Relativization a Framework for Constructing Unified Theories. Hontas F. Farmer February 7, 2016 Adjunct Professor, Malcolm X College, Department of Physical Sciences,1900 W Jackson St, Chicago, IL 60612, United States of America
[email protected] Adjunct Professor, College of DuPage, Department of Math and Physical Sciences, 425 Fawell Blvd., Glen Ellyn IL, 60137, United States of America
[email protected] Abstract Super Symmetry is a powerful and flexible framework for constructing theories which unify General Relativity with quantum field theory. This paper presents a summary and extension of an alternative framework which I call relativization. Websters defines relativization as “the act or result of making relative or regarding as relative rather than absolute”. In the sense of physical theories relativization means modifying a theory so that it comports with the principles of general and special relativity. This paper gives a set of axioms for constructing a relativized quantum theory with space time curvature that changes dynamically with quantum interactions and vice versa. I show how to do this with a toy model, and more significantly with the standard model of particle physics. From the relativized and extended standard model presented I derive testable observational predictions for black hole thermodynamics, gravitational time dilation, and standard model interactions in gravitational fields of varying strength. What is presented here is like SUSY a framework for constructing theories which could yield a unified model from which general relativity and quantum field theory can be derived.
1
Introduction
The theoretical challenge of the mid twentieth and early 21st century has been how to reconcile general relativity with quantum field theory. The standard 1
Special Relativity
Quantum Field Theory
Quantum Mechanics Quantum Field Theory
a
General Relativity
General Relativity Fully Relativized QFT
Quantum Gravity
b
Quantum Field Theory
c
Figure 1: Theoretical family trees. The theory on the left is taken as more fundamental than the theory on the right. The one on the left is sort of the father theory and the one on the right is the mother theory. The “family” the theory belongs to being determined by the father theory. b) QFT sort of “gives his name” to the child theory quantum field theory. c) Instead of a quantization of general relativity I propose relativization [9] of quantum field theory. The resulting child theory is then more fundamentally a relativistic theory, not a quantum theory. [5] approach has been to quantize general relativity. The inspiration for this is that the most successful approach to unifying a relativistic theory with a quantum theory is quantum field theory. That said, in quantum field theory we don’t quantize special relativity we make quantum mechanics more and more relativistic. The analogy I like to use is that of a family tree. In a western Euro-American family the wife takes the name of the husband and to an extent conforms to him. I realize that is an archaic idea of a family but bear with me. In the case of quantum field theory we had quantum mechanics and special relativity. There were efforts to quantize the space-time of special relativity[7]. However in the end we decided that the daddy needed to be special relativity and the mama ought to be quantum mechanics. The child was quantum field theory. The flow charts in 1 explains this. This approach was very successful. quantum field theory gave us quantum electrodynamics, electroweak theory, QCD and now the standard model of particle physics. The only thing it hasn’t been able to easily take in is gravity. In the weak field limit a quantum field theory based on the exchange of gravitons can work. Gravity just won’t have a significant effect until the field is strong and this is where things break down. Never the less when we try to marry QFT to GR we envision a child with certain features. Those features make up what we call “quantum field theory”. This is flow charted in figure1. What I proposed in my papers of 2014 and 2015 is that we have the family tree all wrong. Instead quantum field theory who we traditionally think should be the daddy works better as the mommy. We should treat GR as the daddy and make QFT conform to it. Stated formally instead of quantization of general relativity at least someone should try relativization of quantum field theory. In the end it was relativization of quantum mechanics to conform with special relativity that gave birth to quantum field theory. It may be in the end more simple and natural to relativize quantum field theory.
