Abstract

There are currently three important connected major unanswered questions in physics and astrophysics.

(1) How can the theories of symmetry and the Higgs field be used to calculate the masses of the fundamental particles?

(2) How can dark matter be explained and described?

(3) How can dark energy be explained and described?

A self-consistent theory (called Model 1 in this paper) has been developed that appears to answer questions 2 and 3 quantitatively. In order to derive and justify Model 1, however, it became necessary to calculate the mass-energy of the proton and other fundamental particles that make it up, which answers question 1. This procedure then gave a path for calculating the mass of the Super Particle, which is the primary particle of Model 1. The super particle was found to be a Grand Unified Particle, which unifies the electromagnetic, weak and the strong forces. In investigating the properties of this super particle, it became obvious that it had the properties of dark matter, and when it breaks down, it generates dark energy. In a sequence of related papers (Wald, Model 1-A, Wald, Wald, Model 1-C; Wald, Model 1-D; Wald, Model 1-E; and Wald, Model 1-F), Model 1 is detailed and expanded, but the origin and operation of the Higgs field is still left unclear in these papers. This paper is written to fill this gap in the description of Model 1.

The Problem

The unique features of Model 1 are:

There are two spaces in the universe, low-energy particle space and high-energy quantum vacuum space. A potential barrier separates them. One space contains visible matter, and the other space contains dark matter.

There is a cycling of mass-energy between these spaces through the black holes that connect them. Particles pass from particle space to vacuum space through black holes where they are converted into super particles (energy ~10¹⁷GeV). They are then wrapped with a potential energy barrier shield (~10¹⁹GeV) to stabilize them. The barrier shield forms the boundary of high-energy vacuum space. These stabilized, shielded super particles are then able to escape from the black hole into particle space. The shielded super particles have a low interaction cross-section with ordinary particles except through gravity, and so are observed as dark matter.

Dark matter particles interact with each other and form a slowly building bubble centered on a galaxy that stabilizes its outer edges. Corridors of dark matter are also generated which form a cosmic web between the galaxies. These corridors guide the development of new galaxies.

The super particles can tunnel through the barrier into particle space. Upon reaching particle space, the super particles become unstable and break down into particles (cosmic ray protons) with ultra high kinetic energy (UHECR’s). In doing so, they give up potential energy from their barrier shields into particle space which becomes the dark energy that we observe as the cause of our accelerating, expanding universe.

These features do not define the nature of mass, the Higgs field, and gravitational attraction. Now Schumm points out that mass-energy appears to be the charge associated with the gravitational attraction (Schumm, 10). Shumm also notes that mass can be modeled by a screening effect with the Higgs field (see Shumm, 293-299). Higgs found out that elementary particle mass is connected to a field that permeates space (Kane, 97). The mass comes from the interaction of the Higgs field and the mass charge of the particle. Einstein described a background independent way to connect mass to the curvature of space-time in the field equation (Misner, 431 and 41). In all of this, it seems that there is not an intrinsic mass in the fundamental particles of the nucleus. The mass we observe appears to come from the interaction of the Higgs field and the mass charge of the particle. Further, it was noted that there is a real, scalar field in general relativity that controls space expansion (Peebles, 396).

These facts introduce an important issue, namely, the connection between the Higgs field, the inertial effects of mass, the field that controls the expansion and contraction of space, general relativity, and mass formation. This issue will be pursued here.

The Solution

There are five effects connected with mass that are usually described separately.

The origin of the Higgs field
The inertial effects of mass
The expansion and contraction of space
The gravitational effects of mass
The quantum connection between mass and the Higgs field

These effects are also described with five different theories, but they are closely related, and seem to require a unified theory. Such a unified theory will be described here.

1. Origin of the Higgs Field

We expect a field to have a source-its charge. For example, the electric field has a source-the conserved electric charge Q. This charge results from the invariance of the phase of electromagnetic radiation throughout space (Noether’s theorem). The result is the equation for the electric field (ε):

ε = K_eQ/r²

Now we have already noted that the concept of mass as a charge has been suggested by Schumm (Schumm, 10) and others. If this suggestion is true, we would expect that it will result in the following equation for a field emanating from a mass, which we will call f_h.

