Dirac Particles in a Gravitational Field
aa r X i v : . [ h e p - t h ] S e p Dirac Particles in a Gravitational Field
Pierre Gosselin and Herv´e Mohrbach Institut Fourier, UMR 5582 CNRS-UJF, UFR de Math´ematiques,Universit´e Grenoble I, BP74, 38402 Saint Martin d’H`eres, Cedex, France and Groupe BioPhysStat, ICPMB-FR CNRS 2843, Universit´e Paul Verlaine-Metz, 57078 Metz Cedex 3, France
The semiclassical approximation for the Hamiltonian of Dirac particles interacting with an ar-bitrary gravitational field is investigated. The time dependence of the metrics leads to new con-tributions to the in-band energy operator in comparison to previous works on the static case. Inparticular we find a new coupling term between the linear momentum and the spin, as well ascouplings which contribute to the breaking of the particle - antiparticle symmetry.
PACS numbers:
INTRODUCTION
In this paper we consider the theory of Dirac fermions in an arbitrary curved space-time in the Hamiltonianformulation. To reveal the physical content of the theory it is necessary to perform the diagonalization of theHamiltonian uncoupling the positive and the negative energy states. For a fermion interacting with an electromagneticfield the Foldy-Wouthuysen (FW) transformation based on an approximate scheme valid in the non relativistic limitis often used [1]. This same method was also applied in all the previous studies of Dirac fermions in a gravitationalfield [2]. Here instead, we will consider the fully relativistic regime but in a semiclassical approximation for whichthe de Broglie wave length of the fermion must be much smaller that the characteristic size of the inhomogeneities ofthe external field. A recent semiclassical FW-like transformation used for Dirac particle in a strong electromagneticfield could be adapted to the gravitational problem [3]. But instead we will use another method developed by theauthors which essentially differs from the FW. This method allows us to find the diagonal representation of any kindof matrix valued quantum Hamiltonian as a series expansion in the Planck constant. Here we will directly apply thegeneral formula obtained at the semiclassical (first order in the Planck constant) limit to the case of Dirac fermionsin an arbitrary curved spacetime. This is an extension of previous papers where massless and massive particles in astatic gravitational fields were treated. The extension to time dependent metrics turns out to be non-trivial and leadsto new coupling terms in the in-band energy operator which break the particle-antiparticle symmetry.
ELECTRON IN A GRAVITATIONAL FIELD
A one half spinning particle of mass m coupled to an arbitrary gravitational field is described by 4 − spinor field ψ satisfying the covariant Dirac equation ( i ~ γ α D α − m ) ψ = 0 (1)where we use c = 1, but keep explicit the Planck constant ~ and α = 0 , , , . The covariant spinor derivative is defined as D α = h iα D i with D i = ∂ i − (cid:2) γ α , γ β (cid:3) Γ αβi . The matrices γ α are theusual Dirac matrices, h iα are the the orthonormal vierbein and Γ αβi the spin connection components.Rewriting Eq. (1) under the Schr¨odinger form i ~ ∂ψ∂t = Hψ we obtain the following Hamiltonian H = g h β γ β (cid:0) γ α ¯ P α + m (cid:1) + ~ ε ̺βγ Γ ̺β Σ γ + i ~ β α β (2)where we introduced the notation for the pseudo-momentum ¯ P α = h iα ( P i + ~ ε ̺βγ Γ ̺βi Σ γ ) with the spin matrix Σ γ ,satisfying the relation ε γαβ Σ γ = i ( γ α γ β − γ β γ α ) . We use the conventions of Bjorken and Drell [4] for the Diracmatrices γ α , and α β . Surprisingly, Eq. (2) turns out to be non-hermitian for a time dependent metric. We then follow the approach ofLeclerc [5] who showed that one must add the term i ∂ t ln (cid:0) − gg (cid:1) to make it Hermitian. It will be shown later on,that the presence of this term is also necessary for the diagonalization procedure to work.Therefore the Hamiltonian considered in the following will beˆ H = g h β γ β (cid:0) γ α ¯ P α + m (cid:1) + ~ ε ̺βγ Γ ̺β Σ γ + i ~ β α β + i ∂ t ln (cid:0) − gg (cid:1) (3)The goal of this paper is therefore to diagonalise Eq. (3) to first order in ~ . But before embarking into this, we needfirst to discuss the definitions and properties of the scalar product in a curved space-time. Scalar product.
As said before the Hamiltonian ˆ H is Hermitian. However, the notion of hermiticity is here defined with respect toa scalar product in curved space [5], namely : h ψ ( t ) | ψ ( t ) i U ( t ) = Z ψ +1 √− gh β γ β γ ψ d x = Z ψ +1 U ψ d x (4)where we introduced the notation U = √− gh µ γ µ γ and ψ , ψ two spinors. Therefore an operator O is hermitianfor h|i U ( t ) defined in Eq. (4), if Z ψ +1 √− gh β γ β γ ( Oψ ) = Z ( Oψ ) + √− gh β γ β γ ( ψ ) (5)It means that matricially O + U = U O (6)where ”+” denotes from now, the usual Hermitic conjugation (transposition and complex conjugation), that is, thehermitic conjugate with respect to the scalar product in flat space denoted h|i and defined by h ψ ( t ) | ψ ( t ) i = Z ψ +1 ψ d x (7)Unfortunatly the definition Eq. (5) turns out to be untractable for pratical computations. Actually, for the sake of thediagonalization procedure, we aim at working with matrices which are Hermitian with respect to the usual transposeand complex conjugate operation Eq. (7), so that the diagonalization can be performed through a unitary matrix inthe usual sense. To do so, notice that if O is Hermitian for Eq. (4), then Eq. (6) implies that U OU − is Hermitianfor Eq. (7). Thus, starting with the Hamiltonian ˆ H defined in Eq. (3), U ˆ HU − is Hermitian in the usual sense andcan be diagonalized through a standard unitary matrix (that is unitary for (7)).The Hamiltonian of interest for us will thus be U ˆ HU − . It’s hermiticity in the usual sense allows us to write: U ˆ HU − = 12 U ˆ HU − + 12 (cid:16) U ˆ HU − (cid:17) + = 12 (cid:16) ˆ H + ˆ H + (cid:17) + 12 h U , ˆ H i U − + 12 U − h ˆ H, U i (8)The non unitarity of the transformation U is not problematic here, since it is precisely used to change the metric fromcurved to flat scalar product, and moreover ˆ H and U ˆ HU − have the same spectrum. In the case of a static metric(time independent) and satisfying h µ = f ( R ) δ µ , for a certain position dependent function f ( R ), the transformation U reduces to the multiplication by a function of R . Then, using Eq. (3) for ˆ H , Eq. (8) simplifies easily to: U ˆ HU − = 12 U ˆ HU − + 12 (cid:16) U ˆ HU − (cid:17) + = 12 (cid:16) ˆ H + ˆ H + (cid:17) It is this form that was considered in [6][7]. If in addition the metrics is diagonal, one recovers the transformationstudied in [2].Independently of the practical advantages of the flat scalar product, there is an other and deeper reason to transformthe Hamiltonian to the flat space. Actually, when diagonalizing the Hamiltonian with respect to Eq. (7) the diagonalsubspaces of up and down spinors will appear to be obviously orthogonal. This is of course not the case for the scalarproduct Eq. (4) which mixes both subspaces. As a consequence diagonalizing with respect to the curved space scalarproduct does not lead to a clear separation between particles and antiparticles.
