We reconsider the problem of the interpretation of the Quantum Theory (QT) in the perspective of the entire universe and of Bphr idea that the classical language is the language of our experience and QT acquires a meaning only with a reference to it. We distinguish a classical or macroscopic level, and a quantum or microscopic one that is perceived only through the modifications that it induces in the first. The macroscopic state of the universe is assumed to be specified by a set of variables, a classical energy momentum tensor and some conserved currents, which are supposed to have a well defined value across the entire space-time. To the energy-momentum tensor a classical metric is related by the Einstein equation. The quantum state and dynamics are expressed by the usual QT formalism in terms of a density operator and the ordinary quantum operators in Heisenberg picture. For the macroscopic variables a basic distribution of probability is postulated in terms of a density and the corresponding quantum operators, so in some way their evolution is driven by the underlying QT. Such postulate essentially replaces the usual elfadjoint operators correspondence. For the Universe we adopt a variance of the ΛCDM model with Omega=1, one single inflaton with an Higgs type potential, the initial time at t=minus infinite. The expectation values of all fundamental fields are supposed to vanish for time going to minus infinite. In the framework the scalar fluctuation in the Cosmic Microwave Background are correctly explained giving appropriate calue to the parameters in the potential. As in more conventional models the absence of the tensor fluctuations remains not understood, if even a quantum metric is introduced. This seems to suggest that Gravity is a pure classical phenomenon, what could be consistently accommodated in our formalism by an appropriate even if somewhat ad hoc assumption
We reconsider the problem of the interpretation of the Quantum Theory (QT) in the perspective of the entire universe and of Bphr idea that the classical language is the language of our experience and QT acquires a meaning only with a reference to it. We distinguish a classical or macroscopic level, and a quantum or microscopic one that is perceived only through the modifications that it induces in the first. The macroscopic state of the universe is assumed to be specified by a set of variables, a classical energy momentum tensor and some conserved currents, which are supposed to have a well defined value across the entire space-time. To the energy-momentum tensor a classical metric is related by the Einstein equation. The quantum state and dynamics are expressed by the usual QT formalism in terms of a density operator and the ordinary quantum operators in Heisenberg picture. For the macroscopic variables a basic distribution of probability is postulated in terms of a density and the corresponding quantum operators, so in some way their evolution is driven by the underlying QT. Such postulate essentially replaces the usual elfadjoint operators correspondence. For the Universe we adopt a variance of the ΛCDM model with Omega=1, one single inflaton with an Higgs type potential, the initial time at t=minus infinite. The expectation values of all fundamental fields are supposed to vanish for time going to minus infinite. In the framework the scalar fluctuation in the Cosmic Microwave Background are correctly explained giving appropriate calue to the parameters in the potential. As in more conventional models the absence of the tensor fluctuations remains not understood, if even a quantum metric is introduced. This seems to suggest that Gravity is a pure classical phenomenon, what could be consistently accommodated in our formalism by an appropriate even if somewhat ad hoc assumption
Abstract We reconsider the problem of the interpretation of the quantum theory (QT) in the perspective of the entire universe and of the Bohr's idea that the classical language is the language of our experience and QT acquires a meaning only with a reference to it. We distinguish a classical or macroscopic state, and a quantum or microscopic one that is perceived only through the modifications that it induces in the first. The macroscopic state is specified by a set of variables, a classical energy–momentum tensor and conserved currents, which are supposed to have always a well-defined value. The microscopic state and dynamics are expressed by the usual QT formalism. For the macroscopic variables, a basic distribution of probability is postulated in terms of quantum operators and a statistical operator, what replace the usual probability rule of QT. For the universe, a variance of the ΛCDM model with Ω=1 Ω = 1 is assumed, one inflaton, a Goldstone type potential, initial time at t=−∞ t = - ∞ , expectation values of all fundamental fields vanishing for t→−∞ t → - ∞ . The scalar fluctuations in the cosmic microwave background is correctly explained, but the absence of the tensor fluctuations remains not understood. This seems to suggest that gravity is a pure classical phenomenon what perhaps could be accommodated in our framework.
