APPENDIX E SPIN AND POLARIZATION Nothing shocks me. I m a scientist. Indiana Jones You ve never seen nothing like it, no never in your life. F. Mercury Spin is a fundamental intrinsic property of elementary particles which, despite the analogies with the mechanical phenomenon, exhibits its non-classical nature in several ways, sometimes puzzling for the uninitiated (see Tomonaga (1997) for a history with great insight). The relativistically covariant description of spin was developed in the early times of quantum field theory, but is not often presented in modern introductory particle physics textbooks; being very useful for the understanding of several experiments, in which one rarely works in the particle rest system, we briefly review it here. E.1 Polarization The polarization of a beam of massive particles is defined in their rest frame: a polarized beam has a net angular momentum in such a frame, which can only be due to a non-uniform and asymmetric population of the possible spin states for the particles. The polarization vector is defined as the expectation value of the spin operator S in units of the spin j itself: P = S /j (E.1) e.g. the x component of the polarization is the normalized expectation value of S x, the net angular momentum along the direction x in the particle rest frame. When dealing with non-pure states, i.e. a statistical ensemble, the polarization is the expectation value averaged over the ensemble. The modulus of the polarization vector P = P (usually called the degree of polarization, or simply the polarization) ranges between 0 (unpolarized state) and 1 (fully polarized state); intermediate values correspond to partial polarization.
POLARIZATION 507 Note that a spin 1/2 particle in a pure state is always completely polarized in some direction ( P =1): for example a particle in the pure state S, S z = 1/2, +1/2 has a polarization vector P = ẑ; the state (1/ 2)[ 1/2, +1/2 + 1/2, 1/2 ], with equal weights for the two spin states along ẑ, has no net polarization along such axis but its polarization vector is P = ˆx, again of modulus 1. On the contrary an ensemble of spin 1/2 particles, half of which are in the state 1/2, +1/2 and half in the state 1/2, 1/2 has zero polarization. In general, equal population of the 2j +1 spin states implies a spatially isotropic distribution, while an unequal population singles out a preferred direction in space and can give rise to anisotropies; if the (unequal) relative population of the substates is independent of the sign of S z the system is said to be aligned (determining a privileged axis but not a privileged direction, i.e. possessing a cylindrical symmetry around the spin quantization axis and also a reflection symmetry with respect to a plane orthogonal to such axis), resulting in no net magnetic moment, while if this is not the case the system is said to be polarized, a privileged direction being also defined (and reflection symmetry being lost). Particles with integer spin can have zero polarization also for pure states: a spin 1 particle in the state S, S z = 1, 0 is unpolarized, as is one in the state α[ 1, +1 + 1, 1 ] + β 1, 0 ; in the latter case if α>βthe state has alignment along the z axis. In the case of particles with zero rest mass no rest frame exists, and the polarization must be defined differently: we consider for the moment particles with non-zero mass and come back to massless ones in Section E.4. Note that for a fully polarized spin j state the direction of polarization is the one along which every spin measurement would yield the maximal value j, rather than the direction along which all the spins of an ensemble are pointing (which is actually not defined), as the magnitude of the spin vector is not j but j(j + 1) (the difference being related to the uncertainty principle). Also, the polarization is a property of a state or an ensemble of particles, rather than an observable: for a completely unpolarized beam of spin 1/2 particles individual spin measurements along any given direction would always yield either + /2 or /2, and the same is true even for a beam which is fully polarized in a direction orthogonal to