Continuous wavelet transform in quantum field theory

Altaisky, M. V.; Kaputkina, N. E.

doi:10.1103/physrevd.88.025015

Cited by 25 publications

(29 citation statements)

References 42 publications

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“…Continuous wavelet transform is a generalization of the Fourier transform for the case when the scaling properties of the theory are important. Referring the reader to general reviews on wavelet transform [11,16], and to the original papers devoted to the application of wavelet transform to quantum field theory [1,2,17], below we remind the basic definitions of the wavelet formalism of quantum field theory.…”

Section: Field Theory In Wavelet Representationmentioning

confidence: 99%

“…(In accordance to previous papers [1,2] we use L 1 norm [16,20] instead of usual L 2 to keep the physical dimension of wavelet coefficients equal to the dimension of the original fields). Wavelet coefficients of the Euclidean field φ(x) with respect to the basic wavelet g(…”

Section: Field Theory In Wavelet Representationmentioning

confidence: 99%

“…

It is shown that the renormalization group turns to be a symmetry group in a theory initially formulated in a space of scale-dependent functions, i.e, those depending on both the position x and the resolution a. Such theory, earlier described in [1,2], is finite by construction. The space of scaledependent functions {φa(x)} is more relevant to physical reality than the space of square-integrable functions L 2 (R d ), because due to the Heisenberg uncertainty principle, what is really measured in any experiment is always defined in a region rather than point.

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confidence: 99%

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Unifying renormalization group and the continuous wavelet transform

Altaisky

2016

Phys. Rev. D

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It is shown that the renormalization group turns to be a symmetry group in a theory initially formulated in a space of scale-dependent functions, i.e, those depending on both the position $x$ and the resolution $a$. Such theory, earlier described in {\em Phys.Rev.D} 81(2010)125003, 88(2013)025015, is finite by construction. The space of scale-dependent functions $\{ \phi_a(x) \}$ is more relevant to physical reality than the space of square-integrable functions $\mathrm{L}^2(R^d)$, because, due to the Heisenberg uncertainty principle, what is really measured in any experiment is always defined in a region rather than point. The effective action $\Gamma_{(A)}$ of our theory turns to be complementary to the exact renormalization group effective action. The role of the regulator is played by the basic wavelet -- an "aperture function" of a measuring device used to produce the snapshot of a field $\phi$ at the point $x$ with the resolution $a$. The standard RG results for $\phi^4$ model are reproduced.Comment: LaTeX, 5 pages, 1 eps figur

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Section: Field Theory In Wavelet Representationmentioning

confidence: 99%

Section: Field Theory In Wavelet Representationmentioning

confidence: 99%

“…

It is shown that the renormalization group turns to be a symmetry group in a theory initially formulated in a space of scale-dependent functions, i.e, those depending on both the position x and the resolution a. Such theory, earlier described in [1,2], is finite by construction. The space of scaledependent functions {φa(x)} is more relevant to physical reality than the space of square-integrable functions L 2 (R d ), because due to the Heisenberg uncertainty principle, what is really measured in any experiment is always defined in a region rather than point.

…”

mentioning

confidence: 99%

See 1 more Smart Citation

Unifying renormalization group and the continuous wavelet transform

Altaisky

2016

Phys. Rev. D

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It is shown that the renormalization group turns to be a symmetry group in a theory initially formulated in a space of scale-dependent functions, i.e, those depending on both the position $x$ and the resolution $a$. Such theory, earlier described in {\em Phys.Rev.D} 81(2010)125003, 88(2013)025015, is finite by construction. The space of scale-dependent functions $\{ \phi_a(x) \}$ is more relevant to physical reality than the space of square-integrable functions $\mathrm{L}^2(R^d)$, because, due to the Heisenberg uncertainty principle, what is really measured in any experiment is always defined in a region rather than point. The effective action $\Gamma_{(A)}$ of our theory turns to be complementary to the exact renormalization group effective action. The role of the regulator is played by the basic wavelet -- an "aperture function" of a measuring device used to produce the snapshot of a field $\phi$ at the point $x$ with the resolution $a$. The standard RG results for $\phi^4$ model are reproduced.Comment: LaTeX, 5 pages, 1 eps figur

