1 Introduction

A quantum system evolves according to the Schrödinger equationFootnote 1:

$$\begin{aligned} i \frac{\partial }{\partial t} | \psi \rangle = H | \psi \rangle , \end{aligned}$$
(1.1)

where H is the Hamiltonian operator and \(\vert \psi \rangle \) lies in the Hilbert space \({\mathcal {H}}\) of physical states. Despite the fact that the solution to Schrödinger’s equation is formally given by the time evolution operator \(U_t = \exp (-i H t),\) finding a solution in practice when no analytical solution is available is hampered by the so-called curse of dimensionality: the relevant Hilbert space grows in general exponentially with the system size.

Quantum gauge theories represent such quantum systems for which in general no analytical solution is known. They are attacked by discretizing space and restrict to a finite volume, an approach known as lattice gauge theory. The degrees of freedom are then given by a number of gauge links per lattice site, with an infinite dimensional Hilbert space corresponding to a single link U. The total Hilbert space is eventually given by the tensor product of all these local spaces, growing exponentially with the number of lattice sites.

Even if a digital quantum computer with n qubits was available, it would allow to implement a Hilbert space of size \(2^n\) only, which is always finite. Therefore, the infinite dimensional site-local Hilbert space of a single gauge link must be truncated to a finite dimensional one in order to map it to the space spanned by the qubits. This is even more true if classical computers in combination for instance with tensor network methods are to be used. Such a truncation of \({\mathcal {H}}\) is not unique, and one needs to check that the correct dynamics is restored when the vector space becomes infinite-dimensional [1, 2] while optimizing for efficiency.

For lattice gauge theories an important aspect in the truncation of the Hilbert space is to preserve as much of the gauge symmetry and the fundamental commutation relations as possible.

The relevant Hamiltonian H for a non-Abelian \({\textrm{SU}}(N_c)\) lattice gauge theory [3] is constructed from gauge field operators U and their canonical momenta \(L_a\) and \(R_a\) \((a=1,\ldots , N_c^2-1)\) at each point x and direction \(\mu \) of the lattice. Local gauge symmetry dictates the form of H. While deferring the exact definitions to Sect. 3, the properties the U and \(L_a,\) \(R_a\) need to fulfil read:

  1. 1.

    canonical commutation relations: \([L_a, U] = -\tau _a U\) and \([R_a, U] = U \tau _a\)

  2. 2.

    structure of the Lie algebra: \([L_a, L_b] = i f_{abc} L_c\) and \([R_a, R_b] = i f_{abc} R_c,\)

  3. 3.

    special unitarityFootnote 2: \(U\cdot U^\dagger = U^\dagger \cdot U = 1\) and \(\det (U) = 1,\)

  4. 4.

    the \(L_a,\) \(R_a\) are ultra-local, i.e. the only non-vanishing components couple neighbor points in the discretized manifold.

Here \(\tau _a\) are the generators of \({\textrm{SU}}(N_c)\) and \(f_{abc}\) are the \({\textrm{SU}}(N_c)\) structure constants. Since \({\textrm{SU}}(N_c)\) is parametrized by continuous parameters, any implementation on a quantum device will, as mentioned above, require a truncation of the Hilbert space. An inevitable consequence of such a truncation is that some of the properties 1.–4. cannot be implemented exactly.

This offers a challenge and an opportunity at the same time. One question to ask is how to digitize the \(L_a\) and U operators such that efficient practical simulations are feasible. Another question to ask is which sub-set of the properties 1.–4. can be maximally preserved, i.e. an equivalent to the famous Nielsen–Ninomiya theorem for lattice Dirac operators [4]. One could even speculate whether there is an exact lattice version of local gauge invariance, just like there is an exact version of chiral symmetry on the lattice [5, 6].

There are multiple digitization prescriptions known, see for instance Refs. [7,8,9,10,11]. Most of them try to preserve the canonical commutation relations. For instance, quantum link models [12, 13] enjoy exact properties 1., 2. and 4., while the U operators are no longer unitary but become parts of a larger group. The same is true when a truncated Clebsch–Gordan expansion is used to represent the U operators (see prop. 3 of Ref. [14]).

Only recently we have studied in Ref. [15] an approach where properties 1. and 2. are only approximately fulfilled, while the gauge field operators remain unitary and the \(L_a,R_a\) ultra-local. This approach is based on finite sub-sets of \({\textrm{SU}}(N_c)\) elements and corresponding discretizations of the canonical momenta. In this paper we will show how, by a clever choice of \({\textrm{SU}}(N_c)\) elements in the sub-set and construction of the momenta, properties 2. and 3. are exact and property 1. is exact on a subspace of the total truncated Hilbert space, but the \(L_a\) are no longer ultra-local.

We will work for the special case of a \({\textrm{SU}}(2)\) lattice gauge theory, which shares many properties with the \({\textrm{SU}}(3)\) theory [3, 16], while significantly simpler to simulate. We also discuss how to generalize the arguments presented below to \({\textrm{SU}}(3).\)

The structure of the paper is the following. In Sect. 2 we summarize the main results of this paper, postponing the details to the subsequent sections. Sect. 3 recalls some theoretical background, while the details of our results are discussed in the subsections of Sect. 4. Section 4.1 shows some general arguments on \({\textrm{SU}}(2)\) sampling which are called back later. Section 4.2 gives the definition and properties of what we define as “Discrete Jacobi Transform”, namely the orthogonal transform based on the Jacobi polynomials that relates the electric and magnetic bases. In Sect. 4.3 we give the explicit representation of the gauge links and canonical momenta at finite truncation, together with some discussion on the numerical implementation and advantage of our digitization. Appendix C presents a more expensive approach which however easily generalizes to \({\textrm{SU}}(3).\) Finally, in Sect. 5 we draw our conclusion and give an outlook for future developments.

In order to ease the reading we have also included some appendices, with material that should be fairly standard to the already experts in the field. In Appendix A we recall the main properties of the \({\textrm{SU}}(N_c)\) lattice Hamiltonian degrees of freedom, and in Appendix B we discuss the differential geometry interpretation of the canonical momenta.

2 Main results

In this section we summarize the main results of this paper, which we are going to prove in the following sections. As mentioned already in the introduction, we have discussed in Ref. [15] how to define the canonical momenta \(L_a,\) \(R_a\) for the case of \({\textrm{SU}}(2)\) gauge theories formulated on a finite partitioning of \(S_3,\) which is isomorphic to the group.

