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Introduction

General relativity

Quantum gravity

Dirac’s equation

Path integrals

Representation theory

Methodology

Unification

Standard Model

Evidence for strings?

‘Science is the organized skepticism in the reliability of expert opinion.’ - R. P. Feynman (quoted by Smolin, TTWP, 2006, p. 307).

Mainstream frontier fundamental physics, string theory, isn’t scientific because it can’t ever predict real, quantitative, checkable phenomena, since 10-dimensional superstring and 11-dimensional supergravity yield 10500 ‘possibilities’ based not on observed gravity and particle physics facts, but merely on other unobserved, guesswork speculations - namely, unobserved Planck scale unification and unobserved spin-2 gravitons. Even the AdS/CFT correspondence conjecture in string theory is physically empty, since AdS (anti de Sitter space) requires a negative cosmological constant. So you can’t evaluate the conformal field theory (CFT) of particles with AdS, because AdS isn’t real spacetime!

‘Science n. The observation, identification, description, experimental investigation, and theoretical explanation of phenomena.’ - www.answers.com


Loop quantum gravity is the idea of applying the path integrals of quantum field theory to quantize gravity by summing over interaction history graphs in a network (such as a Penrose spin network) which represents the quantum mechanical vacuum through which vector bosons such as gravitons are supposed to travel in a standard model-type, Yang-Mills, theory of gravitation. This summing of interaction graphs successfully allows a basic framework for general relativity to be obtained from quantum gravity. The model is not as speculative as string theory, which has been actively promoted in the media since 1985 despite opposition from people like Feynman because it fails to predict anything. Despite endless hype, string theory is now in a state called ‘not even wrong’, which is less objective than the wrong theories of caloric, phlogiston, aether, flat earth, and epicycles, which were theories that tried to model some observed phenomena of heat, combustion, electromagnetism, geography, and astronomy.


String theory fails because it postulates that 6 dimensions are compactified into unobservably small manifolds in particles; these 6 unobservable dimensions need about 100 parameters to describe them, and it turns out that there are 10500 or more configurations possible, each describing a different set of particles (different particles within any set arise from the different possible vibration modes or resonances of a given string). This makes it the vaguest, least falsifiable mainstream speculation ever: to make genuine predictions, the state of the extra unobserved 6-dimensions must be known, which means either building a particle accelerator the size of the galaxy and scattering particles to reveal their Planck scale nature, or eliminating the false 10500 guesses, which would take billions of years with supercomputers. But there is some experimental evidence that key stringy assumptions, e.g., spin-2 gravitons and supersymmetry, are false.

For supersymmetry, in the book Not Even Wrong (UK edition), Dr Woit explains on page 177 that - using the measured weak and electromagnetic forces - supersymmetry predicts the strong force incorrectly high by 10-15%, when the experimental data is accurate to a standard deviation of about 3%. Supersymmetry is also a disaster for increasing the number of Standard Model parameters (couping constants, masses, mixing angles, etc.) from 19 in the empirically based Standard Model to at least 125 parameters (mostly unknown!) for supersymmetry. Supersymmetry in string theory is 10 dimensional and involves a massive supersymmetric boson as a partner for every observed fermion, just in order to make the three Standard Model forces unify at the Planck scale (which is falsely assumed to be the grain size of the vacuum just because it was the smallest size dimensional analysis gave before the electron mass was known; the black hole radius for an electron is far smaller than the Planck size).


At first glance, this 10-dimensional superstring theory for supersymmetry contradicts the 11-dimensional supergravity ideas, but this 10/11 dimensional issue was conveniently explained or excused by Dr Witten in his 1995 M-theory, which shows that you can make the case that 10-dimensional superstrings are a brane (a kind of extra-dimensional equivalent surface) on 11-dimensional supergravity, similarly to how an n - 1 = 2 dimensional area is a surface (or mem-brane) on an n = 3 dimensional object (or bulk). 11-dimensional supergravity arises from the old Kaluza-Klein idea, which was debunked and corrected by Lunsford in a peer-reviewed, published paper - see International Journal of Theoretical Physics, Volume 43, Number 1, January 2004 , pp. 161-177(17) for publication details and here for a downloadable PDF file, which was immediately censored from arXiv which seems to be partly influenced in the relevant sections by a string professor at the University of Texas, Austin.


On the speculative nature of conjectures concerning spin-2 (attractive or 'suck') gravitons, Richard P. Feynman points out in The Feynman Lectures on Gravitation, page 30, that gravitons do not have to be spin-2, which has not been observed. Renormalization works in the standard model (for electromagnetic, weak nuclear and strong nuclear charges) because the gauge bosons which mediate force do not interact with themselves to create massive problems. This is not the case with the spin-2 gravitons in general. Spin-2 gravitons, because they have energy, should according to general relativity, themselves be sources for gravity on account of their energy, and should therefore themselves emit gravitons, which usually makes the renormalization technique ineffective for quantum gravity. String theory is supposed to dispense with renormalization problems because strings are not point particles but of Planck-length. The mainstream 11-dimensional supergravity theory includes a superpartner to the unobserved spin-2 graviton, called the spin-3/2 gravitino, which is just as unobserved and non-falsifiable as the spin-2 graviton. The reason is that this supersymmetric scheme gets rid of problems which the spin-2 graviton idea would lead to at unobservably high energy where gravity is speculated to unify with other forces into a single superforce.