φ_h = K_ho m /r²

Let us assume that_φh is the Higgs field, and see the consequences of this assumption.

We expect the Higgs field to be uniform in space, because the mass we measure, which is generated by the Higgs field, is uniform in space, but the above equation indicates that the Higgs field might be lumpy, i.e. concentrated around mass particles. To explain why this conflict does not occur, we must look at three cases.

The Higgs field of particle space. In the particle space in which we live, the Higgs field comes mostly from the total distribution of mass in the universe. Mass distribution is uniform but low in value on a large scale, even though it is not on a small scale. Thus the Higgs field is smooth, but low in value on a large scale and dominated by the average mass density in the universe. Now mass formation is a quantum effect (equation 3, below), and so dominated by thresholds. If the Higgs field is above the threshold needed to form the standard model particles, particles are formed, and the mass achieved does not change until it exceeds the next quantum threshold, where it makes a quantum leap to a new value. A local change in Higgs field around a massive object would not be large enough or long lasting enough to exceed the next quantum threshold. Thus the mass-energy of a particle would remain constant as long as the average value of the Higgs field is above the threshold for particle space, but below the threshold for a higher energy space such as vacuum space.
The Higgs field in vacuum space. In the vacuum bubble that surrounds the super particle, the Higgs field comes mostly from the mass of the bubble shell, which surrounds the super particle, and thus provides a smooth, and very high-energy field density. A significant amount of field also comes from the mass-energy and kinetic energy of the super particle moving within the bubble shell, but it again does not exceed the next (Planck) quantum threshold. Thus the mass-energy in vacuum space would remain constant as long as the average value of the Higgs field is above the threshold for vacuum space, but below the threshold for a higher energy (Planck) space.
The Higgs field in a black hole. Here, the Higgs field is dominated by the average energy of the many particles in the black hole. However, the field is stratified in layers. The closer the layer is to the center of the black hole, the higher the mass-energy density, and so the higher the Higgs field. There exists a layer with enough energy to allow a super particle and its barrier to form, and the mass-energy would remain constant for this layer.

From Noether’s theorem, we expect that if there is an invariance in physical laws in different frames of reference, there must be a conserved quantity or charge connected with it. We have assumed that mass is the charge for the generation of the Higgs field, we have to ask, then, what is the invariance that causes the mass charge. As examples, let us start with the case of dynamics without acceleration. Here, we have observed that dynamics without acceleration follows the same physical laws if we move the frame a distance L in space, and this invariance generates a conserved quantity–momentum. Also, if we rotate the frame an angle θ, there is no change in physical laws, and this generates a conserved quantity—angular momentum. Further, if we move the time forward or backward, the laws do not change. This generates a conserved quantity—energy. Now, let us continue with the case of dynamics with acceleration. Einstein showed that there is no special frame of reference even for masses being accelerated by gravity. Thus we are justified by Noether’s theorem in taking mass as a conserved quantity for this case, and we find by experiment that it is indeed conserved.

We see that we have found that mass satisfies the requirements of a conserved charge that is connected to the Higgs field, and that it can be the source of the Higgs field. Since mass has all the characteristics of the charge and generates a Higgs-like field, we will assume it is the Higgs charge until we find reason to doubt this assumption.

2. The Inertial Effects of Mass Through the Higgs Field

The primary inertial effect of mass is the resistance to acceleration. This effect can be explained by a screening effect from the Higgs field (see Shumm, 293-299) that explains inertial mass formation. To explain this effect, we compare it with the screening effect of electrons on photons. If a photon passes through a medium filled with free electrons, its oscillating electric field oscillates the electron charges, and they generate photons that are out of phase, and so interfere with the oscillation of the original photon field. Thus the electric field in the medium will oppose the constant oscillation of the original photon field, and reduce it. This tendency is called screening, and it gives a finite range to a photon in conducting media. We can explain this effect by saying that the screening electric field generates an “effective mass” for the photon even though the photon has zero mass and infinite range. This screening range is similar to the range of a field of force generated by massive exchange particles, shown in the equation (see Kane, 29).

