Spacetime, Spin, and the Search for Observable Laws

Author: Denis Avetisyan

New research explores how fundamental properties of Majorana fields in curved spacetime can be described through geometric and path integral methods.

This review details empirical laws for Majorana fields, leveraging CAR algebra, holonomy, and the Sorkin density within quantum field theory in curved spacetime.

Establishing empirically verifiable laws in quantum field theory in curved spacetime presents a fundamental challenge due to the absence of a globally defined vacuum state. This paper, ‘The empirical laws for Majorana fields in a curved spacetime’, investigates this issue specifically for Majorana fields, building upon prior work examining the Klein-Gordon equation in similar contexts. We propose that observable quantities can be rigorously defined through representations utilizing the CAR algebra, holonomy, and path integral formalisms, circumventing the need for a conventional vacuum. Ultimately, can these approaches solidify the physical, rather than merely mathematical, status of quantum field theory in curved spacetime and provide a pathway toward testable predictions?

The Relativistic Foundation: Spin as Inevitable Geometry

The advent of relativistic quantum mechanics demanded a significant departure from the Schrödinger equation, which inherently lacked the capacity to accurately describe particles approaching the speed of light. Traditional quantum mechanics treated time and space as separate entities, a treatment incompatible with the principles of special relativity where these are interwoven into a single spacetime continuum. The Dirac equation emerged as a pivotal solution, a relativistic wave equation formulated to reconcile quantum mechanics with special relativity. This equation not only successfully predicted the existence of antimatter – a groundbreaking result – but also fundamentally altered the understanding of particle spin. Unlike the Schrödinger equation which required spin to be imposed as an additional quantum number, spin emerges naturally as a necessary consequence of the Dirac equation’s mathematical structure, demonstrating its power in describing the behavior of particles at relativistic speeds and establishing a cornerstone of modern particle physics.

The successful formulation of solutions to the Dirac equation hinges on the elegant mathematical structure of the Clifford algebra. This algebra isn’t merely a tool, but a fundamental requirement arising from the need to accurately represent relativistic quantum mechanics; it defines the properties of γ matrices, which play a crucial role in expressing the Dirac equation. These γ matrices aren’t simply numbers, but rather objects possessing specific anti-commutation relations – a core feature dictated by the Clifford algebra – that ensure Lorentz invariance and the correct behavior under spatial transformations. Essentially, the algebra provides the rules governing how these matrices interact, dictating the allowed solutions to the equation and, therefore, the possible states of a relativistic fermion. Without this algebraic foundation, consistent descriptions of particles like electrons, accounting for both quantum mechanics and special relativity, would be impossible.

The behavior of fermions – particles like electrons and quarks that obey the Pauli exclusion principle – is fundamentally linked to the principles of the Commutation Relations Algebra, or CAR. This algebra dictates how fermionic creation and annihilation operators interact, ensuring that no two identical fermions can occupy the same quantum state. To fully and naturally describe these systems within the framework of relativistic quantum mechanics, physicists employ DiracSpinorBundles. These bundles aren’t merely a mathematical convenience; they provide the essential geometric and algebraic structure needed to represent fermionic wavefunctions – known as Dirac spinors – and their transformations under Lorentz transformations. Essentially, the DiracSpinorBundle translates the abstract rules of the CAR algebra into a concrete, physically interpretable space, allowing for a rigorous and consistent description of fermionic phenomena, from the behavior of electrons in materials to the interactions within atomic nuclei. The bundle’s mathematical form ensures the proper anti-commutation relations are automatically satisfied, guaranteeing the correct quantum statistics for these fundamental particles.

The description of spin-1/2 particles benefits from two distinct yet interconnected mathematical frameworks: Dirac and Majorana Spinor Bundles. While both address the relativistic quantum mechanical behavior of fermions, they differ in their treatment of particle-antiparticle relationships. Dirac Spinors allow for independent particle and antiparticle states, reflecting the common understanding of matter and antimatter. Conversely, Majorana Spinors posit that a particle is its own antiparticle, a condition requiring specific symmetry properties within the system. This seemingly subtle difference arises from the mathematical properties of the gamma matrices used to construct these bundles, and dictates the types of interactions and decay modes possible for the described particles. Consequently, understanding both Dirac and Majorana spinor bundles is crucial for a complete theoretical description of fermionic systems, and provides a more nuanced understanding of fundamental particle physics.

Propagating Influence: Fields, Connections, and Green’s Operators

The Dirac operator, denoted as $\gamma^\mu \partial_\mu$ , is the central element in determining the temporal evolution of spinor fields, which describe relativistic fermions with intrinsic angular momentum (spin). Acting on a spinor field $\psi(x)$ , the Dirac operator yields the Dirac equation, a relativistic wave equation analogous to the Schrödinger equation in non-relativistic quantum mechanics. Specifically, the equation $i\hbar \partial_t \psi(x) = H \psi(x)$ utilizes the Hamiltonian operator $H$ , which is constructed using the Dirac operator. The solutions to this equation, representing the spinor field’s dynamics, dictate the probability amplitude of finding a particle at a given spacetime point, incorporating both particle and antiparticle components due to the relativistic nature of the equation.

