Wavelet representation of light-front quantum field theory

A formally exact discrete multi-resolution representation of quantum field theory on a light front is presented. The formulation uses an orthonormal basis of compactly supported wavelets to expand the fields restricted to a light front. The representation has a number of useful properties. First, light front preserving Poincar\'e transformations can be computed by transforming the arguments of the basis functions. The discrete field operators, which are defined by integrating the product of the field and the basis functions over the light front, represent localized degrees of freedom on the light-front hyperplane. These discrete fields are irreducible and the vacuum is formally trivial. The light-front Hamiltonian and all of the Poincar\'e generators are linear combinations of normal ordered products of the discrete field operators with analytically computable constant coefficients. The representation is discrete and has natural resolution and volume truncations like lattice formulations. Because it is formally exact it is possible to systematically compute corrections for eliminated degrees of freedom.


I. INTRODUCTION
A discrete multi-resolution representation of quantum field theory on a light front is presented. Light-front formulations of quantum field theory have advantages for calculating electroweak current matrix elements in strongly interacting states in different frames. Lattice truncations have proved to be the most reliable method for nonperturbative calculations of strongly interacting states, but Lorentz transformation and scattering calculations are not naturally formulated in the lattice representation. The purpose of this work is to investigate a representation of quantum field theory that has some of the advantages of both approaches, although this initial work is limited to canonical field theory rather than gauge theories.
In 1939 Wigner [1] showed that the independence of quantum observables in different inertial reference frames related by Lorentz transformations and space-time translations requires the existence of a dynamical unitary representation of Poincaré group on the Hilbert space of the quantum theory. Because there are many independent paths to the future, consistency of the initial value problem requires that a minimum of three of the infinitesimal generators of the Poincaré group are interaction dependent. In 1949 P. A. M. Dirac [2] introduced three "forms of relativistic dynamics" that are characterized by having the largest interaction-independent subgroups.
Dirac's 'front-form dynamics' is the only form of dynamics with the minimal number, 3, of dynamical Poincaré generators. The interaction-independent subgroup is the seven-parameter subgroup that leaves the hyperplane, invariant. The light-front representation of quantum dynamics has several advantages. One is that the kinematic (interaction-independent) subgroup has a three-parameter subgroup of Lorentz boosts. The subgroup property means that there are no Wigner rotations for light-front boosts. A consequence is that the magnetic quantum numbers of the light-front spin are invariant with respect to these boosts. A second advantage is that the boosts are independent of interactions. This means that boosts can be computed by applying the inverse transform to non-interacting basis states. These properties simplify theoretical treatments electroweak probes of strongly-interacting systems, where the initial and final hadronic states are in different Lorentz frames. In light-front quantum field theory [ [14] there are additional advantages. These are consequences of the spectrum of the generator p + = p 0 +n · p ≥ 0 (2) of translations in the direction, tangent to the light-front. The first property is that free fields restricted to the light-front are irreducible. This means that any operator on the free field Fock space can be expressed as a function of fields restricted to the light front. The second advantage is that interactions that commute with the interaction-independent subgroup leave the Fock vacuum invariant. This means that it is possible to express all of the Poincaré generators as operators on the free-field Fock space. There are ultraviolet and infrared (p + = 0) singularities in the light-front Hamiltonian due to local operator products, which could impact these properties, however in an effective theory with ultraviolet and infrared cutoffs the interaction still leaves the Fock vacuum invariant and the light-front Hamiltonian can still be represented as a function of the free fields on the light front.
Having an explicit vacuum along with an expression for the light-front Hamiltonian, in terms of the algebra of fields on the light front means that it is possible to perform non-perturbative calculations by diagonalizing the light-front Hamiltonian in the light-front Fock space. In a given experiment there is a relevant volume and a finite amount of available energy. The available energy limits the resolution of the accessible degrees of freedom. The number of degrees of freedom with the limiting resolution that fit in the experimental volume is finite. It follows that it should be possible to accurately calculate experimental observables using only these degrees of freedom.
Wavelets can be used to represent fields on the light front as linear combinations of discrete field operators with different resolutions. This provides a natural representation to make both volume and resolution truncations consistent with a given reaction. In addition the representation is discrete, which is a natural representation for computations. Finally the basis functions are self-similar, so truncations with different resolutions have a similar form.
There are many different types of wavelets that have been discussed in the context of quantum field theory [15] [40]. These have the property that they are an orthonormal basis of functions with compact support. The price paid for the compact support is that they have a limited smoothness. It is also possible to use a wavelet basis of Schwartz functions that are infinitely differentiable, but these functions do not have compact support.
There are several motivations for considering this approach. These include 1. Volume and resolution truncations can be performed naturally, the resulting truncated theory is similar to a lattice truncation [41] [42], in the sense that it is a theory involving a finite number of discrete degrees of freedom associated with a given volume and resolution.
2. While the degrees of freedom are discrete, the field operators have a continuous space-time dependence. Kinematic Lorentz transformations can be computed by transforming the arguments of the basis functions. While truncations necessarily break kinematic Lorentz invariance, kinematic Lorentz transformations can still be approximated by transforming the arguments of the basis functions.
3. Similarly, even though the truncation could lead to states with energy below the Fock vacuum energy, the error in using the free Fock vacuum as the lowest mass state of the truncated theory is due to corrections that arise from the discarded degrees of freedom.
4. x + is a continuous variable, so there is a natural formulation of Haag-Ruelle [43][44] [45] scattering in this representation.

