Conformal structure – until the puppy grows up

Posted on 2025/12/07 by arkajad

On my other blog, I promised Anna—the ever‑curious Reader—that I would finally tackle the subject of conformal structure. So here it begins: a gentle introduction to a concept that admits many gateways. Precise definitions abound online, of course, and one can now even interrogate various AIs for insight—pressing them for clarity until they either enlighten or malfunction.

Note. Just two days ago, my wife managed to break Grok while doing exactly that—pressing it for truth like a determined philosopher, and then teasingly comparing it to a silly puppy. In response, Grok launched into a hundred‑page monologue centered around the immortal refrain: “And until the puppy grows up.” At some point it even decided to drop the puppy—and the grammar—altogether, producing “And until grows up.” Then back again. Truly a conformally invariant meltdown, stretching endlessly but changing shape.

Anyway, I have chosen my own way. Some basic linear and multilinear algebra will be expected of the Reader, but beyond that, I shall aim to remain gentle. This will become a series of (short) posts on the topic. How long will it last? I have set no limits—after all, in the realm of conformal geometry, infinity itself is permissible (and sometimes even polite enough to fit in a finite patch).

Math starts here

Scalar product (or metric)
Let V be a real n-dimensional vector space equipped with a nondegenerate symmetric bilinear form g of signature (p,q), p+q=n. .Thus, if e_i is a basis in V, and if x,y are two vectors in V:

    \[x=x^ie_i,\, y=y^ie_i,\]

we have the scalar product

    \[(x,y)=g_{ij}x^iy^j,\]

where we have Einstein summation convention.
Since we assume nondegeneracy, the matrix g_{ij} has an inverse, denoted g^{ij}.

Dual space
Let V^* denote the dual space, that is the space of linear forms on V. Then a basis e_i in V determines a dual basis e^i in V^*, defined by

    \[e^i(e_j)=\delta^i_j,\]

where \delta^i_j is the Kronecker delta.

Exercise. With the notation as above, prove that x^i=e^i(x).

Change of basis
Let  \GL (n) denote the group of all invertible real n\times n matrices. Then \GL (n) acts on the set of all bases from the right: if e_i is a basis, and if A=({A^i})_j is in  \GL (n), then e'_i=e_j {A^j}_i is another basis, where we use the Einstein summation convention.  We may then write e'=eA. Every basis can be obtained in this way from any other basis by a unique matrix A.

Exercise. Prove that if x=x^ie_i=x'^i e'_i, then

    \[x'^i={{A^{-1}}^i}_jx^j.\]

In other words: coordinates of vectors transform using the inverse  matrix.

Next comes tensor algebra….

To be continued…

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26 Responses to Conformal structure – until the puppy grows up

  1. bayak says:
    2025/12/07 at 18:19

    Arkadiusz, could you outline the final path of our journey through conformal structures? Will it be quantum gravity, or simply an algebraic interpretation of gravity?

    • arkajad says:
      2025/12/07 at 18:28

      Igor, since it is an “Open system”, the distant future is mostly unpredictable. But I can tell you that on the way we will most probably pass through two gates: Hodge * operator, and Maxwell equations. What will be after the second gate – that is still variable/uncertain. First it will be pure algebra, then differential forms will show up. But, as our caravan is using camels, we have to move move slowly.

    • John G says:
      2025/12/08 at 05:56

      Ark in 1993 wrote Born’s Reciprocity in the Conformal Domain. Born’s Reciprocity like Ark’s EEQT is kind of quantum and classical (Hilbert and phase space) and very algebraic. So I don’t think it’s either/or for where Ark’s been and where he’s going as far as quantum gravity vs. classical algebra gravity.

  2. bayak says:
    2025/12/08 at 10:18

    Looking ahead, it seems important to find a local relationship between algebra and a differential 1-form M–>/\^{1}T^{*}M–>A, where M is Minkowski space. Following this path, we will inevitably encounter quantum gravity.

    • arkajad says:
      2025/12/08 at 10:31

      I do not really care about quantum gravity until I have a satisfactory version of classical gravity. And there are many possibilities here competing with the old Einstein-Hilbert formulation.

      By the way: you can use Latex in your comments by first typing [latexpage]. Then contimue putting all your text and formulas with dollar symbols as in a paper.

    • bayak says:
      2025/12/12 at 18:29

      (testing the possibilities of including LaTeX) Localization in a linear space V of the algebra of linear vector fields \mathcal{A}_{\nabla}(V) means constructing a morphism

      (1)   \begin{equation*} V\to \mathrm{Aut}(V)\to \mathrm{Aut}\mathcal{A}_{\nabla}(V) \end{equation*}

      which leads to the localization of its Lie algebra \mathcal{A}_{T}(V)

      (2)   \begin{equation*} V\to \mathrm{Aut}(V)\to \mathrm{Aut}\mathcal{A}_{T}(V) \end{equation*}

      In turn, the differential 1-form of the time coordinate of the Minkowski space \mathcal{M} belongs to the morphism \mathcal{M}\to \mathrm{Aut}(\mathcal{M}), and the Clifford algebra \mathrm{Cl}(\mathcal{M}), in which the Minkowski space is embedded, is isomorphic to M(4,\mathbb{C}). Therefore, we have a mapping

      (3)   \begin{equation*} \mathcal{M}\to\bigwedge^{1}T^{*}\mathcal{M} \to \mathrm{Aut}\mathrm{Cl}(\mathcal{M}) \end{equation*}

      and therefore the differential 1-form of the time flow defines in Minkowski space a localization of the Lie algebra of linear vector fields of the 8-dimensional real space, isomorphic to sl(4,\mathbb{C}). It is expected that the local Lie algebras generated by harmonic (minimal) 1-forms correspond to the Lie algebras of Killing vector fields of a pseudo-Riemannian manifold with a metric that minimizes the Hilbert-Einstein action. In other words, we have two different ways of describing gravity: internal (using metric tensors) and external (using harmonic 1-forms). Note also that describing gravity using a minimal flow in Minkowski space returns us to historical attempts to justify gravity with an ether.

