Comments about the unified theory of gravitation by Mikko Partanen and Jukka Tulkki

Esa Sakkinen set to me link to an article by Mikko Partanen and Jukka Tulkki proposing a unified theory of gravity (see this). The amusing coincidence is that Jukka Tulkki was an assistant in the Physics Laboratory of the Technological University where I made my thesis more than 4 decades ago. We had nice discussions.

In contrast to the most “breakthrough papers” in this field appearing with rate one per week, this article was coherent and well-written and did not contain the fatal error at the first page so that I decided to try to understand what is said in the article.

The key concepts of the unified theory of gravitation are “8-spinor”, “space-time dimension field”, and “gravity field” as analog of U(2) gauge potential with the labels of components corresponding to space-time coordinates. The claim was that this gives a renormalizable theory. This might be the case, but I dare to be skeptical about the claim that it is a theory of gravitation. 1. Basic notions

1.1 The notion of 8-spinor

1. At the first look, the 8-spinor has nothing to do with spinors as they are usually defined. The components can be, for example, of electromagnetic (em) field or vector potential or even space-time coordinates and some components are identically zero. Normally spinors have definite transformation properties in coordinate transformations and symmetries. Therefore spinors in the standard sense of the word are not in question.

I would be cautious and speak just of an array of 8 numbers with a physical interpretation. What is non-trivial is that Maxwell action can be represented as a bilinear ΨΨ of this 8-spinor.

In physics the spinor is much more than an array of numbers and the notion of spinor structure is a delicate concept: already in relativity. For instance, the spinor structure does not exist for all space-time topologies. This has a key role in TGD.

In the theoretical vision of TGD, octonions and quaternions are central and octospinors emerge (see this). In this framework, the notion of octovector and octospinor could make sense as will be found later. Octovectors, octospinors and their conjugates as representations of SO(1,7) form a kind of holy trinity by triality symmetry.

The U(1) invariance as gauge invariance of Maxwell’s em field holds true for each component of the 8-spinor formed from the em field. The components of 8-spinor formed from a vector potential suffer a non-trivial gauge transformation. This is not the case in non-abelian gauge theory and the 8-spinor components are Lie algebra-valued. This U(1)⁸ symmetry for spinors looks strange to me. In octonionic interpretation the multiplication with quaternions and octonions would define analogs of U(2) gauge transformations.

1.2 The notion of space-time dimension field

The notion of space-time dimension field is new.

1. It is stated that it is not a physical field. The defining equations say that it is a covariant constant. It has a spacetime index “a” labelling space-time dimensions. Its 4 components are exponentiation for each kernel matrix t^a with coefficient X_a depending on the location.
2. The kernel generators t^a are 8×8 matrices, which satisfy commutation relations of quaternionic units and realize U(2) algebra, which is the same as electroweak algebra. The connection with quaternions was not noticed. The exponent of exp(iX_at^a) could be interpreted as a local U(1) gauge transformation generated by t^a. For some reason, more general exponents defining U(2) gauge transformation were not considered. It is claimed that the t^a‘s correspond to the generators of the electroweak gauge group.

In TGD, quaternionic and octonionic structures are central and quaternionization for both M⁴ and CP₂ take place. t^a would correspond to quaternion units and their action on octonions is by multiplication. 8-spinors could correspond to octospinors or octovectors.

2. What calculations were done?

QED and gravitation in lowest order was considered but I do not think that this had much to do with the claimed unified theory of gravitation. I understand that this was essentially QED with coupling to the gravitational field described in standard way in the lowest order. Gravitation was brought in via a connection and claimed to be consistent with general relativity.

3. What unified theory of gravity could mean?

To my view, the second part of the article was the new thing.

1. The non-trivial claim was that quantized gravity is describable as a gauge theory using a compact, finite-dimensional gauge group U(2). Gravitational theories in which the gauge group is the Lorentz group SO(1,3) have been proposed. SO(1,3) finite-dimensional but not compact and this leads to problems. It has been also proposed that gravity is a gauge theory of translation group: also now the gauge group would be non-compact.

