Quantum field theory without divergences

Canonical quantization of quantum field theory models is inherently related to the Lorentz invariant partition of classical fields into the positive and the negative frequency parts u(x) = u + (x) + u − (x), performed with the help of Fourier transform in Minkowski space. That is the commutation relations are being established between non localized solutions of field equations. At the same time the construction of divergence free physical theory requires the separation of the contributions of different space-time scales. In present paper, using the light-cone variables, we propose a quantization procedure which is compatible with separation of scales using continuous wavelet transform, as described in our previous paper [1].

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“…To be concluded, we have wavelet transform is used to construct divergence free Green functions [2,20,21].…”

Section: Discussionmentioning

confidence: 99%

“…The way to rule out the divergences is to separate the contributions of different scales, which can be formally casted in the form [2] u(x) = u a (x) da a ,…”

Section: Introductionmentioning

confidence: 99%

On Quantization in Light-cone Variables Compatible with Wavelet Transform

Altaisky

Kaputkina

2016

Int J Theor Phys

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“…2 below we present the comparison of the "exact" Casimir force between two plates in vacuum (δ = 0), and that calculated according to (7) with δ/a = 0.1. Interestingly, in present experiments the separation between plates is corrected by the factor 1 + δ a 2 , derived from the Taylor expansion of the Casimir force, to account for the r.m.s.…”

Section: Introductionmentioning

confidence: 98%

“…1. The conjecture, relating the quantum fields to the aperture function of the measurement taken with resolution δ was given in [7]. The aperture function g(x) = −xe −x 2 /2 , up to appropriate rescaling, leads to the cutoff function After the choice (4) the regularized Casimir energy is [9]:…”

Section: Introductionmentioning

confidence: 99%

Scale-dependent corrections to Casimir effect

Altaisky

Kaputkina

2011

Proceedings of the International Conference Days on Diffraction 2011

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The Casimir force, which attracts to each other two perfectly conducting parallel plates separated by the distance a in vacuum, is one of the blueprints of the reality of vacuum fluctuations. Following the recent conjecture, that quantum fields should be described in terms of the fields depending on the resolution of measurement (M.V. Altaisky, Phys. Rev. D81(2010)125003), we derive the correction to the Casimir energy depending on the ratio of the plate displacement amplitude to the distance between plates. The effects of random environments to the measurement are also discussed.

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“…This interest is also stimulated by the search for quantum field theory models free of ultra-violet divergences [2]. For instance, the registration of a single photon with the help of photographic plate results in local exposure of the plate, in contrast to theoretical expectation of global exposure for the photon treated as a plane wave.…”

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confidence: 99%

On wavelet transform in Minkowski space

Altaisky

Kaputkina

2012

2012 Proceedings of the International Conference Days on Diffraction

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We reformulate the Poincare wavelet transform, proposed by Gorodnitskiy and Perel in Proc. DD'll [1 J, in the framework of li g ht cone coordinates. The rotation matrix turns to be dia g onal in li g ht cone coordinates.The interest to application of wavelets in quan tum field theory problems is caused by the existence of physical phenomena poorly described in terms of the plane wave decomposition of the wave func tion. This interest is also stimulated by the search for quantum field theory models free of ultra-violet divergences [2]. For instance, the registration of a single photon with the help of photographic plate results in local exposure of the plate, in contrast to theoretical expectation of global exposure for the photon treated as a plane wave.Let us briefly recall the application of wavelet transform to 2D image analysis, specially in cases where the behaviour of the analysed function in dif ferent directions is investigated at different scales.Let G be the affine group in JR d :Having calculated the wavelet coefficients (4) one can analyse the local behaviour of the function f at each point b E JR d at a given scale a in the direction ¢ E SO(d).The most known example is the analysis of im ages. In the case of d = 2 the group of rotations SO( 2) is an abelian group. Its two-dimensional rep resentation is given byThe Haar measure on SO( 2) is dp,(¢) = d¢ and the reconstruction formula is straightforwardwhere M (¢) is a unitary representation of the ro-where tation group SO(d) c G. Let r 2 1 � ( k ) 1 2 Cw = J lR 2 d kl'k'P < 00.(7)be the representation of the affine group (1) in JR d , constructed using a basic function 'I/J E L 2 ( JRd ) , which satisfies the admissibility condition [3] 1 r 2 Cw = 11'l/J112 J e I ('l/J IUI'I/J) I dp,da , b, ¢) < 00 , (3) with dP,L (a, b, ¢) being the left-invariant Haar mea sure on G. Here, for the sake of field theory applica tions, we used L 1 normaliz ation of the group repre sentation ( a I d ) , instead of the usual factor ()/2) of the L2 norm. The wavelet coefficients of a function To analyse anisotropic data in different directions the basic wavelet is chosen in the form stretched along one of the axis, like that shown in Fig. 1. Ro tating the basic wavelet in the range 0 :s; ¢ < 27r the coefficients (4) are obtained. Their depen dences on scale a may be different in different di rections. This is important for example in the anal ysis of anisotropic turbulence data [4]. We would like to have an analysing tool, similar to (6) , for the Poincare group, which includes hyperbolic ro tations of the (t, x ) plane. This rotations are rep resented by the matrix M(¢) = ( co� h¢ -sm h¢ -sinh¢) cosh¢ (8 )

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Quantum field theory without divergences

Cited by 33 publications

References 28 publications

On Quantization in Light-cone Variables Compatible with Wavelet Transform

On Quantization in Light-cone Variables Compatible with Wavelet Transform

Scale-dependent corrections to Casimir effect

On wavelet transform in Minkowski space

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