II. NOTATION
The light front is a three-dimensional hyperplane that is tangent to the light cone. It is defined by the constraint It is natural to introduce light-front coordinates of the four-vector x µ : The componentsx are coordinates of points on the light-front hyperplane. These will be referred to as light-front 3-vectors. In what follows the light front defined byn =ẑ will be used. The contravariant light-front components are and the Lorentz-invariant scalar product of two light-front vectors is For computational purposes it is useful to represent four vectors by 2 × 2 Hermitian matrices. The coordinate matrix is constructed by contracting the four vector x µ with the Pauli matrices and the identity: In this matrix representation Poincaré transformations continuously connected to the identity are represented by The subgroup of the Poincaré group that leaves x + = 0 invariant consists of pairs of matrices (A, B) in (11) of the form where a,c and b ⊥ are complex and b − is real. This is a seven-parameter group. The SL(2, C) matrices with real a represent light-front preserving boosts. They can be parameterized by the light-front components of the four velocity v = p/m: These lower triangular matrices form a subgroup. The inverse light-front boost is given by while the adjoint and the inverse adjoint of these matrices are General Poincaré transformations are generated by 10 independent one-parameter subgroups. Seven of the oneparameter groups leave the light front invariant. The remaining three one-parameter groups map points on the light front to points off of the light front. These are called kinematic and dynamical transformations respectively. The kinematic one-parameter groups in the 2 × 2 matrix representation and the corresponding unitary representations of these groups are related by The corresponding dynamical transformations are Relations (17)(18)(19) define the infinitesimal generators of the kinematic transformations, while (20)(21) define the infinitesimal generators of the dynamical transformations. With these definitions the light-front Poincaré generators are related to components of the angular momentum tensor by The inverse relations are F 1 and F 2 could be replaced by J 1 and J 2 as dynamical generators. The evolution of a state or operator with initial data on the light front is determined by the light-front Schrödinger equation or the light-front Heisenberg equations of motion When P − is a self-adjoint operator the dynamics is well defined and given by the unitary one-parameter group (21). The Poincaré Lie algebra has two polynomial invariants. The mass squared is which gives the light-front dispersion relation The other invariant is the inner product of the Pauli-Lubanski vector, with itself The Pauli-Lubanski vector has components W 0 = P · J W = HJ + P × K (33) or expressed in terms of the light-front Poincaré generators In order to compare the spins of particles in different frames, it is useful to transform both particles to their rest frame using an arbitrary but fixed set of Lorentz transformations parameterized by the four velocity of the particle. The light-front spin is the angular momentum measured in the particle or system rest frame when the particles or system are transformed to the rest frame with the light-front preserving boosts (13) The components of the light-front spin can also be expressed directly in terms of J µν where in (39) the P/M in the Lorentz boosts are operators.

