Can classically branched dynamics associated with spontaneous time-translation-symmetry breaking be quantized without explicitly retaining the branches of the Hamiltonian? We investigate this question using a reduced phase space. formulation in which the Hamiltonian is single-valued, while the critical structure of the classical Legendre map is encoded in a singular, momentum-dependent symplectic geometry. Quantization of the reduced Poisson algebra yields a deformed position-momentum commutator whose singularities determine distinct regular momentum sectors. Using the positive measure induced by the reduced symplectic structure, we identify a bounded central sector and two unbounded outer sectors. In the central sector, states with vanishing mean momentum obey a nonzero lower bound on position uncertainty, subject to the operator-domain assumptions of the representation. We further identify a possible field-theoretic connection with k-essence, where nonlinear scalar-field kinetics can produce finite-velocity degeneracies of the Legendre map. The relationship between the resulting reduced quantum description and branch-matching quantization remains unresolved and requires a comparison of the corresponding Hilbert spaces and operator domains.
AI Summary
Queued: an LLM will read the paper as soon as there is capacity.
Comments
The manuscript has 16 pages, 3 figures, and 1 table
Can classically branched dynamics associated with spontaneous time-translation-symmetry breaking be quantized without explicitly retaining the branches of the Hamiltonian? We investigate this question using a reduced phase space. formulation in which the Hamiltonian is single-valued, while the critical structure of the classical Legendre map is encoded in a singular, momentum-dependent symplectic geometry. Quantization of the reduced Poisson algebra yields a deformed position-momentum commutator whose singularities determine distinct regular momentum sectors. Using the positive measure induced by the reduced symplectic structure, we identify a bounded central sector and two unbounded outer sectors. In the central sector, states with vanishing mean momentum obey a nonzero lower bound on position uncertainty, subject to the operator-domain assumptions of the representation. We further identify a possible field-theoretic connection with k-essence, where nonlinear scalar-field kinetics can produce finite-velocity degeneracies of the Legendre map. The relationship between the resulting reduced quantum description and branch-matching quantization remains unresolved and requires a comparison of the corresponding Hilbert spaces and operator domains.
The algebraic properties of a strict deformation quantization are analyzed on the classical phase space R2. The corresponding quantization maps enable us to take the limit for ħ→0 of a suitable sequence of algebraic vector states induced by ħ-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on R2, defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of ∗ algebras) parametrized by ħ. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in R2 depending on the symmetry of the potential. In addition, since this ∗-algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off ħ. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models.
The algebraic properties of a strict deformation quantization are analyzed on the classical phase space [Formula: see text]. The corresponding quantization maps enable us to take the limit for [Formula: see text] of a suitable sequence of algebraic vector states induced by [Formula: see text]-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on [Formula: see text], defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of [Formula: see text] algebras) parametrized by [Formula: see text]. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in [Formula: see text] depending on the symmetry of the potential. In addition, since this [Formula: see text]-algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off [Formula: see text]. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.
Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.
We consider questions related to a quantization scheme in which a classical variable f:Ω→R on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Γ(fk), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map Γ is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.
We consider questions related to a quantization scheme in which a classical variable f:Ω\to R on a phase space Ωis associated with a semispectral measure E^f, such that the moment operators of E^f are required to be of the form Γ(f^k), with Γa suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in some Hilbert space. In particular, we investigate the situation where the map Γis implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ωis first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R^2, where we determine explicitly the relevant operators Γ(f^k) for certain variables f, in the case where the quantization map Γis implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.