On the line of a recent paper, the problem of interpretation of Quantum Theory (QT) is reconsidered in a cosmological perspective and of Bohr’s idea that the classical language is the language of our experience and QT acquires a meaning only with a reference to it. A classical or macroscopic level is distinguished and a quantum or microscopic one that is perceived only through the modifications that it induces in the first. The macroscopic state is specified by a classical energy-momentum tensor and conserved currents, which are supposed to have always well-defined values. The microscopic dynamics are expressed by the usual QT formalism. Fundamental idea: the usual probability rule of QT, based on an observables self-ajoint operators correspondence, is replaced with a distribution of probability directly for the macroscopic variables, defined in terms of the corresponding quantum operators and a quantum density operator. For the universe, a variance of the standard model with Ω=1 is assumed, one inflaton, a Goldstone type potential, initial time at t =−∞, expectation values of all fundamental fields vanishing for t → −∞. The scalar fluctuations in the cosmic microwave background can be correctly explained giving appropriate values to the potential parameters, but the absence of the tensor fluctuations remains not understood. This seems to suggest that gravity is a pure classical phenomenon, what could be consistently accommodated in our framework but it is inconsistent with the usual formalism.
Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with suggestive names for concepts matching related concepts of experimental culture) and an interpretational halo (trying to make this relation more precise) that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics (see Part II), where many interpretations coexist and compete for the attention of those interested in what quantum physics really means. In this chapter, an axiomatic introduction to the undisputed formal core of quantum physics is given. The axioms described present nonrelativistic quantum statistical mechanics in the Schrödinger picture. The relativistic case is outside the scope of these axioms, as it must be treated by quantum field theory; see Chapter 7. As in any axiomatic setting (necessary for a formal discipline), there are a number of different but equivalent sets of axioms or postulates that can be used to define formal quantum physics. Since they are equivalent, their choice is a matter of convenience. The choice presented here is a formulation featuring three aspects: It emphasises the similarity of quantum mechanics and classical mechanics, gives most direct access to statistical mechanics, and is free from allusions to measurement. The reason for the first is that this similarity serves as a guide to meaning. From the axiomatic setting for quantum mechanics presented below, classical mechanics (in the Koopman setting discussed in Section 7.10) is obtained by simply restricting the operators to be diagonal, so that all operations happen pointwise on the diagonal elements. Thus, multiplication is commutative, and one can identify operators and functions. In particular, the density operator degenerates into a probability density. Therefore, the quantum case appears as a generalization of the classical case allowing for noncommutativity, so that both q-observables and the density operator are (usually infinite-dimensional) operators. The reason for the second is that statistical mechanics is the main tool for applications of quantum physics to the macroscopic systems we are familiar with. The reason for the third is that realistic measurements constitute a complex process involving macroscopic detectors, hence should be explained by quantum statistical mechanics rather than be part of the axiomatic foundations themselves. (This is in marked contrast to other foundations, and distinguishes the present system of axioms.)
Introduction Contemporary cosmology includes inflation as one of its central components. It corresponds to a period of accelerated expansion thought to have occurred very early in cosmic history, which takes the universe from relatively generic post Planckian era conditions to a stage where it is well described (with exponential accuracy in the number of e-folds) by a flat Robertson–Walker space-time (which describes a homogeneous and isotropic cosmology). It was initially proposed to resolve various naturalness problems: flatness, horizons, and the excess of massive relic objects, such as topological defects, that were expected to populate the universe according to grand unified theories (GUT). Nowadays, it is lauded because it predicts that the universe's mean density should be essentially identical to the critical value, a very peculiar situation which corresponds to a spatially flat universe. The existing data support this prediction. However, its biggest success is claimed to be the natural account for the emergence of the seeds of cosmic structure in terms of primordial quantum fluctuations, and the correct estimate of the corresponding microwave spectrum. This represents, thus, a situation which requires the combined application of general relativity (GR) and quantum theory, and that, moreover, is tied with observable imprints left from the early universe, in both the radiation in cosmic microwave background (CMB), and in the large scale cosmological structure we can observe today. It should not be surprising that, when dealing with attempts to apply our theories to the universe as a whole, we should come face to face with some of the most profound conceptual problems facing our current physical ideas. In fact, J. Hartle had noted long ago [28] that serious difficulties must be faced when attempting to apply quantum theory to cosmology, although the situations that were envisioned in those early works corresponded to the even more daunting problem of full quantum cosmology, and not the limited use of quantum theory to consider perturbative aspects, which is what is required in the treatment of the inflationary origin of the seeds of cosmic structure. This led him and his collaborators to consider the Consistent Histories framework, which is claimed to offer a version of quantum theory that does not rely on any fundamental notion of measurements, as the favored approach to use in such contexts. We will have a bit more to say about this shortly.
