Dirac Particles in a Gravitational Field

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Preprint submitted on 10 Sep 2010

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Pierre Gosselin, Herve Mohrbach

To cite this version:

Pierre Gosselin, Herve Mohrbach. Dirac Particles in a Gravitational Field. 2010. �hal-00516606�

Pierre Gosselin¹ and Herv´e Mohrbach²

1Institut Fourier, UMR 5582 CNRS-UJF, UFR de Math´ematiques, Universit´e Grenoble I, BP74, 38402 Saint Martin d’H`eres, Cedex, France and

2Groupe 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. In particular we find a new coupling term between the linear momentum and the spin, as well as couplings 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 Hamiltonian formulation. To reveal the physical content of the theory it is necessary to perform the diagonalization of the Hamiltonian uncoupling the positive and the negative energy states. For a fermion interacting with an electromagnetic field the Foldy-Wouthuysen (FW) transformation based on an approximate scheme valid in the non relativistic limit is often used [1]. This same method was also applied in all the previous studies of Dirac fermions in a gravitational field [2]. Here instead, we will consider the fully relativistic regime but in a semiclassical approximation for which the de Broglie wave length of the fermion must be much smaller that the characteristic size of the inhomogeneities of the external field. A recent semiclassical FW-like transformation used for Dirac particle in a strong electromagnetic field could be adapted to the gravitational problem [3]. But instead we will use another method developed by the authors which essentially differs from the FW. This method allows us to find the diagonal representation of any kind of matrix valued quantum Hamiltonian as a series expansion in the Planck constant. Here we will directly apply the general formula obtained at the semiclassical (first order in the Planck constant) limit to the case of Dirac fermions in an arbitrary curved spacetime. This is an extension of previous papers where massless and massive particles in a static gravitational fields were treated. The extension to time dependent metrics turns out to be non-trivial and leads to 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 massmcoupled to an arbitrary gravitational field is described by 4−spinor field ψ satisfying the covariant Dirac equation

(i~γ^αDα−m)ψ= 0 (1)

where we usec= 1, but keep explicit the Planck constant~and α= 0,1,2,3.

The covariant spinor derivative is defined asDα =hⁱ_αDi withDi =∂i−¹₄ γ^α, γ^β

Γ^αβ_i . The matrices γ^α are the usual Dirac matrices,hⁱ_α are the the orthonormal vierbein and Γ^αβ_i the spin connection components.

Rewriting Eq. (1) under the Schr¨odinger form

i~∂ψ

∂t =Hψ we obtain the following Hamiltonian

H=g00h⁰_βγ^β γ^αP¯α+m +~

4ε̺βγΓ^̺β₀ Σ^γ+i~

4Γ^0β₀ αβ (2)

where we introduced the notation for the pseudo-momentum ¯Pα=hⁱ_α(Pi+ ^~₄ε̺βγΓ^̺β_i Σ^γ) with the spin matrix Σ^γ, satisfying the relation ε^γαβΣγ = ⁱ₈(γ^αγ^β−γ^βγ^α). We use the conventions of Bjorken and Drell [4] for the Dirac matricesγ^α,andα^β.

Surprisingly, Eq. (2) turns out to be non-hermitian for a time dependent metric. We then follow the approach of Leclerc [5] who showed that one must add the term ₂ⁱ∂tln −gg⁰⁰

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 be Hˆ =g00h⁰_βγ^β γ^αP¯α+m

+~

4ε̺βγΓ^̺β₀ Σ^γ+i~

4Γ^0β₀ α^β+ i

2∂tln −gg⁰⁰

(3) The goal of this paper is therefore to diagonalise Eq. (3) to first order in~. But before embarking into this, we need first to discuss the definitions and properties of the scalar product in a curved space-time.

Scalar product.

As said before the Hamiltonian ˆH is Hermitian. However, the notion of hermiticity is here defined with respect to a scalar product in curved space [5], namely :

hψ1(t)|ψ2(t)iU(t)= Z

ψ₁⁺√

−gh⁰_βγ^βγ⁰ψ2d³x= Z

ψ₁⁺U ψ2d³x (4) where we introduced the notation U =√

−gh⁰_µγ^µγ⁰ and ψ1, ψ2 two spinors. Therefore an operator O is hermitian forh|iU(t)defined in Eq. (4), if

Z ψ₁⁺√

−gh⁰_βγ^βγ⁰(Oψ2) = Z

(Oψ1)⁺√

−gh⁰_βγ^βγ⁰(ψ2) (5) It means that matricially

O⁺U =U O (6)

where ”+” denotes from now, the usual Hermitic conjugation (transposition and complex conjugation), that is, the hermitic conjugate with respect to the scalar product in flat space denotedh|iand defined by

hψ1(t)|ψ2(t)i= Z

ψ⁺₁ψ2d³x (7)

Unfortunatly the definition Eq. (5) turns out to be untractable for pratical computations. Actually, for the sake of the diagonalization procedure, we aim at working with matrices which are Hermitian with respect to the usual transpose and complex conjugate operation Eq. (7), so that the diagonalization can be performed through a unitary matrix in the usual sense. To do so, notice that ifO is Hermitian for Eq. (4), then Eq. (6) implies thatU¹²OU⁻¹² is Hermitian for Eq. (7). Thus, starting with the Hamiltonian ˆH defined in Eq. (3),U¹²HUˆ ⁻¹² is Hermitian in the usual sense and can be diagonalized through a standard unitary matrix (that is unitary for (7)).