So a supersymmetric partner for the spin-2 attractive graviton is postulated in mainstream supergravity to make the spin-2 graviton theory work by cancelling out the unwanted effects of the grand unified theory speculations. Hence, you have to add extra speculations to spin-2 gravitons just to cancel out the inconsistencies in the original speculation that all forces should have equal coupling constants (relative charges) at unobservably high energy. The inventing of new uncheckable speculations to cover up inconsistencies in old uncheckable speculations is not new. (It is reminiscent of the proud Emperor who used invisible cloaks to try to cover up his gullibility and shame, at the end of an 1837 Hans Christian Andersen fairytale.) There is no experimental justification for the speculative mainstream spin-2 graviton scheme, nor any way to check it, which is discussed in detail here (discussion of alleged reason for spin-2 gravitons) and here (the stringy landscape of 10500 spin-2 attractive graviton theories really do suck in more ways than one; spin-1 gravitons avert the normal problems of quantum gravity, and make proper predictions without inconsistencies).


String theory predictions are not analogous to Wolfgang Pauli’s prediction of neutrinos, which was indicated by the solid experimentally-based physical facts of energy conservation and the mean beta particle energy being only about 30% of the total mass-energy lost per typical beta decay event: Pauli made a checkable prediction, Fermi developed the beta decay theory and then invented the nuclear reactor which produced enough decay in the radioactive waste to provide a strong source of neutrinos (actually antineutrinos) which tested the theory because conservation principles had made precise predictions in advance, unlike string theory’s ‘heads I win, tails you lose’ political-type, fiddled, endlessly adjustable, never-falsifiable pseudo-‘predictions’. Contrary to false propaganda from certain incompetent string ‘defenders’, Pauli correctly predicted that neutrinos are experimentally checkable, in a 4 December 1930 letter to experimentalists: ‘... Dear Radioactives, test and judge.’ (See footnote on p12 of this reference.)


‘The one thing the journals do provide which the preprint database does not is the peer-review process. The main thing the journals are selling is the fact that what they publish has supposedly been carefully vetted by experts. The Bogdanov story shows that, at least for papers in quantum gravity in some journals [including the U.K. Institute of Physics journal Classical and Quantum Gravity], this vetting is no longer worth much. ... Why did referees in this case accept for publication such obviously incoherent nonsense? One reason is undoubtedly that many physicists do not willingly admit that they don’t understand things.’ - P. Woit, Not Even Wrong, Jonathan Cape, London, 2006, p. 223.


‘Worst of all, superstring theory does not follow as a logical consequence of some appealing set of hypotheses about nature. Why, you may ask, do the string theorists insist that space is nine dimensional? Simply because string theory doesn’t make sense in any other kind of space.’ - Nobel Laureate Sheldon Glashow (quoted by Dr Peter Woit in Not Even Wrong: The Failure of String Theory and the Continuing Challenge to Unify the Laws of Physics, Jonathan Cape, London, 2006, p181).


‘Actually, I would not even be prepared to call string theory a ‘theory’ ... Imagine that I give you a chair, while explaining that the legs are still missing, and that the seat, back and armrest will perhaps be delivered soon; whatever I did give you, can I still call it a chair?’ - Nobel Laureate Gerard ‘t Hooft (quoted by Dr Peter Woit in Not Even Wrong: The Failure of String Theory and the Continuing Challenge to Unify the Laws of Physics, Jonathan Cape, London, 2006, p181).


‘... I do feel strongly that this is nonsense! ... I think all this superstring stuff is crazy and is in the wrong direction. ... I don’t like it that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation ... All these numbers [particle masses, etc.] ... have no explanations in these string theories - absolutely none!’ – Richard P. Feynman, in Davies & Brown, ‘Superstrings’ 1988, at pages 194-195. (Quotation courtesy of Tony Smith.)

Tony Smith’s CERN document server paper, EXT-2004-031, uses the Lie algebra E6 to avoid 1-1 boson-fermion supersymmetry: ‘As usually formulated string theory works in 26 dimensions, but deals only with bosons … Superstring theory as usually formulated introduces fermions through a 1-1 supersymmetry between fermions and bosons, resulting in a reduction of spacetime dimensions from 26 to 10. The purpose of this paper is to construct … using the structure of E6 to build a string theory without 1-1 supersymmetry that nevertheless describes gravity and the Standard Model…’ However, that research was censored off arXiv,apparently because mainstream string theorists are bigoted against 26-dimensional ideas since 10/11-dimensional M-theory was discovered in 1995. They don’t exactly encourage alternatives, even within the general framework of string theory (26-dimensional bosonic string theory is similar to 10-dimensional superstring in having a 2-dimensional spacetime worldsheet, the difference is that conformal field theory requires 24 dimensions in the absense of supersymmetry and 8 dimensions if there is supersymmetry).