2
2
Principles and Axioms
Axiomatic quantum field theory also known as algebraic quantum field theory (AQFT), is key to formulating a model which can comply with general relativity. By this method the algebra of the operators is used to formulate the theory. The details of any underlying space-time do not need to be relevant to the purely QFT parts of the theory. The best way I can describe this is with a set of axioms inspired by the Wightman axioms for QFT in curved space time. A relativized theory in the sense I propose is one which satisfies these axioms. 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 [6] they dealt with a static curved background. What follows will give a formulation which can admit a dynamical background. In summary form these are the axioms. 1: The principle of Relativization: All physical theories must obey the Einstein Equivalence Principle as stated in [2]. Theories must be formulated in a way that is locally Lorentz covariant and generally diffeomorphism covariant. 2: Spectrum condition: All possible states of a QFT will be in the FockHilbert space H. An operator on H must map states to other states in H. 3: Normalization condition: The inner product on H must be in a set isomorphic to the division algebras R,C,H, O such ashψ| ψi = j a with j a ∈ Minkowski , H ∼ = Minkowski and ∀ |ψi ∈ H. [1] (note I am claiming that is an isomorphism not an isometry.) 4: The principle of QFT locality: QFT interactions occur in the locally flat space at the point of interaction long distance propogation is governed by general relativity. 5: Specification condition: Relativized QFT’s are defined by the above, the tensor product of their state space with Minkowski space and the algebra of the operators on Hilbert space. For a theory T, T = {H, H ⊗ M, A (H)} (Inspired by a similar statement in [6].)
2.1
Let us consider the simplest quantum mechanical system.
Consider beams of fermions and how they may effect gravity via their spin. This well known system has a Hilbert space with two eigen vectors and two real eigenvalues. A general vector in this space could be written as, |αi = aγ a |↑i + (1 − a) γ a |↓i |βi = bγ b |↑i + (1 − b) γ b |↓i . 3
𝜙ത
𝜙 Hilbert Space
ത 𝑎𝜙 𝑗 𝑎 = 𝜙𝛾
.
𝑗𝑎
Tangent Minkowski Space-time 𝑎𝜇
𝑅𝑏𝜈
𝜇
𝑗𝜇 = 𝑒𝑎 𝑗 𝑎
𝜇
𝑎𝜇
𝑅𝜈 =𝑗𝑏 𝑅𝑏𝜈 𝑗𝑎
Curved Riemannian Manifold
Figure 2: The relationship between spaces. Mathematical entities can be scalars, vectors, or tensors or combinations of these in different spaces at the same time. Hilbert space vectors like φ combine in an inner product of the form j a =¯φγ a φ. j a is a vector in the tangent Minkowski space time at the point of interaction yet it is by definition a scalar with respect to the Hilbert space. Such quantities ¯ µψ . apear in standard QFT formulations such as the current four vector j µ = ψA That is where standard QFT stops. Relativized QFT includes operators such as aµ the Riemann curvature operator Rbν which depends on the probability current density vectors determined from the relativized QFT. In this manner quantum level interactions can effect the local space time curvature and vice versa.
4
Where a and b are between zero and one. The inner product of these kets will be a tensor of rank two. hα| βi = [2ab − (a + b)] γa γ b The way to get a curvature operator starts with the vierbien formulation as shown in detail in the following section in equations 3 and 4. For this toy example I will make the assumption which while not likely correct is illustrative µa ωνb = |αi hβ| can be substituted into equation 3 to result in this formula relating curvature to the quantum mechanics of spin 1/2, Rνµ = [2ab − (a + b)] γa γ b (d |αi hβ| + |αi hβ| ∧ |αi hβ|) . This last equation gives a plausible Ricci curvature due to the interaction of two beams of neutral spin 1/2 particles such as neutrons or neutrinos. THIS IS JUST A TOY EXAMPLE MODEL. Relativized quantum field theory will be more complex. For example consider standard Quantum Electro Dynamics. We can write the Lagrangian for it, which means we can write a Hamiltonian and from this derive a set of eigenstates. These will comprise a basis for the Fock-Hilbert space of the theory. The tensor product of the Hilbert Space with Minkowski ¯ µ ψ. QED is in the sense of space is hidden in such standard notations as γµ ψA complying with special relativity a relativized model.
3
The Relativized Extended Standard Model.