III. FIELDS
Light-front free fields can be constructed from canonical free fields by changing variables p →p, wherep := (p + , p 1 , p 2 ) are the components of the light front-momentum conjugate tox. The Fourier representation of a free scalar field of mass m and its conjugate momentum operator are where ω m (p) := m 2 + p 2 is the energy of a particle of mass m, p is its three-momentum and x·p := −ω m (p)x 0 +p·x. Changing variables from the three momentum, p, to the light-front components,p = (p + , p 1 , p 2 ), of the four momentum gives the light-front Fourier representation of φ(x): where It follows from and (43) and (44) that The spectral conditions imply that it is possible to independently construct both a(p) and a † (p) from the field φ(x + = 0,x) restricted to the light front. This can be done by computing the partial Fourier transform of the field on the light front: The creation and annihilation operators can be read off of this expressioñ Both operators are constructed directly from the field restricted to the light front without constructing a generalized momentum operator. This means that φ(x) restricted to the light front defines an irreducible set of operators. It follows that any operator O on the Fock space that commutes with φ(x + = 0,x) at all points on the light front must be a constant multiple of the identity: An important observation is that the only place where the mass of the field appears is in the expression for the coefficient of x + . When the field is restricted to the light front, x + → 0, all information about the mass (and dynamics) disappears. This is in contrast to the canonical case because the canonical transformation that relates free canonical fields and their generalized momenta with different masses cannot be realized by a unitary transformation [46]. When these fields are restricted to the light front they become unitarily equivalent [10]. This is because dynamical information that distinguishes the different representations is lost as a result of the restriction.
Since the fields restricted to the light front are irreducible, the canonical commutation relations are replaced by the commutator of the fields at different points on the light front Note that while the x − derivative gives is not the canonical momentum.
Interactions that preserve the light-front kinematic symmetry must commute with the kinematic subgroup. In particular they must be invariant with respect to translations in the x − direction. This means that the interactions must commute with P + , which is a kinematic operator. Since, P + = i P + i , is kinematic, the vacuum of the field theory is invariant with respect to these translations, independent of interactions. This requires that where |0 is the free-field Fock vacuum. This means that V |0 is an eigenstate of P + with eigenvalue 0. Inserting a compete set of intermediate states between V † and V in 0|V † V |0 , the absolutely continuous spectrum of p + i cannot contribute to the sum over intermediate states because or the interactions that preserve the kinematic symmetry leave the free-field Fock vacuum unchanged. The observation that the interaction leaves the vacuum invariant implies that it is an operator on the free-field Fock space. The irreducibility of the light front-Fock algebra means that the interaction can be expressed in terms of fields in this algebra. The Poincaré generators, defined by integrating the + components of the Noether currents that come from Poincaré invariance of the action over the light front, are also linear in this interaction. This means that it should be possible to solve for the relativistic dynamics of the field on the light-front Fock space.
A more careful analysis shows that the interaction, while formally leaving the light front invariant, has singularities at p + = 0, so the formal expressions for the interaction-dependent generators are not well-defined self-adjoint operators on the free field Fock space. This is because the interaction involves products of operator-valued distributions which are not defined. All of these operators are defined on the free field Fock space if infrared and ultraviolet cutoffs are introduced, but they break the Poincaré symmetry. The non-trivial problem is how to remove the cutoffs in a manner that recovers the Poincaré symmetry.
While the solution of this last problem is equivalent to the unsolved problem of giving a non-perturbative definition of the theory, cutoff theories should lead to good approximations for observables on scales where the cutoffs are not expected to be important.