The Hamiltonian of interest for us will thus beU¹²HUˆ ⁻¹². It’s hermiticity in the usual sense allows us to write:

U¹²HUˆ ⁻¹² = 1

2U¹²HUˆ ⁻¹² +1 2

U¹²HUˆ ⁻¹²+

= 1 2

Hˆ + ˆH⁺ +1

2 h

U¹²,Hˆi

U⁻¹² +1 2U⁻¹²h

H, Uˆ ¹²i

(8) The non unitarity of the transformationU is not problematic here, since it is precisely used to change the metric from curved to flat scalar product, and moreover ˆH andU¹²HUˆ ⁻¹² have the same spectrum. In the case of a static metric (time independent) and satisfyingh⁰_µ=f(R)δ⁰_µ, for a certain position dependent functionf(R), the transformation U¹² reduces to the multiplication by a function ofR. Then, using Eq. (3) for ˆH, Eq. (8) simplifies easily to:

U¹²HUˆ ⁻¹² =1

2U¹²HUˆ ⁻¹²+1 2

U¹²HUˆ ⁻¹²+

= 1 2

Hˆ + ˆH⁺

It is this form that was considered in [6][7]. If in addition the metrics is diagonal, one recovers the transformation studied in [2].

Independently of the practical advantages of the flat scalar product, there is an other and deeper reason to transform the Hamiltonian to the flat space. Actually, when diagonalizing the Hamiltonian with respect to Eq. (7) the diagonal subspaces of up and down spinors will appear to be obviously orthogonal. This is of course not the case for the scalar product Eq. (4) which mixes both subspaces. As a consequence diagonalizing with respect to the curved space scalar product 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 evolution operator do not coincide in the flat representation, and in addition, the time evolution operator cannot be made Hermitian. To make this point clearer, consider the Schr¨odinger equation

i~∂

∂tΨ =HΨ Applying the transformationU¹² yields,

∂

∂tΨ^′=

U¹² (t)HU⁻¹²(t)−i~U¹²(t) ∂

∂tU⁻¹² (t)

Ψ^′ with Ψ^′=U¹²Ψ. As a consequence the evolution for Ψ^′, is given by the operator

H^e=U¹²(t)HU⁻¹²(t)−i~U¹²(t) ∂

∂tU⁻¹²(t) (9)

One would likeH^e to be hermitian for the scalar product Eq. (7). However, due to the non unitarity of U¹², the contribution i~U¹²(t)_∂t^∂ U⁻¹²(t) is not. The reason for this non unitarity tracks back to the dependence in t of the scalar product Eq. (4), so that the norm of a wave function is not preserved in time. Actually starting with an initial condition Ψ^′(t0), the solution for the Schroedinger equation is

Ψ^′(t1) =Texp Z t1

t0

H^e(t)dt

Ψ^′(t0) =U¹²(t1)Texp Z t1

t0

H(t)dt

U⁻¹²(t0) Ψ^′(t0)

whereT is the time ordered exponential. Assuming the norm Ψ^′(t0) to be equal to 1, Ψ^′(t1) is easily seen to have a norm (respectively to Eq. (4)) different from one. That can be checked easily on an infinitesimal timeslice,t1=t0+∆t.

Indeed

hΨ^′(t1)| |Ψ^′(t1)i=D

U¹²(t1) exp (−iH(t0) ∆t) Ψ (t0)|U¹²(t1) exp (iH(t0) ∆t) Ψ (t0)E

=hexp (−iH(t0) ∆t) Ψ (t0)|U(t1) exp (iH(t0) ∆t) Ψ (t0)i

=hexp (−iH(t0) ∆t) Ψ (t0)|exp (iH(t0) ∆t) Ψ (t0)iU(t1) (10) and this is different from 1 since,

hexp (−iH(t0) ∆t) Ψ (t0)|exp (iH(t0) ∆t) Ψ (t0)iU(t1)

=hexp (−iH(t0) ∆t) Ψ (t0)|U(t0) exp (iH(t0) ∆t) Ψ (t0)i+

Ψ (t0)| ∂

∂tU(t0) ∆tΨ (t0)

(11)

=hexp (−iH(t0) ∆t) Ψ (t0)|exp (iH(t0) ∆t) Ψ (t0)iU(t0)+

Ψ (t0)| ∂

∂tU(t0) ∆tΨ (t0)

(12) the first term is equal to 1, actually Ψ (t0) is of norm 1 forh|iU(t0)and exp (iH(t0) ∆t) is unitary for this scalar product.