Worse, attempts to explain observed particle physics with string theory result in 10500 or more different vacuum states, each with its own set of particle physics. 10500 solutions is so many it eliminates falsifiability from string theory. This large number of solutions is named the ‘cosmic landscape’ because Professor Susskind claims that each solution exists in a different parallel universe, and when you plot the resulting vacuum ‘cosmological constants’ as a function of two variables, in string theory, you produce a landscape-like three dimensional graph. The reason for the immense ‘cosmic landscape’ is the fact that string theories only ‘work’ (i.e., satisfy the basic criteria for conformal field theory, CFT) in 10 or more dimensions, so the unobserved dimensions have to be ‘compactified’ by a Calabi-Yau manifold, which - conveniently - curls up the extra dimensions in to a small volume, explaining why nobody has ever observed any of them. In superstring theory, two dimensions (one space and one time) form a ‘worldsheet’ and another eight are required for the CFT of supersymmetric particle physics. Sadly, the Calabi-Yau manifold has many parameters (or moduli) describing size and shape of those unobserved conjectured extra dimensions which must have unknown values (since we can’t observe them), so it is the immense number of possible combinations of these unknown parameters which make string theory fail to produce specific results, by producing too many results to ever rigorously evaluate, even given a supercomputer running for the age of the universe. The 10500 figure might not be right: the true figure might be infinity. String theory results depend on many things, e.g., how the moduli are ‘stabilized’ by ‘Rube-Goldberg machines’, monstrous constructions added to the theory just to stop string field properties from conflicting with existing physics! It’s presumably hoped by Dr Witten, discoverer of a 10/11-dimensional superstring-supergravity unification called M-theory, that somehow a way will turn up to pick out the correct solution from the landscape and start making checkable predictions.

However, the best idea of how to go about this is to assume that cosmology is correctly modelled by the Lambda-CDM general relativity solution, which attributes the observed lack of gravitational deceleration in the universe to dark energy, represented by a small positive cosmological constant in general relativity field equations. Then you can try to evaluate parts of the landscape of solutions to string theory which have a suitably small positive cosmological constant. Unfortunately, general relativity does not include quantum gravity, and even the mainstream quantum gravity candidate of an attractive force mediated by spin-2 gravitons, implies that gravity should be weakened over vast distances due to redshift of gravitons exchanged between receding masses, which lowers the energy of the gravitons received in interactions and reduces the coupling constant for gravity. Thus, dark energy may be superfluous if quantum gravity is correct, so it is clear that string theory is really a belief system, a faith-based initiative, with no physics or science of any kind to support it. String theory produces endless research, and inspires new mathematical ideas, albeit less impressively than Ptolemy’s universe, Maxwell’s aether and Kelvin’s vortex atom (e.g., the difficulties of solving Ptolemy’s false epicycles inspired Indian and Arabic mathematicians to develop trigonometry and algebra in the dark ages), but this doesn’t justify Ptolemy’s earth-centred universe, Maxwell’s mechanical aether, Kelvin’s stable vortex atom, and string theory. Another problem of this stringy mainstream research is that it leads to so many speculative papers being published in physics journals that the media and the journals concentrate on strings, and generally either censor out or give less attention to alternative ideas. Even if many alternative theories are wrong, that may be less harmful to the health of physics than one massive mainstream endeavour that isn’t even wrong...


Introduction

General relativity

Quantum gravity

Dirac’s equation

Path integrals

Representation theory

Methodology

Unification

Standard Model

Evidence for strings?


‘It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.’ - R. P. Feynman, The Character of Physical Law, November 1964 Cornell Lectures, broadcast and published in 1965 by BBC, pp. 57-8.


Feynman is here referring to the physics of the infinite series of Feynman diagrams with corresponding terms in the perturbative expansion for interactions with virtual particles in the vacuum in quantum field theory:


‘Given any quantum field theory, one can construct its perturbative expansion and (if the theory can be renormalised), for anything we want to calculate, this expansion will give us an infinite sequence of terms. Each of these terms has a graphical representation called a Feynman diagram, and these diagrams get more and more complicated as one goes to higher and higher order terms in the perturbative expansion. There will be some ... ‘coupling constant’ ... related to the strength of the interactions, and each time we go to the next higher order in the expansion, the terms pick up an extra factor of the coupling constant. For the expansion to be at all useful, the terms must get smaller and smaller fast enough ... Whether or not this happens will depend on the value of the coupling constant.’ - P. Woit, Not Even Wrong, Jonathan Cape, London, 2006, p. 182.


‘For the last eighteen years particle theory has been dominated by a single approach to the unification of the Standard Model interactions and quantum gravity. This line of thought has hardened into a new orthodoxy that postulates an unknown fundamental supersymmetric theory involving strings and other degrees of freedom with characteristic scale around the Planck length. ... It is a striking fact that there is absolutely no evidence whatsoever for this complex and unattractive conjectural theory. There is not even a serious proposal for what the dynamics of the fundamental ‘M-theory’ is supposed to be or any reason at all to believe that its dynamics would produce a vacuum state with the desired properties. The sole argument generally given to justify this picture of the world is that perturbative string theories have a massless spin two mode and thus could provide an explanation of gravity, if one ever managed to find an underlying theory for which perturbative string theory is the perturbative expansion.’ – P. Woit, Quantum Field Theory and Representation Theory: A Sketch (2002), pp51-52.