IV. FORMAL LIGHT-FRONT FIELD DYNAMICS
The Lagrangian density for a scalar field theory is where η µν is the metric tensor with signature (−, +, +, +). The action is Variations of the field that leave the action stationary satisfy the field equation: Changing to light-front variables the partial derivatives become Squaring and subtracting gives It follows that the Lagrangian density (59) and the field equation in light-front variables have the forms and Invariance of the action under infinitesimal changes in the fields and coordinates along with the field equation, leads to the conserved Noether currents where the Noether current is The Noether currents associated with translational and Lorentz invariance of the action are the energy momentum, T µν , and angular momentum M µαβ tensors where for the Lagrangian density (65): Integrating the + component of the conserved current over the light front, assuming that the fields vanish on the boundary of the light front, give the light-front conserved (independent of x + ) charges and These are the conserved four momentum and angular momentum tensors. They are independent of x + and thus can be expressed in terms of fields and derivatives of fields restricted to the light front.
In order to construct the Poincaré generators the first step is to express the + component of the energy momentum tensor and angular momentum tensors in terms of fields on the light front: The Poincaré generators are constructed by integrating these operators over the light front For free fields these operators can be expressed in terms of the light-front creation and annihilation operators (50-51) using the identities Using (90-93) in (83-89) gives the following expressions for the Poincaré generators for a free field in terms of the light-front creation and annihilation operators Since these are independent of x + , the expressions with an explicit x + dependence can be evaluated at x + = 0. These expressions lead to the following identifications

V. WAVELET BASIS
In this section the multi-resolution basis that is used to represent the irreducible algebra of fields on the light front is introduced. Wavelets provide a natural means for exactly decomposing a field into independent discrete degrees of freedom labeled by volume and resolution. In this representation there are natural truncations that eliminate degrees of freedom associated with volumes and resolutions that are expected to be unimportant in modeling a given reaction.
While there are many different types of wavelets, this application uses Daubechies [36][37] L = 3 wavelets. These are used to generate an orthonormal basis of functions with the following desirable properties: (1) all of the basis functions have compact support (2)  In what follows these basis functions will be used to decompose fields restricted to a light front into an infinite linear combination of discrete operators with arbitrarily fine resolutions. The advantage of the light-front representation is that the resulting discrete algebra is irreducible and the vacuum remains trivial.
For Lagrangians that are polynomials in the fields, in the wavelet representation all of the Poincaré generators can be formally expressed as polynomials in the discrete fields on the light front with coefficients that can be computed analytically. While the polynomials are finite degree, there are an infinite number of discrete field operators.
The construction of the wavelet basis starts with the fixed-point solution of the renormalization group equation where   The next step in constructing the wavelet basis is to construct subspaces of L 2 (R) with different resolutions defined by The resolution is determined by the width of the support of these functions, which for L = 3, is 5 × 2 −k . The functions are orthonormal, have compact support on [2 −k n, 2 −k (n + 5)], satisfy and are locally finite partitions of unity The subspace S k is called the resolution 2 −k subspace of L 2 (R). The scale transformation D has the following intertwining properties with translations and derivatives: Applying D k T n to the renormalization group equation, using (111), gives which expresses every basis element of S k as a finite linear combination of basis elements of S k+1 or This means that the lower resolution subspaces are subspaces of the higher resolution subspaces. The orthogonal complement of S k in S k+1 is called W k : Since W k ⊂ S k+1 , orthonormal basis functions w k n (x) in W k are also linear combinations of the s k+1 n (x). These functions are defined by where w(x) is the "mother wavelet" defined by and the coefficients g l are related to the weight coefficients h l by The orthonormal basis functions w k n (x) for W k are called wavelets. Since the w k n (x) are finite linear combinations of the s k+1 n (x) they have the same number of derivatives as s(x). w k n (x) also has the same support as s k n (x). Finally it follows from (105) that Equation (118) is equivalent to the condition (105). Equation (114) means that the wavelet subspace W k consists of functions that increase the resolution of S k from 2 −k to 2 −(k+1) . The inclusions (113) imply a decomposition of S k+n into an orthogonal direct sum of the form which indicates that the resolution of S k can be increased to 2 −k−n by including additional basis functions in the subspaces {W k+n−1 , · · · , W k }. This can be continued to arbitrarily fine resolutions to get all of L 2 (R): Since all of the subspaces are orthogonal, an orthonormal basis for L 2 (R) consists of for any fixed starting resolution 2 −k or The basis (122)

