Inertial and gravitational effects on a geonium atom

In order to confirm predictions from the standard model of particle physics Aoyama et al. (2019, 2020) and/or probe beyond the standard model Giudice et al. (2012); Czarnecki and Marciano (2001), magnetic moments/$g$-factors of fermions have been measured intensively. For instance, measurements for the electron $g$-factor Hanneke et al. (2008); Odom et al. (2006); Van Dyck et al. (1987) and the muon $g$-factor Abi et al. (2021); Bennett et al. (2006); Bailey et al. (1979) have been conducted with very high accuracy. For the case of the electron, one can resolve a one-electron quantum transition in current quantum optical technologies Brown and Gabrielse (1986); D’Urso (2003); Hanneke et al. (2011) and it enables us to measure the electron $g$-factor with remarkable small uncertainty, which is $2.8\times 10^{-13}$ for $g/2$. Considering this current sensitivity, effects of gravity on the $g$-factor measurements may not be negligible. Furthermore, discrepancies between theoretical predictions and experimental results were reported both for the electron Hanneke et al. (2008) and the muon Abi et al. (2021). Therefore it would be important to explore the possibility that effects of gravity could reconcile the discrepancies.

Actually, gravitational effects on the g-factor of the electron or the muon have been studied intensively Morishima et al. (2018); Visser (2018); Nikolic (2018); Venhoek (2018); László and Zimborás (2018); Jentschura (2018); Ulbricht et al. (2019). In order to investigate effects of gravity, it is crucial to consider the equivalence principle appropriately, namely we need to use a coordinate moving with an observer bound on the surface of the Earth Ni and Zimmermann (1978); Ito and Soda (2020). This aspect was emphasized qualitatively in Visser (2018); Nikolic (2018); Venhoek (2018) and investigated quantitatively in László and Zimborás (2018); Notari and Bertacca (2019); Ulbricht et al. (2019). László and Zimborás (2018) and Notari and Bertacca (2019) analyzed some general relativistic effects with the use of the Fermi-Walker transport in equations of motion for the case of the muon. Ulbricht et al. (2019) evaluated an inertial effect, an acceleration relative to (local) inertial frames, which is characterized by $|\bm{a}|=9.81\,{\rm m/s}^{2}$ in the case of the gravity of Earth, in a Hamiltonian of a Dirac particle by taking the non-relativistic limit for the case of the electron. However in Ulbricht et al. (2019), other effects of Earth’s gravity were missed. Moreover, Ulbricht et al. (2019) has not studied a spin-orbit coupling induced by $\bm{a}$, which actually gives rise to a leading correction among corrections from $\bm{a}$ as we will see. In this paper, we study all linear order general relativistic effects, namely inertial effects of the acceleration, the rotation due to the Earth and tidal effects due to weak gravitational fields. In particular, we evaluate magnitude of the general relativistic corrections for the case of the electron g-factor measurements.

To this end, we first take the non-relativistic limit of a Hamiltonian for a Dirac particle (mass $m$) up to the order of $1/m$ in the Foldy-Wouthuysen-like expansion Foldy and Wouthuysen (1950); Bjorken and Drell (1965); Ito and Soda (2020) on a proper reference frame Ni and Zimmermann (1978); Ito and Soda (2020). We mention that analyzing the Hamiltonian is more useful than equations of motion because it enables us to access special effects like a spin-orbit coupling, which can not be derived in equations of motion. Next, we apply the obtained Hamiltonian to the situation of Penning trap experiments where a Dirac particle experiences the cyclotron motion the spin precession in a cavity, i.e., a geonium atom, and estimate magnitude of the effects of gravity. As a result, it turns out that effects of the Earth’s rotation is dominant among the general relativistic corrections. It can be detected if the current sensitivity is improved by 4 orders of magnitude.

The paper is organized as follows. In the section II, we introduce a proper reference frame and consider the Dirac equation in the coordinate. Then a Hamiltonian in the proper reference frame is obtained. In the section III, we take the non-relativistic limit of the Hamiltonian up to the order of $1/m$. This manifests all linear order inertial and gravitational effects on a non-relativistic Dirac particle. In the section IV, we apply the non-relativistic Hamiltonian to the case of Penning trap experiments and analyze the inertial and gravitational effects on the cyclotron motion and the spin precession. In the section V, we consider a particular case of an electron $g$-factor measurement to probe the detectability of the general relativistic corrections. The final section is devoted to the conclusion. In the appendix A, a brief review of Penning trap experiments is given for reference.