With a finite partitioning of \({\textrm{SU}}(2)\) we mean a finite set \({\mathcal {D}}_m\subset {\textrm{SU}}(2)\) of m group elements with the property that \({\mathcal {D}}_m\) becomes asymptotically dense in \({\textrm{SU}}(2)\) when \(m\rightarrow \infty .\) We have given examples for such partitionings in Refs. [15, 17]. These partitionings require one to discretize the differential operators \(L_a,\) \(R_a\) or \(L^2\) directly on these sets. This however leads to a breaking of the properties 1. and 2. mentioned in the introduction (1), and hence of gauge invariance. Both properties are recovered only with \(m\rightarrow \infty .\) On the other hand, at finite m the operators U are unitary and the \(L_a,\) \(R_a\) strictly local, i.e. properties 3. and 4. are preserved.

In this paper we are going to show that by giving up the locality property of the discrete canonical momenta, one can preserve property 1. exactly on a subspace of the truncated Hilbert space, and maintain the Lie algebra structure and unitarity of U. This is done by defining a specific partitioning \({\mathcal {D}}_{N_\alpha }\) of \(S_3\) with \(N_\alpha \in {\mathbb {N}}\) elements based on Euler angles \(\vec \alpha .\) With this specific partitioning defined in the following sections there exists a \(q\in \{n/2, n\in {\mathbb {N}}\}\) and

$$\begin{aligned} N_q\ =\ \frac{1}{3} (4q +3)(q+1)(2q + 1)\ <\ N_\alpha , \end{aligned}$$
(2.1)

with:

  1. (a)

    the \(N_q\) eigenvalues of \(L^2\) with main quantum number \(j\le q\) reproduced exactly.

  2. (b)

    the Lie algebra structure preserved on the subspace spanned by the corresponding \(N_q\) (discretized) eigenvectors of \(L^2.\)

  3. (c)

    the fundamental commutation relations exactly reproduced on the subspace spanned by the discretized eigenvectors corresponding to the smallest \(N_{q-1/2}\) eigenvalues of \(L^2.\)

  4. (d)

    the remaining \((N_\alpha - N_{q-1/2})\)-dimensional part of the truncated Hilbert space possible to be projected to arbitrary energies above the cutoff.

For these properties to hold for a given q,  the partitioning must have at least

$$\begin{aligned} N_\alpha = {\left\{ \begin{array}{ll} (q + 1) \cdot (4q + 1) \cdot (4q + 1),&{}\text {if} \ q \in {\mathbb {N}},\\ (q + 1/2) \cdot (4q + 1) \cdot (4q + 1),&{}\text {otherwise}, \end{array}\right. } \end{aligned}$$
(2.2)

elements, as will also be shown below.

From the above list of properties it should be clear that the proposed scheme is expected to work well if physical states can be approximated by linear combinations of the aforementioned \(N_{q-1/2}\) eigenvectors of \(L^2.\) This expectation is, however, not that far-fetched and underlies in fact all the schemes working in a basis where \(L^2\) is diagonal, truncated at some maximal eigenvalue. So, certainly at large values of the coupling the proposed scheme is expected to work with not too large values of \(N_q.\)

In summary, our approach defines a \({\textrm{SU}}(2)\) effective Hamiltonian, preserving the unitarity of the links. The Lie algebra, equations of motion (and Gauss’ law) are fulfilled below the cutoff. Unitary links allow for a gauge invariant state preparation [3], with the Us being implementable as gates on a quantum device. The direct discretization of the manifold allows for a direct comparison with Lagrangian simulations, and for instance compare to Ref. [17].

An implementation for the construction of the operators described in this work can be found in [18].

3 Theoretical background

3.1 Remarks on the digitization of \({\textrm{SU}}(2)\)

In this work we focus on the digitization of the gauge links and the momenta needed to simulate the standard Wilson lattice Hamiltonian with \(N_c=2\) [19]:

$$\begin{aligned} H= & {} \frac{g^{2}}{4} \sum _{{\vec {x}}} \sum _{\mu =1}^{d-1} \sum _{a=1}^{3} \left[ {(L_a)}_{\mu }^{2}({\vec {x}})+{(R_a)}_{\mu }^{2}({\vec {x}}) \right] \nonumber \\{} & {} -\frac{1}{g^{2}}\sum _{{\vec {x}}} \sum _{\mu =1, \nu <\mu }^{d-1} \, {\textrm{Tr}}[{U}_{\mu \nu }({\vec {x}}) + {U}_{\mu \nu }^\dagger ({\vec {x}}) ]. \end{aligned}$$
(3.1)

The total Hilbert space is given by the tensor product of the spaces corresponding to each pair \(({\vec {x}}, \mu ).\) Therefore, in order to digitize H it is sufficient to address the problem of digitizing the degrees of freedom “pointwise”, i.e. at each point \({\vec {x}}\) and direction \(\mu .\) In the following, when writing U,  \(L_a,\) \(R_a,\) it will be understood that these correspond to a pair \(({\vec {x}}, \mu ).\) These have to fulfil the relations (see Appendix A for a review on these properties for \({\textrm{SU}}(N_c))\):

$$\begin{aligned} {[}L_a, U]= & {} - \tau _a U,\quad [R_a, U]= U \tau _a. \end{aligned}$$
(3.2)
$$\begin{aligned} {[}L_a, R_b]= & {} 0, \quad \sum _a L_a L_a = \sum _a R_a R_a, \end{aligned}$$
(3.3)
$$\begin{aligned} {[}L_a, L_b]= & {} i \epsilon _{abc} L_c, \quad [R_a, R_b] = i \epsilon _{abc} R_c. \end{aligned}$$
(3.4)

The relations in Eq. (3.4) are the standard commutation relations of quantum angular momentum [20]. Using the constraint of Eq. (3.3) we find that the irreps are labeled by 3 half-integer quantum numbers \((j, m_L, m_R)\):

$$\begin{aligned} | j, m_L, m_R \rangle , \,\, 2j \in {\mathbb {N}}, \, |m_L|,|m_R| \le j. \end{aligned}$$
(3.5)

This is the electric basis, and the generators act on its elements as follows:

$$\begin{aligned}&\left( \sum _a R_a^2\right) | j, m_L, m_R \rangle = \left( \sum _a L_a^2\right) | j, m_L, m_R \rangle \nonumber \\&\quad = j(j+1) | j, m_L, m_R \rangle , \end{aligned}$$
(3.6)
$$\begin{aligned}&L_3 | j, m_L, m_R \rangle = m_L | j, m_L, m_R \rangle , \end{aligned}$$
(3.7)
$$\begin{aligned}&R_3 | j, m_L, m_R \rangle = -m_R | j, m_L, m_R \rangle , \end{aligned}$$
(3.8)
$$\begin{aligned}&(L_1 \pm i L_2) | j, m_L, m_R \rangle \nonumber \\&= \sqrt{j(j+1) - m_L (m_L \pm 1)}| j, m_L \pm 1, m_R \rangle , \end{aligned}$$
(3.9)
$$\begin{aligned}&(R_1 \mp i R_2) | j, m_L, m_R \rangle \nonumber \\&= -\sqrt{j(j+1) - m_R (m_R \pm 1)}| j, m_L, m_R \pm 1 \rangle . \end{aligned}$$
(3.10)

