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Geometric phases play an important role in the asymptotic behavior of multicomponent wave fields, such as electromagnetic waves in plasmas or quantum-mechanical spinors, particularly in the problem of Bohr-Sommerfeld quantization. The proper gauge independence of the eigenvalues and asymptotic wave functions can be understood in terms of gauge-invariant but noncanonical coordinates on the classical phase space.PACS numbers: 03.40. Kf, 02.40.+m, 03.65.Sq, 42.20.Cc Multicomponent wave fields and their asymptotic (short wavelength) behavior are important in physics. They include optical waves, electromagnetic waves in plasmas, elastic waves in solids, and various quantummechanical waves, including nuclear wave functions in the Born-Oppenheimer approximation. Aspects of the asymptotic analysis of such waves have been treated by a number of authors, 1 " 5 but essential elements, including the role of geometric phases in the Bohr-Sommerfeld quantization conditions and in the construction of wave functions, have received insufficient attention. It seems, in fact, that a general, geometrically clear, and manifestly gauge-invariant statement of the Bohr-Sommerfeld quantization conditions for such waves has never been made. For reference and to establish notation, we begin by summarizing the asymptotic behavior of scalar wave equations.Consider a scalar wave \f/(x), with x = (x\, . . . ,x"), satisfying a wave equation Dy/^O, where D is a linear operator. D can be regarded as a function of x (= multiplication by x), k = -ied/dx, where e is the WKB ordering parameter. In quantum applications, e is to be identified with h, and k with the momentum p. The classical counterparts, or "symbols," of the operators x, k, and D are the functions x, k, and D(x,k), where the one-to-one correspondence between operators and functions on the classical (x y k) phase space is given by the Weyl correspondence. 6 For example, in the usual Schrodinger equation, we have Z)(x,/?)=/? 2 /2m + K(x) -E. The WKB approximation on y/ proceeds by setting \f/(x) = =A(x)e iS(x)/€ and substituting this into Dy/=0. Expanding in 6, one finds the Hamiltonian-Jacobi equation D(x,k) =0 for S, and the amplitude transport equa-for A, where in both equations D is evaluated at k =dS/dx.It is common to work only to the lowest two orders in e, as we do here.As for multicomponent wave fields, let the wave function be y/ a (x), and let it satisfy the wave equation D a pWp =0. Here a,p are "spinor" indices, indexing the components of the wave field, and Dap is a matrix of "orbital" operators, i.e., functions of x,k. This terminology is useful even for classical waves, such as electromagnetic waves in plasmas. We assume that D a p is Hermitian, (D a pV =Dp a .The multicomponent WKB ansatz is y/ a (x) s=s A a (x)e lS{x)/e , where the generally complex amplitude A a (x) is now a spinor. The derivation of the WKB equations for S and A a is more difficult than in the scalar case, but approaches have been worked out by several authors. 1 " 3The results are the following. Le...