The development of Penrose tensor diagrams has transformed the representation of mathematical structures in contemporary physics, providing a graphical language that unites algebraic abstraction with geometric intuition. Over the past decades, these diagrams have become essential tools in quantum physics, field theory, and spacetime geometry, allowing compact descriptions of tensor contractions and correlations. Authors such as Biamonte, Bergholm, Kissinger, and Coecke have extended this formalism to computational and categorical contexts, establishing a framework of graphical calculus that facilitates the visualization and manipulation of complex physical systems. Within this context, Martín Pomares Calero introduces the concept of Symbolic Physics, an epistemological and ontological approach that interprets Penrose diagrams as symbolic structures of the universe. From this perspective, the symbol is not a mere representational sign but an active component of physical and cognitive processes, capable of encoding the fundamental laws of reality. This approach proposes a synthesis between tensor calculus, scientific semiotics, and visual cognition, oriented toward a physics of meaning. This paper examines the main research lines connecting tensorial thought and symbolic interpretation, emphasizing how the integration between mathematical and symbolic frameworks opens new paths for the theoretical understanding of the universe.
The development of Penrose tensor diagrams has transformed the representation of mathematical structures in contemporary physics, providing a graphical language that unites algebraic abstraction with geometric intuition. Over the past decades, these diagrams have become essential tools in quantum physics, field theory, and spacetime geometry, allowing compact descriptions of tensor contractions and correlations. Authors such as Biamonte, Bergholm, Kissinger, and Coecke have extended this formalism to computational and categorical contexts, establishing a framework of graphical calculus that facilitates the visualization and manipulation of complex physical systems. Within this context, Martín Pomares Calero introduces the concept of Symbolic Physics, an epistemological and ontological approach that interprets Penrose diagrams as symbolic structures of the universe. From this perspective, the symbol is not a mere representational sign but an active component of physical and cognitive processes, capable of encoding the fundamental laws of reality. This approach proposes a synthesis between tensor calculus, scientific semiotics, and visual cognition, oriented toward a physics of meaning. This paper examines the main research lines connecting tensorial thought and symbolic interpretation, emphasizing how the integration between mathematical and symbolic frameworks opens new paths for the theoretical understanding of the universe.
This work introduces the Tensor Diagrammatic Calculus (TDC), a graphical-mathematical language for representing multi-scale tensorial structures through networks of contractions, transformations, and emergence operators. Building on Penrose’s graphical tensor notation, TDC defines four primitive entities: Glyphons (elementary tensorial units), Morphons (transformation operators), Symbions (emergent relational structures), and Indices (connection channels). The framework provides a formal correspondence between algebraic tensor expressions and diagrammatic networks, and introduces two operators—emergence (E) and immersion (I)—to represent transitions between microscopic and macroscopic levels of description. We show that every TDC diagram admits an algebraic translation and that contraction rules preserve index compatibility. The formalism is compatible with existing tensor network notation and category-theoretic approaches to physical processes. Workedexamples from quantum mechanics, statistical mechanics, and classical field theory are provided. TDC is proposed as a representational tool for symbolic physics, not as a replacement for existing formalisms.
High-dimensional data arise naturally in many areas of science and engineering, including machine learning, signal processing, computational physics, and statistics. Such data are often represented as tensors, multi-dimensional generalizations of matrices. While tensors provide a natural representation for multi-modal structure, their direct manipulation quickly becomes challenging as the order grows: the number of parameters increases exponentially, and algebraic expressions involving many indices become difficult to interpret and implement. Tensor networks (TNs) provide an effective framework for addressing these challenges. Originally introduced by Penrose and developed extensively in quantum physics, the graphical language of tensor networks encodes contractions as edges in a graph, reducing notational overhead and revealing structural properties obscured by index notation. Despite the central role of high-dimensional tensors in modern machine learning and numerical analysis, tensor network diagrams remain underutilized outside quantum computing, partly due to the lack of a self-contained mathematical reference accessible to a broad technical audience. This manuscript provides a self-contained guide to tensor networks and their use in tensor algebra. We present the main operations on tensors, contractions, products, and reshaping through, graphical notation, and show how classical tensor decompositions and related computations are naturally expressed in this framework. We also illustrate how tensor networks simplify the derivation of gradients and the manipulation of high-dimensional probability distributions. Throughout, we show that the diagrammatic approach yields genuinely shorter and more transparent proofs of classical identities, rank bounds, and gradient formulas that would otherwise require laborious index manipulation.
