The previous post plus a re-reading of Professor Zee’s Quantum Field Theory in a Nutshell (Princeton, 2003) suggests a new formulation for quantum gravity, the mechanism and mathematical predictions of which were given two posts ago.The sum over histories for real particles is used to work out the path of least action, such as the path of a photon of light which takes the least time to bounce off a mirror. You can do the same thing for the path of a real electron, or the path of a drunkard’s walk. The integral tells you the effective path taken by the particle, or the probability of any given path being taken, from many possible paths.
For gauge bosons or vector bosons, i.e., force-mediating radiation, the role of the path integral is no longer to find the probability of a path being taken or the effective path. Instead, gauge bosons are exchanged over many paths simultaneously. Hence there are two totally different applications of path integrals we are concerned with:
- Applying the path integral for real particles involves evaluating a lot of paths, most of which are not actually taken (the real particle takes only one of those paths, although as Feynman said, it uses a ‘small core of nearby space’ so it can be affected by both of two slits in a screen, provided those slits are close together, within a transverse wavelength or so, so the small core of paths taken overlap both slits).
- Applying the path integral for gauge bosons involves evaluating a lot of paths which are all actually being taken, because the extensive force field is composed of lots of gauge bosons being exchanged between charges, really going all over the place (for long-range gravity and electromagnetism).
In both cases the path taken by a given real particle or a single gauge boson must be composed of straight lines in between interactions (see Fig. 1 of previous post) because the curvature of general relativity appears to be a classical approximation to a lot of small discrete deflections due to discrete interactions with field quanta (sometimes curves are used in Feynman diagrams for convenience, but according to quantum field theory all mechanisms for curvature actually involve lots of little deflections by the quanta of fields).
The calculations of quantum gravity, two posts ago, use geometry to evaluate these straight-line gauge boson paths for gravity and electromagnetism. Presumably, translating the simplicity of the calculations based on geometry in that post into a path integrals will appeal more to the stringy mainstream. Loop quantum gravity methods of summing up a lot of interaction graphs will be used to do this. What is vital are directional asymmetries, which transform a perfect symmetry of gauge boson exchanges in all directions into a force, represented by the geometry of Fig. 1 (below). One way to convert that geometry into a formula is to consider the inward-outward travelling isotropic graviton exchange radiation by using divergence operator. I think this can be done easily because there are two useful physical facts which make the geometry simpler even than appears from Fig. 1: first, the shield area x in Fig. 1 is extremely small so the asymmetry cone can not ever have a large sized base for any practical situation; second, by Newton’s proof the gravity inverse square law force from a lot of little particles spread out in the Earth is the same as you get by mathematically assuming that all the little masses (fundamental particles) are not spread throughout a large planet but are all at the centre. So a path integral formulation for the geometry of Fig. 1 is simple.
Fig. 1: Mechanism for quantum gravity (a tiny falling test mass is located in the middle of the universe, which experiences isotropic graviton radiation – spin 1 gravitons which cause attraction by simply pushing things as this allows predictions as proved in the earlier post – from all directions except that where there is an asymmetry produced by the mass which shields that radiation). By Newton’s 3rd law the outward force of the big bang has an equal inward force, and gravity is equal to the proportion of that inward force covered by the shaded cone in this diagram: (force of gravity) = (total inward force).(cross sectional area of shield projected out to radius R, i.e., the area of the base of the cone marked x, which is the product of the shield’s cross-sectional area and the ratio R2/r2) / (total spherical area with radius R). (Full proof here.)
Weyl’s gauge symmetry principle
A symmetry is anything that doesn’t change as the result of a transformation. For example, the colour of a plastic pen doesn’t change when you rotate it, so the colour is a symmetry of the pen when the transformation type is a rotation. If you transform the plastic pen by burning it, colour is not a symmetry of the pen (unless the pen was the colour of carbon in the first place).
A gauge symmetry is one where scalable quantities (gauges) are involved. For example, there is a symmetry in the fact that the same amount of energy is required to lift a 1 kg mass up by a height of 1 metre, regardless of the original height of the mass above sea level. (This example is not completely true, but it is almost true because the fall in gravity acceleration with height is small, as gravity is only 0.3% weaker at the top of the tallest mountain than it is at sea level.)
The female mathematician Emmy Noether in 1915 proved a great theorem which states that any continuous symmetry leads to a conservation law, e.g., the symmetry of physical laws (due to these laws remaining the same while time passes) leads to the principle of conservation of energy! This particularly impressive example of Noether’s theorem does not strictly apply to forces over very long time scales, because, as proved, fundamental force coupling constants (relative charges) increase in direct proportion to the age of the universe. However, the theorem is increasingly accurate as the time scale involved is reduced and the inaccuracy becomes trivial when the time considered is small compared to the age of the universe.
At the end of Quantum Field Theory in a Nutshell (at page 457), Zee points out that Maxwell’s equations unexpectedly contained two hidden symmetries, Lorentz invariance and gauge invariance: ‘two symmetries that, as we now know, literally hold the key to the secrets of the universe.’
