Looking at quantization of a wave function, from Weber(1961), to signals from wavefunctions at the mouth of a wormhole

: We utilize how Weber in 1961 initiated the process of quantization of early universe fields to the problem of what may be emitted at the mouth of a wormhole. While the wormhole models are well developed, there is as of yet no consensus as to how, say GW or other signals from a wormhole mouth could be quantized, or made to be in adherence to a procedure Weber cribbed from Feynman, in 1961. In addition, we utilize an approximation for the Hubble parameter parameterized from Temperature using Sarkar’s H ~ Temperature relations, as given in the text . Finally after doing this we go to the Energy as E also ~ Temperature, and from there use E (energy) as ~ signal frequency. This gives us an idea of how to estimate frequency generated at the mouth of a wormhole.


Introduction
We bring up this study first a result given by Weber, in 1961 [1] as to getting an initial wavefunction given in [2], which may be able to model behavior of what happens in the mouth of a wormhole if we make the assumption given in [3] that H (Hubbles parameter) is proportional to Temperature, and then go to Energy ~ Temperature. The last part will be enough to isolate, up to first principles a net frequency value The behavior of frequency, versus certain conditions at the mouth of a wormhole may give us clues to be investigated later as to polarization states relevant to the wormhole [4] as well as examining what may be relevant to measurement of signals from a wormhole [5] In doing all of this, the idea is that we are evolving from the Einstein-Rosen bridge to a more complete picture of GR which may entail a new representation of the Visser "Chronology protection" paper as in [6]

Looking at the Weber book as to reformulate Quantization imposed in GR
Using [1] a statement as to quantization for a would be GR term comes straight from The approximation we are making is to pick one index, so as to have'   [2] states a Hartle-Hawking wavefunction which we will adapt for the earlier wavefunction as stated in Eq. (6) so as to read as follows ( )

Examining the behavior of the Earlier wavefunction in Eq. (6)
Here, making use of Sarkar [3], we set, if say g  is the degree of freedom allowed

Methods used in Calculating Eq. (6), with interpretation of the results
We will be considering how, to express Eq. (6) And in doing this we will be looking at having a constant value for Eq. (8). If so then Then using numerical integration, [10], [11], [12] on page 751 of this [12] Notice the terms for the H factor, and from here we will be making our prediction If the energy, E , has the following breakdown 2 The upshot is that we have, in this, a way to obtain a signal frequency by looking at the real part of Eq. (11) above, if we have a small t, initially (small time step)

How to compare with a Kieffer solution and thereby isolate the Cosmological constant contribution
Eq. (11) would imply an initial frequency dependence,. What we are doing next is to strategize as to understand the contribution of the cosmological constant in this sort of problem. I.e. the way to do it would be to analyze a Kieffer "dust solution" as a signal from the Wormhole. i.e. look at [13], where we assume that t, would be in this case the same as in Eq. (10) above. I.e. in this case we will write having If so then we can assume, that the time would be small enough so that If Eq. (13) is of a value somewhat close to t, in terms of general initial time, we can write Here the time t would be proportional to Planck time, and r would be proportional to Planck length, whereas we set Then a preliminary emergent space-time wavefunction would take the form of ( ) Just at the surface of the bubble of space-time, with Planck tt  , and Planck r  This is from a section, page 239 of the 3 rd edition of Kieffer's book, as to a quantum theory of collapsing dust shells, . And so then we have the following procedure as to isolate out the contribution of the Cosmological constant. Namely, take the REAL part of Eq. (16) and compare it with the Real part of Eq. (10) Another way to visualize this situation and this is a different way to interpret Eq,. (15). To do so we examine looking at page 239 of Kieffer, namely [13] where one has an expectation value to energy we can write as What we can do, is to ascertain the last step would be to make the Eq. (16) in a sense partly related to the simple harmonic oscillator,. But we should take into consideration the normalization using that if [15]. If so, then we have that frequency is proportional to 1/t, where t is time. I.e. hence if there is a value of n=0 and making use of the frequency, we then would be able to write Eq. (16) as

Formal bounding of the cosmological constant, in terms of two wavefunctions plus analysis of initial wormhole frequency values
To do this we first of all consider the real part of Eq. (18) when n is set equal to zero, and the state it should be the same as the real part of Eq. (10). Then do the same with the real part of Eq. (19) and compare with Eq. (10) in order to isolate out a bound to the cosmological constant.
Doing so, leads to the following situation, and keep in mind that we are using Eq. (21) for regularization as well as setting If so, then we have the following bounding as far as the value of the cosmological "constant", namely   (24) In doing this, taking into account the Planck units and their normalization, we also need to keep in consideration the frequency, which we will denote here as Whereas what we will be doing , after we obtain a frequency of a signal near the mouth of a wormhole is to use the following scaling of frequency, near Earth Orbit from this wormhole. First if the wormhole is right at the start of the Universe[8], we use ( ) ( ) If we are say far closer to the Earth , or the Solar system, then we would likely see [8] 10 Earth orbit signal wormhole mouth signal Our derivation so far is to obtain the initial signal frequency for Eq.(26) and Eq.(27). Our next task is to obtain some considerations as to the Polarization, of say GW to observe and look for, in conclusion of this document.