In this section, we investigate inertial and gravitational effects on a Dirac particle bound on the surface of the Earth perturbatively. To this end, we use a proper reference frame Ni and Zimmermann (1978); Ito and Soda (2020). A proper reference frame for an observer who is accelerating against the center of the Earth, $|\bm{a}|=9.81\,{\rm m/s}^{2}$, and rotating due to the Earth’s rotation, $|\bm{\omega}|=7.27\times 10^{-5}\,{\rm rad/s}$, relative to (local) inertial frames can be constructed with the use of the Fermi-Walker transport Ni and Zimmermann (1978); Ito and Soda (2020). Then the metric in the frame is obtained perturbatively in powers of $ax\ll 1$, $\omega x\ll 1$ and (the Riemann tensor $\times xx$) $\ll$ 1, where $x$ represents a typical scale of a system. In this paper, we use the term “inertial” for $a_{i}$ and $\omega_{i}$, and “gravitational” for the curvature. Up to the quadratic order for $x$, the metric is given by

	$\displaystyle g_{00}$	$\displaystyle=-1-2a_{i}x^{i}-R_{0i0j}x^{i}x^{j},$
	$\displaystyle g_{0i}$	$\displaystyle=-\omega_{k}\epsilon_{0ijk}x^{j}-\frac{2}{3}R_{0jik}x^{j}x^{k},$		(1)
	$\displaystyle g_{ij}$	$\displaystyle=\delta_{ij}-\frac{1}{3}R_{ikjl}x^{k}x^{l},$

where the anti symmetric tensor is assigned as $\epsilon_{0123}=1$. The Riemann tensor is evaluated at $\bm{x}=0$, so that it only depends on time $x^{0}$. At this occasion, we have not specified the source of the curvature. The origin of the spatial coordinates is set on the center of gravity of a system, which traces a worldline of a freely falling particle in the limit of $\bm{a}=\bm{\omega}=\bm{0}$. In the case of a geonium atom, the origin should be at the center of the cyclotron motion explained in the appendix A.

We now consider the Dirac equation in the proper reference frame by using the metric (1). The Dirac equation in curved spacetime is given by Birrell and Davies (1984)

$$i\gamma^{\hat{\alpha}}e^{\mu}_{\hat{\alpha}}\left(\partial_{\mu}-\Gamma_{\mu}-ieA_{\mu}\right)\psi=m\psi\ ,$$

(2)

where $\gamma^{\hat{\alpha}}$, $e$, $m$, $A_{\mu}$ are the gamma matrices, an electromagnetic charge, a mass and a vector potential, respectively. The tetrad $e^{\mu}_{\hat{\alpha}}$ is defined to satisfy

$$e^{\hat{\alpha}}_{\mu}e^{\hat{\beta}}_{\nu}\eta_{\hat{\alpha}\hat{\beta}}=g_{\mu\nu}\ .$$

(3)

Note that $\eta_{\hat{\alpha}\hat{\beta}}$ is the Minkowski metric of a local inertial frame and hat is used for the frame. More explicitly, for the metric (1), the tetrads are constructed as

$$e^{\hat{\alpha}}_{0}=\delta^{\hat{\alpha}}_{0}\left(1+a_{i}x^{i}\right)-\frac{1}{2}\delta^{\hat{\alpha}}_{\alpha}R^{\alpha}_{\ k0l}\ ,\quad e^{\hat{\alpha}}_{i}=\delta^{\hat{\alpha}}_{0}\omega_{k}\epsilon_{0ijk}x^{j}+\delta^{\hat{\alpha}}_{i}-\frac{1}{6}\delta^{\hat{\alpha}}_{\alpha}R^{\alpha}_{\ kil}x^{k}x^{l}c\ .$$

(4)

The spin connection is defined by

$$\Gamma_{\mu}=-\frac{i}{2}e^{\hat{\alpha}}_{\nu}\sigma_{\hat{\alpha}\hat{\beta}}\left(\partial_{\mu}e^{\nu\hat{\beta}}+\Gamma^{\nu}_{\lambda\mu}e^{\lambda\hat{\beta}}\right),$$

(5)

where $\sigma_{\hat{\alpha}\hat{\beta}}=\frac{i}{4}[\gamma_{\hat{\alpha}},\gamma_{\hat{\beta}}]$ is a generator of the Lorentz group and $\Gamma^{\mu}_{\nu\lambda}$ is the Christoffel symbol. For the metric (1), the spin connection at the linear order for inertial and gravitational terms can be calculated as follows:

	$\displaystyle\Gamma_{0}$	$\displaystyle=-\frac{1}{2}\gamma^{\hat{0}}\gamma^{\hat{i}}a_{i}-\frac{1}{4}\gamma^{\hat{i}}\gamma^{\hat{j}}\omega_{k}\epsilon_{0ijk}-\frac{1}{2}\gamma^{\hat{0}}\gamma^{\hat{i}}R_{0i0j}x^{j}-\frac{1}{4}\gamma^{\hat{i}}\gamma^{\hat{j}}R_{ij0k}x^{k}\ ,\quad$		(6)
	$\displaystyle\Gamma_{i}$	$\displaystyle=-\frac{1}{2}\gamma^{\hat{0}}\gamma^{\hat{j}}\omega_{k}\epsilon_{0ijk}-\frac{1}{4}\gamma^{\hat{0}}\gamma^{\hat{j}}R_{0jik}x^{k}-\frac{1}{8}\gamma^{\hat{j}}\gamma^{\hat{k}}R_{jkil}x^{l}\ .$		(7)

Here we have rewritten $\delta_{\hat{\alpha}}^{\mu}\gamma^{\hat{\alpha}}$ as $\gamma^{\hat{\mu}}$ and we will do so throughout.

On the other hand, the Dirac equation (2) can be rewritten as

	$\displaystyle i\gamma^{0}\partial_{0}\psi$	$\displaystyle=$	$\displaystyle\left[i\gamma^{0}\left(\Gamma_{0}+ieA_{0}\right)-i\gamma^{j}\left(\partial_{j}-\Gamma_{j}-ieA_{j}\right)+m\right]\psi$		(8)
		$\displaystyle=$	$\displaystyle\gamma^{0}H\psi\ ,$

where we defined a Hamiltonian $H$ and the gamma matrices in curved spacetime, $\gamma^{\mu}=e^{\mu}_{\hat{\alpha}}\gamma^{\hat{\alpha}}$, satisfying the relation

$$\{\gamma^{\mu},\gamma^{\nu}\}=-2g^{\mu\nu}\ .$$

(9)

Let us express the Hamiltonian in terms of the gamma matrices of the local inertial frame instead of those of curved spacetime. Because of $\gamma^{0}\gamma^{0}=-g^{00}$, we obtain

$$H=(g^{00})^{-1}\left[ig^{00}\left(\Gamma_{0}+ieA_{0}\right)+i\gamma^{0}\gamma^{j}\left(\partial_{j}-\Gamma_{j}-ieA_{j}\right)-\gamma^{0}m\right]\ .$$

(10)

Using Eqs. (1) and (4), we calculate

	$\displaystyle(g^{00})^{-1}\gamma^{0}\gamma^{j}$	$\displaystyle\simeq$	$\displaystyle-\gamma^{\hat{0}}\gamma^{\hat{j}}\left(1+a_{i}x^{i}\right)+\gamma^{\hat{i}}\gamma^{\hat{j}}\omega_{k}\epsilon_{0ilk}x^{l}-\gamma^{\hat{0}}\gamma^{\hat{j}}-\frac{1}{2}R_{0kjl}x^{k}x^{l}$		(11)
			$\displaystyle-\frac{1}{6}\gamma^{\hat{0}}\gamma^{\hat{a}}R_{jkal}x^{k}x^{l}-\frac{1}{2}\gamma^{\hat{0}}\gamma^{\hat{j}}R_{0k0l}x^{k}x^{l}+\frac{1}{6}\gamma^{\hat{a}}\gamma^{\hat{j}}R_{ak0l}x^{k}x^{l}\ .$

Similarly, we have

$$(g^{00})^{-1}\gamma^{0}\simeq-\gamma^{\hat{0}}\left(1+a_{i}x^{i}\right)-\gamma^{\hat{i}}\omega_{k}\epsilon_{0ilk}x^{l}-\frac{1}{2}\gamma^{\hat{0}}R_{0k0l}x^{k}x^{l}+\frac{1}{6}\gamma^{\hat{a}}R_{ak0l}x^{k}x^{l}\ .$$

(12)