As a consequence, hΨ^′(t1)| |Ψ^′(t1)i differs from one and H^e is non unitary. The reason is clear from Eq. (10): a vector of norm 1 forh|iU(t0)is transported to a vector, that has no more norm 1 forh|iU(t1). During the evolution, the matrixU defining the scalar product has changed too, and the non hermitian connexion term −i~U¹²(t)_∂t^∂U⁻¹²(t) tracks the change of metric betweent andt+ ∆t.

Therefore, the time evolution of the state is non-unitary. In the rest of the paper we focus on the diagonalization of the energy operatorU¹²(t)HU⁻¹²(t) this one being Hermitian, although the diagonalization of −i~U¹²(t)_∂t^∂U⁻¹²(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 explicitlyU¹²HUˆ ⁻¹² which as shown is hermitian in the usual sense. The transformationU is given by

U =√

−g h⁰₀+h⁰_βα^β

=√

−gh⁰₀ 1 +α^βh⁰_β/h⁰₀

One can then deduce

U¹² =f 1 +uβα^β and thus

U⁻¹² = 1

f(1−u²) 1−uβα^β whereu²=uβu^β with

uβ= h⁰_β/h⁰₀

√−gh⁰₀

1 + r

1−^h

0 βh^0β

(^h00)² .

and

f =





√

−gh⁰₀ 1 +

r 1−^h

0 βh^0β

(^h00)² 2







1/2

The greek (lorentzian) indices are assumed now to run only from 1 to 3. As a consequence the Hamiltonian H = U¹²HUˆ ⁻¹² given by Eq. (8) reads

H = 1

2 H+H⁺ +1

2f 1 +uβα^β

H, 1

f(1−u²) 1−uβα^β

−1 2

H⁺, 1−uβα^β 1 f(1−u²)

1 +uβα^β f

which can be rewritten as H =1

2 H+H⁺

−1 2

1 +uβα^β (1−u²)

H, uβα^β +1

2

H⁺, uβα^β 1 +uβα^β (1−u²)

−i~

2 1 +uβα^β

(∇^PH).∇^R

1 f(1−u²)

1−uβα^β +i~

2 1−uβα^β

∇^PH⁺ .∇^R

1 f(1−u²)

1 +uβα^β wherePandRare the canonical momentum and position operator satisfying

Rⁱ, Pj

=i~δ_jⁱ.The Hamiltonian can also be written as

H =1 2

1

(1−u²)H+H⁺ 1 (1−u²)

−1 2

1 (1−u²)

H, uβα^β +1

2

H⁺, uβα^β 1 (1−u²)

−1 2uβα^β

1

(1−u²)H+H⁺ 1 (1−u²)

uβα^β−i~ 2

uβα^β,(∇PH).∇R

1 f(1−u²)

(13) At this level, the computation of the last expression turns out to be quite technical is fully developped in Appendix A. The result is given by the following expression

H= 1

2α^βH˜_βⁱ Pi+~ε̺βγ

Γ˜^̺β_i 4 Σ^γ

! +1

2 Pi+~ε̺βγ

Γ˜^̺β_i 4 Σ^γ

! H˜_βⁱα^β

+βm˜ +1

2gⁱPi+Pigⁱ+~

Γ˜⁰+Γ^e

.Σ+~ g00 h⁰×hⁱ

−u×Hⁱ

(1−u²) .(∇iu)J (14) where from now on, all indicesi,β,ρ...are only spatial and run from 1 to 3, but roman indices are raised and lowered by the mericgij and greek indices by the lorentzian metricsηαβ.The several notations introduced above are given by

the following expressions H˜_βⁱ =H_βⁱ +

2g00

h⁰_δhⁱ_β−h⁰_βhⁱ_δ uδ

(1−u²) Γˆ^̺β_i = Γ^̺β_i −1

4ε^̺β_γ H⁻¹κ

i g00h⁰_δhⁱ_ηε^δη_κΓ^0γ_j Γ˜^̺β_i =

H˜⁻¹η i H_η^jΓˆ^̺β_j Γˆ⁰_γ =1

4

ε̺βγΓ^̺β₀ +g00g⁰ⁱε̺βγΓ^̺β_i +ε^β_νγH_βⁱΓ^0ν_i Γ˜⁰_γ = 1 +u²

δ_γ^η−u^ηuγ

(1−u²) Γˆ⁰_η+1 2

2H_δⁱΓˆ^δ_i+Γ⁰₀−g00g⁰ⁱΓ⁰_i

×u

γ

Γ^e_γ =− u×Hⁱ

γ

∇^Ri f 1−u² f²(1−u²)³

!

