---
abstract: |
    The Schr" odinger picture of the Dirac quantum mechanics is defined in charts with spatially flat Robertson-Walker metrics and Cartesian coordinates. The main observables of this picture are identified, including the interacting part of the Hamiltonian operator produced by the minimal coupling with the gravitational field. It is shown that in this approach new Dirac quantum modes on de Sitter spacetimes may be found analytically solving the Dirac equation.

    Pacs: 04.62.+v
author:
- |
    Ion I. Cotăescu [^1]\
    [*West University of Timişoara,*]{}\
    [*V. P\^ arvan Ave. 4, RO-300223 Timişoara, Romania*]{}
title: 'The Schr" odinger picture of the Dirac quantum mechanics on spatially flat Robertson-Walker backgrounds'
---

The relativistic quantum mechanics of the spin-half particle on a given background can be constructed as the one-particle restriction of the quantum theory of the free Dirac field on this background, considered as a perturbation that does not affect the geometry. The central piece is the Dirac equation whose form depends on the local chart (or natural frame) and the tetrad fields defining the local frames and co-frames. This type of quantum mechanics has two virtues. First of all the charge conjugation of the Dirac field is point-independent indicating that the vacuum of the original field theory is stable in any geometry [@co; @cot]. Therefore, the resulted one-particle Dirac quantum mechanics can be seen as a coherent theory similar to that of special relativity. The second virtue is just the spin which generates specific terms helping us to correctly interpret the physical meaning of principal operators.

In the non-relativistic quantum mechanics the time evolution can be studied in different pictures (e. g., Schr" odinger, Heisenberg, Interaction) which transform among themselves through specific time-dependent unitary transformations. It is known that the form of the Hamiltonian operator and the time dependence of other operators strongly depend on the picture choice. In special and general relativity, despite of its importance, the problem of time-evolution pictures is less studied because of the difficulties in finding suitable Hamiltonian operators for scalar or vector fields. However, the Dirac quantum mechanics is a convenient framework for studying this problem since the Dirac equation can be put in Hamiltonian form at any time.

In this paper we should like to show that at least two different pictures of the Dirac quantum mechanics can be identified in the case of backgrounds with spatially flat Robertson-Walker (RW) metrics. We start with the simple conjecture of the Dirac equation in diagonal gauge and Cartesian coordinates considering that this constitutes the [*natural*]{} picture. Furthermore, we define the Schr" odinger picture such that the kinetic part of the Dirac equation should take the standard form known from special relativity. In this picture we identify the momentum and the Hamiltonian operators pointing out that they represent a generalization of the similar operators we obtained previously on de Sitter spacetimes [@cot].

Let us start denoting by $\{t,\vec{x}\}$ the Cartesian coordinates $x^{\mu}$ ($\mu,\nu,...=0,1,2,3 $) of a chart with the RW line element $$ds^2=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}=dt^2-\alpha(t)^2 (d\vec{x}\cdot d\vec{x})$$ where $\alpha$ is an arbitrary time dependent function. In this chart we introduce the tetrad fields $e_{\hat\mu}(x)$ that define the local frames and those defining the corresponding coframes, $\hat e^{\hat\mu}(x)$ [@SW]. These fields are labeled by the local indices ($\hat\mu,\hat\nu,...=0,1,2,3$) of the Minkowski metric $\eta=$diag$(1,-1,-1,-1)$, satisfy $e_{\hat\mu}(x)\hat
e^{\hat\mu}(x)=1_{4\times4}$ and give the metric tensor as $g_{\mu
\nu}=\eta_{\hat\alpha\hat\beta}\hat e^{\hat\alpha}_{\mu}\hat
e^{\hat\beta}_{\nu}$. Here we consider the tetrad fields of the diagonal gauge that have non-vanishing components [@BD; @SHI], $$\label{tt}
e^{0}_{0}=1\,, \quad e^{i}_{j}=\frac{1}{\alpha(t)}\delta^{i}_{j}\,,\quad \hat
e^{0}_{0}=1\,, \quad \hat e^{i}_{j}=\alpha(t)\delta^{i}_{j}\,,\quad
i,j,...=1,2,3\,,$$ determining the form of the Dirac equation [@BD], $$\label{ED1}
\left(i\gamma^0\partial_{t}+i\frac{1}{\alpha(t)}\gamma^i\partial_i
+\frac{3i}{2}\frac{\dot{\alpha}(t)}{\alpha(t)}\gamma^{0}-m\right)\psi(x)=0\,.$$ This is expressed in terms of Dirac $\gamma$-matrices [@TH] and the fermion mass $m$, with the notation $\dot{\alpha}(t)=\partial_t\alpha(t)$. Thus we obtain the natural picture in which the time evolution is governed by the Dirac equation (\[ED1\]). The principal operators of this picture, the energy $\hat
H$, momentum $\vec{\hat P}$ and coordinate $\vec{\hat X}$, can be defined as in special relativity, $$\label{ON}
(\hat H \psi)(x)=i\partial_t\psi(x)\,,\quad (\hat P^i
\psi)(x)=-i\partial_i\psi_S(x)\,,\quad (\hat X^i \psi)(x)=x^i\psi(x)\,.$$ The operators $\hat X^i$ and $\hat P^i$ are time-independent and satisfy the well-known canonical commutation relations $$\label{com}
\left[\hat X^i, \hat P^j\right]=i\delta_{ij}I\,,\quad \left[\hat H, \hat
X^i\right]=\left[\hat H,\hat P^i\right]=0\,,$$ where $I$ is the identity operator. Other operators are formed by orbital parts and suitable spin parts that can be point-dependent too. In general, the orbital terms are freely generated by the basic orbital operators $\hat X^i$ and $\hat P^i$. An example is the total angular momentum $\vec{J}=\vec{L}+\vec{S}$ where $\vec{L}=\vec{\hat X}\times\vec{\hat P}$ and $\vec{S}$ is the spin operator. We specify that the operators $\hat P^i$ and $J^i$ are generators of the spinor representation of the isometry group $E(3)$ of the spatially flat RW manifolds [@cot]. Therefore, these operators are [*conserved*]{} in the sense that they commute with the Dirac operator [@CML; @ES].

