The discrete canonical commutation relationship

We use non-standard finite differences to propose a quantum momentum operator to be used when the spectrum of the operator is discrete. The defined discrete operator complies with the discrete versions of the properties that the continuous variable operator has. The discrete derivative is exact for its eigenfunction, that is, exponential functions. We obtain the discrete adjoint of the momentum operator. The canonical commutation relationship between conjugate operators for discrete variables is diagonal along a particular direction.


Introduction
An open question is: What is the explicit expression for a quantum momentum operator when the spectrum is discrete, the system is periodic or non-periodic, and with properties similar to those of the momentum operator when the spectrum is continuous?
We show a method to fulfill the canonical commutator relationship between the coordinate and momentum operators for discrete spectrum.In previous works, the authors defined a finite differences derivative with the characteristic that it provides the exact result when acting on exponential functions [1].In this study, we extend the property set of the discrete derivative to fulfill the canonical commutation relationship.We found that the commutator between the derivative and coordinate is constant along some directions.
For functions of continuous variables, there is a well-known relationship concerning the commutator between the continuous variable x and the derivative of functions with respect to that variable: In Quantum Mechanics theory, if Q ˆis the coordinate and P ˆis the momentum operator, then there is a similar equality valid when the spectrum of both coordinate and momentum operators are continuous.However, it would be desirable to have the same relationship between the coordinate and momentum operators when the spectrum of one or both operators is discrete.Several properties of quantum systems are based on this commutator relationship.
When the spectrum of operators is discrete, we must deal with the discrete matrix representation of these operators.In general, there are no matrices that can fulfill the canonical commutator relationship because if we take the trace of equality (2), we obtain a contradicting result [2].But we demonstrate that the commutation relationship is valid in some directions.
The simplest approximation to the derivative of a function f (x) is defined on a grid of equally spaced points

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where f j = f (x j ).In matrix language, the derivation operator is represented as If we denote by Q = diag(x 0 , x 1 , K, x N ) to the matrix of coordinate values, the commutator between these two matrices results in a matrix that shifts the values of the entries of a vector.However, if the vector has the same value in each entry, the commutator between the matrices gives the same vector inside the interval, and the canonical commutation relationship holds when using the finite difference derivative, only in the space of constant functions.Once we know this fact, we develop this idea to define a finite-difference derivative that complies with the canonical commutator for a desired function.
There are several ways to define a finite-difference derivative [6][7][8][9][10][11].In this study, we find finite difference derivative matrices D and vectors u that solve the matrix equation , Q and D are square matrices of dimension (N + 1) × (N + 1).This relationship is an eigenvalue equation for the commutator matrix: Then, in the space spanned by vectors u, matrices Q and D are a pair of canonical conjugate matrices.As we will see, in addition to these eigenvectors, the two other vectors almost satisfy this eigenvalue equation.
A previous attempt to define a discrete quantum momentum operator was based on the requirement that the exponential function be its eigenfunction.However, only the constant diagonal commutator relationship is approximately satisfied.In this study, we add another property to the finite difference derivative so that it also complies with the canonical commutator along particular orientations.
In section 2, we find an infinite family of matrices Y that satisfy the eigenvalue equation We discuss some properties of matrix Y and the commutator with the coordinate matrix.
In section 3, we discuss a particular choice of matrix Y corresponding to a derivative matrix, and we provide an explicit expression for it.We will discuss some properties of the discrete derivative matrix (d-derivative matrix).
In section 4, we explain how to make the discrete derivative exact for a given function.A few examples will be presented to illustrate the quantities involved.We provide interpretations of the improved discrete derivative (dderivative).
The summation by parts theorem is the subject of study in section 5.This result indicates what the discreteadjoint (d-adjoint) of the derivative operator is.
As an interesting application of previous results, we consider the quantum momentum operator in section 6.We found that the discrete version of the momentum operator can be discrete-symmetric (d-symmetric) in a manner similar to as its continuous variables counterpart.
At the end of this paper, there are some concluding remarks.

