{'fraction_non_alphanumeric': 0.05562768008182856, 'fraction_numerical': 0.01156615130414257, 'mean_word_length': 4.0597133757961785, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Wigner-Eckart theorem and Jordan-Schwinger representation for infinite-dimensional representations of the Lorentz group

The Wigner-Eckart theorem is a well known result for tensor operators of SU(2) and, more generally, any compact Lie group. This paper generalises it to arbitrary Lie groups, possibly non-compact. The result relies on knowledge of recoupling theory between finite-dimensional and arbitrary admissible representations, which may be infinite-dimensional; the particular case of the Lorentz group will be studied in detail. As an application, the Wigner-Eckart theorem will be used to construct an analogue of the Jordan-Schwinger representation, previously known only for finite-dimensional representations of the Lorentz group, valid for infinite-dimensional ones.


I. INTRODUCTION
Representation theory of Lie groups has many applications in modern physics, especially in quantum theory. In particular tensor operators for compact Lie groups have been widely used in non-relativistic quantum mechanics [1], mainly because of the fact-known as Wigner-Eckart theorem-that their matrix elements can be expressed as a product of a Clebsch-Gordan coefficient 1 and a factor independent of the particular basis vectors being evaluated.
Although only compact groups are generally considered, one may wonder if these result extends to noncompact ones as well. A generalisation of the Wigner-Eckart theorem to non-compact groups has been considered in [2], but only for tensor operators "transforming" 2 as unitary representations, which are generally infinite-dimensional in the non-compact case; here tensor operators with finitely many components will be considered. The particular case of SL(2, R) was previously studied by the author in [3]; in the present paper that result is generalised to any Lie group. The information provided by the Wigner-Eckart theorem, however, relies on the knowledge of the Clebsch-Gordan decomposition of the product of a finite-dimensional representation and an arbitrary admissible one, which may be infinite-dimensional if the group is non-compact. Since a general result about this decomposition is not available, each case has to be treated individually; in particular the couplings for the Lorentz group-previously unconsidered 3 -will be considered here. Most of the paper will be dedicated on this task.
An application of the theorem will also be considered. An important result of SU(2) representation theory, especially useful in quantum field theory, is the Jordan-Schwinger representation, i.e., the reformulation of the su(2) generators in terms of two quantum harmonic oscillator operators. A similar result for the Lorentz group exists only for finite-dimensional representations, which can be seen as the product of two SU (2) representations (see Section II). It will be shown that, making use of the Wigner-Eckart theorem, the infinite-dimensional representations admit an analogous construction in terms of tensor operators.
The paper is organized as follows: Section II contains a review of representation theory of the Lorentz group and the analysis of the product of a finite-dimensional and an infinite-dimensional representation. Results concerning tensor operators are presented in Section III: definition, Wigner-Eckart theorem and Jordan-Schwinger representation. Finally, a table of notations used throughout the paper, a table of Clebsch-Gordan coefficients for the coupling of finite and infinite-dimensional representations and some results needed in Section II are included as appendices.

II. REPRESENTATION THEORY
This section contains both preliminary notions and new results about the representation theory of the Lorentz group. First the definition of Lorentz group and Lorentz algebra and their irreducible admissible representations will be recalled, then the decomposition of the product of a finite-dimensional and an infinite-dimensional representation will be studied. The reviews of representation theory follows [4][5][6].
A. The Lorentz group and its Lie algebra The proper orthochronous Lorentz group SO 0 (3, 1), henceforth simply referred to as the Lorentz group, is the identity component of the subgroup of GL(4, R) that preserves the quadratic form To allow for spin representations, the double cover Spin(3, 1) ∼ = SL(2, C) R of SO 0 (3, 1) will be used here; moreover, only complex representations will be considered, so that one may work with a complexified Lie algebra. The Lie algebra spin(3, 1) C has 6 generators with commutation relations The J's generate the subalgebra spin(3) ∼ = su(2) (i.e., spatial rotations), while the K's are the generators of boosts. The algebra has two Casimirs which, introducing the operators and making use of (3), can be rewritten as B. Irreducible representations of the Lorentz group We will restrict to admissible Hilbert space representations (which include, in particular, every unitary representation), i.e., those that are unitary when restricted to the maximal compact subgroup SU(2) and such that each irreducible unitary representation of SU(2) appears in it with finite multiplicity.