The natural picture can be changed using point-dependent operators which could be even non-unitary operators since the relativistic scalar product does not have a direct physical meaning as that of the non-relativistic quantum mechanics. We exploit this opportunity for defining the Schr" odinger picture as the picture in which the kinetic part of the Dirac operator takes the standard form $i\gamma^0\partial_t+i\gamma^i\partial_i$. The transformation $\psi(x)\to \psi_S(x)=U_S(x)\psi(x)$ leading to the Schr" odinger picture is produced by the operator of time dependent [*dilatations*]{} $$\label{U}
U_S(x)=\exp\left[-\ln(\alpha(t))(\vec{x}\cdot\vec{\partial})\right]\,,$$ which has the following suitable action $$U_S(x)F(\vec{x})U_S(x)^{-1}=F\left(\frac{1}{\alpha(t)}\vec{x}\right)\,,\quad
U_S(x)G(\vec{\partial})U_S(x)^{-1}=G\left(\alpha(t)\vec{\partial}\right)\,,$$ upon any analytical functions $F$ and $G$. Performing this transformation we obtain the Dirac equation of the Schr" odinger picture $$\label{ED2}
\left[i\gamma^0\partial_{t}+i\vec{\gamma}\cdot\vec{\partial} -m
+i\gamma^{0}\frac{\dot{\alpha}(t)}{\alpha(t)}
\left(\vec{x}\cdot\vec{\partial}+\frac{3}{2}\right)\right]\psi_S(x)=0\,.$$ Hereby we have to identify the specific operators of this picture, the energy $H_S$ and the operators $P^i_S$ and $X^i_S$ that must be time-independent, as in the non-relativistic case. We assume that these operators are defined as $$\label{OS}
(H_S \psi_S)(x)=i\partial_t\psi_S(x)\,,\quad (P^i_S
\psi_S)(x)=-i\partial_i\psi_S(x)\,,\quad (X^i_S \psi_S)(x)=x^i\psi_S(x)\,,$$ obeying commutation relations similar to Eqs. (\[com\]). The Dirac equation (\[ED2\]) can be put in Hamiltonian form, $H_S\psi_S={\cal H}_S\psi_S$, where the Hamiltonian operator ${\cal H}_S={\cal H}_0 + {\cal H}_{int}$ has the standard kinetic term ${\cal H}_0=\gamma^0\vec{\gamma}\cdot \vec{P}_S+\gamma^0
m$ and the interaction term with the gravitational field, $$\label{Hint}
{\cal H}_{int}=\frac{\dot{\alpha}(t)}{\alpha(t)}\left(\vec{X}_S\cdot
\vec{P}_S-\frac{3i}{2}I\right)=\frac{\dot{\alpha}(t)}{\alpha(t)}\left(\vec{\hat
X}\cdot \vec{\hat P}-\frac{3i}{2}I\right)\,,$$ which vanishes in the absence of gravitation when $\alpha$ reduces to a constant.

The sets of operators (\[OS\]) and (\[ON\]) are defined in different manners such that they have similar expressions but in different pictures. For analyzing the relations among these operators it is convenient to turn back to the natural picture. Performing the inverse transformation we find that in this picture the operators (\[OS\]) become new interesting time-dependent operators, $$\begin{aligned}
H(t)&=&U_S(x)^{-1}H_SU_S(x)=\hat H+\frac{\dot{\alpha}(t)}{\alpha(t)}
\vec{\hat X}\cdot\vec{\hat P}\,,\label{Ht}\\
X^i(t)&=&U_S(x)^{-1}X_S^iU_S(x)=\alpha(t) \hat X^i\,,\\
P^i(t)&=&U_S(x)^{-1}P_S^iU_S(x)=\frac{1}{\alpha(t)}\hat P^i\,,\label{Pt}\end{aligned}$$ satisfying the usual commutation relations (\[com\]). Notice that the total angular momentum and the operator (\[Hint\]) have the same expressions in both these pictures since they commute with $U_S(x)$.