‘String theory has the remarkable property of predicting gravity.’ - E. Witten (M-theory originator), Physics Today, April 1996.


‘50 points for claiming you have a revolutionary theory but giving no concrete testable predictions.’ - J. Baez (crackpot Index originator), comment about crackpot mainstream string ‘theorists’ on the Not Even Wrong weblog here.


‘It has been said that more than 200 theories of gravitation have been put forward; but the most plausible of these have all had the defect that they lead nowhere and admit of no experimental test.’ - Sir Arthur Eddington, Space Time and Gravitation, Cambridge University Press, 1921, p64. (Here is a link to checkable quantum gravity framework which made published predictions in 1996 which were confirmed by observations in 1998, but censored out due to the immensely loud noise generators in vacuous string theory.).


Background information

Quantum field theory is the basis of the Standard Model of particle physics and is the best tested of all physical theories, more general in application and better tested within its range of application than the existing formulation of general relativity (which needs modification to include quantum field effects), describing all electromagnetic and nuclear phenomena. The Standard Model does not as yet include quantum gravity, so it is not a replacement yet for general relativity. However, the elements of quantum gravity may be obtained from an application of quantum field to a Penrose spin network model of spacetime (the path integral is the sum over all interaction graphs in the network, and this yields background independent general relativity). This approach, 'loop quantum gravity', is entirely different from that in string theory, which is based on building extra-dimensional speculation upon other speculations, e.g., the speculation that gravity is due to spin-2 gravitons (this is speculative is no experimental evidence for it). In loop quantum gravity, by contrast to string theory, the aim is merely to use quantum field theory to derive the framework of general relativity as simply as possible. Other problems in the Standard Model are related to understanding how electroweak symmetry is broken at low energy and how mass (gravitational charge) is acquired by some particles. There are several forms of speculated Higgs field which may rise to mass and electroweak symmetry breaking, but the details as yet unconfirmed by experiment (the Large Hadron Collider may do it). Moreover, there are questions about how the various parameters of the Standard Model are related, and the nature of fundamental particles (string theory is highly speculative, and there are other possibilities).


There are several excellent approaches to quantum field theory: at a popular level there is Wilczek’s 12-page discussion of Quantum Field Theory, Dyson’s Advanced Quantum Mechanics and the excellent approach by Alvarez-Gaume and Vazquez-Mozo, Introductory Lectures on Quantum Field Theory. A good mathematics compendium introducing, in a popular way, some of maths involved is Penrose's Road to Reality (Penrose's twistors inspired some concepts in an Electronics World article of April 2003). For a very brief (47 pages) yet more abstract or mathematical (formal) approach to quantum field theory, see for comparison Crewther’s http://arxiv.org/abs/hep-th/9505152. For a slightly more ‘stringy’-orientated approach, see Mark Srednicki’s 608 pages textbook, via http://www.physics.ucsb.edu/~mark/qft.html, and there is also Zee's Quantum Field Theory in a Nutshell on Amazon to buy if you want something briefer but with that mainstream speculation (stringy) outlook.


Ryder’s Quantum Field Theory also contains supersymmetry unification speculations and is available on Amazon here. Kaku has a book on the subject here, Weinberg has one here, Peskin and Schroeder's is here, while Einstein's scientific biographer, the physicist Pais, has a history of the subject here. Baez, Segal and Zhou have an algebraic quantum field theory approach available on http://math.ucr.edu/home/baez/bsz.html, while Dr Peter Woit has a link to handwritten quantum field theory lecture notes from Sidney Coleman's course which is widely recommended, here. For background on representation theory and the Standard Model see Woit's page here for maths background and also his detailed suggestion, http://arxiv.org/abs/hep-th/0206135. For some discussion of quantum field theory equations without the interaction picture, polarization, or renormalization of charges due to a physical basis in pair production cutoffs at suitable energy scales, see Dr Chris Oakley's page http://www.cgoakley.demon.co.uk/qft/:

‘... renormalization failed the "hand-waving" test dismally.

‘This is how it works. In the way that quantum field theory is done - even to this day - you get infinite answers for most physical quantities. Are we really saying that particle beams will interact infinitely strongly, producing an infinite number of secondary particles? Apparently not. We just apply some mathematical butchery to the integrals until we get the answer we want. As long as this butchery is systematic and consistent, whatever that means, then we can calculate regardless, and what do you know, we get fantastic agreement between theory and experiment for important measurable numbers (the anomalous magnetic moment of leptons and the Lamb shift in the Hydrogen atom), as well as all the simpler scattering amplitudes. ...