\(m_L\) \((m_R)\) is the left(right) magnetic quantum number, with degeneracy \((2j+1)\) at fixed j and \(m_R\) \((m_L)\). Therefore, the degeneracy of the main quantum number j is \((2j + 1)^2.\) The vacuum \(|0\rangle \) of the electric Hamiltonian is the \(j=0\) state. In fact (recall that \(|m_L|,|m_R| \le 0)\):

$$\begin{aligned} L_3 |0\rangle&= R_3 |0\rangle = 0 |0\rangle = \vec {0}, \end{aligned}$$
(3.11)
$$\begin{aligned} L_\pm |0\rangle&= R_\pm |0\rangle = \vec {0}, \end{aligned}$$
(3.12)

where \(\vec {0}\) is the null vector of the Hilbert space. In other words, \({L_a |0\rangle = R_a |0\rangle = \vec {0} , \, \forall a}.\)

In an infinite dimensional Hilbert space we can write an exact solution for U [8, 14]. As already found in Ref. [3], \(U |0 \rangle \sim |1/2\rangle ,\) where the magnetic quantum numbers \(m_L\) and \(m_R\) are specified by the choice of the matrix element of U in color space. In general, \(U|j\rangle \) will be a combination of the \(|j+1/2\rangle \) and \(|j-1/2\rangle \) states (cf. Eq. (27) of Ref. [8]). In a finite dimensional space however, this ladder-like behavior of U cannot continue indefinitely, resulting in a “boundary effect” on the space of truncated irreps of the su(2) algebra.

We conclude with the following remark. A basis for Gauss’ law invariant states can be obtained by applying gauge invariant operators to the vacuum \(|0\rangle \) [3]. Having unitary links in a quantum simulation allows an initial state preparation that is automatically invariant, using the links as quantum gates. This avoids the presence on unphysical contributions that need to be removed by enforcing Gauss’ law a posteriori with, e.g., a penalty term.

3.2 Asymptotic behavior

In the following we use a basis \(\{ | U \rangle \}\) of the group elements eigenstates. Namely we work with operators that are functionals in the space of the wavefunctions \(\psi (U)\) which are \(L^2\)-integrable with respect to the Haar measure [19]. Since the manifold \(S_3\) is isomorphic to the group \({\textrm{SU}}(2),\) we can use the manifold points p to label the elements of the group: \(|p\rangle \equiv |U(p)\rangle \) and \(\psi (p) \equiv \psi (U(p)).\)

In this formalism, the gauge links U are \({\textrm{SU}}(2)\) matrices in the fundamental representation, while the momenta \(L_a,\) \(R_a\) are differential operators. For instance, the \(L_a\) are represented by (cf. e.g. Refs. [21, 22]):

$$\begin{aligned} L_1&= - i \left( - \sin {\phi } \frac{\partial }{\partial \theta } - \cos {\phi } \cot {\theta } \frac{\partial }{\partial \phi } + \frac{\cos {\phi } }{\sin {\theta }} \frac{\partial }{\partial \psi } \right) , \end{aligned}$$
(3.13)
$$\begin{aligned} L_2&= - i \left( + \cos {\phi } \frac{\partial }{\partial \theta } - \sin {\phi } \cot {\theta } \frac{\partial }{\partial \phi } + \frac{\sin {\phi }}{\sin {\theta }} \frac{\partial }{\partial \psi } \right) , \end{aligned}$$
(3.14)
$$\begin{aligned} L_3&= - i \frac{\partial }{\partial \phi }, \end{aligned}$$
(3.15)

where \(\theta ,\) \(\phi ,\) \(\psi \) are the Euler angles charting \(S_3.\) The operator \(\sum _a L_a L_a\) is, therefore, given by:

$$\begin{aligned} \sum _a L_a L_a= & {} - \cot {\theta } \frac{\partial }{\partial \theta } - \frac{\partial ^{2}}{\partial \theta ^{2}} - \frac{1}{\sin ^2{(\theta )}} \frac{\partial ^{2}}{\partial \phi ^{2}} \nonumber \\{} & {} + 2 \frac{\cos {\theta }}{\sin ^2{\theta }} \frac{\partial ^{2}}{\partial \psi \partial \phi } - \frac{1}{\sin ^{2}{\theta }} \frac{\partial ^{2}}{\partial \psi ^{2}}. \end{aligned}$$
(3.16)

When truncating the Hilbert space to a dimension N,  the states become vectors in a finite-dimensional vector space, and the operators endomorphisms on the latter. This inevitably results in some approximation of the commutation relations. In fact, while it is in principle possible to preserve the Lie algebra (see Eq. (3.4)), the canonical commutation relations of Eq. (3.2) are to be understood in the distributional sense. By taking the Hilbert space trace on the left and right-hand sides, we see that in a finite-dimensional vector space they never hold with unitary links (cf. [23]). Therefore, numerically, the infinite truncation limit has to be verified as a convergence of their action on arbitrary functions \(\Phi \) [15, 24]:

$$\begin{aligned} \left( [L_a, U] + \tau _a U \right) \vec {\Phi } \rightarrow 0, \end{aligned}$$
(3.17)

and analogously for the \(R_a.\) Given a functional \(\Phi (U),\) \(\vec {\Phi }\) is a (normalized) vector whose components converge to the values of \(\Phi \) when \(N \rightarrow \infty .\)

We conclude this section by observing the following. Discretizing a continuous group manifold in general breaks gauge invariance as, e.g., the multiplication of 2 elements of the group may not lie in the set [25]. This can be a non-negligible problem at the renormalization level [26]. In the Hamiltonian formulation this can be solved by a penalty term [27], leading to an effective gauge-invariant theory below the cutoff. In our prescription, the degrees of freedom behave like the continuum manifold ones on a subspace of the truncated Hilbert space, and the projector to the remaining subspace can be used to build such a penalty term in the Hamiltonian.

4 \({\textrm{SU}}(2)\) theory and construction of the momenta

In this section we provide the finite-dimensional representations of the canonical momenta. Section 4.1 discusses some general arguments about \(S_3\) which are used in the subsequent sections. We then provide an explicit construction sharing the aforementioned properties (a)–(d) of Sect. 2. Note that there is an alternative construction based on finite difference operators, which we summarise for completeness in Appendix C.