+ u×∇i Hⁱ

γ

(1−u²) − 1 +u²

δ^ηγ−u^ηuγ

(1−u²) ∇Ri

g00 h⁰×hⁱ

η

where we used vectorial notations h⁰

α=h⁰_α, hⁱ

α=hⁱ_α, as well as the vectors Hⁱ

β ≡H_βⁱ =g00h⁰₀ hⁱ_β−^h

0 βhⁱ₀ h⁰₀

, the following various vectors

Γ˜^δ_iβ

= ˜Γ^δβ_i , Γˆ^δ_iβ

= ˆΓ^δβ_i , Γ⁰_iβ

= Γ^0β_i ,and Γ⁰₀β

= Γ^0β₀ . The matrixJ is given by J =

0 I2×2

I2×2 0

and the effective mass by ˜m=mg00h⁰₀

1+u² 1−u²

as well asgⁱ=g00g⁰ⁱ.The expression Eq.(14) for the energy operatorH 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 with gauge fields resulting from the back reaction of the spin degree of freedom (fast) on the translational momentum which can be treated semiclassically for slowly varying enough inhomogeneities. Indeed, the emergence of gauge fields is a general feature of systems providing fast and slow degrees of freedom. The purpose of ref [9], was to investigate the origin of quantum gauge fields and forces by considering the diagonalization of an arbitrary matrix valued quantum Hamiltonian. To be precise, by diagonalization we mean the derivation of an effective in-band Hamiltonian made of block-diagonal energy subspaces. This approach, based on a new differential calculus on a non-commutative space where~plays the role of running parameter, leads to an in-band energy operator that can be obtained systematically up 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

Ri, P^j

= i~δ^j_i, to obtain the corresponding diagonal representation ε(r,p) to order ~², in terms of non-canonical coordinates and momentum (r,p) defined later and commutators between gauge fields. The method is quite involved, so that in the present paper we restrict ourself to the semiclassical approximation (order~).

The mathematical difficulty in performing the diagonalization of H comes from the intricate entanglement of noncommuting operators due to the canonical relation

Ri, P^j

=i~δ_i^j. In [8] starting with a very general but time independentH(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) (whereX≡(R,P)) to their classical expressions through integro- differential operators, i.e. ε(X) =Ob(ε(X0)) and V(X) =Nb(V(X0)), where in the matricesε(X0) andV(X0),the dynamical operatorsXare replaced by classical commuting variablesX0= (R0,P0).

The only requirement of the method is therefore the knowledge of V(X0) which gives the diagonal form ε(X0).

Generally, these equations do not allow to find directlyε(X),V (X), however, they allow us to produce the solution recursively in a series expansion in~.With this assumption that bothεandV can be expanded in power series of~, we determined in [8], the explicitn-th in band energy to order~for an arbitrary given Hamiltonian

εn(r,p) =ε0,n(r,p) +i~ 2 Pn

ε0(r,p),A^Rl

A^Pl−

ε0(r,p),A^Pl

A^Rl +O(~²) (15) whereAR=i~V∇PV⁺ andAP =−i~V∇RV⁺ withV the diagonalizing matrix. The operatorPn has the meaning of the projection on then-th energy subspace. The new non-canonical dynamical operatorsrandpdepend on gauge fields AR =Pn(A^R) and AP=Pn(A^P) similarly to electromagnetism, as we haver=i~∂p+AR andp=P+AP.

These gauge invariant quantities not only are emerging naturally but are also necessary to have a gauge invariant energy Eq. (15). The operator ε0(r,p) is the diagonal energy obtained at zero order (~⁰) in which the classical variableR0andP0 are replaced by new non-commuting operatorsr,p. Therfore the first step consists in finding the matrixV(X0) which diagonalizes the ”classical” HamiltonianH(X0).

Zero Order Diagonalization.

For practical purpose we introduce the three dimensional effective metric G^ij = ˜H_αⁱH˜_β^jδ^αβ as well as the gravity coupled momentum ˜Pα= ˜H_αⁱ(Pi+^~₄ε̺βγΓ˜^̺β_i 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 F0 for Foldy- Wouthuysen)

F0(P) =˜ D

E0+ ˜m+c1 2β

α.P˜+P˜⁺.α +N

/p

2E0(E0+ ˜m)

withE0=r

α.P+˜ P˜⁺.α 2

2

+ ˜m²,N =^~₄^iα.(^P˜×(^˜Γ0+~Γe))

E0 , andD= 1 + ^~₄β(^P˜×(^Γ˜0+~Γe))^×P˜

2E₀²(E0+ ˜m) . Indeed one can easily check that

F0HF0+=β q

PiG^ijPj+~εαβγΓ˜^αβ_j Σ^γG^ijPi+ ˜m²

+ ~

4E0

˜Γ⁰+Γ^e .



mΣ˜ +

Σ.H˜ⁱPi

H˜ⁱPi

(E0+ ˜m)



+1

2gⁱPi+1 2Pigⁱ +F0

g00 h⁰×hⁱ

−u×Hⁱ

(1−u²) .(∇ⁱu)J

!