Unitarity
We will end up this section by stressing the fact that for a non-static metric the Hamiltonian and the time evolutionoperator do not coincide in the flat representation, and in addition, the time evolution operator cannot be madeHermitian. To make this point clearer, consider the Schr¨odinger equation i ~ ∂∂t Ψ = H ΨApplying the transformation U yields, ∂∂t Ψ ′ = (cid:18) U ( t ) HU − ( t ) − i ~ U ( t ) ∂∂t U − ( t ) (cid:19) Ψ ′ with Ψ ′ = U Ψ. As a consequence the evolution for Ψ ′ , is given by the operator H e = U ( t ) HU − ( t ) − i ~ U ( t ) ∂∂t U − ( t ) (9)One would like H e to be hermitian for the scalar product Eq. (7). However, due to the non unitarity of U , thecontribution i ~ U ( t ) ∂∂t U − ( t ) is not. The reason for this non unitarity tracks back to the dependence in t of thescalar product Eq. (4), so that the norm of a wave function is not preserved in time. Actually starting with an initialcondition Ψ ′ ( t ), the solution for the Schroedinger equation isΨ ′ ( t ) = T exp (cid:18)Z t t H e ( t ) dt (cid:19) Ψ ′ ( t ) = U ( t ) T exp (cid:18)Z t t H ( t ) dt (cid:19) U − ( t ) Ψ ′ ( t )where T is the time ordered exponential. Assuming the norm Ψ ′ ( t ) to be equal to 1, Ψ ′ ( t ) is easily seen to have anorm (respectively to Eq. (4)) different from one. That can be checked easily on an infinitesimal timeslice, t = t +∆ t .Indeed h Ψ ′ ( t ) | | Ψ ′ ( t ) i = D U ( t ) exp ( − iH ( t ) ∆ t ) Ψ ( t ) | U ( t ) exp ( iH ( t ) ∆ t ) Ψ ( t ) E = h exp ( − iH ( t ) ∆ t ) Ψ ( t ) | U ( t ) exp ( iH ( t ) ∆ t ) Ψ ( t ) i = h exp ( − iH ( t ) ∆ t ) Ψ ( t ) | exp ( iH ( t ) ∆ t ) Ψ ( t ) i U ( t ) (10)and this is different from 1 since, h exp ( − iH ( t ) ∆ t ) Ψ ( t ) | exp ( iH ( t ) ∆ t ) Ψ ( t ) i U ( t ) = h exp ( − iH ( t ) ∆ t ) Ψ ( t ) | U ( t ) exp ( iH ( t ) ∆ t ) Ψ ( t ) i + (cid:28) Ψ ( t ) | ∂∂t U ( t ) ∆ t Ψ ( t ) (cid:29) (11)= h exp ( − iH ( t ) ∆ t ) Ψ ( t ) | exp ( iH ( t ) ∆ t ) Ψ ( t ) i U ( t ) + (cid:28) Ψ ( t ) | ∂∂t U ( t ) ∆ t Ψ ( t ) (cid:29) (12)the first term is equal to 1, actually Ψ ( t ) is of norm 1 for h|i U ( t ) and exp ( iH ( t ) ∆ t ) is unitary for this scalar product.As a consequence, h Ψ ′ ( t ) | | Ψ ′ ( t ) i differs from one and H e is non unitary. The reason is clear from Eq. (10): avector of norm 1 for h|i U ( t ) is transported to a vector, that has no more norm 1 for h|i U ( t ) . During the evolution, thematrix U defining the scalar product has changed too, and the non hermitian connexion term − i ~ U ( t ) ∂∂t U − ( t )tracks the change of metric between t and t + ∆ t .Therefore, the time evolution of the state is non-unitary. In the rest of the paper we focus on the diagonalization ofthe energy operator U ( t ) HU − ( t ) this one being Hermitian, although the diagonalization of − i ~ U ( t ) ∂∂t U − ( t )is provided for the sake of completness in appendix B. Transformation to the flat space
We thus now focus on the Hamiltonian Eq. (3), and compute explicitly U ˆ HU − which as shown is hermitian inthe usual sense. The transformation U is given by U = √− g (cid:0) h + h β α β (cid:1) = √− gh (cid:0) α β h β /h (cid:1) One can then deduce U = f (cid:0) u β α β (cid:1) and thus U − = 1 f (1 − u ) (cid:0) − u β α β (cid:1) where u = u β u β with u β = h β /h √− gh (cid:18) r − h β h β ( h ) (cid:19) . and f = √− gh r − h β h β ( h ) / The greek (lorentzian) indices are assumed now to run only from 1 to 3. As a consequence the Hamiltonian H = U ˆ HU − given by Eq. (8) reads H = 12 (cid:0) H + H + (cid:1) + 12 f (cid:0) u β α β (cid:1) (cid:20) H, f (1 − u ) (cid:0) − u β α β (cid:1)(cid:21) − (cid:20) H + , (cid:0) − u β α β (cid:1) f (1 − u ) (cid:21) (cid:0) u β α β (cid:1) f which can be rewritten as H = 12 (cid:0) H + H + (cid:1) − (cid:0) u β α β (cid:1) (1 − u ) (cid:2) H, u β α β (cid:3) + 12 (cid:2) H + , u β α β (cid:3) (cid:0) u β α β (cid:1) (1 − u ) − i ~ (cid:0) u β α β (cid:1) ( ∇ P H ) . ∇ R (cid:18) f (1 − u ) (cid:19) (cid:0) − u β α β (cid:1) + i ~ (cid:0) − u β α β (cid:1) (cid:0) ∇ P H + (cid:1) . ∇ R (cid:18) f (1 − u ) (cid:19) (cid:0) u β α β (cid:1) where P and R are the canonical momentum and position operator satisfying (cid:2) R i , P j (cid:3) = i ~ δ ij . The Hamiltonian canalso be written as H = 12 (cid:18) − u ) H + H + − u ) (cid:19) −