The big picture, polarization of signals from a wormhole mouth may affect GW astronomy investigations
We will be referencing [16] and [17]. i.e. for [16] we have a rate of production from the worm hole mouth we can quantify as Whereas we have from [17] a probability for "scalar" particle production from the wormhole given by Whereas if we assume that there is a "negative temperature in Eq. (28) and say rewrite Eq. (29) as obeying having This is specifying a rate of particle production from the wormhole. And so then : Whereas what we are discussing in Eq. (28) and Eq. (29) is having a rate of , from a wormhole mouth, presumably from graviton production. If as an example, we are examining the mouth of a wormhole as being equivalent of a linkage between two black holes, or a black hole -white hole pair, we are presuming a release from the mouth of the wormhole commensurate with looking at [18] with an eye to "white holes" for a black hole model as of probability for "scalar" particle production given as, if M is the mass of the black(white) hole, m is the mass of an emitted "particle",  is frequency of emitted particles, Whereas we define the parameter  via a modified energy expression, as in [18] given by E as a modified energy expression in [18] , [19] ( ) ( ) 22 1 Our equation (28) and (29) , which are for wormholes, should encompass the same information of Eq. (31) which would be consistent with a white hole [20] [21`]at the mouth of a worm hole, as would be expected from Eq. (31), whereas reviewing a linkage between black holes and white holes as may be for forming a wormhole may give more credence to the information loss criteria as given in [22] Our next step is to ask if this permits speaking of say GW polarization in the mouth of a worm hole. To do this, first of all, note that in [23] that the simplest version of a worm hole is one of two universes connected by a "throat" of the form of a 'ball' given by 2 b  , whereas the term b, is in a diagram, consigned to be the radius, or shape of the initial "ball" joining two "universes" In the case of extending b to become the "shape" of the mouth of a wormhole, we would likely be using [24] for what is called by Visser the "shape" function of the wormhole [25], whereas what we are referring to in Eq. (33) comes straight from [23] ( ) ( ) Whereas we need to keep in mind the equation of state for pressure and density of [24] ( ) pr  = (34) The long and short of it is as follows. Following [24] we have that Whereas the b coefficient in the case of NON commutative geometry is chosen [26] ( )  (38) The idea would be that there would be n additional dimensions, as given in Eq,. (38) which would then lay the door open to investigating [29] and [30] in terms of applications, with [30] of additional polarization states to be investigated, as to signals from the mouth of the wormhole. We will next then go into some predictions into first , the strength of the signals, the frequency range, and several characteristics as to the production rate of Planck sized black holes which conceivably could get evicted by use of Eq. (28) , in terms of what could be observed via instrumentation.

A First order guess as to the rate of production of Planck sized black holes through a wormhole, using Eq.(28)
In order to do this, we will be estimating that the temperature would be of the order of Planck temperature, i.e. using ideas from [30] and [31] Some of the considerations given in this could be related to [32] as an afterthought whereas the author in [33] estimated for an LHC that there would be about 3,000 gravitons produced per second. Assuming a figure from [34] as to the percentage of black hole mass decaying into gravitons, i.e. [34], i.e,. 1/1000 of the mass of a Planck sized black hole would delve into gravitons, so if one had 3000 gravitons produced per second, as measured on Earth, one would likely have 2-3 black holes, of mass of about 10^-5 grams per black hole, producing say 10^57 gravitons, produced per black hole of mass about 10^-62 grams per black hole [35] Having said, that what about frequencies ? Here, if we have a wormhole throat of about 2-3 Planck lengths in diameter, with a frequency of emitted gravitons of about 10 19 GHz initially, it is realistic, using the following, to expect in many cases a redshift downscaling of frequencies of about 10^-18, if the worm holes are close to the initial near singularity, so then that we could be looking at approximately 10 to 12 GHz, on Earth, for frequencies, of initially about 10^19 GHZ . So then note at inflation we have ( ) ( )  [40] for additional clues and insights to consider while commencing this investigation. As well as details given in [41]