F0+ (16)

with ˜Γiγ =ε̺βγΓ˜^̺β_i . All contributions are block-diagonal except the last term which was not present before in the case of a static metrics [7]. The proof of this block diagonalization relies on the simple fact that for classical variables X0,the matrices ˜H_αⁱ and ˜Γ^αβ_i are independent of both the momentum and position,β andα.P˜ anticommute and in the Taylor expansion ofE0 all terms commute withβ andα.P˜+P˜⁺.α.

As said before, the last term in Eq.(16) is non diagonal and must treated specifically. Actuallt, one can apply a second unitary transformation that will cancel the non diagonal contributions ofF0((^g00(^h0×hi)^−u×Hi)

(1−u²) .(∇iu)J)F0+

without affecting the rest of the diagonalized Hamiltonian to the first order in~. The explicit form of this transfor- mation is :

F₀^′ = 1− P⁻ F0

g00 h⁰×hⁱ

−u×Hⁱ

(1−u²) .(∇iu)JF0+

! β 2p

PiG^ijPj+ ˜m²

P⁻ being the projection outside the diagonal. One can check that F₀^′ −1 is antihermitian so that F₀^′ is unitary to the first order. As a consequence, the composition of the two unitary transformations yields the following diagonal energy operator :

ε0(R,P,t) =F₀^′F0Hˆ0F0+F₀^′+ =β q

PiG^ijPj+~εαβγΓ˜^αβ_j Σ^γG^ijPi+ ˜m² + ~

4E0

Γ˜⁰+Γ^e .



mΣ˜ +

Σ.H˜ⁱPi

H˜ⁱPi

(E0+ ˜m)



+1

2gⁱPi+1 2Pigⁱ +P+ F0

g00 h⁰×hⁱ

−u×Hⁱ

(1−u²) .(∇iu)J

! F0+

!

(17)

The last term is explicitely given by:

P+ F0

g00 h⁰×hⁱ

−u×Hⁱ

(1−u²) .(∇iu)J

! F0+

!

=DP⁺

E0+ ˜m+cβ

α.P˜ (E0+V(r)m)J+cβ

Σ.P˜~ g00 h⁰×hⁱ

−u×Hⁱ .(∇ⁱu) 2E0(E0+ ˜m) D⁺

=~cβ g00 h⁰×hⁱ

−u×Hⁱ .(∇iu)

2E0(1−u²) P.Σ˜ =~cβ (f−1)g00 h⁰×hⁱ .(∇iu) 2f E0(1−u²) P.Σ˜ Ultimately, the diagonalization process yields:

ε0(R,P,t) =F₀^′F0Hˆ0F0+F₀^′+

=β q

PiG^ijPj+~εαβγΓ˜^αβ_j Σ^γG^ijPi+ ˜m² +~

Γ˜⁰+Γ^e .



mΣ˜ +

Σ.H˜ⁱPi

H˜ⁱPi

(E0+ ˜m)



+1

2gⁱPi+1

2Pigⁱ+~cβ (f−1)g00 h⁰×hⁱ .(∇iu) 2f E0(1−u²) P.Σ˜

(18) where for the momentRandPare 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 subspaceε0,+. The semiclassical energy is given by Eq. (15), whereε0,+corresponds to the positive energy subspace Eq. (18) in which the classical variablesR,Pare replaced by the quantum covariant onesr=i~∂p+~AR andp=P+~AP.The explicit computation for the Berry connectionsAR=P⁺(A^R) andAP =P⁺(A^P) withA^R=P⁺(i~(F₀^′F0)∇^P(F₀^′F0)⁺) and A^P=P⁺(i~(F₀^′F0)∇^R(F₀^′F0)⁺) gives the components

ARk=c²ε^αβγH˜_γⁱP˜αΣβ

2E(E+ ˜m) gik+o(~) (19)

APk=−c²ε^αβγP˜αΣβ(∇RkP˜γ)

2E(E+ ˜m) +o(~) (20)

whereE is the same asE0above, but nowRis an operator and ˜H_γ^k is the inverse matrix of ˜H_βⁱ.