Therefore using Eqs. (6), (7), (11) and (12) in the Hamiltonian (10), we obtain

	$\displaystyle H$	$\displaystyle=$	$\displaystyle-\frac{i}{2}\gamma^{\hat{0}}\gamma^{\hat{i}}\left(a_{i}+R_{0i0j}x^{j}\right)-\frac{i}{4}\gamma^{\hat{i}}\gamma^{\hat{j}}R_{0ikj}x^{k}-\frac{i}{8}\gamma^{\hat{0}}\gamma^{\hat{i}}\gamma^{\hat{j}}\gamma^{\hat{k}}R_{jkil}x^{l}-eA_{0}$		(13)
			$\displaystyle+\Big{[}\gamma^{\hat{0}}\gamma^{\hat{i}}\Big{(}\delta^{j}_{i}\left(1+a_{i}x^{i}\right)+\theta^{j}_{i}\Big{)}-\gamma^{\hat{i}}\gamma^{\hat{j}}\left(\omega_{k}\epsilon_{0ilk}x^{l}+\frac{1}{6}R_{ik0l}x^{k}x^{l}\right)+\frac{1}{2}R_{0kjl}x^{k}x^{l}\Big{]}\left(-i\partial_{j}-eA_{j}\right)$
			$\displaystyle+\left[\gamma^{\hat{0}}\left(1+a_{i}x^{i}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)-\gamma^{\hat{i}}\left(\omega_{k}\epsilon_{0ijk}x^{j}+\frac{1}{6}R_{ik0l}x^{k}x^{l}\right)\right]m\ ,$

where we defined

$$\theta^{j}_{i}=\frac{1}{2}\delta^{j}_{i}R_{0k0l}x^{k}x^{l}+\frac{1}{6}R_{jkil}x^{k}x^{l}\ .$$

(14)

The above Hamiltonian is a 4$\times$4 matrix and contains both the fermion and the anti-fermion. What we want to consider is the fermion particle with a non-relativistic velocity. In order to take the non-relativistic limit of the Hamiltonian for the fermion, we need to separate the fermion and the anti-fermion while expanding the Hamiltonian in powers of $1/m$. In the next section, we will explicitly show how to perform this.

In the previous section, the (non-relativistic) Hamiltonian of a Dirac field in the proper reference frame was derived. We take the non-relativistic limit of the Hamiltonian (13) on the assumption that a fermion has a velocity well below the speed of light, which is usual in experiments on the Earth like the electron g-factor measurements Hanneke et al. (2008); Odom et al. (2006).

The Hamiltonian (13) can be divided into the even part, the odd part and the terms multiplied by $m$:

	$\displaystyle H$	$\displaystyle=$	$\displaystyle-\frac{i}{2}\alpha^{i}\left(a_{i}+R_{0i0j}x^{j}\right)+\frac{i}{8}\alpha^{i}\alpha^{j}\alpha^{k}R_{jkil}x^{l}+\alpha^{j}\Big{(}\delta^{j}_{i}\left(1+a_{i}x^{i}\right)+\theta^{j}_{i}\Big{)}\Pi_{j}$		(15)
			$\displaystyle-eA_{0}-\frac{i}{4}\alpha^{i}\alpha^{j}\left(\omega_{k}\epsilon_{0ijk}-R_{0ikj}x^{k}\right)+\bigg{[}\frac{1}{2}R_{0kjl}x^{k}x^{l}+\alpha^{i}\alpha^{j}\left(\omega_{k}\epsilon_{0ilk}x^{l}+\frac{1}{6}R_{ik0l}x^{k}x^{l}\right)\bigg{]}\Pi_{j}$
			$\displaystyle+\left[\beta\left(1+a_{i}x^{i}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)-\beta\alpha^{i}\left(\omega_{k}\epsilon_{0ijk}x^{j}+\frac{1}{6}R_{ik0l}x^{k}x^{l}\right)\right]m$
		$\displaystyle=$	$\displaystyle\mathcal{O}+\mathcal{E}+\left[\beta\left(1+a_{i}x^{i}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)-\beta\alpha^{i}\left(\omega_{k}\epsilon_{0ijk}x^{j}+\frac{1}{6}R_{ik0l}x^{k}x^{l}\right)\right]m\ ,$