This creates questions: How would the quaternionic automorphism group U(2) of quaternions act as a gauge group and produce a theory of gravitation? What happens to the general coordinate invariance? If GCI is present, what happens to the Poincare symmetry? The group U(2) is not the Lorentz group. Does one really obtain the metric theory of gravitation? It was claimed that this is the case.

Consider now a possible number theoretic interpretation in terms of octonions and quaternions.

1. There are two indices: the indices a for the t^a and the indices μ for the space-time coordinates. “a” would correspond to the vierbein indices in general relativity. The four quaternionic units as labels of gauge Lie algebra generators would correspond to the labels vielbein vectors in spacetime.
2. One way to proceed is to assume that space-time allows quaternionic structure and quaternion units allow to define the analog of vielbein. The U(2) gauge group would act as quaternionic multiplication on octonionic 8-vectors and 8-spinos: they were proposed to be 8-spinors but in the very weird sense in which they actually are not spinors.
3. Could electric and magnetic fields allow an interpretation as octo-vectors? This would explain why the octospinors have identically vanishing components: they would correspond to octonionic and quaternionic units vanishing for 3-vectors.
4. It is argued that this gives a gauge theory of gravity in which the kernel generators t^a, identifiable as quaternion units in TGD, correspond to the generators of the gauge group U(2). This would guarantee renormalizability. It can do that, but it is difficult to see how the emerging theory could describe gravitation.

4. Some critical comments

At least following critical comments can be raised.

1. Is the space-time dimension field really needed?
2. The gravitational field was identified as a gauge potential in non-Abelian U(2). However, the gauge transformations and covariant derivatives were defined as if t^a would generate the Abelian Lie algebra U(1)⁴. This I fail to understand. I would allow the quaternionic action as multiplication as symmetries.
3. An action density for the gravitational field was introduced. The proposed action was instanton density E*B rather than E²-B². The inner product E*B is a total divergence and it cannot serve as an action.

To sum up, octonions and quaternions could make it possible to formulate the notion of 8-spinor mathematically. Octovectors, octospinors and their conjugates would form a triad. If one transforms tensor indices to vielbein indices, one can associate octovectors with various vectors and antisymmetric tensors having only spatial indices.

5. Analogies with number theoretic vision of TGD

In the number theoretic vision of TGD, quaternionic and octonionic structures play a central role.

1. M⁸-H, where one has H=M⁴×CP₂ duality as analog of momentum-position duality means that M⁸ has interpretation in terms of octonions with Minkowski scalar product defined as Re(o₁o₂). At the level of M⁸ 4-D associative surfaces are the counterparts of space-time surfaces in H.
2. Complexified octonionic spinors are 8-component spinors and quaternionic spinors are an associative 4-D subspace of them. The octo-spinor property gives rise to spin and electroweak spin. These have a central role in the twistorialization of TGD. The non-associativity of course brings some delicacies and Associativity is proposed to be the dynamic principle of number theoretic TGD defined in octonionic M⁸. Octonionic automorphism group G₂ and its subgroup SU(3) identified as color group in TGD play a key role in TGD and quaternion multiplication corresponds to the action of electroweak U(2).
3. What happens to the Lorentz symmetry violated by the replacement of the Lorentz group with U(2)? This was also a longstanding problem of M⁸-H duality but is now solved. The first wrong guess was that the surface Y⁴ in M⁸ is quaternionic and has Minkowskian signature. The guess was wrong. The surfaces Y² in M⁸ have Euclidean number theoretic induced metric and their normal spaces are quaternionic and Minkowskian. This is also required by the associativity as a dynamic principle

For the almost most recent summary of TGD see this and this.

For the holography= holomorphy vision and its relationship to an analog of Langlands duality relating geometric and number theoretic visions of TGD see this and this.

The most recent summary of twistorialization of TGD in both H and M⁸ can be found at here.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/05/comments-about-unified-theory-of.html