Path Integral for Spin

Transcription

1 Path Integral for Spin Altland-Simons have a good discussion in 3.3 Applications of the Feynman Path Integral to the quantization of spin, which is defined by the commutation relations [Ŝj, Ŝk = iɛ jk lŝl, ɛ 123 = ɛ 12 3 = +1, Ŝ Ŝ = iŝ. These are the Lie algebra commutation relations of su(2) and the Lie group SU(2), the covering group of rotations in 3-dimensional space. Spin versus Angular Momentum Angular momentum in classical mechanics is a 3-dimensional vector valued function of coordinates and velocites in 6N-dimensional phase space L = N m j r j v j. j=1 Orbital angular momentum was quantized by Bohr and Sommerfeld using the old quantum theory. This semi-classical quantization actually gave the exact eigenvalue spectrum L = 0, 1, 2,, L z = L, L + 1,, +L. PHY /16/2013

2 In 1924, Pauli introduced Spin as a new internal degree of freedom to explain the doubling of spectral lines. In the new quantum mechanics of Schrödinger, Heisenberg, et al., a total angular momentum was defined as the vector sum Ĵ = ˆL + Ŝ, [ ˆLk, Ŝl = 0, Ĵ Ĵ = iĵ. A consistent quantum mechanics of electron spin was discovered in 1928 by Dirac, who showed that the spin operator S µν = i 4 [γµ, γ ν, with exactly the right properties postulated by Pauli emerged from combining quantum mechanics with relativity. Oscillator Path Integral A quantum oscillator path integral q f e iĥt/ q i = [ˆq, ˆp = i, Dq exp Ĥ(ˆq, ˆp) = ˆp2 2m + mω2ˆq 2 2 [ i t dt L(q(t ), q(t )) 0 is a sum over continuous paths q(t) in 1-dimensional spacetime M 1,0. =, Dq exp [ i S[q, PHY /16/2013

3 Spin Path Integral Let us attempt to construct a path integral for spin using the oscillator analogy. In addition to the spin commutation relations, a Hamiltonian is needed to generate classical trajectories. The simplest Hamiltonian is the Pauli coupling to an applied magnetic field: Ŝ Ŝ = i Ŝ, Ĥ(Ŝ) = B Ŝ. PHY /16/2013

4 Note that this is not a normal Hamiltonian system with an equal number of coordinate and conjugate momentum variables. There are only three spin variables and there are no obvious canonically conjugate pairs. A natural way to define a classical spin trajectory is to use the orientation of a rigid rotator and the Euler angles representation of the rotation group SU(2) = g(φ, θ, ψ) = e iφŝ3 e iθŝ2 e iψŝ3, 0 φ, ψ < 2π, 0 θ < π, where g is an element of the rotation group. The classical coordinates φ, θ, ψ of the spin variable define a point on the surface of the 2-sphere S 2 up to a gauge rotation by the angle ψ, which leaves the Hamiltonian invariant. Haar Measure and Sum Over Paths The Haar measure in group theory provides a completeness relation 1 = C dg g g where C is a constant, analogous to the complete sets of eigenstates of ˆq, ˆp for the quantum oscillator. The measure is uniquely defined by the invariance properties h SU(2) : dg f(gh) dg f(hg) = dg f(g), PHY /16/2013

5 which is invariant under left- and right-multiplication by any rotation. The propagator for a small time step t = t j+1 t j can be approximated g j+1 e i tb Ŝ g j g j+1 g j i t g j+1 B Ŝ g j + 1 g j g j [ exp g j+1 g j g j g j i t g j+1 B Ŝ g j Taking the limit of an infinite number of time slices gives the path integral representation of the propagator g j+1 e i tb Ŝ g N [ N 1 { j = lim dg n exp i t g } j+1 g j g j g j + g j+1 B N t Ŝ g j j=0 k=0 [ t = Dg exp i dt { t g g + g B S g } 0 [ t = Dg exp is dt { (1 cos θ(t ))i t φ(t ) + B cos θ(t )}. 0 Altland-Simons have a good discussion of the physical interpretation of this expression for the path integral. PHY /16/2013