Quantum Fields on dS and AdS

Moschella, Ugo

doi:10.1007/s00023-003-0925-y

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…In Figs. (1)(2)(3)(4) we present the spectra of the mesons of interest. Tables 1 and 2 contain the results of the least square parameter fit.…”

Section: Degeneracy Of High-lying Unflavored Mesonsmentioning

confidence: 99%

“…The fact is that according to the gauge-gravity duality conjecture [1], a string theory at the boundary of AdS 5 appears dual to a conformal field theory in (1+3) dimensions, associated with QCD [2]. The AdS 5 geometry is defined as a surface embedded in a (2+4)dimensional flat ambient space of two time-like (X 0 , X 5 ) and four space-like (X 1 , X 2 , X 3 , X 4 ) dimensions, according to [3]…”

Section: Introductionmentioning

confidence: 99%

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Conformal symmetry breaking and degeneracy of high-lying unflavored mesons

Kirchbach

Pallares-Rivera

Compean

et al. 2012

J. Phys.: Conf. Ser.

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Abstract. We show that though conformal symmetry can be broken by the dilaton, such can happen without breaking the conformal degeneracy patterns in the spectra. Our argumentation goes as follows: We departure from the gauge-gravity duality which predicts on the boundaries of the AdS5 geometry a conformal theory, associated with QCD at high temperatures, and consider S 1 × S 3 slicing. The inverse radius, R, of S 3 relates to the temperature of the deconfinement phase transition and has to satisfy,hc/R ≫ ΛQCD. On S 3 , whose isometry group is SO(4), we then focus on the eigenvalue problem of the conformal Laplacian there, given bywith K 2 standing for the Casimir invariant of the so(4) algebra. This eigenvalue problem describes the spectrum of a scalar particle, to be associated with a qq system. Such a spectrum is characterized by a (K + 1) 2 -fold degeneracy of its levels, with K ∈ [0, ∞). We then break the conformal S 3 metric, ds 2 = dχ 2 + sin 2 χ(dθ 2 + sin 2 θdϕ 2 ) -in polar χ, θ, and azimuthal ϕ coordinates-according to, d s 2 = e −bχ (1 + b 2 /4)dχ 2 + sin 2 χ(dθ 2 + sin 2 θdϕ 2 ) , and attribute the symmetry breaking scale bh 2 c 2 /R 2 to the dilaton. Next we show that the above metric deformation is equivalent to a breaking of the conformal curvature of S 3 by a term proportional to b cot χ, and that the perturbed conformal Laplacian is equivalent to K 2 + cK , with cK a representation constant, and K 2 being again an so(4) Casimir invariant, but this time in a representation unitarily inequivalent to the 4D rotational one. As long as the spectra before and after the symmetry breaking happen to be determined each by eigenvalues of a Casimir invariant of an so(4), no matter whether or not in a representation that generates the orthogonal group SO(4) as a subgroup of the conformal group SO(2, 4), the degeneracy patterns remain unaltered though the conformal symmetry breaks at the level of the representation of the algebra. We fit the S 3 radius and theh 2 c 2 b/R 2 scale to the high-lying excitations in the spectra of the unflavored mesons, as reported by the Crystal Barrel collaboration, and observe the correct tendency of thehc/R = 373 MeV value to notably exceed ΛQCD. The size of the symmetry breaking scale is calculated ashc √ b/R = 673.7 MeV.

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“…In Figs. (1)(2)(3)(4) we present the spectra of the mesons of interest. Tables 1 and 2 contain the results of the least square parameter fit.…”

Section: Degeneracy Of High-lying Unflavored Mesonsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Conformal symmetry breaking and degeneracy of high-lying unflavored mesons

Kirchbach

Pallares-Rivera

Compean

et al. 2012

J. Phys.: Conf. Ser.

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“…The rôle of Wightman axioms ( [10]) for a QFT in Minkowski are replaced by other axioms of similar nature ( [3], [4] and [9]) where the spectral condition is replaced by the notion of minimal analyticity. Specifically the axioms to be satisfied are (for a quasi-free theory):…”