We also denote, for the rest of the paper,p˜to be the same expression asP˜ in whichRand Phave been replaced byrandp, namely:

˜

pα= ˜H_αⁱ(pi+~

4ε̺βγΓ˜^̺β_i σ^γ) To complete the diagonalization we need to evaluate the quantity

M+= i 2P+

nhε0(X),A^R0^l

iA^P0^l−h

ε0(X),A^P0^l

iA^R0^l

o+O(~²)

which being on all point similar to the one given in [6] or [7] is simply stated : M+= 1

E 1

2Σ−(ARטp)

.B− 1

2E∇m(r).˜ (˜p×Σ)

where the ”magnetotorsion field” B is defined through a three dimensional effective torsion tensor Bγ =

−¹2PδT^δαβεαβγ where the effective torsionT^δαβ is defined as T^αβδ= ˜H_k^δ

H˜^lα∂lH˜^kβ−H˜^lβ∂lH˜^kα

+ ˜H^lαΓ˜^βδ_l −H˜^lβΓ˜^αδ_l

The physical origin of the termM has been discussed in [6] and [7] for the static case. Note that for the static case (g⁰ⁱ = 0) we retrieve the true torsionT^αβδ=H_k^δ H^lα∂lH^kβ−H^lβ∂lH^kα

+H^lαΓ^βδ_l −H^lβΓ^αδ_l .

Considering similarly the anti-particle subspace, it turns out that the coordinate operators are identical with the particule ones and thatM⁻=−M+ therefore the full energy operator with both particles and anti particles can be cast in the form

ε(p,r) =βε+e 1 2gⁱpi+1

2pigⁱ+ ~ 4E

Γ˜⁰+Γ^e .



mΣ˜ +

Σ.H˜ⁱPi

H˜ⁱPi

(E+ ˜m)



+β~c (f−1)g00 h⁰×hⁱ .(∇iu)

2f E(1−u²) ˜p.Σ+β~M (21) whereM =M+ and

e ε=c

s pi+ ~

4EΓ˜i.

˜

mΣ+ (Σ.˜p)˜p (E+ ˜m)

G^ij

pi+ ~

4E˜Γi.

˜

mΣ+ (Σ.˜p)˜p (E+ ˜m)

+ ˜m² with the vectorΓ˜i defined in terms of its components as

Γ˜i

γ ≡εαβγΓ˜^αβ_i (r). Equation (21) is the main result of this paper.

The coordinates operators in Eq. (21) satisfy a non commutative algebra as

[ri, rj] =i~²Θ^rr_ij (22)

[pi, pj] =i~²Θ^pp_ij (23)

[pi, rj] =−i~gij+i~²Θ^pr_ij (24) where Θ^αβ_ij =∂αiAβj−∂βjAαi+ [Aαi,Aβj] the so-called Berry curvature. An explicit computation gives :

Θ^rr_kl = −1 2E³



mΣ˜ γ+

Σ^δP˜δ

P˜γ

(E+ ˜m)



ε^αβγH˜_αⁱH˜_β^jgikgjl

Θ^pp_ij = −1 2E³



mΣ˜ γ+ Σ^δP˜δ

P˜γ

(E+ ˜m)



∇^riP˜α∇^rjP˜βε^αβγ+ 1 2E³

h

∇^rim˜

Σ×P˜

.∇^rjP˜

− ∇^rjV(r)

Σ×P˜

.∇^riP˜i

Θ^pr_ij = 1 2E³



mΣ˜ γ+ Σ^δP˜δ

P˜γ

(E+ ˜m)



ε^αβγ∇^riP˜αH_β^lgjl− 1

2E³∇^rim˜

Σ×P˜

j (25)

There are also other Berry curvature mixing coordinates and spin Θ^rΣ_ij = [ri,Σj] =ic²−pjΣi+p.Σδij

E(E+ ˜m) Θ^pΣ_ij = [pi,Σj] =−ic²−pjΣl+p.Σδlj

E(E+ ˜m) H˜_l^γ∇rip˜γ (26) Interestingly Eq. (21) can be rewritten as

ε=cβ s

pi−~ 2Γ˜i.Θ˜^rr

G^ij

pi−~

2Γ˜i.Θ˜^rr

+ ˜m²(r) +1

2gⁱpi+1 2pigⁱ−~

2

Γ˜⁰+Γ^e .Θ˜^rr +β~c (f −1)g00 h⁰×hⁱ

.(∇iu)

2f E(1−u²) ˜p.Σ+β~M (27)

where ˜Θ^rr_γ = ⁻_2E¹

˜

mΣγ+(^ΣδP˜δ)^P˜γ

(E+ ˜m)

is the ”rescaled” Berry curvature. This formula clearly shows that the spin connection couples only to the Berry curvature.

We note in Eq. (27) the presence of the term −^~2

Γ˜⁰+Γ^e

.Θ˜^rr + ¹₂gⁱpi + ¹₂pigⁱ not proportional to β and independant of the particle charge, that is discrimnating between particles and anti particles. These terms give

different energy levels for the Dirac particles and antiparticles as ε+ 6= −ε⁻. The coupling ^−~₂

Γ˜⁰+Γ^e .Θ˜^rr proportional to the spin will survive in the case of the non-diagonal static gravitational field but cancels for a diagonal metrics as studied in [2]. On the other hand the term ¹₂gⁱpi+¹₂pigⁱ vanishes for a static metrics since in this case g⁰ⁱ = ∂⁰g^ij = 0 so that gⁱ = 0. Therefore the symmetry between particle and antiparticle is restablished only for static diagonal metrics.