To derive the relativized extended standard model of particle physics I will start with the standard model written as a quantum field theory in curved space time. Using this starting point goes a long way toward meeting the first condition. √
1 ab 2 a ¯ L = −g − F Fab + iψγ Da ψ + ψi gij ψi φ + h.c. + |Da φ| − V (φ) + R . 4 (1) QFT in curved space time is not enough since it does not allow for the curvature to be effected by interactions. The scalar curvature in the above is just a parameter not a field or an operator. According to the specification condition a theory is defined by the algebra of its operators. The Lagrangian I am looking for will be defined in terms of algebra and operators. To write interaction terms I need an inner product like the one described in axiom 3 In my previous paper [4] I was mainly interested in inner products of the form 1 1 ¯ a φn γ φm + φ¯n γ a φm (2) 2 2 Quantities such as this are a set of four vectors which exist in the tangent space of an interaction. This subspace H ⊗ M consist of the probability current four vectors, j a . The next step is to use these expressions to formulate a operator hφn |φm i =
5
which connects one tangent space to another. I conjecture that the way to do this is by way of the curvature of space time if I can derive an operator to describe it.
3.1
A Curvature Operator
I used the vierbein formulation of general relativity [3] . Start from the Cartan structure equations using the gamma matrices as a basis Rba = dωba + ωca ∧ ωbc
(3)
T a = dγ a + ωba ∧ γ b .
(4)
Note the Greek indices are suppressed. Then solve for the Cartan connection. T a = dγ a + ωba ∧ γ b ⇒ (T a − dγ a ) ∧ γb = ωba
(5)
Set the torsion equal to the probability current for a scalar field such as the Higgs field |φi, j a = hφ|φi, and simplify. This results in the exterior derivative of the Cartan connection. dωba = dj a ∧ γb
(6)
Next we can solve for the Riemann curvature in terms of Dirac gamma matrices and probability currents. Rba = dj a ∧ γb + j a ∧ γc ∧ j c ∧ γb
(7)
a
To get a valid operator on the Hilbert space use j = hφ|φi. Also use the outer product |φihφ|. Then I define Rba (hφ|). also d R (8) ab = (dhφ|φi ∧ γb + hφ|φi ∧ γc ∧ hφ|φi ∧ γb ) hφ|. Equation 8 relates the curvature of space time near a point to the probability four currents due to local non-gravitational fields. Equation 8 provides eigenvalues and eigenstates of space-time curvature. d R ab |φm i = Rabm |φm i .
(9)
If I use this expression for the Higgs field equation 11. For the remainder of this paper I will drop the hat and bra-ket notation for operators. Assume a quantity is an operator unless otherwise specified explicitly ¯ a φ. or by convention. So for example j a = hφ|φi would be notated as j a = φγ In this notation equation8 becomes equation 10. ¯ a φ ∧ γb + φγ ¯ a φ ∧ γc ∧ φγ ¯ c φ ∧ γb φφ. ¯ Rab = dφγ
6
(10)
φ = Φeika x
a
(11)
Knowing the curvature operator equation 8, and the form of the Higgs field, equation 11, I can find the curvature eigenvectors, equation 12. ¯ a Φ ∧ γc ∧ Φγ ¯ c Φ ∧ γb ΦΦΦe ¯ ika xa Rab φ = Φγ
(12)
As well as the eigenvalues, rab , ¯ a Φ ∧ γc ∧ Φγ ¯ c Φ ∧ γb ΦΦ. ¯ rab = Φγ
(13)
a
One could write Rab = rab Φeika x . Now I can 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 Hsm ⊗M with all indicies summed over. The standard model fields will satisfy the axioms when written as local operators just as they are in established QFT in curved space time.
LRESM =
√
1 ¯ a Da ψ + ψi gij ψi φ + h.c. + |Da φ|2 − V (φ) −g − F ab Fab + iψγ 4 ¯ a Rab φγ b . (14) +R − φγ
Equation 14 is a form of the relativized extended standard model. The term ¯ a Rab φγ b includes the standard Einstein gravity term with a correction due R−φγ to higgs-graviton interactions. Including standard Einstein gravity means this model predicts everything that GR does. The graviton-higgs interaction will be the source of new predictions. What is required is a algebraic construction of the relativized extended standard model.
4
Gravity Mediated Interaction Cross-sections.
When I presented a talk based on this model at the 2015 April APS conference in Baltimore, Maryland the question was asked, does this model solve the problem of UV divergences. To answer this question rigorously I will use a very abstract formulation then I will derive a Feynman diagram expansion for graviton-graviton scattering in the RESM. This will lead directly to a simple experiment in gravitational time dilation and to consequences for black hole thermodynamics which may be observed.
4.1
via Generating Functional Methods.