Section: De Sitter Geometry and The Massive Scalar Fieldmentioning

confidence: 99%

Massless Scalar Field in a Two-dimensional de Sitter Universe

Bertola¹,

Corbetta²,

Moschella³

2007

Rigorous Quantum Field Theory

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We study the massless minimally coupled scalar field on a two-dimensional de Sitter space-time in the setting of axiomatic quantum field theory. We construct the invariant Wightman distribution obtained as the renormalized zero-mass limit of the massive one. Insisting on gauge invariance of the model we construct a vacuum state and a Hilbert space of physical states which are invariant under the action of the whole de Sitter group. We also present the integral expression of the conserved charge which generates the gauge invariance and propose a definition of dual field.

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Perturbing free motions on hyperspheres without degeneracy lift

Pallares-Rivera¹,

Rosales-Aldape²,

Kirchbach³

2014

J. Phys. A: Math. Theor.

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We consider quantum motion on S 3 perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, ℓ, the ordinary orbital angular momentum value, and another, continuous parameter, b, through which an external scale is introduced. We argue that a loss of the geometric hyper-spherical so(4) symmetry of the free motion occurs that leaves intact the unperturbed hydrogen-like degeneracy patterns characterizing the spectrum under discussion. The argument is based on the observation that the expansions of the Scarf I wave functions for fixed ℓ-values in the basis of properly identified so(4) representation functions are power series in the perturbation parameter, b, in which carrier spaces of dimensionality (K + 1) 2 with K varying as K ∈ [ℓ, N − 1], and N being the principal quantum number of the Scarf I potential problem, contribute up to the order O b N −1−K . Nonetheless, the degeneracy patterns can still be interpreted as a consequence of an effective so(4) symmetry, i.e. a symmetry realized at the level of the dynamic of the system, in so far as from the perspective of the eigenvalue problem, the Scarf I results are equivalently obtained from a Hamiltonian with matrix elements of polynomials in a properly identified so(4) Casimir operator. The scheme applies to any dimension d. Symmetry and degeneracy: Introductory remarksSymmetry and degeneracy are two concepts which one traditionally associates with the basics of the quantum mechanics teachings. One may think of the degeneracy with respect to the magnetic quantum number, m, of a quantum level of a given angular momentum ℓ which is (2ℓ + 1)-fold, and typical for the states bound within all central potentials. A more advanced example would be the degeneracy of the states within the levels describing the quantum motion within the Coulomb potential of an electron without spin, which is N 2 -fold with N standing for the principal quantum number of the Coulomb potential problem. In the first case, the degeneracy is due to the rotational invariance of the three-dimensional position space, which requires conservation of angular momentum, and demands the total wave functions of the central potentials to be simultaneously eigenfunctions of L 2 , and L z , with L standing for the angular momentum pseudo-vector, and L z for its z-component. As long as L 2 acts as the Casimir invariant of the so(3) algebra, the degeneracy with respect to the magnetic quantum number (the L z eigenvalues m ∈ [−ℓ, +ℓ]) is attributed to the rotational invariance of the Hamiltonian. The second case is bit more involved in so far as in order to explain the larger N 2 -fold degeneracy, one needs to invoke the higher so(4) symmetry algebra underlying the Coulomb potential problem by accounting for the constancy of the Runge-Lenz vector next to that of angular momentum [1]. 1

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Quantum Fields on dS and AdS

Abstract: We give a short account of general approach to de Sitter and anti-de Sitter Quantum Field Theories that is based on the assumption of analyticity properties of the n-point correlation functions. We briefly discuss the status of the AdS/CFT and dS/CFT correspondences in this context.

Cited by 3 publications

References 18 publications

Conformal symmetry breaking and degeneracy of high-lying unflavored mesons

Conformal symmetry breaking and degeneracy of high-lying unflavored mesons

Massless Scalar Field in a Two-dimensional de Sitter Universe

Perturbing free motions on hyperspheres without degeneracy lift

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