Now the problem is to select the set of operators with a good physical meaning. We observe that in the moving charts with RW metrics of the de Sitter spacetime the operator (\[Ht\]) is time-independent (since $\dot{\alpha}/\alpha
=$const.) and [*conserved*]{}, corresponding to the unique time-like Killing vector of the $SO(4,1)$ isometries [@cot]. This is an argument indicating that the correct physical observables are the operators (\[Ht\])-(\[Pt\]) while the operators (\[ON\]) may be considered as auxiliary ones. In fact these are just the usual operators of the relativistic quantum mechanics on Minkowski spacetime where is no gravitation. The examples we worked out [@cot; @C; @C1] convinced us that this is the most plausible interpretation even though this is not in accordance with other attempts [@Guo].

The Schr" odinger picture we defined above may offer one some technical advantages in solving 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 ${\cal
H}_0$ directly by the Schr" odinger kinetic term $\frac{1}{2m}{\vec{P}_S}^2$. Thus we obtain the Schr" odinger equation $$\label{Sc}
\left[-\frac{1}{2m}\Delta -i\,
\frac{\dot{\alpha}(t)}{\alpha(t)}\left(\vec{x}\cdot \vec{\partial
}+\frac{3}{2}\right)\right]\phi(x)=i\partial_t \phi(x)\,,$$ for the wave-function $\phi$ of a spinless particle of mass $m$. Moreover, using standard methods one can derive the next approximations in $1/c$ producing characteristic spin terms. It is remarkable that the non-relativistic Hamiltonian operators obtained in this way are Hermitian with respect to the usual non-relativistic scalar product.

In the particular case of the de Sitter spacetime, the Schr" odinger picture will lead to important new results for the Dirac and Schr" odinger equations in moving charts with RW metrics and spherical coordinates. In these charts where $H=H(t)$ is conserved both the mentioned equations, namely Eqs. (\[ED2\]) and (\[Sc\]), are analytically solvable in terms of Gauss hypergeometric functions and, respectively, Whittaker ones [@coco]. Therefore, in this picture it appears the opportunity of deriving new Dirac quantum modes determined by the set of commuting operators $\{H, {\vec{J}\,}^2,
J_3, K\}$ where $K=\gamma^0(2\vec{L}\cdot\vec{S}+1)$ is the Dirac angular operator. We specify that common eigenspinors of this set of operators were written down in [*static*]{} central charts with spherical coordinates [@C] but never in moving charts. We remind the reader that in moving charts with spherical coordinates one knows only the Shishkin’s solutions of the Dirac equation [@SHI] derived in the natural picture. We have shown [@C1] that there are suitable linear combinations of these solutions representing common eigenspinors of the set $\{{\vec{\hat P}\,}^2,{\vec{J}\,}^2, J_3, K\}$. Taking into account that the operators $H$ and ${\vec{\hat P}\,}^2$ do not commute with each other we understand the importance of the new quantum modes that could be showed off grace to our Schr" odinger picture.

Finally we note that the quantum mechanics developed here is a specific approach working only in spatially flat RW geometries. This may be completed with new pictures (as the Heisenberg one) after we shall find the general mechanisms of time evolution in the relativistic quantum mechanics. However, now it is premature to look for general principles before to carefully analyze other significant particular examples.

Acknowledgments {#acknowledgments .unnumbered}
---------------

We are grateful to Erhardt Papp for interesting and useful discussions on closely related subjects.

[20]{}

I. I. Cotăescu, [*Int. J. Mod. Phys. A*]{} [**19**]{}, 2117 (2004).

I. I. Cotăescu, [*Phys. Rev. D*]{} [**65**]{}, 084008 (2002).

S. Weinberg, [*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*]{} (Wiley, New York, 1972).

A. O. Barut and I. H. Duru, [*Phys. Rev. D*]{} [**36**]{}, 3705 (1987).

G. V. Shishkin, [*Class. Quantum Grav.*]{} [**8**]{}, 175 (1991).

B. Thaller, [*The Dirac Equation*]{}, Springer Verlag, Berlin Heidelberg, 1992

B. Carter and R. G. McLenaghan, [*Phys. Rev. D*]{} [**19**]{}, 1093 (1979).

I. I. Cotăescu, [*J. Phys. A: Math. Gen.*]{} [**33**]{}, 1977 (2000).

I. I. Cotăescu, [*Mod. Phys. Lett. A*]{} [**13**]{}, 2991 (1998).

I. I. Cotăescu, Radu Racoceanu and Cosmin Crucean, Mod. Phys. Lett. A [**21**]{}, 1313 (2006).

H.-Y. Guo, C.-G. Huang, Z. Xu and B. Zhou, [*Mod. Phys. Lett. A*]{} [**19**]{}, 1701 (2004); id. [hep-th/0311156]{}; E. A. Tigrov, [gr-qc/0011011]{}.

I. I. Cotăescu, in preparation.

[^1]: E-mail:   cota@physics.uvt.ro
