Few-Body Systems manuscript No. (will be inserted by the editor)

W. N. Polyzou · W. Glöckle · H. Witała

Spin and dynamics in relativistic quantum theories

Received: date / Accepted: date

Abstract The role of relativity and dynamics in defining the spin and orbital angular momentum content of hadronic systems is discussed.

Keywords spin · orbital angular momentum · relativity

1 Introduction

There is a great deal of interest in the distribution of spin and orbital angular momentum in hadronic systems. In general the underlying dynamics of partons in hadrons is relativistic. The treatment of spin in relativistic systems is different than it is in non-relativistic systems. In a relativistic system the spin of a parton is identified with the angular momentum of the parton in its rest frame while the spin of the hadron is defined as the angular momentum of the hadron in its rest frame. Transforming a parton from its rest frame to the hadrons rest frame, where the spins can be coupled, involves boosts which generate dynamical rotations. These rotations transform the parton spins and also impact the relative orbital angular momentum before they can be coupled. The spin of the constituents, the internal orbital angular momentum and the spin of the system are related by Clebsch-Gordan coefficients of the Poincaré group[1][2][3] . The Poincaré group Clebsch-Gordan coefficients are labeled by eigenvalues of mass and spin Casimir operators, which are dynamical operators.

An additional complication is that boosts to the rest fame are not unique; a boost to the rest frame followed by a momentum dependent rotation is a different boost to the rest frame. There are as many different kinds of boosts as there are momentum dependent rotations. Each boost defines a different spin observable. For example, there are distinct boosts that are used to define the helicity, light front spin or canonical spin. These are three specific choices, that are distinguished by useful properties, out of an infinite number of possibilities. In many-body systems there is another relevant spin, which I call the constituent spin[3], which is distinguished by the property that spins and orbital angular momenta can be combined using ordinary SU(2) coupling methods to get the hadronic spin.

While all of the spins satisfy SU(2) commutation relations, the different spins observables are related by the momentum dependent (Melosh) rotations[4] that relate different boosts. Because different spin observables differ by momentum-dependent rotations, partial derivatives with respect to momentum that hold one spin observable constant will not commute with a different spin observable. This means not only are there an infinite number of possible spin observables, but each one is associated with a

Research supported by the US DOE Office of Science, under grant number No. DE-FG02-86ER40286

W. N. Polyzou The University of Iowa Tel.: +319-335-1856 Fax: +319-335-1753 E-mail: polyzou@uiowa.edu

different quantity that can be identified with an orbital angular momentum. As a result the spin and orbital angular momentum content of a hadron is dynamical and representation dependent. In what follows I discuss some of the relevant issues.

The dependence of the spin on the choice of boost is seen in the relations between the spin and angular momentum

j_{x}^{l} := \frac{1}{2}ϵ^{lmn}B_{x}^{−1}(p)^{m}_{μ}B_{x}^{−1}(p)^{n}_{ν}J^{μν}  (1)

where B_{x}^{−1}(p)^{m}_{μ} is a boost that maps p to its rest frame.

This definition can be equivalently expressed in terms of the polarization vectors that are three space-like vectors e_{x}^{m}_{μ}(p) that are orthogonal to the four momentum, that can be identified with the space-like rows of the boost B_{x}^{−1}(p)^{m}_{μ}:

j_{x}^{l} = \frac{1}{2}ϵ^{lmn}e_{x}^{m}_{μ}(p)e_{x}^{n}_{ν}(p)J^{μν}.  (2)

Spins constructed using different boosts are related by momentum-dependent Melosh rotations

j_{x}^{l} = R_{xy}(p)^{l}_{m}j_{y}^{m}  where  R_{xy}(p)^{l}_{m} = B_{x}^{−1}(p)^{l}_{μ}B_{y}(p)_{m}^{μ}.  (3)

Because the different types of spin observables differ by momentum-dependent rotations, “Position operators” [5][6][2]that involve partial derivatives with respect to momentum need to specify which kind of spins is being held constant during the differentiation,

