On quantization in light-cone variables compatible with wavelet ...

On quantization in light-cone variables compatible with wavelet transform M.V.Altaisky

arXiv:1509.03659v1 [hep-th] 4 Sep 2015

Space Research Institute RAS, Profsoyuznaya 84/32, Moscow, 117997, Russia∗ N.E.Kaputkina National Technological University ”MISIS”, Leninsky prospect 4, Moscow, 119049, Russia† (Dated: Sep 3, 2015)

Abstract 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]. PACS numbers: 03.70.+k

I.

The quantization procedure itself is then

INTRODUCTION

based on the commutators of the energyThe construction of quantum field theory momentum tensor components T 0µ with the models is inherently related to the Lorentz positive and the negative frequency field opinvariant partition of classical fields into the erators u± (k), k ∈ R3 , with the field equations being used to eliminate the redundant

positive and the negative frequency parts

component of the quantum fields. The calculation of the n-point Green func+

−

u(x) = u (x) + u (x),

x∈R

1,3

tions hu(x1 ) . . . u(xn )i, the functional derivatives of the generating functional, is well

∗

Electronic address: [email protected] † Electronic address: [email protected]

known to suffer from loop divergences in both 1

UV and IR domains of momentum space. because in any measurement are not accessiThe way to rule out the divergences is to ble exactly in a given point x: to localize a separate the contributions of different scales, particle in an interval ∆x the measuring dewhich can be formally casted in the form [2] vice requests a momentum transfer of order Z da ∆p ∼ ~/∆x. If ∆x is too small the field u(x) u(x) = ua (x) , a at a fixed point x has no experimentally verwhere the ”scale component” ua (x) is not yet ifiable meaning. At the same time establishwell defined. The most known way to sepaing of canonical commutation relations berate the scales is the renormalization group tween the field operators is essentially based technique [3, 4] the less known is the wavelet on Fourier transform (3). It is intuitively transform in quantum field theory [1]. clear that the commutator [ua1 (b1 ), ua2 (b2 )] The consideration would be straightforis a function of aa21 , which vanish if log aa12 is ward for Euclidean quantum field theory, significantly different from zero. This fact is where the projection of an arbitrary function well known in radiophysics : if a field (sysu(x) ∈ L2 (Rd ) onto the scale a is given by tem) is localized in a region of size a1 centered the convolution at point b1 , it may be detected by other field   Z 1 x−b ua (b) := u(x)dd x, (1) with significant probability only in case when g¯ d a a d R so that the function u(x) can be recon- a1 and a2 have the same order. If the window structed from the set of its wavelet coeffi- width a2 is too narrow or too wide in comcients {ua (b)} by the inverse wavelet trans- parison to a1 the probability of detection is low.

form [5] u(x) =

The

1 Cg

Z

1 g ad

R+ ×Rd

analyzing



dadd b x−b ua (b) a a (2)

function



g(x),

satisfy-

ing rather loose admissibility condition R |˜g(ak)|2 da = Cg < ∞ is usually referred to a as a basic wavelet.

The continuous wavelet transform (CWT) is a feasible alternative to the usual Fourier

In the remainder of this paper we present the derivation of the canonical commutation relations between the field operators describing massive scalar field that depend on both

transform u(x) =

Z

ıkx

e

dd k u˜(k) (2π)d

position and resolution in R1,3 Minkowski (3) 2

space.

II.

CONTINUOUS

WAVELET space Rd

TRANSFORM A.

G : x′ = aR(θ)x + b,

Basics of the continuous wavelet

x, b ∈ Rd , a ∈ R+ , θ ∈ SO(d),

transform

(5)

Let H be a Hilbert space of states for a

where R(θ) is the rotation matrix. We define

quantum field |φi. Let G be a locally com-

unitary representation of the affine transform

pact Lie group acting transitively on H, with

(5) with respect to the basic wavelet g(x) as

dµ(ν), ν ∈ G being a left-invariant measure

follows:

on G. Then, similarly to representation of a vector |φi in a Hilbert space of states H as a linear combination of an eigenvectors of R momentum operator |φi = |pidphp|φi, any

  x−b 1 −1 . (6) U(a, b, θ)g(x) = d g R (θ) a a

|φi ∈ H can be decomposed with respect to (We use L1 norm [8, 9] instead of usual L2 to

keep the physical dimension of wavelet coefa representation U(ν) of G in H [6, 7]: Z ficients equal to the dimension of the original 1 U(ν)|gidµ(ν)hg|U ∗(ν)|φi, (4) |φi = Cg G fields).

