How the QFT-GRT limit of TGD differs from QFT and GRT?
Yesterday evening I got an intereting idea related to both the definition and conservation of gauge charges in non-Abelian theories. First the idea popped in QCD context but immediately generalized to electro-weak and gravitational sectors. It might not be entirely correct: I have not yet checked the calculations.
I have been working with possible TGD counterparts of so called chiral magnetic effect (CME) and chiral separation effect (CSE) proposed in QCD to describe observations at LHC and RHIC suggesting relatively large P and CP violations in hadronic physics associated with the deconfinement phase transition. See the recent article About parity violation in hadron physics).
The QCD based model for CME and CSE is not convincing as such. The model assumes that the theta parameter of QCD is non-vanishing and position dependent. It is however known that theta parameter is extremal small and seems to be zero: this is so called strong CP problem of QCD caused by the possibility of istantons. The axion hypothesis could make θ(x) a dynamical field and θ parameter would be eliminated from the theory. Axion has not however been however found: various candidates have been gradually eliminated from consideration!
What is the situation in TGD? In TGD instantons are impossible at the fundamental space-time level. This is due to the induced space-time concept. What this means for the QFT limit of TGD?
- Obviously one must add to the action density a constraint term equal to Lagrange multiple θ times instanton density. If θ is constant the variation with respect to it gives just the vanishing of instanton number.
- A stronger condition is local and states that instanton density vanishes. This differs from the axion option in that there is no kinetic term for θ so that it does not propagate and does not appear in propagators.
- The variation with respect to θ(x) gives the condition that instanton density rather than only instanton number vanishes for the allowed field configurations. This guarantees that axial current having instanton term as divergence is conserved if fermions are massless. There is no breaking of chiral symmetry at the massless limit and no chiral anomaly which is mathematically problematic.
- The field equations are however changed. The field equations reduce to the statement that the covariant divergence of YM current - sum of bosonic and fermionic contributions - equals to the covariant divergence of color current associated with the constraint term. The classical gauge potentials are affected by this source term and they in turn affect fermionic dynamics via Dirac equation. Therefore also the perturbation theory is affected.
- The following is however still uncertain: This term seems to have vanishing ordinary total divergence by Bianchi identities - one has topological color current proportional to the contraction of the gradient of θ and gauge field with 4-D permutation symbol! I have however not checked yet the details.
If this is really true then the sum of fermionic and bosonic gauge currents not conserved in the usual sense equals to a opological color current conserved in the usual sense! This would give conserved total color charges as topological charges - in spirit with "Topological" in TGD! This would also solve a problem of non-abelian gauge theories usually put under the rug: the gauge total gauge current is not conserved and a rigorous definition of gauge charges is lost.
- What the equations of motion of ordinary QCD would mean in this framework? First of all the color magnetic and electric fields can be said to be orthogonal with respect to the natural inner product. One can have also solutions for which θ is constant. This case gives just the ordinary QCD but without instantons and strong CP breaking. The total color current vanishes and one would have local color confinement classically! This is true irrespective of whether the ordinary divergence of color currents vanishes.
- This also allows to understand CME and CSE believed to occur in the deconfinement phase transition. Now regions with non-constant θ(x) but vanishing instanton density are generated. The sum of the conserved color charges for these regions - droplets of quark-gluon plasma - however vanish by the conservation of color charges. One would indeed have non-vanishing local color charge densities and deconfinement in accordance with the physical intuition and experimental evidence. This could occur in proton-nucleon and nucleon-nucleon collisions at both RHIC and LHC and give rise to CME and CSE effects. This picture is however essentially TGD based. QCD in standard form does not give it and in QCD there are no motivations to demand that instanton density vanishes.
The analog of θ (x) is present also at the QFT limit of TGD in electroweak sector since instantons must be absent also now. One would have conserved total electroweak currents - also Abelian U(1) current reducing to topological currents, which vanish for θ(x)= constant but are non-vanishing otherwise. In TGD the conservation of em charge and possibly also Z0 charge is understood if strong form of holography (SH) is accepted: it implies that only electromagnetic and possibly also Z0 current are conserved and are assignable to the string world sheets carrying fermions. At QFT limit one would obtain reduction of electroweak currents to topological currents if the above argument is correct. The proper understanding of W currents at fundamental level is however still lacking.
It is now however not necessary to demand the vanishing of instanton term for the U(1) factor and chiral anomaly for pion suggest that one cannot demand this. Also the TGD inspired model for so called leptohadrons is based on the non-vanishing elecromagnetic instanton density. In TGD also M4 Kähler form J(CD) is present and same would apply to it. If one applies the condition empty Minkowski space ceases to be an extremal.
Could this generalize also the GRT limit of TGD? In GRT momentum conservation is lost - this one of the basic problems of GRT put under the rug. At fundamental level Poincare charges are conserved in TGD by the hypothesis that the space-time is 4-surface in M4 × CP2. Space-time symmetries are lifted to those of M4.
What happens at the GRT limit of TGD? The proposal has been that covariant conservation of energy momentum tensor is a remnant of Poincare symmetry. But could one obtain also now ordinary conservation of 4- momentum currents by adding to the standard Einstein-YM action a Lagrange multiplier term guaranteing that the gravitational analog of instanton term vanishes?
- First objection: This makes sense only if vier-bein is defined in the M4 coordinates applying only at GRT limit for which space-time surface is representable as a graph of a map from M4 to CP2.
- Second objection: If metric tensor is regarded as a primary dynamical variable, one obtains a current which is symmetry 2-tensor like T and G. This cannot give rise to a conserved charges.
- Third objection: Taking vielbein vectors eAμ as fundamental variable could give rise to a conserved vector with vanishing covariant divergence. Could this give rise to conserved currents labelled by A and having interpretation as momentum components? This does not work. Since eAμ is only covariantly constant one does not obtain genuine conservation law except at the limit of empty Minkowski space since in this case vielbein vectors can be taken to be constant.
- Curvature tensor is indeed essentially a gauge field in tangent space rotation group when contracted suitably by two vielbein vectors eAμ and the instanton term is formally completely analogous to that in gauge theory.
- The situation is now more complex than in gauge theories due to the fact that second derivatives of the metric and - as it seems - also of vielbein vectors are involved. They however appear linearly and do not give third order derivatives in Einstein's equations. Since the physics should not depend on whether one uses metric or vielbein as dynamical variables, the conjecture is that the variation states that the contraction of T-kG with vielbein vector equals to the topological current coming from instanton term and proportional to the gradient of θ
(T-kG)μν eAν =jAμ.
The conserved current jAμ would be contraction of the instanton term with respect to eAμ with the gradient of θ suitably covariantized. The variation of the action with respect to the the gradient of eAμ would give it. The resulting current has only vanishing covariant divergence to which vielbein contributes.
- The covariantly conserved energy momentum current would be sum of parts corresponding to matter and gravitational field unlike in GRT where the field equations say that the energy momentum tensors of gravitational field and matter field are identical. This conforms with TGD view at the level of many-sheeted space-time.
- In GRT one has the problem that in absence of matter (pure gravitational radiation) one obtains G=0 and thus vacuum solution. This follows also from conformal invariance for solutions representing gravitational radiation. Thanks to LIGO we however now know that gravitational radiation carries energy! Situation for TGD limit would be different: at QFT limit one can have classical gravitational radiation with non-vanishing energy momentum density
thanks the vanishing of instanton term.
See the article About parity violation in hadron physics
For background see the chapters New Physics Predicted by TGD: Part I.
For a summary of earlier postings see Latest progress in TGD.