12 1(1 − u ) (cid:2) H, u β α β (cid:3) + 12 (cid:2) H + , u β α β (cid:3) − u ) − u β α β (cid:18) − u ) H + H + − u ) (cid:19) u β α β − i ~ (cid:20) u β α β , ( ∇ P H ) . ∇ R (cid:18) f (1 − u ) (cid:19)(cid:21) (13)At this level, the computation of the last expression turns out to be quite technical is fully developped in AppendixA. The result is given by the following expression H = 12 α β ˜ H iβ P i + ~ ε ̺βγ ˜Γ ̺βi γ ! + 12 P i + ~ ε ̺βγ ˜Γ ̺βi γ ! ˜ H iβ α β + β ˜ m + 12 g i P i + P i g i + ~ (cid:16) ˜Γ + Γ e (cid:17) . Σ + ~ (cid:0) g (cid:0) h × h i (cid:1) − u × H i (cid:1) (1 − u ) . ( ∇ i u ) J (14)where from now on, all indices i , β , ρ... are only spatial and run from 1 to 3, but roman indices are raised and loweredby the meric g ij and greek indices by the lorentzian metrics η αβ . The several notations introduced above are given bythe following expressions˜ H iβ = H iβ + 2 g (cid:16) h δ h iβ − h β h iδ (cid:17) u δ (1 − u )ˆΓ ̺βi = Γ ̺βi − ε ̺βγ (cid:0) H − (cid:1) κi g h δ h iη ε δη κ Γ γj ˜Γ ̺βi = (cid:16) ˜ H − (cid:17) ηi H jη ˆΓ ̺βj ˆΓ γ = 14 (cid:16) ε ̺βγ Γ ̺β + g g i ε ̺βγ Γ ̺βi + ε βνγ H iβ Γ νi (cid:17) ˜Γ γ = (cid:0) u (cid:1) δ ηγ − u η u γ (1 − u ) ˆΓ η + 12 (cid:16)(cid:16) H iδ ˆΓ δi + Γ − g g i Γ i (cid:17) × u (cid:17) γ Γ eγ = − (cid:0) u × H i (cid:1) γ ∇ R i (cid:0) f (cid:0) − u (cid:1)(cid:1) f (1 − u ) ! + (cid:0) u ×∇ i (cid:0) H i (cid:1)(cid:1) γ (1 − u ) − (cid:0) u (cid:1) δ ηγ − u η u γ (1 − u ) ∇ R i (cid:16) g (cid:0) h × h i (cid:1) η (cid:17) where we used vectorial notations (cid:0) h (cid:1) α = h α , (cid:0) h i (cid:1) α = h iα , as well as the vectors (cid:0) H i (cid:1) β ≡ H iβ = g h (cid:16) h iβ − h β h i h (cid:17) , the following various vectors (cid:16) ˜Γ δi (cid:17) β = ˜Γ δβi , (cid:16) ˆΓ δi (cid:17) β = ˆΓ δβi , (cid:0) Γ i (cid:1) β = Γ βi , and (cid:0) Γ (cid:1) β = Γ β . The matrix J is given by J = (cid:18) I × I × (cid:19) and the effective mass by ˜ m = mg h (cid:16) u − u (cid:17) as well as g i = g g i . The expression Eq.(14) forthe energy operator H will be the one to diagonalize in the next section. SEMI-CLASSICAL ENERGY
The semiclassical diagonalization the Hamiltonian Eq. (14) is expected to lead to an effective Hamiltonian withgauge fields resulting from the back reaction of the spin degree of freedom (fast) on the translational momentum whichcan be treated semiclassically for slowly varying enough inhomogeneities. Indeed, the emergence of gauge fields is ageneral feature of systems providing fast and slow degrees of freedom. The purpose of ref [9], was to investigate theorigin of quantum gauge fields and forces by considering the diagonalization of an arbitrary matrix valued quantumHamiltonian. To be precise, by diagonalization we mean the derivation of an effective in-band Hamiltonian made ofblock-diagonal energy subspaces. This approach, based on a new differential calculus on a non-commutative spacewhere ~ plays the role of running parameter, leads to an in-band energy operator that can be obtained systematicallyup to arbitrary order in ~ . Particularly important for our purpose, it has been possible, for an arbitrary Hamiltonian H ( R , P ) with the canonical coordinates and momentum (cid:2) R i , P j (cid:3) = i ~ δ ji , to obtain the corresponding diagonalrepresentation ε ( r , p ) to order ~ , in terms of non-canonical coordinates and momentum ( r , p) defined later andcommutators between gauge fields. The method is quite involved, so that in the present paper we restrict ourself tothe semiclassical approximation (order ~ ).The mathematical difficulty in performing the diagonalization of H comes from the intricate entanglement ofnoncommuting operators due to the canonical relation (cid:2) R i , P j (cid:3) = i ~ δ ji . In [8] starting with a very general but timeindependent H ( R , P ) and by considering ~ as a running parameter, we related the in-band Hamiltonian V HV + = ε ( X ) and the unitary transforming matrix V ( X ) (where X ≡ ( R , P )) to their classical expressions through integro-differential operators, i.e. ε ( X ) = b O ( ε ( X )) and V ( X ) = b N ( V ( X )), where in the matrices ε ( X ) and V ( X ) , thedynamical operators X are replaced by classical commuting variables X = ( R , P ) . The only requirement of the method is therefore the knowledge of V ( X ) which gives the diagonal form ε ( X ) . Generally, these equations do not allow to find directly ε ( X ), V ( X ), however, they allow us to produce the solutionrecursively in a series expansion in ~ . With this assumption that both ε and V can be expanded in power series of ~ , we determined in [8], the explicit n -th in band energy to order ~ for an arbitrary given Hamiltonian ε n ( r , p ) = ε ,n ( r , p ) + i ~ P n (cid:8)(cid:2) ε ( r , p ) , A R l (cid:3) A P l − (cid:2) ε ( r , p ) , A P l (cid:3) A R l (cid:9) + O ( ~ ) (15)where A R = i ~ V ∇ P V + and A P = − i ~ V ∇ R V + with V the diagonalizing matrix. The operator P n has the meaningof the projection on the n -th energy subspace. The new non-canonical dynamical operators r and p depend on gaugefields A R = P n ( A R ) and A P = P n ( A P ) similarly to electromagnetism, as we have r = i ~ ∂ p + A R and p = P + A P .These gauge invariant quantities not only are emerging naturally but are also necessary to have a gauge invariantenergy Eq. (15). The operator ε ( r , p ) is the diagonal energy obtained at zero order ( ~ ) in which the classicalvariable R and P are replaced by new non-commuting operators r , p . Therfore the first step consists in finding thematrix V ( X ) which diagonalizes the ”classical” Hamiltonian H ( X ) . Zero Order Diagonalization.
