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From regular quantum mechanics we have the momentum operator;

$\hat{p} = -i\hbar\nabla = -i\hbar(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}) $

When transitioning to relativistic particle mechanics, we define coordinates

$x^{\mu} =(ct,x,y,z)$

Which I would presume that in index notation for the spatial components

$P^{\mu} = -i\hbar \partial_\mu = -i\hbar \frac{\partial}{\partial x^\mu}$

And therefore the spatial components of $p^\mu = -p_\mu$

And so $P_\mu = i\hbar \partial_\mu$

However my notes say that.

$P^\mu = -i\hbar\frac{\partial}{\partial x_\mu}$

$P_\mu = i\hbar\frac{\partial}{\partial x^\mu}$

So my question is,

What does the momentum relation in QM imply for the index form, are we finding the covariant or contravariant component of momentum, and where my did my notes get the p^mu expression from, where they take a derivative wrt a covariant quantity?

Qmechanic
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jensen paull
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1 Answers1

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Which I would presume that in index notation for the spatial components

$P^{\mu} = -i\hbar \partial_\mu = -i\hbar \frac{\partial}{\partial x^\mu}$

This equation doesn't make sense because the upper index in $P^\mu$ cannot match the lower index in $-i\hbar \partial_\mu$. Instead it should be $$P_{\mu} = -i\hbar \partial_\mu = -i\hbar \frac{\partial}{\partial x^\mu}$$ and everything will be consistent.

Thomas Fritsch
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