Each admissible representation induces an admissible (g, K)-module 4 , with g = spin(3, 1) and K = SU (2), which as an algebraic object is easier to work with; although not true in general, for Spin(3, 1) admissible irreducible representations of the group and admissible irreducible (g, K)-modules are in one-to-one correspondence 5 .
The general irreducible admissible (g, K)-module, labelled by a pair (λ, ρ) ∈ Z/2 × C, is the algebraic direct sum 6 4 An algebraic object that, given a Lie group G with Lie algebra g and a maximal compact subgroup K, is both a Lie algebra representation of g and a group representation of K, with the appropriate compatibility conditions [6]. Here a module is a vector space equipped with a linear action of one of more rings on it (e.g., g and K). 5 Compare the list of admissible group representations in [5, chap. VIII] and the list of admissible (g, K)-modules in [6, chap. 4]. 6 i.e., only sums of finitely many vectors are considered. of unitary irreducible SU(2)-modules V j λ,ρ , where the sum is in integer steps and, depending on the values of λ and ρ, it is either j max ∈ |λ| + N 0 or j max = ∞ (see later discussion). The (complex) vector space V j λ,ρ is spanned by the basis on which the su(2) C generators act as with i.e., they are eigenvectors for J 0 and J 2 ; since SU(2) is simply connected, its action on V j λ,ρ is completely determined by the corresponding Lie algebra action. The (g, K)-module will be given an inner product by requiring the SU(2)-modules to be orthogonal to each other and the vectors in (8) to be orthonormal.
The possible matrix elements of the boost generators are where and The Casimirs act on V λ,ρ as The values j can take have an upper bound j max ∈ |λ| + N 0 if and only if P + λ,ρ (j max ) = 0 and P + λ,ρ (j) = 0 ∀j < j max , i.e., ρ = ±(j max + 1).
It follows that V λ,ρ is finite-dimensional when ρ ∈ ±(|λ| + N) and it is infinite-dimensional in all other cases.
Remark (isomorphic modules). The values of the Casimirs and of P λ,ρ (j), P + λ,ρ (j) and P − λ,ρ (j) are invariant under the change (λ, ρ) → (−λ, −ρ); moreover, whether the module is finite-dimensional and the eventual value of j max are unaffected by the change as well. It follows that the modules V λ,ρ and V −λ,−ρ are isomorphic.

Finite-dimensional modules
It was shown that the (g, K)-module with ρ = B(ω + 1), ω ∈ |λ| + N 0 , B = ±1 is finite-dimensional. We will assume, for finite-dimensional modules (and for those only), that λ ≥ 0. It is then easy to check that Finite-dimensional modules can be given an alternative construction using the fact that i.e., by changing to the basis with commutation relations one can easily show that, for finite-dimensional modules, so that each M A a is self-adjoint, i.e., the action of each su(2) subalgebra is unitary. From su(2) representation theory we know that, if V j is the (2j + 1)-dimensional unitary irreducible su(2)module, Since J = M (−) + M (+) , one can then change the basis to where j, m|j 1 , m 1 ; j 2 , m 2 are the su(2) Clebsch-Gordan coefficients 7 and (26) 7 The Condon-Shortley convention [7] is used here.