where we have defined $\beta=\gamma^{\hat{0}}$, $\alpha^{i}=\gamma^{\hat{0}}\gamma^{\hat{i}}$ and $\Pi_{j}=-i\partial_{j}-eA_{j}$ for brevity. The even part, $\mathcal{E}$, means that the matrix has only block diagonal elements and the odd part, $\mathcal{O}$, means that the matrix has only block off-diagonal elements. Literally, a product of two even (odd) matrices is even and a product of even and odd matrices becomes odd. In order to take the non-relativistic limit of the Hamiltonian, we need to diagonalize the Hamiltonian (15) and expand the upper block diagonal part in powers of $1/m$. More precisely, $1/m$ stands for two dimensionless parameters, $(mx)^{-1}$ and $v/c$. Here, $x$ represents a typical length scale of the system, $v$ is the velocity of the fermion particle and $c$ denotes the speed of light. Assuming $1/mx\ll 1$ and $v/c\ll 1$, which hold in the electron g-factor measurements Hanneke et al. (2008); Odom et al. (2006), we will perform the $1/m$ expansion. This can be carried out both in flat spacetime Foldy and Wouthuysen (1950); Bjorken and Drell (1965) and in curved spacetime Ito and Soda (2020) by repeating unitary transformations order by order in powers of $1/m$. We will follow the procedure shown in Ito and Soda (2020).

A unitary transformation to a spinor field is

$$\psi^{\prime}=e^{iS}\psi\ ,$$

(16)

where $S$ is a time-dependent Hermitian 4 $\times$ 4 matrix. Observing that

	$\displaystyle i\frac{\partial\psi^{\prime}}{\partial t}$	$\displaystyle=$	$\displaystyle i\frac{\partial}{\partial t}\left(e^{iS}\psi\right)$		(17)
		$\displaystyle=$	$\displaystyle e^{iS}\left(i\frac{\partial\psi}{\partial t}\right)+i\left(\frac{\partial}{\partial t}e^{iS}\right)\psi$
		$\displaystyle=$	$\displaystyle\left[e^{iS}He^{-iS}+i\left(\frac{\partial}{\partial t}e^{iS}\right)e^{-iS}\right]\psi^{\prime}\ ,$

we see that the Hamiltonian after the unitary transformation is

$$H^{\prime}=e^{iS}He^{-iS}+i\left(\frac{\partial}{\partial t}e^{iS}\right)e^{-iS}\ .$$

(18)

Taking $S$ to be proportional to powers of $1/m$, the transformed Hamiltonian (18) can be expanded in powers of $S$ up to arbitrary order of $1/m$:

	$\displaystyle H^{\prime}$	$\displaystyle=$	$\displaystyle H+i\big{[}S,H\big{]}-\frac{1}{2}\big{[}S,\big{[}S,H\big{]}\big{]}-\frac{i}{6}\big{[}S,\big{[}S,\big{[}S,H\big{]}\big{]}\big{]}+\cdots$		(19)
			$\displaystyle-\dot{S}-\frac{i}{2}\big{[}S,\dot{S}\big{]}+\cdots\ .$

First, we eliminate the off-diagonal part of the Hamiltonian (15) at the order of $m$ by a unitary transformation. Then we will drop the higher order terms with respect to $a_{i}$, $\omega_{i}$ and the Riemann tensor. We assume that the time derivative of the Riemann tensor is small enough to neglect. Notice that the time derivative of the spatial coordinate $x^{i}$ is a higher order of $v/c$ and thus we also neglect it¹¹1 The Hermiticity of the non-relativistic Hamiltonian is guaranteed when the metric is time independent Huang and Parker (2009). . To cancel the last term in the square bracket of (15), we take

$$S=-\frac{i}{2m}\beta\left[-\beta\alpha^{i}\left(\omega_{k}\epsilon_{0ijk}x^{j}+\frac{1}{6}R_{ik0l}x^{k}x^{l}\right)m\right]\ .$$

(20)

We then obtain

	$\displaystyle i\big{[}S,H\big{]}$	$\displaystyle\simeq$	$\displaystyle\beta\alpha^{i}\left(\omega_{k}\epsilon_{0ijk}x^{j}+\frac{1}{6}R_{ik0l}x^{k}x^{l}\right)m-\frac{1}{2}\big{[}\alpha^{i},\alpha^{j}\big{]}\left(\omega_{k}\epsilon_{0ilk}x^{l}+\frac{1}{6}R_{ik0l}x^{k}x^{l}\right)\Pi_{j}$		(21)
			$\displaystyle+\frac{i}{2}\alpha^{i}\alpha^{j}\left(-\omega_{k}\epsilon_{0jik}+\frac{1}{3}R_{0ikj}x^{k}+\frac{1}{6}R_{0jik}x^{k}\right)\ .$