This paper systematically discusses the core role and computational methods of tensor networks in the study of geometry-physics duality. By analyzing the unified mathematical structure of tensor networks in multi-scale entanglement renormalization, lattice field theory compression, and algebraic equation solving, we establish their position as computational bridges in AdS/CFT duality, gauge theory, and string theory compactification. We focus on exploring the intrinsic connections between matrix product states, tensor contraction algorithms, and geometric invariant computations, providing a unified theoretical framework and numerical implementation pathways for future research. This paper emphasizes tensor networks as a mathematical language that effectively connects continuous geometric structures with discrete physical models, offering a new computable perspective for understanding quantum gravity and gauge duality.
The content of this paper can be roughly organized into a three-level hierarchy of generality. At the first, most general level, we introduce a new language which allows us to express various categorical structures in a systematic and explicit manner in terms of so-called 2-schemes. Although 2-schemes can formalize categorical structures such as symmetric monoidal categories, they are not limited to this, and can be used to define structures with no categorical analogue. Most categorical structures come with an effective graphical calculus such as string diagrams for symmetric monoidal categories, and the same is true more generally for interesting 2-schemes. In this work, we focus on one particular non-categorical 2-scheme, whose instances we refer to as tensor types. At the second level of the hierarchy, we work out different flavors of this 2-scheme in detail. The effective graphical calculus of tensor types is that of tensor networks or Penrose diagrams, that is, string diagrams without a flow of time. As such, tensor types are similar to compact closed categories, though there are various small but potentially important differences. Also, the two definitions use completely different mechanisms despite both being examples of 2-schemes. At the third level of the hierarchy, we provide a long list of different families of concrete tensor types, in a way which makes them accessible to concrete computations, motivated by their potential use in physics. Different tensor types describe different types of physical models, such as classical or quantum physics, deterministic or statistical physics, many-body or single-body physics, or matter with or without symmetries or fermions.
Contemporary physics developed in the twentieth century through a succession of revolutions that upset its epistemological status with the prospect of a new ontology and a new gnoseology. From the physics of chaos to the theories of relativity, from quantum physics to quantum-relativistic field theory and the theory of the S matrix ("scattering matrix"). Recently, the transversal scientific paradigm of information has been definitively establishing itself: above all, through its concretization in the realization of machines, as widespread as personal computers, capable of processing and transmitting information up to an even greater ability to virtually simulate any type of physical reality. A new informational conception of Nature emerged, of the Universe as a potentially infinite computer. However, there is still a lack of a mathematical formulation of physics and cosmology in terms of information. In recent years, the research of Eduardo R. Caianiello (1921-1993), a Neapolitan physicist and cybernetist, has been linked to this goal, who, in 1980, formulated quantum theory in an 8-dimensional XA (x, ct; p, p4 = E/c) non-Euclidean, curve relativistic phase space, which has only the form of a geometry but is a dynamic written in a pseudo-geometric form. The operators of Heisenberg algebra (also position and time) are expressed in a quantization by means of covariant derivatives and the quantum commutation relations are interpreted as components of the curvature tensor. The geometric curvature thus expresses the quantum uncertainty in terms of informational entropy (it is a complex information geometry: the ds2 coincides with the cross-entropy). The algebra is that of the octonions. Caianiello reformulated not only quantum mechanics, but also general relativity, thermodynamics and geometry itself on the basis of information theory and systems theory, opening a way to the greatest revolution in physics: in philosophical terms it is an informational transformation of a transcendental phenomenological hermeneutics.
Abstract Quantum Mechanics (QM) stands alone as a (very) successful physical theory, but the meaning of its variables and the status of many quantities in the mathematical formalism is obscure. This unique situation prompted the need for attribution of a physical meaning to the latter, a procedure known as interpretation. On the other hand, the study of QM is usually presented, even to future scientists, within the only framework developed by Bohr and the Copenhagen researchers, known as the Copenhagen interpretation. As a contribution to the understanding and teaching of Quantum Mechanics, aimed to a broader and deeper appreciation of its fundamentals, including contemplating alternatives and updated interpretations for physicists and philosophers interested in the study of exact sciences (through Ontology, Epistemology, Logic or the Theory of Knowledge), we present a set of Conceptual Diagrams elaborated and designed to expose and facilitate the visualization of elements intervening in any interpretation of Quantum Mechanics and apply them to several well‐developed cases of the latter.