Abstract
A new approach to reconciling General Relativity with Quantum Field Theory is Relativization, the act of making a physical model which obeys the principles of special relativity and General Relativity. This approach immediately yields results that no other approach has. I have established the foundations and fundamentals of relativization via a set of axioms. Expressions such as appear $P^{\mu}\leftp\right\rangle\,=k^{\mu}\leftp\right\rangle$, or $\left\langle p\rightP^{\mu}\leftp\right\rangle\,=\,\left\langle p\rightk^{% \mu}\leftp\right\rangle\,=k^{\mu}\left\langle p\right.\leftp\right\rangle\,=% k^{\mu}$. Such expressions appear in textbooks and papers but they are given a clearer interpretation in this model. Using this new approach to the problem, I will formulate the standard model as a relativized model in a curved space time with a locally valid gravitonHiggs interaction. This interaction will lead to a renormalized perturbation theory that can be summed up exactly to give the amplitude of gravitongraviton interaction as approximately $Cosh(p)$. I will solve a Schrodinger equation for a gravitationally bound system and get theoretical predictions relating to the thermodynamics of Planck scale black holes. Relativization has already provided a finite quantitative prediction for the quantum corrections to the local gravitational field, and gives results compatible with established black hole thermodynamics. Further research will certainly yield new insights.
1 Introduction
Relativization, the process of making something relative, specifically putting a theory of physics into a form that obeys the principles of Special and General Relativity. In my last paper (7) I explained the relativization of the KleinGordon field. Thus uniting Quantum Field Theory (QFT) and General Relativity (GR) with one formalism in one special case. In this paper I will formalize the fundamentals of relativization with a set of axioms, and apply them to the standard model of particle physics.
Models which accomplish this unification one must use Hilbert space and Minkowski space at the same time. Expressions of the form $P^{\mu}\leftp\right\rangle\,=k^{\mu}\leftp\right\rangle$ appear in text books on Quantum Field Theory, and String/M Theory such as (12; 2) in papers on QFT in curved space time, and in textbooks that dare raise the matter (3). Expressions which mix elements of Hilbert space and elements of Minkowski space, as above, are textbook parts of any modern relativistic quantum theory. If expressions such as $P^{\mu}\leftp\right\rangle\,=k^{\mu}\leftp\right\rangle$ make no sense to you, stop reading this paper and study something else.
Instead of treating the quantum theory as more physically fundamental, I am treating the principles of General Relativity as the more fundamental Physics. By restating the problem as one of making QFT comply with the prescripts of General Relativity my hypothesis opens a new path which can lead in a interesting new direction. In this paper I will present not only a hypothesis but a mathematical theory which makes concrete physical predictions.
2 Axioms of Relativization
The fundamentals of this model are encapsulated in the following axioms. These axioms are not all my own invention. These are based on the knowledge built up by the work of many, cited where appropriate, for how nature should behave in light of relativity and quantum field theory. The principle of relativization seems to have no precedent, and while other axiomatic formulations for QFT in curved space time have been published such as (11) they dealt with a static curved background. What follows will give a formulation which can admit a dynamical background.
F1The principle of Relativization: All physical theories must obey the Einstein Equivalence Principle. “that for an infinitely small fourdimensional region, the relativity theory is valid in the special sense when the axes are suitably chosen.” (6) In other words physical theories must be formulated in a way that is locally Lorentz covariant and globally diffeomorphism covariant. Stated with equations.
$x^{\mu}=e_{a}^{\mu}x^{a}$ 
$x^{a}$ is a vector in the locally flat space near a point.
$e_{a}^{\mu}$ is a vielbien flat space to the curved manifold.
$x^{\mu}$ is a vector in the curved space time manifold.
F2:Spectrum condition: All possible states of a QFT will be in the FockHilbert space $\mathcal{H}$. An operator on $\mathcal{H}$ must map states to other states in $\mathcal{H}$.
Nomalization condition: The inner product on $\mathcal{H}$ must be in a set isomorphic to the division algebras $\mathbb{R}$,$\mathbb{C}$,$\mathbb{H}$, $\mathbb{O}$.(1) For example an inner product on $\mathcal{H}$ of the form $\left\langle\psi\right\left.\psi\right\rangle=j^{a}$ with $j^{a}\in\mathcal{M}$ and $\forall$$\left\psi\right\rangle\in\mathcal{H}.$
The principle of QFT locality: QFT interactions occur in the locally flat space at the point of interaction. The propagation of particles between interactions is governed by Relativity.
Specification condition: Relativized QFT’s are defined by the above and the tensor product of their state space with Minkowski space. For a theory T, $T=\left\{\mathcal{H},\mathcal{H\otimes\mathcal{M}},A\left(\mathcal{H}\right)\right\}$ (Inspired by a similar statement in (11).)
The following sections show how each of these axioms are applied.
3 Hilbert and Fock Spaces.
A typical textbook explanation of QFT starts with Fock Space (12). Fock space is the Hilbert space completion composed of taking the tensor products of the Hilbert spaces of zero particles, one particle, two particles, … and so on up to n particles subspaces. As if every point in space was equipped with a quantum harmonic oscillator.
$\mathcal{F}_{\nu}(H)=\bigoplus_{n=0}^{\infty}\mathcal{H}^{\otimes n}=\left% \langle 0\right.\left0\right\rangle\oplus\mathcal{H}\oplus\left(\left(% \mathcal{H}\otimes\mathcal{H}\right)\right)\oplus\left(\left(\mathcal{H}% \otimes\mathcal{H}\otimes\mathcal{H}\right)\right)\oplus\ldots$  (1) 
The Hilbert space of a quantum field theory is in fact a Fock space(12). From here (12) introduces the quantum fields, usually KleinGordon for it’s simplicity. As it turns out a simple scalar field, the Higgs may be of fundamental importance to this issue.
3.1 Hilbert Space
I will take a different approach than (12). This approach based on the algebra of the field operators $A\left(\mathcal{H}\right)$ as prescribed by the specification condition. Instead of building up the space from prehilbert spaces let us consider the KleinGordon equation as if it were any other differential equation. The solutions of any differential equation of order 2 or greater form a space of solutions. This is certainly true of KleinGordon. I can show this by rewriting it in terms of particle creation and annihilation operators.
$\displaystyle\hat{H}\left\phi_{n}\right\rangle\,=\int\frac{d^{3}p}{(2\pi)^{3}% }\omega_{p}\left(a_{p}^{\dagger}a_{p}+\frac{1}{2}\left[a_{p},a_{p}^{\dagger}% \right]\right)\left\phi_{n}\right\rangle$  (2)  
$\displaystyle=\omega\left(N+\frac{1}{2}\left[a,a^{\dagger}\right]\right)\left% \phi_{n}\right\rangle.$  (3) 
The solutions $\phi_{n}$ to the above equation are vectors in the solution space associated with this equation. To find the conjugate solutions we need the conjugate equation
$\displaystyle\left\langle\phi_{n}\right\hat{H}=\left\langle\phi_{n}\right% \int\frac{d^{3}p}{(2\pi)^{3}}\omega_{p}\left(a_{p}^{\dagger}a_{p}+\frac{1}{2}% \left[a_{p},a_{p}^{\dagger}\right]\right)$  (4)  
$\displaystyle=\left\langle\phi_{n}\right\omega\left(N+\frac{1}{2}\left[a,a^{% \dagger}\right]\right).$  (5) 
To make a Hilbert space out of the solution space I need to define a inner product $<>$ with certain properties. I define the inner product as
$\left\langle\phi_{n}\right.\left\phi_{m}\right\rangle\,=\frac{1}{2}\left(\bar% {\phi_{n}}\circ\phi_{m}+\bar{\phi_{m}}\circ\phi_{n}\right).$  (6) 
In the above $\circ$ denotes the composition of functions. Does this inner product behave properly under the hermitian conjugation?
$\overline{}$
$\displaystyle\overline{\left\langle\phi_{n}\right.\left\phi_{m}\right\rangle}$  $\displaystyle=\overline{\frac{1}{2}\left(\bar{\phi_{n}}\circ\phi_{m}+\bar{\phi% _{m}}\circ\phi_{n}\right)}$  (7)  
$\displaystyle=\frac{1}{2}\left(\bar{\phi_{m}}\circ\phi_{n}+\bar{\phi_{n}}\circ% \phi_{m}\right)$  
$\displaystyle=\left\langle\phi_{n}\right.\left\phi_{m}\right\rangle$ 
it does.
This product has to be linear in its first argument. To show this I need to define a couple of values
$a=\left\langle\phi_{n}\right.\left\phi_{n}\right\rangle$ and $b=\left\langle\phi_{m}\right.\left\phi_{m}\right\rangle$.
$\displaystyle\left\langle a\phi_{1}+b\phi_{2}\right.\left\phi_{m}\right% \rangle\,=\frac{1}{2}\left(\overline{\left(a\phi_{1}+b\phi_{2}\right)}\circ% \phi_{m}+\bar{\phi_{m}}\circ\left(a\phi_{1}+b\phi_{2}\right)\right)$  (8)  
$\displaystyle=\left(a\bar{\phi_{1}}\circ\phi_{m}+b\bar{\phi_{2}}\circ\phi_{m}+% a\bar{\phi_{m}}\circ\phi_{1}+b\bar{\phi_{m}}\circ\phi_{2}\right)$  
$\displaystyle=\left(a\bar{\phi_{1}}\circ\phi_{m}+a\bar{\phi_{m}}\circ\phi_{1}+% b\bar{\phi_{2}}\circ\phi_{m}+b\bar{\phi_{m}}\circ\phi_{2}\right)$  
$\displaystyle=a\left\langle\phi_{1}\right.\left\phi_{m}\right\rangle+b\left% \langle\phi_{2}\right.\left\phi_{m}\right\rangle.$  (9) 
Last but not least there is the condition that the inner product of an element of the space with itself needs to be positive definite. This ensures that we may normalize probabilities to one
$\left\langle\phi_{n}\right.\left\phi_{m}\right\rangle\geq 0.$  (10) 
For this inner product the result is
$\displaystyle\left\langle\phi_{n}\right.\left\phi_{n}\right\rangle\,=\frac{1}% {2}\left(\bar{\phi_{n}}\circ\phi_{n}+\bar{\phi_{n}}\circ\phi_{n}\right)$  (11)  
$\displaystyle=\frac{1}{2}\left(2\bar{\phi_{n}}\circ\phi_{n}\right)$  
$\displaystyle=\bar{\phi_{n}}\circ\phi_{n}\geq 0$  (12) 
which satisfies the condition. Zero in the above would simply be the zero element in any of the appropriate division algebras which are isomorphic to $\mathbb{R}$,$\mathbb{C}$,$\mathbb{H}$, or $\mathbb{O}$.
This proves that the solution space of the KleinGordon equation forms a Hilbert space. I have demonstrated this fact of mathematics without recourse to the specifics of any particular division algebra.
For a complete treatment of Hilbert space using geometric algebra, let me define the following outer product or ketbra, $><$, as it is usually called in physics. This product gives as it’s result an operator in $\mathcal{H}$ ((5; 9) (in which it is represented with a wedge $\wedge$)
$\left\phi_{n}\right\rangle\left\langle\phi_{m}\right=\frac{1}{2}\left(\phi_{% n}\circ\bar{\phi_{m}}\phi_{m}\circ\bar{\phi_{n}}\right).$  (13) 
With this product and the inner product one may define the Clifford product (it is always denoted without a symbol) which will always be associative(9; 5).
$\phi_{n}\phi_{m}=\,\left\langle\phi_{n}\right.\left\phi_{m}\right\rangle+% \left\phi_{n}\right\rangle\left\langle\phi_{m}\right$  (14) 
Quantum Mechanical theories have to use an algebra of scalars which is isomorphic to one of these four $\mathbb{R}$,$\mathbb{C}$,$\mathbb{H}$,$\mathbb{O}$(1). Complex numbers are not the only real option (9), and for a truly relativized quantum field theory they may not be mathematically rich enough, hence the need for Clifford Algebra and Lie Algebra etc. Quantum Field Theory includes quantities that are 4vectors in Minkowski space which leads to special considerations. To satisfy F5, the specification condition requires a word on the space time algebra.
3.2 SpaceTime Algebra
The spacetime algebra is covered in great detail in a number of references (5; 10; 8; 4; 13; 1). The following is a summary of those materials.
Minkowski space $\mathcal{M}$, is constructed from the set of mathematical objects which vary, like vectors, under Lorentz transformations. These are four component objects which are, as a set, isomorphic to quaternions ($\mathcal{M}$is isomorphic to $\mathbb{H}$). When working with quaternions we often take advantage of this fact and write them as if they were a sort of four vector. The one to one and onto correspondence between them should be clear
$\left(a,b,c,d\right)\Longleftrightarrow\left(x^{0},x^{1}x^{2},x^{3}\right).$ 
The Minkowski norm is not the quaternion norm so $\mathcal{M}$ is not isometric to $\mathbb{H}$.
$x^{\mu}x_{\mu}=x^{0}x_{0}+x^{i}x_{i}.$  (15) 
This norm could not fulfill the requirement of being positive definite. Therefore Minkowski space is not a Hilbert space. This is not a barrier to using the algebra associated with it as the set of scalars over which one may write a Hilbert space.
It is possible to divide four vectors. There exist zero divisors and idempotent elements in the algebra of space time. These elements mean that certain operations that would be permissible in pure quantum theory are not in even standard relativistic quantum theory (12). Minkowski space time is a pseudodivision algebra. However that is not really a problem since in the approach taken in this paper quantum mechanics, and quantum theory are subordinate to relativity. The results will be relativistically correct, that is what matters.
3.3 Hilbert space over Minkowski Space.
Define the Hilbert space of KleinGordon as being the set of all eigenfunctions of the KleinGordon Hamiltonian with function composition $\circ$
$\mathcal{H}=\left(\left\{\left\phi_{n}\right\rangle\leftH\left\phi_{n}% \right\rangle=\omega\left(N+\frac{1}{2}\left[a,a^{\dagger}\right]\right)\left% \phi_{n}\right\rangle\right.\right\},\circ\right).$  (16) 
Then define Minkowski space as the set of all vectors that transform as vectors under Lorentz transformation with the Clifford product (which has no symbol). As above I choose to use the basis elements of the space, which in this case are non other than the gamma matrices.
$\mathcal{M}=\left(\left\{\gamma^{a}\gamma^{a}\gamma^{b}+\gamma^{a}\gamma^{b}=% 2\eta^{ab}\right\},\right).$  (17) 
With these definitions of the Hilbert space and Minkowski space we may finally combine them to get a “Hilbert space over Minkowski space”. This is the tensor product.
$\mathcal{H}\otimes\mathcal{M}=\left(\left\{\left\phi_{n}\right\rangle\otimes x% ^{a}(\gamma^{a})\leftH\left\phi_{n}\right\rangle=\omega\left(N+\frac{1}{2}% \left[a,a^{\dagger}\right]\right)\left\phi_{n}\right\rangle\,\,,\,\,\right.% \gamma^{a}\gamma^{b}+\gamma^{a}\gamma^{b}=2\eta^{ab},x^{\mu}=e_{a}^{\mu}x^{a}% \right\},\circ\otimes\right)$  (18) 
The tensor product of the relevant Hilbert Space with Minkowski space is defined as the set of all possible tensor products of the elements of H and M. The elements of the relevant Hilbert Space will satisfy the KleinGordon equation. The elements of M, $x^{\mu}$, will have as their basis set $\gamma^{\mu}$ which will satisfy the equation which defines the algebra of space time.