Quantum theory plays an increasingly significant role in contemporary early-universe cosmology, most notably in the inflationary origins of the fluctuation spectrum of the microwave background radiation. I consider the two main strategies for interpreting (as opposed to modifying or supplementing) standard quantum mechanics in the light of cosmology. I argue that the conceptual difficulties of the approaches based around an irreducible role for measurement - already very severe - become intolerable in a cosmological context, whereas the approach based around Everett's original idea of treating quantum systems as closed systems handles cosmological quantum theory satisfactorily. Contemporary cosmology, which indeed applies standard quantum theory without supplementation or modification, is thus committed - tacitly or explicitly - to the Everett interpretation.
\begin{abstract} We propose a structural modification of the non-relativistic Schrödinger equation obtained by introducing an additional operator-valued potential term that couples nontrivially to both kinetic and potential contributions. The modified dynamical equation takes the form \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\Phi+ m\delta\Phi- \frac{\hbar^2}{2m}\,\delta\Phi\,\nabla^2\right]\psi,\end{equation} where δΦ represents an additional scalar field or structural correction that deforms the standard Hamiltonian operator. Unlike perturbative potential additions, this term modifies the kinetic structure itself, effectively generating a position-dependent renormalization of the Laplacian operator and inducing a nontrivial coupling between propagation and background geometry. We analyze the mathematical consistency of the modified Hamiltonian, deriving conditions for self-adjointness on L2(R3) and establishing criteria under which unitary time evolution and probability conservation are preserved. The deformation leads to a modified dispersion relation and alters the spectral structure of the system, suggesting an interpretation in terms of emergent or dynamically modified configuration-space geometry. Beyond its formal properties, this framework raises foundational questions regarding the ontological status of the kinetic operator in quantum mechanics. By altering the generator of time evolution itself, the proposal challenges the assumption that the Laplacian represents a fixed Euclidean background structure. Instead, quantum evolution may reflect a deeper geometric or structural layer, in which mass, potential energy, and propagation are relational rather than primitive quantities. This formulation provides a mathematically consistent extension of the Schrödinger framework and opens avenues for interpreting quantum dynamics as arising from modified geometric or higher-structural principles. \end{abstract} \noindent\textbf{Keywords:} Modified Schrödinger Equation; Operator Deformation; Quantum Geometry; Self-Adjoint Hamiltonians; Foundations of Quantum Mechanics. \section{Spacetime–Induced Modification of the Schrödinger Equation} \subsection{1. Introduction} In conventional quantum mechanics, uncertainty is treated as a fundamental property of wavefunctions. In this work we explore an alternative interpretation: that quantum uncertainty may emerge from small-scale spacetime moduli perturbations. We begin in standard 4-dimensional spacetime and then extend the formulation to a constrained multi-time geometry. \subsection{2. Quantum Dynamics in Curved Spacetime} \paragraph{Explanation of the Curved-Spacetime Klein–Gordon Equation.} The equation (□g+ℏ2m2c2)Φ=0describes the dynamics of a scalar quantum field Φ propagating in a curved spacetime background. Here, □g denotes the covariant d'Alembert operator associated with the metric tensor gμν, defined as □gΦ=−g1∂μ(−ggμν∂νΦ), where g=det(gμν) is the determinant of the spacetime metric. The factor −g ensures covariance under general coordinate transformations and accounts for the local volume element of curved spacetime. Physically, this equation generalizes the flat-spacetime Klein–Gordon equation by replacing ordinary derivatives with covariant derivatives determined by the geometry. The metric tensor gμν encodes gravitational effects, meaning that spacetime curvature directly modifies wave propagation. Thus, particle dynamics are influenced not only by mass m but also by geometric structure. When weak metric perturbations are assumed, gμν=ημν+hμν,∣hμν∣≪1,the geometry deviates slightly from Minkowski space ημν. In this regime, gravitational effects enter as small corrections to kinetic terms, enabling a perturbative analysis of how spacetime moduli fluctuations influence quantum evolution. We begin from the relativistic Klein–Gordon equation in curved spacetime: \begin{equation}\left( \Box_g + \frac{m^2 c^2}{\hbar^2} \right)\Phi = 0\end{equation} where \begin{equation}\Box_g \Phi = \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g} g^{\mu\nu} \partial_\nu \Phi \right)\end{equation} Assume weak metric