The Dirac equation describes relativistic spin-1/2 particles, and its consistent formulation in curved spacetime necessitates the inclusion of a SpinConnection, also known as the spin affine connection. This connection accounts for the change in the basis vectors of the tangent space as one moves between points in the curved manifold, ensuring that spinor fields transform covariantly. Specifically, the SpinConnection appears in the covariant derivative acting on the spinor field ψ, modifying the standard derivative to include terms that compensate for the curvature of spacetime. Without the SpinConnection, the Dirac equation would not be generally covariant-meaning its form would change under coordinate transformations, and physical predictions would become ill-defined. The SpinConnection effectively introduces a notion of parallel transport for spinors, preserving their properties as they move along curves in the curved manifold.

Green’s Operators provide a means of determining the solution to the Dirac equation given a specified source distribution. Functionally, they represent the propagator, detailing how a spinor field evolves and propagates through spacetime originating from a localized source. Mathematically, a Green’s Operator $G$ satisfies the equation $D G = \delta^{(4)}(x-y)$ , where $D$ is the Dirac operator and $\delta^{(4)}(x-y)$ is the four-dimensional Dirac delta function. Knowing $G$ allows the solution $\psi(x)$ to be constructed via integration: $\psi(x) = \in t G(x, y) J(y) d^4y$ , where $J(y)$ represents the source current density. The specific form of the Green’s Operator is dependent on the spacetime geometry and the boundary conditions imposed on the solution.

Green’s functions, utilized to solve the Dirac equation and determine spinor field propagation, are not independent entities but are fundamentally derived from the Dirac operator and the SpinConnection. Specifically, the mathematical form of the Green’s function-often expressed as an inverse of the Dirac operator $G = (\slashed{D})^{-1}$ , where $\slashed{D}$ represents the Dirac operator-explicitly incorporates both the metric tensor defining spacetime geometry, and the SpinConnection, which accounts for covariant derivatives in curved spacetime. Changes to either the Dirac operator (due to altered spacetime geometry) or the SpinConnection directly affect the Green’s function, altering the predicted field propagation. This interdependency necessitates a self-consistent solution where the Green’s function, Dirac operator, and SpinConnection are simultaneously determined to ensure a physically valid solution to the Dirac equation.

The Quantum Landscape: Path Integrals and Measurement’s Geometry

The Path Integral formulation of quantum mechanics calculates the probability of a quantum event by summing contributions from all possible histories, or field configurations, of the system. Rather than focusing on a single trajectory, as in classical mechanics, the path integral considers every conceivable path between an initial and final state. Each path is assigned a complex weight, proportional to $e^{iS/ħ}$ , where $S$ is the classical action for that particular path and $ħ$ is the reduced Planck constant. The probability amplitude is then obtained by summing (integrating) these complex weights over all possible paths, and the square of the magnitude of this sum yields the probability of the event. This approach is particularly useful in quantum field theory, where the “paths” represent possible field configurations throughout spacetime, and the summation becomes a functional integral.

The Sorkin density, denoted as $ρ2(𝖢1, 𝖢2) = 2 + 2ℜhol(𝖢1∙𝖢2-1)$ , functions as a foundational element in defining the probability measure within the path integral formulation of quantum gravity. Here, $𝖢1$ and $𝖢2$ represent complex-valued cylinder functions, and $ℜhol$ denotes the holomorphic part of the expression. This density is not simply a weighting factor; it directly influences the calculation of probabilities associated with different field configurations. Its specific form, incorporating the holomorphic part of the cylinder function product, ensures that the resulting probability measure is well-defined and consistent with the principles of quantum mechanics, particularly in the context of summing over all possible spacetime geometries.

Incorporating the stress-energy tensor, denoted as $T_{\mu\nu}$ , into the path integral formalism allows for the calculation of quantum amplitudes in curved spacetime and, consequently, accounts for gravitational effects. This is achieved by modifying the action within the path integral; the original action, $S$ , is augmented with a term proportional to the stress-energy tensor, effectively coupling matter and gravity at the quantum level. This modified action then dictates the weight of each field configuration in the path integral summation, thereby influencing the calculated probabilities for quantum processes occurring in a gravitational field. The inclusion of $T_{\mu\nu}$ is essential for describing scenarios where gravity is not merely a classical background, but an active participant in quantum dynamics, such as in cosmology or near black holes.

The accurate definition of quantum measurements and their associated probabilities necessitates a detailed understanding of spacetime geometry, specifically utilizing the framework of Kahler manifolds. These complex manifolds, possessing a compatible Riemannian metric, allow for the consistent definition of probability measures in quantum field theory. A subspace of dimension $2n$ is particularly relevant, as it provides the minimal phase space required for a complete description of the system’s dynamics and allows for the consistent application of geometric quantization techniques. The Kahler potential, inherent to these manifolds, dictates the metric and facilitates the calculation of path integral measures, ensuring the probabilities are properly normalized and physically meaningful. This geometric approach is crucial for addressing issues of observer dependence and defining unambiguous measurement outcomes in quantum gravity scenarios.