VI. WAVELET REPRESENTATION OF QUANTUM FIELDS
In what follows the basis (121) is used to expand quantum fields restricted to a light front. It is useful to think of the starting scale 2 −k in (121) as the resolution that is relevant to experimental measurements. The higher resolution degrees of freedom are used to represent shorter distance degrees of freedom that couple to experimental-scale degrees of freedom.
The basis (121) can be used to get a formally exact representation of the field operators of the form where the ξ l are the basis functions In what follows the short-hand notation is used . (125) With this notation (123) has the form which gives a discrete representation of the field as a linear combination of discrete operators with different resolutions on the light front. Each discrete field operator, φ n (0), is associated with a degree of freedom that is localized in a given volume on the light-front hyperplane. In addition, there are an infinite number of these degrees of freedom that are localized in any open set on the light front.
While the fields are operator valued distributions, that does not preclude the existence of operators constructed by smearing with functions that have only one derivative. Note that the support condition implies that the Fourier transform of the basis functions are entire.

VII. KINEMATIC POINCARÉ TRANSFORMATIONS OF FIELDS IN THE WAVELET REPRESENTATION
Since this representation is formally exact, kinematic Poincaré transformations on the algebra of fields restricted to the light-front can be computed by acting on the basis functions. This follows from the kinematic covariance of the field This shows that kinematic transformations can be computed exactly by transforming the arguments of the expansion functions.
The transformation property of the discrete field operators restricted to a light front follows from the orthonormality of the basis functions (128): where the matrix is a discrete representation of the light front kinematic subgroup. This identity implies that in the wavelet representation kinematic Lorentz transformations on the fields can be computed either by transforming the arguments of the basis functions or by transforming the discrete field operators.

VIII. STATES IN THE WAVELET REPRESENTATION
Because the algebra of free fields restricted to the light front is irreducible and kinematically invariant interactions leave the Fock vacuum unchanged, the Hilbert space for the dynamical model can be generated by applying functions of the discrete field operators, φ n (x + = 0), to the Fock vacuum.
Smeared light-front fields can be represented in the discrete representation as linear combinations of the discrete field operators Equation (131) can be expressed as States can be expressed as polynomials in the smeared fields applied to the light-front Fock vacuum This representation can be re-expressed as a linear combination of products of discrete fields applied to the Fock vacuum The inner product of two vectors of this form is a linear combination of n-point functions. For the free field algebra, the n-point functions are products of two-point functions. The two-point functions have the form This integral is logarithmically divergent if the Fourier transforms of the smearing functions do not vanish at p + = 0. Since p + = 0 corresponds to infinite 3-momentum, this requirement is that the smearing functions need to vanish for infinite 3-momentum. From (132) and (136) it follows that the inner product above is a linear combination of two-point functions in the discrete fields, φ n (x + = 0).
The basis functions ξ m (x) have compact support which implies that their Fourier transforms are entire functions of the light-front momentap. This means that they cannot vanish in a neighborhood of p + = 0, however they can have isolated zeroes at p + = 0. For the wavelet basis functions, w l m (x), the vanishing (118) of the first three moments of the L = 3 wavelets implies that Since the Fourier transforms are entire this means that they have the formw l m (p + ) = (p + ) 3 f l m (p + ) where f l m (p + ) is entire. For the scaling function basis functions, s k m (x), the normalization condition (110) gives These results imply that is singular if both basis functions have scaling functions in the x − variable, but are finite if at least one of the basis functions has a wavelet in the x − variable.
Since the smearing functions, f (p), should all vanish at p + = 0, the discrete representation will involve linear combinations of wavelets and scaling functions whose Fourier transforms all vanish at p + = 0. In computing these quantities the linear combinations of scaling functions should be summed before performing the integrals. This can alternatively be done by including a cutoff near p + = 0, doing the integrals, adding the contributions and then letting the cutoff go to zero.