‘As long as I have known about it I have argued the case against renormalization. [What about the physical mechanism of virtual fermion polarization in the vacuum, which explains the case for a renormalization of charge since this electric polarization results in a radial electric field that opposes and hence shields most of the core charge of the real particle, and this shielding due to polarization occurs wherever there are pairs of charges that are free and have space to become aligned against the core electric field, i.e. in the shell of space around the particle core that extends in radius between a minimum radius equal to the grain size of the Dirac sea - i.e. the UV cutoff - and an outer radius of about 1 fm which is the range at which the electric field strength is Schwinger's threshold for pair-production (i.e. the IR cutoff)? This renormalization mechanism has some physical evidence in several experiments, e.g., Levine, Koltick et al., Physical Review Letters, v.78, no.3, p.424, 1997, where the observable electric charge of leptons does indeed increase as you get closer to the core, as seen in higher energy scatter experiments.] ...

‘[Due to Haag’s theorem] it is not possible to construct a Hamiltonian operator that treats an interacting field like a free one. Haag's theorem forbids us from applying the perturbation theory we learned in quantum mechanics to quantum field theory, a circumstance that very few are prepared to consider. Even now, the text-books on quantum field theory gleefully violate Haag's theorem on the grounds that they dare not contemplate the consequences of accepting it.

‘... The next paper I wrote, in 1986, follows this up. It takes my 1984 paper and adds two things: first, a direct solving of the equal-time commutators, and second, a physical interpretation wherein the interaction picture is rediscovered as an approximation.

‘With regard to the first thing, I doubt if this has been done before in the way I have done it3, but the conclusion is something that some may claim is obvious: namely that local field equations are a necessary result of fields commuting for spacelike intervals. Some call this causality, arguing that if fields did not behave in this way, then the order in which things happen would depend on one's (relativistic) frame of reference. It is certainly not too difficult to see the corollary: namely that if we start with local field equations, then the equal-time commutators are not inconsistent, whereas non-local field equations could well be. This seems fine, and the spin-statistics theorem is a useful consequence of the principle. But in fact this was not the answer I really wanted as local field equations lead to infinite amplitudes. It could be that local field equations with the terms put into normal order - which avoid these infinities - also solve the commutators, but if they do then there is probably a better argument to be found than the one I give in this paper. ...

‘With regard to the second thing, the matrix elements consist of transients plus contributions which survive for large time displacements. The latter turns out to be exactly that which would be obtained by Feynman graph analysis. I now know that - to some extent - I was just revisiting ground already explored by Källén and Stueckelberg4.

‘My third paper [published in Physica Scripta, v41, pp292-303, 1990] applies all of this to the specific case of quantum electrodynamics, replicating all scattering amplitudes up to tree level. ...

‘Unfortunately for me, though, most practitioners in the field appear not to be bothered about the inconsistencies in quantum field theory, and regard my solitary campaign against infinite subtractions at best as a humdrum tidying-up exercise and at worst a direct and personal threat to their livelihood. I admit to being taken aback by some of the reactions I have had. In the vast majority of cases, the issue is not even up for discussion.

‘The explanation for this opposition is perhaps to be found on the physics Nobel prize web site. The five prizes awarded for quantum field theory are all for work that is heavily dependent on renormalization. ...

‘Although by these awards the Swedish Academy is in my opinion endorsing shoddy science, I would say that, if anything, particle physicists have grown to accept renormalization more rather than less as the years have gone by. Not that they have solved the problem: it is just that they have given up trying. Some even seem to be proud of the fact, lauding the virtues of makeshift "effective" field theories that can be inserted into the infinitely-wide gap defined by infinity minus infinity. Nonetheless, almost all concede that things could be better, it is just that they consider that trying to improve the situation is ridiculously high-minded and idealistic. ...

‘The other area of uncertainty is, to my mind, the ‘strong’ nuclear force. The quark model works well as a classification tool. It also explains deep inelastic lepton-hadron scattering. The notion of quark "colour" further provides a possible explanation, inter alia, of the tendency for quarks to bunch together in groups of three, or in quark-antiquark pairs. It is clear that the force has to be strong to overcome electrostatic effects. Beyond that, it is less of an exact science. Quantum chromodynamics, the gauge theory of quark colour is the candidate theory of the binding force, but we are limited by the fact that bound states cannot be done satisfactorily with quantum field theory. The analogy of calculating atomic energy levels with quantum electrodynamics would be to calculate hadron masses with quantum chromodynamics, but the only technique available for doing this - lattice gauge theory - despite decades of work by many talented people and truly phenomenal amounts of computer power being thrown at the problem, seems not to be there yet, and even if it was, many, including myself, would be asking whether we have gained much insight through cracking this particular nut with such a heavy hammer.’

The humorous and super-intelligent (no joke intended) Professor Warren Siegel has an 885 pages long free textbook, Fields http://arxiv.org/abs/hep-th/9912205, the first chapters of which consist of a very nice introduction to the technical mathematical background of experimentally validated quantum field theory (it also has chapters on speculative supersymmetry and speculative string theory toward the end).

Gerard ’t Hooft has a brief (69 pages) review article, The Conceptual Basis of Quantum Field Theory, here, and Meinard Kuhlmann has an essay on it for the Stanford Encyclopedia of Philosophy here.