Before I proceed I will rewrite the Lagrangian LRESM in terms of exterior algebra and treat the operators in it as differential forms. I will also use the geometric product as defined in [8] (which is easy to confuse for an inner product as in the standard notation both lack a symbol). The result is as follows 7
LRESM = T ab Rab + T ab ∧ Rab + LSM = T ab Rab + LSM
(15)
. In the equation above T ab = η ab ∧ 1 + 16 γ a φ¯ ∧ γ b φ ∧ 1. One could also object on the grounds that we don’t have a formula for the standard model in terms of algebraic quantum field theory. I don’t need one for the calculation of interest and formulating the standard model as an AQFT in concrete terms would be a paper or even a book unto itself. Such a formulation must exist we just don’t know it yet. The next step is to compute the generating functional for this model Z0 = 1 eγ S . (In this framework it makes sense to use the gamma matrices as a quaternion basis. Using the gamma matrices in this way has several geometric algebraic advantages.) For now I will drop LSM because its generating functionals are well known. To ´ compute the generating functional I need to simplify this integral γ 1 S = γ 1 d4 xT ab Rab . After integration by parts and full simplification the integral works out to γ 1 S = γ 1 η ab 2 + 23 δ (x − x0 ) Rab . Which can ´ 4 ab g g ab R ab = be rewritten as γ 1 S = γ 1 T d xT . Therefore the generating ab and T functional will be given by. Z0 =
1 ab −1 γ Tf eγ Tf Rab ab = γ 1 Tf e 1 Rab ab g 1 ab γ T
(16)
Consider equation 16 if the curvature goes to infinity then the generating functional oscillates rapidly. If the separation between points x and x’ is zero, i.e. a system of zero length the generating functional remains finite. Put another way, even with a wavelength of zero the amplitude of the oscillation remains finite. Following a similar procedure of integration by parts one can find Z1 . 1 ab −1 γ Tf eγ Tf ∧Rab ab Z1 = = γ 1 Tf e 1 ∧Rab ab g 1 ab γ T
(17)
As per the axioms of relativization this quantity is a quaternion, which is one of the groups that a quantum mechanics must have as it’s set of “scalars”. To get the observed cross section for graviton graviton interaction in this model I need to take some functional derivatives with respect to Tf ab . This works out to a simple expression in terms of the scalar of the stress energy tensor. δ 1 δ −1 −γ1 Z → 2 2 + γ1 T 2 − 1 (18) Z0 −γ1 f f T δ Tab δ Tab By examination one can see that this equation will not give an infinite answer for a finite input. One could set T equal to the scalar stress energy of the whole universe and get a large, but finite answer. One could set it equal to the cosmological vacuum energy and get a finite answer. Using this framework one may compute the gravitationally corrected interaction cross sections for all
8
standard model particles by adding the stress energy due to the standard model, f f Tf ab = Tab + TabSM , and computing the following. −1 σ = Z0SM −i
4.2
δ δJSM
−i
δ δJSM
2
ZSM Z1
(19)
Feynman diagram rules.