For practical purpose we introduce the three dimensional effective metric G ij = ˜ H iα ˜ H jβ δ αβ as well as the gravitycoupled momentum ˜ P α = ˜ H iα ( P i + ~ ε ̺βγ ˜Γ ̺βi σ γ ) . As shown in [7], the classical block-diagonalization of the Hamiltonian Eq. (14), but without the last term (cor-responding to the static case), can be performed by the following unitary FW-like matrix (denoted F for Foldy-Wouthuysen) F ( ˜P ) = D (cid:18) E + ˜ m + c β (cid:16) α. ˜P + ˜P + .α (cid:17) + N (cid:19) / p E ( E + ˜ m )with E = r(cid:16) α. ˜P + ˜P + .α (cid:17) + ˜ m , N = ~ iα. ( ˜P × ( ˜Γ + ~ Γ e )) E , and D = 1 + ~ β ( ˜P × ( ˜Γ + ~ Γ e )) × ˜P E ( E + ˜ m ) .Indeed one can easily check that F HF = β q P i G ij P j + ~ ε αβγ ˜Γ αβj Σ γ G ij P i + ˜ m + ~ E (cid:16) ˜Γ + Γ e (cid:17) . ˜ m Σ + (cid:16) Σ . ˜H i P i (cid:17) ˜H i P i ( E + ˜ m ) + 12 g i P i + 12 P i g i + F (cid:0) g (cid:0) h × h i (cid:1) − u × H i (cid:1) (1 − u ) . ( ∇ i u ) J ! F (16)with ˜Γ iγ = ε ̺βγ ˜Γ ̺βi . All contributions are block-diagonal except the last term which was not present before in thecase of a static metrics [7]. The proof of this block diagonalization relies on the simple fact that for classical variables X , the matrices ˜ H iα and ˜Γ αβi are independent of both the momentum and position, β and α. ˜P anticommute and inthe Taylor expansion of E all terms commute with β and α. ˜P + ˜P + .α .As said before, the last term in Eq.(16) is non diagonal and must treated specifically. Actuallt, one can apply asecond unitary transformation that will cancel the non diagonal contributions of F ( ( g ( h × h i ) − u × H i ) (1 − u ) . ( ∇ i u ) J ) F without affecting the rest of the diagonalized Hamiltonian to the first order in ~ . The explicit form of this transfor-mation is : F ′ = 1 − P − F (cid:0) g (cid:0) h × h i (cid:1) − u × H i (cid:1) (1 − u ) . ( ∇ i u ) JF ! β p P i G ij P j + ˜ m P − being the projection outside the diagonal. One can check that F ′ − F ′ is unitary tothe first order. As a consequence, the composition of the two unitary transformations yields the following diagonalenergy operator : ε ( R , P ,t ) = F ′ F ˆ H F F ′ +0 = β q P i G ij P j + ~ ε αβγ ˜Γ αβj Σ γ G ij P i + ˜ m + ~ E (cid:16) ˜Γ + Γ e (cid:17) . ˜ m Σ + (cid:16) Σ . ˜H i P i (cid:17) ˜H i P i ( E + ˜ m ) + 12 g i P i + 12 P i g i + P + F (cid:0) g (cid:0) h × h i (cid:1) − u × H i (cid:1) (1 − u ) . ( ∇ i u ) J ! F ! (17)The last term is explicitely given by: P + F (cid:0) g (cid:0) h × h i (cid:1) − u × H i (cid:1) (1 − u ) . ( ∇ i u ) J ! F ! = D P + (cid:16) E + ˜ m + cβ (cid:16) α. ˜P (cid:17)(cid:17) (cid:16) ( E + V ( r ) m ) J + cβ (cid:16) Σ . ˜P (cid:17)(cid:17) ~ (cid:0) g (cid:0) h × h i (cid:1) − u × H i (cid:1) . ( ∇ i u )2 E ( E + ˜ m ) D + = ~ cβ (cid:0) g (cid:0) h × h i (cid:1) − u × H i (cid:1) . ( ∇ i u )2 E (1 − u ) ˜P . Σ = ~ cβ (cid:0) ( f − g (cid:0) h × h i (cid:1)(cid:1) . ( ∇ i u )2 f E (1 − u ) ˜P . Σ Ultimately, the diagonalization process yields: ε ( R , P ,t ) = F ′ F ˆ H F F ′ +0 = β q P i G ij P j + ~ ε αβγ ˜Γ αβj Σ γ G ij P i + ˜ m + ~ (cid:16) ˜Γ + Γ e (cid:17) . ˜ m Σ + (cid:16) Σ . ˜H i P i (cid:17) ˜H i P i ( E + ˜ m ) + 12 g i P i + 12 P i g i + ~ cβ (cid:0) ( f − g (cid:0) h × h i (cid:1)(cid:1) . ( ∇ i u )2 f E (1 − u ) ˜P . Σ (18)where for the moment R and P are treated as classical commuting quantities. First order in ~ diagonalization From expression Eq. (18) we can deduce the diagonal energy operator for, let say, the particule subspace ε , + . Thesemiclassical energy is given by Eq. (15), where ε , + corresponds to the positive energy subspace Eq. (18) in which theclassical variables R , P are replaced by the quantum covariant ones r = i ~ ∂ p + ~ A R and p = P + ~ A P . The explicitcomputation for the Berry connections A R = P + ( A R ) and A P = P + ( A P ) with A R = P + ( i ~ ( F ′ F ) ∇ P ( F ′ F ) + ) and A P = P + ( i ~ ( F ′ F ) ∇ R ( F ′ F ) + ) gives the components A R k = c ε αβγ ˜ H iγ ˜ P α Σ β E ( E + ˜ m ) g ik + o ( ~ ) (19) A P k = − c ε αβγ ˜ P α Σ β ( ∇ R k ˜ P γ )2 E ( E + ˜ m ) + o ( ~ ) (20)where E is the same as E above, but now R is an operator and ˜ H kγ is the inverse matrix of ˜ H iβ .We also denote, for the rest of the paper, ˜p to be the same expression as ˜P in which R and P have been replacedby r and p , namely: ˜ p α = ˜ H iα ( p i + ~ ε ̺βγ ˜Γ ̺βi σ γ )To complete the diagonalization we need to evaluate the quantity M + = i P + nh ε ( X ) , A R l i A P l − h ε ( X ) , A P l i A R l o + O ( ~ )which being on all point similar to the one given in [6] or [7] is simply stated : M + = 1 E (cid:18)(cid:18) Σ − ( A R × ˜p ) (cid:19) . B − E ∇ ˜ m ( r ) . ( ˜p × Σ ) (cid:19) where the ”magnetotorsion field” B is defined through a three dimensional effective torsion tensor B γ = − P δ T δαβ ε αβγ where the effective torsion T δαβ is defined as T αβδ = ˜ H δk (cid:16) ˜ H lα ∂ l ˜ H kβ − ˜ H lβ ∂ l ˜ H kα (cid:17) + ˜ H lα ˜Γ βδl − ˜ H lβ ˜Γ αδl The physical origin of the term M has been discussed in [6] and [7] for the static case. Note that for the static case( g i = 0) we