The Schr\" odinger picture of the Dirac quantum mechanics on spatially flat Robertson-Walker backgrounds
aa r X i v : . [ g r- q c ] A ug The Schr¨odinger picture of the Dirac quantum mechanics onspatially flat Robertson-Walker backgrounds
Ion I. Cot˘aescu
West University of Timi¸soara,V. Pˆarvan Ave. 4, RO-300223 Timi¸soara, Romania
Abstract
The Schr¨odinger picture of the Dirac quantum mechanics is defined in charts with spatially flatRobertson-Walker metrics and Cartesian coordinates. The main observables of this picture areidentified, including the interacting part of the Hamiltonian operator produced by the minimalcoupling with the gravitational field. It is shown that in this approach new Dirac quantum modeson de Sitter spacetimes may be found analytically solving the Dirac equation.Pacs: 04.62.+v natural picture. Fur-thermore, we define the Schr¨odinger picture such that the kinetic part of the Dirac equationshould take the standard form known from special relativity. In this picture we identify themomentum and the Hamiltonian operators pointing out that they represent a generalizationof the similar operators we obtained previously on de Sitter spacetimes [2].Let us start denoting by { t, ~x } the Cartesian coordinates x µ ( µ, ν, ... = 0 , , ,
3) of a chartwith the RW line element ds = g µν ( x ) dx µ dx ν = dt − α ( t ) ( d~x · d~x ) (1)where α is an arbitrary time dependent function. In this chart we introduce the tetrad2elds e ˆ µ ( x ) that define the local frames and those defining the corresponding coframes,ˆ e ˆ µ ( x ) [3]. These fields are labeled by the local indices (ˆ µ, ˆ ν, ... = 0 , , ,
3) of the Minkowskimetric η =diag(1 , − , − , − e ˆ µ ( x )ˆ e ˆ µ ( x ) = 1 × and give the metric tensor as g µν = η ˆ α ˆ β ˆ e ˆ αµ ˆ e ˆ βν . Here we consider the tetrad fields of the diagonal gauge that have non-vanishing components [4, 5], e = 1 , e ij = 1 α ( t ) δ ij , ˆ e = 1 , ˆ e ij = α ( t ) δ ij , i, j, ... = 1 , , , (2)determining the form of the Dirac equation [4], (cid:18) iγ ∂ t + i α ( t ) γ i ∂ i + 3 i α ( t ) α ( t ) γ − m (cid:19) ψ ( x ) = 0 . (3)This is expressed in terms of Dirac γ -matrices [6] and the fermion mass m , with the notation˙ α ( t ) = ∂ t α ( t ). Thus we obtain the natural picture in which the time evolution is governedby the Dirac equation (3). The principal operators of this picture, the energy ˆ H , momentum ~ ˆ P and coordinate ~ ˆ X , can be defined as in special relativity,( ˆ Hψ )( x ) = i∂ t ψ ( x ) , ( ˆ P i ψ )( x ) = − i∂ i ψ S ( x ) , ( ˆ X i ψ )( x ) = x i ψ ( x ) . (4)The operators ˆ X i and ˆ P i are time-independent and satisfy the well-known canonical com-mutation relations h ˆ X i , ˆ P j i = iδ ij I , h ˆ H, ˆ X i i = h ˆ H, ˆ P i i = 0 , (5)where I is the identity operator. Other operators are formed by orbital parts and suitablespin parts that can be point-dependent too. In general, the orbital terms are freely generatedby the basic orbital operators ˆ X i and ˆ P i . An example is the total angular momentum ~J = ~L + ~S where ~L = ~ ˆ X × ~ ˆ P and ~S is the spin operator. We specify that the operatorsˆ P i and J i are generators of the spinor representation of the isometry group E (3) of thespatially flat RW manifolds [2]. Therefore, these operators are conserved in the sense thatthey commute with the Dirac operator [7, 8].The natural picture can be changed using point-dependent operators which could be evennon-unitary operators since the relativistic scalar product does not have a direct physicalmeaning as that of the non-relativistic quantum mechanics. We exploit this opportunity fordefining the Schr¨odinger picture as the picture in which the kinetic part of the Dirac operatortakes the standard form iγ ∂ t + iγ i ∂ i . The transformation ψ ( x ) → ψ S ( x ) = U S ( x ) ψ ( x )3eading to the Schr¨odinger picture is produced by the operator of time dependent dilatations U S ( x ) = exp h − ln( α ( t ))( ~x · ~∂ ) i , (6)which has the following suitable action U S ( x ) F ( ~x ) U S ( x ) − = F (cid:18) α ( t ) ~x (cid:19) , U S ( x ) G ( ~∂ ) U S ( x ) − = G (cid:16) α ( t ) ~∂ (cid:17) , (7)upon any analytical functions F and G . Performing this transformation we obtain the Diracequation of the Schr¨odinger picture (cid:20) iγ ∂ t + i~γ · ~∂ − m + iγ ˙ α ( t ) α ( t ) (cid:18) ~x · ~∂ + 32 (cid:19)(cid:21) ψ S ( x ) = 0 . (8)Hereby we have to identify the specific operators of this picture, the energy H S and theoperators P iS and X iS that must be time-independent, as in the non-relativistic case. Weassume that these operators are defined as( H S ψ S )( x ) = i∂ t ψ S ( x ) , ( P iS ψ S )( x ) = − i∂ i ψ S ( x ) , ( X iS ψ S )( x ) = x i ψ S ( x ) , (9)obeying commutation relations similar to Eqs. (5). The Dirac equation (8) can be put inHamiltonian form, H S ψ S = H S ψ S , where the Hamiltonian operator H S = H + H int has thestandard kinetic term H = γ ~γ · ~P S + γ m and the interaction term with the gravitationalfield, H int = ˙ α ( t ) α ( t ) (cid:18) ~X S · ~P S − i I (cid:19) = ˙ α ( t ) α ( t ) (cid:18) ~ ˆ X · ~ ˆ P − i I (cid:19) , (10)which vanishes in the absence of gravitation when α reduces to a constant.The sets of operators (9) and (4) are defined in different manners such that they have sim-ilar expressions but in different pictures. For analyzing the relations among these operatorsit is convenient to turn back to the natural picture. Performing the inverse transforma-tion we find that in this picture the operators (9) become new interesting time-dependentoperators, H ( t ) = U S ( x ) − H S U S ( x ) = ˆ H + ˙ α ( t ) α ( t ) ~ ˆ X · ~ ˆ P , (11) X i ( t ) = U S ( x ) − X iS U S ( x ) = α ( t ) ˆ X i , (12) P i ( t ) = U S ( x ) − P iS U S ( x ) = 1 α ( t ) ˆ P i , (13)4atisfying the usual commutation relations (5). Notice that the total angular momentumand the operator (10) have the same expressions in both these pictures since they commutewith U S ( x ).Now the problem is to select the set of operators with a good physical meaning. Weobserve that in the moving charts with RW metrics of the de Sitter spacetime the operator(11) is time-independent (since ˙ α/α =const.) and conserved , corresponding to the uniquetime-like Killing vector of the SO (4 ,
1) isometries [2]. This is an argument indicating thatthe correct physical observables are the operators (11)-(13) while the operators (4) may beconsidered as auxiliary ones. In fact these are just the usual operators of the relativisticquantum mechanics on Minkowski spacetime where is no gravitation. The examples weworked out [2, 9, 10] convinced us that this is the most plausible interpretation even thoughthis is not in accordance with other attempts [11].The Schr¨odinger picture we defined above may offer one some technical advantages insolving problems of quantum systems interacting with the gravitational field. For example,in this picture we can derive the non-relativistic limit (in the sense of special relativity)replacing H directly by the Schr¨odinger kinetic term m ~P S . Thus we obtain the Schr¨odingerequation (cid:20) − m ∆ − i ˙ α ( t ) α ( t ) (cid:18) ~x · ~∂ + 32 (cid:19)(cid:21) φ ( x ) = i∂ t φ ( x ) , (14)for the wave-function φ of a spinless particle of mass m . Moreover, using standard methodsone can derive the next approximations in 1 /c producing characteristic spin terms. It is re-markable that the non-relativistic Hamiltonian operators obtained in this way are Hermitianwith respect to the usual non-relativistic scalar product.In the particular case of the de Sitter spacetime, the Schr¨odinger picture will lead toimportant new results for the Dirac and Schr¨odinger equations in moving charts with RWmetrics and spherical coordinates. In these charts where H = H ( t ) is conserved both thementioned equations, namely Eqs. (8) and (14), are analytically solvable in terms of Gausshypergeometric functions and, respectively, Whittaker ones [12]. Therefore, in this pictureit appears the opportunity of deriving new Dirac quantum modes determined by the setof commuting operators { H, ~J , J , K } where K = γ (2 ~L · ~S + 1) is the Dirac angularoperator. We specify that common eigenspinors of this set of operators were written downin static central charts with spherical coordinates [9] but never in moving charts. We remindthe reader that in moving charts with spherical coordinates one knows only the Shishkin’s5olutions of the Dirac equation [5] derived in the natural picture. We have shown [10] thatthere are suitable linear combinations of these solutions representing common eigenspinors ofthe set { ~ ˆ P , ~J , J , K } . Taking into account that the operators H and ~ ˆ P do not commutewith each other we understand the importance of the new quantum modes that could beshowed off grace to our Schr¨odinger picture.Finally we note that the quantum mechanics developed here is a specific approach work-ing only in spatially flat RW geometries. This may be completed with new pictures (asthe Heisenberg one) after we shall find the general mechanisms of time evolution in therelativistic quantum mechanics. However, now it is premature to look for general principlesbefore to carefully analyze other significant particular examples. Acknowledgments
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