4.1 Frequencies on \(S_3\)

As mentioned previously, we are working in a basis of eigenstates of the (unitary) operator U,  discretized by means of a partitioning \({\mathcal {D}}_{N_\alpha }.\) For this we have to express the operators \(L^2\) and \(L_a\) \((R^2,\) \(R_a)\) in this basis while trying to preserve as much of the continuum properties of the operators mentioned in the introduction (1) as possible. The handle we can use to optimise for the continuum properties is the choice of the element of \({\mathcal {D}}_{N_\alpha }\) as well as the construction of the momenta. More specifically, the strategy is to choose these elements such that the lowest \(N_q\) (see Eq. (2.1)) continuum eigenvectors of \(L^2\) (and \(R^2)\) can be uniquely represented on this partitioning.

In the continuum manifold limit, we can use the isomorphy of \({\textrm{SU}}(2)\) to \(S_3\) [21], labelling the elements of the partitioning by the three Euler angles \(\theta , \phi , \psi \) with the following convention:

$$\begin{aligned} 0 \le \theta \le \pi , \quad 0 \le \phi \le 4\pi , \quad 0 \le \psi \le 4\pi . \end{aligned}$$
(4.1)

In the continuum manifold, the irreducible representations of the su(2) algebra are labelled by 3 half-integers \((j, m_L, m_R),\) where \(2j \in {\mathbb {N}}\) and \(m_L, m_R=-j,\ldots ,j,\) and the eigenfunctions of \(L^2\) are the Wigner functions \(D^j_{m_L m_R}\) [21]:

$$\begin{aligned} D^{j}_{m_L m_R}(\vec {\alpha }) \equiv D_{m_L m_R}^{j}(\phi ,\theta ,\psi ) = e^{i m_L \phi }d_{m_L m_R}^{j}(\theta )e^{i m_R \psi },\nonumber \\ \end{aligned}$$
(4.2)

with \(d_{m_L m_R}^j\) the so-called Wigner d-functions [28]

$$\begin{aligned} d_{m_L m_R}^{j}(\theta )= & {} \left( \frac{(j-m_L)!(j+m_L)!}{(j-m_R)!(j+m_R)!}\right) ^{\frac{1}{2}}(1-x)^{\frac{m_L + m_R}{2}}\nonumber \\{} & {} \times (1+x)^{-\frac{m_L-m_R}{2}} J^{m_R - m_L, m_R + m_L}_{j - m_R} (x), \quad \nonumber \\{} & {} x = \cos {(\theta )}, \end{aligned}$$
(4.3)

where J is a Jacobi polynomial. One can interpret \((j,m_L,m_R)\) as the Fourier frequencies for \(\theta , \phi , \psi \) respectively, as functions on \(S_3\) can be spectrally decomposed by means of the \(D^J_{m_L m_R}\) (see Sec. 4.10 of Ref. [29]):

$$\begin{aligned} f(\vec {\alpha })= & {} f(\phi ,\theta ,\psi )\nonumber \\= & {} \sum _{j=0}^{\infty } \sum _{m_L, m_R = -j}^{j} a^j_{m_L m_R} D_{m_L m_R}^{j}(\phi ,\theta ,\psi ). \end{aligned}$$
(4.4)

This is analogous to the spherical harmonics transform on the sphere \(S_2\) [30].

According to the Nyquist–Shannon (or Whittaker–Kote- lnikov–Shannon) sampling theorem [31], the sampling rate for each direction must be at least twice the bandwidth to be able to uniquely reconstruct functions up to the corresponding maximal frequency. Thus, on a finite partitioning, if \(N_q\) is the number of modes with \(j\le q,\) in general we will need a significantly larger \(N_\alpha \) to be able to represent these \(N_q\) modes exactly on \({\mathcal {D}}_{N_\alpha }.\) This property is the analogous of the sampling theorems for \(S_2\) [32]. From the physical point of view we are saying that frequencies higher than a threshold \(j=q\) have to be treated as unphysical. However, since we regularize the gauge theory with a cutoff, one can chose q large enough, such that the modes with \(j>q\) are above this cutoff.

4.2 A discrete Jacobi transform on \(S_3\)

In this section we discuss how to chose the elements of \({\mathcal {D}}_{N_\alpha }\) in order to be able to represent the modes with \(j\le q\) exactly. The construction is based on Jacobi polynomials, and can be viewed as a generalization of the discrete orthogonal transform for \(S_2\) found in Ref. [33]. We will use it to build a finite-dimensional representation of the canonical momenta in Sect. 4.3. We start with defining partitionings with the help of polynomials:

Definition 4.1

(Polynomial partitioning of the circle) Consider a set of orthogonal polynomials \(\{p_k(x), \, k = 0, \ldots , n , \, x=\cos {(\theta )}\}\) such that \(p_n\) has n roots (in \([-1, 1])\). We call the set of these roots, \(\{\theta _s, \, s = 1, \ldots , n\},\) a polynomial partitioning of the circle.

An example of such a partitioning many are familiar with is the one induced by Chebyshev polynomials of 1st kind: \(p_k(x) = \cos {(k\theta )}.\) The roots of \(p_N\) are \(\theta _s = \frac{2\pi }{(2j+1)} s , \, s=-j,\ldots ,j, N=2j+1.\)

Definition 4.2

(Polynomial partitioning of the 3-sphere) Let \(\{\theta _s\}\) be a polynomial partitioning from Definition 4.1. A polynomial partitioning of \(S_3\) is a set of angular coordinates \(\vec {\alpha }_k=(\theta _a, \phi _b, \psi _c)\) such that:

$$\begin{aligned}&\theta _a ,\quad a = 1, \ldots , N_\theta \end{aligned}$$
(4.5)
$$\begin{aligned}&\phi _b = \frac{4\pi }{N_\phi } b,\quad b = 1, \ldots , N_\phi , \end{aligned}$$
(4.6)
$$\begin{aligned}&\psi _c = \frac{4\pi }{N_\psi } c,\quad c = 1, \ldots , N_\psi , \end{aligned}$$
(4.7)

\(k=1,\ldots ,N_\alpha ,\) where \(N_\alpha =N_\theta N_\phi N_\psi \) is the total number of points on the sphere.

For instance, the Legendre-partitioning will be such that the \(\theta _s\) are the roots of the \(N_\theta \)-th Legendre polynomial, while the \(\phi \) an \(\psi \) will always be evenly distributed along the corresponding circles.

