Sunday, March 22, 2015

Second quantisation of Kähler-Dirac action

Second quantization of Kähler-Dirac action is crucial for the construction of the Kähler metric of world of classical worlds as anticommutators of gamma matrices identified as super-symplectic Noether charges. To get a unique result, the anticommutation relations must be fixed uniquely. This has turned out to be far from trivial.

The canonical manner to second quantize fermions identifies spinorial canonical momentum densities and their conjugates as Πbar= ∂ LKD/∂Ψ= ΨbarΓt and their conjugates. The vanishing of Kähler-Dirac gamma matrix Γt at points, where the induced Kähler form J vanishes can cause problems since anti-commutation relations are not internally consistent anymore. This led me to give up the canonical quantization and to consider various alternatives consistent with the possibility that J vanishes. They were admittedly somewhat ad hoc. Correct (anti-)commutation relations for various fermionic Noether currents seem however to fix the anti-commutation relations to the standard ones. It seems that it is better to be conservative: the canonical method is heavily tested and turns out to work quite nicely.

Consider first the 4-D situation without the localization to 2-D string world sheets. The canonical anti-commutation relations would state {Πbar, Ψ}= δ3(x,y) at the space-like boundaries of the string world sheet at either boundary of CD. At points where J and thus K-D gamma matrix ΓTt vanishes, canonical momentum density vanishes identically and the equation seems to be inconsistent.

If fermions are localized at string world sheets assumed to always carry a non-vanishing J at their boundaries at the ends of space-time surfaces, the situation changes since Γt is non-vanishing. The localization to string world sheets, which are not vacua saves the situation. The problem is that the limit when string approaches vacuum could be very singular and discontinuous. In the case of elementary particle strings are associated with flux tubes carrying monopole fluxes so that the problem disappears.

It is better to formulate the anti-commutation relations for the modes of the induced spinor field. By starting from

{Πbar (x),Ψ (y)}=δ1(x,y)

and contracting with Ψ(x) and Π (y) and integrating, one obtains using orthonormality of the modes of Ψ the result

{bm,bn} = γ0 δm,n

holding for the modes with non-vanishing norm. At the limit J→ 0 there are no modes with non-vanishing norm so that one avoids the conflict between the two sides of the equation.

Quantum deformation introducing braid statistics is of considerable interest. Quantum deformations are essentially 2-D phenomenon, and the condition that it indeed occurs gives a further strong support for the localization of spinors at string world sheets. If the existence of anyonic phases is taken completely seriously, it supports the existence of the hierarchy of Planck constants and TGD view about dark matter. Note that the localization also at partonic 2-surfaces cannot be excluded yet.

I have wondered whether quantum deformation could relate to the hierarchy of Planck constants in the sense that n=heff/h corresponds to the value of deformation parameter q=exp(i2π/n). The quantum deformed anti-commutation relations

bb+q-1bb= q-N

are obtained by posing the constraints that the eigenvalues of bb and bb are Nq (1-N)q. Here N=,1 is the number of fermions in the mode (see this). The modification to the recent case is obvious.


Anonymous said...

Matti, can you talk about "Huygens principle" in TGD?

I see this post from

which got me all excited because of

"We analyze the implications of the violations of the strong Huygens principle in the transmission of information from the early universe to the current era via massless fields. We show that much more information reaches us through timelike channels (not mediated by real photons) than it is carried by rays of light, which are usually regarded as the only carriers of information. "

a totally new kind of "two-way radio?" ? :)

--Stephen (that was me that asked about Markov... yes)

Matti, why has someone not consulted you about LHC happenings? Well, I guess the machine is pretty boring, massive and noisy anyway... who cares said...

Dear Stephen,

I answered the question about Huygens in blog post since the answer was too long to serve as a comment. To my opinion time like channels provide also information in the sense that ordinary stable and massive particles propagate along them. Difficult to say how significant this information

I learned from some source that wave equations in curved background indeed predict that signals propagate classically inside future light-cone rather than alone its boundary: this can be understood perturbation theoretically by treating the deviation from flat metric as perturbation serving as a source of secondary waves. said...

Dear Stephen,

concerning your question about LHC and missing consultations. CERN is an institution, a very big institution, and it is well known that the intelligence of of organism/institution is inversely proportional to its size - dinosaur effect. Academic researchers are willing to use their brains only if they do not get research money. CERN and also the academic institutions have a clearly formulated policy concerning academic dissidents: no contacts with those who are academic outsiders. Pretend that they do not exist.

I am certainly such an outsider. Even worse: I am danger for the respectability of the academic community since I have been working for almost forty years without a single coin of research money, and as it is now clear, the work has been highly successful while the entire GUT-super-symmetry-super string paradigm has suffered monumental and humiliating failure. The only manner to deal with this kind of harmful individuals is zombieing. One finnish colleague formulated this policity in layman terms: finnish physicists would not touch to my work even with a long stick.