Witten about mass gap
Witten has a nice talk about mass gap problem in 3-D (mostly) and 4-D gauge theories demonstrating how enormous his understanding and knowledge about mathematical physics is. Both Peter Woit and Kea have commented it.
In 3-D case the coupling strength g2 has the dimension of inverse length and therefore it would not be surprising if mass gap would emerge. Witten argues that by adding to the theory a Chern-Simons term the theory could reduce in long length scales to non-trivial topological QFT at the IR limit. This would be also a nice manner to resolve the IR difficulties of 3-D gauge theories. Could one imagine effective reduction to topological QFT in long length scales also in 4-D case as a solution to IR divergences?
In D=4 the situation is much more difficult since the gauge coupling is dimensionless. My un-educated is opinion is that the proper question is whether the theory actually exists mathematically and my equally un-educated guess is that it does not - unless one brings in the mass scale somehow by hand. The standard Muenchausen trick to bring the scales in perturbation theory is via UV and IR cutoffs. This is going outside what one means with gauge theory strictly mathematically. In order to make progress, one must bring in the new physics and mathematics. A rigorous mathematical formulation of 4-D gauge theory is not enough: it simply does not exist since something very important is missing.
TGD view about the mass gap problem
TGD is one proposal for what this new physics and mathematics could be. I do not try to re-explain in any detail what this new physics and mathematics might be since I have done this explaining for 6 years in this blog. The basic statement is however that the fundamental UV length scale must be present explicitly in the definition of the theory and must have concrete geometric interpretation rather than being a dimensional number like string tension. In TGD framework it corresponds to the "radius" of CP2, which is fixed from simple symmetry arguments as the only possible choice. This scale is not an outcome of some conceptually highly questionable procedure like spontaneous compactification, which has paralyzed theoretical physics for more than two decades and led to the landscape problem and the proposal to bring anthropic principle to physics - something extremely uninviting for anyone who has spent few minutes by trying to understand what one can say about consciousness as a physicist and mathematician.
To my rebellionary view the mathematics of standard gauge theories is not enough.
- Quite a far reaching generalization is needed besides the replacement of the recent view about space-time with the identification of space-times as 4-surfaces.
- The usual positive energy ontology having its roots in Newtonian mechanics based on absolute time (Hamiltonian approach especially) must be replaced with zero energy ontology which is natural in the relativistic context.
- A further generalization is number theoretical universality requiring that the physics in different number fields must be unified to single coherent whole.
- In some of the latest postings I have explained how the number theoretical universality would be realized in terms of quantum arithmetics - something also missing from ordinary gauge theory but for the existence of which there exist indications (quantum groups, inclusions of hyper-finite factors, and Shnoll effect on experimental side). In particular, the size scales of CDs coming as powers of 2 correspond to p=2 quantum arithmetics, which is very special in the sense that for odd integers it is just the ordinary arithmetics. For other primes p the corresponding p-adic length scale is in preferred position since the states with mass squared proportional to p as almost massless states giving an almost pole to the propagator which is given in terms of M2 momentum. I dare to hope that these observations finally answer the question why p-adic primes near powers of two are favored physically.
Further comments about mass gap
I cannot avoid the temptation to represent some further comments related to how the mass gap - or rather a hierarchy of mass gaps defined by p-adic mass scales in turn expressible in terms of p-adic prime and the fundamental mass scale defined by CP2 mass - emerges in TGD.
- One of the surprises of zero energy ontology was that that all - including those associated with virtual particles - braid strands carrying fermion numbers are on massless on mass shell states with possibly negative sign of energy so that wormhole contact can have space-like virtual net momentum. This leads to extremely powerful restrictions of loops integrals and guarantees finiteness and with certain additional natural assumptions deriving from ZEO the number of contributing diagrams is finite (discussed in recent postings: see this and this), and therefore guarantees algebraic universality (sum of infinite number of rationals (rational functions) need not be rational (rational function)!).
- Quite recently I have learned finally to accept that for generalized Feynman diagrams the presence of preferred M2⊂M4 having interpretation in terms of quantization axis of energy and spin is unavoidable (see for instance this). Of course, also propagators for on mass shell massless states are literally infinite unless one restricts the momentum in propagator to its M2 projection. There is integral over different choices of M2 so that Poincare invariance is not lost. Also number theoretic vision forces M2 with an interpretation as commutative subspace of complexified octonions. The last posting about Very Special Relativity and TGD gave one additional justification M2.
- Witten talks about 3-D gauge theories with emphasis on Chern-Simons term and the idea that in long length scales one obtains a non-trivial topological QFT for non-trivial mass gap. In TGD framework effective 2-dimensionality - or strong form of holography - follows from strong form of General Coordinate Invariance and for preferred extremals of Kähler action the action reduces to Chern-Simons terms if weak form of electric-magnetic duality holds true at the space-like 3-D surfaces at the ends of space-time sheet and at wormhole throats.