The inner product on this tensor product space ($<,>$)is a combination of the inner products on each of the subspaces like so (shown using the basis elements of each space)
$\left\langle\left(\left\phi_{n}\right\rangle\otimes\gamma^{a}\right),\right.% \left.\left(\left\phi_{m}\right\rangle\otimes\gamma^{b}\right)\right\rangle\,% =\,\left\langle\phi_{n}\right.\left\phi_{m}\right\rangle\gamma^{a}\eta_{ab}% \gamma^{b}=\frac{1}{2}\left(\bar{\phi_{n}}\circ\phi_{m}+\bar{\phi_{m}}\circ% \phi_{n}\right)\gamma^{a}\gamma_{b}.$  (19) 
The inner product on $\mathcal{H}$ and the inner product on $\mathcal{M}$ can be multiplied by each other to result in an object which is a scalar in both the Hilbert space and the Minkowski space. This need not be so. Suppose one wanted to compute the inner product on the Hilbert space yet have as a result a vector in the Minkowski space. The physical reason one would want this would be to find a probability current four vector.
$\left\langle\left(\left\phi_{n}\right\rangle\otimes\gamma^{a}\right),\right.% \left.\left(\left\phi_{m}\right\rangle\otimes\left(\gamma^{0}\right)^{2}% \right)\right\rangle\,\,=\frac{1}{2}\left(\bar{\phi_{n}}\circ\phi_{m}+\bar{% \phi_{m}}\circ\phi_{n}\right)\gamma^{a}\eta_{ab}.$  (20) 
The result of that equation is a quantity which is a scalar in $\mathcal{H}$ yet also a vector in $\mathcal{M}$. It is a subtle one. Instead of just stating this let me prove it by taking a Lorentz transformation.
$\displaystyle\left(1\otimes\Lambda^{ab}\right)\left\langle\left(\left\phi_{n}% \right\rangle\otimes\gamma^{a}\right)\right.,\left.\left(\left\phi_{n}\right% \rangle\otimes\left(\gamma^{0}\right)^{2}\right)\right\rangle$  $\displaystyle=\left(1\otimes\Lambda^{ab}\right)\left(\bar{\phi_{n}}\circ\phi_{% n}\right)\gamma_{b}\left(\gamma^{0}\right)^{2}$  (21)  
$\displaystyle=\left(\bar{\phi_{n}}\circ\phi_{n}\right)\otimes\Lambda^{ab}% \gamma_{b}\left(\gamma^{0}\right)^{2}$  
$\displaystyle=\left(\bar{\phi_{n}}\circ\phi_{n}\right)\gamma^{a}$  (22) 
This shows that it is clearly possible for an object in a Hilbert space over Minkowski space to be a scalar in Hilbert space while being a vector in Minkowski space. So when anyone writes expressions such as $P^{\mu}\leftp\right\rangle\,=k^{\mu}\leftp\right\rangle$, or $\left\langle p\rightP^{\mu}\leftp\right\rangle\,=\,\left\langle p\rightk^{% \mu}\leftp\right\rangle\,=k^{\mu}\left\langle p\right.\leftp\right\rangle\,=% k^{\mu}$ as was done in (7) it is perfectly valid physically and mathematically.
3.4 Discussion
At this point I have justified the foundations and fundamentals of relativization. I have cited supporting papers and textbooks, and rederived these expressions from first principles. Any socalled reviewer who is not satisfied with this must show where there is an actual error not simply state their own confusions as facts and preferences as principles. If one still does not get this read the following literature (12; 2; 3; 5; 10; 8; 4; 13; 1) then reread (7), then reread the present paper and view the accompanying PowerPoint and Mathematica files.
4 Theoretical Predictions and Consequences.
In this section I will relativize the standard model of particle physics and include gravity. From this testable predictions will be derived.
4.1 Relativization of The Standard Model of Particle Physics
In my previous paper (7) I was mainly interested in products of the form
$\left\langle\left(\phi_{n}\otimes\gamma^{a}\right),\right.\left\left(\phi_{m}% \otimes\left(\gamma^{0}\right)^{2}\right)\right\rangle\,=\bar{\phi_{n}}\gamma^% {a}\phi_{m}+\bar{\phi_{n}}\gamma^{a}\phi_{m}$  (23) 
Quantities such as this are of special interest for Quantum Field Theory. This subspace of $\mathcal{H}\otimes\mathcal{M}$ consist of the probability current four vectors, $j^{\mu}$. Knowing the probability four vectors leads us to the currents due to a QFT interaction. In the terms used in this paper one may write down the Riemann curvature operator
$\widehat{R_{ab}}=\left(d\langle\phi\phi\rangle(\gamma^{0})^{2}\wedge\gamma_{b% }+\langle\phi\phi\rangle(\gamma^{0})^{2}\wedge\gamma_{c}\wedge\langle\phi% \phi\rangle(\gamma^{0})^{2}\wedge\gamma_{b}\right)\langle\phi.$  (24) 
In equation 24 I have chosen to indicate that those brakets have their 4vector components explicitly using the $(\gamma^{0})^{2}$ which is an identity matrix. $\widehat{R_{ab}}$ will act on the Higgs field and gives an eigenvalue equation
$\widehat{R_{ab}}\left\phi_{m}\right\rangle\,=R_{abm}\left\phi_{m}\right\rangle.$  (25) 
$R_{abm}$ is the m${}^{th}$ curvature eigenvalue of the Riemann curvature operator, $\widehat{R_{ab}}$, for the eigenvector $\left\phi_{m}\right>$.
To incorporate the standard model into this theory the simplest path is to use the Lagrangian formulation. So I will write the standard model as a QFT in curved space time in terms of a invariant Lagrangian. In other words it will be a scalar in $\mathcal{H}_{sm}\otimes\mathcal{M}$ with all indicies summed over.
$\mathcal{L}=\sqrt{g}\left(\frac{1}{4}F^{ab}F_{ab}+i\bar{\psi}\gamma^{a}D_{a}% \psi+\psi_{i}g_{ij}\psi_{i}\phi+h.c.+\leftD_{a}\phi\right^{2}V(\phi)+R\bar% {\phi}\gamma^{a}R_{ab}\phi\gamma^{b}\right).$  (26) 
The term $R\phi\gamma^{a}R_{ab}\phi\gamma^{b}$ includes the standard Einstein gravity term with a correction due to Higgsgraviton interactions. Including standard Einstein gravity means this model predicts everything that GR does. The gravitonhiggs interaction will be the source of new predictions.
The Higgs interacts with gravitons in a way which will moderate (not mediate) the gravitational interaction terms. Consider the Feynman diagram rules which would relate directly to the graviton and HiggsHiggs graviton interactions. At every loop order this hypothesis introduces terms which will balance out the graviton loops (see fig 1) . This means that the perturbation series for gravitongraviton interaction will not run away. This series of diagrams can be summed, at least approximately. The amplitude of the gravitongraviton interaction will be approximately
$\overline{\leftM_{GG}\right}=\frac{1}{2}\left(\bar{\phi}\gamma^{a}R_{ab}\phi% +\bar{\phi}\gamma^{a}R_{ab}\phi\right)\gamma_{a}\gamma^{b}\approx R_{0}\,\,% Cosh(\hbar p).$  (27) 
The gravitational correction to the amplitude of the probability 4current for HiggsHiggs interactions would be tiny. The gravitational interaction can be safely ignored in any particle physics experiment at currently accessible energies. However at cosmological scales we may see the effects of these interactions.
4.2 States enclosed in a gravitationally induced event horizon.
The gravitongraviton self interaction is very interesting because it is this interaction which would dominate any region where gravity is strong enough to create a relativistic horizon, such as the interior of a black hole, or the whole of the universe. In classical GR nothing can escape a black holes horizon, and nothing may escape the universe. While boundary effects may cause Hawking radiation at any relativistic horizon due in part to the fact that even these astronomical objects ,while huge, are not of infinite extent.
With equation 27 providing the shape of the potential well one may attempt to set up and solve a Schroedinger equation to obtain a set of energy eigenstates for the gravitationally bound system. I did this in momentum space where the momentum is $p=\sqrt{E^{2}m_{0}^{2}}$. I assumed the mass of the system was proportional to the Planck mass and $\hbar=c=1$.
The command I gave Mathematica was (In which $m=m_{p}$ as in the Planck mass).
$\text{Solve}\left[\frac{p^{2}}{2m}\psi[p]+\text{Cosh}[p]==0,\psi[p]\right]$  (28) 
The output was, verbatim
$\left\{\left\{\psi[p]\to\frac{2\text{mp}\lambda\text{Cosh}[p]}{p^{2}}\right\}% \right\}.$  (29) 
Written in traditional form
$\psi\left(p\right)=\frac{2m_{p}\lambda Cosh\left(p\right)}{p^{2}}.$  (30) 
Interpretation of these last few equations is that the state function $\psi(p)$, as seen in figure 2, goes to negative infinity at zero momentum. Where will the momentum be zero in a black hole? The location of zero momentum is the event horizon. Black holes may be quantum mechanically all surface and no inside.
The region where the squared wave function is outside of the potential well represents a range of momenta and energies over which Hawking Radiation may be emitted. This sets a theoretical maximum on the momentum of the particles being emitted of about eight times the Planck Momentum
$p_{Hawking\,Max}=8m_{p}=56\,kg\,m\,s^{1}.$  (31) 
The maximum energy will be approximately 1500 times the Planck energy
$E_{Hawking\,max}=1500\,m_{p}=2.934\times 10^{12}J.$  (32) 
If these numbers are per Planck time then the following equation for the intensity of the radiation emitted will be true.
$I_{Hawking}=\frac{2.934\times 10^{12}J}{(4/3)\pi 4G^{2}m_{p}{}^{2}n^{2}}$  (33) 
I have arrived at a relationship between the mass of a black hole an the intensity of the Hawking radiation. Black hole thermodynamics a topic worthy of separate discussion follows from here.
5 Discussion
What does it mean for a space to be defined over another space? The first thing to realize is that a Hilbert space and the normed division algebra it is defined over are separate things. For example the Hilbert space of spin 1/2 particles is used to learn the basics in graduate school usually represented with 2x2 complex matrices, it could be represented with quaternions and 4x4 matrices with elements that are all real (8).
The underlying normed division algebra need not itself be a Hilbert space. The underlying normed division algebra need not consist of mathematical objects representable by 1x1 matrices. i.e. a quaternion which can be written as if it were a four vector, can be a scalar with respect to a Hilbert space. In a sense the Hilbert space is where all the actual physics takes place which is separated from the normed division algebra, which is also separate from the particular representation we choose for the operators and vectors in the Hilbert space, or the representation of the algebra.
However, we need those tools in order to translate abstract statements into concrete predictions. In the case of QFT we need to combine $\mathcal{H}$ with $\mathcal{M}$ to get the QFT we recognize. Then we need to combine $\mathcal{H}$ with $\mathcal{M}$ with a Riemannian manifold to get the General Relativized theory.
Physically equations 25 and 26 make perfect sense. Interactions involving the Higgs field give mass to the particles of the standard model. Gravity couples into the mass of a particle.The inclusion of the gravitonHiggs interaction leads to a finite number of counter terms at every loop order. They sum to a well known exact function. The gravitational effect will be proportional to the mass, so said both Newton and Einstein. It makes sense then that the field that gives mass could be an eigenvector of the gravitational field with a tensorial Eigenvalue. While the effect of this on the Lagrangian and the Feynman diagrams, as in figure 1) is to moderate and control the divergences in the perturbation expansion in terms of Feynman diagrams. The result is the amplitude for the graviton graviton interaction 27.
The graviton graviton interaction figure 1 will dominate in regions that are bound primarily by gravity, not only black holes but the whole universe, especially near the big bang. Figure 2 shows the shape of the gravitational potential. It is a well with infinitely high walls, this theory comports with the simple observation that on may not escape th universe, or the interior of a black hole. One may even make a hypothesis about the nature of the big bang, perhaps it was a transition from one excited state to another within this potential well. That analysis leads to the prediction of equation 33.
This model removes the classical big bang/ blackhole singularity and replaces it with a semirelativized quantum field theoretical reality. This is all only possible when one relativizes the standard model with the sensible assumption that the Higgs field which gives mass to all things also plays a special role in gravity.
All of that said I do not propose that what is exposed in this paper is a fundamental and final model. The portion dealing with the standard model is not fully relativized, it does not explain just how the standard model arises, it does not predict what other particles will arise between the Higgs scale and the Planck scale. Any fundamental theory will derive the standard model as one of its consequences. There is a program which can provide the tools to do such a thing, Mtheory. Only MTheory once formulated in a background free, diffeomorphism covariant way , could provide a final relativized theory of everything.
Relativization has already provided a finite quantitative prediction for the quantum corrections to the local gravitational field, and gives results compatible with established black hole thermodynamics. Further research will certainly yield new insights.
References
 Division Algebras and Quantum Theory. Foundations of Physics 42, pp. 819–855. External Links: Document, 1101.5690 Cited by: F3:, 3.1, 3.2, 3.4.
 String theory and mtheory:a modern introduction. Vol. 1, Cambridge University Press. External Links: Document Cited by: 1, 3.4.
 Spacetime and geometry. An introduction to general relativity. External Links: Document Cited by: 1, 3.4.
 Quaternions and special relativity.. Journal of Mathematical Physics 37, pp. 2955–2968. External Links: Document, hepth/9508011 Cited by: 3.2, 3.4.
 States and operators in the spacetime algebra. Foundations of Physics 23, pp. 1239–1264. External Links: Document Cited by: 3.1, 3.1, 3.2, 3.4.
 The Foundation of the Generalised Theory of Relativity. Annalen der Physik 7 (354), pp. 769–822. Note: In this Wikisource edition of Bose’s translation, his notation was replaced by Einstein’s original notation. Also some slight inaccuracies were corrected, and the omitted references were included and translated from the German original. External Links: Link Cited by: F1.
 Quantum gravity by relativization of quantum field theory.. The Winnower. External Links: Document, Link Cited by: 1, 3.3, 3.4, 4.1.
 Principle of General Q Covariance. Journal of Mathematical Physics 4, pp. 788–796. External Links: Document Cited by: 3.2, 3.4, 5.
 Imaginary numbers are not realthe geometric algebra of spacetime. Foundations of Physics 23 (9), pp. 1175–1201 (English). External Links: Document, ISSN 00159018, Link Cited by: 3.1, 3.1, 3.1.
 The Algebraic Approach to Spacetime Geometry. Acta Cosmologica 16, pp. 53. Cited by: 3.2, 3.4.
 Quantum fields in curved spacetime. ArXiv eprints. External Links: 1401.2026, Link Cited by: F5:, 2.
 An Introduction to Quantum Field Theory. Westview Press. Cited by: 1, 3.1, 3.2, 3.4, 3, 3.
 Geometric algebra and the causal approach to multiparticle quantum mechanics. Journal of Mathematical Physics 40, pp. 3327–3340. External Links: Document Cited by: 3.2, 3.4.
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Showing 11 Reviews