He then argues that Maxwell’s long-hand differential equations masked these symmetries and it took Einstein’s genius to uncover them (special relativity for Lorentz invariance, general relativity for the tensor calculus with the repeated-indices summation convention, e.g., mathematical symbol compressions by defining notation which looks something like: Fab = 2dAab = daAb – dbAa). This is actually a surprisingly good point to make.
Zee, judging from what his Quantum Field Theory in a Nutshell book contains, does not seem to be aware how useful Heaviside’s vector calculus is (Heaviside compressed Maxwell’s 20 equations into 4 field equations plus a continuity equation for conservation of charge, while Einstein merely compressed the 4 field equations into 2, a less impressive feat but one leading to less intuitive equations; divergence and curl equations in vector calculus describe simple divergence of radial electric field lines which you can picture, and simple curling of electric or magnetic field lines which again are easy to picture). In addition, the way relativity comes from Maxwell’s equations is best expressed non-mathematically, just because it is so simple: if you move relative to an electric charge you get a magnetic field, if you don’t move relative to an electric charge you don’t see the magnetic field.
Zee adds: ‘it is entirely possible that an insightful reader could find a hitherto unknown symmetry hidden in our well-studied field theories.’
Well, he could start with the insight that U(1) doesn’t exist, as explained in the previous post. There are no single charged leptons about, only pairs of them. They are created in pairs, and are annihilated as pairs. So really you need some form of SU(2) symmetry to replace U(1). Such a replacement as a bonus predicts gravity and electromagnetism quantitatively, giving the coupling constants for each and the complete mechanism for each force.
Just to be absolutely lucid on this, so that there can be no possible confusion:
- SU(2) correctly asserts that quarks form quark-antiquark doublets due to the short-range weak force mediated by massive weak gauge bosons
- U(1) falsely asserts that leptons do not form doublets due to the long-range electromagnetic force mediated by mass-less electromagnetic gauge bosons.
The correct picture to replace SU(2)xU(1) is based on the same principle for SU(2) but a replacement of U(1) by another effect of SU(2):
- SU(2) correctly asserts that quarks form quark-antiquark doublets due to the short-range weak force mediated by massive weak gauge bosons.
- SU(2) also correctly asserts that leptons form lepton-antilepton doublets (although since the binding force is long-range electromagnetism instead of short-range massive weak gauge bosons, the lepton-antilepton doublets are not confined in a small place because the range over which the electromagnetic force operates is simply far greater than that of the weak force).
Solid experimentally validated evidence for this (including mechanisms and predictions of gravity and electromagnetism strengths, etc., from massless SU(2) gauge boson interactions which automatically explain gravity and electromagnetism): here. Sheldon Glashow’s early expansion of the original Yang-Mills SU(2) gauge interaction symmetry to unify electromagnetism and weak interactions is quoted here. More technical discussion on the relationship of leptons to quarks implies by the model: here.
However, innovation of a checkable sort is now unwelcome in mainstream stringy physics, so maybe Zee was joking, and maybe he secretly doesn’t want any progress (unless of course it comes from mainstream string theory). This suggestion is made because Zee on the same page (p457) adds that the experimentally-based theory of electromagnetic unification (unification of electricity and magnetism) was a failure to achieve its full potential because those physicists: ‘did not possess the mind-set for symmetry. The old paradigm “experiments -> action -> symmetry” had to be replaced in fundamental physics by the new paradigm “symmetry -> action -> experiments,” the new paradigm being typified by grand unified theory and later by string theory.’ (Emphasis added.)
Problem is, string theory has proved an inedible, stinking turkey (Lunsford both more politely and more memorably calls string ‘a vile and idiotic lie’ which ‘has managed to slough itself along for 20 years, leaving a shiny trail behind it’). I’ve explained politely why string theory is offensive, insulting, abusive, dictatorial ego-massaging, money-laundering pseudoscience at my domain http://quantumfieldtheory.org/.
Zee needs to try reading Paul Feyerabend’s book, Against Method. Science actually works by taking the route that most agrees with nature, regardless of how unorthodox an idea is, or how crazy it superficially looks to the prejudiced who don’t bother to check it objectively before arriving at a conclusion on its merits; ‘science,’ when it does occasionally take the popular route that is a total and complete moronic failure, e.g., mainstream string, temporarily becomes a religion. String theorists are like fanatical preachers, trying to dictate to the gullible what nature is like ahead of any evidence, the very error Bohr alleged Einstein was making in 1927. Actually there is a strong connection between the speculative Copenhagen Interpretation propaganda of Bohr in 1927 (Bohr in fact had no solid evidence for his pet theory of metaphysics, while Einstein had every causal law and mechanism of physics on his side; today we all know from high-energy physics that virtual particles are an experimental physics fact and they cause indeterminancy in a simple mechanical way on small distance scales), and string. Both rely on exactly the same mixture of lies, hype, coercion, ridicule of factual evidence, etc. Both are religions. Neither is a science, and no matter how much physically vacuous mathematical obfuscation they use, it’s failure to cover-up the gross incompetence in basic physics remains as perfectly transparent as the Emperor’s new clothes. Unfortunately, most people see what they are told to see, so this farce of string theory continues.