‘In loop quantum gravity, the basic idea is to use the standard methods of quantum theory, but to change the choice of fundamental variables that one is working with. It is well known among mathematicians that an alternative to thinking about geometry in terms of curvature fields at each point in a space is to instead think about the holonomy [whole rule] around loops in the space. The idea is that in a curved space, for any path that starts out somewhere and comes back to the same point (a loop), one can imagine moving along the path while carrying a set of vectors, and always keeping the new vectors parallel to older ones as one moves along. When one gets back to where one started and compares the vectors one has been carrying with the ones at the starting point, they will in general be related by a rotational transformation. This rotational transformation is called the holonomy of the loop. It can be calculated for any loop, so the holonomy of a curved space is an assignment of rotations to all loops in the space.’ - P. Woit, Not Even Wrong, Jonathan Cape, London, 2006, p189. (Emphasis added.)

‘Plainly, there are different approaches to the five fundamental problems in physics.’ – Lee Smolin, The Trouble with Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next, Houghton Mifflin, New York, 2006, p254.

The major problem today seems to be that general relativity is fitted to the big bang without applying corrections for quantum gravity which are important for relativistic recession of gravitational charges (masses): the redshift of gravity causing gauge boson radiation reduces the gravitational coupling constant G, weakening long range gravitational effects on cosmological distance scales (i.e., between rapidly receding masses). This mechanism for a lack of gravitational deceleration of the universe on large scales (high redshifts) has counterparts even in alternative push-gravity graviton ideas, where gravity - and generally curvature of spacetime - is due to shielding of gravitons (in that case, the mechanism is more complicated, but the effect still occurs).

Professor Carlo Rovelli’s Quantum Gravity is an excellent background text on loop quantum gravity, and is available in PDF format as an early draft version online at http://www.cpt.univ-mrs.fr/~rovelli/book.pdf and in the final published version from Amazon here. Professor Lee Smolin also has some excellent online lectures about loop quantum gravity at the Perimeter Institute site, here (you need to scroll down to 'Introduction to Quantum Gravity' in the left hand menu bar). Basically, Smolin explains that loop quantum gravity gets the Feynman path integral of quantum field theory by summing all interaction graphs of a Penrose spin network, which amounts to general relativity without a metric (i.e., background independent). Smolin also has an arXiv paper, An Invitation to Loop Quantum Gravity, here which contains a summary of the subject from the existing framework of mathematical theorems of special relevance to the more peripherial technical problems in quantum field theory and general relativity.

However, possibly the major future advantage of loop quantum gravity will be as a Yang-Mills quantum gravity framework, with the physical dynamics implied by gravity being caused by full cycles or complete loops of exchange radiation being exchanged between gravitational charges (masses) which are receding from one another as observed in the universe. There is a major difference between the chaotic space-time annihilation-creation massive loops which exist between the IR and UV cutoffs, i.e., within 1 fm distance from a particle core (due to chaotic loops of pair production/annihilation in quantum fields), and the more classical (general relativity and Maxwellian) force-causing exchange/vector radiation loops which occur outside the 1 fm range of the IR cutoff energy (i.e., at lower energy than the closest approach - by Coulomb scatter - of electrons in collisions with a kinetic energy similar to the rest mass-energy of the particles).

‘Light ... ‘smells’ the neighboring paths around it, and uses a small core of nearby space. (In the same way, a mirror has to have enough size to reflect normally: if the mirror is too small for the core of nearby paths, the light scatters in many directions, no matter where you put the mirror.)’ - R. P. Feynman, QED, Penguin, 1990, page 54.

‘When we look at photons on a large scale ... there are enough paths around the path of minimum time to reinforce each other, and enough other paths to cancel each other out. But when the space through which a photon moves becomes too small ... these rules fail ... The same situation exists with electrons: when seen on a large scale, they travel like particles, on definite paths. But on a small scale, such as inside an atom, the space is so small that there is no main path, no ‘orbit’; there are all sorts of ways the electron could go, each with an amplitude. The phenomenon of interference [due to pair-production of virtual fermions in the very strong electric fields (above the 1.3*1018 v/m Schwinger threshold electric field strength for pair-production) on small distance scales] becomes very important, and we have to sum the arrows to predict where an electron is likely to be.’- R. P. Feynman, QED, Penguin, 1990, page 84-5.

‘... the ‘inexorable laws of physics’ ... were never really there ... Newton could not predict the behaviour of three balls ... In retrospect we can see that the determinism of pre-quantum physics kept itself from ideological bankruptcy only by keeping the three balls of the pawnbroker apart.’ – Dr Tim Poston and Dr Ian Stewart, ‘Rubber Sheet Physics’ (science article) in Analog: Science Fiction/Science Fact, Vol. C1, No. 129, Davis Publications, New York, November 1981.