retrieve the true torsion T αβδ = H δk (cid:0) H lα ∂ l H kβ − H lβ ∂ l H kα (cid:1) + H lα Γ βδl − H lβ Γ αδl . Considering similarly the anti-particle subspace, it turns out that the coordinate operators are identical with theparticule ones and that M − = − M + therefore the full energy operator with both particles and anti particles can becast in the form ε ( p , r ) = β e ε + 12 g i p i + 12 p i g i + ~ E (cid:16) ˜Γ + Γ e (cid:17) . ˜ m Σ + (cid:16) Σ . ˜H i P i (cid:17) ˜H i P i ( E + ˜ m ) + β ~ c (cid:0) ( f − g (cid:0) h × h i (cid:1)(cid:1) . ( ∇ i u )2 f E (1 − u ) ˜p . Σ + β ~ M (21)where M = M + and e ε = c s(cid:18) p i + ~ E ˜Γ i . (cid:18) ˜ m Σ + ( Σ . ˜p ) ˜p ( E + ˜ m ) (cid:19)(cid:19) G ij (cid:18) p i + ~ E ˜Γ i . (cid:18) ˜ m Σ + ( Σ . ˜p ) ˜p ( E + ˜ m ) (cid:19)(cid:19) + ˜ m with the vector ˜Γ i defined in terms of its components as (cid:16) ˜Γ i (cid:17) γ ≡ ε αβγ ˜Γ αβi ( r ). Equation (21) is the main result ofthis paper.The coordinates operators in Eq. (21) satisfy a non commutative algebra as[ r i , r j ] = i ~ Θ rrij (22)[ p i , p j ] = i ~ Θ ppij (23)[ p i , r j ] = − i ~ g ij + i ~ Θ prij (24)where Θ αβij = ∂ α i A β j − ∂ β j A α i + [ A α i , A β j ] the so-called Berry curvature. An explicit computation gives :Θ rrkl = − E ˜ m Σ γ + (cid:16) Σ δ ˜ P δ (cid:17) ˜ P γ ( E + ˜ m ) ε αβγ ˜ H iα ˜ H jβ g ik g jl Θ ppij = − E ˜ m Σ γ + (cid:16) Σ δ ˜ P δ (cid:17) ˜ P γ ( E + ˜ m ) ∇ r i ˜ P α ∇ r j ˜ P β ε αβγ + 12 E h ∇ r i ˜ m (cid:16)(cid:16) Σ × ˜P (cid:17) . ∇ r j ˜P (cid:17) − ∇ r j V ( r ) (cid:16)(cid:16) Σ × ˜P (cid:17) . ∇ r i ˜P (cid:17)i Θ prij = 12 E ˜ m Σ γ + (cid:16) Σ δ ˜ P δ (cid:17) ˜ P γ ( E + ˜ m ) ε αβγ ∇ r i ˜ P α H lβ g jl − E ∇ r i ˜ m (cid:16) Σ × ˜P (cid:17) j (25)There are also other Berry curvature mixing coordinates and spinΘ r Σ ij = [ r i , Σ j ] = ic − p j Σ i + p . Σ δ ij E ( E + ˜ m )Θ p Σ ij = [ p i , Σ j ] = − ic − p j Σ l + p . Σ δ lj E ( E + ˜ m ) ˜ H γl ∇ r i ˜ p γ (26)Interestingly Eq. (21) can be rewritten as ε = cβ s(cid:18) p i − ~ ˜Γ i . ˜Θ rr (cid:19) G ij (cid:18) p i − ~ ˜Γ i . ˜Θ rr (cid:19) + ˜ m ( r ) + 12 g i p i + 12 p i g i − ~ (cid:16) ˜Γ + Γ e (cid:17) . ˜Θ rr + β ~ c (cid:0) ( f − g (cid:0) h × h i (cid:1)(cid:1) . ( ∇ i u )2 f E (1 − u ) ˜p . Σ + β ~ M (27)where ˜Θ rrγ = − E (cid:18) ˜ m Σ γ + ( Σ δ ˜ P δ ) ˜ P γ ( E + ˜ m ) (cid:19) is the ”rescaled” Berry curvature. This formula clearly shows that the spinconnection couples only to the Berry curvature.We note in Eq. (27) the presence of the term − ~ (cid:16) ˜Γ + Γ e (cid:17) . ˜Θ rr + g i p i + p i g i not proportional to β andindependant of the particle charge, that is discrimnating between particles and anti particles. These terms givedifferent energy levels for the Dirac particles and antiparticles as ε + = − ε − . The coupling − ~ (cid:16) ˜Γ + Γ e (cid:17) . ˜Θ rr proportional to the spin will survive in the case of the non-diagonal static gravitational field but cancels for a diagonalmetrics as studied in [2]. On the other hand the term g i p i + p i g i vanishes for a static metrics since in this case g i = ∂ g ij = 0 so that g i = 0. Therefore the symmetry between particle and antiparticle is restablished only forstatic diagonal metrics. THE STATIC GRAVITATIONAL FIELD
This case is caracterized by the following time independent metric: g ij ≡ g ij ( R ), g ≡ g ( R ), g i = 0. In thatcase expressions simplify greatly. Actually, the transformation matrix U is U = √− gh and U = p √− gh = f .The effective quantities reduce to: (cid:16) ˜H i (cid:17) β = ˜ H iβ = H iβ = g h h iβ = √ g h iβ ˜Γ ̺βi = ˆΓ ̺βi = Γ ̺βi ˆΓ γ = ˜Γ γ = 14 n ε ̺βγ Γ ̺β + ε βνγ H iβ Γ νi o = − g (cid:0) h (cid:1) ε βνγ h iβ h νl h λi Γ klλ Γ e = 0 G ij = g g ij ˜ m = mg h as u β = 0where Γ klλ stands for the Christoffel symbol. In this case Eq. (21) reduces to ε ≃ cβ s(cid:18) p i + ~ E Γ i ( r ) . (cid:18) e m Σ + ( Σ . ˜p ) ˜p ( E + e m ) (cid:19)(cid:19) G ij ( r ) (cid:18) p i + ~ E Γ i ( r ) . (cid:18) e m Σ + ( Σ . ˜p ) ˜p ( E + e m ) (cid:19)(cid:19) + e m + ~ βM + ~ E ˜Γ γ (cid:18) e m Σ γ + ( Σ . ˜p ) ˜ p γ ( E + e m ) (cid:19) (28)with ˜ p α = √ g h iα ( p i + ~ ε ̺βγ Γ ̺βi σ γ ) and the vector Γ i defined in terms of its components as ( Γ i ) γ ≡ ε αβγ Γ αβi ( r ) . Then even for this case, particles and antiparticles avec a different in-band energy operator because of term ~ E ˜Γ γ (cid:16) e m Σ γ + ( Σ . ˜p )˜ p γ ( E + e m ) (cid:17) breaking the symmetry ε + = − ε − . Although this term was already in previous studies [6][7]this essential point has not been pointed out. For a diagonal metric ˜Γ = 0, and the symmetry particles/anti-particlesis recovered. ULTRARELATIVISTIC LIMIT
It is interesting to look at the ultrarelativistic limit mc →
