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Not having been trained in physics at all, my intuitive search for the right description of physics repeatedly lands me into the circle of the picture of physics that originates in Williamsburg Brooklyn in 2008: that Ricci curvature of any three dimensional hypersurface M \sub S^4(1/h) can be interpreted as Einstein’s equation with stress-energy tensor the second fundamental form and cosmological constant O(h^2) and that the source of all quantum phenomena should be due to this ambient scaled 4-sphere.  But since this time I had been slowly trying to learn more established physics and its historical trajectory in slow steps in order to find ways of testing my ideas in more orthodox educated framework.  Steven Weinberg is the most articulate of the great physicists of the day and most familiar with the movements of physics from QED to 4 July 2012 discovery of Standard Model Higgs so it’s always a great pleasure to try to learn from the master.  Here are his reminisces.  To put things in context, I find parallel between the time of SW’s account of the period between 1950s pinnacle of QED and the chaos for weak and strong forces to the 1967 paper on weak forces which is one part of the pinnacle today of established physics the other being QCD.  I think my intuitive picture produces a simple complement to this development of quantum field theory by considering the great missing link not in the intricacy of the quantum field theory in the small but by the question of geometry of spacetime.  Dark energy and dark matter are both easy to explain as geometric error for example:  dark energy is but the ambient curvature of a scaled four-sphere within which our physical universe is constrained to lie; the solution of the hierarchy problem is also clear in this context, and one can even re-interpret the redshift-distance of Hubble as pure geometric error in a curved universe.  Nima Arkani-Hamed has made clear that Einstein’s, Dirac’s and Yang-Mill’s equations cannot be improved upon; and they are the wrong thing to look at for a perfection of physics.   It is in the large scale geometry where physics is going wrong and not in the small so much.  In a compact universe mass gap for Yang-Mills is obvious and there might be a re-interpretation of ‘reality’ of quarks when stable matter must have three principal curvatures and perhaps the issue of confinement is completely the wrong issue if quarks are not really particles, so maybe what we are calling ‘strong force’ is something of the strength of geometry of matter formation in the hypersurface-of-four-sphere universe.

 

 

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Historically, Richard Feynman (1963) made the first breakthrough in quantization of Yang-Mills fields by introducing fictitious particles with opposite statistics to the gauge fields to cancel the unphysical degrees of freedom and showed how this could be implemented for one-loop diagrams.  Bryce DeWitt (1964) then extended this to higher orders.  The prevalent method of quantization is the functional integral approach, done originally by DeWitt (1967) and Faddeev and Popov (1967).

Leonard Parker and David Toms, Quantum Field Theory in Curved Spacetime offers a geometric approach which is quite similar to the idea that I had yesterday.  The idea is to focus on the symplectic structure of the classical solutions of the Yang-Mills equations modulo gauge transformations as a finite dimensional manifold and consider Hamiltonian mechanics on this manifold and quantize position-momentum in this manifold.  In the specific case of gauge group SU(2) the structure of Yang-Mills moduli spaces has been extensively studied for arbitrary closed simply connected four-manifolds in Donaldson theory, so this method of quantization leads to a clear interpretation of quantum Yang-Mills fields in a strictly finite dimensional context without any mathematical ambiguity.  In particular, this approach avoids introducing unphysical particles as the original approach of Feynman.

In Donaldson theory there is a well-developed description of moduli space of anti-self dual connections modulo gauge transformations for a closed simply connected manifold X.  One evaluates polynomials in H_2(X) on the fundamental class of the moduli space to obtain smooth invariants.  This machinery can be turned around for quantization of gauge theories as follows:  the moduli space is stratified finite dimensional space which can be handled more or less smoothly and there is an Uhlenbeck compactification.  So quantization of Yang-Mills theory is equivalent to quantization of classical mechanics on the cotangent space of this moduli space, which is more or less standard and can be done in many ways such as geometric quantization.  The key point is that quantum fields do not need infinite dimensional analysis beyond that needed to get this moduli space; the analysis for this is already done in good detail for Donaldson theory.  So quantization of Yang-Mills theories is a finite dimensional problem.  Maybe this issue is considered well-known but this strikes me as extremely useful because quantization strategies and interpretation of quantum fields themselves seem quite obscure geometrically when one throws the machinery of C* algebras at them.  The quantization of finite dimensional symplectic manifolds on the other hand seem geometrically more intuitive.

First of all, Jackiw lays out clearly that there should be nagging doubts about QCD as a quantum Yang-Mills so this is not just me being contrarian — great paper here:Jackiw-YangMills-50 where he raises questions.  To me it seems that confinement is not explained by this model as parsimoniously as a simpler idea that matter is second fundamental form, extrinsic curvature of space as hypersurface of a 4-sphere ambient universe and quarks may not be particles at all but principal curvatures of relatively stable formations.  But my nagging doubts extend more extensively to quantum field theory itself as not fundamental.  Here too one can bring Dirac himself to my aid as he was not happy with quantum field theory.  Of course in my cause, I am still convinced that quantization is a geometrical fact about our universe that is a consequence of space being a hypersurface of a 4-sphere with a rigid radius.

The solution that I have in mind is very simple: the idea is that there is no strong force per se — spacetime is globally hyperbolic with physical universe modeled as a hypersurface of a 4-sphere whose radius depends on Planck’s constant and is fixed.  Matter arises as the second fundamental form of the embedding into the 4-sphere — this can explain both the energy-momentum tensor of Einstein’s gravity as well as Dirac’s matter for particles that are eigenspinors of the hypersurface Dirac operator which picks up a second fundamental form.  Now the idea is that quarks are not really particles but principal curvatures of nuclear particles.  This type of model would have confinement automatically and would explain the triplet of quarks associated with most of the nuclear particles.

My solution to confinement is that quarks are not particles at all but principal curvatures of an embedding of the physical universe M into a scaled 4-sphere S4(1/h) where nuclear particles have three principal curvatures. In this solution, you can’t see quarks in isolation but you could smash particles and find traces of the dissolving geometric structure.

I don’t think Tesla knew the ‘mind of God’ laws of nature although he was clearly a brilliant inventor and understood electromagnetism as a genius electical engineer. I think the fundamental problem IN physics today is that the sophistication of approaches to a unification of gravity and quantum mechanics is based on some very bad foundations of cosmology not to mention a cut-throat medieval institution of kings, kingdoms, ego. It’s suicidal for young people to go into areas where innovative and revolutionary thinking is necessary because you’ll be ostracized by the ‘cool kids’ doing the hypersophisticated baroque mathematical physics say string theory and to a lesser extent loop quantum gravity. I think the key step beyond what Gerhardt has done is to get a sharper sense of marrying Einstein’s matter and Dirac’s matter in static globally hyperbolic spacetime with hypersurface of S4 as space, a clear clean and beautiful geometrical merge as a shiny new physics that starts getting rid of the obscurantism. Boy, string theory, what an enormously massive overambitious project that sucks young talented theorists like a vacuum leading them like lemmings into a career in a monastary without any possibility of producing experimental results.

For globally hyperbolic models with compact slices quantization of gravity and unified gravity, Yang-Mills, and Higgs is a solved problem with mass gap:
Gerhardt-Unified1 and Gerhardt-Unified2.  So the real issue is whether we can make sense of whether our actual universe can be described by the S4 picture of hypersurfaces of a 4-sphere and more importantly whether the quantization phenomena in our universe is ultimately due to this geometry (which is what I believe) and whether there is a more concrete and simpler set of concepts that can describe laws of nature that begins with a concrete realization of unified physics and so on.