generates states of a system that are adiabatically or linearly derived from a stable state. This stable state is generally an exact solution of the equations describing the system at hand. However, the system can be perturbed in a way that the exact solution, associated with the stable state, is no longer valid to model the changes in the system. Perturbation theory has been formulated in various domains, beyond quantum mechanics, using small quantities in order to describe the perturbed states, degenerated from the stable state. This perturbation shows up as a broadening of the initial energy quantity corresponding to the system in stable state [

In previous works (i.e., [

This work is a discussion between the energy parallax and the perturbation theory developed in quantum mechanics. In the next section, we recall this theory. An overview of the energy parallax together with the formal theory developed in [

In this work, several symbols are used. The set of integer numbers

Following [

However, perturbation theory can only estimate solution close to the exact solution. The addition of the small quantities to the exact solution can be expressed in (power) series (i.e. Taylor series around a nominated quantity―time, position in space, ... ). If the quantity becomes large, the series can diverge and the resulting solution is not valid to describe the perturbed system. In the example of our simple system described above, we have the condition that it exists N in

In quantum mechanics, those states are associated with intermediate levels of energy. These intermediate states are also solving the equations describing the system [

perturbation can be written such as

Hamiltonian operator driven by small perturbations [

It is important to recall for the following that the perturbations for integrable Hamiltonian system are described by the Kolmogorov-Arnold-Moser (KAM) theorem [

This section starts with a short summary of the energy space theory, which defines the energy parallax. Readers can refer to [

The concept of multiplicity of the solutions was developed in [

Furthermore, the author defines in [

To recall [

Definition 1. [

Thus, all the solutions are here defined in

The theorem of (Multiplicity of Solutions in

Theorem 1. If

1) (General condition to be a solution)

2) (Solutions in

3) (Multiplicity of the solutions) If

4) (Superposition of solutions and energy conservation ) If

Readers can refer to the appendices in [

Furthermore, let us recall a proposition first stated in [

Proposition 1. If for

then

is a convergent series.

In areas where the solutions are described via a set of PDEs, the perturbation theory can be rather complicated to use. Instead, the energy parallax shows that the variations of energy quantity lead to define solutions according to the spaces associated with the energy quantities (or energy spaces as defined in Definition 1). Theorem 1―the multiplicity of the solutions―lays the basis to define the solutions associated with the perturbed system, every time the energy increases in small quantities. Let us recall the definition of energy parallax [

Definition 2. Considering a linear PDE with some solutions in

The validity of this approach is only guaranteed if the power series of the energy is converging. The convergence properties is essential, because of the assumptions of small variations of energy. Due to those small energy variations, a limited number of energy subspaces are taken into account and thus a limited number of additional solutions are introduced in the considered system. This idea is written mathematically in the superposition of solutions and energy conservation, property 4 in Theorem 1. That is why the fundamental work in [

To illustrate the energy parallax, let us come back to our simple system example used in the explaination of the perturbation theory. The energy of this system

the solutions of PDEs describing this system is

Let us consider that

Let remind us of an example in functional analysis first shown in [

As a simple case of linear PDE, the wave equation with the particular solutions of the form of evanescent waves, was already discussed in Section 6 of [

c is the speed of light. Note that the values of t and r are restricted to some interval, because it is conventional to solve the equation for a restricted time interval in

finite energy functions, decaying for large values of r and t. It was previously underlined in [

solution does not belong to

In

Let us consider the form of solutions which propagates in a closed cavity (e.g., closed wave guide [

Here the symbol “

Now, let us do a hypothesis that

and then,

To recall that

Discussion (1): With the above example, we can now expose some common features between the energy parallax formulation and the perturbation theory. Firstly, one can emphasize the Lindset series of the energy (

of the corresponding energy

The energy spaces

Note that our formulation of the energy parallax is at the moment restricted to functions in

Perturbation theory may be difficult to implement when the system is described by a set of PDEs. One area in particular is the area of field theory such as Electro Magnetic (EM) field with EM waves as solutions of those PDEs. The term field is first coined by M. Faraday in 1849. The work of J. C. Maxwell leaded to the discovery of the propagation of EM waves [

Let us recall an example of variation of EM energy density in the skin layer of a conductor. The theory of energy space is now applied to the possible variations of electromagnetic energy density due to, for example, skin depth effect [

o is the Landau notation to omit higher order quantities. Note that at the first order

EM waves inside the skin layer of the copper plate are evanescent waves and thus functions in the Schwartz space (

