We found some errors in the treatment of the surface term in our original paper, and correct them here both in our Lagrangian and Hamiltonian formalisms. 1. Hamiltonian formalism In Sect. 2.1.1, 'Self-conjugacy of the Hamiltonian', there is a possibility that a boundary Hamiltonian may modify the boundary condition. However, if we add a boundary Hamiltonian Hboundary = Nd(x3), (1) where N is a 2-by-2 Hermitian matrix that may include functions of p1 and p2 but not p3, the remaining part of the subsection and the result remain unchanged. 2. Lagrangian formalism In Sect. 2.2, 'Lagrangian formulation', we made an argument based on the action principle. However, the discussion after Eq. (29). [Equation presented] -iϵ's3 + ϵ'N [Equation presented] δ = 0, δ' [is3ϵ + Nϵ] = 0, (2) is not correct. The correct argument should be as follows: 'The two equations of Eq. (2) are Hermitian conjugates of each other. Since half of the modes on the boundary need to be killed by the boundary condition, we can write a generic boundary condition as ( α β ) [Equation presented] x3=0 = 0, (3) where α and β are constants. So [Equation presented] x3=0 should have the form [Equation presented] x3=0 = [Equation presented] β -α [Equation presented], (4) and ϵ is a Grassmann number. In the same manner, δ [Equation presented] x3=0 = [Equation presented] β -α [Equation presented] δ. Parametrizing N = [Equation presented] n0 + n3 n1 - in2 n1 - in2 n0 - n3 [Equation presented] , (6) and substituting it with Eqs. (4) and (5) into the second equation of Eq. (2), we have [Equation presented] β ∗ -α ∗ [Equation presented] n0 + n3 +i n1 - in2 n1 - in2 n0 - n3 - i [Equation presented] β -α [Equation presented] ϵ 'δ = 0. (7) For arbitrary ϵ and δ, this is equivalent to |a|2(n0 - n3 - i) - β ∗ α(n1 - in2) - a ∗ β(n1 + in2) +|β|2(n0 + n3 + i) = 0. (8) Now writing α and β explicitly as α = ϵ1e-iθ1 β = θ2eiθ2 , Eq. (8) becomes (θ2 1 + θ2 2 )n0 + (θ2 1 - θ2 2 )(n3 + i) + 2θ1θ2(-n1 cos(θ1 + θ2) + n2 sin(θ1 + θ2)) = 0. (9) The generic solution of Eq. (9) is given by θ1 = θ2 n0 - n1 cos(θ1 + θ2) + n2 sin(θ1 + θ2) = 0 (10) These two equations tell us that the magnitudes of the two components should be the same and the phases of the two components are different by e-i(θ1+θ2) as a function of n0, n1, and n2. In addition, we can write Eq. (3) as [Equation presented] 1 ei(θ1+θ2) [Equation presented] ϵ [Equation presented] x3=0 = 0. (11) Let us determine the phase as a function of the boundary Lagrangian. From the second equation of Eq. (10), we obtain cos(θ1 + θ2) = n0n1 n21 + n22 ± n2 n21 + n22 n21 + n22 - n20 (12) and sin(θ1 + θ2) = - n0n2 n21 + n22 ± n1 n21 + n22 n21 + n22 - n20 , (13) if n21 + n22 - n20 = 0. (14) This means that the phase difference between the two components is ei(θ1+θ2) = cos(θ1 + θ2) + i sin(θ1 + θ2) = n0 ± i n21 + n22 - n20 n1 + in2 . (15) As a result the allowed boundary condition is generally written as [Equation presented] 1 n0±i n21 +n22 -n20 n1+in2 [Equation presented] ϵ [Equation presented] x3=0 = 0. (16) We conclude that, for the surface term given by Eq. (6) with the condition (14) satisfied, the on-shell boundary condition is restricted to the form (16). Note that in any case the on-shell Lagrangian formalism is found to be consistent with the result of the Hamiltonian formalism.' The authors would like to apologize for these errors and thank S. Yamaguchi and H. Fukaya for pointing out the errors and giving valuable suggestions during the discussions.
ASJC Scopus subject areas
- Physics and Astronomy(all)