[∇ ∇ ∇_{P_{|_{jx}}} , j_{x}] = 0 ⇒ [∇ ∇ ∇_{P_{|_{jx}}} , j_{y}] ≠ 0.  (4)

These partial derivatives can be written in terms of the Poincaré generators with V = P/M by [6][2]

X_{x}^{k} = i∇ ∇ ∇_{P_{|_{jx}}} = −\frac{1}{2}\{H^{−1}, K^{k}\} + iH^{−1}C_{1x}^{kl}(V)j_{x}^{l}.  (5)

In terms of these operators the spins and angular momentum are related by

J^{j} = (X_{x} × P)^{j} + C_{2x}^{jk}(V)j_{x}^{k}  (6)

where the operators C_{1x}^{jk}(V) and C_{2x}^{jk}(V) are the following functions of the Poincaré generators:

C_{1x}^{jk}(V) = \frac{1}{2}Tr[B_{x}(V)^{−1}σ_{j}B_{x}(V)σ_{k}] − V^{0}Tr[B_{x}(V)^{−1} \frac{∂}{∂V_{l}} B_{x}(V)σ_{m}]  (7)

C_{2x}^{jk}(V) = \frac{1}{2}Tr[B_{x}(V)^{−1}σ_{j}B_{x}(V)σ_{k}] + iϵ_{jlm}Tr[B_{x}(V)^{−1} \frac{∂}{∂V_{l}} B_{x}(V)σ_{m}]  (8)

and B_{x}(V) is the SL(2, ℂ) representation of the x-boost. The quantity X_{x} ×P is the associated orbital angular momentum [6][2]

Three components of the four momentum and the projection of any of these spin observables on a given axis are labels for vectors in irreducible subspaces. Products of two such irreducible representations can be expressed as direct integrals of composite irreducible representations using the Clebsch-Gordan coefficients for the Poincaré group. Like any set of Clebsch-Gordan coefficients, the actual coefficients depend on the choice of irreducible basis. The Poincaré group Clebsch-Gordan coefficients for a basis labeled by the x-type spin are

x⟨(M_{1}, j_{1})P_{1}, μ_{1}(M_{2}, j_{2})P_{2}, μ_{2}|k, j(M_{1}, j_{1}, M_{2}, j_{2})P, μ, l, s_{12}⟩_{x} =

∑_{μ′_{1},μ′_{2},μ′′_{1},μ′′_{2},μ_{s},m} δ(P − P_{1} − P_{2})\frac{δ(k − k(P_{1}, P_{2}))}{k^{2}}×

\sqrt{\frac{ω_{M_{1}}(k)ω_{M_{2}}(k)}{ω_{M_{1}}(P_{1})ω_{M_{2}}(P_{2})}}\sqrt{\frac{ω_{M_{1}}(P_{1}) + ω_{M_{2}}(P_{1})}{ω_{M_{1}}(k) + ω_{M_{2}}(k)}}×

D_{μ_{1}μ′_{1}}^{j_{1}}[R_{wx}(B_{x}(V), k_{1})]D_{μ′_{1}μ′′_{1}}^{j_{1}}[R_{xc}(k_{1})]×

D_{μ_{2}μ′_{2}}^{j_{2}}[R_{wx}(B_{x}(V), k_{2})]D_{μ′_{2}μ′′_{2}}^{j_{2}}[R_{xc}(k_{2})]×
Y_{m}^{l}(k̂(P_{1}, P_{2}))⟨j_{1}, μ′′_{1}, j_{2}, μ′′_{2}|s_{12}, μ_{s}⟩⟨l, m, s_{12}, μ_{s}|j, μ⟩  (9)

These involve two types of spin rotations. There are Wigner rotations R_{wx}(B_{x}(V), k_{i}) that arise from the x-boosts that relate the system and parton rest frames and generalized Melosh rotations, R_{xc}(k_{2}), that transform the resulting spins to the canonical spin representation where all of the spins and orbital angular momenta Wigner rotate together so they can be added using ordinary SU(2) spin addition.

