Equivalent Forms of Dirac Equations in Curved Space-times and Generalized de Broglie Relations

Abstract

One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved space-time. This canonical form is needed to apply Whitham’s Lagrangian method. The latter method, unlike the Wentzel–Kramers–Brillouin method, places no restriction on the magnitude of Planck’s constant to obtain wave packets and furthermore preserves the symmetries of the Dirac Lagrangian. We show by using canonical Dirac fields in a curved space-time that the probability current has a Gordon decomposition into a convection current and a spin current and that the spin current vanishes in the Whitham approximation, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck’s constant. We further discuss the classical-quantum correspondence in a curved space-time based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved space-time are a direct consequence of Whitham’s Lagrangian method and not just a physical hypothesis as introduced by Einstein and de Broglie and by many quantum mechanics textbooks.

Appendix: Proof of Theorem 1

To prove Theorem 1 of Section 2, we will use the following corollary of a deep theorem of linear hyperbolic partial differential equations:

Theorem 0

Let M ₁ , M ₂ , ⋯ , M _n be complex d×d matrix functions that depend smoothly on n + 1 independent real variables t , x ₁ , x ₂ , ⋯ , x _n in a slab − T ≤ t ≤ T , x ∈ R ⁿ , denoted as I×R ⁿ . Suppose that M ₀ , M ₁ , M ₂ , ⋯ , M _n are Hermitian matrices and furthermore assume that M ₀ is positive definite. Let $F = F\left( S \right)$ be a homogeneous linear function of complex d×d matrices S , as well as having explicit dependence on t , x ₁ , x ₂ , ⋯ , x _n . Then the complex linear hyperbolic system

$$ M_{0} \frac{\partial S}{\partial t} + \sum _{j=1}^{n} M_{j} \frac{\partial S}{ \partial x_{j} } = F\left( S \right) \label{GrindEQ__73_} $$

(81)

has a smooth d×d complex matrix-valued solution $S {\bf :} {\bf I}\times {\bf R}^{{\rm n}} \to {\it M}\left( {\it C} {\it ,} {\it d} \right)$ which equals the identity matrix at t = 0 , as its prescribed smooth initial data.

Proof

Theorem 0 is Corollary 3 in Ref. [20], Sect. 6, which is based on a theorem of Lax [33] (see also Ref. [34]).□

Proof of Theorem 1 of Section 2

First note from (4) that the matrices B ^μ ≡ Aγ ^μ are Hermitian matrices. Then, note that the conditions stated in Theorem 1 for the metric components g _μν imply that B ⁰ ≡ Aγ ⁰ is a positive definite matrix by Theorem 6 of Ref. [22], Appendix B. By Theorem 3 of Ref. [21], Sect. 3.4, Eq. (54), a local similarity transformation T of the second kind, takes a generalized Dirac equation of the form (11) into a Dirac equation of the normal form (14), if and only if T satisfies the following partial differential equation:

$$ B^{\mu } D_{\mu } T= -\frac{1}{2} \left( D_{\mu } B^{\mu } \right) T . \label{GrindEQ__74_}$$

(82)

If we can solve (82) for such a local similarity transformation T , then the transformed coefficient fields will be as in (5) and (8):

$$\begin{array}{lll} \widetilde{\gamma }^{\mu } &=& T^{-1} \gamma ^{\mu } T, \\ \widetilde{A}\quad &=&\quad T^{+} A T, \\ \widetilde{\Gamma }_{\mu } &=& \Gamma _{\mu } . \end{array} $$

(83)

Now from (18), a local similarity transformation S of the first kind takes a Dirac equation of the normal form (14) into a normal QRD–0 equation (16), if and only if S satisfies the following partial differential equation:

$$\widetilde{B}^{\mu } \partial _{\mu } S= -\widetilde{B}^{\mu } \widetilde{\Gamma }_{\mu } S, \label{GrindEQ__76_}$$

(84)

where $\widetilde{B}^{\mu } \equiv \widetilde{A}\widetilde{\gamma }^{\mu }$. The matrices B ^μ and $\widetilde{B}^{\mu }$ are Hermitian, and moreover, the matrices B ⁰ and $\widetilde{B}^{0}$ are positive definite. Thus, the two systems (82) and (84) have same form as the complex linear hyperbolic system in (81).

Let χ : U →R ×R ³ , mapping $X \to \left( t , {\bf x} \right)$, be the coordinate chart that we assume to be defined on U . Then, consider the projection map π : R ×R ³ →R taking $\left( t , {\bf x} \right) \to t$. Let X ₀ ∈ U and let $t_{0} = \pi \circ \chi \left( X_{0} \right) \in {\bf R}$. Let ${\bf M} = \left( \pi \circ \chi \right)^{-1} \left( t_{0} \right)$. Note that X ₀ ∈ M ⊂ U . It will suffice to prove that there exist nonsingular solutions T and S of the systems (82) and (84) that are both defined in a common open neighborhood $\widetilde{{\bf W}}$ of X ₀ ∈ U.

By Theorem 0, the Cauchy problem for (82) with the smooth initial data T| _M  = 1 ₄ has a smooth solution T in an open neighborhood W ^# of X ₀ . Denote by W the open subset of W ^# in which T is a nonsingular matrix, so that T ^− 1 exists. Note that X ₀ ∈ W since X ₀ ∈ M and T| _M  = 1 ₄ . Thus, W is an open neighborhood of X ₀ such that the local similarity transformation T (and its inverse T ^− 1 ) is well defined on W , and hence, from Eq. (83), the complex linear hyperbolic system (84) is well defined on W .

By Theorem 0, the Cauchy problem for (84) with the smooth initial data S| _M ∩ W  = 1 ₄ has a smooth solution S in an open neighborhood $\widetilde{{\bf W}}^{{\bf \# }} \subset {\bf W}$ of X ₀ . Denote by $\widetilde{{\bf W}}$ the open subset of $\widetilde{{\bf W}}^{{\bf \# }}$ in which S is a nonsingular matrix so that S ^− 1 exists. Note that $X_{0} \in \widetilde{{\bf W}}$ since X ₀ ∈ M ∩ W and S| _M ∩ W  = 1 ₄ . Thus, $\widetilde{{\bf W}} \subset \widetilde{{\bf W}}^{{\bf \# }} \subset {\bf W}$ is an open neighborhood of X ₀ such that both the local similarity transformation S and the local similarity transformation T (and their inverses S ^− 1 and T ^− 1 ) are well defined on $\widetilde{{\bf W}}$, and therefore the local similarity transformation T ∘ S (and its inverse S ^− 1 ∘ T ^− 1 ) is also well defined on $\widetilde{{\bf W}}$.□