φ=φ_o /4πr exp(-mr) (natural units)

Note that the field is significant in size only out to a range of r ~ 1/m (natural units), where m is the “effective mass” of the exchange particles.

In a similar manner, we can explain the inertial mass of a particle with mass charge (quark, electron, Higgs) by the operation of a scalar (Higgs) field that “drags” on the quark as it accelerates. If an accelerating particle passes through a medium (vacuum) filled with mass-charged particles that create the field, the accelerating particle generates a Higgs particle that is out of phase, and so interferes with the Higgs field in the vacuum. This interference reduces the relative acceleration toward zero. Thus the ambient Higgs field will oppose the acceleration, and slow it down toward a constant velocity. We can then explain this effect by saying that the screening Higgs field generates an “effective mass” for the accelerating particle even though the particle has zero mass and infinite range. There is no evidence of any free inertial mass in a particle other than the “effective” mass generated by the Higgs field, so we will again assume that the Higgs field accounts for the inertial effects of mass until we find reason to doubt this assumption.

3. The Expansion and Contraction of Space

The expansion and contraction of space in high- energy zones are controlled by the following field density (r_f) and pressure (p) equations from general relativity (see Peebles, 396) In order to deal with the expansion and contraction of space, Peebles postulates the existence of a new real scalar field, and develops the equations for the field density r_f, and the pressure p in space from general relativity:

r_φ =φ’ ²/2+ V = field density

p =φ’ ²/2 – V = pressure (1)

Where:

V = a potential energy density

φ = a new real scalar field = the real part of the Higgs field

φ’= the time rate of change of the field

φ’ ²/2 = a field kinetic energy term

Also, from the field equation of general relativity, Peebles develops the cosmological equation for the time evolution of the expansion parameter (a(t)) due to average mass-energy density (r_m), pressure (p) and the cosmological constant (Λ) (see Peebles, 75)

ä/a = -4/3πG (r_m+ 3p) + Λ (2)

= acceleration of the cosmological expansion parameter

Where:

m = particle mass

r_m= Σm/vol

vol = volume of space containing the particles

Note from the field pressure equation (1), that if the potential energy density exceeds the field kinetic energy, the pressure is negative. The field kinetic energy term is slowly varying, however, because it depends primarily on the total mass of the universe, so the potential energy controls the pressure. Then, if the potential energy increases enough, the negative pressure term in equation (2) can become large enough to exceed the mass density term in equation (2), and the acceleration of the cosmological expansion parameter (ä/a) turns positive, and space will expand. If the potential energy V is small compared to the field kinetic energy term, however, the field pressure term is positive, and if Λ is small, the acceleration ä/a becomes negative, and space will contract. Even if the (ä/a) term is negative, however, space will continue to expand for a while, thus maintaining its prior state, but eventually, expansion velocity will reduce below zero and space will contract. Thus, if ä/a is positive, space will expand faster and faster as time goes on. On the other hand, if ä/a is negative, space will expand slower and slower until the expansion velocity reverses sign, and then space will contract.

It is important to understand that the field described here as a real, scalar field is associated with spatial curvature as is mass-energy, and has the general characteristics of the real part of the Higgs field. So the real part of the Higgs field accounts for the expansion and contraction of space, and mass-energy also accounts for the expansion and contraction of space. Thus we have reason to ascribe this effect to the Higgs field.

4. Gravitational Attraction from Mass-energy

Einstein described a background independent way to show how the curvature of space-time can be related to the local distribution of mass-energy in the field equation (Misner, 42). He then used the field equation to show that gravitation is a local phenomenon of massive particles in free fall in curved space-time, and does not require a force of attraction acting at a distance as indicated by Newton. Only the local distribution of mass-energy is important. The mass-energy curves space-time and gives an apparent attraction between massive objects that we call gravitational attraction.