Feynman points out in that book QED that there is a simple physical explanation via Feynman diagrams and path integrals for why the mathematics of electron orbits and photon paths is classical on large scales and chaotic on small ones:

‘... when seen on a large scale, they [electrons, photons, etc.] travel like particles, on definite paths. But on a small scale, such as inside an atom, the space is so small that there is no main path, no ‘orbit’; there are all sorts of ways the electron could go, each with an amplitude. The phenomenon of interference [from quantum interactions, each represented by a Feynman diagram] becomes very important, and we have to sum the arrows [amplitudes] to predict where an electron is likely to be.’

- R. P. Feynman, QED, Penguin, 1990, page 84-5.

So according to Feynman, an electron inside the atom has a chaotic path which is physically the result of the small scale involved, which prevents individual virtual photon exchanges from statistically averaging out the way they do on large scales. For analogy, think of the different effects of the impacts of air molecules on a micron sized dust particle - i.e. chaotic Brownian motion - and on a football, where such large numbers of impacts [are] involved that they can be accurately represented by the classical approximation of 'air pressure'.

But Feynman uses integration (requiring non-quantized continuous variables) to average out the effects of these many paths or interaction histories, where strictly speaking he should be using discrete (sigma symbol) summation of all individual (quantum) interactions.

If you look at general relativity and quantum field theory (QFT), both represent fields using calculus: they both use differential equations describing continuous variables to represent fields which should strictly be sigma sums for the action in discrete interactions. This is why differential QFT leads to perturbative expansions with an infinite number of terms, each term corresponding to a Feynman diagram:

‘It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.’

- R. P. Feynman, The Character of Physical Law, BBC Books, 1965, pp. 57-8.

Maybe this effect is what Prof. John Baez was thinking about in his comment at http://www.math.columbia.edu/~woit/wordpress/?p=615.


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Solution to a problem with general relativity: A Yang-Mills mechanism for quantum field theory exchange-radiation dynamics, with prediction of gravitational strength, space-time curvature, Standard Model parameters for all forces and particle masses, and cosmology, including comparisons to other research and experimental tests


Acknowledgement


Professor Jacques Distler of the University of Texas inspired recent reformulations by suggesting in a comment on Professor Clifford V. Johnson’s discussion blog that I’d be taken more seriously if only I’d only use tensor analysis in discussing the mathematical physics of general relativity.


Part 1: Summary of experimental and theoretical evidence, and comparison of theories

Part 2: The mathematics and physics of general relativity [Currently this links to a paper by Drs. Baez and Bunn]

Part 3: Quantum gravity approaches: string theory and loop quantum gravity [Currently this links to Dr Rovelli's Quantum Gravity]

Part 4: Quantum mechanics, Dirac’s equation, spin and magnetic moments, pair-production, the polarization of the vacuum above the IR cutoff and it’s role in the renormalization of charge and mass [Currently this links to Dyson's QED introduction]

Part 5: The path integral of quantum electrodynamics, compared with Maxwell’s classical electrodynamics [Currently this links to Siegel's Fields, which covers a large area in depth, one gem for example is that it points out that the 'mass' of a quark is not a physical reality, firstly because quarks can't be isolated and secondly because the mass is due to the vacuum particles in the strong field surrounding the quarks anyway]

Part 6: Nuclear and particle physics, Yang-Mills theory, the Standard Model, and representation theory [Currently this links to Woit's very brief Sketch showing how simple low-dimensional modelling can deliver particle physics, which hopefully will turn into a more detailed, and also slower-paced, technical book very soon]

Part 7: Methodology of doing science: predictions and postdictions of the theory based purely on empirical facts (vacuum mechanism for mass and electroweak symmetry breaking at low energy, including Hans de Vries’ and Alejandro Rivero’s ‘coincidence’) [Currently this links to Alvarez-Gaume and Vazquez-Mozo, Introductory Lectures on Quantum Field Theory]

Part 8: Riofrio’s and Hunter’s equations, and Lunsford’s unification of electromagnetism and gravitation [Currently this links to Lunsford's paper]

Part 9: Standard Model mechanism: vacuum polarization and gauge boson field mediators for asymptotic freedom and force unification [Currently this links to Wilczek's brief summary paper]

Part 10: Evidence for the ‘stringy’ nature of fundamental particle cores? [Currently links to Dr Lubos Motl's list of 12 top superstring theory 'results', with literature references]




‘I like Feynman’s argument very much (although I have not thought about the virtual charges in the loops bit bit). The general idea that you start with a double slit in a mask, giving the usual interference by summing over the two paths... then drill more slits and so more paths... then just drill everything away... leaving only the slits... no mask. Great way of arriving at the path integral of QFT.’ - Prof. Clifford V. Johnson's comment, here


‘The world is not magic. The world follows patterns, obeys unbreakable rules. We never reach a point, in exploring our universe, where we reach an ineffable mystery and must give up on rational explanation; our world is comprehensible, it makes sense. I can’t imagine saying it better. There is no way of proving once and for all that the world is not magic; all we can do is point to an extraordinarily long and impressive list of formerly-mysterious things that we were ultimately able to make sense of. There’s every reason to believe that this streak of successes will continue, and no reason to believe it will end. If everyone understood this, the world would be a better place.’ – Prof. Sean Carroll, here