IX. DYNAMICS
The dynamical problem involves diagonalizing P − on the free field Fock space or solving the light-front Schrödinger (27) or Heisenberg equations (28). The two dynamical equations can be put in integral form where Ψ(x + = 0) and O(x + = 0) are operators in the light-front Fock algebra.
The formal iterative solution of these equations has the structure of a linear combination of products of discrete fields, φ n (0), in the light front Fock algebra with x + -dependent coefficients. What is needed to perform this iteration are the initial operators Ψ(x + = 0) and O(x + = 0) expressed as polynomials in the φ n (0), the expression for P − as a polynomial in the φ n (0), and an expression for the commutator, [φ m (0), φ n (0)], of the discrete fields on the light front.

X. THE COMMUTATOR
It follows from (54) that the commutator of the discrete fields is Unlike the inner product, the commutator is always finite since both ξ m − (x − ) and ξ n − (y − ) have compact support. The commutator (143) can be computed exactly using the renormalization group equations. The computation involves three steps. The first step is to express ξ m − (x − ) and ξ n − (y − ) as linear combinations of scaling functions on a sufficiently fine common scale. The second step is to change variables so the commutator is expressed as a linear combination of commutators involving integer translates of the fixed point s(x − ) solution of the renormalization group equation. The last step is to use the renormalization group equation to construct a finite linear system relating the commutators involving integer translates of the s(x − ).
Applying D k T n to the renormalization group equation and the expression for w(x) gives and These equations express s k n (x) and w k n (x) as linear combinations of the s k+1 n (x) : and While the matrices H n;m and G n;m are formally infinite, for each fixed n these are 0 unless 2n ≤ m ≤ 2L − 1 + 2n. Using powers of the matrices and G nl the basis function can be represented as finite linear combinations of finer resolution scaling functions where the sums in (149) and (150) are finite. Using these identities all of the integrals can be reduced to finite linear combinations of integrals involving a pair of scaling functions, s k n (x) = 2 k/2 s(2 k x − n), on a common fine scale, 2 −k . What remains is linear combinations of products of integrals of the form Changing variables this becomes where can be expressed as a difference of two integrals while the normalization condition (106) gives Adding (157) and (158) gives: If the support of s(x − + n) is to the right of the support of s(y − ), the integral is 1 while if the support of s(x − + n) is to the left of the support of s(y − ) the integral is −1. Thus for L = 3 basis functions The I[n] for n ∈ [−4, 4] are related by the renormalization group equations where a n := 2 The numbers a n will appear again. The a[n] are rational numbers [47][48][49] [50]. For L=3 the non-zero a n are The 9 × 9 matrix A mn := a n−2m (−4 ≤ m, n ≤ 4) has the following rational eigenvalues λ = 2, 1, 1 2 , 1 4 , ± 1 8 , 1 16 , 9 32 , − 9 64 , so it is invertible The non-trivial I[n] are solutions of the linear system The solution of (169) is 34201389e + 00, n = −4 8.33333333e + 00, n = −3 −1.79796007e + 01, n = −2 1.94444444e + 01, n = −1 0.00000000e − 00, n = 0 −1.94444444e + 01, n = 1 1.79796007e + 01, n = 2 −8.33333333e + 00, n = 3 3.34201389e + 00, n = 4 While (171) is a numerical solution, the exact solution is rational since both A mn and d n are rational. This solution along with (160) can be used to construct the commutator of any of the discrete field operators using (144-154).
The general structure of the commutators is Note that while this commutator looks very non-local, if the scaling functions in (143) are replaced by wavelets with supports that are sufficiently separated, the integrals vanish because the moments of wavelets vanish. This will also be true of linear combinations of scaling functions that represent functions that vanish at p + = 0.

