<i>SU</i>(2) particle sigma model: the role of contact symmetries in global quantization

Aldaya, V.; Guerrero, Julio Becerra; López-Ruiz, Francisco F.; Cossío, Francisco González de

doi:10.1088/1751-8113/49/50/505201

Cited by 9 publications

(25 citation statements)

References 28 publications

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“…so that dΘ/Z L (π4) reproduces (10). The Noether invariants are also regained by contracting Θ with the rightinvariant vector fields (see [1]).…”

Section: Brief Report On Group-theoretical Quantizationmentioning

confidence: 97%

“…The next basis will be that of common eigenstates of the commuting operators Ĥ ,Ĵ 2 ,Ĵ 3 . It is convenient to resort to hyperspherical coordinates (2) where the required eigen-problem can be easily solved with the result [1]:…”

Section: A12 Basis Of Eigenstates Of the Hamiltonianmentioning

confidence: 99%

“…We finally conclude with some comments on future applications and or extensions of the present research. Several Appendices are given at the end with some technical material; we shall also take the opportunity of completing some further details which where absent from the basic study in the previous paper [1].…”

Section: Introductionmentioning

confidence: 99%

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$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

Guerrero

López-Ruiz

Aldaya

2020

J. Phys. A: Math. Theor.

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We perform the momentum-space quantization of a spin-less particle moving on the SU (2) group manifold, that is, the three-dimensional sphere S 3 , by using a non-canonical method entirely based on symmetry grounds. To achieve this task, non-standard (contact) symmetries are required as already shown in a previous article where the configuration-space quantization was given. The Hilbert space in the momentum space representation turns out to be made of a subset of (oscillatory) solutions of the Helmholtz equation in four dimensions. The most relevant result is the fact that both the scalar product and the generalized Fourier transform between configuration and momentum spaces deviate notably from the naively expected expressions, the former exhibiting now a non-trivial kernel, under a double integral, traced back to the non-trivial topology of the phase space, even though the momentum space as such is flat. In addition, momentum space itself appears directly as the carrier space of an irreducible representation of the symmetry group, and the Fourier transform as the unitary equivalence between two unitary irreducible representations.

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“…so that dΘ/Z L (π4) reproduces (10). The Noether invariants are also regained by contracting Θ with the rightinvariant vector fields (see [1]).…”

Section: Brief Report On Group-theoretical Quantizationmentioning

confidence: 97%

Section: A12 Basis Of Eigenstates Of the Hamiltonianmentioning

confidence: 99%

See 1 more Smart Citation

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

Guerrero

López-Ruiz

Aldaya

2020

J. Phys. A: Math. Theor.

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“…For the sake of simplicity, we assume that such system is free of gauge symmetries from the beginning. There always exists a set of symmetries whose Noether charges parametrize the space of initial conditions [18], and hence the whole space of solutions which we call S. These symmetries are in general complicated symmetries that cannot be found explicitly; they can only be found for integrable systems. Knowing that they exist is enough for our purposes.…”

Section: A Mechanism For the Emergence Of Gauge Symmetriesmentioning

confidence: 99%

Toward a Mechanism for the Emergence of Gravity

et al. 2021

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One of the main problems that emergent-gravity approaches face is explaining how a system that does not contain gauge symmetries ab initio might develop them effectively in some regime. We review a mechanism introduced by some of the authors for the emergence of gauge symmetries in [JHEP 10 (2016) 084] and discuss how it works for interacting Lorentz-invariant vector field theories as a warm-up exercise for the more convoluted problem of gravity. Then, we apply this mechanism to the emergence of linear diffeomorphisms for the most general Lorentz-invariant linear theory of a two-index symmetric tensor field, which constitutes a generalization of the Fierz–Pauli theory describing linearized gravity. Finally we discuss two results, the well-known Weinberg–Witten theorem and a more recent theorem by Marolf, that are often invoked as no-go theorems for emergent gravity. Our analysis illustrates that, although these results pinpoint some of the particularities of gravity with respect to other gauge theories, they do not constitute an impediment for the emergent gravity program if gauge symmetries (diffeomorphisms) are emergent in the sense discussed in this paper.

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“…The standard classical approach to a particle moving on a Riemann manifold with metric g i j (x) is established by the Lagrangian (see [38,39] and references therein):…”

Section: Particle Moving On Su (2): Pnlσ Mmentioning

confidence: 99%

Structure of gauge theories

Aldaya

2021

Eur. Phys. J. Plus

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Elementary interactions are formulated according to the principle of minimal interaction although paying special attention to symmetries. In fact, we aim at rewriting any field theory on the framework of Lie groups, so that, any basic and fundamental physical theory can be quantized on the grounds of a group approach to quantization. In this way, connection theory, although here presented in detail, can be replaced by “jet-gauge groups” and “jet-diffeomorphism groups.” In other words, objects like vector potentials or vierbeins can be given the character of group parameters in extended gauge groups or diffeomorphism groups. As a natural consequence of vector potentials in electroweak interactions being group variables, a typically experimental parameter like the Weinberg angle $$\vartheta _W$$ ϑ W is algebraically fixed. But more general remarkable examples of success of the present framework could be the possibility of properly quantizing massive Yang–Mills theories, on the basis of a generalized Non-Abelian Stueckelberg formalism where gauge symmetry is preserved, in contrast to the canonical quantization approach, which only provides either renormalizability or unitarity, but not both. It proves also remarkable the actual fixing of the Einstein Lagrangian in the vacuum by generalized symmetry requirements, in contrast to the standard gauge (diffeomorphism) symmetry, which only fixes the arguments of the possible Lagrangians.

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SU(2) particle sigma model: the role of contact symmetries in global quantization

Cited by 9 publications

References 28 publications

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

Toward a Mechanism for the Emergence of Gravity

Structure of gauge theories

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SU(2) particle sigma model: the role of contact symmetries in global quantization

Cited by 9 publications

References 28 publications

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S 3

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S 3

Toward a Mechanism for the Emergence of Gravity

Structure of gauge theories

Contact Info

Product

Resources

About

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³