Jordan-Schwinger Map in the Theory of Angular Momentum

Jordan-Schwinger map in the theory of angular momentum

V´aclav Zatloukal∗

Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Bˇrehov´a7, 115 19 Praha 1, Czech Republic

(Dated: February 21, 2020) The Jordan-Schwinger representation of the su(2) algebra utilizes ladder operators to eﬃciently handle su(2) representations with arbitrary spin. In these notes we point out the usefulness of this technique for calculating the Clebsch-Gordan coeﬃcients when two angular momenta are being composed.

I. JORDAN-SCHWINGER REPRESENTATION: GENERIC CASE

Let the n × n matrices A1,..., AN form a representation (typically fundamental) of a Lie algebra g:

k [Ai, Aj] = cijAk, (1)

k where cij are the structure constants of g, and summation over k = 1,...,N is implied. † † Consider n pairs of creation and annihilation operatorsa ˆ1,..., aˆn anda ˆ1,..., aˆn with usual bosonic commutation relations

† † † [âα, aˆβ] = δαβ , [âα, aˆβ] = 0 , [âα, aˆβ] = 0. (2) Then, the operators

ˆ † Ai =a ˆα(Ai)αβaˆβ, (3) where (Ai)αβ is the (α, β)-th entry of the matrix Ai, and summation over α, β = 1, . . . , n is implied, form again a representation of the Lie algebra g. Indeed,

ˆ ˆ † † [Ai, Aj] = (Ai)αβ(Aj)γδ [âαaˆβ, aˆγ aˆδ] † † † † = (Ai)αβ(Aj)γδ (âα[âβ, aˆγ ]âδ +a ˆγ [âα, aˆδ]âβ) † † = (Ai)αβ(Aj)βδ aˆαaˆδ − (Ai)αβ(Aj)γα aˆγ aˆβ † = (AiAj − AjAi)αδ aˆαaˆδ k † = cij(Ak)αβaˆαaˆβ k ˆ = cijAk. (4) ˆ The map Ai 7→ Ai from matrices to operators (on an abstract Hilbert space) is referred to as the Jordan-Schwinger map [1].

∗Electronic address: [email protected]; URL: http://www.zatlovac.eu 2

II. JORDAN-SCHWINGER REPRESENTATION: su(2)

In particular, we will be concerned with the angular momentum Lie algebra su(2), whose σi fundamental representation is spanned by the 2 × 2 matrices Ji = 2 , i = 1, 2, 3, which fulﬁl the commutation relations

[Ji, Jj] = i εijk Jk. (5)

Here σi denote the standard Pauli matrices 0 1 0 −i 1 0 σ = , σ = , σ = . (6) 1 1 0 2 i 0 3 0 −1

For the case of su(2), the Jordan-Schwinger map yields the following operators: 1 Jˆ = (â†aˆ +a ˆ†aˆ ), 1 2 2 1 1 2 i Jˆ = (â†aˆ − aˆ†aˆ ), 2 2 2 1 1 2 1 Jˆ = (â†aˆ − aˆ†aˆ ). (7) 3 2 1 1 2 2

From now on we shall omit the ‘hat’, writing simply Ji instead of Jˆi, and aα instead ofa ˆα. The angular momentum ladder operators J± = J1 ± iJ2 assume a particularly simple form

† † J+ = a1a2 ,J− = a1a2. (8)

~2 2 2 2 The angular momentum squared, J = J1 + J2 + J3 , reads 1 J~2 = J 2 + (J J + J J ) 3 2 + − − + 1 1 = (a†a − a†a )2 + (a†a a a† + a†a a a†) 4 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 = (N 2 − 2N N + N 2) + (N N − N + N N − N ) 4 1 1 2 2 2 1 2 1 1 2 2 N N = + 1 , (9) 2 2 where, in passing, we have denoted by N1, N2, and N the number operators

† † N1 = a1a1 ,N2 = a2a2.,N = N1 + N2. (10)