4
I had been corresponding with Hontas via email, with the ultimate aim of helping her see one of the simplest, fundamental errors/inconsistencies she has consistently had within both of her papers.
We discussed many things, then, finally, I tried to focus the discussion on that simple, fundamental issue. Perhaps, if I had simply discussed it with her as an abstract discussion of Hilbert spaces and things like them, maybe, just maybe, we could have gotten further. I don't know. All I do know is that whenever I got too close to the point she would either insist upon changing notation, or try to change the subject. Finally, she simply refused to engage, at least via our email exchange.
So, I am posting that particular thread of our discussions here, so as to abide by her oft expressed wish that discussions of her papers would be here, on the Winnower, so they can be seen with the paper.
This will also allow all readers to judge for themselves concerning this exchange.
So, what follows is that email thread. I hope people can follow it, since the formatting didn't carry over.On Oct 6, 2014, at 2:52 AM, David Halliday wrote:Hontas:Thank you for finally responding. (Unfortunately, with absolutely no indication you are willing to continue the discussion.)What you seem to completely miss in your interpretation(s) of the (self consistent)* literature is that, with regard to "Hilbert space(s)"**, such fall into three categories: I shall refer to these categories as C, D, and F.*** All of these classes can include expressions involving the Dirac matrices/operators. Only class C may include something (at least) like the Dirac "matrices" as scalars of the "Hilbert space". The other two will include them only as operators that may operate upon elements of the "Hilbert space", though the results of such operations may or may not be elements of the same "Hilbert space".Additionally, you fail to recognize that what you have is what I will refer to as class I, for Internally Inconsistent. This is (primarily) all that I and others have been trying to get you to recognize.I have no problem with Higgs et al.'s work, but you, my friend, are no Higgs (to warp a quote), at least not in this instance.I will check your "updated paper" before making this thread public. You have until then to choose to continue the discussion in a reasonable manner, with real intent.David* No inconsistent literature should make it through any good peer review, just as our peer review is calling you out on your internal inconsistencies.** I will use quotes around any use of the term that may not adhere to all of the axioms of an actual Hilbert space, since any other use will disallow certain operations that would be otherwise legal, and cannot rely upon all of the theorems of an actual Hilbert space.(Incidentally, another thing you seem to not recognize is that any finite dimensional vector space with an inner product is also a Hilbert space [a finite dimensional one, of course]. However, such is not all that important in the discussion of your papers.)*** Only those in class F are actual Hilbert spaces. However, completely general treatments of classes C and D include all of class F. Furthermore, completely general treatments of classes C and D have additional overlap over and above inclusion of class F.Sent from my iPadOn Oct 5, 2014, at 11:44 PM, Hontas Farmer wrote:I am not reticent. I have responded to your inquiries privately and now with an updated paper.What I don’t get is why you keep insisting there are fundamental issues when I have cited for you a long list of papers in which things I have used in this model are used by others up to and including the surprisingly controversial combination of Hilbert space and Minkowski space elements. (what else would a relativistic quantum theory have? )Listen. I hate to do this sort of think because it ranks high on the crackpot index, but my model now has some concrete prdictions to it’s name. At some point one has to take the attitude of Higgs and say with all due respect you do not understand. Higgs and collaborators had to say that to the likes of I beliveThat paper was another scientific milestone, but Higgs had to have a certain mental strength to complete it. His little gang of six maverick scientists who believed in his elusive particle were being stamped on from a great height.Two of them, Gerald Guralnik and Carl Hagen, had accepted an invitation that summer to a scientific conference organised by Werner Heisenberg in a German lake resort. Heisenberg, famously the author of the "uncertainty principle", was a scientific god, and this was a chance to impress their heroes.Guralnik and Hagen came back crushed. The mighty had viewed them with total disdain, and Heisenberg was the most vicious. He took it upon himself to tell the pair this stupid idea cooked up by them and Higgs was total "junk".If one wants to be like Higgs one has to be willing to tell Werner Heisenberg he’s wrong, and just leave it at that. With all due respect Dr. Halliday, your just not getting it and that does not mean I’m wrong.In either state so what… it’s not as if one can’t do 95% of the physics that matters without these hifaluitin ideas anways.Sent from Windows MailFrom: David HallidaySent: Saturday, October 4, 2014 5:04 PMTo: Hontas FarmerHontas:I truly don't understand your apparent reticence to discuss such a simple and fundamental issue with both of your papers. It would even be quite easy for you to correct this issue, and see how the correction trickles through your work. (You could even claim that this corrected approach was actually what you always meant. That we simply didn't understand that this is what you really meant. ;) )If we cannot continue this discussion, then, I suppose, it will be time for me to post at least this thread to the Winnower (that, as you have often requested, this discussion can be kept with your paper), and to Science 2.0, so all may judge for themselves.I will wait until midnight, CDT, Sunday, October 5th. If I haven't heard from you by then (12:00:01 am, CDT, Monday, October 6th) with an indication of when we will continue this discussion, I shall prepare to post at least this discussion thread.I'm hopeful that you will choose the more fruitful approach toward learning by continuing this discussion. However, it's all up to you.DavidSent from my iPadOn Oct 2, 2014, at 8:06 PM, David Halliday wrote:Hontas:Are we going to continue this conversation?Or are you just too busy, right now?I hope Windows 10 has been OK for you. (That's if you have installed the Preview.)DavidSent from my iPadOn Wednesday, October 1, 2014, David Halliday wrote:Hontas:Did you not read what I wrote?"Oh. It's not at all anything that 'caused the most [or any] problems for [me]'. I am absolutely familiar with many such equations. I have used such for three decades. ..."I have absolutely no problem whatsoever with "expressions like P\phi>". After all, I was the one that was using such notation (like what you were using in your first paper) when you asked me to use different notation (like what you use in your second paper). I was the one that first brought up the "eigen vectors of the momentum operator" that you had such a strong reaction to (September 20, 2014, 4:52 PM): "Eigenvectors of the momentum operator? What?"Why are you insisting upon "puting words in my mouth" that I have never said, or attributing feelings or thoughts to me that I do not have?If what I am "trying to teach" is "leading nowhere" then there can be only one reason. I have been moving slowly only due to your own apparent skidishness, your apparent reluctance to accept larger chunks, such as my email dated Saturday, September 27, 2014 4:00 PM, below.So, "Are you [truly] ready to go into one of the simplest such issues [with both your papers]?"Sent from my iPadOn Tuesday, September 30, 2014, Hontas Farmer wrote:David,I appreciate what you are trying to teach, but this is leading nowhere. I can cite for you where other thorist use the same terminology, and math but it won’t matter.We can discuss other things, but if expressions like P\phi> disturb you theories like this are just not going to make sense.Sent from SurfaceFrom: David HallidaySent: Tuesday, September 30, 2014 9:17 PMTo: Hontas FarmerNow. Back to what you have in your second paper.Let's assume that the far lefthand side of your equation (9), of your second paper, is actually equal to the far righthand side of your equation (10):<a\phi_1 + b\phi_2,\phi_m> = a<\phi_1,\phi_m> + b<\phi_2,\phi_m>Where \phi_1, \phi_2, and \phi_m are elements of H (as we have already agreed upon, right? [See your last email of Sunday, September 28, 2014, below]).Now, in your second paper, you set a and b to things that look like simple products of elements of H (presumably using the "Clifford product" that you don't define until your equation (15)). However, for equation(s) (9)(10) to have anything like full meaning, a and b should be arbitrary elements of the scalars of H, which may not be given by restricting such to inner products of elements of H with themselves (just as how Hilber spaces over the Complex numbers will not yield the full set of Complex numbers using only inner products of elements with themselves [such are only the Real subset]).So, let's define a and b as more arbitrary inner products of elements of H, as in:a = <\phi_n,\phi_p> and b = <\phi_q,\phi_r>Is that acceptable to you?Sent from my iPadOn Tuesday, September 30, 2014, David Halliday wrote:Oh. It's not at all anything that "caused the most [or any] problems for [me]". I am absolutely familiar with many such equations. I have used such for three decades. On the other hand, they have far different meaning than some that you seem to claim. However, I am not going to go into such things, here, since we are supposed to be talking about what you do have within your papers (with particular emphasis on your second paper, as per your wishes).Sent from my iPadOn Tuesday, September 30, 2014, Hontas Farmer wrote:I thought I had replied to this but maybe it did not get through… Yes I see that equation has the same type of configuration in particular it shows a four vector momentum  which is also an operator in the hilbert space acting on a ket in the hilbert space. The result of which is a vector k of momentum eigenvalues for each component. (I am still wondering if we really can’t measure that for some fundamental reason or just a limtation of instrumentation.)Taking that equation and using their braket notation. P\phi> = k\phi>if we take the braket. <\phiP\phi> = <\phik\phi>= k<\phi\phi>does k act as a scalar in that Hilbert space while being a vector in M yes it does. The concept I’m using, and which has caused the most problems for you is really not exclusive to me or my invention. It’s one that those who study quantum gravity, QFT in curved space time, Algebraic Geomotires, and String/M theory find useful and fundamental.Sent from SurfaceFrom: David HallidaySent: Monday, September 29, 2014 7:36 AMTo: Hontas FarmerWhen you ask "Did you read the M theory text I referee you too", are you referring to equation (2.60) on the bottom of page 36? Did you notice that that is concerning the very "eigenvvectors of the momentum operator" that you seemed to have so much trouble with, earlier?Sent from my iPadOn Monday, September 29, 2014, Hontas Farmer wrote:If that's how I used them in the paper.Did you read the M theory text I referee you too. They actually use an identical notion mixing 4vectors and kets as I do.On Sep 29, 2014 7:02 AM, David Halliday wrote:I'm also glad you stated that "Latin letters are in M".So, you agree that we may use elements such as a, b, c_1, and c_2 as elements of M. Correct?Sent from my iPadOn Sunday, September 28, 2014, Hontas Farmer wrote:Not being non committal. I would just prefer the notation of the second paper. Greek letters with Latin indicies are in H. Latin letters are in M ( Vielbiens will have Greek and Latin indicies and connect M to the underlying manifold) . :)On Sep 28, 2014 10:41 PM, David Halliday wrote:Ah. I see. You want to be as noncommittal as possible. ;)May we take \phi_1, \phi_2, \phi_m, \phi_n, \phi_p, \phi_q, and \phi_r as elements of your Hilbert space, of your second paper (call it H, or \superH, or whatever you want to call your Hilbert space of solutions of the KleinGordon equation, with inner product <,>)?Sent from my iPadOn Sunday, September 28, 2014, Hontas Farmer wrote:I say what I intend in the second paper which is a restatement of Wald and Hollands (cited earlier) with use of the geometric algebra language of the Cambridge group.Wald and Hollands show that one can set up a Hilbert space without prior reference to these or those scalars (R C H O). One looks at the operators and their products.I did not make any of that up that much is in the literature but I guess one would need to read those papers in order to appreciate what I'm writing now.On Sep 28, 2014 3:17 PM, David Halliday wrote:Hontas:I have no problem with bra/ket notation, functionlooking notation, operator notation, or even some new notation you made up, so long as everything is properly defined, including all necessary relationships.Unfortunately, when you go to something like H\otimes M we no longer have something that even looks like a "Hilbert space" with scalars taken from M, unless we do a lot of setup work.I'm assuming, based upon what you have stated within your papers, that the structure of M is the usual 3+1 vector space over the Real numbers with the (pseudo)inner product having three (3) spacial signs and one (1) temporal sign (technically, such are expressed in terms of the signs of eigenvalues of the quadratic form [think metric, expressed as a matrix], so we aren't restricted to a particular form of basis, but we'll assume that that's what is meant). (Whether one uses the [+++] or the [+] convention, it matters not to me.)Is that what you intend?Now, as for your "H": Is that a Hilbert space? If so, what set is being used for its scalars? (There can be many more pieces of information that may be important, but let's just start with that, for now.) If not, what is it?Sent from my iPadOn Sunday, September 28, 2014, Hontas Farmer wrote:In the first paper I used bras and kets to stand for vectors in H.In the second paper I switched to a more generic notation for inner products. <\phi,\psi> I wanted to deemphasize bra ket notation because it seemed to be a source of confusion.In terms used in the second paper:Are \psi >, \phi > \element H (or whatever you want to call your "Hilbert space" that permits elements of Minkowski space to be used as its scalars)?Yes denoted as in the second paper just \psi and \phi are in HIs u^a \element MYes they are vectors in M(or whatever you want to call your Minkowski space, whose elements you are able to use as the scalars of your "Hilbert space")?Not quite as I clarified in my second paper these act as scalars with respect to H, but remain vectors in M. I demonstrated this by explicity writing down the tensor product H\otimesM and it’s inner product.Is v^a\psi_1 + y^a\psi_2> \element H?It would be in H\otimesM and the relevant parts would be vectors in either H or M. (Remember in the second paper I’ve dropped the bra ket since it made people think QM instead of QFT.)v^a\psi_1 + y^a\psi_2 = v^a\otimes\psi_1 + y^a\otimes\psi_2Sent from SurfaceFrom: David HallidaySent: Saturday, September 27, 2014 5:10 PMTo: Hontas FarmerI'm using your equations and notation.Are \psi >, \phi > \element H (or whatever you want to call your "Hilbert space" that permits elements of Minkowski space to be used as its scalars)?Is u^a \element M (or whatever you want to call your Minkowski space, whose elements you are able to use as the scalars of your "Hilbert space")?Are \psi_1>, \psi_2>, \phi> \element H?Are v^a, y^a \element M?Is v^a\psi_1 + y^a\psi_2> \element H?Sent from my iPadOn Saturday, September 27, 2014, Hontas Farmer wrote:Not if those equations lead to the same confusions between M and H and which parts are in M and which are in H. I rewrote them explicitly pulling those pieces apart for a reason.Sent from SurfaceFrom: David HallidaySent: Saturday, September 27, 2014 4:00 PMTo: Hontas FarmerHontas:Is it completely OK if we assume the following are true?1) your very first equation in your first paper (the second equation in your second paper): <\psi \phi > = u^a \element M, for \psi >, \phi > \element H (Note: I did rename the vector element of M, here, just to avoid potential conflicts with the following.)2) the far right and lefthand parts of equation (5) in you first paper (similar to equations (9) and (10) in your second paper): <v^a\psi_1 + y^a\psi_2\phi > = v^a<\psi_1\phi > + y^a<\psi_2\phi >, for \psi_1>, \psi_2>, \phi> \element H, and v^a, y^a \element M3) let v^a\psi_1 + y^a\psi_2> = \psi> \element HSent from my iPadOn Saturday, September 27, 2014, Hontas Farmer wrote:As long as they are things I can’t site supporting literature for.Sent from SurfaceFrom: David HallidaySent: Saturday, September 27, 2014 2:59 PMTo: Hontas FarmerI was referring to fundamental issues, not practical issues.You have fundamental issues, long before any consideration of "predicted effects". Issues sufficiently fundamental that they call into question any possible "predicted effects".So, "Are you [truly] ready to go into one of the simplest such issues?"Sent from my iPadOn Saturday, September 27, 2014, Hontas Farmer wrote:Sure tell me which issue you see.The real issue I see is that the predicted effects are so small they may never be observed in any practical Earthly experiment. A prediction, once confirmed renders moot all issues.Sent from SurfaceFrom: David HallidaySent: Saturday, September 27, 2014 12:58 PMTo: Hontas FarmerYou are correct that, with regard to your ideas behind your papers, "It will take time to work it all out." However, there is a "shortcut" when it comes to issues one doesn't seem to truly recognize in one's own work, but at least someone else does.I'm offering you that "shortcut", if you are willing.As I said, "Are you ready to go into one of the simplest such issues?"Sent from my iPadOn Saturday, September 27, 2014, Hontas Farmer wrote:Yes, byt as you pointed out the same can be said of M theory. It will take time to work it all out.Sent from SurfaceFrom: David HallidaySent: Saturday, September 27, 2014 12:35 PMTo: Hontas FarmerHontas:You do recall that there are still unaddressed, fundamental issues with your papers, don't you? These are almost certainly issues that "reading up on M theory" is unlikely to help you with.Are you ready to go into one of the simplest such issues?Sent from my iPadOn Friday, September 26, 2014, Hontas Farmer wrote:Have you ever noticed how, with practically every scientific revolution, some "long cherished belief" has "gone by the wayside"? I'm certain the same will have to be the case with merging QFT and GR. The only question is which "long cherished belief" will need to be sacrificed: The Spacetime Continuum? Space (spacetime) being "based" on the Real numbers? Human observation as a basis for reality? That reality does not "allow" for "singularities"**? Or something else? (What other "long cherished beliefs" are left?)Yes. It may be that e need to give up both quantum and relativity. Another problem with theories of everything is that they often are also theories of anything.I’m going to wait months for comments before I edit the last paper. Meanwhile I’m reading up on M theory.Sent from SurfaceFrom: David HallidaySent: Thursday, September 25, 2014 9:23 AMTo: Hontas FarmerThank you, for the link to the "Axiomatic quantum field theory in curved spacetime" paper.I have yet to read it, but from the title (and the abstract) it appears they are trying to work toward an axiomatic formulation of QFT in curved Spacetime. I find that rather ironic, since people have been trying to formulate an axiomatic quantum mechanics (let alone, one for QFT in flat spacetime) for a number of decades (five or more?) with only limited success.I do think that attempting to Relativize Quantum Mechanics (QM), and, especially QFT, is an interesting and worthwhile endeavor. I also think that, perhaps, we need "to try and find something more fundamental than QM QFT or GR to truly describe the universe", or at least something quite different.I have often commented that there appear to be two primary camps: 1) those that believe QM is fundamental, and GR is just an approximation; vs. 2) those that believe "Einstein was right, that QM stuff is a bunch of huey" and try to get GR to include something that is "more complete" than QM. There is very little, it seems, that is being done in between. However, I'm reasonably certain both are wrong together: I think "reality" is something quite different from both camps, but with separate approximations that yield either QFT or GR.(Superstring theory [SST], and its descendants, such as M theory seems somewhat close to being in between, though, in my opinion, it is far too close to actually being in the first camp. LQG also appears somewhat in between, and, perhaps, somewhat closer to the second camp than SST. However, the worst thing, in my opinion, about SST, and its descendants, is that such seem to be "everything including the kitchen sink" theories. It's as though they have never run into an idea that they haven't tried to incorporate. Yet, they are still stuck in a background dependent formulation that violates the Relativistic fundamentals of GR.)My doctoral dissertation, a couple of decades ago, was on generalizing the Dirac equation, and other related things such as the integral that is the basis of the Hilbert space inner product (including what conditions operators have to fulfill in order to be placed therein), in conjunction with GR (not just on curved spacetime manifolds, but also taking into account the Dirac contribution to the energymomentumstress tensor that is the source term for the spacetime curvature).* So I have some direct experience in some of these matters. (I also found some unexpected features concerning the submanifold[s] over which the integral is allowed to be taken over. Unfortunately, I was not able to prove nor disprove whether the integral could be taken over a lightlike manifold, even though I was able to prove that it can be taken over any arbitrary spacelike manifold, even ones that are arbitrarily close to lightlike manifolds.)About that time, and since, I have come to wonder about spacetime itself.Have you ever noticed how, with practically every scientific revolution, some "long cherished belief" has "gone by the wayside"? I'm certain the same will have to be the case with merging QFT and GR. The only question is which "long cherished belief" will need to be sacrificed: The Spacetime Continuum? Space (spacetime) being "based" on the Real numbers? Human observation as a basis for reality? That reality does not "allow" for "singularities"**? Or something else? (What other "long cherished beliefs" are left?)David* Incidentally, I did include the Torsion and its couplings to both the Dirac field and the metric. I also included all forms of YangMills fields (electromagnetism being the abelian case): They come out as naturally as the connection coefficients and curvature tensor of GR. (I also found that the Dirac equation, itself, can be generalized in such a way as to allow Righthanded particles to be introduced without the Left, or vice versa. This negates the need for the use of projection operators to get rid of the ones we don't observe. However, this means that the algebra need not be the "Spacetime algebra", but something a little smaller.)