where |gi ∈ H is referred to as an admissible

Thus the wavelet coefficients of the func-

vector, or basic wavelet, satisfying the admis- tion u(x) ∈ L2 (Rd ) with respect to the basic sibility condition wavelet g(x) in Euclidean space Rd can be Z 1 written as Cg = |hg|U(ν)|gi|2dµ(ν) < ∞. 2 kgk G   Z The coefficients hg|U ∗ (ν)|φi are referred to as 1 x−b −1 u(x)dd x. ua,θ (b) = g R (θ) d a a d wavelet coefficients. R (7) If the group G is abelian, the wavelet The wavelet coefficients (7) represent the retransform (4) with G : x′ = x + b′ coincides sult of the measurement of function u(x) at with Fourier transform. the point b at the scale a with an aperture function g rotated by the angle(s) θ [10]. B.

Euclidean space

The function u(x) can be reconstructed The next to the abelian group is the group from its wavelet coefficients (7) using the forof the affine transformations of the Euclidean mula (4): 3

1 u(x) = Cg

Z

  1 dadd b x−b −1 u (b) g R (θ) dµ(θ) aθ ad a a

The normalization constant Cg is readily The

is

0 , so that the sum over all possible k0 |˜g(ak Cg

scales is k0 . It is impossible however to do such separation in Minkowski space R1,3 in space-time coordinates (t, x, y, z).

For isotropic wavelets Z ∞ Z dd k 2 da Cg = |˜ g (ak)| = |˜ g (k)|2 , a Sd |k|d 0 2π d/2 Γ(d/2)

per octave

)|2

evaluated using Fourier transform: Z ∞ da Cg = |˜ g (aR−1 (θ)k)|2 dµ(θ) a 0 Z d d k = |˜ g (k)|2 d < ∞. |k|

where Sd =

partial momentum

(8)

is the area of unit sphere

in Rd .

C.

Minkowski space

To

construct

wavelet

transform

in

It is helpful to rewrite continuous wavelet Minkowski space it is convenient to turn from transform in Fourier form: the space-time coordinates xµ = (t, x, y, z) Z ∞ Z dd k ıkx da 1 e g˜(ak)˜ ua (k), to the light-cone coordinates: u(x) = Cg 0 a (2π)d t±x u(k). u˜a (k) = g˜(ak)˜ xµ = (x+ , x− , y, z), x± = √ , x⊥ = (y, z). 2 The wavelet function g˜(ak) works as a (10) band-pass filter, which injects a part of the This is the so-called infinite momentum energy carried by the k-mode of the function frame. The advantage of the coordinates (10) u(x) into the ”detector” of scale a, depending for the calculations in quantum field theory is on how the product |ak| is different from the significant simplification of the vacuum strucunity.

ture [11, 12]. The metrics in the light-cone

Indeed, taking the plane wave φ(x) = coordinates becomes (2π)−d exp(ık0 x) as an example of free particle with momentum k0 , so that Pˆ φ(x) = k0 φ(x), Pˆ = −ı∂x , we get ˜ φ(k) = δ d (k − k0 ),

gµν

φ˜a (k) = g¯˜(ak)δ d (k − k0 )

and hence

 0  1  = 0  0

1 0

0



 0  . 0 −1 0   0 0 −1 0 0

The rotation matrix – the Loretnz boosts in φa (b) = eık0 b g¯˜(ak0)

(9) x direction and the rotations in (y, z) plane 4

affine group

– has a block-diagonal form   eη 0 0 0    0 e−η 0 0    M(η, φ) =  ,  0 0 cos φ sin φ    0 0 − sin φ cos φ

x′ = aM(η, φ)x + b, with the representation written in the same form as that of wavelet transform in Eu-

so that M −1 (η, φ) = M(−η, −φ). Hyperbolic rotation in (t, x) plane is determined by the hyperbolic rotation angle – the rapidity η.

clidean space Rd , viz:   1 x−b −1 U(a, b, η, φ)u(x) = 4 u M (η, φ) , a a

The rotations in the transverse plane, not af- defined in L1 norm in accordance to [1, 9]. fected by the Lorentz contraction, are deter- So, we have straightforward generalization of mined by the rotation angle φ.