that. If it is possible to determine (e.g. by repeated measurements on an ensemble of identical particles) the probabilities P(±; k) of measuring either a positive or a negative value for the spin component along direction k then the corresponding component of the polarization is P k =[P(+; k) P( ; k)]/[p(+; k) + P( ; k)]. The complete information on the polarization of the ensemble is then obtained by performing such measurement along three mutually orthogonal directions to determine P. For the description of mixed (non-pure) spin states the formalism of the density matrix is used (see e.g. Messiah (1961)): in this case the matrix has dimension (2j + 1) (2j + 1) and involves (2j + 1) 2 1 real parameters. As an example, in the case of spin 1/2 particles the 2 2 density matrix can be written in general with
508 APPENDIX E SPIN AND POLARIZATION the help of the Pauli matrices σ i as 170 ρ = 1 2 [1 + P σ ] (E.2) which has Tr(ρ) = 1, as can be verified using eqn (E.1) and S =Tr(Sρ). For scattering reactions in which one of the particles has non-zero spin many spin-dependent observables exist. The simplest is the analysing power A: in the scattering of a spin 0 beam particle from a spin 1/2 target particle this is defined as A(θ) = 1 N L N R (E.3) P T N L + N R where N L,R denote the number of events in which the particle is scattered at an angle θ to the left or to the right, and P T is the polarization of the target. Instead of using two spectrometers placed at symmetric angles with respect to the forward direction, A can be measured by using a single one at angle θ and comparing the number of events N ± with the target polarization in opposite directions (e.g. up and down): A(θ) = 1 P T N + N N + + N (E.4) which is much more cost effective; errors related to the geometry of the two spectrometers in the first approach are traded for errors related to the equality of the target polarization in the two opposite directions, and possible time differences (e.g. efficiency, normalization) between the two measurements. A difficulty arising in the analysis of polarized target data is due to the fact that only the protons in hydrogen atoms contained in the target material are polarized, while any remaining nuclei are not. More observables can be measured by using both polarized beams and targets. E.2 Covariant polarization In order to know the polarization in other frames of reference, the transformation properties of P must be known. According to Ehrenfest s theorem (see e.g. Messiah (1961)) the expectation value of a quantum-mechanical observable follows a classical equation of motion: in the case of spin S, for a particle with magnetic moment µ = gµ 0 S (µ 0 = Q/2m for a charged particle with charge Q, see eqn (4.91)) in 170 The set of four numbers (I, P), with I the intensity, are called Stokes parameters.
COVARIANT POLARIZATION 509 presence of a magnetic field B this gives (in the particle rest frame) d S = µ B = gµ 0 S B (E.5) dp = gµ 0 P B (E.6) see (4.112). Consider an ensemble of spin 1/2 particles described by a density matrix (E.2) in a region with a magnetic field B; the time evolution of the polarization vector, which in this case is P = σ, can be obtained from the time evolution of the density matrix i ρ/ t =[H(t), ρ(t)] with the Hamiltonian being H = µ B (see eqn (4.93)): i dp k = i d σ k which is just eqn (E.6) in vector form. = i t Tr(ρσ k) = i Tr ( ) ρ t σ k = Tr([H, ρ] σ k ) = Tr([σ k, H] ρ) = gµ 0 2 B j Tr([σ k, σ j ] ρ) = gµ 0 4 B [ j Tr([σk, σ j ]) + P l Tr([σ k, σ j ] σ l ) ] = gµ 0 iɛ kjl B j P l The polarization vector can be considered as the space component of a covariant polarization vector s µ (s 0, s), defined by s µ R = (0, P) in the particle rest frame (E.7) s 2 = P 2 (E.8) (one also needs to check that the quantity s µ as defined actually transforms as a four-vector, which indeed it does). Note that the Lorentz-invariant product of s µ with the four-velocity v µ = (γ, γ β) vanishes s µ v µ = s 0 v 0 s v = 0 (E.9) as can be verified by evaluating it in the rest frame; this allows one to compute the time derivative of the s 0 component in this frame: ds µ v µ = s µ dv ( µ ds 0 ) = (E.10) R where the last equality holds in the rest frame. The unique covariant generalization of the equations of motion for P (E.6) and s 0 (E.10) is (Bargmann et al., 1959) ds µ dτ = gµ [ 0 sν F νµ (s ν v ρ F νρ ) v µ] dv (s ρ ) ρ v µ (E.11) dτ
