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arXiv:2412.05997 · Quantum

Doubly Quantum Mechanics

Vincenzo Esposito, Giuseppe Fabiano, Domenico Frattulillo, Flavio Mercati

Published Apr 24, 2025• 1 citation
arXivHTMLPDF

Abstract

Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the geometrical configurations of physical systems, measurement apparata, and reference frame transformations are themselves quantized and described by ''geometry'' states in a Hilbert space. We develop the formalism for spin- 12 measurements by promoting the group of spatial rotations SU(2) to the quantum group SUq(2) and generalizing the axioms of Quantum Theory in a covariant way. As a consequence of our axioms, the notion of probability becomes a self-adjoint operator acting on the Hilbert space of geometry states, hence acquiring novel non-classical features. After introducing a suitable class of semi-classical geometry states, which describe near-to-classical geometrical configurations of physical systems, we find that probability measurements are affected, in these configurations, by intrinsic uncertainties stemming from the quantum properties of SUq(2) . This feature translates into an unavoidable fuzziness for observers attempting to align their reference frames by exchanging qubits, even when the number of exchanged qubits approaches infinity, contrary to the standard SU(2) case.

DOI
10.22331/q-2025-04-24-1721

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Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the geometrical configurations of physical systems, measurement apparata, and reference frame transformations are themselves quantized and described by ''geometry'' states in a Hilbert space. We develop the formalism for spin- 12 measurements by promoting the group of spatial rotations SU(2) to the quantum group SUq(2) and generalizing the axioms of Quantum Theory in a covariant way. As a consequence of our axioms, the notion of probability becomes a self-adjoint operator acting on the Hilbert space of geometry states, hence acquiring novel non-classical features. After introducing a suitable class of semi-classical geometry states, which describe near-to-classical geometrical configurations of physical systems, we find that probability measurements are affected, in these configurations, by intrinsic uncertainties stemming from the quantum properties of SUq(2) . This feature translates into an unavoidable fuzziness for observers attempting to align their reference frames by exchanging qubits, even when the number of exchanged qubits approaches infinity, contrary to the standard SU(2) case.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum jjj and introduce the transformations between them by upgrading the Euler angles occurring in classical SO(3)\text{SO}(3)SO(3) spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large jjj, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonians (such as that of the Heisenberg model) are invariant under a larger group of quantum reference frame transformations. Our result is a development of the quantum reference frame formalism for a non-Abelian group.
Non-commutative spacetime and quantum groups have been argued to capture non-classical features of spacetime and its symmetries in quantum gravity. In this letter, we show that employing the SUq(2)SU_q(2)SUq​(2) quantum group to describe rotational symmetry for spin-12\frac{1}{2}21​ systems and Stern-Gerlach apparatuses leads to the description of probabilities of outcomes of spin measurements in terms of non-commuting operators. As a result, we obtain an uncertainty principle between different probability operators, realising a notion of indefinite probabilities. This is then reflected in the non-commutativity of the entries of the rotation matrix relating the reference frames of two observers, hence fundamentally preventing them from sharply measuring their relative orientation.
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.
Abstract Quantum gravity is expected to introduce quantum aspects into the description of reference frames. Here we begin exploring how quantum gravity induced deformations of classical symmetries could modify the transformation laws among reference frames in an effective regime. We invoke the quantum group SUq(2) as a description of deformed spatial rotations and interpret states of a representation of its algebra as describing the relative orientation between two reference frames. This leads to a quantization of one of the Euler angles and to an aspect of agency dependence: space is reconstructed as a collection of fuzzy points, exclusive to each agent, which depends on their choice of reference frame. Each agent can choose only one direction in which points can be sharp, while points in all other directions become fuzzy in a way that depends on this choice. Two agents making different choices will thus observe the same points with different degrees of fuzziness.
A geometric framework for quantum mechanics arose during the mid 1970s when authors such as Cantoni explored the notion of generalized transition probabilities, and Kibble promoted the idea that the space of pure quantum states provides a natural quantum mechanical analogue for classical phase space. This central idea can be seen easily since the projection of Schrödinger's equation from a Hilbert space into the space of pure spaces is a set of Hamilton's equations. Over the intervening years considerable work has been carried out by a variety of authors and a mature description of quantum mechanics in geometric terms has emerged with many applications. This current offering would seem ideally placed to review the last thirty years of progress and relate this to the most recent work in quantum entanglement. Bengtsson and Zyczkowski's beautifully illustrated volume, Geometry of Quantum States (referred to as GQS from now on) attempts to cover considerable ground in its 466 pages. Its topics range from colour theory in Chapter 1 to quantum entanglement in Chapter 15—to say that this is a whirlwind tour is, perhaps, no understatement. The use of the work 'introduction' in the subtitle of GQS , might suggest to the reader that this work be viewed as a textbook and I think that this interpretation would be incorrect. The authors have chosen to present a survey of different topics with the specific aim to introduce entanglement in geometric terms—the book is not intended as a pedagogical introduction to the geometric approach to quantum mechanics. Each of the fifteen chapters is a short, and mostly self-contained, essay on a particular aspect or application of geometry in the context of quantum mechanics with entanglement being addressed specifically in the final chapter. The chapters fall into three classifications: those concerned with the mathematical background, those which discuss quantum theory and the foundational aspects of the geometric framework, and applications of the geometric approach. The first four chapters contain the standard mathematics required to understand the rest of the material presented: specific areas in colour theory, set theory, probability theory, differential geometry and projective geometry are all covered with an eye to the material that follows. Chapter 5 starts the first real discussion of quantum theory in GQS and serves as an elegant, succinct introduction to the geometry which underlies quantum theory. This may be the most worthwhile chapter for the casual reader who wants to understand the key ideas in this field. Chapter 6 builds on the discussion in Chapter 5, introducing a group theoretic approach to understand coherent states and Chapter 7 describes a geometric tool in the form of an approach to complex projective geometry called 'the stellar representation'. Chapter 8 returns to a more purely quantum mechanical discussion as the authors turn to study the space of density matrices. This chapter completes the discussion which started in Chapter 5. Chapter 9 begins the part of the book concerned with applications of the geometric approach. From this point on the book aims, specifically, to prepare the reader for the material in Chapter 15 beginning with a discussion on the purification of mixed quantum states. In the succeeding chapters a definite choice has been made to present a geometric approach to certain quantum information problems. For example, Chapter 10 contains an extremely well formulated discussion of measurement and positive operator-valued measures with several well illustrated examples and Chapter 11 reopens the discussion of density matrices. Entropy and majorization are again revisited in Chapter 12 in much greater detail than in previous chapters. Chapters 13 and 14 concern themselves with a discussion of various metrics and their relation to the problem of distinguishing between probability distributions and their suitability as probability measures.