{
} of the interval [q 0 , q N ] of equally spaced points, we consider complex functions evaluated at these points, for instance, ), where T stands for the transpose operation.

Given a vector
with non vanishing entries, the matrix that we consider is where Δ = q j+1 − q j for j = 0, 1, L , N − 1, and y 0 , y 1 , y 2 ,K, y N are arbitrary complex numbers.The commutator between the above matrix and matrix Q with the coordinate values in its diagonal, Q = diag(q 0 , q 1 ,K, q N ), results in: This matrix causes a shift to the left and rescales when it acts on a vector, regardless of the values y i .For convenience, we can choose the values of y i to obtain a derivative matrix.The solutions to the eigenvalue equation for the commutator are the subspaces in which the commutator is constant.The eigenvalues of the commutator matrix are all zero, and the corresponding eigenvectors are ¼ 1,0, , 0 T ( ) and N null vectors.At first sight, only the delta function u j = δ j,1 can solve eigenvalue equation (6).However, in addition to these eigenvectors, vectors h and = -¼ - ) are almost eigenvectors (the last entry of the resulting vector becomes zero), with eigenvalues 1, − 1 respectively, of the commutator matrix.Then, the discrete commutator is constant in the subspaces of dimension one.This result is valid for any values y 0 , y 1 , y 2 ,K, y N , which appear in the diagonal of the matrix Y.
The vectors h and h ˜can be eigenvectors of the commutator when the last point of the lattice is placed at a different location, but we will not have a lattice with equally spaced points.

Defining a discrete derivative
Among the possible matrices Y, we are interested in a matrix that represents a discrete derivative.Therefore, we replace matrix Y with where ( ) The determinant of the discrete derivative matrix D vanishes.Then, there is no inverse to the derivative matrix.
Determining the eigenvalues and eigenvectors of the derivative matrix is straighforward.The eigenvalues of the derivative matrix are: and the corresponding eigenvectors are These vectors are linearly independent.
When the discrete derivative matrix is applied to the left, to a vector , we obtain where , 0 .14 Therefore, we approximated the derivative of the function defined on the mesh.In the second equality (in the limit part) above, we used the Taylor series expansion which appear in 1/ξ j and 1/ξ j−1 .
When the discrete derivative matrix is applied to the right, to a vector = ¼ g g g g , , , where is a discrete derivative of vector g because from equation (10), in the limit Δ → 0, we have that ξ aproaches Δ asymptotically.This finite difference derivative is the primary definition of the finite difference derivative used in this study.

Example Case in which h
), where Î c , a constant.The ratios h j /h j+1 are equal to one, which yields that ξ j = Δ, and the finite differences approximations to the derivative introduced in this study, such as in equations (14), coincide with the usual expressions in equation (3) for the finite differences derivative of a function [5].
Thus, we found a family of approximations to the derivative operator depending on a given vector h with the property that the commutator with the coordinate matrix equals one inside the interval when it is applied to h.In the next section, we provide a method for selecting vector h for convenience.

Making the discrete derivative exact for a desired function
Because h is any vector with non-vanishing entries, we choose it so that the finite differences derivatives (16) are exact for a particular function u(q), evaluated at the mesh points u j = u(q j ), 0 j N.
Let us consider the finite differences derivative , 0 , 17 where When these derivatives are applied to function u(q) at q j , they provide the exact result ( ( ) ) ( ).We converted the finite differences derivative of the previous section to an exact finite difference derivative of the function u(q) with a proper choice of ξ j (u), sometimes called a 'denominator function' [11].Now, by combining equations ( 10) and (18), we obtain the following recursion equation for h j where 0 j < N. The proportionality factor in the above equation is the ratio between the usual finite-difference derivative and exact derivative.This factor is also ratio of ξ j to Δ.The ratio h j /h j+1 is an adjustment to the usual finite-difference derivative to obtain an exact derivative for u(q).Thus, the discrete derivative of a vector g is This approximation of the derivative of a function uses the function u(q) as a reference function because the discrete derivative of a function g(q), at q j , is written as Some properties of this discrete derivative, for the case of the exponential function, are discussed in [1,[12][13][14].
Thus far, we have a discrete derivative that complies with the constant commutator equality when applied to vector h.This is exact when applied to vector u.We now consider the adjoint of the discrete derivative operation.