it is assumed that M (−) and M (+) only act respectively on |j 1 , m 1 and |j 2 , m 2 . The dimension of the new basis is as expected. The choice of j 1 , j 2 depends on the sign of ρ: in fact, one has but also so that (28) is consistent if and only if Conversely, one can show that every product of su(2)-modules gives rise to a Lorentz (g, K)-module with As a consequence, every finite-dimensional irreducible (g, K)-module can be specified by a pair (j 1 , j 2 ) ∈ N 0 /2 × N 0 /2; is it customary to use the pair to denote the module itself. Examples of finite-dimensional modules are • (0, 0): the scalar module (trivial representation); • ( 1 2 , 0) and (0, 1 2 ): respectively the left and right Weyl spinor modules; : the Dirac spinor module (not irreducible). It is not difficult to infer the decomposition of the product of two finite-dimensional modules from su(2) results; one has and, in particular, We will refer to modules of the kind (j, 0) and (0, j) respectively as left and right modules; it follows from (34) that any other irreducible module can be constructed from the product of a left and right one. To allow to easily specify if a module is left or right, the notation will be used in the following sections. A basis for F A j is given by

C. Decomposition of the product of finite and infinite-dimensional modules
We are interested in the tensor product of a non-trivial finite-dimensional module (necessarily non-unitary) and an infinite-dimensional one (either unitary or non-unitary). In light of the consequences of (34) mentioned above, it will suffice to restrict ourselves to couplings of the kind F A γ ⊗ V λ,ρ , where γ ≥ 1 2 and λ, ρ are such that The goal is to study the Clebsch-Gordan decomposition of such a product, i.e., wheter it is possible to rewrite such a module (generally reducible) as a sum of irreducible ones. Equivalently, one can ask if it is possible to simultaneously diagonalise the two Casimirs in the product module 8 , where the generators act as on the basis elements Instead of working with an infinite dimensional vector space, one can decompose the product space into a sum of finite-dimensional spaces: this is accomplished by diagonalising J 0 and J 2 first. Using su(2) recoupling theory, one finds that the vectors (41) provide an orthonormal basis of (J 0 , J 2 )-eigenvectors for the product space. The eigenspace V J M , defined by is spanned by the basis vectors so that note that, when |λ| ≥ γ, it is always true that J ≥ ||λ| − γ| = |λ| − γ, so that the case J < ||λ| − γ| only needs to be considered when |λ| < γ. The set of possible values of J is It will prove useful in the following to build J (λ, γ) in a different way. First notice that as Let then simple counting arguments show that where χ is the indicator function Since the Casimirs commute with both J 0 and J 2 , one can work with their restriction on the finite-dimensional subspaces V J M and diagonalise those; moreover, it suffices to consider the restrictions to V J := V J J thanks to the following so that Since J ± commutes with the Casimirs, given a C a -eigenvector |α a ∈ V J M with eigenvalue α a ∈ C one has whenever V J M ±1 is defined, so that each V J M has the same eigenvalues.
The action of the Casimirs on the basis vectors of V J is given by where it is implicitly assumed that |(j) J = 0 if j ∈ Ω J (λ, γ). Note that the matrix form of each C a is tridiagonal (see Appendix C) and that, for the subdiagonal entries, it follows from Proposition C2 that the eigenspaces of C a are all 1-dimensional, so that it is diagonalisable if and only if it has dim V J distinct eigenvalues. Explicitly, the Casimirs are simultaneously diagonalisable on V J if and only if there is a basis with such that and for every (Λ, P), (Λ , P ) ∈ C J (λ, ρ, γ, A) note that at this stage Λ is allowed to be any complex number, to ensure that any pair of eigenvalues of the Casimirs can be written as in (59). The coefficients of the change of basis A{γ A ; (λ, ρ) j|(Λ, P) J} will be called Clebsch-Gordan coefficients, in analogy with su(2) representation theory. Conversely, one has the inverse change of basis where the B{(Λ, P) J|γ A ; (λ, ρ) j} will be called inverse Clebsch-Gordan coefficients. As a consequence of Solving the eigenvalue equations for arbitrary γ is not an easy task: instead, we will solve explicitly the case γ = 1 2 and proceed by induction for the other cases. When γ = 1 2 , one has and it can be explicitly checked that, when λ = −Aρ, the corresponding Clebsch-Gordan coefficients can be found in Table II (Appendix B). When ρ = −Aλ the eigenvalues for J ≥ |λ| + 1 2 coincide, so that, as pointed out earlier, the Casimirs cannot be diagonalised. As the eigenvalues do not depend on J and the eigenvectors can be extended to an eigenbasis of the whole product space, where (49) ensures that the counting is consistent. One can check that, for all eigenvalue pairs, These modules are never both unitary: a list of the possible pairs (λ, ρ) such that there is one unitary module in the decomposition can be found in have a unitary one in the decomposition. A generalisation of the case γ = 1 2 to arbitrary γ is provided by the following Propositions: Lemma 1. Let γ ∈ N/2, A = ±1 and (λ, ρ) ∈ Z/2 × C, with ρ ∈ ±(|λ| + N). Then Proof. Let µ = ν ∈ M γ . The statement reduces to which is equivalent to The possible values the sum µ + ν can take are so that (71) is true if and only if ρ + Aλ ∈ (−2γ, 2γ) ∩ Z.