** Even the strings or 'brains of SST/Mtheory are singularities. Any submanifold of a manifold, with lower dimensionality compared to the full manifold, is a singularity.Sent from my iPadOn Sep 23, 2014, at 10:58 AM, Hontas Farmer wrote:So true. I often wonder how much of what we know about the physical world is simply an artifact of our type of brain trying to understand the world. Vectors may in a sense be as imaginary as i.On the issue of QFT in curved space time. I cited for Ed Henry a good paper. The issues raised in which you may be familiar with. http://arxiv.org/abs/0803.2003. I cited that paper in response to his review as he named off certain… axioms of quantum mechanics which are not all valid in a curved space time, even when we consider locally flat little patches near where particles interact.As for the theory of Relativizaion let me say for the record I am not married to it. It looks interesting to me is all. I am starting to think someone has to try and find something more fundamental than QM QFT or GR to truly describe the universe. MTheory is a good try, maybe the only try. I may just abandon this to start studying M theory.Sent from SurfaceFrom: David HallidaySent: Tuesday, September 23, 2014 10:02 AMTo: Hontas FarmerWow. This is some wonderful (though, frankly, not truly expected) progress!Unfortunately, I didn't have time to respond, when I came in late last night, and I don't have much time this morning to do your reply justice. But I wanted to let you know that I appreciate your thoughtful response.In a sense, it does seem rather disconcerting that we, humans/scientists, have been building up all this vector knowledge base for all these centuries, and, yet, perhaps, we humans can never actually measure such directly.I'm reasonably certain that every case, where we humans seem to observer vectorlike quantities, moreorless directly, involve only displacementlike quantities (usually in space). One of the troubles is such quantities are the first to cease to be actual vectors in any curved space (or spacetime).On the other hand, this is not to say that we don't have vector spaces, and the like,Sent from my iPadOn Sep 22, 2014, at 12:13 PM, Hontas Farmer wrote:Ah, but the relativistic four current is not the same as the probability four current density! One is taken after the Hilbert space inner product is taken (hence, no longer a probability density), the other is taken afterward.Good point. I'm pretty sure we actually observe vectors though I am having trouble thinking of one where we really truly simeltaneously observe all the components. If not an electrical current…then I’m stumped. If what you say is right then your instinct that all we ever observe are the projected components of vectors… scalars that we then put together into vectors.That is the sort of foundational thing that makes peoples heads hurt. Kind of like weather we can quantize gravity, if the difficulty means it’s not possible, or that there is something more fundamental that we are missing.Sent from SurfaceFrom: David HallidaySent: Monday, September 22, 2014 7:16 AMTo: Hontas FarmerAh, but the relativistic four current is not the same as the probability four current density! One is taken after the Hilbert space inner product is taken (hence, no longer a probability density), the other is taken afterward.Sent from my iPadOn Sep 22, 2014, at 7:11 AM, Hontas Farmer wrote:The relativistic 4 current is just the relativistic 4 current. Even in quantum electrodynamics.On Sep 22, 2014 6:51 AM, David Halliday wrote:>> We are getting far afield, far off topic.>> Of course, you have made no indication as to whether you are referring to the probability four current of (classical, though Special Relativistic [Dirac]) Quantum Mechanics, or of QED. In the latter case, all observables are obtained from scalars in Hilbert space (with operators on Hilbert space inserted within the inner product), so there is never an observable "probability four current" like "thing" (though there are probabilities of observing different eigenvalues of such operators, however, even that is based upon projection operators within the Hilbert space).>> I would hope you know that when you move well into the realm where you have a very large number of particles, especially when you have a large number of interactions, you have moved into the realm where classical approximations work great.>> Such is the case with your lightning example. Such is also the case with a two slit interference experiment when the light source is good and bright (so you no longer see anything like single particle instances).>> In the case of the two slit interference, classical waves yield the same answer as (first) quantization of the electromagnetic field (which QED treats as its "concept" of a classical field). As far as we know, this is a good approximation to what QED would yield if one were to actually carry out the calculations for such.>> In the case of lightning, classical E&M with Statistical Mechanics (aka Thermodynamics) is a great approximation to classical QM wave mechanics (though it, to, is usually thrown in with Stat. Mech.). It is even supposed that, again, this would be a good approximation to what QED would yield if one were to actually carry such calculations (though it is debatable as to whether one would need to include Sta. Mech.).>> (Now, when it comes to Stat. Mech., one should keep in mind that that branch of Physics has yet to be brought into line with Special Relativity [SR], let alone General Relativity [GR]. However, there are people working on the problem, and they have made some progress. Of course, so long as Stat. Mech. is formulated on point particles, bringing it into line with SR will, practically automatically, bring it in line with GR.)>> Sent from my iPad>> On Sep 21, 2014, at 8:20 AM, Hontas Farmer wrote:>>> When you look up at a lightning bolt what do your eyes observe? The light from a strong current of electrons passing throygh the air. The four current of QED is simply the pure probability four current times e times n.>>>> Sent from Surface>>>> From: David Halliday>> Sent: Saturday, September 20, 2014 3:43 PM>> To: Hontas Farmer>>>> By the way "[t]he probability current four vector j is" never "one we observe" even "…as in electricity." We don't even observe components of such.>>>> Of course, since you had already acknowledged that "all the quantum ness is summed over / integrated out , however you like to put it", j is not actually "[t]he probability current four vector" but something more like the expectation value of "[t]he probability current four vector". Actually, something more like the expectation value of something proportional to the electromagnetic "current four vector".>>>> That's something we can observe components of.>>>> Sent from my iPad>>>> On Sep 20, 2014, at 11:17 AM, Hontas Farmer wrote:>>>>>> By the way, you are quite correct when you say "Four vectors exist in Minkowski space which is itself not a Hilbert space... But a relativistically correct theory must use them for the observable quantities." However, it turns out, one is allowed to do some things, with such quantities, in Special Relativity (with its universal tangent space that is identifiable with the base manifold), that are quite disallowed within General Relativity. One of those that doesn't work well (though one may be able to "get away with", if one is particularly careful) is Hamiltonian like equations. Another that simply cannot work, because there is no meaningful definition for such, are nonscalar valued integrals.>>>>>>>>> Exactly, the four vectors aren’t in H. So one cannot operate on them with an operator in H. They are at best in H \otimes M. An operator defined in H\otimes M could perhaps operate on them….but the operators in QED are not really like that. Even as QED is a mixture of both Quantum and Special Relativity Hilbert Space and Minkowski space…. there are these separate parts that play off eachother in a way that seems to work.>>>>>>>>>>>> Actually, when you really think about it, are we ever really able to directly observe anything besides scalar quantities? Even when we do measure higher order quantities, don't we actually, directly, only measure scalar quantities formed via "projections", and the like, to scalar quantities? (Could this lie at the heart of such phenomena as only being able to measure a single component of spin [along with magnitude], at any one time? Just a thought.)>>>>>>>> So when we are using laboratory tools are we ever actually measuring a vector? That’s an interesting question…. I think the answer would depend on weather we mean classical or quantum physics.>>>>>>>>> Consider this>>> I had my students do a lab where they had two pucks on an air table and collided them a few different ways. While one did the collisions the other taped it, with their cameras, from above. They had meter sticks along the sides of this table so they could look at them in the video and measure the distances traveled, the video would also time the interactions so they could measure time using the video players clock.>>>>>> Looking at the video in motion it seems to have measured vectors but did it? The whole video together, the series of pictures 1/60th of a second apart looks like the video measures a vector…. On the other hand all it really is is a bunch of observations of moments in time, with the position along one axis and the position on the other. The camera measures both at the same time, so it measures position vectors. The answer for classical physics seems to be a yes.>>>>>>>>> Quantum mechanics and QFT would be a bit more complicated. We can write things like a position operator>>>>>> (for the sake of less typing I’ll refer to the equations here>>>>>> In theory I think of why we could not measure all the components of r with some sort of Camera….. but in real life I don’t think so.>>>>>>>>> On the other hand, when we actually look at those “pictures” we end up taking a look at the scale on one axis and the scale on another axis and the time and doing a bunch of calculations to put together a vector. :/ I ‘m not sure what the answer is to that.>>>>>> Where this comes into the model we are discussing is in a relativsitic theory whatever we observe has to obey relativity, so four vectors, and scalars in M(If we take the Minkowski inner product of the resulting four vectors) .>>>>>>> Ok. So, <h1,h2> had already had the special dependence integrated out, as must be for any usual Hilbert space of QED. Correct?>>>>>>>> If so, then operating on <h1,h2>m2 with the Dirac Hamiltonian is not well defined. (We can simply treat it as a member of the Minkowski space, multiplied by the scalar, <h1,h2>, which is an element of whatever [mathematical] field the Hilbert space is taken over [exclusively the Complex numbers, if we are talking about QED]. Then the question of operating on "J" with the Dirac Hamiltonian simply boils down to operating on an element of the Minkowski space, which will all depend on how one wishes to define such an operation.)>>>>>>>>>>>> Yes, all the quantum ness is summed over / integrated out , however you like to put it. The probability current four vector j is one we observe…as in electricity.>>>>>> Dirac Hamiltonian on j in M… I could call it D>>> (D \otimes I)( n>\otimes j) =(N+1/2)n>I j>>> =(n+1/2)j>>>>>> j is in a sense along for the ride like a scalar would be.>>>>>>>>>>>>>>>>>>>>>>>> Sent from Surface>>>>>> From: David Halliday>>> Sent: Saturday, September 20, 2014 9:43 AM>>> To: Hontas Farmer>>>>>> By the way, you are quite correct when you say "Four vectors exist in Minkowski space which is itself not a Hilbert space... But a relativistically correct theory must use them for the observable quantities." However, it turns out, one is allowed to do some things, with such quantities, in Special Relativity (with its universal tangent space that is identifiable with the base manifold), that are quite disallowed within General Relativity. One of those that doesn't work well (though one may be able to "get away with", if one is particularly careful) is Hamiltonian like equations. Another that simply cannot work, because there is no meaningful definition for such, are nonscalar valued integrals.>>>>>> The latter would seem to preclude many "observational quantities". Fortunately, since one can use "laboratory" (or "instrument") quantities to obtain scalars, from what would otherwise be higher order tensors, we do have a way to obtain something very much like actual, real world, "observational quantities".>>>>>> Actually, when you really think about it, are we ever really able to directly observe anything besides scalar quantities? Even when we do measure higher order quantities, don't we actually, directly, only measure scalar quantities formed via "projections", and the like, to scalar quantities? (Could this lie at the heart of such phenomena as only being able to measure a single component of spin [along with magnitude], at any one time? Just a thought.)>>>>>> Sent from my iPad>>>>>> On Sep 20, 2014, at 5:30 AM, Hontas Farmer wrote:>>>>>>> J is the product of the inner product on H with the gamma matricies. How can it not be a scalar in H?>>>>>>>> You don't claim that the four vectors that appear in QED are vectors in the Hilbert space? Right? Four vectors exist in Minkowski space which is itself not a Hilbert space... But a relativistically correct theory must use them for the observable quantities.>>>>>>>> On Sep 19, 2014 10:52 PM, David Halliday wrote:>>>> Oh, there are a lot of elements in QED that involve a number of the elements of the algebra of the gamma "matrices"/operators (as well as the electromagnetic four potential), nestled within wave function solutions of the Dirac equation (whether expressed as a Hamiltonian or not).>>>>>>>> Now, if the J^\mu you expressed below is actually a scalar in the Hilbert space (H), then operating on it with the Dirac Hamiltonian is not at all well defined.>>>>>>>> If, on the other hand, it is not a scalar in H, then we can operate on it, and the result will be something like a temporal partial derivative of the four probability density current. However, even that is not all that well defined in Special Relativity. (The fact is that Hamiltonians are anachronistic within Special Relativity, and most especially problematic from the standpoint of General Relativity.)>>>>>>>> So, let me ask you: Given two elements of the Hilbert space of QED, say x and y, and a linear combination, ax + by, that is also an element of the Hilbert space of QED, with scalars a and b; what "space" can the scalars (a and b) come from, and still have the linear combination (ax + by) also be an element of the Hilbert space of QED? (Remember, the elements of the Hilbert space of QED are the solutions of the Dirac equation.)>>>>>>>> Sent from my iPad>>>>>>>> On Sep 19, 2014, at 9:20 PM, Hontas Farmer wrote:>>>>>>>>> "it is, by no means, any indication that QFT (or even, more specifically, QED) uses elements of the Minkowski space (or even something related to it) as the scalars of its Hilbert space.">>>>>>>>>>>>>>>>>>>> This is the crux of the matter. Let me ask you this.>>>>>>>>>> We agree that quantities like the 4current and the EM 4potential do exist in QED. So suppose we take the Dirac Hamiltonian and operate on J^\mu with that What do we get?>>>>>>>>>>>>>>> ________________________________>>>>> From: David Halliday>>>>> Sent: Friday, September 19, 2014 9:17 PM>>>>> To: Hontas Farmer>>>>> Subject: Re: So, what's your verdict? (was: Re: [Science 2.0] Your email in the system is still...)>>>>>>>>>> You respond to my question>>>>>>>>>>>> “Now, is this the use of "a Hilbert space over ..." that you intend, except where the field is replaced by something related to a Minkowski space?”>>>>>>>>>> with>>>>>>>>>>>> Basically.>>>>>>>>>> This could be interpreted as agreement, or as some equivocation.>>>>>>>>>> Then, your very next sentence is>>>>>>>>>>>> As I keep citing standard QFT has many such terms in it.>>>>>>>>>> and then you continue with>>>>>>>>>>>> My favorite example the fourcurrent.>>>>>>>>>>>> j^\mu = \bar{\psi}\gamma^\mu\psi>>>>>>>>>>>>>>> Now, while (so long as there is an implied integral in that expression) it is true that>>>>>>>>>>>> That quantity is a scalar in H…but a vector in M.>>>>>>>>>> it is, by no means, any indication that QFT (or even, more specifically, QED) uses elements of the Minkowski space (or even something related to it) as the scalars of its Hilbert space.>>>>>>>>>> See, this is one of the communication issues that has been cropping up a lot in your exchanges.>>>>>>>>>> This is why I asked the question (that you didn't even address)>>>>>>>>>>>>>> [D]o you still hold that the question of whether something like a Hilbert space can be defined, as you propose with that very fist equation, so the vectors of Minkowski space are used as the scalars of such a "Hilbert space", is new/novel?>>>>>>>>>> After all, if you think that question is new and/or novel, then it cannot have already been a part of QFT (or even, more specifically, QED). On the other hand, if you truly do think that it has already been a part of QFT (or even, more specifically, QED), then I would have to ask why has no one done any of the other things you have in your paper, wouldn't such have been just as obvious to so many people over so many years?>>>>>>>>>> Sent from my iPad>>>>>>>>>> On Sep 19, 2014, at 8:39 PM, Hontas Farmer wrote:>>>>>>>>>>> “Now, is this the use of "a Hilbert space over ..." that you intend, except where the field is replaced by something related to a Minkowski space?”>>>>>>>>>>>> Basically. As I keep citing standard QFT has many such terms in it. My favorite example the fourcurrent.>>>>>>>>>>>> j^\mu = \bar{\psi}\gamma^\mu\psi>>>>>>>>>>>> That quantity is a scalar in H…but a vector in M. I find it odd that people have trouble with that. The inner product given in the first paper was designed so that if one takes the product of a vector in H… like \psi with itself it will be a probability current four vector.>>>>>>>>>>>> Sent from Surface>>>>>>>>>>>> From: David Halliday>>>>>> Sent: Friday, September 19, 2014 8:16 PM>>>>>> To: Hontas Farmer>>>>>>>>>>>> Hontas:>>>>>>>>>>>> You didn't actually answer, or even address my question.>>>>>>>>>>>> Now, while a Hilbert space over the (mathematical) field F is, in a fundamental sense, a tensor product of an infinite vector space and the field F, there are many additional aspects, such that we almost never have to continue using a tensor product, but can treat it as a simple multiplication (this has particularly to do with relationships between the "tensor product" "multiplication" vs. the multiplication in the field, as well as the relationship between addition in the field vs. addition in the vector space). Just as with any vector space over a field, the elements of the field (F) are the scalars of the Hilbert space: All linear combinations of elements of the Hilbert space are formed from sums (in the Hilbert space) of products of elements of the Hilbert space with elements of the field (F). (This also implies that the results of inner products are elements of the same field.)>>>>>>>>>>>> Now, is this the use of "a Hilbert space over ..." that you intend, except where the field is replaced by something related to a Minkowski space? Or are you intending something with a different structure (such as having the base domain of the function space basis of the Hilbert space be the Minkowski space)? (After all, tensor products between "spaces" need not have any particular structure. One has a lot of leeway in defining the rest of the structure.)>>>>>>>>>>>> Sent from my iPad>>>>>>>>>>>> On Sep 19, 2014, at 9:02 AM, Hontas Farmer wrote:>>>>>>>>>>>>> I state in the paper that it is a Hilbert space over minkowski. In a followup paper I posted to the winnower I explain exactly what that means.>>>>>>>>>>>>>> H tensor product with M>>>>>>>>>>>>>> so a quantity can be a scalar in H vector nm and exist in the tensor product of those two spaces. When I have learned this in school we were referred to that as Hilbert space over Minkowski space.>>>>>>>>>>>>>> On Sep 19, 2014 8:27 AM, David Halliday wrote:>>>>>>> So, are you refusing to engage at all?>>>>>>>>>>>>>> I have very simple questions for you to honestly consider and engage with. They are absolutely fundamental to your premises.>>>>>>>>>>>>>> You say, in the very fist sentence of the introduction of your paper:>>>>>>> "The motivating question for this study was can Hilbert space can be defined such that the>>>>>>>>>>>>>> <ψϕ>=va∈M.">>>>>>>>>>>>>> Ignoring grammatical issues with that sentence, do you still hold that the question of whether something like a Hilbert space can be defined, as you propose with that very fist equation, so the vectors of Minkowski space are used as the scalars of such a "Hilbert space", is new/novel?>>>>>>>>>>>>>> Sent from my iPad>>>>>>>>>>>>>> On Sep 19, 2014, at 7:48 AM, Hontas Farmer wrote:>>>>>>>>>>>>>>> Context is key. The In the first one where I mention Latex I am responding>>>>>>>> on this blog here where someone has decided to take the issue where I could>>>>>>>> not let that go unchallenged. How could I not engage that article?>>>>>>>>>>>>>>>> On my own article I have some control. I can say this is now becoming a>>>>>>>> discusison of the paper itself which should be held where people can look at>>>>>>>> it with full context and make up their own minds. That is the whole point of>>>>>>>> OA OPR.>>>>>>>>>>>>>>>> I know it is crackpotish to do so but I'm sure having a PhD you know very>>>>>>>> well that theories in science are rarely accepted when they are first>>>>>>>> revealed. With the exception of Newton's Laws (and maybe not even those... I>>>>>>>> believe he had some serious controversy with Hooke over the Inverse square>>>>>>>> law). in physics no new idea is ever ever accepted right away. That is how>>>>>>>> it should be since most of them are wrong. I may well be wrong. Continuing>>>>>>>> to discuss it around in circles for another few months will not prove>>>>>>>> anything.>>>>>>>>>>>>>>>>>>>>>>>> Sent from Surface>>>>>>>>>>>>>>>> From: David Halliday>>>>>>>> Sent: Friday, September 19, 2014 7:39 AM>>>>>>>> To: Hontas Farmer>>>>>>>>>>>>>>>> So, Hontas, what's your verdict? Where do you really want to engage in a dialogue about your paper?>>>>>>>>>>>>>>>> David>>>>>>>>>>>>>>>> Sent from my iPad>>>>>>>>>>>>>>>> On Sep 19, 2014, at 7:05 AM, Hontas Farmer wrote:>>>>>>>>>>>>>>>>> Strange I got that on but when I replied to it…>>>>>>>>>>>>>>>>>> Sent from Surface>>>>>>>>>>>>>>>>>> From: David Halliday>>>>>>>>> Sent: Friday, September 19, 2014 7:03 AM>>>>>>>>> To: Hontas Farmer>>>>>>>>>>>>>>>>>> I'm sorry for the inconvenience. Did you receive my email (sent through your Science 2.0 contact form) letting you know that I have updated my email in my Science 2.0 profile?>>>>>>>>>>>>>>>>>> This email account, from which I am sending this, is the best one to use, these days.>>>>>>>>>>>>>>>>>> David>>>>>>>>>>>>>>>>>> Sent from my iPad>>>>>>>>>>>>>>>>>> > On Sep 18, 2014, at 8:35 PM, hfarmer wrote:>>>>>>>>> >>>>>>>>>> > Halliday,>>>>>>>>> >>>>>>>>>> > Hfarmer has sent you a message via your contact form at Science 2.0.>>>>>>>>> >>>>>>>>>> > If you don't want to receive such emails, you can change your settings at http://www.science20.com/profile/david_halliday.>>>>>>>>> >>>>>>>>>> > Message:>>>>>>>>> >>>>>>>>>> > My diresct email replies to you bounce with the reply>>>>>>>>> >>>>>>>>>> > mx01.nic. rejected your message to the following email addresses:>>>>>>>>> > ()>>>>>>>>> > The email address you entered couldn't be found or is invalid. It may be due to a bad entry in your Outlook or Outlook Web App recipient AutoComplete cache. Use the steps below to clear the entry from the cache:>>>>>>>>> > Click New mail.>>>>>>>>> > In the To field, start typing the recipient's name or email address until the recipient appears in the dropdown list.>>>>>>>>> > Use the DOWN ARROW and UP ARROW keys to select the recipient, and then press the DELETE key.>>>>>>>>> > Delete and retype the recipient's address, then try sending it again.>>>>>>>>> > For more tips on how to resolve this issue, see DSN code 5.1.1 in Exchange Online.>>>>>>>>> > mx01.nic. gave this error:>>>>>>>>> > <>: Recipient address rejected: User unknown

