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I am working on the Dirac equation on curved spacetime. A Foldy-Wouthuysen transformation was applied to obtain the semiclassical limit of the equation to study the dynamics of the spin of the electron in curved spacetime.

In the semiclassical limit up to order $\hbar$ the Hamiltonian then takes a familiar form:

$H =\; ... \,+\, \hbar \, \Sigma \cdot \, \Omega^{(1)}$,

where $\Sigma$ is the spin vector and $\Omega$ is the precession frequency due to gravitational effects. If I look closer to $\Omega$, I find that it contains the metric $g_{\mu \nu}(x)$ and its first derivatives $\partial_\lambda g_{\mu\nu}$.

As far as I understand, one can always find an appropriate coordinate system in which the first derivatives of the metric can be made vanish at a certain point $x$ (so called normal coordinates). To my knowledge, this is actually good because the principle of equivalence states that gravitational effects (in this case the spin precession of the electron) must be able to be set zero at one point.

Q1: The Dirac equation is certainly a local theory. Therefore, should gravitational effects only appear as fictitious forces? (Should they only contain first derivatives of the metric)?

Now I calculated the semiclassical limit to an higher order $\hbar^2$ (I did this by evaluating the commutators in $\Omega^{(1)}$ to a higher precision.). The Hamiltonain looks like this:

$H =\; ... \,+\, \hbar \, \Sigma \cdot \, \Omega^{(1)} \; + \hbar^2 \, \Sigma \cdot \, \Omega^{(2)}$

Looking at $\Omega^{(2)}$, I find second derivatives of the metric. This is a bit more confusing to me, since the Riemann curvature tensor is made of second derivatives and since curvature is the "objective ingredient" in general relativity, there exists no coordinate system in which curvature effects can be "gauged away".

Q2: Is the appearance of these second derivative terms problematic in any way? Do they mean that the principle of equivalence could be violated here (even if only at the order $\hbar^2$)?

This is my ongoing research so any input would be very welcome.

This is my first post. Please excuse me, if my question is not asked optimally.

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In your first order equation, do you also have partial derivatives on the Dirac spinor? If so, you should probably not be worried about the first partial derivatives of the metric. The Christoffel symbols are sums of derivatives of the metric, so what you find could be just the covariant derivative. –  Robin Ekman May 22 at 13:08
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See also this question. –  Ruslan May 22 at 14:17
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