The spins obtained by applying these two rotations to the hadronic spins are the constituent spins mentioned earlier. These are the spins associated with the magnetic quantum numbers μ′′_{i} in (9). It is apparent from this equation that when these spins are combined with the orbital angular momentum using spherical harmonics and SU(2) Clebsch-Gordan coefficients the result is the total spin.

The Poincaré group Clebsch-Gordan coefficients (9) simplify in special bases. If the spins are defined using the standard rotationless boosts there are no Melosh rotations, if the rotationless boost is replaced by a light-front boost there are no Wigner rotations, and if the rotationless boost is replaced by a helicity boost the Wigner rotations become multiplication by a phase.

One result of the momentum-dependence of the rotations is that the momentum dependence of the hadronic wave function affects the expectation values of both the spins and orbital angular momentum. Since the structure of the hadronic wave function is dynamical, the dynamics enters in spin coupling when the the Poincaré Clebsch-Gordan coefficients are integrated against the hadronic wave functions.

When one couples two interacting subsystems, one has to ask whether the masses in the Poincaré Clebsch-Gordan coefficients are the physical masses of the subsystems or the invariant masses of their constituents. For example, the mass of a meson or the invariant mass of a quark antiquark pair? So far we have treated them as invariant masses of the constituents. Cluster properties suggest that one should really use the physical mass operators of the subsystems. Fortunately there is a unitary transformation that removes the interaction dependence from the hadronic spin[7][8]. In this representation the spins can be coupled by sequential coupling using the Clebsch-Gordan coefficients of the Poincaré group as if the particles were not interacting. This unitary transformation changes the Hamiltonian, generating many-body interactions. It also changes the representation of the wave function in a way that preserves probabilities, expectation values, as well as scattering observables. In this case all of the dynamical spin effects can be absorbed by changing the representation of the wave function.

A second ambiguity with spin has to do with whether the dynamics formulated using Poincaré covariant or Lorentz covariant bases. The unitary representation of the Poincaré group on positivemass positive-energy irreducible basis states has the form

U(Λ, 0)|(M, j)P, μ⟩_{x} = ∑ |(M, j)Λ Λ ΛP, ν⟩_{x}D_{νμ}^{j}(R_{wx}(Λ, P)).  (10)

The Wigner rotation can be decomposed into the composition of a boost followed by a Lorentz transformation followed by an inverse boost with the transformed four momentum

R_{wx}(Λ, P) = B_{x}^{−1}(ΛP)ΛB_{x}(P).  (11)

The group representation property can be used to split the Wigner function apart. The finite dimensional representations of SU(2) are related to finite dimensional representation of SL(2, ℂ) by analytic continuation[9][3], so we can still use the group representation property. Absorbing the Wigner functions of the boosts into the states gives the Lorentz spinor representation of the states:

|(m, j)P, b⟩ := ∑_{μ} |(m, j)P, μ⟩_{x}D_{μb}^{j}[B_{x}^{−1}(P/M)].  (12)

Here the boosts are represented by 2 × 2 SL(2, ℂ) transformations. These spinor basis states (12) have the following Lorentz covariant transformation property

U(Λ, 0)|(m, j)P, b⟩ = ∑_{b′} |(m, j)Λ Λ ΛP, b′⟩D_{b′b}^{j}[Λ].  (13)

The price paid for using the covariant representation is that the Hilbert space inner product becomes dynamical

⟨ψ|ϕ⟩ =

∫ ⟨ψ|(m, j)P, b⟩d^{4}P θ(P^{0})δ(P^{2} + M^{2})D_{bb′}^{j}[P^{μ}σ_{μ}/M]⟨(m, j)P, b′|ϕ⟩  (14)

where we have used the hermiticity of the SL(2, ℂ) representation of the rotationless boost which gives

B_{x}(V)B_{x}^{†}(V) = B_{c}(V)R_{cx}(V)R_{cx}^{†}B_{c}^{†}(V) = B_{c}(V)B_{c}^{†}(V) = B_{c}^{2}(V) = P^{μ}σ_{μ}/M  (15)

independent of the type (x) of boost. Here the dynamics is contained in the mass-shell condition, which makes the Wigner function into a positive matrix. The inner product (14) identical the original Poincaré covariant inner product.