A detailed and elementary derivation of the Feynman diagram rules for this model is given in[3]. starting from equation 13, contraction of the appropriate indexes allows me to find the quantized Ricci tensor and quantized Ricci scalar. Then the EinsteinHilbert action can be written down. ˆ
q 1 ¯ a rab φγ b S= γ a γ b rab − φγ − ηab γµa γbν d4 x (20) 2κ S=
1 2κ
ˆ
q ¯ rab − η b γ a γ ν d4 x γ a γ b 1 − ΦΦ a µ b
(21)
The Higgs interacts with gravitons in a way which will moderate (not mediate) the gravitational interaction terms. From equation 21, the Feynman diagram expansion can be read off. The term linear in the Ricci curvature will produce the diagrams associate with standard Einstein gravity. This means this model inherits all the predictions of classical general relativity. The term which is quadratic in the Higgs produces a vertex which introduces counter terms at every loop order. The Higgs therefore moderates not mediates gravitational interaction, figure 3. This removes the UV divergences in any perturbation expansion. Consider the graviton-graviton scattering crossection in terms of Feynman diagrams under the RESM, figure 4. The tree level diagrams are the familiar diagrams from standard quantum field theory for a graviton. This means that this theory predicts everything we know from classical general relativity as far as we have been able to measure such results. The RESM differs when we get to one loop corrections and beyond. The vertex, figure 3, provides counter terms at every loop order. So instead of a infinite progression of diagrams which converges we get a series which will converge. The series converges to approximately hyperbolic cosine of the momentum (or position). ¯ a Rab γ b φ |MGG | = φγ 2 4 rab 4 4 rab rab rab 2 ¯ ≈ 2 + x + 1 − ΦΦ x ∼ 2 Cosh (x) (22) lp lp2 lp2 lp If I take the derivative of this with respect to x, which is really x = γ a xa , the result is. 9
=
𝑖 𝑝2 − 𝑚ℎ2 + 𝑖𝜖
=−
=
𝜆 2
1 −𝛾𝑎 𝛾𝑏 −𝛾𝑐 𝛾𝑑 + (−𝛾𝑎 𝛾𝑐 )(−𝛾𝑑 𝛾𝑏 ) 2 𝑝2 + 𝑖𝜖
=−
1 =− 𝜆 4
𝛾𝑎 𝛾𝑏 𝑟 𝑎𝑏 2
= −𝛾𝑎 𝛾𝑏 𝑟 𝑎𝑏
ഥ 𝑎𝑏 = −ΦΦ𝑟 Figure 3: Only one vertex is unique to the relativized extended standard model. This shows a higgs-higgs-graviton interaction. The problem of having two spin zero particles interact to create a spin two particle is addressed by the third axiom . This vertex introduces terms at every loop order to counter the diagrams that cause UV divergence in standard quantum gravity. With this vertex Feynman diagrams may be used.
𝑀𝐺𝐺 = + −
−
+
−
+
+
−
+
− −
+⋯
Figure 4: Graviton-Graviton scattering in the relativized extended standard model. All the diagrams from standard quantum field theory gravity are reproduced. .
10
d|MGG | ≈
R0 ¯ 4 2 2 + 4 1 − ΦΦ lp
R0 lp2
!
4 x
2
x∼
rab Sinh (x) lp2
(23)
The quantity in parentheses defines an operator which will give the time dilation as a function of distance from a gravity source in this model. ! 4 R0 4 2 R0 ¯ 4τ ≡ 2 2 + 4 1 − ΦΦ x (24) lp lp2 This suggest a simple experiment. The decay of a sample of radioactive material should decay more quickly at higher altitude from a gravitating source. Atomic clock experiments done to test classical time dilatation back this up. More sensitive atomic clock experiments will either support or refute this prediction. There are some limits to using Feynman diagrams in the framework of this model. Such calculations can’t be truly relativized the way that the algebraic calculations. The advantage is they can lead simply and intuitively to measurable observable consequences.
5
Black Holes and Hawking radiation in the Framework of Relativization.
