UPDATE III is a rather important one. There is a group at Cambridge which does some similar work to mine. Great minds think alike, I guess. I have posted and revised a few times a paper on my new hypothesis for a theory of gravity which unifies Quantum Mechanics and General Relativity. Whenever someone writes "I have posted and revised a few times a paper on my new hypothesis for a theory of gravity which unifies Quantum Mechanics and General Relativity." The crank alarms should be flashing red alert, but I assure you all Einstein was quite right, but then so was Feynman and those who followed him.
First some background on Quantum Gravity research:
What falls to my generation of theoretical physicist is the problem of how to combine General Relativity with Quantum mechanics. The question is usually phrased as "how to quantize General Relativity". There are two well studied very popular approaches to quantizing General Relativity.
One is String/M-theory in which our four dimensional universe is merely a hyper surface or "D-brane" in a 11 dimensional super space. The theory contains 1 dimensional objects called string. Vibrations in these strings or D-branes are the particles which compose our reality. The graviton is just one of those particles.
The other alternative is Loop Quantum Gravity. In LQG there are no higher dimensional branes and it does not attempt to speak to forces other than gravity. In LQG general relativity is cast in terms of new variables called Ashtekar variables. Then in terms of these variables an operator which gives quantums of area is derived. Another variation of this cast the theory as a quantum field theory over a lie group instead of over Minkowski space.
I studied both of these for ten years and even wrote a book with my own little take on how to quantize general relativity. My approach was similar to LQG but instead I posulated that length was quantized. Then derived the consequences from it. A problem with it is why should one assume a particular quantum of length.
The two major approaches described above LQG and M Theory have their successes and their problems. The problems of M theory derive from the fact it requires extra dimensions which we can't observe, and super symmetry. The problems of LQG originate from the fact that it does not contain a graviton as we understand it from quantum field theory. Yet linearized gravity, which we know how to quantize, and which works in weak gravity situations works as well as any other QFT.
My new thought was instead of how to quantize General Relativity, instead we can try to relativize quantum field theory. To relativize means to take a theory which does not obey General Relativity and to recast it in a form which agrees with the principles of relativity. The problem may be that QFT was formulated with flat space time in mind. That made sense since in situations we can test in particle accelerators gravity is negligible.
QFT and the Standard Model are 100% correct, as far as we can measure, and any resulting theory should give the same results as those theories in the weak gravity limit.
Quantum gravity by relativization of Quantum Field Theory.
Abstract
The question is how can we make quantum field theory part of General Relativity instead of how we can quantize gravity. Here it will be shown that a Hilbert space can be defined such that the bra-ket is a four vector in ℳinkowski space-time, ψ∣∣ϕ>=va∈ℳ. Similar to a Hilbert space over the field of quarternions. Minkowski space is the tangent space at an arbitrary point on a Riemannian manifold. It will then be shown that the Riemann curvature connects these spaces by operating on the probability density 4-current ja of the local QFT. Choosing the Klein-Gordon field as a simple example QFT, the quantized Einstein-Hilbert action will then be derived. From there the expected Feynman diagrams for General Relativity can be read off. In this way one may calculate the gravitational effect due to a quantum field theoretical event.
1 Introduction
The motivating question for this study was can Hilbert space can be defined suchthat the ψ∣∣ϕ>=va∈ℳ. What kind of physics can be modeledwith such a structure. My thought was that QFT could be incorporated intoGeneral Relativity, and by that method a new hypothesis of short length scalegravity or possibly quantum gravity could be proposed. In this model QFT wouldbe exactly correct in the tangent spaces of a Riemannian manifold, while gravitywould physically connect these spaces.
Quantum field theory has been the most successful program of theoretical physicssince General Relativity. The one failure of Quantum Field theory has been aninability to incorporate General Relativity or handle dynamically curving spacetime. The idea expressed in this short paper is that one could try incorporatingQFT into GR instead of the other way around. Instead of quantizing GeneralRelativity we relativize QFT by demanding strict adherence to the EinsteinEquivalence Principle (2).
That for aninfinitely small four-dimensional region, the relativity theory is valid in thespecial sense when the axes are suitably chosen.
Each point on acurved space time has a flat Minkowski space time ℳ in which specialrelativity is exactly valid. With an appropriate inner product in the Hilbertspace, four vectors may act as scalars in the Hilbert space. Thus quantum fieldtheory would automatically obey the Einstein Equivalence Principle.
What is proposed in this paper is not completely without precedent. Standardquantum mechanics is formulated in terms of operators on a Hilbert space overthe field of complex scalars. It has been shown that the set of quaternions canalso be used as the field of scalars over which the Hilbert space isdefined(3). The idea that the key to quantum gravity liesin the formulation of the theory in terms of new variables can be found in theliterature (1). There are also many works on the subjectof algebraic local quantum field theory(4).
What is new in this paper is the concept of ℋ over the fieldℳ and as a consequence a new approach to quantum gravity. Instead ofassuming QFT is fundamental and incorporating gravity into it, we couldincorporate QFT into Relativity.
2 Hilbert space over the field of 4-vectors
Latin indicies will indicate vectors in the tangent space at each point on aRiemannian manifold. Greek indicies will indicate vectors in the curved globalspace-time of the manifold and ket’s will be vectors in Hilbert space. In thefollowing the vierbein formulation of General Relativity will be used.
- Let
-
ℋ be a vector space with elements∣∣ψ> and ∣∣ϕ> over the field of 4-vectors with va∈ℳ andthe inner product defined as follows.
ψ∣∣ϕ>=12(ψ¯γaϕ+ϕ¯γaψ)
|
(1) |
Intrigued? Want to object and say I'm an Idiot? Want to supply a detailed cirtique of this hypothesis. Read more at the winnower. (Self compiled PDF of the paper)
UPDATE on my Kickstarter:
For now my Kickstarter project is standing at $55. Two of the publishers are giving me a discounted rate so I don't need so much money. However if the kickstarter succeeds, I will use those monies to publish and to support my research. I am thinking I could buy a telescope to use for my Astronomy classes at the City Colleges of Chicago and the College of DuPage.
I hate having to do this as this is a really good discussion but I must insist that if one wishes to review the paper we do it on The winnower. All the equations are there ,all the context is there, and all the discussion will be kept with the paper once it's finalized.
I look forward to having the paper torn to shreds, spending a while, weeks or months patching it up, then finalizing it.
UPDATE II
A version of the paper modified based on the feedback received here is now available on "The Winnower". Hontas Farmer, Quantum gravity by relativization of Quantum Field Theory., The Winnower1:e140751.17561 (2014). DOI:10.15200/winn.140751.17561
UPDATE III
My idea might not be as novel as I thought. A worse problem than any caught by any review so far. I'm pretty sure my formulation and implementation are unique. The very concept of doing quantum mechanics over an algebra instead of a set like R C H or O is NOT NEW. I knew a QFT could be formulated over a lie group instead of Minkowski space as in Group Field Theory. It turns out there is a whole research group at Cambridge that does largely similar work! http://www.mrao.cam.ac.uk/~clifford/publications/
Comments now closed to ensure that any discussion of substance occurs with the paper. This blog post was intended to direct review type commentary to the paper, in the same place as the paper so it can be saved with the paper. They are part of the record in a post pub peer review and commenting system. To ensure any further such comments go there, comments here are now closed.