1 (Note that now J3 = 2 (N1 − N2).) Normalized states with occupation numbers n1, n2 (i.e., the simultaneous eigenstates of operators N1, N2) read

† n1 † n2 (a1) (a2) |n1, n2i = √ √ |0i , (11) n1! n2! where |0i is the abstract vacuum state, and n1, n2 = 0, 1, 2,.... These are also eigenstates |j, mi 2 of J3 and J~ :

† 1 Therefore, adding a ‘quantum’ with a1 increases the spin j and the spin projection m by 2 , † 1 1 whereas adding a ‘quantum’ with a2 increases j by 2 , but decreases m by 2 (see Fig.). Note that the operators Ji of Eq. (7) preserve the total occupation number n1 + n2, and hence † † also the value of spin j. The Fock space generated by a1, a2 then decomposes into subspaces, 1 1 labelled by j = 2 (n1 + n2) = 0, 2 , 1,..., which are invariant under the Ji (and, of course, under 2 the derived operators J± and J~ ). Moreover, within the spin-j subspace, m = −j, −j − 1, . . . , j, as follows from the inequalities j + m = n1 ≥ 0 and j − m = n2 ≥ 0. Let us remark that one can realize the abstract Fock space as a space of functions in two complex variables f(z1, z2), and the abstract creation and annihilation operators as multiplicative and diﬀerential operators

† † ∂ ∂ a1 ' z1 , a2 ' z2 , a1 ' , a2 ' . (13) ∂z1 ∂z2 The vacuum state |0i is identiﬁed with 1, and the scalar product can be deﬁned via a two-fold integral over the complex plane

1 Z 2 2 ∗ −|z1| −|z2| ∗ ∗ hf|gi = 2 f (z1, z2) g(z1, z2) e dz1dz1 dz2dz2 . (14) π 2 C The states |j, mi are then realized by the polynomials

zj+m zj−m |j, mi ' 1 2 . (15) p(j + m)! p(j − m)!

A. Spin coherent states

Let us deﬁne a spin coherent state by the formula

µJ |j, µi = e − |j, m = ji , µ ∈ C. (16)

† 2j µa†a (a1) |j, µi = e 2 1 |0i p(2j)! † 2j µa†a (a1) −µa†a = e 2 1 e 2 1 |0i p(2j)!

1 µa†a † −µa†a 2j = (e 2 1 a e 2 1 ) |0i p(2j)! 1 (a† + µa†)2j = 1 2 |0i . (17) p(2j)! To prove the last equality, observe that

d µa†a † −µa†a † µa†a † −µa†a † (e 2 1 a e 2 1 ) = a e 2 1 [a , a ]e 2 1 = a . (18) dµ 1 2 1 1 2 In passing we note that Eq. (17) implies

(J )` 1 2j 2j1/2 − |j, ji = (a†)2j−`(a†)` |0i = |j, j − `i . (19) `! p(2j)! ` 1 2 ` 4

III. COMPOSITION OF TWO ANGULAR MOMENTA

The Jordan-Schwinger representation for a system of two independent angular momenta (labelled a and b) utilizes four pairs of creation and annihilation operators, and identiﬁes 1 1 J a = (σ ) a† a ,J b = (σ ) b† b ,J tot = J a + J b. (20) i 2 i αβ α β i 2 i αβ α β i i i

a b (Explicit expressions are analogous to those of Eq. (7).) Since [Ji ,Jj ] = 0 for all i, j = 1, 2, 3, the tot composed angular momentum operators Ji satisfy the su(2) commutation relations, Eq. (5). Our task is now to build out of the tensor product states |ja, mai |jb, mbi = |ja, mai ⊗ |jb, mbi ~a 2 a ~b 2 b (i.e., eigenstates of the operators (J ) ,J3 , (J ) ,J3 ) linear combinations that are eigenstates of ~a 2 ~b 2 ~tot 2 tot operators (J ) , (J ) , (J ) ,J3 . We shall denote the latter states by |ja, jb, jtot, mtoti, and look for their expansion in terms of |ja, mai |jb, mbi. The coeﬃcients in this expansion are the Clebsch-Gordan coeﬃcients. First, we realize that

