Ordinary Fock space, defined as concatenated combination of three-dimensional spaces is the main workhorse of the canonical version of quantum field theory. It is also fundamental to the modern version of quantum field theory without observers, as in the theory of Everett, Wheeler and Deutsch, which led to the theory of the universal quantum computer. It also has many practical uses, from quantum optics to the closure of turbulence and the derivation of the emergent behavior of space-time dynamical systems. But many of us believe that time and space are interchangeable to some degree, and that methods based on 4D Fock space are needed. This letter describes two simple mathematical tools to help make that possible. It discusses how computing the emergent statistics of such systems is a proper generalization of value function estimation in machine learning, which may therefore carry the same complexities with it.
Ordinary Fock space, defined as concatenated combination of three-dimensional spaces is the main workhorse of the canonical version of quantum field theory. It is also fundamental to the modern version of quantum field theory without observers, as in the theory of Everett, Wheeler and Deutsch, which led to the theory of the universal quantum computer. It also has many practical uses, from quantum optics to the closure of turbulence and the derivation of the emergent behavior of space-time dynamical systems. But many of us believe that time and space are interchangeable to some degree, and that methods based on 4D Fock space are needed. This letter describes two simple mathematical tools to help make that possible. It discusses how computing the emergent statistics of such systems is a proper generalization of value function estimation in machine learning, which may therefore carry the same complexities with it.
We derive machine learning algorithms from discretized Euclidean field theories, making inference and learning possible within dynamics described by quantum field theory. Specifically, we demonstrate that the ${\ensuremath{\phi}}^{4}$ scalar field theory satisfies the Hammersley-Clifford theorem, therefore recasting it as a machine learning algorithm within the mathematically rigorous framework of Markov random fields. We illustrate the concepts by minimizing an asymmetric distance between the probability distribution of the ${\ensuremath{\phi}}^{4}$ theory and that of target distributions, by quantifying the overlap of statistical ensembles between probability distributions and through reweighting to complex-valued actions with longer-range interactions. Neural network architectures are additionally derived from the ${\ensuremath{\phi}}^{4}$ theory which can be viewed as generalizations of conventional neural networks and applications are presented. We conclude by discussing how the proposal opens up a new research avenue, that of developing a mathematical and computational framework of machine learning within quantum field theory.
Relativistic quantum field theory (QFT) is commonly formulated in terms of operators, asymptotic states, and covariant amplitudes, a perspective that tends to obscure the real-time origin of field dynamics and correlations. Here we formulate quantum fields in a real-time Schrödinger-picture framework, in which fields evolve as probability ensembles on the space of field configurations. Within this formulation, the wavefunctional Ψ[φ,t] encodes a first-order, causal ensemble dynamics on configuration space. Interactions appear as couplings between configuration-space directions, while propagators arise as derived correlation structures rather than as fundamental postulates. Entanglement, scattering amplitudes, and conformal field theory correlators emerge as distinct projections of the same underlying ensemble evolution, corresponding to equal-time, asymptotic, and symmetry-organized observables. Standard operator, diagrammatic, and path-integral formulations are recovered as computational representations of this single real-time dynamics. This organization makes explicit the distinction between fundamental dynamical structure and representational tools in QFT, and clarifies the scope within which ensemble-averaged correlators account for quantum fluctuations, while also delineating the level at which questions associated with individual realizations and randomness would arise beyond the correlator-based field-theoretic description.
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.
The Fock space Hamiltonian H has a simpler structure than its projection Hn to n-particle Hilbert space. It is therefore recommended to diagonalize H—to the extent that this is possible—before one specifies n. Diagonalization of H is possible if one defines the diagonal and nondiagonal parts of an operator appropriately. It is shown that the diagonalized Fock space Hamiltonian L, called an energy operator, contains all information about the eigenvalues of H in a simply coded form. For a spinfree Hamiltonian, spin can be completely eliminated and all interesting quantities are expressible in terms of spinfree excitation operators (generators of the unitary group). The construction of the wave operator W and the energy operator L is formulated in terms of perturbation theory both in the strictly degenerate and the quasidegenerate version. Three variants are discussed that differ in the normalization of W, namely, a intermediate normalization, b and c unitary normalization with two additional conditions, b: WD=WD+, c:σD=(ln W)D=0. The transformation of operator products to normal product form is achieved by means of a generalization of Wick’s theorem to spinfree quantum chemistry. The contributions to the various orders of L, W, and σ are illustrated by means of diagrams. Only variant c guarantees a connected-diagram expansion of the Fock space energy operator L. None of the three variants satisfies a generalization of the Wigner (2n+1) rule, but with a slight change in the normalization from b to b′ or c to c′ (that does, in the case of c′, not affect the connectedness of the diagram expansion) L(2n) and L(2n+1) are expressible in terms of the W(k), k⩽n. A generalization of the variation principle for the Fock space energy operator is given and its consequences are discussed. Generalized Fock space Brillouin conditions are derived and a variational scheme for the construction of the energy operator is proposed. We finally compare the present Fock space approach with the traditional degenerate perturbation theory in n-particle Hilbert space. If the diagonal part ΩD of an operator Ω is defined in the spirit of the theory of the universal wave and energy operators, then the two approaches lead to the same results (for the same normalization), but the Fock space approach looks simpler and more transparent.
A process of idea filtration in two distinct streams of physics i.e., 1) The dimensionality perspective of spacetime, and 2) The quantum perspective leads us to an understanding of what might be a true reality of all that we perceive. The conclusions arrived at in this paper are a bit perplexing in the sense that our perceived reality could be a manifestation of a combination of 4D + n (n > 0) flat space-time, universal wave function, and cognizance. The work is based on a review and analysis of the main concepts in quantum theory, relativistic physics, and cosmology. Key ideas and conclusions are filtered and logically connected to arrive at what might be a view of the true reality. A significant part of the paper is dedicated to the concept of the “observer” and the “ability of cognizance” that should accompany the “observer”. Though the “observer” is central to modern physics, it is not known what constitutes observation, and the term observer, often open to interpretations, does not have a standard definition and hence, is lacking in clarity. In our analysis, we have argued that the environment, in which the observer-observed system is embedded, emerges as an all-knowing, cognizant, and ideal observer that has the knowledge of the observer-observed system. At a philosophical level, we link to the fundamentals of physics, “consciousness” or “ability of cognizance” as an unavoidable and key element in not only carrying out the observation but perhaps, as believed by many, having a role in shaping the reality perceived. In the review and analysis of another stream of physics, that of General Theory of Relativity (GTR) and cosmology, we examine the question of reality from a cosmological and dimensionality perspective. Research on the 4D and 5D constructs of the universe indicates that the “reality” perceived in 4D spacetime as matter, distance, time, etc., is a manifestation of a higher dimensional 4D + n (n > 0) reality. Theoretical research on this front points towards a 4D-spacetime embedded in a 5D or higher dimensional flat space with matter and energy being a manifestation of the higher dimensions. The flow of logic in this paper leans towards a view of an ultimate true reality that is flat 4D + n (n > 0) space combined with cognizance, universal wave function, and the environment.