For any candidate theory of quantum gravity formulating black hole thermodynamics in that model is key. Any serious contender for a unification of general relativity and quantum field theory must reproduce the thermodynamic results of Bekenstein and Hawking. With that in mind I did set up and solve the problem in (cite) and corrected it in (cite). The result was part of a presentation at the 2015 April APS conference (cite). By working with the a Lorentz invariant x, x = γ a xa , instead of the invariant momentum p the nature of the equations is more easily revealed. Each value of x, x = γ a xa , represents a set of values of x,y,z and t for which x is the same. A set of isometric four dimensional surfaces. This locally Lorentz invariant x is a parameter which specifies a set of equivalent geometries. Each x defines a set of physically equivalent four dimensional surfaces. In the language used in relativization theory x is a locally Lorentz invariant parameter used to write a Schrodinger equation for the Hilbert space H over the Minkowski space. In stringy M-Theory language x is a fifth dimension in which these D4 branes may move. The important thing to remember is these functions which define vectors in Hilbert space depend on their geometry of the underlying local Minkowski space which is tangent to the curved Spacetime manifold. So this “position” with the dimension of “length” is a bit deceptive! Yet this simplifies the analysis considerably. 11
What follows is the reasoning which will lead to the exact solutions for the position like basis quantum states for a relativistically bound system. Bound in such a way that there is an event horizon. This would include systems such as black holes of any size. This model could also apply to the whole of the universe near the time of the big bang. The key to setting the length/energy scale via the parameter L which stands for length here and not a momentum operator. L needs to be chosen so as to encompass the magnitude of whatever system is of interest. In the case of the universe L would best be the Hubble length. In the case of “quantum” black holes L should be the Planck length. In the following it will be shown that for the solutions to a Schrodinger type equation with the potential derived in the previous paper. L could even be zero. This model will work and give reasonable results at any imaginable length scale. I will enter the equations in a dimensionless form by introducing the simplest combination of parameters to cancel out the dimensionality. NOTE: this math was done with the computer algebra system Mathematica. The Mathematica notebook can be furnished on request or will be available online with this paper. To find the general solution first I will set the potential x v(x)= cosh L . Then I have the computer solve the following equation, −
~2 R 0 L L ~2 ψ 00 (x) L + v(x)ψ(x) − En ψ(x) = 0 2M hc 2M hc hc
. The solution output contains undetermined coefficients for the even and odd solutions, 8EnL2 M ix 8EnL2 M ix 2 2 ψ(x) = c1 MathieuC − , −2L R , −c MathieuS − , −2L R , 0 2 0 ~2 2L ~2 2L . With the eigenfunctions known I will have the computer solve for the corresponding eigenvalues. The result for the even states is, ~2 an+ 12 −2L2 R0 . En = − 8L2 M . Likewise for the odd eigenvalues, ~2 bn+ 12 −2L2 R0 En = − . 8L2 M .
12
For symmetric solutions ψ’[0]=0. In the periodic Fouquet-Bloch form of 1 the solution ψ[L]=ei(n+ 2 ) where µ is the Mathieu Characteristic exponent. The equation to solve is as before with the following mixed boundary conditions, 1 ψa0 [0] = 0, ψa (L) = ei(n+ 2 ) . For the even states the eigenstates are given by equation 25. h i 1 ix ei(n+ 2 ) MathieuC an+ 12 −2L2 R0 , −2L2 R0 , 2L h i ψa (x) = MathieuC an+ 12 (−2L2 R0 ) , −2L2 R0 , 2i
(25)
The odd, anti symmetric states are a bit different the difference being the 1 boundary conditions, ψb [0] = 0, ψb (L)h = ei(n+ 2 ) . i 1 ix ei(n+ 2 ) MathieuS bn+ 12 −2L2 R0 , −2L2 R0 , 2L i h (26) ψb (x) = MathieuS bn+ 12 (−2L2 R0 ) , −2L2 R0 , 2i The full solution for an arbitrary black hole eigenstate will be27 . h i 1 ix ei(n+ 2 ) MathieuC an+ 21 −2L2 R0 , −2L2 R0 , 2L i h ψ(x) = λa MathieuC an+ 12 (−2L2 R0 ) , −2L2 R0 , 2i i h 1 ix ei(n+ 2 ) MathieuS bn+ 12 −2L2 R0 , −2L2 R0 , 2L h i + λb MathieuS bn+ 21 (−2L2 R0 ) , −2L2 R0 , 2i
(27)
The set of eigenvalues are, En = − λa
~2 an+ 12 −2L2 R0
8L2 M
+ λb
~2 bn+ 12 −2L2 R0
!
8L2 M
.
