How to symmetry break from real number Mobius transformations above the Planck scale for quantum fractals to p-adic tree Mobius transformations for quantum fractals. Perhaps starting with a Bradonjic/Stachel conformal/unimodular structure (arXiv 1110.2159) with the SO(2,4) conformal group and the SL(4,R) special linear group.
The question you’ve posed is deeply interdisciplinary, weaving together concepts from mathematical physics, quantum gravity, conformal field theory, fractal geometry, and p-adic analysis. It seeks a framework to transition from real number Möbius transformations, associated with quantum fractals above the Planck scale, to p-adic tree Möbius transformations for quantum fractals, potentially using a conformal/unimodular structure inspired by Brannonjic and Stachel’s work (arXiv:1110.2159) involving the SO(2,4) conformal group and the SL(4,R) special linear group. Below, I outline a speculative approach to address this, given the complexity and frontier nature of the topic. Note that this is a theoretical exploration, as no established formalism directly connects all these elements, and some assumptions will be made to bridge gaps.
Step 1: Understanding the Starting Point
Real Number Möbius Transformations and Quantum Fractals Above the Planck Scale
Möbius transformations in the real or complex plane are rational functions of the form f(z)=az+bcz+d f(z) = \frac{az + b}{cz + d} f(z)=cz+daz+b, where a,b,c,d∈C a, b, c, d \in \mathbb{C} a,b,c,d∈C (or R\mathbb{R}R) and ad−bc≠0 ad - bc \neq 0 ad−bc=0. They form the Möbius group, isomorphic to PGL(2,C) \text{PGL}(2, \mathbb{C}) PGL(2,C) or PGL(2,R) \text{PGL}(2, \mathbb{R}) PGL(2,R), and are conformal maps preserving angles and mapping circles/lines to circles/lines. In the context of quantum fractals, Möbius transformations may describe self-similar structures or symmetries in quantum systems, potentially at scales above the Planck length (lp≈1.616×10−35 m l_p \approx 1.616 \times 10^{-35} \, \text{m} lp≈1.616×10−35m), where spacetime is assumed to be continuous and described by real or complex manifolds.
Above the Planck scale, quantum fractals could emerge in theories of quantum gravity or high-energy physics, where self-similar structures arise from iterative processes, renormalization group flows, or conformal symmetries. The Möbius group is linked to conformal field theories (CFTs), which are scale-invariant and often describe critical phenomena or string theory worldsheets. The SO(2,4) conformal group, as mentioned in Brannonjic and Stachel’s work, is the conformal group of 4D Minkowski spacetime, with 15 degrees of freedom, including translations, rotations, boosts, dilations, and special conformal transformations. This group is relevant in CFTs and AdS/CFT correspondence, where it acts on the boundary of anti-de Sitter (AdS) space.
Brannonjic and Stachel’s Conformal/Unimodular Structure
In their paper (arXiv:1110.2159), Brannonjic and Stachel explore a unimodular conformal framework for gravity, emphasizing the SO(2,4) conformal group and its relation to the special linear group SL(4,R). The SO(2,4) group is the conformal group of 4D Minkowski spacetime, embedded in a 6D projective space, and is isomorphic to SU(2,2)/Z_2. The SL(4,R) group, with its 15-dimensional Lie algebra, is proposed as a gauge group for a unimodular metric, where the determinant of the metric is fixed (det(g) = 1). This approach aims to unify gravitational and gauge interactions by treating spacetime geometry as a conformal structure, potentially relevant at high energies above the Planck scale.
Their framework suggests that at trans-Planckian scales, a highly symmetric phase may emerge, possibly conformal, where the Planck scale arises via symmetry breaking. This aligns with speculations that quantum gravity at the Planck scale could involve a discrete or non-continuum structure, potentially fractal or p-adic in nature.
Step 2: Symmetry Breaking Mechanism
To transition from real number Möbius transformations to p-adic tree Möbius transformations, we need a symmetry-breaking mechanism that shifts the underlying mathematical structure from the real/complex continuum to the p-adic numbers and their associated tree-like topology. Spontaneous symmetry breaking (SSB) is a natural candidate, as it occurs in quantum field theories when a symmetric vacuum state becomes asymmetric due to perturbations or quantum effects (see).