2
You write: "Any socalled reviewer who is not satisfied with this must show where there is an actual error not simply state their own confusions as facts and preferences as principles."
Let's look at what the textbooks say about Hilbert spaces.
page 195Edwin Hewitt, Karl Stromberg, Real and Abstract Analysis===================
(13.6) Definition. Let H be a linear space over K having an inner product(x,y) > <x,y> in Kmapping H x H into K such that<x+y,z> = <x,z> + <y,z>,<a x,y> = a <x,y> for a in K,conjugate( <x,y> ) = <y,x>,<x,x> > 0 if x != 0....Then H is called an inner product space, or a preHilbert space. For x in H, definex = sqrt( <x,x> )....If H is complete relative to this norm, then H is called a Hilbert space.===================This is enough to derive the requirement that any space you wish to define a Hilbert space over, must be closed under addition and multiplication. Your space is not closed under multiplication, and thus you do not have a Hilbert space.
The definition of a Hilbert space is directly relevant if you are going to claim to have a Hilbert space. The textbook backs the points David, I and others have been making.
Now shifting focus to another mistake you made.
You still have not handled the issues with "function composition" that I brought up earlier. You somehow mistook this as merely a complaint about formatting. It was actually a complaint about content. So here's a direct question (please stop avoiding these, they are the quickest way to resolve math issues)
DIRECT QUESTION] If you want to use function composition, then please state precisely: What is the domain and codomain of the function \phi_n ?
Currently in the paper you state: "let us consider the KleinGordon equation as if it were any other differential equation. The solutions of any differential equation of order 2 or greater form a space of solutions." So that would make function have the domain R^4, and the codomain C. So you are using function composition on functions whose codomain and domain do not line up, so this is math gibberish. If you wish these functions to have some other domain and codomain, please answer my direct question above and make this clear in the paper.
So Eq 6 is ill defined mathematically, and leads to Eq 13 which you claim "derives" some result but really just requires you to assume your result as it does not prove function composition maps onto the positive real line. If instead of functions, you wished to interpret \phi_n as operators (guessing based on some of your references) and use operator composition, this still would not make sense, as now the inner product is operator valued.
There are of course plenty of other issues, but the required properties of a Hilbert space and an inner product would be a great place to start.
You write: "I’ve, cited supporting papers and textbooks, and rederived these expressions from first principles."
But you have not supported your claims. You just claim it, and link to a paper or cite a book, that doesn't actually support your claims. For example Liu Wei's request for a citation. You claim Carrol's book supports you, but it doesn't. You are misreading books, and then demanding we prove a negative. He backed his statements instead with logic. And furthermore the burden shouldn't be on him. If you make a ridiculous statement that you claim is supported by a common textbook, then you need to back it up. Show all of us precisely where you believe the book states the claims you are making.
You also have not "derived these expressions from first principles". You just write equations like Eq 21, as if you have defined something valid, and then proceed blindly forward. Where is your mistake? Your starting equations. For instance, your eq 21 is claiming that in your "Hilbert space" the right and left arguments of inner product are from different spaces. This is not an inner product then. And no, this is not a notation or formatting issue.