the definition of wavelet coefficients of a func-

The Poincare group can be extended by tion f (x) ∈ L2 (R1,3 ) with respect to the basic

the scale transformations x′ = ax to the wavelet g [1, 13]

Wa,b,η,φ [f ] =

Z

  1 x−b −1 f (x+ , x− , x⊥ ). dx+ dx− d x⊥ 4 g M (η, φ) a a 2

(11)

The difference from calculations in Eu- We assume the following definition of the clidean space R4 is that the basic wavelet g(·) Fourier transform in light cone coordinates: Z cannot be defined globally on R1,3 . Instead, f (x+ , x− , x⊥ ) = eık− x+ +ık+ x− −ık⊥ x⊥ × it should be defined in four separate domains dk+ dk− d2 k⊥ × f˜(k− , k+ , k⊥ ) . impassible by Lorentz rotations: (2π)4 Substituting the Fourier images into the def-

A1 : k+ > 0, k− < 0;A2 : k+ < 0, k− > 0;

inition (11) we get Z i Wabηφ = eık− b+ +ık+ b− −ık⊥ b⊥ f˜(k− , k+ , k⊥ ) √ x . Four = ω±k 2 A

A3 : k+ > 0, k− > 0;A4 : k+ < 0, k− < 0, where k is wave vector, k±

i

separate wavelets should be defined in these four domains [14, 15]: Z d4 k , gi (x) = eıkx g˜(k) (2π)4 Ai

g˜(aeη k− , ae−η k+ , aR−1 (φ)k⊥ )

dk+ dk− d2 k⊥ . (2π)4 (13)

i = 1, 4. (12) 5

Similarly to the Rd case, the reconstruction

formula is [13]:

f (x) = =

  Z ∞ Z Z 2π Z ∞ 4 X 1 ξ−b da 1 i 2 −1 Wabηφ dη db+ db− d b⊥ 4 gi M (η) dφ 4 C a a a gi −∞ M1 0 0 i=1 Z Z Z Z 4 ∞ 2π ∞ X 1 dk+ dk− d2 k⊥ ık− x+ +ık+ x− −ık⊥ x⊥ da dη e × dφ 4 C a (2π) g A −∞ 0 0 i i i=1 ˜ aηφ (k)˜ W g (ak− eη , ak+ e−η , aR−1 (φ)k⊥ )

×

III.

After the substitution of integration variable

QUANTIZATION

Same as in standard quantum field theory

k → −k in integration over A4 , the decomposition of u(x) takes the form

in Minkowski space we ought to use the massshell delta function to get rid of redundant degrees of freedom [16]. Let us consider the massive scalar field in R1,3 Minkowski space Z d4 k u(x) = eıkx 2πδ(k 2 −m2 )˜ u(k− , k+ , k⊥ ) (2π)4 (14) The Lorentz invariant scalar product and the invariant volume in k-space are

 2 k⊥ + m2 u(x) = [e u˜ , k+ , k⊥ + 2k+   2 k⊥ + m2 −ıkx , −k+ , −k⊥ ]× + e u˜ − 2k+ dk+ d2 k⊥ × θ(k+ ) 2k+ (2π)3 Z  ıkx +  ≡ e u˜ (k) + e−ıkx u˜− (k) × Z

×

ıkx



θ(k+ )

dk+ d2 k⊥ 2k+ (2π)3

(15)

kx ≡ k0 x0 − kx = k− x+ + k+ x− − k⊥ x⊥

dk0 dkx dky dkz dk− dk+ d2 k⊥ d4 k = = . (2π)4 (2π)4 (2π)4

Both u˜+ (k) and u˜− (k) are defined on a

3 For a massive scalar field because of the mass hemisphere in R and can be decomposed

shell delta function δ(2k+ k− − k2 − m2 ) only into scale components by continuous wavelet

two domains A3 and A4 for which k+ k− is transform in Euclidean space. The straight-

positive will contribute to the decomposition forward way to quantize the fields in the of u(x). The integration over the k− variable light-cone representation is to use the formal analogy between the decomposition (15) and

with the mass shell delta function gives

the positive/negative frequency decomposi-

2 k⊥ + m2 k− = 2k+

tion in usual coordinates (t, x) in the equal6

time quantization scheme commutator one gets   ∂L = ıδ 3 (x − y), (16) u(t, x),  −  ∂ u(t, ˙ y) t=0 u˜ (k), u ˜+ (q) = 2k+ (2π)3 δ 3 (k − q). (19)

where the curly brackets stand for the

Poisson brackets substituted by commutator The latter equation is different from the stan(anti-commutator) for Bose (Fermi) quantum dard commutation relation by changing the fields.