THE STATIC GRAVITATIONAL FIELD

This case is caracterized by the following time independent metric: gij ≡gij(R),g00≡g00(R),gi0= 0. In that case expressions simplify greatly. Actually, the transformation matrixU is U =√

−gh⁰₀ and U¹² =p√

−gh⁰₀ =f. The effective quantities reduce to:

H˜ⁱ

β= ˜H_βⁱ =H_βⁱ =g00h⁰₀hⁱ_β=√g00hⁱ_β Γ˜^̺β_i = ˆΓ^̺β_i = Γ^̺β_i

Γˆ⁰_γ = ˜Γ⁰_γ =1 4

nε̺βγΓ^̺β₀ +ε^β_νγH_βⁱΓ^0ν_i o

=−1

4g00 h⁰₀2

ε^β_νγhⁱ_βh^ν_lh^λ_iΓ^kl_λ Γ^e= 0

G^ij =g00g^ij

˜

m=mg00h⁰₀ asu^β= 0

where Γ^kl_λ stands for the Christoffel symbol. In this case Eq. (21) reduces to

ε≃cβ s

pi+ ~

4EΓi(r).

e

mΣ+ (Σ.˜p)˜p (E+m)e

G^ij(r)

pi+ ~

4EΓi(r).

e

mΣ+ (Σ.˜p)˜p (E+m)e

+me²+~βM + ~

4EΓ˜⁰_γ

e

mΣ^γ+(Σ.˜p) ˜p^γ (E+m)e

(28) with ˜pα=√g00hⁱ_α(pi+ ^~₄ε̺βγΓ^̺β_i σ^γ) and the vectorΓi defined in terms of its components as (Γi)_γ ≡ εαβγΓ^αβ_i (r). Then even for this case, particles and antiparticles avec a different in-band energy operator because of term

~ 4EΓ˜⁰_γ

e

mΣ^γ+^(Σ.˜_(E+^{p) ˜}_m)e^pγ

breaking the symmetryε+6=−ε⁻.Although this term was already in previous studies [6][7]

this essential point has not been pointed out. For a diagonal metricΓ˜0= 0, and the symmetry particles/anti-particles is recovered.

ULTRARELATIVISTIC LIMIT

It is interesting to look at the ultrarelativistic limitmc²→0. One readily obtain ε≃βεe+cβ (f−1)g00 h⁰×hⁱ

.(∇ⁱu)

2f(1−u²) λ˜+βλg˜ 00

2E B.˜p

˜ p +λ˜

4

˜ p^γ

Γ˜⁰_γ+Γ^e_γ

˜

p +1

2gⁱpi+1

2pigⁱ (29) with εe= cr

pi+^λ˜₄^Γ˜i(r).p_p G^ij

pj+^˜λ₄^˜Γj(r).p_p

and ˜λ = ~p.Σ/˜ p˜ a biased helicity, that is not projected on the momentum pbut rather onp. This fact is not astonishing. Actually, as we saw in the diagonalization process, the˜ particle is submitted to the action of an effective gravitationnal field, which differs slightly from the initial field. This effective metric is responsible for considering the momentum ˜prather than as a dynamical variable P. Nicely this energy can be expressed in terms of the helicity which is the relevant variable for massless particles and not in term of Σ. As shown in [6], Eq. (29) is also valid for photon with the one-half spin matrixΣreplaced with spin one matrix S.

Here also we see that photons and anti-photons do not have the same energy spectrum. The symmetry is again only restored for a static diagonal metric.

THE TIME DEPENDENT SYMMETRIC GRAVITATIONAL FIELD

A typical example of such a metric is the Schwarzschild space-time in isotropic coordinates. This case, studied in a different manner in [2]and [3] for a time independent metric, received a full independent treatment within our formalism in [6][7]. We now present the equivalent results for a time dependent metric, completed with the spin matrix dynamics. For a symmetric metric, the semiclassical Hamiltonian has the following form [3]

H0=1

2(α.P F(R, t) +F(R, t)α.P) +βmV(R, t) (30) corresponding to the metric gij =δij

V(R,t) F(R)

2

, gi0= 0 and g00=V²(R, t). We will defineφ= ^V_F. In that context the relevant quantities for the diagonalization appear to be :

h^β_i =φδ^β_i, hⁱ_β= 1 φδ_βⁱ h^β₀ =V(R, t)δ₀^β, h⁰_β= 1

V(R, t)δ⁰_β and

Hⁱ

β=H_βⁱ =g00h⁰₀hⁱ_β =V hⁱ_β H˜ⁱ

β = ˜H_βⁱ =H_βⁱ Γ˜^̺β_i = ˆΓ^̺β_i = Γ^̺β_i =

∂^ρφh^β_i −∂^βφh^ρ_i

φ² Γ^0ν_i =~∂⁰φh^ν_i

φV Γ˜⁰_γ = ˆΓ⁰_γ= ~

4ε̺βγ F(R)