Read the references given in the paper. Problem #1 with what you say above is you are applying axioms of QM to a work that deals with QFT. One of many problems.

I quoted from a textbook the mathematical properties _defining_ a Hilbert space. If you have some mathematical object and it does not have those properties, it is not a Hilbert space. Are you claiming Hilbert spaces in QFT do not have these properties? That is wrong, and NONE of your references make such a claim.

I'm starting to think you are purposely avoiding these issues because while you are utterly convinced of your "understanding" of QFT, you also confusingly realize you can't seem to find any statements directly backing your claims, but are unwilling to admit it. I challenge you to actually show precisely where ANY of your references claim Hilbert spaces in QFT do not have the very mathematical properties that define a Hilbert space as specified in the textbook quote. Hint: you won't be able to, as your references do not actually back your claim.



2Since Hontas seems unwilling to discuss the simple, yet fundamental issue she has had with both of her papers: her apparent, fundamental misunderstanding of Hilbert spaces, and things that are almost Hilbert spaces (which I will refer to as Hilbert modules, rather than using quotation marks like "Hilbert spaces"), which don't adhere to all of the Hilbert space axioms.Edward "Henry" Brown also tried to help her see, essentially, the same issue. With Henry, as people can see, here on the Winnower, Hontas "shot down" his first five (5) points (along with his somewhat separate sixth point), which were all about Hilbert spaces, saying:"As to your six points. While those are true in quantum mechanics this is quantum field theory. Not all of those are true in Quantum Field Theories in flat space time, much less those that involve curved space time. See here http://arxiv.org/abs/0803.2003"Yet, she later returns with a quote of QFT axioms, from the same source "(http://arxiv.org/pdf/0803.2003v1.pdf)", stating that "All of those are true in my model." Yet, half of the axioms she quoted (3 out of 6) refer to Hilbert space, thus depending upon the very things Henry was bringing up in his first five (5) points.Unfortunately, Hontas is either showing a gross misunderstanding of QFT, and, most particularly of the concepts of Hilbert spaces/modules that QFT deeply depends upon, or she is doing an extremely poor job of communicating on these subjects.However, for the sake of those that may be wondering what this is all about, I am going to address this here, whether or not Hontas even pays attention.Hontas is correct "that one can set up a Hilbert space without prior reference to" the particular space of its scalars (though such must adhere to all of the axioms of mathematical fields). It is even true that one may do similarly with Hilbert modules, provided one is very careful about the (mathematical) field axioms that are violated by the scalars. (If anyone wants to see any details on such things, simply post a comment to this with your questions.) So, we shall obtain the Hilbert space/module scalars from the inner product, <,>, of the Hilbert space/module, and shall use Latin letters whenever we then refer to such elements.It really doesn't matter what we call the Hilbert space/module, whether we call it H, \superH, H \otimes M, or whatever. So, I shall use H, and Greek letters for elements of H.Hilbert spaces/modules fulfill the following axioms (or ones very much like them):For all φ and ψ elements of H, φ + ψ = ψ + φ are elements of H.For all a = <φ,ψ>, where φ, ψ, and θ are arbitrary elements of H, aθ is also an element of H.For all a = <φ,ψ>, and b = <δ,θ>, where φ, ψ, δ, and θ are arbitrary elements of H, there exist at least one pair of elements ε, and σ of H, such that ab = c = <ε,σ>.For all a = <φ,ψ>, where φ, and ψ are arbitrary elements of H, and all arbitrary elements δ, and θ of H, a(δ + θ) = aδ + aθ.For all a = <φ,ψ>, where φ, and ψ are arbitrary elements of H, and all arbitrary elements δ, and θ of H, <aδ,θ> = a<δ,θ>. To quote Henry, this is "linear in the first term to match [Hantas'] choice of convention, which is the opposite of the usual physics convention  there's no problem in that, just be aware of it".(There are additional conditions of closure and completeness, but we shall "ignore" such, for now.)The important distinctions between Hilbert spaces and Hilbert modules are that while aθ = θa and ab = ba are true for Hilbert spaces, such may not be true for Hilbert modules. In fact, for certain Hilbert modules, there can be other additional issues involving products. (The truth is, if one wants to be using some noncommutative algebra or mathematical ring built from something like the Dirac gamma "matrices"/operators, then one almost certainly does not want commutators, such as the above, to be true.)From axioms 1 and 2, above, we see that for all a = <φ,ψ>, and b = <δ,θ>, where φ, ψ, δ, and θ are arbitrary elements of H, for all arbitrary elements β, and γ of H, aβ + bγ is also an element of H (call it ω). Now, combining this with the Hilbert space/module inner product, we have <ω,χ> = <aβ + bγ,χ> = a<β,χ> + b<γ,χ>, by the linearity of the first argument of the inner product (axiom 5).Since all inner products of elements of the Hilbert space/module H are scalars, we may let c = <ω,χ>, e = <β,χ>, and f = <γ,χ>. Therefore we have the seemingly obvious c = ae + bf. However, this has significant implications that seem to have escaped the notice of Hontas Farmer: Regardless whether the scalars form a (mathematical) field, or not, the scalars must be closed under multiplication as well as addition. (This is also a consequence of axiom 3, above. This was the focus of John Lee's review of Hontas's first paper.)You see, Hantas' Minkowski space, M (or however she may wish to label it) is not closed under multiplication. On the other hand, one may take the closure, in the Clifford algebra, say, and use that as the scalars! That is not M, but something significantly larger!The thing is, it is painfully obvious from other expressions Hantas tries to use in other parts of her paper, that this larger algebra (a mathematical ring, actually) is truly what Hantas "wants" to be using, but, apparently, she hasn't, yet, realized this fact, and all we have been trying to do is educate her about this fact.

Unfortunately, I didn't notice, before submitting, that the five (5) numbered paragraphs of the axioms didn't keep there numbers! I apologize, and I hope people will still be able to understand the five (5) axioms.

I don't discuss the issues because I can instead cite published peer reviewed papers where these foundational issues are hashed out by well established scientist who do nothing but theory. Read those papers and get an understanding of the topic. With all due respect you are just not getting it and mistaking that for me being wrong.

As Liu Wei has pointed out, you quite misunderstand the literature, then. As another test case, provide references to published peer reviewed literature that you believe refutes anything I have written in the above review. Again, not just generalities, but complete references with relevant page numbers. As Luis Wei said, "Really read carefully and with scientific honesty in your pursuit of learning... If after your careful reading, you still believe your claims, then give me the sentence or paragraph you believe supports your claims. Don't cite an entire book [or paper, or whatever]. Show me precisely what in your best effort of reading ability leads you to believe your claim."



