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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.
‘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.
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:
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]:
‘and Witten has further commented:
‘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.
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