Next we recall the following property of orthogonal polynomials (cf. theorem (3.6.12) of Ref. [34]):

Theorem 4.1

Let \(\langle \cdot , \cdot \rangle \) be a scalar product on the linear space \(L^2[a,b]{:}\)

$$\begin{aligned} \langle f, g \rangle = \int _{a}^{b} {\textrm{d}}x \, \omega (x) f(x) g(x), \end{aligned}$$
(4.8)

where \(\omega (x)\) is the weight function. Now let \(\{p_k(x)\}_{k=0,\ldots ,n}\) be a set of orthogonal polynomials,  and \(x_1,\ldots ,x_n\) the roots of \(p_n(x).\) If the weights \(w_1,\ldots ,w_n\) are the solution of the (non-singular) system of equations : 

$$\begin{aligned} \sum _{s=1}^{n} p_k(x_s) w_s = {\left\{ \begin{array}{ll} \langle p_0, p_0 \rangle &{} k=0, \\ 0 &{} k = 1, \ldots , n-1. \end{array}\right. } \end{aligned}$$
(4.9)

Then \(w_s > 0, \, s=1,\ldots ,n\) and : 

$$\begin{aligned} \sum _{s=1}^{n} p(x_s) w_s = \int _{a}^{b} {\textrm{d}}x \, \omega (x) p(x), \end{aligned}$$
(4.10)

where p(x) is any polynomial of degree less than \(2n+1.\)

The weights \(w_s\) are often called Gaussian weights, since they can be used to integrate numerically \(e^{-x^2}\) by Taylor expanding it to a polynomial of finite degree.

Using Theorem 4.1 we now formulate the following lemma:

Lemma 4.1

Let \(d^{j}_{m_L m_R}(\theta )\) be the Wigner d-functions introduced in Eq. (4.3). We assume \(2j \in {\mathbb {N}}, \, j \le q\) and \(|m_L|,|m_R| < j.\) Let also \(w_s\) be the weights of Theorem 4.1 with \(n > 2q\) and weight function \(\omega (x)=1.\) Then if \(j_1\) and \(j_2\) are both integers or both half integers : 

$$\begin{aligned} \sum _{s=1}^{n} w_s \, d^{j_1}_{m_L m_R}(\theta _s) d^{j_2}_{m_L m_R}(\theta _s) = \frac{1}{{j_1 + 1/2}} \delta _{j_1 j_2}. \end{aligned}$$
(4.11)

Proof

We recall that under a sign change of \(m_L\) and \(m_R\) we have \(d^{j}_{m_L, m_R}(\theta ) = (-1)^{m_L + m_R} d^{j}_{-m_L, -m_R}(\theta )\) (see [28]). Therefore after some algebra we get,

$$\begin{aligned}{} & {} \sum _{s} w_s \, d^{j_1}_{m_L, m_R}(\theta _s) d^{j_2}_{m_L m_R}(\theta _s)\nonumber \\{} & {} \quad = (-1)^{m_L + m_R} \, \sum _{s} w_s d^{j_1}_{-m_L, -m_R}(\theta _s) d^{j_2}_{m_L m_R}(\theta _s) \nonumber \\{} & {} \quad = (-1)^{m_L + m_R} \, \sum _{s} w_s \, J^{-m_R + m_L, -m_R - m_L}_{j_1+m_R}(x_s)\nonumber \\{} & {} \qquad \times J^{m_R -m_L, m_R+m_L}_{j_2-m_R}(x_s). \end{aligned}$$
(4.12)

Now, whether j is integer or half-integer, \(j \pm m_R\) is always integer valued. Second, the product of two polynomials of degree \(n_1\) and \(n_2\) is \(n_1+n_2,\) therefore the product

$$\begin{aligned} J^{-m_R + m_L, -m_R - m_L}_{j_1+ m_R}(x_s)\,\cdot \, J^{ m_R -m_L, m_R+m_L}_{j_2- m_R}(x_s) \end{aligned}$$

is a polynomial of degree \(j_1+j_2,\) which is also integer valued. Finally, since \(j_1 + j_2 \le 2q < n\) by definition, we can replace the weighted sum with the integral:

$$\begin{aligned}{} & {} (-1)^{m_L + m_R} \sum _{x_s} w_s \, J^{-m_R + m_L, -m_R - m_L}_{j_1+m_R}\nonumber \\{} & {} \qquad (x_s)J^{m_R -m_L, m_R+m_L}_{j_2-m_R}(x_s) \nonumber \\{} & {} \quad = (-1)^{m_L + m_R} \int _{-1}^{1} {\textrm{d}}x J^{-m_R + m_L, -m_R - m_L}_{j_1+m_R}\nonumber \\{} & {} \qquad (x)J^{m_R -m_L, m_R+m_L}_{j_2-m_R}(x).\nonumber \\ \end{aligned}$$
(4.13)

We now use again the \(m_L, m_R\) sign change property of the d-functions, in the other direction, to conclude the proof:

$$\begin{aligned}{} & {} (-1)^{m_L + m_R} \, \int _{-1}^{1} {\textrm{d}}x \, J^{-m_R + m_L, -m_R - m_L}_{j_1+m_R}\nonumber \\{} & {} \quad (x)J^{m_R -m_L, m_R+m_L}_{j_2-m_R}(x) \nonumber \\{} & {} \quad = \int _{-1}^{1} {\textrm{d}}x \, d^{j_1}_{m_L, m_R}(\arccos {x}) d^{j_2}_{m_L, m_R}(\arccos {x})\nonumber \\{} & {} \quad = \frac{1}{{j_1 + 1/2}} \delta _{j_1 j_2}. \end{aligned}$$
(4.14)

In the last step we have used the well known orthogonality property [28]

$$\begin{aligned} \int _{-1}^{1} {\textrm{d}}x \, {\mathcal {J}}_{j_1}^{m_L, m_R}(x) \, {(j_1 + 1/2)} \, {\mathcal {J}}_{j_2}^{m_L, m_R}(x) = \delta _{j_1 j_2}. \end{aligned}$$
(4.15)

of the Algebraic Jacobi Polynomials

$$\begin{aligned} {\mathcal {J}}^{m_R, -m_L}_{j}(\cos {(\theta )}) = d^{j_1}_{m_L, m_R}(\theta ). \end{aligned}$$

\(\square \)

Now we can define the discrete transform anticipated above:

Definition 4.3

(Discrete Jacobi transform) Let \(\vec {\alpha }_k\) be a Polynomial Jacobi-partitioning of \(S_3.\) The following equation defines the Discrete Jacobi Transform (DJT) of a function f on \(S_3\):

$$\begin{aligned} f(\vec {\alpha }_k)= & {} f(\theta , \phi , \psi ) \nonumber \\= & {} \sum _{j=0}^{q} \sum _{m_L, m_R = -j}^{j} {({\text {DJT}})}^j_{m_L, m_R}(\vec {\alpha }_k) \hat{f}(j, m_L, m_R),\nonumber \\ \end{aligned}$$
(4.16)

where:

$$\begin{aligned} {({\text {DJT}})}^j_{m_L, m_R}(\vec {\alpha }_k) = (j + 1/2)^{1/2} \sqrt{\frac{w_s}{N_\phi N_\psi }} D^j_{m_L, m_R}(\vec {\alpha }_k).\nonumber \\ \end{aligned}$$
(4.17)

We recall that \(D^j_{m_L, m_R}\) are the Wigner D-functions of Eq. (4.2). If we list all the values of f on \(S_3\) in a vector of size \(N_\alpha ,\) and all the moments of the distribution \(\hat{f}\) in a vector of size

$$\begin{aligned} N_{q}=\sum _{j=0}^{q} (2j+1)^2 = \frac{1}{6} (4q+3)(2q+2)(2q+1),\nonumber \\ \end{aligned}$$
(4.18)

Eq. (4.16) can be understood in the matrix sense:

$$\begin{aligned} \vec {f}_i = {({\text {DJT}})}^{i}_{b} \vec {\hat{f}}_b, \end{aligned}$$
(4.19)

where the indices i and b are the checkerboard indices of the \(\vec {\alpha }\) s and the \((j,m_L,m_R)\) triplets, respectively. One important property of the \({\text {DJT}}\) is the following:

Theorem 4.2

If \(N_\theta > q,\) \(N_\phi > 4q\) and \(N_\psi > 4q,\) the DJT matrix has orthonormal columns,  i.e. : 

$$\begin{aligned} {({\text {DJT}})}^\dagger {({\text {DJT}})} = {1}_{N_q \times N_q} \end{aligned}$$
(4.20)

Proof

We need to show that \(({({\text {DJT}})}^\dagger {({\text {DJT}})})_{b_1 b_2} = \delta _{b_1 b_2} = \delta _{j_1 j_2} \delta _{{m_L}_1 {m_L}_2} \delta _{{m_R}_1 {m_R}_2}.\) From the explicit expression of \({({\text {DJT}})}\) we find:

$$\begin{aligned}{} & {} ({({\text {DJT}})}^\dagger {({\text {DJT}})})_{b_1 b_2}= {{({\text {DJT}})}^{*}}_{b_1}^{k} {({\text {DJT}})}^{k}_{b_2}\nonumber \\{} & {} \quad = \sum _{k} (v^{j_1}_{{m_L}_1 {m_R}_1})^* (D^{j_1}_{{m_L}_1 {m_R}_1}(\vec {\alpha }_k))^* v^{j_2}_{{m_L}_2 {m_R}_2} D^{j_2}_{{m_L}_2 {m_R}_2}(\vec {\alpha }_k) \nonumber \\{} & {} \quad = \sum _{\theta , \phi ,\psi } [(j_1 + 1/2)(j_2 + 1/2)]^{1/2}\nonumber \\{} & {} \qquad \times \frac{w_s}{N_\phi N_\psi } e^{i({m_L}_2-{m_L}_1)\phi } e^{i({m_R}_2-{m_R}_1)\psi }\nonumber \\{} & {} \qquad \times d^{j_1}_{{m_L}_1 {m_R}_1}(\theta ) d^{j_2}_{{m_L}_2 {m_R}_2}(\theta ) \nonumber \\{} & {} \quad = \delta _{{m_L}_1 {m_L}_2} \delta _{{m_R}_1 {m_R}_2} [(j_1 + 1/2)(j_2+1/2)]^{1/2}\nonumber \\{} & {} \qquad \times \sum _{s} w_s d^{j_1}_{{m_L}_1 {m_R}_1}(\theta _s) d^{j_2}_{{m_L}_1 {m_R}_1}(\theta _s) \nonumber \\{} & {} \quad = \delta _{{m_L}_1 {m_L}_2} \delta _{{m_R}_1 {m_R}_2} \frac{[(j_1 + 1/2)(j_2+1/2)]^{1/2}}{{j_1 + 1/2}} \delta _{j_1 j_2} \nonumber \\{} & {} \quad = \delta _{{m_L}_1 {m_L}_2} \delta _{{m_R}_1 {m_R}_2} \delta _{j_1 j_2}. \end{aligned}$$
(4.21)

In the intermediate step we have used the known relation \(({N,k \in {\mathbb {N}},\, N>k})\):

$$\begin{aligned} \sum _{\ell =0}^{N-1} e^{i \frac{2\pi }{N} \ell k} = N \delta _{k, 0}, \end{aligned}$$
(4.22)

which applies here by the definition of the partitioning and the lower bounds on \(N_\phi \) and \(N_\psi .\) The \(\delta _{{m_L}_1,{m_L}_2}\) and \(\delta _{{m_R}_1 {m_R}_2}\) ensure also that \(j_1\) and \(j_2\) are either both integers or both half-integers. The final step is done using the Lemma 4.1. \(\square \)

A possible choice (and also the one we will use from now on) is, therefore:

$$\begin{aligned} N_\alpha= & {} N_\theta N_\phi N_\psi \nonumber \\= & {} {\left\{ \begin{array}{ll} (q + 1/2) \cdot (4q + 1) \cdot (4q + 1) , \\ \qquad \text {if} \ q \notin {\mathbb {N}} \wedge 2q \in {\mathbb {N}},\\ (q + 1) \cdot (4q + 1) \cdot (4q + 1) , &{}\text {if} \ q \in {\mathbb {N}}. \end{array}\right. }\nonumber \\ \end{aligned}$$
(4.23)

4.3 Canonical momenta with the DJT

For \({\textrm{SU}}(2),\) classically each gauge link is parametrized by 2 complex numbers a, b:

$$\begin{aligned} {\mathcal {U}} = \begin{pmatrix} a &{} b \\ -b^* &{} a^* \end{pmatrix} , \, |a|^2 + |b|^2 = 1. \end{aligned}$$
(4.24)