XI. POINCARÉ GENERATORS
The other quantity needed to formulate the dynamics is an expression for P − or one of the other dynamical Poincaré generators expressed in terms of operators in the irreducible algebra. Since the generators are conserved Noether charges, they are independent of x + , so the generators can be expressed in terms of fields on the light front. The discrete representations of the generators can be constructed by replacing the fields on the light front by the discrete representation (123,126) of the fields. The integrals over the light front become integrals over products of basis functions and their derivatives. This section discusses the computation of these integrals using renormalization group methods.
A scalar φ 4 (x) theory is used for the purpose of illustration. In this case the problem is to express all of the generators as linear combinations of products of discrete fields.
The construction of the Poincaré generators from Noether's theorem was given in section IV. Using the discrete representation of fields the light-front Poincaré generators (83-89) have the following forms where where where and where and Setting x + = 0 this becomes For the remaining generators where where where and where and All of these operators have the structure of linear combinations of normal products of discrete fields evaluated at x + = 0 times constant coefficients, P + n1,n2 , P i n1,n2 , P − n1,n2 , P − n1,n2,n3,n4 , K 3 n1,n2 , J 3 n1,n2 , E i n1,n2 , F i n1,n2 , F i n1,n2,n3,n4 , which are integrals involving products of basis functions and their derivatives. The three-dimensional integrals that need to be evaluated to compute these coefficients are products of three one-dimensional integrals that have one of the following eight forms: In what follows it is shown how all of these integrals can be computed using the renormalization group equation (103). The integrals (194)(195)(196) are products of basis functions which may be scaling functions with scale 2 −k or wavelets of scale 2 −k−l for l ≥ 0. The same methods that were used in the computation of the commutator function, (144-150), can be used to express the integrals (194)(195)(196) as linear combinations of integrals involving scaling functions on a common scale fine scale, 2 −l .
After expressing the integrals in terms of scaling functions, s l n (x), and their derivatives, the one-dimensional integrals (194)(195)(196) can be expressed in terms of integrals involving products of the s n (x). A variable change, x → x ′ = 2 −l x can be used to express all of the integrals in terms of translates of the original fixed point s(x). The scale factors for each type of integral are shown below: These identities express all of the integrals involving scale 2 −l scaling functions in terms of related integrals involving the s n (x). The compact support of the functions s n (x) means the these integrals are identically zero unless the indices and the absolute values of their differences are less than 2L − 2 which is 4 for L = 3. The integrals of the right side of (198-204), are the following integrals: The renormalization group equation in the form and a variable change x → x ′ = 2x leads to the following linear equations relating the non-zero values of these integrals where a m a m := 2 where with normalization (234) and

XII. TRUNCATIONS
The value of the wavelet representation is that, while it is formally exact, it also admits natural volume and resolution truncations in the light-front hyperplane. Truncations define effective theories that are expected to be good approximations to the theory for reactions associated with a volume and energy scale corresponding to the volume and resolution of the truncations. The simplest truncation discards degrees of freedom smaller than some limiting fine resolution, 2 −l as well as degrees of freedom with support outside of some volume on the light front.
In this regard it has similar properties to a lattice truncation. Unlike a lattice truncation, because the theory is formally exact it is straightforward to systematically include corrections associated with finer resolution or larger volumes. Some other appealing features are that the truncated fields have a continuous space-time dependence and can be differentiated, so there is no need to use finite difference approximations. Finally it is possible to take advantage of some of the advantages of the light-front quantization.
One problem that is common to lattice truncations of field theory is that they break symmetries. The vacuum of the formally exact theory is the trivial Fock vacuum. The lowest mass eigenstate of the truncated P − is not necessarily the Fock vacuum, however the Fock vacuum states should become the lowest mass state in the infinite-volume, zero-resolution limit. This suggests that using trivial Fock vacuum might still be a good approximation. Similarly, truncations break the kinematic covariance. The consequence is that transforming the truncated field covariantly using (127) is not the same as transforming the truncated field using the matrix (129) and truncating the result. The difference between these two calculations is due to the discarded degrees of freedom, which should be small for a suitable truncation. This suggests that kinematic Lorentz transformations can be approximated by using (127) with the truncated fields.
In the wavelet representation the fields are all smeared. Products of fields at the same point are replaced by infinite linear combinations of products of smeared fields. In the wavelet representation the ultraviolet singularities