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arXiv:2602.13648 · Physical Review Letters

Comment on "Evolution Operator Can Always Be Separated into the Product of Holonomy and Dynamic Operators"

Adam Fredriksson, Erik Sjöqvist

quant-phSubmitted Feb 14, 2026• Announced Feb 17, 2026• 3 citations
arXivHTMLPDF

Abstract

We show that the claim in Ref. [PRL 131, 200202 (2023)], that the quantum time evolution always can be written as a product of a holonomy operator and a dynamic operator, is false, as it is based on a circular use of the time evolution operator.
Journal reference
Phys. Rev. Lett. 136, 068901 (2026)
DOI
10.1103/d5rt-jpjk
Comments
Comment on arXiv:2311.09597

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Comment on "Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution"

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I point out that if one defines the operator $U_R(t)$ as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of $U_R(t)$ as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator $H$ but an operator $H'$ which differs from $H$ if the metric operator is time-dependent. In the latter case this effective Hamiltonian $H'$ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.

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In arXiv:0710.5653v1 M. Znojil claims that he has found and corrected an error in my paper: [Phys. Lett. B \textbf{650}, 208 (2007), arXiv:0706.1872v2] and that it is possible to escape its main conclusion, namely that the unitarity of the time-evolution and observability of the Hamiltonian imply time-independence of the metric operator. In this note I give a very short calculation showing that the analysis given by M. Znojil also leads to the same conclusion as of my above-mentioned paper. This is actually a reconfirmation of the validity of the results of the latter paper.

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Reply to ‘Comment on “Overcoming misconceptions in quantum mechanics with the time evolution operator”’

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We contend that the statement attributing to us the implication that the wave functions ?I(x, t) and ?II(x, t) are different cannot not be logically supported. On the contrary, throughout our paper ?Overcoming misconceptions in quantum mechanics with the time evolution operator? (Garcia Quijas and Ar?valo Aguilar 2007 Eur. J. Phys. 28 147) we constructed an argument to give support to the statement that the two wave functions ?I(x, t) and ?II(x, t) are the same. In other words, in our paper we really did appreciate that methods I and II lead to identical results.

European Journal of Physics•Apr 22, 2013•4 citations
I point out that if one defines the operator UR(t)U_R(t)UR​(t) as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of UR(t)U_R(t)UR​(t) as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator HHH but an operator H′H'H′ which differs from HHH if the metric operator is time-dependent. In the latter case this effective Hamiltonian H′H'H′ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.
In arXiv:0710.5653v1 M. Znojil claims that he has found and corrected an error in my paper: [Phys. Lett. B \textbf{650}, 208 (2007), arXiv:0706.1872v2] and that it is possible to escape its main conclusion, namely that the unitarity of the time-evolution and observability of the Hamiltonian imply time-independence of the metric operator. In this note I give a very short calculation showing that the analysis given by M. Znojil also leads to the same conclusion as of my above-mentioned paper. This is actually a reconfirmation of the validity of the results of the latter paper.
The Unitary Dynamical Group in Schrödinger's Picture Up to now we have been concerned with the description of a physical system at a fixed instant of time. The time evolution of quantum systems constitutes a many-faced, wide issue which, however, is somewhat tangential to the main aim of this volume. We shall touch upon it only briefly, restricting ourselves to the more familiar aspects of the problem. We assume, to begin with, that the physical system has no superselection rules, so that states and physical quantities correspond one-to-one to density operators and self-adjoint operators, respectively, on the Hilbert space. To describe the dynamical evolution of the system one has to specify the way the representatives of states and physical quantities evolve in, which is fixed in time, since the physical system is supposed to preserve its identity. There is an old tenet according to which there are two kinds of time evolution: the discontinuous, nondeterministic evolution undergone by the quantum system when a measurement is made on it, and the continuous, deterministic evolution caused by the interaction with external forces or with other quantum systems. The first kind of evolution will be dealt with in Chapter 8, where we shall find it less alarming than is often claimed.
We develop a framework for quasi-holonomy in non-adiabatic quantum time evolution of subspaces along loops in a complex Grassmannian. By factoring the Schrödinger evolution into dynamical and connection-induced contributions in a moving basis, we obtain an effective geometric generator that depends explicitly on the dynamical propagator. This quasi-connection does not define a genuine connection on the original Grassmann bundle, since its gauge transformation law acquires a history-dependent, nonlocal term. Other ways of factoring the Schrödinger evolution are briefly discussed. All these approaches suffer from the same type of history-dependence, thereby defining transport of subspaces in which geometric and dynamical effects are generally intertwined, just as in the case of the quasi-holonomy. Our work sheds light on the issue of separating quantum evolution of subspaces into holonomic and dynamical parts from an essentially gauge-theoretic perspective.
The geometric phase is a fundamental quantity characterizing the holonomic feature of quantum systems. It is well known that the evolution operator of a quantum system undergoing a cyclic evolution can be simply written as the product of holonomic and dynamical components for the three special cases concerning the Berry phase, adiabatic non-Abelian geometric phase, and nonadiabatic Abelian geometric phase. However, for the most general case concerning the nonadiabatic non-Abelian geometric phase, how to separate the evolution operator into holonomic and dynamical components is a long-standing open problem. In this Letter, we solve this open problem. We show that the evolution operator of a quantum system can always be separated into the product of holonomy and dynamic operators. Based on it, we further derive a matrix representation of this separation formula for cyclic evolution, and give a necessary and sufficient condition for a general evolution being purely holonomic. Our finding is not only of theoretical interest itself, but also of vital importance for the application of quantum holonomy. It unifies the representations of all four types of evolution concerning the adiabatic/nonadiabatic Abelian/non-Abelian geometric phase, and provides a general approach to realizing purely holonomic evolution.
We contend that the statement attributing to us the implication that the wave functions ?I(x, t) and ?II(x, t) are different cannot not be logically supported. On the contrary, throughout our paper ?Overcoming misconceptions in quantum mechanics with the time evolution operator? (Garcia Quijas and Ar?valo Aguilar 2007 Eur. J. Phys. 28 147) we constructed an argument to give support to the statement that the two wave functions ?I(x, t) and ?II(x, t) are the same. In other words, in our paper we really did appreciate that methods I and II lead to identical results.