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Geometrical Formulation of Quantum Mechanics
General Relativity and Quantum Cosmology · Submitted 23 Jun 1997
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim More importantly, it suggests generalizations of quantum mechanics.
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The abstract, sentence by sentence
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have, in addition, a compatible Riemannian metric), observables are represented by certain real-valued functions on this space and the Schr\"odinger evolution is captured by the symplectic flow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features---such as uncertainties and state vector reductions---which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric---a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semi-classical considerations and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | States of a quantum mechanical system are represented by rays in a complex Hilbert space. | background 1.00 | 0.07 | 0.04 | asserted 1.00 |
| s1 | The space of rays has, naturally, the structure of a K\"ahler manifold. | background 0.67 | 0.16 | 0.06 | asserted 1.00 |
| s2 | This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. | contribution 0.79 | 0.67 | 0.34 | asserted 1.00 |
| s3 | In particular, states are now represented by points of a symplectic manifold (which happens to have, in addition, a compatible Riemannian metric), observables are represented by certain real-valued functions on this space and the Schr\"odinger evolution is captured by the symplectic flow generated by a Hamiltonian function. | contribution 0.70 | 0.59 | 0.29 | asserted 1.00 |
| s4 | There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. | interpretation 0.92 | 0.50 | 0.09 | asserted 0.99 |
| s5 | Features---such as uncertainties and state vector reductions---which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric---a structure which is absent in classical mechanics. | interpretation 0.42 | 0.59 | 0.10 | asserted 0.96 |
| s6 | The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semi-classical considerations and the WKB approximation. | interpretation 0.56 | 0.64 | 0.12 | asserted 0.88 |
| s7 | More importantly, it suggests generalizations of quantum mechanics. | interpretation 0.71 | 0.73 | 0.41 | hedged possibility 0.97 |
| s8 | The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. | result 0.65 | 0.40 | 0.09 | asserted 0.99 |
| s9 | The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. | contribution 0.63 | 0.66 | 0.22 | asserted 0.98 |
| s10 | Finally, it also suggests directions in which more radical generalizations may be found. | future work 0.88 | 0.65 | 0.16 | hedged possibility 0.99 |
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Original abstract
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have, in addition, a compatible Riemannian metric), observables are represented by certain real-valued functions on this space and the Schr\"odinger evolution is captured by the symplectic flow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features---such as uncertainties and state vector reductions---which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric---a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semi-classical considerations and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.
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References
How later papers used it
46 later papers cite it
1 later paper by other authors is listed below: 1 used its method or tool.
Uses its method or tool 1 of 2 papers with this label
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Used properties of this paper's geometric reformulation of quantum mechanics to write down equations of motion for the expectation values of quantum mechanical operators.
Constructing Dualities from Quantum State Manifolds (2015) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
48 arXiv papers cite this in 95 sentences
Papers citing it · 47 sentences without a confident label
Builds on it 1 paper
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In the current situation we will consider the geometric structure of $$ for infinite dimensional cases [ref] .
Spread complexity as classical dilaton solutions · Geometry of quantum state manifolds · 0.81
Uses its method or tool 2 papers
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The quantum algebra thus obtained allows to construct a quantum phase space coordinatised by expectation values and moments of the relevant operators, according to standard techniques [ref] .
Effective cosmology from one-body operators in group field theory · Introduction · 0.99
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By using properties of the geometric reformulation of quantum mechanics [ref] we are able to write down equations of motion for the expectation values of quantum mechanical operators.
Constructing Dualities from Quantum State Manifolds · Variables · 1.00
Compares with it 1 paper
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The geometric formulation of quantum mechanics on Kähler manifolds discussed in [ref] differs from our approach on linear Kähler spaces.
Notes on Real Quantum Mechanics in a Kahler Space · Introduction · 0.98
Cites it as support 2 papers
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Now, as shown in [ref] , the quantum evolution of the quantum system as prescribed by Schrödinger's equation corresponds with the Hamiltonian flow of $F_H$ as defined above.
Complexity measures in QFT and constrained geometric actions · Quantum action, Hamilton-Jacobi and the semiclassical limit · 1.00
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In [ref] it was shown that appropriate geometric structures allow a given quantum mechanical model to be reformulated entirely in terms of these structures.
Constructing Dualities from Quantum State Manifolds · Summary · 0.98
Cites it as background 27 papers
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A more thorough review can be found in [ref] .
Corrections to the Friedmann Equations from LQG for a Universe with a Free Scalar Field · Framework for Effective Theories · 0.96
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Nambu QM generalizes canonical QM via the geometric formulation of QM [ref] .
Constraints on a Generalization of Geometric Quantum Mechanics from Neutrino and $B^0$-$\overline{B^0}$ Oscillations · Introduction · 0.88
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[ref] The symplectic form on $P()$ is given by [equation] Since the expectation value $A(ψ)$ in ([ref]) does not depend on the choice of element of span{$ψ}$, the quantum observable $a:P() →ℝ$ is well-defined as [equation] with corresponding hamiltonian vector field $X_a$ on $P(ℋ)$ defined by $ι_X_aΩ=da$.
The uncertainty principle and the energy identity for holomorphic maps in geometric quantum mechanics · Quantum mechanics as a Hamiltonian dynamical system · 0.94
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