Proposition 2. Consider the product F A γ ⊗ V λ,ρ , with γ ≥ 1 2 and V λ,ρ infinite-dimensional. When ρ + Aλ ∈ (−2γ, 2γ) ∩ Z the Casimirs are simultanously diagonalisable, with Proof. The proof proceeds by induction on γ ∈ N/2. Assume that the statement is true for γ − 1 2 , and consider the product F A γ ⊗ V λ,ρ , γ > 1 2 . It is known from su(2) representation theory that so that in particular Consider now the J 2 -eigenspace V J , J ≥ |λ| + γ, so that J − γ ∈ Ω J (λ, γ) and the vector exists. Using (75), |(J − γ) J can be rewritten as where the inductive hypothesis and the fact that were used. Since in particular ρ = −Aλ, the results of the case γ = 1 2 can be used, so that where [σ] keeps track of the fact that (λ + τ + σ, ρ + Aτ + Aσ) comes from (λ + τ, ρ + Aτ ). There are exactly 4γ (independent) vectors on the rhs of (79), namely with 2γ + 1 ≡ dim V J distinct eigenvalue pairs (see Lemma 1). As shown in Appendix B, when J ≥ |λ| + γ the Clebsch-Gordan coefficients satisfy where α is fixed by the normalisation convention and is independent of J. Using this formula and the γ = 1 2 Clebsch-Gordan coefficients from Table II B{(Λ + σ, P + Aσ) J| 1 it is possible to write where the vectors on the rhs are defined (up to a normalisation factor) as As these vectors live in different (C 1 , C 2 )-eigenspaces, they are necessarily independent. Moreover, they form a basis of V J : in fact, we know from (55) that with so that Since |(J − γ) J is a linear combination of the |(Λ, P) J vectors, it follows recursively that It follows from Proposition 1 that all the results obtained for V J hold for each V J M , M ∈ M J . One can then extend them to every J ∈ J (λ, γ) by defining recursively (once the Clebsch-Gordan coefficients and the vectors have been appropriately normalised, so that the generators act on the (C 1 , C 2 )-eigenvectors as (11)) for J ≤ |λ| − γ, which are trivially still eigenvectors. Since a basis of eigenvectors has been constructed for the whole product space, it follows that the Casimirs are diagonalisable.
Proof. Consider the J 2 -eigenspace V J , J ≥ |λ| + γ, and let which is continuous in ρ ∈ C, as it is a product of continuous function of ρ (see eq. (55)). From Proposition 2 we have that, for ρ + Aλ ∈ (−2γ, 2γ) ∩ Z, it follows from continuity that, for each fixed λ ∈ Z/2, n ∈ {−2γ + 1, . . . , 2γ − 1}, From Lemma 1 we know that there are at most 2γ distinct eigenvalues in this case, while dim V J = 2γ + 1. As pointed out earlier, the matrix form of C a | V J satisfies the assumptions of Proposition C2, so that it has at most 2γ eigenvectors, i.e., it is not diagonalisable on V J (and hence on the whole product space).
Proof. As already noted, one has (γ 1 , γ 2 ) ∼ = F − γ1 ⊗ F + γ2 , so that we may diagonalise the Casimirs in F + γ2 ⊗ V λ,ρ first, and then, for each resulting eigenspace V λ,ρ , in F − γ1 ⊗ V Λ,P . We can distinguish 3 cases: 1. if ρ + λ ∈ (−2γ 2 , 2γ 2 ) ∩ Z the product F + γ2 ⊗ V Λ,P admits a decomposition. The second decomposition exists if and only if, for each ν ∈ M γ2 , and decompose the product F − γ1 ⊗ V Λ,P first. Following the same reasoning of the previous case, the product of each resulting submodule with F + γ2 will not be decomposable, as for each ν ∈ M γ1 The only results we have are on the product of F A γ with irreducible modules, so we are not in a position to say anything in this case. However, one can check, following the same procedures of Proposition 2, that F + γ2 ⊗ V λ,ρ has at least one submodule V λ+γ2,ρ+γ2 . Since the product F − γ1 ⊗ V λ+γ2,ρ+γ2 is not decomposable, the product (γ 1 , γ 2 ) ⊗ V λ,ρ ⊇ F − γ1 ⊗ V λ+γ2,ρ+γ2 will be indecomposable as well.
