2 different ways of doing quantum field theory in curved space time? | PhysicsOverflow
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  2 different ways of doing quantum field theory in curved space time?

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Deriving the massless Dirac equation using differential geometry 


Beginning with the line element:

$$ ds^2 = (c dt)^2 - dr^2$$

Taking the square root of the metric and using the gamma matrices:

$$ ds =  \gamma^0 c dt -  \gamma^i \cdot dr_i$$ 

For time like geodesics: $ ds = 0$. Taking the dual basis:

$$ 0 =  \frac{1}{c}\gamma^0 \partial _t - \gamma^i \cdot \nabla_i$$ 

To convice yourself of the above formula use $\langle dx^i | \partial_j \rangle = \delta^{i}_j $


Multiplying $i \hbar$ both sides and the wave function we obtain: 

$$ 0=i\hbar \gamma^\mu \partial_\mu \psi $$

Using the same technique for static spherical symmetrical line element:
 

$$ds^2 = -(1-\frac{2M}{r}) dt^2 + (1-\frac{2M}{r})^{-1}  dr^2 $$  

Taking square root and setting $ds = 0$ (as done above):

$$ 0 = \gamma^0 dt - \gamma^i \cdot (1-\frac{2M}{r})^{-1} dr $$

Again using the dual basis and multiplying $\psi$:

$$ 0 = \gamma^0 \partial_t \psi- \gamma^i \cdot (1-\frac{2M}{r}) \nabla \psi $$

Multiplying $i \hbar$ we have:

$$ i \gamma^0 \hat E \psi =  \gamma^i\cdot (1-\frac{2M}{r}) \nabla \psi$$

Question

I then saw the wikipedia entry and saw a different equation. Why is my method wrong?

P.S: I am a postgraduate who tried his hand at quantum field theory at curved spacetime and genuinely want to know why I'm wrong? (I am not advocating my position)

asked May 2 in Theoretical Physics by Asaint (30 points) [ revision history ]
edited May 2 by Asaint

You mess too much with square roots, time like geodesics, multiplying by $\hbar$, $\psi$, etc. Although $\hbar$ is a small number, you must multiply by zero instead. Then everything will be alright.

This not about quantum field theory but about a single particle in curved space.

For a modern view of quantum field theory in curved space, I recommend Fredenhagen & Rejzner, QFT on curved spacetimes : axiomatic framework and examples.

This derivation is a pile of errors. First, $dr$ in $ds^2$ is a particle position increment while moving, i.e., it is implied $dr(t)=v(t)dt$. The "square root" with gamma-matrices is not proportional to a unit $4\times 4$ matrix $I$. The third equation contains a gradient whose coordinates are implied to be time-independent derivatives; this operator equation is wrong. Etc., etc. The final step transforms $\partial_t$ into $\hat{E}$, but it does not change the right-hand side. Complete rubbish.

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