The special features of light-like 3-surfaces and boundaries of CDs is that they allow an extension of 2-D conformal invariance by their metric 2-dimensionality: this actually raises 4-D space-time and 4-D Minkowski space in completely unique position mathematically. An extremely simple and profound discovery, whose communication has turned out to be impossible- I think that even my cat is able to understand its significance-: what is wrong with these bright-minded colleagues in their academies;-)? An interesting question raised by Witten's talk is whether also TGD as almost topological TGD in some sense reduces to topological QFT at long length scales for given p-adic length scale. Exponential decrease of correlation function as function of distance might imply this but what happens on light-like boundaries of CDs?
- There are also open questions. For instance, should one assign different M2 to each sub-CD of CD or to each propagator line connecting the 3-vertices? One can be even more general and also consider a local choices of M2 defined by an integrable distribution of M2⊂ M4 defining the analog of string world sheet.
The special role of M2⊂M4 in relation to mass gap
The special role of M2⊂M4 in the construction of generalized Feynman diagrams deserves additional comments.
- What is remarkable that gauge conditions generalize in the sense that it is M2 momentum that appears in gauge conditions so that also the third polarization for gauge bosons creeps into the spectrum and even photon, gluons, and gravitons would receive small mass given in terms of IR cutoff given by the largest causal diamond in the hierarchy of causal diamonds defining the experimental range of length scales about which experimentalist can gain information. This is of course extremely natural from the view point of experimentalist.
- Physical particles are bound states of massless states with parallel M2 momenta assigned with the wormhole throats of the same wormhole contact. Also this bring in the overall important IR cutoff - and thus mass gap - not present in gauge theories. UV cutoff given by the size of the smallest CD gives UV cutoff. As already mentioned, there is no breaking of Poincare invariance.
- The highly non-trivial question is how the p-adic mass calculations can be consistent with the massless of braid strands. How can wormhole throat satisfy stringy mass formula if it is massless? One of the latest realizations is that that it is not the full M4mass squared but the longitudinal M2 mass squared, which is quantized by the stringy mass formula! The modified Dirac equation indeed strongly suggests that M2 momenta have integer valued components. I could not however decide whether only hyper-complex primes should be accepted: it now seems that integers coming as multiple of given hyper-complex integer, whose modulus square is prime, must be allowed. Particles would get longitudinal mass squared by p-adic thermodynamics and this mass would be the observed mass. Mass gap again but only in longitudinal degrees of freedom
- There is also experimental support for the necessity of introducing M2. In QCD one characterizes partons with M2 momentum and this again brings to gauge theory as a purely mathematical construct - something which really is not there! The great experimental question is whether Higgs exists or not. In TGD Higgs mechanism is replaced by a microscopic mechanism based on p-adic thermodynamics and identification of the mass squared as longitudinal mass squared (in Lorentz invariance manner since one averages over different M2:s). The natural prediction is that instead of Higgs there is entire M89 hadron physics to be discovered. If Higgs really is there as some bloggers have already revealed to us;-), profound re-interpretation of TGD is necessary.
New physics in non-perturbative sector
Asymptotically free gauge theories can handle the UV divergences by using renormalization group approach bringing in a scale analogous to QCD Lambda. Lambda defines the IR scale identified as the length scale associated with hadronization and confinement. One gets rid of UV scale altogether (but not from the mathematically tedious and ugly procedures removing UV infinities). In perturbative gauge theories IR remains however a source of difficulties since one really does not know how to calculate anything since the proposed expression for IR scale is non-analytic function of coupling constant strength (expressible in terms of exp(-8π2ℏ/g2), I hope I remember correctly). Also twistor approach is plagued by IR divergences. These difficulties are of course the reason for arranging a conference about mass gap problem! I do not however believe that the mass gap is a mathematical problem within the framework of 4-D gauge theories. One must go outside the system.
In TGD framework magnetic flux tubes are the concrete classical space-time correlate for non-perturbative aspects of quantum theory and appear in all applications from primordial cosmology to biology to elementary particle physics. They are not present in gauge theories. They are obtained as deformations of what I call cosmic strings, which are Cartesian products of string world sheets in M2 with 2-D complex sub-manifolds of CP2. In this case one cannot anymore speak about space-time as a small deformation of Minkowski space. The quantized size scale of the complex sub-manifolds brings in the scale via string tension. Wormhole contacts themselves are magnetic monopoles and thus homologically non-trivial surfaces of CP2 with quantized area so that again the fundamental mass scale creeps in. Note that the Kähler action for the magnetic flux tubes and also for deformations CP2 vacuum extremals contains a power of exp(-8π2ℏ/g2) giving rise to non-analyticity in g. In gauge theories classical action for instantons would give rise kind of factor.
Note: The address of my homepage has changed and the links to my homepage from the earlier postings will fail. The cure of the problem is the replacement of tgd.wippiespace or tgdq.wippiespace in the address with tgdtheory.