The Clebsch-Gordan coefficients of the product (γ 1 , γ 2 ) ⊗ V λ,ρ will be denoted by where α keeps track of the multiplicity of V Λ,P in the decomposition.

III. TENSOR OPERATORS
This sections contains the main results of the paper, i.e., the generalisation of the Wigner-Eckart theorem to arbitrary groups and in particular to the infinite-dimensional representations of the Lorentz group. The section is organized as follows: first tensor operators for group representations and (g, K)-modules of arbitrary Lie groups are introduced, then Wigner-Eckart theorem will be presented. Lastly, as an application, the theorem will be used to generalise the Jordan-Schwinger representation to infinite-dimensional (g, K)-modules of the Lorentz group.

A. Definition of tensor operators
Tensor operators for a Lie group G are, roughly speaking, a particular class of operators that transform as a vector in a representation of G under the action of the group. Explicitly Definition 1 (Strong tensor operator). Let V 0 , V and V be arbitrary (topological) G-modules of a Lie group G, with V 0 finite dimensional. A strong tensor operator for G is an intertwiner between V 0 ⊗ V and V , i.e. a continuous linear map If V 0 is irreducible T is called an irreducible strong tensor operator.
A weaker definition can be used when working with (g, K)-modules instead of group representations: Definition 2 (tensor operator). Let V 0 , V and V be arbitrary (g, K)-modules of a Lie group G, with V 0 finite dimensional. A tensor operator for G is an intertwiner between V 0 ⊗ V and V , i.e., a linear map where g and K act on the product module as If V 0 is irreducible T is called an irreducible tensor operator.
The two definitions of tensor operator are related to each other; one has, in fact Proposition 4. An intertwiner T : V → V between G-modules is also an intertwiner between the corresponding (g, K)-modules. As a consequence, a strong tensor operator is also a tensor operator.
Proof. The subspace V K ⊆ V of K-finite vectors, i.e., the set of all vectors v such that span{k v, k ∈ K} is finite-dimensional, is the (g, K)-module associated to V , with One has, since T commutes with the action of K ⊆ G, where the fact that T is continuous was used. It follows that T | V K is an intertwiner between the (g, K)-modules V K and V K .
It is often preferable to have operators between V and V : this can be achieved by defining the "components" of a (strong) tensor operator T in a basis e i ∈ V 0 , i ∈ I as the definitions of strong tensor operator and tensor operator become respectively and [X, where ·, · is the dual pairing of V * 0 and V 0 and e j ∈ V * 0 , j ∈ I is the dual basis defined by e j , e i = e j (e i ) = δ j i .
The definition of tensor operator can be simplified when K is connected, since one can simply require the the operator commutes with every element of g. In fact one has 9 Proposition 5. If K is connected a linear map T : V → V is a (g, K)-module homomorphism if Proof. Let k ⊆ g be the Lie algebra of K. For any X ∈ k, v ∈ V , u ∈ V * one has Since the derivative vanishes, it must be so that However, if K is connected, exp(k) ⊆ K generates K [9], hence

B. Wigner-Eckart theorem
One of the most useful property of irreducible tensor operators is the so-called Wigner-Eckart theorem, originally proved for compact groups [10] and later extended to non-compact groups [2] for the particular case of tensor operators transforming as (infinite-dimensional) unitary representations, which we don't consider.