0. One readily obtain ε ≃ β e ε + cβ (cid:0) ( f − g (cid:0) h × h i (cid:1)(cid:1) . ( ∇ i u )2 f (1 − u ) ˜ λ + β ˜ λg E B . ˜p ˜ p + ˜ λ p γ (cid:16) ˜Γ γ + Γ eγ (cid:17) ˜ p + 12 g i p i + 12 p i g i (29)with e ε = c r(cid:16) p i + ˜ λ i ( r ) . p p (cid:17) G ij (cid:16) p j + ˜ λ j ( r ) . p p (cid:17) and ˜ λ = ~ ˜p . Σ / ˜ p a biased helicity, that is not projected on themomentum p but rather on ˜p . This fact is not astonishing. Actually, as we saw in the diagonalization process, theparticle is submitted to the action of an effective gravitationnal field, which differs slightly from the initial field. Thiseffective metric is responsible for considering the momentum ˜p rather than as a dynamical variable P . Nicely thisenergy can be expressed in terms of the helicity which is the relevant variable for massless particles and not in termof Σ. As shown in [6], Eq. (29) is also valid for photon with the one-half spin matrix Σ replaced with spin one matrix S . Here also we see that photons and anti-photons do not have the same energy spectrum. The symmetry is againonly restored for a static diagonal metric.0
THE TIME DEPENDENT SYMMETRIC GRAVITATIONAL FIELD
A typical example of such a metric is the Schwarzschild space-time in isotropic coordinates. This case, studiedin a different manner in [2]and [3] for a time independent metric, received a full independent treatment within ourformalism in [6][7]. We now present the equivalent results for a time dependent metric, completed with the spinmatrix dynamics. For a symmetric metric, the semiclassical Hamiltonian has the following form [3] H = 12 ( α.P F ( R , t ) + F ( R , t ) α.P ) + βmV ( R , t ) (30)corresponding to the metric g ij = δ ij (cid:16) V ( R ,t ) F ( R ) (cid:17) , g i = 0 and g = V ( R , t ). We will define φ = VF . In that contextthe relevant quantities for the diagonalization appear to be : h βi = φδ βi , h iβ = 1 φ δ iβ h β = V ( R , t ) δ β , h β = 1 V ( R , t ) δ β and (cid:0) H i (cid:1) β = H iβ = g h h iβ = V h iβ (cid:16) ˜H i (cid:17) β = ˜ H iβ = H iβ ˜Γ ̺βi = ˆΓ ̺βi = Γ ̺βi = (cid:16) ∂ ρ φh βi − ∂ β φh ρi (cid:17) φ Γ νi = ~ ∂ φh νi φV ˜Γ γ = ˆΓ γ = ~ ε ̺βγ F ( R ) V ( R ) ∂ g ̺β + ~ g h F ( R ) V ( R ) (cid:18) ∂ (cid:18) V ( R ) F ( R ) (cid:19)(cid:19) ε βνγ h iβ h νi = 0 Γ e = 0 G ij = V g ij Similar computations to the ones performed in the previous section lead to the following expressions for the dynamicalvariables and the diagonalized Hamiltonian r = R − ~ F ( R , t ) Σ × P E ( E + mV ( R )) , p = P (31)and the diagonal energy becomes: ε = β p F ( r ,t ) P + P F ( r ,t ) + mV ( r ,t ) − F ( r ,t )2 E m ~ β ∇ φ ( r ,t ) . ( P × Σ ) (32)The Berry curvatures are given by:Θ rrij = − ~ F ( r , t ) ε ijk E (cid:18) mφ ( r , t ) Σ k + F ( r , t ) ( Σ . P ) P k E + mV ( r ) (cid:19) (33)Θ prij = − ~ F ( r , t )2 E m ∇ i φ ( r ) ( Σ × P ) j (34)Θ ppij = 0 (35)and Θ r Σ ij being unchanged, Θ p Σ ij = 0, and e ε = p F ( r ,t ) P + P F ( r ,t ) + mV ( r ,t ) . Note here that the magnetotorsion field B = 0. From Appendix B, we can also get the non Hermitian contributionsthe time evolution operator which in this case reads i ~ ∂∂t ln f = ∂∂t ln √− gh .One can check, after developing r as a function of R and the Berry phase, that our Hamiltonian coincides, inthe weak field approximation,with the one given in [3] when considering the semiclassical limit (order ~ ). This alsoconfirms the validity of the Foldy Wouthuysen approach asserted in [3].1 ONLY TIME DEPENDENT METRIC
The metric tensor and the vierbein only depend on time. Therefore we have (cid:0) H i (cid:1) β = H iβ = g h h iβ − h β h i h !(cid:16) ˜H i (cid:17) β = ˜ H iβ = H iβ + 2 g (cid:16) h δ h iβ − h β h iδ (cid:17) u δ (1 − u )ˆΓ ̺βi = Γ ̺βi − ~ ε ̺βγ (cid:0) H − (cid:1) κi g h δ h iη ε δη κ Γ γj ̺βi = (cid:16) ˜ H − (cid:17) ηi H jη ˆΓ ̺βj ˆΓ γ = 14 (cid:16) ε ̺βγ Γ ̺β + g g i ε ̺βγ Γ ̺βi + ε βνγ H iβ Γ νi (cid:17) ˜Γ γ = (cid:0) u (cid:1) δ ηγ − u η u γ (1 − u ) ˆΓ η + (cid:18)(cid:18) H iδ ˆΓ δi + Γ − g g i Γ i (cid:19) × u (cid:19) γ Γ eγ = 0so that the energy becomes ε ( p , r ,t ) = β ˜ ε + ~ E ˜Γ . ˜ m Σ + (cid:16) Σ . ˜H i p i (cid:17) ˜H i p i ( E + ˜ m ) + 12 g i ( t ) p i + 12 p i g i ( t )+ β ~ M (36)with ˜ ε = c s(cid:18) p i + ~ E Γ i ( r , t ) . (cid:18) e m Σ + ( Σ . ˜p ) ˜p ( E + e m ) (cid:19)(cid:19) G ij ( r , t ) (cid:18) p i + ~ E Γ i ( r , t ) . (cid:18) e m Σ + ( Σ . ˜p ) ˜p ( E + e m ) (cid:19)(cid:19) + e m which show that particles and antiparticles have a different energy spectrum in this graviational field. CONCLUSION
The semiclassical limit for Dirac particles interacting with a fully general gravitational field was investigated througha first order in ~ diagonalization of the Dirac Hamiltonian. This work extends previous ones where only static metricswere considered. The time dependence of the metrics leads to new contributions of the in-band energy operator. Inparticular we found a coupling term between the linear momentum and the spin, and terms which in general willbreak the particle - antiparticle symmetry.As already found by other authors, the time dependence leads also to special features like the non-unitarity of theevolution operator, whose origin can be tracked back to the notion of scalar product in the Hilbert space of wavefunctions for a time dependent metric. This non-unitarity is unavoidable but we could nevertheless diagonalize thefull evolution operator, even though our main focus was to obtain the block-diagonal form of the energy, this oneturning out to be Hermitian. In addition, to the very general semiclassical diagonal energy operator, we providedseveral physically relevant examples. [1] L. L. Foldy and S. A. Wouthuysen, Phys. Rev. (1950) 29.