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

+ 1 like - 0 dislike

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:

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


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, 2018 in Theoretical Physics by Asaint (90 points) [ revision history ]
edited May 2, 2018 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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