Mass Gap Problem Solution in the Superfluid Quantum Space Model

A given problem in physics can be solved if it is well formulated. Well formulated means that it has a bijective correspondence to physical reality. Mass Gap Problem has no bijective correspondence with the physical reality and is that’s why not solvable mathematically. It can be solved in the frame of quantum mechanics by the formulation of the photon’s mass accordingly to the Planck-Einstein relation.


Introduction
To solve the Yang-Mills Mass Gap Problem [1] we have to understand well what mass is. We propose a bijectivity solution for the mass-gap problem where every element in the model of physical reality has exactly one correspondent model in physical reality. The model of physical reality is set Y and physical reality is set X. Every element in set Y has exactly one element in the set X. For example, element mass in the physical universe has correspondent element mass in the model of the universe. They are related by the bijective function of set theory: : → (1). (2).
Every object with mass is existing in space we call today a superfluid quantum space (SQS).
SQS is not "empty", it is the fundamental energy of the universe. Elementary particles are different structures of SQS [2]. In the space-time model of Special Relativity, the fourth coordinate 4 is imaginary, 1 , 2 , 3 are real coordinates. In the time-invariant space model, the 4th coordinate 4 is a complex coordinate as the other three coordinates 1 , 2 , 3 : "Timeinvariant superfluid quantum space (SQS) has a general n-dimensional complex structure ℂ n ; every point of it has complex coordinates: (xi, yi) (i = 1, ... , n) is an ordered n-uple of real numbers ((xi, yi)∈ℝ n ); for the purpose of this paper, we consider its subset ℂ 4 where all elementary particles are different structures of ℂ 4-SQS and have four complex dimensions zi " [2]. The model of the superfluid quantum space is close to the model of 4D superfluid quantum space-time, also named "superfluid ether" which uses quaternions as its mathematical background [3].
The idea of "empty" space as something real is harming physics for more than 100 years. It is time we demolish the idea of empty space and introduce ether back into physics.
Michelson-Morley's experiment has given null results because it was carried out by the proposition that Earth is moving through a stationary ether. Ether around the physical object is moving and rotating with the physical object. Masanory research suggested that on the distance of 20000 km above the Earth ether drift could be measured: "The satellites in the higher orbit (in the yellow region) have a possibility to detect the ether-drift. The evidence of the ether-drift can be proven by the fact that the ECI coordinate system does not work well. Of course, these experiments have not been carried out yet. The discussions of the ether-drift and framedragging were carried out more than 100 years ago. I have not carried out any calculation of the height of the frame-dragging using the theory of general relativity. At this stage, I consider that the height of the ether-drift detected is more than 20,000 km from the ground level" [4].
Mass is in physics an element with the attribute of energy . It is false to think that this element mass can exist in space that has no attribute of energy . The equation (4) below is false: .

SQS and Planck-Einstein relation
Mass of a given physical object is related to the energy of SQS accordingly to the well-known physical law of homogenous distribution of energy. Every physical system tends that the energy of the system is distributed in a homogeneous way; because of this every physical object with mass is diminishing the energy density of SQS exactly for the amount of its energy : where is the energy density of the SQS infinitely far away from the physical object surface, is the energy density of the SQS in the centre of a given physical object and is the volume of the object. In this way energy density of SQS remains uniform. In the Newtonian perspective, the area of space with a higher density is asserting a given pressure towards the area of space with a lower density. From a quantum physics perspective, universal space has the vector orientation towards lower energy density, or in space are quantum fluctuations towards lover energy density of space.
Equation (5) is valid from the scale of the proton to the scale of AGN. It can describe the process of the formation of the jets in the AGNs. The energy density of SQS there is so low that atoms become unstable and form jets that are spreading into intergalactic space [5].
Equation (5) is the extension of the mass-energy equivalence principle on the universal space that has its origin in time-invariant superfluid quantum space SQS and can solve the mass gap problem. The variable energy density of SQS is giving the origin to the inertial mass and gravitational mass as follows in Eq. (6) below: where is the energy density of the space in interstellar space; is the energy density of the space in the centre of the proton, and is the volume of the proton. Equation (6) shows that inertia and gravity are the results of the dynamics between given mass and a variable energy density of SQS. It is false to think that inertial mass is the same phenomenon as the rest mass 0 as the amount of energy Equation (7) below is false: We can combine two fundamental equations = 2 and equation = ℎ and we get: Photon is massless in the sense that it has no inertial mass . But photon has energy and so it has correspondent mass . Equation (9) is showing the mass of the photon related to its frequency. Combining (5) and (9) we get: Out of (10) follows: The space energy density difference − we can express as = − and we get: Eq. (12) shows that the value is constant. When frequency is increasing delta energy density is also increasing: For Yang-Mills Mass Gap Solution is required: "It must have a "mass gap;" namely there must be some constant ∆ > 0 such that every excitation of the space has energy at least ∆ " [1]. Eq. (12) shows this constant Δ > 0 is Planck constant ℎ. Namely, every particle can be seen as the excitation of the SQS and is defined by the difference of vacuum pressure , volume V and frequency ν of a given particle. These three elements are defining the Planck constant ℎ. A given excitation of SQS is producing a given elementary particle with energy .
We know the origin of the Planck constant (Eq. 12) and we see that the minimal "mass gap" in the excitation of SQS is defined by the Planck constant ℎ. In physical terms "mass gap" would be better named as "mass-energy gap", because Einstein told us already that mass and energy are made out of the same "stuff". What's this "stuff" is being clearly explained by Erving Schrodinger who also regarded space as the fundamental energy of the universe: "What we observe as material bodies and forces are nothing but shapes and variations in the structure of

The existence of imaginary time in Euclidean Green's functions is questionable
In 1 − ct by x1, x2, x3, x4, we also obtain the result that ds 2 = dx1 2 + dx2 2 + dx3 2 + dx4 2 is independent of the choice of the body of reference. We call the magnitude ds the "distance" apart of two events or four-dimensional points. Thus, if we choose as time variable the imaginary variable 1 − ct instead of the real quantity t, we can regard the continuum space-time, in accordance with the special theory of relativity, as an "Euclidean" four-dimensional continuum, a result following by the consideration of the preceding section" [8]. In the above citation, Einstein suggestion that we can choose the time variable t as the imaginary variable can be written as follows:  [6] has no bijective correspondence with the physical world and cannot contribute to the solution of the "Mass Gap". Osterwalder's and Schrader's model has internal inconsistency because it considers the existence of an imaginary time . The universe exists in time-invariant SQS, also mathematics that we humans developed exists in time-invariant SQS. Keeping time in mathematical models as a dimension will not give good results. We have to develop models of physical reality where time is the duration of changes running in a time-invariant SQS and is the result of the measurement from the side of the observer.
We proposed in this article the solution of the "mass gap" problem based on the Planck-Einstein relation that occurs in the time-invariant SQS. The result is that in the view of the mass-energy equivalence principle, the minimal change of "mass-energy" excitation and so the minimal mass gap in the superfluid quantum space and is defined by Planck constant ℎ.

Conclusions
The mathematical solution of the "Mass Gap Problem" might not be found because by keeping imaginary time, Euclidean green's functions do not have bijective correspondence with physical reality. Our proposal for the solution of the "Mass Gap Problem" is based on the variable energy density of superfluid quantum space and on the Planck-Einstein relation.