energy (k0 ) to the momentum (k+ ). The role

Using the Lagrangian

of energy is played by k− in the light-cone

2

L=

∂u ∂u 1 m 2 − (∂⊥ u)2 − u, ∂x+ ∂x− 2 2

we can infer that the x+ =

t+x √ 2

(17)

coordinates. Substituting the inverse wavelet transform

variable can

be considered as ”time” on the light-cone [17]. In analogy to common case the Pois-

1 u˜ (k+ , k⊥ ) = Cg ±

Z

0

∞

g˜(ak)˜ u± a (k)

da , (20) a

son bracket can be then casted in the form ) ( where u˜± ˜(ak)˜ u± (k), k ≡ (k+ , k⊥ ), a (k) = g ∂u = ıδ 3 (x − y) into the equality (19), and assuming an u(x+ = 0, x− , x⊥ ), ∂y− y+ =0 (18) isotropic basic wavelet g(·) for simplicity, we Substituting decomposition (15) into the derive the commutation relations for the scale bracket (18) and changing the bracket to components

 −  3 2 u˜a1 (k), u˜+ a2 (q) = 16π Cg a1 δ(a1 − a2 )k+ δ(k+ − q+ )δ (k⊥ − q⊥ ).

(21)

The commutation relation (21) meets the k ∈ R+ ⊗ R2 it is easier for practical calgeneral

form

of

wavelet

transform

of culations.

the canonical commutation relations in

Introducing the vacuum state Φp with the momentum p we get

Minkowski space, eq.(18) of [1] + ′ ′ ′ [u− iaη (k), uja′ η′ (k )] = aδ(a − a )δ(η − η )×

P n u˜+ (k)Φp =

(pn + k n )˜ u+ (k)Φp ,

δij Cgi [u− (k), u+ (k ′ )],

P n u˜− (k)Φp =

(pn − k n )˜ u− (k)Φp .

×

defined on four Lorentz-invariant domains In the latter equations u˜± (k) can be subAi , i = 1, 4.

However, being defined on jected to wavelet transform so that u˜± (k) is 7

expressed by (20) with k having only 3 in- construct the multiscale Fock space of states dependent components. In this way we can

Φ=

XZ

Fs(···j··· ) (a1 , k1 , . . . , as , ks )˜ u+ ˜+ j1 a1 (k1 ) . . . u js as (ks )

j,s

das dks+ d2 ks⊥ da1 dk1+ d2 k1⊥ . . . Φ0 , a1 Cg 16k1+ π 3 as Cg 16ks+ π 3 (22)

where ki = (ki+ , ki⊥ ) are three dimensional parameter Λ, which is either cutoff momenvectors, j denote all other indices of the tum, or renormalization scale. From funcquantum states, and Φ0 is a vacuum state tional analysis point of view, this may suggest u− i (x)Φ0 = 0.

the use of space of functions which explicitly depend on both the position and the reso-

IV.

CONCLUSIONS

lution. being operator-valued functions they certainly require commutation relations. The

To be concluded, we have developed a use of light-cone coordinates enables this conquantization scheme suitable for applications struction. The massive scalar field quantizain quantum theory of fields ua (x), which ex- tion was choosen as a simple example. Perplicitly depend on both position x and the haps the same technique can be used in genscale (resolution) a. It is not suprising, that eral problems of quantum field theory, when such fields can form a prospective framework wavelet transform is used to construct diverfor analytic calculations in quantum chro- gence free Green functions [2, 20, 21]. modynamics, where most approved results are obtained either numerically lattice sim-

Acknowledgement

ulations [18], or analytically, with perturbation expansion being corrected by renormal-

The work was supported in part by RFBR

ization group methods [19]. In the latter case projects 13-07-00409, 14-02-00739 and by the the obtained results, viz., process amplitudes, Ministry of Education and Science of the parton distribution functions, nucleon form Russian Federation in the framework of Infactors, tacitly depend on some formal scale crease Competitiveness Program of MISiS.

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