Therefore, from Eqs. (19) and (21), we have the transformed Hamiltonian with accuracy mentioned above:

	$\displaystyle H^{\prime}$	$\displaystyle\simeq$	$\displaystyle H+i\big{[}S,H\big{]}$		(22)
		$\displaystyle=$	$\displaystyle\mathcal{O}+\mathcal{E}^{\prime}+\beta\left(1+a_{i}x^{i}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)m\ ,$

where we have used the relation $\big{\{}\alpha^{i},\alpha^{j}\big{\}}=2\delta^{ij}$ and $\mathcal{E}^{\prime}$ is defined by

	$\displaystyle\mathcal{E}^{\prime}=$	$\displaystyle-$	$\displaystyle eA_{0}+\frac{i}{8}[\alpha^{i},\alpha^{j}]\omega_{k}\epsilon_{0ijk}+\omega_{k}\epsilon_{0ijk}x^{j}\Pi_{i}$		(23)
		$\displaystyle+$	$\displaystyle\frac{i}{3}R_{0iki}x^{k}+\frac{2}{3}R_{0kil}x^{k}x^{l}\Pi_{i}-\frac{i}{8}[\alpha^{i},\alpha^{j}]R_{ijk0}x^{k}\ .$

One can see that only even terms remain at the order of $m$, as expected.

Next, let us focus on the order of $m^{0}$ and eliminate the odd terms by a unitary transformation. In order to do so, we choose the Hermitian operator to be

$$S^{\prime}=-\frac{i}{2m}\beta\left[\mathcal{O}-\alpha^{i}\left(a_{j}x^{j}\Pi_{i}-\frac{i}{2}a_{i}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\Pi_{i}-\frac{i}{2}R_{0k0i}x^{k}\right)\right]\ .$$

(24)

First of all,

$$i\big{[}S^{\prime},H^{\prime}\big{]}\simeq-\mathcal{O}+i\big{[}S^{\prime},\mathcal{O}\big{]}+i\big{[}S^{\prime},\mathcal{E}^{\prime}\big{]}\ ,$$

(25)

Furthermore, up to the order of $1/m$, we find

$$-\frac{1}{2}\big{[}S^{\prime},\big{[}S^{\prime},H^{\prime}\big{]}\big{]}\simeq-\frac{i}{2}\big{[}S^{\prime},\mathcal{O}\big{]}\ ,$$

(26)

and

$$-\dot{S}^{\prime}\simeq\frac{i}{2m}\beta\dot{\mathcal{O}}-\frac{i}{m}\beta\alpha^{i}a_{j}x^{j}e\dot{A}_{i}\ .$$

(27)

Therefore, the unitary transformed Hamiltonian is given by

	$\displaystyle H^{\prime\prime}$	$\displaystyle\simeq$	$\displaystyle H^{\prime}+i\big{[}S^{\prime},H^{\prime}\big{]}-\frac{1}{2}\big{[}S^{\prime},\big{[}S^{\prime},H^{\prime}\big{]}\big{]}-\dot{S}^{\prime}$		(28)
		$\displaystyle\simeq$	$\displaystyle H^{\prime}+\frac{i}{2}\big{[}S^{\prime},\mathcal{O}\big{]}+i\big{[}S^{\prime},\mathcal{E}^{\prime}\big{]}-\dot{S}^{\prime}$
		$\displaystyle\simeq$	$\displaystyle\frac{i}{m}\alpha^{i}a_{j}x^{j}eE_{i}-\frac{i}{4m}\beta\alpha^{j}R_{0k0l}x^{k}x^{l}eE_{j}+\frac{1}{2m}\beta\left(\big{[}\mathcal{O},\mathcal{E}^{\prime}\big{]}+i\dot{\mathcal{O}}\right)$
			$\displaystyle+\mathcal{E}^{\prime}+\frac{1}{2m}\beta\mathcal{O}^{2}+\frac{i}{2m}\beta\left(a^{i}+R_{0k0i}x^{k}\right)\Pi_{i}+\frac{i}{8m}\beta\big{[}\alpha^{i},\alpha^{j}\big{]}\left(a^{i}+R_{0k0i}x^{k}\right)\Pi_{j}$
			$\displaystyle-\frac{1}{2m}\beta\alpha^{i}\alpha^{j}\left(a_{k}x^{k}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)\Pi_{i}\Pi_{j}+\frac{1}{8m}\beta R_{0i0i}$
			$\displaystyle+\beta\left(1+a_{i}x^{i}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)m$
		$\displaystyle=$	$\displaystyle\mathcal{O}^{\prime}+\mathcal{E}^{\prime\prime}+\beta\left(1+a_{i}x^{i}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)m\ ,$

where $E_{j}\equiv\partial_{j}A_{0}-\dot{A}_{j}$ is an electric field. We see that $\mathcal{O}^{\prime}$ consists of only terms of the order of $1/m$, so that odd terms at the order of $m^{0}$ have been eliminated correctly.