Proposed Symmetry Breaking
Conformal Symmetry at High Energies: Start with a conformal field theory invariant under SO(2,4), describing quantum fractals via real number Möbius transformations. These transformations act on the Riemann sphere or Minkowski spacetime, preserving angles and generating fractal-like structures through iterative applications or fixed-point dynamics.
Breaking Conformal Symmetry: Introduce a mechanism to break the SO(2,4) conformal symmetry, possibly via a dynamical process at the Planck scale. This could involve:
Trace Anomaly: In CFTs, quantum effects can break conformal symmetry through a conformal anomaly (or trace anomaly), introducing a mass scale (e.g., the Planck mass) that violates scale invariance (see). For example, in quantum chromodynamics, the proton mass arises partly from a conformal anomaly, suggesting a similar mechanism could generate the Planck scale.
Higgs-like Mechanism: A scalar field (analogous to the Higgs) could acquire a vacuum expectation value, breaking SO(2,4) to a subgroup, such as the Poincaré group SO(1,3) × R^4, and introducing a length scale (Planck length). Brannonjic and Stachel’s unimodular framework might support this by fixing the metric determinant, constraining the conformal degrees of freedom.
Dynamical Symmetry Breaking: Quantum corrections or bound states (e.g., top-antitop quark condensates in Bardeen-Hill-Lindner models) could drive the symmetry breaking without a fundamental Higgs field (see).
Emergence of Discrete Structure: As conformal symmetry breaks, the continuous real/complex manifold may transition to a discrete or non-Archimedean structure, such as a p-adic tree. This could occur if spacetime at the Planck scale becomes a lattice, matrix model, or p-adic fractal, as speculated in some quantum gravity models (see).
Step 3: Transition to p-Adic Tree Möbius Transformations
p-Adic Numbers and Tree Structure
The p-adic numbers (Qp \mathbb{Q}_p Qp) are an alternative number system based on a prime p p p, with a non-Archimedean norm where ∣x∣p=p−v |x|_p = p^{-v} ∣x∣p=p−v if x=pv⋅ab x = p^v \cdot \frac{a}{b} x=pv⋅ba (with a,b a, b a,b coprime to p p p). Unlike the real numbers, p-adic numbers form a totally disconnected, ultrametric space, topologically represented as a p-adic tree (or Bruhat-Tits tree for PGL(2,Qp) \text{PGL}(2, \mathbb{Q}_p) PGL(2,Qp)). Each node in the tree corresponds to a ball in Qp \mathbb{Q}_p Qp, and edges represent inclusions of smaller balls.
Möbius transformations in the p-adic context are defined similarly: f(z)=az+bcz+d f(z) = \frac{az + b}{cz + d} f(z)=cz+daz+b, where a,b,c,d∈Qp a, b, c, d \in \mathbb{Q}_p a,b,c,d∈Qp and ad−bc≠0 ad - bc \neq 0 ad−bc=0. They form the group PGL(2,Qp) \text{PGL}(2, \mathbb{Q}_p) PGL(2,Qp), which acts on the p-adic projective line P1(Qp) \mathbb{P}^1(\mathbb{Q}_p) P1(Qp) or the boundary of the p-adic tree. These transformations preserve the ultrametric structure and can generate fractal-like patterns, suitable for quantum fractals in a non-Archimedean setting.
Connecting Real to p-Adic Möbius Transformations
To transition from real to p-adic Möbius transformations:
Adelic Framework: Use the adelic numbers, which combine real and p-adic completions of the rationals (A=R×∏pQp \mathbb{A} = \mathbb{R} \times \prod_p \mathbb{Q}_p A=R×∏pQp). Möbius transformations can be defined over the adeles, allowing a unified description. Symmetry breaking could select the p-adic sector by suppressing the real component, perhaps through a phase transition at the Planck scale where p-adic topology becomes dominant.