We will present here a version of the theorem working for a generic Lie group G. In order to prove it, the following Lemma is needed: K)-modules for a Lie group G, with V 0 finite-dimensional. If a Clebsch-Gordan decomposition into irreducible modules for V 0 ⊗ V exists, a non-zero intertwiner is possible if and only if V appears (at least once) in the decomposition. If T is the vector space of all such intertwiners, dim T equals the multiplicity of V in the decomposition, and a basis is provided by the projections in each of the submodules W α ⊆ V 0 ⊗ V , W α ∼ = V , with α keeping track of the multiplicities.
Proof. Let T : V 0 ⊗V → V be a (g, K)-module homomorphism. Shur's Lemma for irreducible (g, K)-modules [6] guarantees that, if W ⊆ V 0 ⊗ V is a submodule, It is then trivial to see that any such T can be written as a linear combinations of the independent maps T α that project We can now state the theorem, which trivially follow from the Lemma.
Theorem (Wigner-Eckart). Let T : V 0 ⊗ V → V be an irreducible tensor operator, with V , V irreducible. If a Clebsch-Gordan decomposition for V 0 ⊗ V exists, T is a linear combination of the projections T α : Note that, although the theorem works for a generic group, no information is available when V 0 ⊗ V is not decomposable; moreover, knowledge of the decomposition is needed in order to gain any real information about the operator: it is for this reason that for non-compact groups, where the non-trivial decomposition of the product of a finite-dimensional and an infinite-dimensional is needed, one has to study it case by case.
We obtain in particular, for Spin(3, 1), the following Corollaries, where the conditions for the decomposition to exist were studied in Section II C. Corollary 2. Let T : (γ 1 , γ 2 ) ⊗ V λ,ρ → V λ ,ρ be a tensor operator for Spin (3,1), with V λ,ρ , V λ ,ρ infinitedimensional. If a Clebsch-Gordan decomposition for (γ 1 , γ 2 ) ⊗ V λ,ρ exists, the matrix elements of T satisfy where α keeps track on the multiplicity of V λ ,ρ in the decomposition and the proportionality factors N α do not depend on the particular vectors being evaluated. In particular, if V λ ,ρ does not appear in the Clebsch-Gordan decomposition, T must identically vanish.
Corollary 3. Let T : F A γ ⊗V λ,ρ → V λ ,ρ be a tensor operator for Spin (3,1), with V λ,ρ , V λ ,ρ infinite-dimensional. If a Clebsch-Gordan decomposition for F A γ ⊗ V λ,ρ exists, the matrix elements of T satisfy where the proportionality factor does not depend on the particular vectors being evaluated. In particular, if V λ ,ρ does not appear in the Clebsch-Gordan decomposition, T must identically vanish.
The operators a i , a † i , i = 1, 2 and 1 act on the algebraic direct sum γ∈N0/2 F γ as the (complexified) 5dimensional Heisenberg algebra h 2 (R) C , making it a unitary irreducible h 2 (R)-module.
A proof can be found in [3]. The a i , a † i operators are extensively studied in quantum mechanics, where they are known as quantum harmonic oscillators [1].
The extension of this result to finite-dimensional Spin(3, 1) representations trivially follows from the fact that spin(3, 1) C ∼ = su(2) C ⊕ su(2) C (see Section II). A generalisation to infinite-dimensional (g, K)-modules can be obtained by making use of tensor operators as follows.

IV. CONCLUDING REMARKS
Although the Wigner-Eckart theorem was generalised to arbitrary Lie groups, the actual information it provides still relies on the knowledge of the appropriate Clebsch-Gordan decomposition. The author hopes that the techniques developed in this paper for the Lorentz group can be used as a guide in the study of other groups. A possible topic for future research is the generalisation of the theorem to quantum groups, in particular to deformed enveloping algebras U q (sl(2, R)) and U q (spin(3, 1)); in this context, the tensor operators defined here admit a simple generalization to Hopf algebras by requiring them to intertwine Hopf algebra representations [12].
One topic that remains to be investigated is the nature of the Heisenberg algebra representations provided by the tensor operators considered for the Jordan-Schwinger representation; the author leaves it for further analysis.
The Wigner-Eckart theorem and the Jordan-Schwinger representation for the special case of SL(2, R) have found an application in quantum gravity [13]; it is hoped the generalisations provided here will have applications to physics as well.