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“…, a n ), then the integration in all internal lines is to be performed within the range ∞ A dai ai . The theory defined in this way is finite by construction [27,28,41]. In contrast to standard means of regularization, such as introduction of cutoff momenta, our method provides an exact conservation of momentum in each vertex.…”

Section: Multiscale Theory Of Turbulence In Wavelet Basismentioning

confidence: 99%

“…Its Fourier transform isg 1 (k) = −ıke − k 2 2 , with C g1 = 1 2 . Due to the limited range of integration over the scale variables da; for g 1 wavelet cutoff factor is f g1 (x) = e −x 2 , see [27,28] for details.…”

Section: One-loop Corrections To Viscositymentioning

confidence: 99%

Renormalization of viscosity in wavelet-based model of turbulence

2018

Self Cite

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Statistical theory of turbulence in viscid incompressible fluid, described by the Navier-Stokes equation driven by random force, is reformulated in terms of scale-dependent fields ua(x), defined as wavelet-coefficients of the velocity field u taken at point x with the resolution a. Applying quantum field theory approach of stochastic hydrodynamics to the generating functional of random fields ua(x), we have shown the velocity field correlators ua 1 (x1) . . . ua n (xn) to be finite by construction for the random stirring force acting at prescribed large scale L. The study is performed in d = 3 dimension. Since there are no divergences, regularization is not required, and the renormalization group invariance becomes merely a symmetry that relates velocity fluctuations of different scales in terms of the Kolmogorov-Richardson picture of turbulence development. The integration over the scale arguments is performed from the external scale L down to the observation scale A, which lies in Kolmogorov range l ≪ A ≪ L. Our oversimplified model is full dissipative: interaction between scales is provided only locally by the gradient vertex (u∇)u, neglecting any effects or parity violation that might be responsible for energy backscatter. The corrections to viscosity and the pair velocity correlator are calculated in one-loop approximation. This gives the dependence of turbulent viscosity on observation scale and describes the scale dependence of the velocity field correlations.

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Spinor Structure and Internal Symmetries

Варламов

2015

Int J Theor Phys

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Spinor structure and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It is shown that tensor products of biquaternion algebras are associated with the each irreducible representation of the Lorentz group. Space-time discrete symmetries $P$, $T$ and their combination $PT$ are generated by the fundamental automorphisms of this algebraic background (Clifford algebras). Charge conjugation $C$ is presented by a pseudoautomorphism of the complex Clifford algebra. This description of the operation $C$ allows one to distinguish charged and neutral particles including particle-antiparticle interchange and truly neutral particles. Spin and charge multiplets, based on the interlocking representations of the Lorentz group, are introduced. A central point of the work is a correspondence between Wigner definition of elementary particle as an irreducible representation of the Poincar\'{e} group and $SU(3)$-description (quark scheme) of the particle as a vector of the supermultiplet (irreducible representation of $SU(3)$). This correspondence is realized on the ground of a spin-charge Hilbert space. Basic hadron supermultiplets of $SU(3)$-theory (baryon octet and two meson octets) are studied in this framework. It is shown that quark phenomenologies are naturally incorporated into presented scheme. The relationship between mass and spin is established. The introduced spin-mass formula and its combination with Gell-Mann--Okubo mass formula allows one to take a new look at the problem of mass spectrum of elementary particles.Comment: 44 page

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Continuous wavelet transform in quantum field theory

Cited by 25 publications

References 42 publications

Unifying renormalization group and the continuous wavelet transform

Unifying renormalization group and the continuous wavelet transform

Renormalization of viscosity in wavelet-based model of turbulence

Spinor Structure and Internal Symmetries

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