2
The manner in which this paper was written makes it apparent it is attempting to respond to criticisms from the author's previous paper on this subject in this journal. However after reading the original paper, reviews, and comments, it is clear the author does not understand the complaints regarding the foundations of the paper and therefore fails to address them appropriately.
Terms and phrases such as Hilbert space, Clifford algebra, inner product, scalar in a Hilbert space, function composition, and so on, have well defined mathematical meaning. A Hilbert space is a well defined mathematical object. The definitional properties of a Hilbert space are not dependent on what physics you wish to describe with a Hilbert space. Either a misunderstanding of the author regarding several mathematical concepts, or a mistaken belief that some of these have been appropriately 'extended' in these two papers, is leading to communication issues with the reviewers.
In multiple cases here in the comments or choices of references in the paper, the misunderstandings of the mathematical terminology and physics has clearly lead the author to misread textbooks and other papers, thereby providing a reinforcement of these fundamental misunderstandings. I realize this is a harsh statement, but until the author is able to break this cycle, it is clear to this reviewer that no amount of explanation will help. The pursuit of science is hindered by a defensive ego. Science requires honesty, adherence to logic, and the humility to accept that one can be mistaken:
*"In science it often happens that scientists say, "You know, that's a really good argument, my position is mistaken," and then they actually change their minds, and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day."*  Carl Sagan
So I strongly encourage the author to listen to the patient reviewers, put honest effort into responding to their direct line of queries, and if from basic definitions logical conclusions lead to a contradiction with your current understanding, then be humble enough to realize that it is not the definitions you can throw away as not being applicable, but should instead be your current understanding that you modify.
.............................................
For instance, the belief that somehow \bar{\psi}\gamma^\mu\psi is "a scalar in Hilbert space" in QED is not correct in the sense you are using the term. For a free dirac spinor field, let \psi be a vector of the Hilbert space. With the inner product < , > the value < \psi, \psi > is a scalar of the Hilbert space. For any operator A on the Hilbert space (maps vectors of the Hilbert space to vectors of the same Hilbert space), < \psi, A \psi > will also be a scalar of the Hilbert space. A dirac gamma will act on a spinor, yielding another spinor, potentially mixing up the spinor components. So the dirac gammas are operators in this Hilbert space. So a term such as < \psi, \gamma^0 \psi > is a scalar of the Hilbert space. So is the term < \psi, \gamma^1 \psi >, and so on. Together the four Hilbert space scalars <\psi, \gamma^\mu \psi > transform as components of a fourvector. Note that these Hilbert space scalars are not Lorentz (Minkowski space) scalars (not that anyone here claimed they were). I point this out because the term "scalar" is very overloaded in both math and physics, and often its meaning must be implied from context if not clarified. It appears at least one fundamental issue here is arising because the existence of tensor notation in QED led the author to mistakenly think \bar{\psi}\gamma^\mu\psi represents a (singular object) Hilbert space scalar, when actually the Hilbert space of QED is a Hilbert space over the field of complex numbers. To try to generalize the Hilbert space to make the fourvector objects themselves be scalars in the Hilbert space is not as simple as just declaring them so. The author's attempt to do so leads to mathematical inconsistencies showing that the Hilbert space as proposed does not mathematically exist.
In addition to misuse of terms, the paper suffers from very sloppy logic. Care must be taken whenever generalizing mathematical objects, for instance if it is desired to extend the concept of a Hilbert space to be over something other than a (mathematical) field. As an example of being careless, the author assumes multiplication commutes. This is not true even in the simpler case of a Hilbert space over elements of the algebra Cl_{0,2}(R), the quaternions. (While not a truly proper Hilbert space this can be viewed as a module equipped with a suitably generalized inner product, and in context can be called a Hilbert space without confusion.) For instance, what does it mean for the inner product to be linear? (does <a x, y> = a <x,y> ... or ... <a x, y> = <x,y> a ) With care, such constructions have been made. As David Halliday is clearly hinting, in a similar manner you could likely define a generalized Hilbert space over the elements of Cl_{1,3}(R), but you cannot define it over the fourvector subset as this is not a subalgebra. Futhermore, your choice of inner product violates the positive definiteness requirement of a Hilbert space. Also, to use the Hilbert space in a quantum theory, you will need to be able to divide by these scalars and this is prevented by the lack of unique multiplicative inverses (due to zero divisors, unlike with quaternions or octonions).
I feel I should also note that while the reviewers have (rightfully) focused their specific examples on the foundations of this proposal, there are numerous other issues throughout the two papers. For example the author appears to believe the Cartan structural equations are equivalent to General Relativity with torsion, or possibly believes they are once including the Torsion source equation the author proposed. This and the discussion in the comments makes it clear the author does not understand current quantum and gravity theories, which should be a prerequisite before proposing a theory of quantum gravity.
Your supposed expertise is belied when you say that the Cartan Structure equations of Einstein  Cartan theory are not General Relativity. In response I cite Sean Caroll's book on the subject. The "foundational" issues show that you need to go back to school.

The Cartan Structure equations are not equivalent to general relativity with torsion. If they were, you could derive the field equations of EinsteinCartan gravity from the structure equations, and therefore in the nontorsion limit also obtain the usual Einstein field equations for GR. However this is not possible as the structure equations do not relate the matter content to the spacetime geometry.
But you really believe the Cartan structure equations are equivalent to GR, and that Sean Carrol's book backs this claim of yours? This is exactly the issue here, your misunderstanding of the material seems to be comprehensive enough to affect your reading comprehension and reinforces your misunderstanding instead of learning from books or articles.

So this is a test case. Try to find a passage in your copy of Carrol's book which backs your claim that the Cartan structure equations are equivalent to general relativity. Really read carefully and with scientific honesty in your pursuit of learning: the book never directly makes the claim you are making, so at best one has to infer it, and it is this inference step in which all your misunderstanding is coming into play. If you follow the derivation of the structure equations you will see there is never reference to or assumptions about the matter content. The structure equations are independent of the matter content, and therefore cannot be equivalent to the field equations of general relativity. If after your careful reading, you still believe your claims, then give me the sentence or paragraph you believe supports your claims. Don't cite an entire book. Show me precisely what in your best effort of reading ability leads you to believe your claim.

Look in the appendix of the book, the vierbien formulation if GR is given there.

I can only guess you are referring to appendix J, as this discusses noncoordinate bases and the Cartan structure equations. But no where does it state that the Cartan structure equations are equivalent to GR. It also never writes the EinsteinField equations of GR, nor the field equations of EinsteinCartan gravity, nor even discusses the stress energy tensor. As explained earlier, since the Cartan Structure equations are independent of any matter distribution, they cannot possibly be equivalent to GR. You keep claiming books and papers support you, but nothing in them actually back any of your claims. Multiple times now reviewers have asked you directly to give exact quotes that you believe demonstrate your sources back any of your claims. For some reason you continue to refuse. So I ask again: Give me the sentence or paragraph you believe supports your claim. Show me precisely what in your absolutely BEST EFFORT of reading ability leads you to believe Sean Carrol's book claims the Cartan Structure equation is equivalent to the field equations of EinsteinCartan theory (and thus GR in the nontorsion limit). If you give it your best effort, I still hope you will be able to see your error here. The Cartan structure equations are not GR written in vierbien notation.


This may help Hontas Farmer track down her misunderstanding: http://physics.stackexchange.com/questions/61328/cartanequationsversuseinsteinequationsinclassicalgravity It discusses exactly the issue Liu Wei brought up, and even refers to Carrol's book.

I cite papers and books and your counter is a message board....you really don't see the problem with that?

You don't cite papers and books, you just mention anything you can find that contains similar words. That is not how it works. You need to actually read the material, in which case you'd see that your own references don't support you. You might as well be "citing" the yellowpages. The link I gave was directly related to the discussion and I hoped you'd read it for the content.

Maybe this will help you realize the extent to which you are misunderstanding your "sources". I emailed Sean Carrol, the author of the book you cite to support your claim that the Cartan Structure equations are equivalent to General Relativity. He agreed with me that you are misreading his textbook if you are getting that out of it, and went on to explain "Cartan's equations are statements about differential geometry, independent of physics. Einstein's equation adds the additional (crucial) input that spacetime curvature is related to the energymomentum tensor."
Lui Wei is correct, and yet you dismissed him outright because you thought your textbooks disagreed with him. In reality, as Lui Wei pointed out, you are allowing your misunderstanding to cloud your reading comprehension. At least admitting your fault here and learning from this tendency could go a long way in allowing you to actually learn from your mistakes here.



2
Hontas:
You have a lot of work ahead of you before you should even contemplate making this paper "final".
Neglecting a lot of the more technical issues, for now, you have a number of formatting issues (at least) with your equations. For instance, you have a number of cases where a right angle bracket ">" appears to have "merged" with an equal sign "=" to form a greaterthan or equal sign "≥". I suppose this is due to some overzealous ligature algorithm, either here on the Winnower, or built into the LaTeX package they have authors use.
Equation (4) is missing your "ϕ_n" on the righthand side.
I believe your equation (5) is not intended to have the subscripted "p"s (or else equation (3) is missing them, but since your equations (2) and (4) are integrating over the three momentum, p, I'm going with the "p" subscripts are not intended).
Are you functions actually such that
Your equations (9) and (10) never use your "definitions" of "a" and "b". (That doesn't even touch on the issue of whether your functions are or are not linear, which bears on whether you can have your function composed over a linear combination be a linear combination of function compositions.)
You set, in equation (19) has x^μ on the left of the vertical bar (the "such that" of the set definition) while it has no x anywhere on the right of the vertical bar, just gammas.
I'm sorry, but I don't have time to enumerate everything.
David
P.S. "Quality of Figures" is being used for "Use of Equations" and/or quality thereof.
OOPS. I didn't realize I left the beginning of a sentence ("Are you functions actually such that") in my review when I submitted it. My bad. :/

I also don't know why, in the last several months to a year, I have been missing the final "r" on some of my "your"s. Lazy typing? (I have been noticing a tendency toward other mistyping where I hit a key next to the one I intended, but I think that's mainly due to my keyboard setup.)

No problem. When one types as much as we do an error here or there is excusable.


Yes the formatting errors, it looks like where ever I had =... it was interpreted into a \leq or \geq sign. As for equations 9 and 10 and the defintions of a and b, I included those to show that those quantities can be defined in terms of the elements of the hilbert space itself without choosing a representation or specific algebra R,C,H, or O. Equation 19 with x mu on one side and gammas on the other is a typing error. I forgot to change the x to a gamma.

This is a very good and useful review filled with good observations. The formatting issues I will have to take up with the winnower. Looking at the PDF... I will have to look at my Latex, and fix what I can on my end. For now, I will post as a resource a version of the PDF processed on my end for now.

This is to let you know I haven't ignored your emails but I have instead responded by posting an updated version. I am going to have to insist that any and all conversation regarding these papers take place on the Winnower. I am fully willing to discuss them just here on the Winnower with full context.


1Hontas:Since you didn't allow for comments on your "The Fundamentals of Relativization, and The Purpose of Peer Review After Publication" 'blog post, I can only assume that you want comments here, on the Winnower.Within that 'blog post at http://www.science20.com/quantum_gravity/blog/the_fundamentals_of_relativization_and_the_purpose_of_..., you make the claim:"... the reviewers don't seem to get what the purpose of reviews on a platform such as TheWinnower or anything similar to it, like say ScienceOpen Research, is. They are reviewing a product, just because of bad initial reviews the product does not vanish." (Emphasis added)and"I think overall the reviews have been fair but that the reviewers don't seem to get the point of reviewing on a platform like TheWinnower. Reviews are intended to express a point of view on the work, ways to improve the work, or ways that the work may just be wrong. Then the author has to respond. I have responded with a new paper. Then I modified my paper in response to all nine reviews. Now if the reviewers don't feel that I have convinced them well...this isn't a PhD committee we are equals discussing ideas among equals. They and every reader has a right to personally dismiss my ideas, however nothing gives them the right to suppress them and keep them from general circulation." (Emphasis added)and further"All in all I would say that Open Access Post Publication Peer Review can work as long as the authors have a strong stomach for a degree anonymous trolling, welcome serious discussions, and are willing to be publicly wrong and publicly corrected. Likewise, those who review papers need to accept that what they do there is more like a book review or movie review. No matter how long their bad review is the paper under review will not simply go away. ..." (Emphasis added)Do I see a pattern here? Where did you get the idea that your reviewers of your two papers wish to simply make your papers "go away", "vanish", or wish to "suppress them and keep them from general circulation"?While I certainly cannot speak for the other reviewers, I have never wished any of that. Additionally, I certainly have never gotten the impression that Edward Brown, Liu Wei, or John Lee wished any of the above, either.While we have all tried to help you see your errors, and hope that once you have seen your errors that you will make the appropriate corrections. We simply want you to "welcome serious discussions, and [be] willing to be publicly wrong and publicly corrected." (Again, speaking only for myself, you need not be "publicly wrong and publicly corrected." That's why I even tried the email approach. Unfortunately, you did not "welcome serious discussions" even thenif they came too close to fundamental issues in your papers. [As long as they were peripheral to your papers, you seemed fine, but once they came "too close to home" you would "bolt".])You go on with:"Lets consider the ideas in circulation. There is an article on this websites [Science 2.0] news feed about MTheory re positioning itself as now being more fundamental than quantum mechanics. That's ok, but my different thinking gets attacked by some. Why? There is no logical reason for it! Even if someone thinks I am wrong why get personal?" (Emphasis added)Again, other than pointing out the flawsself contradictions and internal inconsistenciesI haven't seen these alleged "attacks". Certainly not "personal" "attacks". Remember, attacking ideas is absolutely valid science.As for the "logical reason for it [the 'attacks' on the flaws]", it is purely good science to point out any and all flawsespecially unacknowledged self contradictions and internal inconsistencies. That's how science progresses. That's also how the Winnower got its name: Winnowing out the chaff from the grain.Maybe you just need to "welcome serious discussions, and [be] willing to be ... wrong and ... corrected."David

1
Hontas:
F1: The vielbien guarantee nothing about general covariance. So just because you may use such, you have no guarantees (except local Lorentz invariance, but such can be easily guaranteed without even using such a crutch).F2: Your FockHilbert space is not signified by your H, or hadn't you noticed that (see your equation (1)).F3: You say that "The inner product on H must be in a set isomorphic to the division algebras R,C,H, O." Yet, you go on to say "For example an inner product on H of the form <ψψ>=j^a with j^a∈M and ∀ψ>∈H."
Your M, is not at all "isomorphic to the division algebras R,C,H, O", and you really don't seem to want it to be, anyway. Do you care nothing for your internal inconsistencies?
F4: Where did you get the notion that "QFT interactions occur in the locally flat space at the point of interaction"? Wishful thinking? Certainly not from QFT. (Perhaps your misinterpretation of Feynman diagrams?)
Your F5 doesn't even deserve comment, at this time, due to its intimate dependence upon the above inconsistencies and outright misunderstandings.