This requires one real number and 2 phases. We can use the three Euler angles of Eq. (4.1):

$$\begin{aligned} {\mathcal {U}}(\vec {\alpha })= & {} e^{-i \phi \tau _3} e^{-i \theta \tau _2} e^{-i \psi \tau _3}\nonumber \\= & {} \begin{pmatrix} \cos ({\theta }/{2}) e^{-i(\phi +\psi )/2} &{} -\sin ({\theta }/{2}) e^{-i(\phi -\psi )/2} \\ \sin ({\theta }/{2}) e^{i(\phi -\psi )/2} &{} \cos ({\theta }/{2}) e^{i(\phi +\psi )/2} \end{pmatrix} \nonumber \\= & {} \begin{pmatrix} D^{1/2}_{-1/2, -1/2}(\vec {\alpha }) &{} -D^{1/2}_{-1/2, +1/2}(\vec {\alpha }) \\ -D^{1/2}_{+1/2, -1/2}(\vec {\alpha }) &{} D^{1/2}_{+1/2, +1/2}(\vec {\alpha }) \end{pmatrix} . \end{aligned}$$
(4.25)

\(\tau _a = \sigma _a/2\) are the generators of \({\textrm{SU}}(2)\) in the fundamental irrep \(j=1/2\) (see Appendix A), and \(D^j_{m_L, m_R}\) are the Wigner D-functions of Eq. (4.2). The generalization of Eq. (4.25) to higher irreps is obtained by replacing the \(\tau _a\) with the generators \({(T_j)_a}\) of the j-th irrep (cf. e.g. [3]). \({\mathcal {U}}\) is invariant under the simultaneous transformation \({\phi \rightarrow \phi +2\pi }\) and \({\psi \rightarrow \psi +2\pi }.\) Therefore we need \(0 \le \phi \le 2\pi \) (see e.g. appendix A of Ref. [21]) in order to avoid a double counting of the elements. However, even if in our discretization we have chosen \(0 \le \phi \le 4\pi ,\) the double counting does not happen. In fact, \(N_\phi \) and \(N_\psi \) are both odd numbers (see Eq. (4.23)), and the above transformation is never realized in the partitioning.

When we quantize the theory, the eigenstates of the quantum operator U are such that:

$$\begin{aligned} {U} | \vec {\alpha } \rangle = {\mathcal {U}}(\vec {\alpha }) | \vec {\alpha } \rangle . \end{aligned}$$
(4.26)

This means that we can work in the basis where the links are unitary and diagonal in the Hilbert space:

$$\begin{aligned} {U}{} & {} = \begin{pmatrix} U_{-1/2, -1/2} &{} U_{-1/2, 1/2} \\ U_{1/2, -1/2} &{} U_{+1/2, +1/2} \end{pmatrix} = \sum _{k=1}^{N_\alpha } |\vec {\alpha }\rangle \,{\mathcal {U}}(\vec {\alpha })\,\langle \vec {\alpha }| \nonumber \\{} & {} \quad \dot{=} \, \begin{bmatrix} {\mathcal {U}}(\vec {\alpha }_1) &{} 0 &{} \cdots &{} 0 \\ 0 &{} {\mathcal {U}}(\vec {\alpha }_2) &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} \cdots &{} {\mathcal {U}}(\vec {\alpha }_{N_{\alpha }}) \\ \end{bmatrix} . \end{aligned}$$
(4.27)
Fig. 1
figure 1

Numerical spectrum of the canonical momenta obtained with the DJT, limited to the first \(N_q\) eigenstates with \(j \le q.\) For purely illustrative purposes we consider \(q=3/2.\) The horizontal axes are label indices, while the eigenvalues \(\lambda _i\) are on the vertical axes. Left panel: Spectrum of the \(L_a.\) The eigenvalues are slightly shifted as a function of a for better visualization, but are equal to \(m_L = -j,\ldots ,j\) (for each j) up to machine precision. Right panel: Spectrum of \(\sum _a L_a L_a.\) We reproduce exactly the eigenvalues \(j(j+1),\) for \(j=0,\ldots ,q.\) The data have been produced with the implementation of Ref. [18]

Full size image

Using the Definition 4.3 of the discrete Jacobi transform in matrix form with the notation \(V = {\text {DJT}}\) and the property proven in Theorem 4.2, we can now give our representations of the truncated momentum operators in the magnetic basis:

$$\begin{aligned} L_a&= V \hat{L}_a V^\dagger , \end{aligned}$$
(4.28)
$$\begin{aligned} R_a&= V \hat{R}_a V^\dagger . \end{aligned}$$
(4.29)

\(\hat{L}_a,\) \(\hat{R}_a\) are the matrix representations of the generators in the electric basis truncated at \(j=q.\) They have the following properties, which we anticipated in the introduction:

Proposition 4.1

The truncated momentum operators in the magnetic basis preserve the group algebra : 

$$\begin{aligned}{}[L_a, L_b] = i f_{abc} L_c. \end{aligned}$$
(4.30)

Proof

This can be proven using \(V^\dagger V = \textrm{1}\):

$$\begin{aligned}{}[L_a, L_b]= & {} L_a L_b - L_b L_a = V \hat{L}_a V^\dagger V \hat{L}_b V \nonumber \\{} & {} \quad - V \hat{L}_b V^\dagger V \hat{L}_a V \nonumber \\= & {} V \hat{L}_a \hat{L}_b V^\dagger - V \hat{L}_b \hat{L}_a V^\dagger = V [\hat{L}_b, \hat{L}_a] V^\dagger \nonumber \\= & {} i f_{abc} V \hat{L}_c V^\dagger = i f_{abc} L_c. \end{aligned}$$
(4.31)

The proof for the \(R_a\) is identical. \(\square \)

Proposition 4.2

The first \(N_q\) eigenvalues of \(\sum L_a^2\) are reproduced exactly,  while the remaining \({N_r = N_\alpha - N_{q}}\) dimensional subspace belongs to the kernel of \(\sum L_a^2.\)

Proof

This is true because the columns of V are eigenvectors of \(\sum L_a^2=\sum R_a^2,\) \(L_3\) and \(R_3.\)

Without loss of generality, we prove this only for \(\sum L_a^2\) and \(L_3\) since for the \(R_a\) the steps are identical. If a is the checkerboard index of \((j_a, {m_L}_a, {m_R}_a),\) the vector \(\vec {v}^{j_a}_{{m_L}_a {m_R}_a} = \vec {v}_a\) with components \(V^i_a, \, i=1,\ldots ,N_\alpha \) satisfies:

$$\begin{aligned} \left[ \left( \sum _b L_b^2\right) \vec {v}_k\right] ^i&= \sum _{a_1, a_2, k} {V}^{i}_{a_1} \left( \sum _b \hat{L}_b^2 \right) _{a_1 a_2} {V^\dagger }^{k}_{a_2} {V}^{k}_a\nonumber \\&= \sum _{a_1, a_2} V^{i}_{a_1} j_{a_1} (j_{a_1} + 1) \delta _{a_1 a_2}\delta _{a_2 a} \nonumber \\&= j_a(j_a + 1) [\vec {v}_k]^{i} \end{aligned}$$
(4.32)

and

$$\begin{aligned} \left[ L_3 \vec {v}_k\right] ^i= & {} \sum _{a_1, a_2, k} {V}^{i}_{a_1} (L_3)_{a_1 a_2} {V^\dagger }^{k}_{a_2} {V}^{k}_a \nonumber \\= & {} \sum _{a_1, a_2} V^{i}_{a_1} m_{a_1} \delta _{a_1 a_2} \delta _{a_2 a} = {m_L}_a [\vec {v}_k]^{i}. \end{aligned}$$
(4.33)