perturbation: \begin{equation}g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},\qquad |h_{\mu\nu}| \ll 1\end{equation} We separate fast rest energy oscillation: \begin{equation}\Phi(x,t) = e^{-i m c^2 t / \hbar} \psi(x,t)\end{equation} Taking the non-relativistic limit yields a modified Schrödinger equation: \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m} \left(1 + \frac{2\phi}{c^2}\right)\nabla^2+ m\phi\right] \psi\end{equation} where ϕ is the weak gravitational potential. \subsection{3. Introducing Spacetime Moduli Perturbations} We decompose: \begin{equation}\phi(x,t) = \bar{\phi}(x,t) + \delta\phi(x,t)\end{equation} where δϕ represents small-scale spacetime moduli fluctuations. The Hamiltonian becomes: \begin{equation}H = H_0 + \delta H\end{equation} with \begin{equation}H_0 = -\frac{\hbar^2}{2m}\nabla^2 + m\bar{\phi}\end{equation} and \begin{equation}\delta H \approx m\,\delta\phi(x,t)-\frac{\hbar^2}{m c^2}\delta\phi(x,t)\nabla^2\end{equation} Thus the modified Schrödinger equation is: \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\bar{\phi}+ m\delta\phi-\frac{\hbar^2}{m c^2}\delta\phi\nabla^2\right]\psi\end{equation} \textbf{Interpretation:}Spacetime fluctuations enter quantum evolution as both an effective potential term and a modulation of kinetic energy. \paragraph{Physical Interpretation and Motivation.} The decomposition ϕ(x,t)=ϕˉ(x,t)+δϕ(x,t)separates spacetime structure into a smooth background component ϕˉ and a small fluctuating component δϕ. The background term represents the classical gravitational potential arising from large-scale geometry, while δϕ encodes microscopic spacetime moduli perturbations. These perturbations may originate from quantum geometric effects, discrete spacetime structure, or Planck-scale fluctuations. Physically, this decomposition allows us to isolate how small-scale spacetime irregularities influence quantum dynamics. By writing the Hamiltonian asH=H0+δH,we distinguish between ordinary quantum evolution in a smooth geometry (H0) and additional corrections induced by spacetime fluctuations (δH). The termδH≈mδϕ−mc2ℏ2δϕ∇2shows that spacetime perturbations modify both the potential energy and the kinetic structure of the particle. \paragraph{Intuitive Picture for a Non-Expert.} A helpful analogy is to imagine a wave traveling across the surface of a calm lake. If the water surface is perfectly smooth, the wave propagates cleanly. However, if the surface contains tiny ripples, the wave becomes distorted and slightly scattered. In this analogy, the smooth lake corresponds to ϕˉ, while the small ripples correspond to δϕ. Similarly, a quantum particle is described by a wavefunction. If spacetime itself contains microscopic fluctuations, then the wavefunction does not evolve in a perfectly smooth background. Instead, it experiences tiny distortions in its phase and amplitude. Over time, these small distortions accumulate and can lead to spreading of momentum and position distributions. Thus, the modified Schrödinger equation iℏ∂tψ=[−2mℏ2∇2+mϕˉ+mδϕ−mc2ℏ2δϕ∇2]ψ mathematically encodes the idea that spacetime itself actively influences quantum states. Rather than treating uncertainty as purely intrinsic, this framework allows us to explore the possibility that part of quantum indeterminacy arises from dynamical geometric perturbations at very small scales. \subsection{4. Stochastic Model of Spacetime Perturbation} \paragraph{Physical Motivation for the Stochastic Model.} In the previous section, spacetime perturbations δϕ(x,t) were introduced as small deviations from a smooth geometric background. To proceed analytically, we model these perturbations as stochastic fluctuations. This assumption does not imply that spacetime is fundamentally random; rather, it reflects our effective description after coarse-graining microscopic geometric degrees of freedom. In approaches such as Loop Quantum Gravity, spacetime geometry is fundamentally discrete at the Planck scale. When averaged over many microscopic quantum geometric states, these discrete structures manifest as effective fluctuations at larger scales. The condition⟨δϕ(x,t)⟩=0states that there is no preferred bias in the fluctuations: on average, spacetime remains centered around the smooth background ϕˉ. The two-point correlation function⟨δϕ(x,t)δϕ(x′,t′)⟩=C(x−x′)δ(t−t′)encodes how strongly spacetime fluctuations at one location are related to those at another. The spatial function C(x−x′) determines the correlation length, while the temporal delta function represents rapid (short-time) fluctuations. We introduce the interaction HamiltonianHint(t)=m∫d3xδϕ(x,t)∣x⟩⟨x∣to capture how spacetime perturbations couple locally to the mass-energy of the particle. The use of a second-order cumulant expansion is justified when fluctuations are weak but rapidly varying; it allows us to derive an effective evolution equation for the ensemble-averaged density matrix ρˉ. The resulting term∂tρˉ(x,x′,t)=−Γ(x−x′)ρˉ(x,x′,t)describes decoherence: of