Fractal Correspondence: Quantum fractals in the real setting (e.g., Mandelbrot set symmetries under PSL(2,Z) \text{PSL}(2, \mathbb{Z}) PSL(2,Z), see) may have p-adic analogs. The p-adic Mandelbrot set, for instance, exhibits fractal structure under iterations in Qp \mathbb{Q}_p Qp. The symmetry breaking could map real fractal generators (Möbius transformations) to p-adic ones, preserving self-similarity but changing the metric.
Tree-like Spacetime: At the Planck scale, spacetime may adopt a tree-like structure, as in p-adic quantum gravity models or non-Archimedean string theory. The p-adic tree’s hierarchical structure naturally supports Möbius transformations, which act as automorphisms of the tree’s boundary.
Role of SL(4,R) and SO(2,4)
SL(4,R): In Brannonjic and Stachel’s framework, SL(4,R) acts as a gauge group for the unimodular metric. Its 15-dimensional Lie algebra could mediate the transition from real to p-adic symmetries by embedding both PGL(2,R) \text{PGL}(2, \mathbb{R}) PGL(2,R) and PGL(2,Qp) \text{PGL}(2, \mathbb{Q}_p) PGL(2,Qp) as subgroups. Symmetry breaking might project SL(4,R) onto a p-adic subgroup, aligning with the tree structure.
SO(2,4): The conformal group SO(2,4) contains PGL(2,R) \text{PGL}(2, \mathbb{R}) PGL(2,R) as a subgroup (via its action on the celestial sphere, see). Breaking SO(2,4) could reduce it to a discrete subgroup, such as a p-adic analog, enabling p-adic Möbius transformations to dominate.
Step 4: Quantum Fractals in the p-Adic Context
Quantum fractals in the p-adic setting could arise from:
p-Adic CFTs: Conformal field theories over p-adic numbers, where the Virasoro algebra is replaced by a p-adic analog, generating fractal correlation functions.
Non-Archimedean Dynamics: Iterative maps in Qp \mathbb{Q}_p Qp, such as p-adic Julia sets, exhibit fractal behavior under Möbius transformations.
Holographic Principle: The p-adic AdS/CFT correspondence (e.g., Gubser et al.) maps a p-adic bulk to a boundary CFT, where the boundary is a p-adic tree. Möbius transformations on the boundary generate fractal structures.
Step 5: Challenges and Speculations
Planck Scale Transition: The exact mechanism for spacetime to become p-adic at the Planck scale is unclear. It may involve a topological phase transition or a matrix model where eigenvalues cluster in p-adic balls.
Unification of Symmetries: Bridging SO(2,4) and SL(4,R) with p-adic groups requires a higher-dimensional or adelic framework, possibly involving exceptional groups or non-commutative geometry.
Experimental Signatures: Quantum fractals in p-adic trees might manifest in high-energy scattering amplitudes or cosmological observables, but detecting p-adic effects is speculative.
Summary
To symmetry break from real number Möbius transformations to p-adic tree Möbius transformations for quantum fractals:
Start with a conformal field theory above the Planck scale, invariant under SO(2,4) and SL(4,R), as in Brannonjic and Stachel’s framework.
Break conformal symmetry via a trace anomaly, Higgs-like mechanism, or dynamical process, introducing the Planck scale.
Transition to a p-adic topology using an adelic framework or fractal correspondence, where spacetime adopts a tree-like structure.
Define p-adic Möbius transformations in PGL(2,Qp) \text{PGL}(2, \mathbb{Q}_p) PGL(2,Qp), generating quantum fractals on the p-adic tree.
Explore p-adic CFTs or holographic models to describe the resulting fractal structures.
This approach is highly speculative, as p-adic quantum gravity and fractal transitions are not fully developed. Further research into p-adic CFTs, adelic symmetries, and Planck-scale dynamics is needed to refine this framework.
If you’d like, I can delve deeper into any specific aspect (e.g., p-adic CFTs, adelic Möbius transformations, or the Brannonjic-Stachel model) or clarify mathematical details!