Finally, from the rank-nullity theorem [35] for \(V^\dagger ,\) there exist \(N_r\) states \({\{|r_k\rangle \}_{k=1,\ldots ,N_r}}\) such that \(V^\dagger |r_k \rangle = \vec {0}.\) The \(|r_k \rangle \) are the \(N_r\) Wigner functions with \(j > q,\) and satisfy:

$$\begin{aligned} L_a \vec {v}^{j}_{m_L, m_R} = R_a \vec {v}^{j}_{m_L, m_R} = \vec {0} , \quad j > q. \end{aligned}$$
(4.34)

\(\square \)

We remark that the residual \(N_r\) states behave like the vacuum, but are not the same as \(|0\rangle ,\) since the action of U will not necessarily mix them with \(j=1/2\) only. Figure 1 shows explicitly how our implementation reproduces exactly the \(N_q\) eigenvalues of the \(L_a\) and \(\sum _a L_a L_a.\)

Let us now discuss the eigenstates of the discretised operators.

Proposition 4.3

Consider the first (linearly independent) \(N_q\) eigenstates of \(\sum _a L_a L_a,\) \(L_3,\) \(R_3.\) As \(q \rightarrow \infty ,\) they approach the naive discretization of the eigenfunctions of the continuum manifold operators, namely the Wigner D-functions.

Proof

The columns of V are not only some eigenstates with the correct eigenvalues (with Eq. (4.28) this would be the case for any invertible matrix V), but from Lemma D.1 we know that in the \(q \rightarrow \infty \) limit their components are the values of the Wigner D-functions stacked into a vector of \(N_\alpha \) components. From the continuum manifold formalism we also know that the electric field operators are represented by differential operators [21, 36] and the eigenfunctions corresponding to the states \(|j, m_L, m_R\rangle \) of Eqs. (3.6) to (3.10) are indeed the Wigner D-functions:

$$\begin{aligned} \langle \vec {\alpha } | j, m_L, m_R \rangle= & {} \langle \theta , \phi , \psi | j, m_L, m_R \rangle \ \nonumber \\\propto & {} D^{j}_{m_L m_R}(\theta , \phi , \psi ). \end{aligned}$$
(4.35)

This implies that, up to a normalization factor, for \(q \rightarrow \infty \) the columns of V become the continuous manifold eigenfunctions sampled at the points of the \(S_3\) partitioning.

The linear independence (at any q) follows from the fact that they are eigenvectors with distinct eigenvalues. \(\square \)

Finally, we are able to also show that the canonical commutation relations are exactly reproduced on a subspace of the discretised Hilbert space.

Proposition 4.4

The canonical commutation relations are reproduced exactly on the first \({N_{q'} = N_{q-1/2}}\) eigenvectors of \(\sum L_a^2=\sum _a R_a^2,\) \(L_3\) and \(R_3.\)

The proof follows immediately from the Lemma D.2 and Proposition B.3, by replacing \(\psi (U)\) by an element \(\vec {\Phi }\) in the space spanned by the first \(N_{q'}\) vectors \(\vec {w}^j_{m_L m_R}\) of Lemma D.2.

Let us make the following remarks:

  1. 1.

    The states with \(q < j \le q'\) have the correct eigenvalues but don’t fulfill the canonical commutation relations.

  2. 2.

    We can always project the residual \(N_r = N_\alpha -N_q\) (or \(N_{r'} = N_\alpha - N_{q'})\) states to whatever energy above the cutoff of the theory. If \(P_g\) is the projector to this “garbage space”, this is implemented as \({L_a \rightarrow L_a + \kappa P_g},\) where:

    $$\begin{aligned} P_g{} & {} = \sum _{j> q_t} | j, m_L, m_R \rangle \langle j, m_L, m_R | \,\nonumber \\{} & {} \dot{=} \, \sum _{j > q_t} \vec {v}^{j}_{m_L, m_R} \left( {\vec {v}}^{j}_{m_L, m_R} \right) ^\dagger , \end{aligned}$$
    (4.36)

    for some \(\kappa \gg 1\) and target truncation \(q_t.\)

  3. 3.

    One can show (e.g. by induction) that \(N_r = N_\alpha -N_q\) is always even. For, if \(P_g\) projects to the first \(N_r\) states, we can also preserve the Lie algebra while projecting above the cutoff:

    $$\begin{aligned} L_a \rightarrow L_a + P_g \left[ {\mathbbm {1}}_{N_q \times N_q} \oplus (\tau _a \otimes {\mathbbm {1}}_{\frac{N_r}{2} \times \frac{N_r}{2}}) \right] P_g.\nonumber \\ \end{aligned}$$
    (4.37)

5 Conclusion and outlook

In this paper we have discussed a specific approach to the digitisation of the \({\textrm{SU}}(2)\) lattice gauge theory Hamiltonian, which is needed for tensor network or quantum computer based simulations of lattice gauge theories. The digitisation scheme is formulated in a so-called magnetic basis, where the gauge field operator is diagonal and unitary, while gauge symmetry is preserved exactly on a subspace of the truncated Hilbert space. This comes at the price of a dense matrix representation for the canonical momentum operators.

The approach is based on specific partitioning of the sphere \(S_3\cong {\textrm{SU}}(2)\) and a discrete Jacobi transform with the main property that the \(N_q\) continuum eigenfunctions with main quantum number \(j \le q\) of the electric part \(L^2\) in the Hamiltonian can be exactly and uniquely represented. The remaining states can be shifted above an energy cutoff and interpreted as integrated out.

It remains to be seen whether this formulation performs more efficiently than other formulations on the market, for instance compared to the proposal from Ref. [10]. Most importantly, it needs to be investigated in how far the residual gauge symmetry breaking spoils simulation results and renormalisability.

Including fermionic fields in this formulation is unproblematic. First numerical results have been presented at Lattice 2023 for a 2-sites \(1+1\) dimensional \({\textrm{SU}}(2)\) Schwinger type model [37,38,39], for which the discretization introduced in the previous sections reproduces the spectrum exactly.