Here f is either the electric or magnetic field (i.e. the absolute norm of

Now considering the wave equation, the electric field and magnetic fields are solutions and belong to the subspace

Finally one can write the relationship with the energy density following (14) and the previous Taylor series development for the electric and/or magnetic field:

Therefore, taking into account the second order term of the energy density

We are taking the example of the variation of EM energy density inside a copper wall due to planar waves reflecting and refracting on it [

with the principle of charge conservation:

Now, the variation of energy density (14) together with the equation of charge conservation is formulated such as:

using the equalities

We can separate in three groups,

The Poynting vector is defined as

The last line is the contribution from only the fields

Finally, the creation of the wave defined by the EM field (

Uncertainty principle is generally known from the Heisenberg’s relationship in quantum mechanics. In a broad sense, uncertainty principles are a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties describing a system, known as complimentary variables (e.g. position and momentum of a particle), can be known [

So far in our comparison between the energy parallax and the perturbation theory, the development is based on the mathematical properties of the functions in

Let us define the electric field E function in

defining

Associating the quantities

with the relationship (modified Parseval-Dirichlet equality)

Thus, the variation of the energy quantity

Finally, if we want to look at the inequality involving the position x, one needs to use the wave-particle duality and consider the wave as a photon. In this case, we can use the Heisenberg uncertainty principle in quantum mechanics to state the relationship between x and the moment p [

The Woodward effect, also referred to as a Mach effect, is part of a hypothesis proposed by James F. Woodward in 1990 [

The Woodward effect is based on a formula which the author implicitly assumed that the rest mass of the piezoelectric material via the famous Einstein's relation in special relativity

If we define the mass density such as

Let us define the the rest energy

In some particular cases such as an EM cavity, we assume that the variation in time of the rest energy is equal to the variation of EM energy density u (i.e.

The EM energy density u follows the general definition of the sum of energy density from the electric (

Discussion (2): The above equation shows that the variation of mass density is a linear relationship with the first and second derivative of the EM energy density. To recall Example 2 in Section 3.2.2, we underline the relationship between the order of the derivatives of the EM energy density and the energy spaces. As we are dealing with evanescent waves (functions in

This work is a discussion on the energy parallax and the comparison with the perturbation theory. One of the motivation is that the energy parallax is based on the multiplicity of the solutions (i.e. Theorem 1) developed by [

The perturbation theory is well defined when the system can be described with an operator (e.g., Hamiltonian) such as in quantum mechanics. However, complex systems using multiple operators or various PDEs may be best described in terms of the variation of the total energy. In this way, the energy parallax can be seen as an alternative. In the first example, the energy parallax is applied to the evanescent waves in the skin layer of a dielectric material (i.e. EM fields). We also show the possible application of this concept with the Woodward effect for the special case of the asymmetrical cavities. The energy parallax is used with the higher order derivatives of the EM energy density.

The authors would like to acknowledge people who have been involved in developing the presented model during the past two years via discussions or various feedbacks including Dr. José Rodal and Prof. Heidi Fearn (California State Fullerton University).

Montillet, J.-P. (2018) Discussion on the Energy Parallax and the Relationship to Perturbation Theory in Mathematical Physics. Journal of Modern Physics, 9, 479-499. https://doi.org/10.4236/jmp.2018.93034

In this section, we derive the Woodward effect for a particle moving along a world line in a varying electric field. It is a simplistic model of a so called “relativistic” capacitor, due to the variation of mass only dependent of the

variation of charge

Let us call

using the same development as the Euler-Lagrange equation (and the assumption that

Note that the momentum

The idea is to use the Lagrangian for a particle inside an EM field subject to a Lorentz force, but with a varying charge in time

Knowing that

with the assumptions that

Thus, from this equation, we can see that the first group of terms in

and reciprocally with the assumption

and by definition

As the “relativistic” capacitor model is a particle moving along a world line, one can follow the same way that Prof. Woodward used to establish the Woodward effect (See Appendix A of [

which ends up in

This equation is the particle accelerated by a Lorentz force with only the electric field. We can qualify it as the macroscopic view of the system. One can

then define a force

with

or

Let us define the potential

The infinitesimally variation

with

Note that it is possible to consider higher order derivatives in time if we consider the variations of the quantity

Also, it is worth to underline that equation (42) is established when considering only an electrical potential in the Lagrangian formalism of a point mass particle moving in an electrical field (i.e. Equation (34)). According to [

vector potential

do the same assumption as with the electric field in order to get an equation similar to equation (47). That is why, for the sake of the example, the addition of the magnetic potential is not so important.