Geometric Echoes: Holonomy, Spin, and the Fabric of Reality

The Path Integral formulation of quantum mechanics leverages the mathematical concept of holonomy to precisely track how physical fields evolve as they are carried around closed loops within the fabric of spacetime. This isn’t merely a descriptive tool; it reveals that the transformation of a field isn’t solely determined by the path taken, but by the geometry of the path itself. Holonomy effectively captures the cumulative effect of parallel transport, accounting for any curvature or twist in spacetime that alters the field’s initial state. This geometric phase, encoded within the holonomy, becomes a crucial element in calculating quantum amplitudes, influencing the probabilities of various outcomes. By considering all possible paths and their associated holonomies, the Path Integral provides a comprehensive picture of quantum evolution, linking the seemingly abstract world of quantum mechanics to the tangible geometry of the universe.

The SpinGroup plays a crucial role in describing how quantum states transform during parallel transport around closed loops in spacetime, effectively safeguarding the intrinsic angular momentum, or spin, of particles. Unlike traditional rotation groups, the SpinGroup incorporates the concept of ‘spinors’ – mathematical objects that transform differently from vectors – and guarantees that the spin structure remains consistent even as fields are moved along complex paths. This preservation is not merely a mathematical convenience; it’s fundamental to the very fabric of quantum reality, preventing the loss of crucial information about a particle’s identity and ensuring that quantum measurements yield consistent and predictable results. The group’s action dictates precisely how these spin states evolve, revealing a deep connection between the geometry of spacetime and the inherent properties of quantum particles.

The measurable results within quantum processes aren’t arbitrary; they are fundamentally linked to the geometric properties of spacetime, specifically how paths intertwine and relate to one another. This relationship is elegantly captured by the equation $𝖢1∙𝖢2-1$ , which describes the non-commutative effect of traversing two curves, $𝖢1$ and $𝖢2$ , in sequence and then reversing the process. Essentially, the order in which a quantum system explores different paths matters, and this geometric interplay directly influences probabilities and observable phenomena. This isn’t merely a mathematical curiosity; it suggests that the very fabric of spacetime dictates the outcomes of quantum events, offering a pathway to understand how quantum mechanics arises from deeper geometric principles.

The interwoven relationship between quantum mechanics, geometry, and the fundamental laws of physics emerges as a central tenet of this theoretical framework. It suggests that the seemingly disparate realms of quantum behavior and geometric structure are, in fact, intimately connected – quantum phenomena aren’t simply occurring within a geometric space, but are fundamentally shaped by its properties. This isn’t merely a mathematical curiosity; the preservation of spin, dictated by the SpinGroup, and the holistic understanding of field transformations via Holonomy, demonstrate that geometric principles aren’t just descriptive tools, but active participants in quantum processes. The resulting interplay suggests a universe where the laws governing the very small are deeply rooted in the large-scale structure of spacetime, offering a path toward a more unified and complete description of physical reality, quantified by relationships such as $𝖢1∙𝖢2-1$ .

The pursuit of empirical laws, as detailed in this work concerning Majorana fields, resembles less a construction project and more the tending of a garden. One anticipates eventual decay, even as one attempts to cultivate order from the inherent chaos. It’s a system destined for entropy, a prophecy etched into every equation. As Epicurus observed, “It is not the desire for pleasure and the avoidance of pain that defines happiness, but the attainment of a life free from disturbance.” This echoes the challenge of defining observable quantities in curved spacetime – a quest not for perfect prediction, but for a stable, predictable baseline amidst inevitable fluctuations, a minimization of ‘disturbance’ within the system.

The Horizon of Understanding

The pursuit of empirical laws for fields in curved spacetime, as demonstrated by this work, isn’t the construction of a fortress against complexity – it’s merely charting the inevitable erosion. Every attempt to define observables, to anchor them to holonomy and path integrals, is a temporary reprieve. The CAR algebra offers a language, yes, but languages evolve, and the universe speaks in dialects of failure. This isn’t a limitation of the mathematics; it’s a consequence of attempting to impose order on a fundamentally chaotic system.

The Sorkin density, promising a background-independent formulation, feels less like a solution and more like a carefully constructed illusion. It postpones the problem of defining measurement without resolving it. Future work will inevitably grapple with the limitations of this approach, perhaps shifting focus from precise definitions of observables to probabilistic descriptions of their emergence. The true challenge isn’t finding the right law, but understanding why any law holds, even briefly.

One suspects the field will eventually confront a deeper truth: that the observable universe isn’t governed by laws discovered, but by constraints imposed – by the very act of observation. The architecture of reality isn’t a blueprint; it’s a palimpsest, layers of failed attempts overwritten by the persistent, messy reality of existence. And each new formalism, however elegant, will add another faded layer to the manuscript.

Original article: https://arxiv.org/pdf/2602.16907.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Relativistic Foundation: Spin as Inevitable Geometry

Propagating Influence: Fields, Connections, and Green’s Operators

The Quantum Landscape: Path Integrals and Measurement’s Geometry

Geometric Echoes: Holonomy, Spin, and the Fabric of Reality

The Horizon of Understanding

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