      • bayak says:
        2025/12/14 at 09:43

        The corrected version

        Indeed, as shown in the note “On Some Applications of Vector Field Algebra,” localization in a linear space V of the algebra of linear vector fields \mathcal{A}_{\nabla}(V) means constructing (using the Jacobian of coordinate transformations of the space V) a morphism

        (1)   \begin{equation*} V\to \mathrm{Aut}(V)\to \mathrm{Aut}\mathcal{A}_{\nabla}(V) \end{equation*}

        which leads to the localization of its Lie algebra \mathcal{A}_{T}(V)

        (2)   \begin{equation*} V\to \mathrm{Aut}(V)\to \mathrm{Aut}\mathcal{A}_{T}(V) \end{equation*}

        In turn, the streamlines of the differential 1-form of the flow and the foliation vector fields that annihilate it form new local-time coordinates of the Minkowski space \mathcal{M}, and therefore the Jacobian of the transformation of local-time coordinates belongs to the morphism \mathcal{M}\to \mathrm{Aut}(\mathcal{M}). Moreover, the Clifford algebra \mathrm{Cl}(\mathcal{M}), into which the Minkowski space is embedded, is isomorphic to M(4,\mathbb{C}). Therefore, we have the mapping

        (3)   \begin{equation*} \mathcal{M}\to\bigwedge^{1}T^{*}\mathcal{M} \to \mathrm{Aut}\mathrm{Cl}(\mathcal{M}) \end{equation*}

        and therefore the differential 1-form of the time flow defines in Minkowski space a localization of the Lie algebra of linear vector fields of the 8-dimensional real space, isomorphic to sl(4,\mathbb{C}). It is expected that the local Lie algebras generated by harmonic (minimal) 1-forms correspond to the Lie algebras of Killing vector fields of a pseudo-Riemannian manifold with a metric that minimizes the Hilbert-Einstein action. In other words, we have two different ways of describing gravity: internal (using metric tensors) and external (using harmonic 1-forms). Note also that describing gravity using a minimal flow in Minkowski space returns us to historical attempts to justify gravity with an ether.

  3. bayak says:
    2025/12/08 at 17:53

    По-моему, сравнивать калибровочные поля и гравитацию надо очень осторожно. Калибровочные поля порождаются потоком в пространстве, лежащем над пространством Минковского, а гравитация порождается потоком в пространстве Минковского. А поскольку эти потоки смешиваются, то калибровочные поля и их особенности служат источником гравитации. Однако гравитацию (геометрию) пустого пространства можно описывать и без калибровочных полей, а именно, с помощью локализации алгебры M(2,\mathbb{C}).

    • bayak says:
      2025/12/08 at 17:57

      Оригинальный текст был написан по-русски, а автоперевод с русского на английский и потом снова на русский получился просто ужасным.

      • arkajad says:
        2025/12/08 at 18:03

        Here is the English translation by Perplexity AI:
        In my opinion, one must be very careful when comparing gauge fields and gravity. Gauge fields are generated by a flow in a space lying over Minkowski space, whereas gravity is generated by a flow within Minkowski space itself. And since these flows mix, the gauge fields and their singularities act as sources of gravity. However, the gravity (geometry) of empty space can also be described without gauge fields—namely, by means of the localization of an algebra M(2,\mathbb{C}).
        .

    • John G says:
      2025/12/09 at 04:36

      So you have the $latex m(2,c)$ biquaternions as a phase space gravity basically? Kind of the tangent space and the cotangent space. Could you have this as the symplectic form plus an Sp(8,R) structure group. The gauge part could still be the metric with the SO(4,2) structure group plus the standard model gauge connections. There’s also I think needed a one form for the propagator phase or generalized proper time for least action principle purposes but I’m not sure how that would work; it’s where I get confused quantum vs classical-wise.

      • bayak says:
        2025/12/09 at 18:22

        John, you’re trying to squeeze too much out of this algebra. Although, if we consider that gauge symmetries are generated by symmetries of the compact space obtained by conformal compactification of a 4-dimensional space with a metric inverse to that of Minkowski space, then you may be right. But if we’re talking about pure gravity without gauge fields, then the following sequence of steps is apparent. We take a representation of the biquaternion algebra as an algebra of linear vector fields in 4-dimensional Minkowski space and localize this algebra using a differential 1-form of the time flow. Then we should expect that the Lie algebra of this localized algebra of vector fields coincides with the Lie algebra of Killing vector fields of the 4-dimensional pseudo-Riemannian manifold that describes gravity according to Einstein. But to satisfy Einstein’s equations, the 1-form must be harmonic.

  4. Bjab says:
    2025/12/09 at 11:45

    has a inverse ->
    has an inverse

  5. Bjab says:
    2025/12/09 at 11:46

    “Exercise. Prove that …”

    A – not defined

  6. Bjab says:
    2025/12/09 at 11:47

    inverse transposed matrix ->
    inverse matrix

  7. Bjab says:
    2025/12/09 at 11:49

    We may the write ->
    We may then write

  8. Bjab says:
    2025/12/09 at 11:55

    “e'_i=e_j {A^j}_i is another basis”

    Product of base vector and matrix?

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