Always remember, however, that mass is not the only thing that locally controls space-time curvature. There is a real, scalar field embedded in space-time that controls the shape of space as well (see Peebles, 396), and this field appears to be the real part of the Higgs field.

5. The Quantum Expression for Particle Mass Formation from the Higgs field

In Wald, Model 1-A, an expression for the formation of particle mass was derived based on a complex Higgs field and quantum principles. The expression for total particle mass that resulted is as follows:

m_pη= K_ohc m_tmu (3)

With:

m_t= [(S_o) n_o^2m’(1- λ_t(4-m)/4)) + (S_s)n_sp^2m’(1- λ_t((8-m)/8))]

And:

p = (s+z)-q

Where:

(S_o) = orbital symmetry factor = (1),(1×2),(1x2x3),(1x2x3x4)

(S_s) = spin symmetry factor = (0),(1/2),(1),(3/2)

K_ohc = 1/2 (r_e/a_e)^z f ^st^p

λ_t= self interaction coefficient

(r_e/a_e)^z = 0.534×10^-4for z = 0,1

f ^s= (0.9909)^s, for s = 1,2,3,4

t^p = (10³)^p, for p = 1,2,3

n_o = 1,2,3,4…

n_sp = +1/2, +1, +3/2, -3/2, -1, -1/2

m’(down) = m = 1, 2, 3, 4, 6, 10.

m’(up) = (m+(m-1)) = 1, 3, 5, 7,11,19

mu = eV

q_r = charge ratio number

= q_e /q_eo= -1 for electric charge

= q_i/q_eo= 1 for the isospin charge

= q_c/ q_eo= 1 for the color charge

= m_p/ q_eo= 1 for the mass charge

This expression was found to predict the mass-energy of all of the basic particles tried to within the experimental error of the measurement.

Summary

We see that a complex Higgs field originating from the mass-energy of the particles in space can account for all the effects of mass in both particle space and quantum vacuum space. Thus:

Mass generates the Higgs field.
The real part of the Higgs field generates the inertial effects of mass
The real part of the Higgs field works with the mass to generate the curvature of space. The curvature then generates the expansion and contraction of space.
The mass generated curvature of space works with the massive particles to generate the gravitational force.
The complex Higgs field works with particle charge to generate the mass-energy of the basic particles.

In a simpler statement, we may say that gravitational force charge (mass-energy) generates the Higgs field, the Higgs field generates mass-energy, and the mass energy generates the curvature of space and thus the gravitational force. Inertia and expansion-contraction of space are side effects of the process.

Conclusion

We see that the real part of the Higgs field contains the dynamics of space, and the imaginary part is required to generate particle mass. Together the real and imaginary parts of the Higgs field contain the total effect of mass in the universe.

References

L. H. Wald, Model 1-A “Mass and Function of the Standard Model Particles” www.Aquater2050.com/2017/01/
L. H. Wald, Model 1-C “Shaping the Dark Matter Cloud” www.Aquater2050.com/2017/02/
L. H. Wald, Model 1-D “The Recycling Universe” www.Aquater2050.com/2017/02/
L. H. Wald, Model 1-E “The Super Particle as a Cosmic Ray” www.Aquater2050.com/2017/03/
L. H. Wald, Model 1-F “How to Prove a Theory’s Correctness” www.Aquater2050.com/2017/03/
B. A. Schumm, Deep Down Things, Baltimore, Johns Hopkins University Press, 2004.
G. Kane, Modern Elementary Particle Physics, Ann Arbor, Michigan, Perseus Publishing, 19
Misner, Thorne and Wheeler, Gravitation, New York, Freeman and Co., 1973.
P. J. E. Peebles, Principles of Physical Cosmology, Princeton, New Jersey, Princeton University Press.