Examination of h(x) Real Field of Higgs Boson as Originating in Pre-Planckian Space-Time Early Universe

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1. Introduction

We begin this inquiry with a Higgs Boson scalar field along the lines of [1]

(1)

Here, the expression we wish to find is the change in the real field h(x) in time, whereas we have spatially

(2)

Our supposition is to change, then the evolution of this real field as having an initial popup value in a time, such that

(3)

The potential field we will be working with, is assuming a unitary gauge for which

(4)

The above, potential energy system, is defined as a minimum by having reference made to [1] Equation (4) as

(5)

And the quantum of the Higgs field, will be ascertained by [1] as having

(6)

Our abbreviation as to how the real valued Higgs field h(x) behaves is as follows

(7)

With, if is the inflaton, as given by [2] [3] , part of the modified Heisenberg U.P., as in [3] with specified by [3] [4]

(8)

The above eight equations will be what is used in terms of defining the change in the real Higgs field, h(x) in the subsequent work done in this paper. With the inflaton defined via [2] and the energy defined through [5] .

(9)

2. Analyzing Equation (7) and Equation (8) and Equation (9) to Ascertain Dh

We will be using by [2]

(10)

Note that the last line of Equation (10) is for the potential of the inflaton. We will be using, the first two lines for Equation (7), Equation (8) and Equation (9) in order to ascertain.

Leading to

(11)

Using the CRC abbreviation of the expansion of the Logarithm factor [6] , we have, with H.O.T. higher order terms

(12)

If we set coefficients in the above so that

(13)

Then, Equation (11) takes the form

(14)

To put it mildly, Equation (11) and Equation (14) are wildly nonlinear Equations for. What we can do to though is comment upon the equation for and also consider what if we consider

(15)

Equation (14) and Equation (15) lead to a different dynamic as given as to which is commented upon below.

3. What If We Look at a Time Step Dt as Real Valued, Due to Dh?

In doing this we are examining Equation (14) as a way to isolate an equation in and to ascertain what inputs of are effective in giving real value solutions to

We will re write Equation (14) as follows, to get powers of

(16)

To put it mildly, this will give cubic equation values for and according to [6] only one of the three roots for this would avoid having complex time solutions for. Accordingly, we have come up with an approximation to the energy, which would be a potential way out of this problem.

4. Using Nonlinear Electrodynamics, for a Value of the DE

What we are doing is finding a way to avoid having cubic roots, and worse for the and values. To do this we will make the following approximation based upon [7] , namely consider the energy density from a nonlinear Magnetic field, i.e. in this case set the E (electric) field as zero, and then

(17)

The scale factor

Here, we have that the Lagrangian defined by [7]

(18)

If so then the Equation (7) above, with this input into Equation (7) from Equation (17) will lead to using

(19)

Then going to put it together

(20)

If the right hand side of Equation (20) is chosen to be a constant, it fixes a value for the initial magnetic field which in turn fixes which in turn fixes a value for. Once this fixing of the term occurs, we have then

(21)

Equation (21) in terms of solving for is tractable, in terms of numerical input, depending upon defacto finding a minimum value of which could be obtained by taking the derivative of both sides of Equation (21) to obtain

(22)

It would then be a straightforward matter to take the quadratic equation for

(23)

This is assuming that we find a special and an initial configuration of the magnetic field for which we can write

(24)

5. Conclusion: Is NLED, Really That Important Here for h(x)?

Frankly the answer is that the author does not know. i.e. the idea is that NLED would enable the formation of Equation (24) which may be sufficient in the Pre-Planckian to Planckian regime to form Equation (23) which may be in initial configuration a first ever creation of the real valued Higgs field from Pre-Planckian space-time physics considerations.

Like many simple black board experiments, the frank answer is that the author does not know the answer, but finds that the above presented blackboard exercise intriguing and worth sharing with an audience.

The author hopes that additional extensions of this exercise may enable ties in with [8] below.

It is very important to note that in [9] [10] the foundations of nonlinear electrodynamics as outlined for cosmological implications for an initial scale factor less than zero is made a function of electromagnetic fields, and this will undoubtedly with additional study be in tandem with the inflaton physics details as outlined in this text.

Furthermore, in [11] , there is a proof that NLED (nonlinear electrodynamics) also is vital for the purpose of black hole physics, to avoid singularities, as well.

We do, indeed, have ample reason to suppose that nonlinear electrodynamics also ties into the h(x) field given and this tie in is part of a general modus operandi we are referencing in this paper.

Acknowledgements

This work is supported in part by National Nature Science Foundation of China grant No. 11375279.

References

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