† 2ja † 2jb (a1) (b1) |ja, jb, ja + jb, ja + jbi = |ja, jai |jb, jbi = p p |0i (21) (2ja)! (2jb)! are common eigenstates for both sets of operators. From these we will generate all the other tot eigenstates |ja, jb, jtot, mtoti using the ladder operator J− , which lowers the eigenvalue mtot, and the operator [2]

† † † † † S = a2b1 − a1b2, (22) which fulﬁls the following commutation relations:

a,b † a,b tot † tot † [N ,S ] = N , [J3 ,S ] = 0 , [J± ,S ] = 0. (23) Moreover, by the last two relations, and the ﬁrst line in Eq. (9),

[(J~tot)2,S†] = 0. (24)

† 1 That is, the operator S raises (simultaneously) the eigenvalues ja and jb by 2 , while preserving jtot and mtot:

† 1 1 S |ja, jb, jtot, mtoti = α ja + 2 , jb + 2 , jtot, mtot . (25) To determine the factors α, we realize that the operator SS† can be cast as

N a + N b N a + N b SS† = + 1 + 2 − (J~tot)2. (26) 2 2 Hence, choosing α real and positive, we ﬁnd p α(ja, jb, jtot) = (ja + jb + 1)(ja + jb + 2) − jtot(jtot + 1) p = (ja + jb + 1 − jtot)(ja + jb + 2 + jtot), (27) and after repeated application of S† we obtain

† k 1/2 (S ) 2(ja + jb) + k + 1 k k |ja, jb, ja + jb, mtoti = ja + , jb + , ja + jb, mtot . (28) k! k 2 2 5

0 0 Now, Eqs. (28) and (19) give (writing for the moment ja, jb instead of ja, jb) (J tot)` (S†)k − |j0 , j0 , j0 + j0 , j0 + j0 i = `! k! a b a b a b 0 0 1/2 0 0 1/2 2(ja + jb) + k + 1 2(ja + jb) 0 k 0 k 0 0 0 0 = j + , j + , j + j , j + j − ` . (29) k ` a 2 b 2 a b a b At the same time, the left-hand side is equal to † k ` (S ) 1 d a b µ(J−+J−) 0 0 0 0 ` e |ja, jai |jb, jbi = k! `! dµ µ=0 † † † † † † 0 † † 0 k ` 2ja 2jb (a2b1 − a1b2) d (a1 + µa2) (b1 + µb2) = |0i , (30) k! `! dµ` p 0 p 0 µ=0 (2ja)! (2jb)! where we have made use of Eq. (17). In order to ﬁnd the expansion of a state |ja, jb, jtot, mtoti in terms of |ja, mai |jb, mbi we set j − j + j −j + j + j k = j + j − j , ` = j − m , j0 = a b tot , j0 = a b tot , (31) a b tot tot tot a 2 b 2 and equate the right-hand sides of Eqs. (29) and (30): 1/2 1/2 ja + jb + jtot + 1 2jtot |ja, jb, jtot, mtoti = ja + jb − jtot jtot − mtot † † † † † † † † (a b − a b )ja+jb−jtot djtot−mtot (a + µa )ja−jb+jtot (b + µb )−ja+jb+jtot = 2 1 1 2 1 2 1 2 |0i . jtot−mtot p p (ja + jb − jtot)!(jtot − mtot)! dµ µ=0 (ja − jb + jtot)! (−ja + jb + jtot)! (32) The right-hand side is a sum of terms of the form † a † a † b † b n1 n2 n1 n2 (a1) (a2) (b1) (b2) 1 a a 1 a a |0i = ja = (n + n ), ma = (n − n ) . (33) p a p a p b p b 2 1 2 2 1 2 n1! n2! n1! n2! A general explicit expression is relatively complicated so we merely illustrate the calculations with a simple example.