510 APPENDIX E SPIN AND POLARIZATION where τ t/γ denotes proper time and F µν is the electromagnetic field strength tensor. 171 Indeed since the above equation is manifestly covariant and reduces in the rest frame to the equations (E.6) and (E.10) it is the correct generalization of those. In a homogeneous field the equation of motion for a charged particle with charge Q and mass m is dv µ /dτ = (Q/m)F µρ v ρ, so that setting µ 0 = Q/2m eqn (E.11) reduces to ds µ dτ = Q [ gsν F νµ (g 2)(S ν v ρ F νρ ) v µ] (E.12) 2m For a neutral particle instead ds µ dτ = gµ [ 0 Sν F νµ (S ν v ρ F νρ )v µ] (E.13) The expression of s µ in any frame is now easily obtained: in the laboratory frame in which the particle moves with velocity βc one has ( s µ L = γ β P, P + β γ 2 ) γ + 1 β P (E.14) The three-vector s L in the laboratory frame does not have any immediate interpretation: both its magnitude and [ direction depend on β: ] s 2 L = P2 1 + β 2 γ 2 cos 2 θ R (E.15) where θ R is the angle between the polarization direction (in the rest frame) and the boost direction β; eqn (E.14) indicates that at relativistic velocities (β 1) s L becomes either parallel or anti-parallel to β, their relative angle being so that the helicity is cos θ L = γ 2 cos 2 θ R 1 + β 2 γ 2 cos 2 θ R (E.16) λ = γ P cos θ R (E.17) Note that the transverse component of the polarization is not affected by the Lorentz boost: s L = s R = P if cos θ R = 0. Taking as an example the π µν decay discussed in Chapter 2, in the rest frame of the pion the muon has a longitudinal polarization (due to the violation of parity symmetry); if the pion is moving in the laboratory the muon will also have a transverse polarization component there. 171 It was implicitly assumed that the particle has no electric moments nor magnetic moments higher than the dipole one (see Bargmann et al. (1959) and Section 4.4.3 for the case of an electric dipole moment).
TIME EVOLUTION 511 E.3 Time evolution The degree of polarization P is a Lorentz-invariant quantity (E.8) which is conserved in interactions with electromagnetic fields, 172 since using the equation of motion (E.11) and recalling eqn (E.9) ds 2 dτ = 2s ds µ µ dτ = gµ [ 0 sµ s ν F νµ (s ν v ρ F νρ ) s µ v µ] (s ρ dv ρ dτ ) s µ v µ = 0 What is most relevant is therefore the change in direction of the polarization vector, and in particular how the angle between the polarization (in the rest frame) and the direction of motion changes. Two unit vectors ˆl and ˆn are introduced in the laboratory frame: the first parallel to the direction of motion of the particle β = β ˆl, and the second orthogonal to it in the plane containing ˆl and the vector s L. The covariant polarization vector in the laboratory frame (E.14) can be written as s µ L = P ( L µ L cos θ R + N µ L sin θ ) R (E.18) having defined the two four-vectors L µ, N µ which in the laboratory frame have components L µ L =γ(β, ˆl) N µ L =(0, ˆn) Note that L 2 = N 2 = 1 and L µ N µ = L µ v µ = N µ v µ = 0; note also that the expression (E.18) is also valid for the covariant polarization vector in the rest frame, where L µ R =(0, ˆl) N µ R =(0, ˆn) and in particular the angle between s µ and L µ from (E.18) is just θ R, the angle between P and the direction of motion ˆl in the rest frame. Inserting (E.18) into the equation of motion (E.12) for the case of homogeneous fields and evaluating the derivatives of L µ and N µ explicitly one finally obtains dθ R = ( gµ 0 β Q mβ ) E ˆn + ( gµ 0 Q m ) ˆl B ˆn (E.19) in terms of the electric and magnetic fields E, B in the laboratory frame; for Q = 0 the above equation can be written as dθ R = Q [ (g 2) g/γ 2 ] E ˆn + (g 2) ˆl B ˆn (E.20) 2m β 172 In presence of inhomogeneous fields the trajectory of a particle depends on its polarization, so that beam particles might be deflected differently and experience different fields, thus affecting the overall degree of polarization (Good, 1962).