‘Part of the reason string theory makes no new predictions is that it appears to come in an infinite number of versions. ... With such a vast number of theories, there is little hope that we can identify an outcome of an experiment that would not be encompassed by one of them. Thus, no matter what the experiments show, string theory cannot be disproved. But the reverse also holds: No experiment will ever be able to prove it true. ... if string theorists are wrong, they can’t be just a little wrong. If the new dimensions and symmetries do not exist, then we will count string theorists among science’s greatest failures, like those who continued to work on Ptolemaic epicycles while Kepler and Galileo forged ahead. Theirs will be a cautionary tale of how not to do science, how not to let theoretical conjecture get so far beyond the limits of what can rationally be argued that one starts engaging in fantasy.’ - Professor Lee Smolin, The Trouble with Physics: The Rise of String Theory, the Fall of a Science and What Comes Next, Haughton Mifflin Company, New York, 2006, pp. xiv-xvii.


THE ROAD TO REALITY: A COMPREHENSIVE GUIDE TO THE LAWS OF THE UNIVERSE by Sir Roger Penrose, published by Jonathan Cape, London, 2004. The first half of the 1094 pages hardback book (2.5 inches/6.5 cm thick) briefly summarises fairly well known mathematics of background importance to the subject at issue. The remaining half of the book deals with quantum mechanics and attempts to unify it with general relativity. On page 785, Penrose neatly quotes his co-author Professor Stephen Hawking:


  • ‘I don’t demand that a theory correspond to reality because I don’t know what it is. Reality is not a quality you can test with litmus paper. All I’m concerned with is that the theory should predict the results of measurements.’ [Quoted from: Stephen Hawking in S. Hawking and R. Penrose, The Nature of Space and Time, Princeton University Press, Princeton, 1996, p. 121.]

  • But acidity is a reality which you can, indeed, test with litmus paper! On page 896, Penrose analyses those who use string ‘theory’ as an obfuscation (or worse) of the meaning of ‘prediction’:

    ‘In the words of Edward Witten [E. Witten, ‘Reflections on the Fate of Spacetime’, Physics Today, April 1996]:

  • ‘String theory has the remarkable property of predicting gravity,

  • ‘and Witten has further commented:

  • ‘the fact that gravity is a consequence of string theory is one of the greatest theoretical insights ever.

  • ‘It should be emphasised, however, that in addition to the dimensionality issue, the string theory approach is (so far, in almost all respects) restricted to being merely a perturbation theory …’


    Hence, string ‘theory’ as hyped up by genius Witten in 1996 as predicting gravity, is misleading, really. String ‘theory’ has no proof of a physical mechanism and predicts nothing checkable, not even the strength of gravity, unlike the causal mechanism! (In apt words of exclusion-principle proposer Wolfgang Pauli, string ‘theory’ is in the class of belief junk, ‘not even wrong’.)


    On page 1020 of chapter 34 ‘Where lies the road to reality?’, 34.4 Can a wrong theory be experimentally refuted?, Penrose says: ‘One might have thought that there is no real danger here, because if the direction is wrong then the experiment would disprove it, so that some new direction would be forced upon us. This is the traditional picture of how science progresses. Indeed, the well-known philosopher of science [Sir] Karl Popper provided a reasonable-looking criterion [K. Popper, The Logic of Scientific Discovery, 1934] for the scientific admissability [sic; mind your spelling Sir Penrose or you will be dismissed as a loony: the correct spelling is admissibility] of a proposed theory, namely that it be observationally refutable. But I fear that this is too stringent a criterion, and definitely too idealistic a view of science in this modern world of "big science".’


    Penrose identifies the problem clearly on page 1021: ‘We see that it is not so easy to dislodge a popular theoretical idea through the traditional scientific method of crucial experimentation, even if that idea happened actually to be wrong. The huge expense of high-energy experiments, also, makes it considerably harder to test a theory than it might have been otherwise. There are many other theoretical proposals, in particle physics, where predicted particles have mass-energies that are far too high for any serious possibility of refutation.’


    On page 1026, Penrose points out: ‘In the present climate of fundamental research, it would appear to be much harder for individuals to make substantial progress than it had been in Einstein’s day. Teamwork, massive computer calculations, the pursuing of fashionable ideas – these are the activities that we tend to see in current research. Can we expect to see the needed fundamentally new perspectives coming out of such activities? This remains to be seen, but I am left somewhat doubtful about it. Perhaps if the new directions can be more experimentally driven, as was the case with quantum mechanics in the first third of the 20th century, then such a "many-person" approach might work.’


    ‘Science is the belief in the ignorance of [the speculative consensus of] experts.’ - R. P. Feynman, The Pleasure of Finding Things Out, 1999, p187.


    Introduction

    General relativity

    Quantum gravity

    Dirac’s equation

    Path integrals

    Representation theory

    Methodology

    Unification

    Standard Model

    Evidence for strings?

    Last Updated: 28 October 2007. If your computer doesn't have the maths symbols and Greek fonts available for the equations to display properly, see this PDF file.