1Hontas:I see you have (partially) returned to the bra and ket notation. (I actually prefer that notation, though I have no trouble either way, as long as you are consistent.) Unfortunately, now portions of all equations that use "function composition" are "orphaned" and with even less meaning.I see you finally fixed equation (equations?) (4)(5). I also see that you made a bad b, of equation (9)(10) slightly more general. (I still recommend making them completely independent of all other Hilbert space/module elements used therein, since they must be able to be completely arbitrary.)Did you actually intend equation (11) to involve the inner product of two different Hilbert space/module elements? This makes no sense in any Hilbert space, let alone any Hilbert module.Now, as for the definition of the "ketbra" in equation (14), the usual ketbra is not, necessarily, antisymmetrized. Perhaps you need to explicitly antisymmetrized the m an n labels.While your definitions of H in equations (17) and (19) agree, your definitions of M in equations (18) and (19) do not. Your definition of M within equation (19) is, arguably, superior to that given in equation (19), though the use of superscripted vs. subscripted indices is inconsistent within the part defining x^μ as a_{μν} γ^ν. Besides, there is no need for the superscript on x, at least if you actually intend the gamma matrices to be the basis of your Minkowski space, M. (The only problem, I see, is in notationally distinguishing an element of the vector space, M, from something that looks like a simple number. One could use bold face for the gamma basis elements and all vectors formed as linear combinations of such, as in x = x_μ γ^μ. There are other choices as well, of course.)I'm reasonably certain you do not actually want to define the part of your inner product, defined in equation (20), the way you did. Do you know what γ^μ η_{μν} γ^ν = γ^μ γ_μ equals, given your definition that γ^μ γ^ν + γ^ν γ^ν = 2η^{μν}? (I strongly recommend that you "do the math" and see.)Well, that's all the time I have, for now.

In the first sentence of the last paragraph, "define the part of your inner product" should have read "define the part of your inner product, acting on M".

I also noticed that auto correct had turned "a and b" into "a bad b". OOPS. I hope that doesn't throw anybody off. (Especially, not "off the deep end".)

Another OOPS: "Your definition of M within equation (19) is, arguably, superior to that given in equation (19)" obviously doesn't make sense. It should have been "Your definition of M within equation (19) is, arguably, superior to that given in equation (18)", of course.



1
I'll hold off on any rating until the paper is finalized. There is misuse of terminology and notation here, so in pursuit of clarity it would be best to fix that up first.
For instance, you instruct us to consider the KleinGordon equations as if it were any other ordinary differential equation, and then looking at the solutions which are eigenstates of the number operator. (As a side note, you never explicitly write it as an ordinary differential equation, nor follow up on how (if at all) the conjugate solutions differ. If this section is for explanatory purposes, you might as well have the math follow your explanation closer and give more detail.) Then you try to build a Hilbert space with these. The first issue is you seem to imply the Hilbert space is the set of these eigenvectors (and you write this explicitly later in eq 17). This may be a basis for your Hilbert space, but this cannot be the full set of vectors in the Hilbert space. Another problem is your attempt to define an inner product with "the composition of functions". This doesn't make sense here.
If we consider ϕ as a solution to the KleinGordan equation as an ordinary differential equation, then we have the function ϕ: R^4 > C. In other words, given 4 real values (for example for momentum space) it will map to a complex value.
So given two solutions, ϕn and ϕm, the composition of functions ϕn ∘ ϕm doesn't make sense because the codomain of ϕm is not a subset of the domain of ϕn. It is not at all clear how you wish us to interpret this, either your point is nonsense or your are misusing the terminology and notation so badly that it obscures your point.
Then when trying to prove some relations with this inner product, you are basically just assuming the answer instead of proving anything. For instance, with ordinary multiplication of complex numbers (or quaternions, or numbers represented as matrices, etc) we have:conjugate(ab) = conjugate(b)conjugate(a)so you seem to assume that function composition satisfies the following,conjugate( f o g ) = conjugate(g) o conjugate(f)But you don't prove this assumption, and it actually is not true in general. For a really obvious counter example, just consider: f is a function R^3 > C, and g is a function R^2 > R^3. Then conjugate(f o g) is well defined, while (conjugate(g) o conjugate(f)) is nonsense. So you cannot just assume that.
Similarly, there is no way to make sense of eq 13. in terms of checking if an inner product is positive definite. Even if you could write a composition of functions from this function space, the resulting function space does not have a linear ordering. How do you intend to define a linear ordering of a function space? Is the function cosine greater or less than the function sine?
To get to the other terminology and notation issues, and also the most important technical issues, it would be helpful to agree on some basic things about Hilbert spaces and scalars of Hilbert spaces. Do you agree with the following?
For a Hilbert space H over some scalars S
1) addition of vectors in the space H exists
for every x,y in H, x+y is also a vector in H
2) addition of scalars in the space S exists
for every c,d in S, c+d is also an element in S
3) scalar product in the vector space of H exists
in particular
3a) for every x in H, and d in S, dx is also a vector in H.
3b) for every x,y in H, and d in S, d(x+y) = dx+dy.
I would also expect the other properties listed here to hold as well
https://en.wikipedia.org/wiki/Scalar_multiplication#Properties
I realize multiplication in your scalars is contentious, so if you don't agree to any of those properties, please let us know. For this discussion I think we can make do with just the 3a, 3b if necessary.
4) the inner product of any two vectors in H will be a scalar in S:
for any x,y in H, <x,y> is an element of S.
5) the inner product is linear (linear in the first term to match your choice of convention, which is the opposite of the usual physics convention  there's no problem in that, just be aware of it):
for any x,y in H, and d in S, <dx,y> = d <x,y>
probability in quantum mechanics
6) If the system is in a state x (a vector in the hilbert space), and a measurement is made according to some operator V (with eigen vectors v_k), then the probability of the measurement resulting in A is:
first normalizing the eigenvectors, define w_k = v_k / sqrt(<v_k,v_k>)
Probability(A) = ( Sum over eigen vectors w_n with corresponding eigen value = A, <x,w_n><w_n,x> ) / (<x,x>)
If we can agree on these basic properties, it will give a good foundation to discussing the main technical issues. It is very clear from your comments in this paper that you are not understanding the technical complaints, so we need to agree on some basics first.
Some of what you say regarding notations and equations here is because of typesetting issues with the Winnower's software. Anyplace where I had >= or =ed on my end, attached under, additional resources, in which that issue does not happen.

NONE of what I said is because of typesetting issues with TheWinnower. Before posting my review, I checked with the PDF you put up for David (thank you by the way) to ensure the issues were not caused by TheWinnower.


As to your six points. While those are true in quantum mechanics this is quantum field theory. Not all of those are true in Quantum Field Theories in flat space time, much less those that involve curved space time. See here http://arxiv.org/abs/0803.2003

Please just state explicitly which points you agree with and which you do not. Do not make me guess or assume your position, just tell me. Even if we do not agree on every point, hopefully there will still be enough foundation to build upon.

None of those points apply, this s QFT in curved space time. The axioms of which are given in the reference provided. If you think QM is mire fundamental than Relativity feel free to say so. The whole point of this theory is that relativity is more fundamental.

You are using a Hilbert space. Points 15 are mathematical properties of Hilbert spaces, and so are DIRECTLY applicable. Furthermore, the postulates of quantum mechanics formulated with Hilbert spaces is the same for nonrelativistic particle QM and for QFT, the only thing that changes is the content of the Hilbert spaces and therefore the operators on this space. The actual postulates of quantum mechanics, how the Hilbert space and operators relates to physical systems and measurements, remains the same. What you linked doesn't support your claims that any of my 6 points are wrong. That paper is trying to axiomatically build up field theory in quantum mechanics, it however does _NOT_ give the postulates of QM, because their goal is to axiomatically build a well defined and well behaved Hilbert space for QFT, _NOT_ propose different postulates for QM. For instance, the entire document doesn't even contain the word "measurement", and doesn't give postulates on how to calculate the probabilities of measurements.

Let's try this again. I won't want to fork off on a tangent. I want to build a foundation of mathematical ground truth we can agree on, so that I can build up from there. Points 15 are mathematical properties of Hilbert spaces, and sufficient to build up some conclusions from. You are using a Hilbert space, and until you can agree on some definitional properties of Hilbert spaces, it will remain difficult to discuss the technical issues with your paper. You closed all conversation on your blog and told us to have discussion here. So let's move this discussion forward here. Please state whether you agree or disagree if a Hilbert space has the mathematical properties 15.


These are a well known set of Axioms of QFT in minkowski space time. Quote. The Wightman axioms of quantum field theory in Minkowski spacetime are generally believed to express the fundamental properties that quantum fields possess. In essence, these axioms require that the following key properties hold: (1) The states of the theory are unit rays in a Hilbert space, H , that carries a unitary representation of the Poincare group.
(2) The 4momentum (defined by the action of the Poincare group on the Hilbert space) is positive, i.e., its spectrum is contained within the closed future light cone (“spectrum condition”). (3) There exists a unique, Poincare invariant state (“the vacuum”). (4) The quantum fields are operatorvalued distributions defined on a dense domain D ⊂ H that is
both Poincare invariant and invariant under the action of the fields and their adjoints. (5)The fields transform in a covariant manner under the action of Poincare transformations. (6) At spacelike separations, quantum fields either commute or anticommute. Unquote any theory that satisfies these axioms is a QFT in flat space time. (http://arxiv.org/pdf/0803.2003v1.pdf) All of those are true in my model.


0
I have always a doubt on these all equations based on only space ,relativity not only handle space but time also , but unfortunately no framework has done on higher dimension time.what is the reason behind it , in fact there is no mathematical tool or treating time as a simple dimension only..

0
Hontas:
Are you aware that your equation (29) cannot be the Mathematica output of the Mathematica expression in equation (28)? You must have mistyped something, or you have a copy/paste error.
Furthermore, what do you mean by "The region where the squared wave function is outside of the potential well"? Just because the amplitude of the "squared wave function" is higher or lower than the amplitude of the "potential well" has nothing to do with "inside" vs. "outside". (Besides, this "potential well" has no true "outside". This is to say nothing of the lack of normalization of your, so called, "wave function".)
Additionally, you cannot talk about spatial positions when you are referring to a function of momentum space. (You do know that it is this very complementarity of these two spaces that leads to the Heisenberg uncertainty principle.)
And where did you get the notion that "The location of zero momentum is the event horizon"? From particles being "trapped" by the event horizon? If so, you only betray your lack of understanding of General Relativity.
Open the mathematica CDF file to see how equations 28 and 29 are related. As to your other point... you are thinking of this wrongly. 1.) The state is given as a function of momentum not position. 2.) the momentum in there is the norm of the 4vector momentum.

You have now added the expression m = m_p (though, in reality, your Mathematica code uses the equivalent of m = m_p λ, which actually does work, once you produced a TeX version that showed "mp" as m_p, since TeX doesn't distinguish between a variable "mp" and the product of "m" and "p"). Fortunately, your paper doesn't make the mistake of your Mathematica CDF file, wherein you asserted that "the momentum p should be interpreted as a locally lorentz invariant quantity. p=\[Sqrt]p.p" (your point #2, in your message). You really need to work harder at keeping separate concepts distinct, rather than mixing them up, apparently even within your own mind. Additionally, having a Quantum state "function" of momentum, rather than position, does not obviate the normalization requirement (see for instance your equation, within your abstract, that requires = 1). After all, remember, momentum and position spaces are complementary spaces, within QFT (in flat spacetime)!


License
This article and its reviews are distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and redistribution in any medium, provided that the original author and source are credited.
A.) Thanks for posting what appears to be all of this if you are going to post it at all.
B.) It is unethical to post publicly what is clearly intended to be private communication. This just shows that the so called reviewers of this article so far have lacked scruples in general. As I said in the last reply to you. Even if you are Werner Heisenberg or Paul Dirac, your inability to get something does not mean it is incorrect. Learn the difference.
Go back to climate science or whatever it is you do then.
You were the one that said "I am going to have to insist that any and all conversation regarding these papers take place on the Winnower. I am fully willing to discuss them just here on the Winnower with full context."
Additionally, I forewarned you that this was going to be posted, and gave you ample opportunity to not have it be so. It is now available to all, so all may judge for themselves.
I have responded to your fundamental misunderstanding of not only this model but of quantum field theory with a short power point presentation which is attached as an extra resource to this paper. As for your posting this here... you know darn well I would not mean "post confidential emails online". Be thankful I have better things to do than sue you for it.
It should be obvious to all that there is nothing truly "confidential" in our email exchange.
Other than your claimed "Results", on slide 5, there is nothing truly controversial within your presentation. The reason the "Results" are controversial is due to the utterly sloppy way you have misued the concepts of QFT, and the concepts such depend upon. (To say nothing of notational sloppiness.) Furthermore, you appear to have a fundamental misunderstanding of "vectors" vs. "components of vectors". Things like P^μ, k^μ, x^μ, x_μ, and such, are not vectors (whether "covariant" or "contravariant"), but only */components/* of vectors. (Liu Wei was referring to this same fact with his references to the use of \gamma^\mu really referring to the components \gamma^0, ... , \gamma^3.) I know this isn't entirely your fault, since all too many authors on Special and General Relativity brush over such distinctions to get to the "good stuff". (I find it seems to be even worse for those that like the spinoral/vierbein/tetrad notational choice.) Perhaps you need to check out Misner, Thorne, and Wheeler, just to suggest one source that helps clarify the distinction.