. For a black hole the lambda’s would have to be 1/2. This would make the state of the hole a state of maximum entropy in accordance with established theory. Thus I can write down the relativized quantum state of a black hole in this framework as a function of x which in this paper is x = γ a xa , as in equation 28. h i i(n+ 21 ) ix 2 2 1 R , −2L R , e MathieuC a −2L 0 0 2L n+ 2 1 h i ψBH (x) = i 2 MathieuC an+ 12 (−2L2 R0 ) , −2L2 R0 , 2 h i i(n+ 21 ) ix 2 2 1 e MathieuS b −2L R , −2L R , 0 0 n+ 2L 1 2 h i + i 2 2 2 MathieuS b 1 (−2L R ) , −2L R , n+ 2
13
0
0 2
(28)
Enbh = −
! 2 2 2 2 1 ~ an+ 12 −2L R0 1 ~ bn+ 12 −2L R0 + 2 8L2 M 2 8L2 M
(29)
In the framework of relativization the form of the problem of Hawking radiation is that of barrier penetration and tunneling. There are two ways to approach this. One is to compute the probability current density vectors using the Inner product on a relativized Hilbert space (if this were M theory this would be an inner product on the world sheet of the brane describing the black hole). Instead of doing that I will use the WKB approximation. It is a simple and straight forward calculation that one hopes Mathematica can easily automate . This transmission coefficient times the value of the energy eigenvalue divided by the area of the black hole and a unit of time, say the Planck time, will give the Luminosity of Hawking radiation for a black hole. 1 ~2 0.0625an+0.5 −4.53072 × 10−52 L2 + 0.0625bn+0.5 −4.53072 × 10−52 L2 LBH = 4π L4 M tp ´ q 2 x ~ (−0.0625an+0.5 (−4.53072×10−52 L2 )−0.0625bn+0.5 (−4.53072×10−52 L2 )+1.13268×10−52 L2 cosh( L )) L dx L2 M ×exp 0 ~ (30)
That is the fully Mathematica simplified result of this calculation. Mathematica assumes a number can in general be at least complex. Let us consider each possible “OR” ∨ condition the software generates. ! an+ 12 −2L2 R0 + bn+ 12 −2L2 R0 < 4 + 4e2 , L2 R0 which is telling us that when e times the above number is greater than 4+4e2 that Hawking radiation will be possible when this is true. The last condition is more mysterious, an+ 21 −2L2 R0 + bn+ 12 −2L2 R0 ∈ / R. L2 R0 14
Temperature (Kelvin) T thispaper 4. × 10-18 T Hawking 3. × 10-18 2. × 10-18 1. × 10-18
2 × 1016
4 × 1016
6 × 1016
8 × 1016
Mass (kg) 1 × 1017
Figure 5: This plot shows how the Hawking temperature computed from the accepted Hawking result compares to the temperature computed by the methods shown in this paper. This was originally published in the form of an APS talk. [5] This condition is saying that when the integral is not a real number, that it is at least a complex number, and under those circumstances tunneling is mathematically possible from the center of the black hole to its surface. Since all the values input will be real and the Mathieu characteristic functions have real output this situation need not be considered physical. The exponential term which will appear is a complicated oscillatory function of L. L would be the Schwarzschild radius of the black hole. Hawking and Bekenstein’s theory would predict a perfectly smooth variance in the Luminosity of the black hole with mass and Schwarzschild radius. This theory predicts that the black hole will radiate with a slightly and rapidly varing intensity as it looses mass. For a black hole which is in a stable or metastable state the luminosity of the black hole due to Hawking radiation simplifies to... ~2 a 1 −8G2 M 2 R + b 1 −8G2 M 2 R 0 0 n+ 2 n+ 2 1 (31) LBH ≈ 4 5 16 G M tp For a hypothetical one kilogram black hole the temperature due to hawking radiation would be 55 billion Kelvin. For a stellar mass black hole the Hawking radiation would be 1.4 × 10−44 Kelvin. For Sagittarius A* this corresponds to a temperature of5.9 × 10−54 K. A run of the mill stellar mass black hole, and the super-massive Sagittarius A* are orders of magnitude colder than the cosmic microwave Background. Right now there should be a net inflow of radiation into any astrophysical black hole. There should be no observable Hawking radiation from a stellar mass black hole. 15
Temperature (Kelvin) 8. × 10-53
6. × 10-53
4. × 10-53
2. × 10-53
0
2 × 1036
4 × 1036
6 × 1036
8 × 1036
Mass (kg)
Figure 6: This plot shows a feature in the temperature mass relationship for black holes in my model which differs substantially from the Hawking prediction. If actual black hole Hawking temperatures are cataloged someday they may provide observational support for my model. That is not likely anytime soon since both my model and Hawking’s predict black holes will be cooler than the CMB. Perhaps a high resolution survey in the microwave range could tease out these temperatures thus allowing for a simple test of the RESM. This was originally published in the form of an APS talk. [5] The predictions of my model derived by very different means are very close to those of standard Hawking radiation as shown in figure 5 This model differs from the Hawking model in one other way to see how examine figure .