[2] Y. N. Obukhov, Phys. Rev. Lett (2001) 192; Fortschr. Phys. (2002) 711; Phys. Rev. Lett (2002) 068903.[3] A. J. Silenko and O. V. Teryaev, Phys. Rev. D (2005) 064016.[4] J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, New York: McGraw- Hill, (1965).[5] M. Leclerc, Class. Quant. Grav. (2006) 4013. [6] P. Gosselin, A. B´erard, H. Mohrbach, Phys. Rev. D, 75 (2007) 084035.[7] P. Gosselin, A. B´erard, H. Mohrbach, Phys. Lett. A, 368 (2007) 356.[8] P. Gosselin and H. Mohrbach, Eur. Phys. J. C 64, (2009) 495, P. Gosselin, J. Hanssen and H. Mohrbach, Phys. Rev. D 77(2008) 085008, P. Gosselin, A. B´erard and H. Mohrbach, Eur. Phys. J. B 58 (2007) 137.[9] P. Gosselin and H. Mohrbach, J. Phys. A: Math. Theor. 43 (2010) 354025. APPENDIX.Derivation of Eq. ( ) . Let us start with the following development for ˆ H :ˆ H = g h β γ β γ α ˆ P α + ~ ε ̺βγ Γ ̺β Σ γ + i ~ β α β + g h β γ β m + i ∂ t ln (cid:0) − gg (cid:1) = g g i ( P i + ~ ε ̺βγ Γ ̺βi Σ γ + i ~ Γ γi α γ )+ ig (cid:0) h δ h iη ε δηκ (cid:1) Σ κ ( P i + ~ ε ̺βγ Γ ̺βi Σ γ + i ~ Γ γi α γ )+ g h α β h iβ ( P i + ~ ε ̺βγ Γ ̺βi Σ γ + i ~ Γ γi α γ ) − g α β h β h i ( P i + ~ ε ̺βγ Γ ̺βi Σ γ + i ~ Γ γi α γ )+ ~ ε ̺βγ Γ ̺β Σ γ + i ~ β α β + g h β γ β m + i ∂ t ln (cid:0) − gg (cid:1) and remark that we can rewrite the four first terms in the following form : g h α β h iβ − h β h i h ! ( P i + ~ ε ̺βγ Γ ̺βi Σ γ + i ~ Γ γi α γ )+ g g i ( P i + ~ ε ̺βγ Γ ̺βi Σ γ + i ~ Γ γi α γ )+ ig (cid:0) h δ h iη ε δηκ (cid:1) Σ κ ( P i + ~ ε ̺βγ Γ ̺βi Σ γ ) − ~ g (cid:0) h δ h iη ε δηκ (cid:1) α κ Γ γi Σ γ = g h α β h iβ − h β h i h ! × P i + ~ ε ̺βγ Γ ̺βi Σ γ − h iβ − h β h i h ! − κi (cid:18) ~ h (cid:0) h δ h jη ε δηκ (cid:1)(cid:19) Γ γj Σ γ + i ~ Γ γi α γ + g g i ( P i + ~ ε ̺βγ Γ ̺βi Σ γ + i ~ Γ γi α γ )+ ig (cid:0) h δ h iη ε δηκ (cid:1) Σ κ ( P i + ~ ε ̺βγ Γ ̺βi Σ γ )= α β H iβ P i + ~ ε ̺βγ ˜Γ ̺βi Σ γ + i ~ Γ γi α γ ! + g g i ( P i + ~ ε ̺βγ Γ ̺βi Σ γ + i ~ Γ γi α γ )+ ig (cid:0) h δ h iη ε δηκ (cid:1) Σ κ ( P i + ~ ε ̺βγ Γ ̺βi Σ γ )3where the effective dreibein and spin connection : H iβ = g h h iβ − h β h i h ! ˆΓ ̺βi = Γ ̺βi − ~ ε ̺βγ (cid:0) H − (cid:1) κi g (cid:0) h δ h jη ε δη κ (cid:1) Γ γj H + ˆ H + , which is the first contribution in Eq.(8) (up to the factors − u β u β ) that will be skippedconstantly in this section for the sake of readability, and reintroduced ultimately) .12 (cid:16) ˆ H + ˆ H + (cid:17) = 12 α β H iβ P i + ~ ε ̺βγ ˆΓ ̺βi Σ γ ! + 12 P i + ~ ε ̺βγ ˆΓ ̺βi Σ γ ! α β H iβ + 12 g g i P i + 12 P i g g i + g g i ( ~ ε ̺βγ Γ ̺βi Σ γ ) − ~ ∇ R i (cid:0) g (cid:0) h δ h iη ε δηκ (cid:1)(cid:1) Σ κ − g (cid:0) h δ h iη ε δηκ (cid:1) ε γνκ ( ~ ε ̺βγ Γ ̺βi Σ ν − ~ ε βγν H iβ Γ γi Σ ν + ~ ε ̺βγ Γ ̺β Σ γ + g h βm = 12 α β H iβ P i + ~ ε ̺βγ ˆΓ ̺βi Σ γ ! + 12 P i + ~ ε ̺βγ ˆΓ ̺βi Σ γ ! α β H iβ + g h βm + (cid:16) ˆΓ − ~ ∇ R i (cid:0) g (cid:0) h δ h iη ε δηγ (cid:1)(cid:1)(cid:17) . Σ + 12 g g i P i + 12 P i g g i with : (cid:16) ˆΓ (cid:17) γ = ~ ε ̺βγ Γ ̺β + ~ g g i ε ̺βγ Γ ̺βi ~ g (cid:0) h δ h iη ε δηκ (cid:1) ε νκγ ( ε ̺βν Γ ̺βi ~ ε βνγ H iβ Γ νi (cid:0) Γ (cid:1) γ = Γ γ = ε ̺βγ Γ ̺β , (cid:0) Γ i (cid:1) γ = g g i ~ ε ̺βγ Γ ̺βi , and use the following expression for thespin connection Γ ̺βi = h ̺µ h βν (cid:16) h ξi Γ µνi − ∇ µ h νi (cid:17) to show that ~ g (cid:0) h δ h iη ε δηκ (cid:1) ε νκγ ( ε ̺βν Γ ̺βi ) = 0. As a consequence, we are left with ˆΓ = ~ (cid:0) Γ + g g i Γ i (cid:1) + ~ H i × Γ i U ˆ HU − in Eq. (8), we need to find the expressions for − u β α β ( H + H + ) u β α β , − [ H,u β α β ] + [ H + ,u β α β ] , and − i ( u β α β ) ∇ P H ∇ R ( u β u β ) − u β u β )) (cid:0) − u β α β (cid:1) . We start with the computa-tion of u β α β (cid:16) ˆ H + ˆ H + (cid:17) u β α β by decomposing (cid:16) ˆ H + ˆ H + (cid:17) as: (cid:16) ˆ H + ˆ H + (cid:17) (cid:16) ˆ H ′ + ˆ H ′ + (cid:17) (cid:16) ˆΓ γ − ~ ∇ R i (cid:0) g (cid:0) h × h i (cid:1)(cid:1)(cid:17) Σ γ + g h βm + 12 g g i P i + 12 P i g g i (cid:16) ˆ H ′ + ˆ H ′ + (cid:17) α β H iβ P i + ~ ε ̺βγ ˆΓ ̺βi Σ γ ! + 12 P i + ~ ε ̺βγ ˆΓ ̺βi Σ γ ! α β H iβ For the sake of the computations, we will denote ˆ P α = H iα (cid:16) P i + ~ ε ̺βγ ˆΓ ̺βi Σ γ (cid:17) ≡ H iα ˆ P i .so that u β α β (cid:16) ˆ H ′ + ˆ H ′ + (cid:17) u β α β = 12 u β α β α β H iβ ˆ P i u β α β + 12 u β α β ˆ P i α β u β α β H iβ = 12 u γ α γ α β H iβ ˆ P i u γ α γ + 12 u γ α γ ˆ P i α β u γ α γ H iβ = 12 √ g u γ α γ α β u γ α γ H iβ ˆ P i + 12 ˆ P i H iβ u γ α γ α β u γ α γ + 12 u γ α γ α β H iβ h ˆ P i , u γ α γ i + 12 h u γ α γ , ˆ P i i α β u γ α γ H iβ √ g = 12 u γ α γ α β u γ α γ H iβ ˆ P i + 12 ˆ P i H iβ u γ α γ α β u γ α γ + 12 iu γ ε γβδ (cid:16)h Σ δ H iβ h ˆ P i , u γ α γ i + H iβ h ˆ P i , u γ α γ i Σ δ i(cid:17) = 12 α β u H iβ ˆ P i + 12 ˆ P i H iβ u α β + iu γ ε γβδ Σ δ u γ α γ H iβ ˆ P i − H iβ ˆ P i u γ α γ iu γ ε