1 A. Example: ja = jb = 2

1 In the case ja = jb = 2 and jtot = 0, mtot = 0, and Eq. (32) gives: √ 1 1 † † † † 1 1 1 1 1 1 1 1 2 2 , 2 , 0, 0 = (a2b1 − a1b2) |0i = 2 , − 2 2 , 2 − 2 , 2 2 , − 2 . (34) 1 In the case ja = jb = 2 , jtot = 1, Eq. (32) simpliﬁes as follows:

1/2 1−mtot 2 1 1 1 d † † † † , , 1, mtot = (a + µa )(b + µb ) |0i . (35) 2 2 1−mtot 1 2 1 2 1 − mtot (1 − mtot)! dµ µ=0

This yields for mtot = −1, 0, 1 1 1 † † 1 1 1 1 , , 1, −1 = a b |0i = , − , − , √ 2 2 2 2 2 2 2 2 1 1 † † † † 1 1 1 1 1 1 1 1 2 2 , 2 , 1, 0 = (a2b1 + a1b2) |0i = 2 , − 2 2 , 2 + 2 , 2 2 , − 2 , 1 1 † † 1 1 1 1 2 , 2 , 1, 1 = a1b1 |0i = 2 , 2 2 , 2 . (36) 6

APPENDIX A: REPRESENTATIONS ON VECTORS AND ON OPERATORS

For operators deﬁned in Eq. (3) we now show that, for any N-tuple of parameters θi ∈ C,

ˆ ˆ † θiAi θj Aj † −θkAk aˆα (e )αβ = e aˆβ e , (A1) which can be further multiplied by a vector (vβ) from the representation space to ﬁnd a double- sided action on the corresponding operator vβaˆβ (the right-hand side). To this end, deﬁne operator-valued functions

ˆ ˆ ˆ τθj Aj † −τθkAk Oβ(τ) = e aˆβ e , (A2)

ˆ † † and calculate, with a help of [Ai, aˆβ] =a ˆα(Ai)αβ,

d ˆ ˆ Oˆ (τ) = eτθj Aj [θ Aˆ , aˆ† ] e−τθkAk dτ β i i β

τθj Aˆj † −τθkAˆk = θi e aˆα(Ai)αβ e ˆ = Oα(τ)(θiAi)αβ. (A3) ˆ † Integration of this diﬀerential equation, observing the initial condition Oβ(0) =a ˆβ, yields

ˆ † τθiAi Oβ(τ) =a ˆα(e )αβ, (A4) therefore proving relation (A1) upon setting τ = 1.

APPENDIX B: FERMIONIC OPERATORS

† Instead of the bosonic operators ai, ai , let us consider n pairs of fermionic operators f1, . . . , fn † † and f1 , . . . , fn with (canonical) anticommutation relations ˆ ˆ† ˆ ˆ ˆ† ˆ† {fα, fβ} = δαβ , {fα, fβ} = 0 , {fα, fβ} = 0, (B1) and deﬁne, for we every Ai, an operator ˆ ˆ† ˆ Fi = fα(Ai)αβfβ. (B2) Due to the identity [AB, CD] = A{B,C}D − AC{B,D} + {A, C}DB − C{A, D}B, (B3) which holds for arbitrary operators A, B, C, D, the operators Fˆi form again a representation of the Lie algebra g: ˆ ˆ k ˆ [Fi, Fj] = cijFk. (B4) Moreover, since [AB, C] = A{B,C} − {A, C}B, (B5) we have an analogue of Eq. (A1), namely,

ˆ ˆ ˆ† θiAi θj Fj ˆ† −θkFk fα (e )αβ = e fβ e . (B6) 7

[1] J. Schwinger, On Angular Momentum, Unpublished Report, Number NYO-3071 (1952). [2] J. Bellissard and R. Holtz, Composition of coherent spin states, J Math Phys 15, 1275 (1974).