Unlike representations of SU(2), the representations of SL(2, C) are not equivalent to the complex conjugate representations. This means that we could alternatively replace (12) by

|(m, j)P, ḃ⟩ := ∑_{μ} |(m, j)P, μ⟩_{x}D_{μḃ}^{j}[B_{x}^{†}(P/M)].  (16)

and (13) by

U(Λ, 0)|(m, j)P, ḃ⟩ = ∑_{b′} |(m, j)Λ Λ ΛP, ḃ′⟩D_{ḃ′ḃ}^{j}[((Λ)^{†})^{−}1)].  (17)

This gives a representation of the scalar product that has the same form as (14) with the replacement

D_{bb′}^{j}[P^{μ}σ_{μ}/M] → D_{ḃḃ′}^{j}[P^{μ}σ_{2}σ_{μ}^{∗}σ_{2}/M].  (18)

While the inner products in all three representations are identical, the Lorentz covariant and its complex conjugate representations are related by space reflection. Space reflection changes the kernel of the Hilbert-space scalar product in the covariant representations. Space reflection can be represented as an operator on states by replacing the representation (12) and (16) by a direct sum of both representations. In the direct sum representation the wave function becomes a 2 × (2j + 1) component spinor

ψ(P, b) → ( \begin{matrix}ξ(P, b) \\ χ(P, ḃ) \end{matrix} )  (19)

and the kernel of the inner product becomes

d^{4}P θ(P^{0})δ(P^{2} + M^{2}) ( \begin{matrix}D_{bb′}^{j}[P^{μ}σ_{μ}/M] & 0 \\ 0 & D_{ḃḃ′}^{j}[P^{μ}σ_{2}σ_{μ}^{∗}σ_{2}/M] \end{matrix} ) .  (20)

.

One desirable feature of the covariant representation is that the basis-dependent features are hidden in the wave functions. To see this note that for rotations the upper and lower components have identical transformations laws

U(R, 0)|(M, j)P, b⟩ = ∑_{ḃ′} |(M, j)RP, b′⟩D_{b′b}^{j}[R].  (21)

and

U(R, 0)|(M, j)P, ḃ⟩ = ∑_{ḃ′} |(M, j)RP, ḃ′⟩D_{ḃ′ḃ}^{j}[R].  (22)

which is the standard rotational transformation law that leads to standard relation

J = X × P + j  (23)

in the covariant representation.

In this representation the relation between the spin, angular momentum, and orbital angular momentum add very much like the corresponding non-relativistic quantities. The price paid for this simplification is that the Hilbert space inner product has a non-trivial kernel. This kernel contains all of the dynamics effects discussed in the context of Poincare irreducible spins. The covariant spin is related to the Poincaré irreducible spin of a particle by a boost. For spin 1/2 particles the usual u and v spinors are direct sum representations of a Lorentz boost. The choice (helicity, canonical, light front spin) appear in the representation of these spinors.

In the end there are many different kinds of spin observables. In order to measure the spin we need to know how the various spin operators couple to the electroweak current operators. This will be different for each type of spin observable.

The conclusion is that for relativistic systems the coupling of spin is more complicated than it is for non-relativistic systems. It requires determining the relation between the irreducible representation of the Poincaré group associated with the hadron and the irreducible representations of the Poincaré group associated with the partons. In all cases the relation between the hadronic and parton spins depends on the momenta of the partons in the hadron, which is determined by the dynamics.

In addition, there are many different spin and orbital angular momentum observables. How there different quantities contribute to the hadronic spin is representation dependent.

References

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