This model predicts the same general dependence of black hole temperature on black hole mass as Hawking’s semi-classical model. Furthermore, this model predicts that a solar mass black hole will not be warmer than the current CMB temperature. What this model predicts that isn’t predicted by any other odel is a temperature fluctuation at about 1 × 1036 kg or 500,000 . This could be verified or refuted by currently impractical but not impossible observations of black holes in that mass range.
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Discussion
Relativization is not a theory or a hypothesis but a framework for constructing hypotheses which can unite a quantum mechanical description of a system with a General Relativisic description by altering the quantum theory to fit into the framework of general relativity. I have shown a number of examples of how this 16
may be done. Starting with a simple toy model relating the spin on an electron beam to the curvature of space-time 2.1. The axioms listed in section 2 were applied to the standard model of particle physics to produce the Relativized Extended Standard Model, equation 14. Iit was shown that interaction cross sections can be computed which will not have ultraviolet divergences using the RESM. This was shown to be exact using the framework of generating functionals. This was also shown via a Feynman diagram method. From the Feynman diagram method it was shown that gravitational time dilation is explicitly predicted in this model. From the “potential” found from the Feynman diagram method, it was shown that the RESM can produce Hawking radiation and a dependence between the mass and temperature of a black hole that comports with established theory and all known observations. Yet this is not a trivial agreement since it differs slightly and in a manner that could in principle be observed. All of the above not withstanding the RESM cannot be a final model. Equation 19 which I call the three inch equation can be used to compute the probability of any standard model interaction along with its gravitational consequences. However, this model does not explain where the RESM comes from? Why should that model work at all? A a richer more fundamental theory is needed. Perhaps this framework could be applied to M-theory or a M-Therory like model. A relativized M-Theory of some sort instead of a SUSY M-Theory.
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Conclusion.
Relativization provides a framework for constructing theories that unify general relativity with Quantum field theory by making the QFT conform to the principles of general relativity. As such this is really an alternative to SUSY which does the same thing by replacing general relativity with 11 D supergravity and QFT with String theories related by various dualities. SUSY is a powerful flexible and well studied framework which may well lead to a final theory. SUSY is way ahead in terms of testing, and community support. SUSY has M-Theory which can predict numerous possible Lagrangian. The problem with SUSY that lead me to set down the axioms of relativization is that so far the minimally supersymmetric standard model particles have not been detected. SUSY as such cannot be “ruled out”. However, so far predictions made based on SUSY haven’t panned out. The goal of this paper was to summarize my work on relativization of 2014 and 2015 and to show that this framework is also a powerful tool for constructing theories. I set out to show how theories such as the Lagrangian of the relativized extended standard model LRESM can be constructed. I set out to calculate the measurable observable predictions of this model. These predictions for the thermal behavior of black holes, time dilation, and importantly via the equation for the gravitational corrected cross section, equation 19. This paper provides three testable predictions unique to this model. The results for
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black holes and in a more general sense any system enclosed in an event horizon, were found computationally. A a pure theory paper this is where theory stops and experimentation and Astronomical observation (in the case of black hole predictions) which are beyond the scope of theory, and beyond my means to do, take up.
References [1] J. C. Baez. Division Algebras and Quantum Theory. Foundations of Physics, 42:819–855, July 2012. [2] A. Einstein. The Foundation of the Generalised Theory of Relativity. Annalen der Physik, 7(354):769–822, 1916. 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. [3] H. F Farmer. Fundamentals of relativization. The Winnower, 12 2014. [4] H. F Farmer. Quantum gravity by relativization of quantum field theory. The Winnower, 08 2014. [5] H. F Farmer. Quantum Gravity: Have We Been Asking The Right Question? In APS Meeting Abstracts, page 13006, April 2015. [6] S. Hollands and R. M. Wald. Quantum fields in curved spacetime. ArXiv e-prints, January 2014. [7] H. S Snyder. Quantized space-time. Phys. Rev., 71:38–41, Jan 1947. [8] S. Somaroo, A. Lasenby, and C. Doran. Geometric algebra and the causal approach to multiparticle quantum mechanics. Journal of Mathematical Physics, 40:3327–3340, July 1999. [9] Websters. Websters Online Dictonary. Webseters, 2015.
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