Summation by parts theorem
We intend to obtain a matrix that acts as the inverse operation of the discrete derivative.
The matrix that performs a sesquilinear form operation between two vectors is The sesquilinear form that defines this matrix is The vectors in this definition are not normalized but they can be normalized.This definition is an extension of an inner product since the scalar 〈f|f〉 u in general is a complex number due to ξ j .For ξ j all real, we have the usual definition of an inner product.
The summation S and derivative D matrices are inverse operations of each other meaning that that is, a discrete version of the fundamental theorem of calculus is fulfilled for each subinterval [q j , q j+1 ] because and in the entire interval [q 0 , q N ] since where 1 T = (1, 1,K,1).When the matrix SD acts to the left, We obtain the negative of the differences of the entries of the vector f * for each subinterval plus boundary terms, and their sum gives the value zero: where is the boundary matrix.We obtained a finite-difference version of the integration by parts theorem.Point-bypoint, the adjoint of the discrete derivative acts as follows: which is another approximation of the derivative of a function in terms of the finite-difference.In the limit Δ → 0, these discrete derivatives coincide.
The eigenvalues of the d-adjoint operator D ˜are m These vectors are linearly independent.The N + 1 dimensional vector space is the domain of the d-derivative matrix and of its d-adjoint.
Besides the eigenvectors of the d-adjoint derivative matrix, the exponential function is again a type of almost eigenfunction of the finite-difference adjoint derivative.The first entry of the resulting vector is zero.Using =   u e j pq j , we obtain The eigenfunctions of the derivative matrix differ from those of the d-adjoint derivative matrix.We won't find the known problem of vanishing average of the commutator between the eigenfunctions of the derivative operation or coordinate matrix.The integration by parts theorem is relevant to the Quantum Mechanics Theory.Its discrete counterpart is discussed in the next section.

The discrete quantum momentum operator
In quantum mechanics theory, the momentum operator is defined as = - P i d dq ˆ, intended to operate on functions of continuous variable q [15].The discrete version of the quantum momentum operator where u p (q) = e ± ipq/ ÿ , e ± pq/ ÿ .The functions to be considered to define the momentum operator are the real and complex exponential functions, that is, the eigenfunctions of the derivative operation.
For the real exponential functions e ± pq/ ÿ , the recursion relationship for the intermediate function h This function becomes large for a large magnitude of p, but equals one at q = 0.For the case of the complex exponential function e ± ipq/ ÿ we have that The discrete momentum operator is said to be d-symmetric if the boundary term gBf * vanishes, that is, when f N = e i θ f 0 and g N = e i θ g 0 , θ ä [0, 2π).The momentum operator and its d-adjoint coincide in the limit Δ → 0.

Concluding remarks
We propose a derivative-type operator for discrete variables that complies with a discrete version of the canonical commutation relationship with the coordinate along a particular direction.In addition, the exponential functions are eigenfunctions.The discrete momentum operator complies with a discrete version of the operator's properties for continuous variables [15].
The discrete momentum operator is applicable when the derivative with respect to energy is required because the spectrum of a Hamiltonian operator can be discrete.Thus, we have a discrete operator that is related to the time variable in quantum mechanics [1].
The results in this paper can be used in quantum gravity because it is assumed that there is a minimum length in space-time.This assumption leads to a mesh in space.Then, it would be necessary to use a discrete derivative and commutator [16,17].
Our method can be used on periodic or non-periodic systems.
adjoint derivative matrix (d-adjoint derivative matrix), and The magnitude of this function becomes small for large magnitude of p, and is equal to one at q = 0.The rewriting of the summation by parts equality equation (29) results in