γβδ Σ δ + 12 iu γ ε γβδ (cid:16)h Σ δ H iβ h ˆ P i , u γ α γ i + H iβ h ˆ P i , u γ α γ i Σ δ i(cid:17) = 12 α β u H iβ ˆ P i + 12 ˆ P i H iβ u α β + h(cid:0) u Σ β − ( uΣ ) u β (cid:1) H iβ ˆ P i − H iβ ˆ P i (cid:0) u Σ β − ( uΣ ) u β (cid:1)i + 12 iu γ ε γβδ (cid:16)h Σ δ H iβ h ˆ P i , u γ α γ i + H iβ h ˆ P i , u γ α γ i Σ δ i(cid:17) Now, given that: iu γ ε γβδ (cid:16)h Σ δ H iβ h ˆ P i , u γ α γ i + H iβ h ˆ P i , u γ α γ i Σ δ i(cid:17) = − ~ u γ ε γκδ " Σ δ H iκ −∇ i u γ α γ + ˆΓ ̺βi Σ β u ̺ ! + H iκ −∇ i u γ α γ + ˆΓ ̺βi Σ β u ̺ ! Σ δ = − ~ u γ ε γκδ H iκ (cid:16) − J ∇ i u δ + ˆΓ ̺δi u ̺ (cid:17) one thus has, u β α β (cid:16) ˆ H ′ + ˆ H ′ + (cid:17) u β α β = 12 α β u ˆ P β + 12 ˆ P β u α β − g h ~ u γ ε γκδ H iκ (cid:16) − J ∇ i u δ + ˆΓ ̺δi u ̺ (cid:17) = 12 α β u ˆ P β + 12 ˆ P β u α β − g h ~ u γ ε γκδ H iκ ( − J ∇ i u δ )since, by an argument already used, u γ ε γκδ H iκ ˆΓ ̺δi u ̺ ∝ (cid:0) h × h i (cid:1) δ ˆΓ ̺δi u ̺ = 0.5To complete the computation of u β α β ( H + H + ) u β α β , we need to calculate the following contribution: u γ α γ (cid:18) g h βm + 12 g g i P i + 12 P i g g i + (cid:16) ˆΓ γ − ~ ∇ R i (cid:0) g (cid:0) h × h i (cid:1)(cid:1) γ (cid:17) Σ γ (cid:19) u γ α γ = − g h β u m + 12 g g i u P i + 12 u P i g g i + u (cid:16) ˆΓ γ − ~ ∇ R i (cid:16) g (cid:0) h × h i (cid:1) γ (cid:17)(cid:17) Σ γ − u (cid:16) ˆΓ γ − ~ ∇ R i (cid:16) g (cid:0) h × h i (cid:1) γ (cid:17)(cid:17) Σ γ + u . Σ (cid:16) ˆΓ γ − ~ ∇ R i (cid:16) g (cid:0) h × h i (cid:1) γ (cid:17)(cid:17) u γ = − g h β u m + 12 g g i u P i + 12 u P i g g i + u . Σ (cid:16) ˆΓ γ − ~ ∇ R i (cid:16) g (cid:0) h × h i (cid:1) γ (cid:17)(cid:17) u γ − u (cid:16) ˆΓ γ − ~ ∇ R i (cid:16) g (cid:0) h × h i (cid:1) γ (cid:17)(cid:17) Σ γ Now, we turn our attention toward the second contribution in Eq. (8), namely − [ ˆ H,u β α β ] + [ ˆ H + ,u β α β ] . To do so, andsince − h ˆ H, u β α β i + h ˆ H + , u β α β i − h ˆ H − ˆ H + , u β α β i we only need to concentrate on the anti hermitian part of H . Given that,ˆ H = α β H iβ P i + ~ ε ̺βγ ˆΓ ̺βi Σ γ + i ~ Γ γi α γ ! + g g i i ~ Γ γi α γ + ig (cid:0) h δ h iη ε δηκ (cid:1) Σ κ ( P i + ~ ε ̺βγ Γ ̺βi Σ γ ) + i ~ β α β we can write thatˆ H − ˆ H + i ~ ∇ i (cid:0) α β H iβ (cid:1) + iH iβ ~ Γ βi g g i i ~ Γ βi α β − i ~ H i̺ α β ˆΓ β̺i + i g (cid:0) h δ h iη ε δηκ (cid:1) Σ κ P i + i P i g (cid:0) h δ h iη ε δηκ (cid:1) Σ κ + ig (cid:0) h δ h iη (cid:1) ~ Γ δηi i ~ β α β and − h ˆ H, u β α β i + h ˆ H + , u β α β i − ~ ∇ i (cid:0) H iβ (cid:1) + g g i Γ βi β − H i̺ ˆΓ β̺i ! ε βγκ u γ Σ κ + g (cid:0) h δ h iη − h η h iδ (cid:1) u δ α η P i + P i g (cid:0) h δ h iη − h η h iδ (cid:1) u δ α η − g ~ (cid:0) h δ h iη ε δηκ (cid:1) J ∇ i u κ Ultimately, we need the third contribution to Eq. (8): − i ~ (cid:20) u β α β , (cid:16) ∇ P ˆ H (cid:17) . ∇ R (cid:18) f (1 − u β u β ) (cid:19)(cid:21) = i ~ " u β α β , α β H iβ . ∇ R i (cid:0) f (cid:0) − u β u β (cid:1)(cid:1) [ f (1 − u β u β )] ! = − ~ Σ . (cid:0) u × H i (cid:1) ∇ R i (cid:0) f (cid:0) − u (cid:1)(cid:1) [ f (1 − u )] ! We can now gather all these terms to obtain the expression of U ˆ HU − .Define, as in text, the vectors ˜Γ δi , Γ i by: (cid:16) ˜Γ δi (cid:17) β = ˜Γ δβi , (cid:0) Γ i (cid:1) β = Γ βi , (cid:0) Γ (cid:1) β = Γ β H iβ : ˜ H iβ = H iβ + 2 g (cid:16) h δ h iβ − h β h iδ (cid:17) u δ (1 − u )˜Γ ̺βi = (cid:16) ˜ H − (cid:17) ηi H jη ˆΓ ̺βj Ultimately, reintroducing the factors − u ) when needed in Eq. (8), U ˆ HU − can be written in a compact form : U ˆ HU − = 12 α β ˜ H iβ P i + ~ ε ̺βγ ˜Γ ̺βi γ ! + 12 P i + ~ ε ̺βγ ˜Γ ̺βi γ ! ˜ H iβ α β + g h (cid:0) u (cid:1) (1 − u ) βm + 12 g g i P i + 12 P i g g i + (cid:16) ˜Γ γ + Γ e (cid:17) . Σ − ~ (cid:0) g (cid:0) h × h i (cid:1) + g h u × H i (cid:1) (1 − u ) . ( ∇ i u ) J where we have defined ˜Γ γ by :˜Γ γ = (cid:0) u (cid:1) δ ηγ − u η u γ (1 − u ) ˆΓ η + ~ (cid:18)(cid:18) H iδ ˆΓ δi + Γ − g g i Γ i (cid:19) × u (cid:19) γ Γ eγ = − ~ (cid:0) u × H i (cid:1) γ ∇ R i (cid:0) f (cid:0) − u (cid:1)(cid:1) f (1 − u ) ! + ~ (cid:0) u ×∇ i (cid:0) H i (cid:1)(cid:1) γ (1 − u ) − ~ (cid:0) u (cid:1) δ ηγ − u η u γ (1 − u ) ∇ R i (cid:16) g (cid:0) h × h i (cid:1) η (cid:17) Non Hermitian contributions of the time evolution operator
We compute here the contributions to the diagonalization due to the non-hermitian term − i ~ U ( t ) ∂∂t U − ( t ) .First we obain − i ~ U ( t ) ∂∂t U − ( t ) = i ~ ∂∂t (cid:2) f (cid:0) − u (cid:1)(cid:3) f (1 − u ) + i ~ (cid:0) u β ∂∂t u β (cid:1) (1 − u ) + i ~ ∂∂t u β α β (1 − u ) − ~ (cid:0) u × ∂∂t u (cid:1) . Σ (1 − u ) (37)and then, the contributions of the terms in Eq.(37) to the diagonalization are computed by applying the transfor-mation matrix and then projecting on the diagonal blocks. We are left with: U (cid:16) ˜P (cid:17) (cid:20) − i ~ U ( t ) ∂∂t U − ( t ) (cid:21) U (cid:16) ˜P (cid:17) + = i ~ ∂∂t (cid:2) f (cid:0) − u (cid:1)(cid:3) f (1 − u ) + i ~ (cid:0) u . ∂∂t u (cid:1) (1 − u ) + i ~ cβ ∂∂t u . ˜P (1 − u ) E − ~ c (cid:16) ˜P × (cid:16) ˜P × (cid:0) u × ∂∂t u (cid:1)(cid:17) . Σ (cid:17) (1 − u ) E ( E + ˜ mm