V²(R)∂⁰g^̺β+~g00h⁰₀ F(R) V²(R)

∂⁰

V(R) F(R)

ε^β_νγhⁱ_βh^ν_i = 0 Γ^e= 0

G^ij =V²g^ij

Similar computations to the ones performed in the previous section lead to the following expressions for the dynamical variables and the diagonalized Hamiltonian

r=R−~ F²(R, t)Σ×P

2E(E+mV(R)), p=P (31)

and the diagonal energy becomes:

ε=βp

F²(r,t)P²+P²F²(r,t) +mV²(r,t)−F³(r,t)

2E² m~β∇φ(r,t).(P×Σ) (32) The Berry curvatures are given by:

Θ^rr_ij =−~F³(r, t)ε^ijk 2E³

mφ(r, t)Σk+F(r, t) (Σ.P)Pk

E+mV(r)

(33) Θ^pr_ij =−~F³(r, t)

2E³ m∇ⁱφ(r) (Σ×P)_j (34)

Θ^pp_ij = 0 (35)

and Θ^rΣ_ij being unchanged, Θ^pΣ_ij = 0, andεe=p

F²(r,t)P²+P²F²(r,t) +mV²(r,t).

Note here that the magnetotorsion fieldB = 0. From Appendix B, we can also get the non Hermitian contributions the time evolution operator which in this case readsi~^∂

∂tlnf =¹_2∂t^∂ ln√

−gh⁰₀.

One can check, after developing r as a function of R and the Berry phase, that our Hamiltonian coincides, in the weak field approximation,with the one given in [3] when considering the semiclassical limit (order~). This also confirms the validity of the Foldy Wouthuysen approach asserted in [3].

ONLY TIME DEPENDENT METRIC The metric tensor and the vierbein only depend on time. Therefore we have

Hⁱ

β =H_βⁱ =g00h⁰₀ hⁱ_β−h⁰_βhⁱ₀ h⁰₀

!

H˜ⁱ

β = ˜H_βⁱ =H_βⁱ + 2g00

h⁰_δhⁱ_β−h⁰_βhⁱ_δ uδ

(1−u²) Γˆ^̺β_i = Γ^̺β_i −~ε^̺β_γ H⁻¹κ

i g00h⁰_δhⁱ_ηε^δη_κΓ^0γ_j 4 Γ˜^̺β_i =

H˜⁻¹η i H_η^jˆΓ^̺β_j Γˆ⁰_γ =1

4

ε̺βγΓ^̺β₀ +g00g⁰ⁱε̺βγΓ^̺β_i +ε^β_νγH_βⁱΓ^0ν_i

Γ˜⁰_γ = 1 +u²

δ_γ^η−u^ηuγ

(1−u²) Γˆ⁰_η+

H_δⁱΓˆ^δ_i+Γ⁰₀

2 −g00g⁰ⁱΓ⁰_i 2

×u

γ

Γ^e_γ = 0 so that the energy becomes

ε(p,r,t) =βε+˜ ~ 4EΓ˜⁰.



mΣ˜ +

Σ.H˜ⁱpi

H˜ⁱpi

(E+ ˜m)



+1

2gⁱ(t)pi+1

2pigⁱ(t)+β~M (36) with

˜ ε=c

s pi+ ~

4EΓi(r, t).

e

mΣ+ (Σ.˜p)p˜ (E+m)e

G^ij(r, t)

pi+ ~

4EΓi(r, t).

e

mΣ+ (Σ.˜p)p˜ (E+m)e

+me² 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 through a first order in~diagonalization of the Dirac Hamiltonian. This work extends previous ones where only static metrics were considered. The time dependence of the metrics leads to new contributions of the in-band energy operator. In particular we found a coupling term between the linear momentum and the spin, and terms which in general will break the particle - antiparticle symmetry.

As already found by other authors, the time dependence leads also to special features like the non-unitarity of the evolution operator, whose origin can be tracked back to the notion of scalar product in the Hilbert space of wave functions for a time dependent metric. This non-unitarity is unavoidable but we could nevertheless diagonalize the full evolution operator, even though our main focus was to obtain the block-diagonal form of the energy, this one turning out to be Hermitian. In addition, to the very general semiclassical diagonal energy operator, we provided several physically relevant examples.

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[2] Y. N. Obukhov, Phys. Rev. Lett86(2001) 192; Fortschr. Phys.50(2002) 711; Phys. Rev. Lett89(2002) 068903.

[3] A. J. Silenko and O. V. Teryaev, Phys. Rev. D71(2005) 064016.

[4] J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, New York: McGraw- Hill, (1965).

[5] M. Leclerc, Class. Quant. Grav.23(2006) 4013.