512 APPENDIX E SPIN AND POLARIZATION The above equations simplify in specific cases: an interesting one is that of a magnetic field orthogonal to the particle trajectory, E = 0 and B = B ˆn ˆl, when dθ R = ( gµ 0 Q ) m B = Q B (g 2) 2m (E.21) so that the angle between the polarization vector and the direction of motion varies at a constant rate if the particle has an anomalous magnetic moment (g = 2), independently of the particle velocity: this is actually what is exploited for the measurement of g 2. The component of the polarization along the direction of motion is called longitudinal, and the one orthogonal to it transverse. Longitudinal polarization cannot be produced by scattering (if parity symmetry holds) nor measured from scattering asymmetries; however longitudinal and transverse polarization can often be transformed into each other: if the Larmor frequency (precession of a magnetic dipole) and the cyclotron frequency (revolution of a charged particle) are different (as is the case for the proton) such a transformation can be performed by having the particle passing through a suitably oriented magnetic field, in which the momentum and the spin rotate at different rates (the momentum does not rotate at all for a neutral particle). For an electron such frequencies are almost the same so that the above technique is not usable, but one can rather rotate only the momentum (in an electric field, at low energies) or only the spin (in crossed electric and magnetic fields). E.4 Massless particles For particles with zero mass (m = 0) the construction of a covariant polarization vector presented above is not possible, since no rest frame is defined; indeed for γ the four-vector of eqn (E.14) diverges. In this case it is possible to consider instead the four-vector W µ = ms µ, which has a finite limit for m 0: W µ s R cos θ R (E, E ˆl) = λp µ where P µ is the four-momentum of the massless particle. The quantity λ is obviously Lorentz-invariant and can be called the degree of polarization for the particle in this case. The direction of polarization is always parallel (λ >0) or anti-parallel (λ <0) to the direction of motion; in this sense the properties of the massless spin 1 photon are closer to those of a massive spin 1/2 particle, in that there are only two possible polarization states. In this case the degree of polarization is the same in any reference frame, i.e. the helicity is always ±λ, and there is no need for an equation of motion for the polarization.
MASSLESS PARTICLES 513 The above fact can also be seen from eqn (E.16): as the particle speed increases the longitudinal component of s L also increases in modulus; for a massless particle, which has no rest frame and moves at the speed of light, the angle between s L and the direction of motion becomes either 0 or π. Thus the fact that a massless particle has only two possible polarization states is not due to some property of spin but is just a consequence of Lorentz transformations (Wigner, 1939). Actually, since for a massless particle the helicity is a Lorentz invariant quantity, only one polarization state would be required by special relativity (either λ>0orλ<0): the second one is actually necessary only because of parity symmetry, which implies that left-handed polarized light must exist if right-handed one does. For mixed states an expression similar to (E.2) can be used for photons, which also have two possible polarization states (e.g. linear horizontal and vertical, or circular right and left): ρ = 1 2 [1 + ξ σ ] (E.22) although in this case ξ is not the photon polarization vector but rather a vector in what is called Poincaré space : 173 if the two states defining the basis of the 2 2 space are chosen to be those of vertical and horizontal polarization respectively, then the components of ξ correspond to: ξ 1 =+1( 1): full vertical (horizontal) linear plane polarization; ξ 2 =+1( 1): full right (left) circular polarization (helicity or longitudinal polarization); ξ 3 =+1( 1): full 45 (135 ) linear plane polarization; while if ξ < 1 the photon beam is partially polarized. This description of photon polarization is invariant with respect to Lorentz transformations between reference systems with relative velocities along the direction of the photon momentum, so that there is no need to work in any special frame. More details can be found e.g. in Hagedorn (1963). 173 The four numbers (I, ξ), with I being the intensity, are the Stokes parameters for the photon.