Finally, again, we can eliminate the odd term $\mathcal{O}^{\prime}$ by an appropriate unitary transformation. The resultant Hamiltonian consists of only even terms up to the order of $1/m$, which we want to get. Thus, up to the order of $1/m$, we have

$$H^{\prime\prime\prime}\simeq\mathcal{E}^{\prime\prime}+\beta\left(1+a_{i}x^{i}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)m\ ,$$

(29)

where $\mathcal{E}^{\prime\prime}$ is

	$\displaystyle\mathcal{E}^{\prime\prime}$	$\displaystyle=$	$\displaystyle-eA_{0}-\frac{i}{8}[\alpha^{i},\alpha^{j}]\omega_{k}\epsilon_{0ijk}-\omega_{k}\epsilon_{0ijk}x^{j}\Pi_{i}+\frac{i}{3}R_{0iki}x^{k}+\frac{2}{3}R_{0kil}x^{k}x^{l}\Pi_{i}-\frac{i}{8}[\alpha^{i},\alpha^{j}]R_{ijk0}x^{k}$		(30)
			$\displaystyle+\frac{1}{2m}\beta\mathcal{O}^{2}+\frac{i}{2m}\beta\left(a^{i}+R_{0k0i}x^{k}\right)\Pi_{i}+\frac{i}{8m}\beta\big{[}\alpha^{i},\alpha^{j}\big{]}\left(a^{i}+R_{0k0i}x^{k}\right)\Pi_{j}$
			$\displaystyle-\frac{1}{2m}\beta\alpha^{i}\alpha^{j}\left(a_{k}x^{k}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)\Pi_{i}\Pi_{j}+\frac{1}{8m}\beta R_{0i0i}\ .$

Moreover, the first term in the second line of Eq. (30) can be evaluated as

	$\displaystyle\frac{1}{2m}\beta\mathcal{O}^{2}$	$\displaystyle\simeq$	$\displaystyle\frac{1}{2m}\beta\Big{(}\delta_{ij}\left(1+2a_{k}x^{k}\right)+2\theta_{ij}\Big{)}\Pi_{i}\Pi_{j}+\frac{i}{8m}\beta\big{[}\alpha^{i},\alpha^{j}\big{]}\epsilon_{0ilm}eB^{m}\Big{(}\delta_{lj}(1+2a_{k}x^{k})+2\theta_{lj}\Big{)}$		(31)
			$\displaystyle-\frac{i}{4m}\beta\big{[}\alpha^{i},\alpha^{j}\big{]}\left(a_{i}\Pi_{j}+\frac{1}{4}R_{lmji}+\delta^{l}_{j}R_{0i0m}\right)x^{m}\Pi_{l}+\frac{i}{12m}\beta R_{kikj}x^{j}\Pi_{i}-\frac{i}{m}\beta R_{0i0j}x^{i}\Pi_{j}$
			$\displaystyle-\frac{1}{4m}\beta R_{0i0i}-\frac{1}{16m}\beta\alpha^{i}\alpha^{j}\alpha^{k}\alpha^{l}R_{ijkl}+\frac{i}{16m}\beta\big{\{}\alpha^{i},\alpha^{j}\alpha^{k}\alpha^{l}\big{\}}R_{kljm}x^{m}\Pi_{i}\ ,$

where $B^{i}\equiv\frac{1}{2}\epsilon_{0ijk}(\partial_{j}A_{k}-\partial_{k}A_{j})$ is a magnetic field²²2In general, an external magnetic field itself would be modified by inertial and gravitational effects as was explicitly shown for a simple system like a Hydrogen atom Parker (1980a, b); Perche and Neuser (2020). We ignore such corrections since there is no way to evaluate them model-independently, namely they depend on detail of a mechanism for creating an external magnetic field. . Using Eqs. (30), (31) and the relation, $\big{[}\alpha^{i},\alpha^{j}\big{]}=2i\epsilon_{0ijk}\sigma^{k}$, in the transformed Hamiltonian (29), we finally arrive at the Hamiltonian for a non-relativistic fermion/anti-fermion up to the order of $1/m$:

	$\displaystyle H^{\prime\prime\prime}$	$\displaystyle=$	$\displaystyle\beta\left(1+a_{i}x^{i}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)m$		(32)
			$\displaystyle-eA_{0}-\omega_{k}\epsilon_{0ijk}x^{i}\Pi_{j}-\omega_{i}S^{i}+\frac{1}{3}R_{0kil}\left(\Pi_{i}x^{k}x^{l}+x^{k}x^{l}\Pi_{i}\right)+\frac{1}{2}\epsilon_{0ijl}S^{l}R_{ijk0}x^{k}$
			$\displaystyle-\frac{e}{m}\beta S^{i}B^{j}\left[\delta_{ij}\left(1+a_{k}x^{k}+\frac{1}{2}R_{0k0l}x^{k}x^{l}+\frac{1}{6}R_{mkml}x^{k}x^{l}\right)-\frac{1}{6}R_{ikjl}x^{k}x^{l}\right]$
			$\displaystyle+\frac{1}{2m}\beta\Pi_{i}\left[\delta_{ij}\left(1+a_{k}x^{k}+\frac{1}{2}R_{0k0l}x^{k}x^{l}\right)+\frac{1}{3}R_{jkil}x^{k}x^{l}\right]\Pi_{j}$
			$\displaystyle+\frac{1}{2m}\beta\epsilon_{0ijk}S^{k}a^{i}\Pi^{j}+\frac{1}{4m}\beta\epsilon_{ijk}S^{k}\left(R_{ijlm}+2\delta_{jm}R_{0i0l}\right)x^{l}\Pi_{m}$
			$\displaystyle-\frac{1}{8m}\beta R_{0i0i}+\frac{1}{24m}\beta R_{ijij}\ ,$

where we have replaced the canonical momentum in flat spacetime by one in curved spacetime Gneiting et al. (2013): $\Pi_{i}\rightarrow\Pi_{i}+i\frac{1}{\sqrt{-g}}\partial_{i}\sqrt{-g}=\Pi_{i}-\frac{i}{6}R_{jijk}x^{k}$ ($g$ is the determinant of the metric). Also a spin has been defined by $S^{i}=\sigma^{i}/2$ with the Pauli matrices $\sigma^{i}$. Note that if one wants to consider an anti-fermion in the above Hamiltonian, one needs to take charge conjugation for the wave function of the lower two compoents. The terms for inertial effects coincide with an earlier work Singh and Papini (2000). On the other hand, several terms for gravitational effects are updated compared with the previous work Ito and Soda (2020) where there was a miscalculation. The first parenthesis represents the rest mass and its corrections due to inertial and gravitational effects. The inertial one is recognized as an usual inertial force and the gravitational one is the leading order gravitational modification to a particle trajectory as we will see later. The third term Werner et al. (1979) corresponds to the Coriolis force. The fourth term is the spin-rotation coupling Mashhoon (1988), which modifies the magnetic moment, i.e., g-factor. In the fifth term, we can clearly see that a correct operater ordering has been derived automatically. Interestingly, the sixth term represents the effect of a dipole structure, namely the spin angular momentum, on the trajectry of a freely falling particle in curved spacetime as can be seen in the Mathisson-Papapetrou-Dixon equation³³3 Strictly speaking, we should have taken the term into account when we constructed a proper reference frame. Indeed, the effect can be treated as a linear acceleration of a paticle and can be included in $a_{i}$ in the metric (1). Then the sixth term disappers.. The third line represents corrections of the magnetic moment due to gravity. The fourth line shows that also the kinetic term is modified by inertial and gravitational effects. In the fifth line, we find the inertial spin-orbit coupling Hehl and Ni (1990) and the gravitational spin-orbit coupling Ito and Soda (2020), respectively. The sixth line consists of energy shifts due to gravity at the order of $1/m$.

In this section, we investigate the inertial and the gravitational effects in the Hamiltonian (32) to trajectories and spin kinematics of a Dirac particle in the presence of an external magnetic field. In the sections IV.1 and IV.2, we will show that the cyclotron and the Larmor frequencies are corrected according to modifications of particle trajectories and spin kinematics, respectively. In the section IV.3, it will turn out that the spin-orbit couplings modify the cyclotron and the Larmor frequencies simultaneously.