Quantum field theory, gravity and cosmology in a fractal universe (1001.0571v2)

Abstract: We propose a model for a power-counting renormalizable field theory living in a fractal spacetime. The action is Lorentz covariant and equipped with a Stieltjes measure. The system flows, even in a classical sense, from an ultraviolet regime where spacetime has Hausdorff dimension 2 to an infrared limit coinciding with a standard $D$-dimensional field theory. We discuss the properties of a scalar field model at classical and quantum level. Classically, the field lives on a fractal which exchanges energy-momentum with the bulk of integer topological dimension D. Although an observer experiences dissipation, the total energy-momentum is conserved. The field spectrum is a continuum of massive modes. The gravitational sector and Einstein equations are discussed in detail, also on cosmological backgrounds. We find ultraviolet cosmological solutions and comment on their implications for the early universe.

Quantum Field Theory, Gravity, and Cosmology in a Fractal Universe

The paper "Quantum Field Theory, Gravity, and Cosmology in a Fractal Universe" by Gianluca Calcagni introduces a model addressing the integration of quantum field theory with fractal spacetime considerations. It proposes a framework that allows for a renormalizable field theory operating within a fractal spacetime structure, which diverges in significant ways from conventional theories that rely on integer-dimensional spacetimes. This investigation into fractal geometry is motivated by the pursuit of a viable theory of quantum gravity, posing alternatives to prominent approaches such as string theory and loop quantum gravity.

Core Concepts

Calcagni's approach hinges on the concept of fractal spacetimes, wherein the spacetime exhibits a Hausdorff dimension that evolves with scale: from a lower value (two dimensions) in the ultraviolet (UV) regime to a standard integer-dimensional field theory in the infrared (IR) sector. This scale dependence introduces a fractal structure to the universe, challenging the traditional view of a homogeneous spacetime fabric.

Key elements of the theoretical model include:

• Fractal Spacetime: The use of fractional calculus to describe non-integer dimensions, allowing for the definition of fractional field theories where measures are defined using Stieltjes integrals.
• Lorentz Invariance: Unlike some other proposals like Hořava-Lifshitz gravity, this model retains Lorentz invariance while maintaining a different geometrical structure.
• Energy-Momentum Conservation: Despite the presence of dissipation perceived by an observer due to fractal characteristics, total energy-momentum remains conserved.

Mathematical Formulation

The framework employs Stieltjes measures to redefine integral calculus within this fractal construct. This influences the physical interpretation of phenomena from the classical to the quantum levels. For example:

• Propagator in Configuration Space: The model derives the propagator for the theory, demonstrating its consistency with known solutions in integer dimensions while adapting to the fractional context.
• Renormalization: The power-counting renormalizability is assessed by considering the scaling dimensions of fields and couplings, crucially dependent on the fractal dimensionality introduced.
• Flow from UV to IR: The model proposes a flow from a two-dimensional effective theory in the UV to a standard four-dimensional field theory in the IR, engaging with speculative cosmological scenarios at high energies.

Implications and Future Prospects

The implications of this research are manifold, extending from theoretical physics to potential cosmological applications:

• Quantum Gravity: The fractal structure provides a novel avenue in understanding the interplay between gravity and quantum mechanics, potentially leading to insights beyond the scope of traditional formulations.
• Cosmology: Implications for early universe cosmology are explored, where the model suggests unique behaviors such as modified inflationary dynamics and peculiar equations of state for the cosmological fluid.
• Extra Dimensions and Holography: Speculatively, the model opens dialogue with theories of extra dimensions and holographic principles, suggesting fertile ground for future exploration.

Calcagni's work, while abstract, is a valuable contribution to theoretical physics, posing a distinctive method to reconcile quantum field theories with the complex structure of spacetime at quantum scales. The proposal, by embedding a fractal geometry into the very framework of spacetime, may offer insights into resolving the discord between general relativity and quantum mechanics, opening new channels for theoretical and observational investigation in the universe's fractal-nature at different scales.