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What are the underlying principles for choosing signs for the momentum operators in QM/QFT?

Let's say, for the $(+,-,-,-)$ metric convention we have $\partial^\mu = (+\partial_0,-\nabla)$. Why not $\partial^\mu = (-\partial_0,+\nabla)$? Is it due to the Canonical Commutation Relations $[p^\mu,x^\nu]=ig^{\mu\nu}$?

How then does one decide on the momentum operator components in $(-,+,+,+)$ metric? Should we fix CCR as $[p^\mu,x^\nu]=-ig^{\mu\nu}$ or $[p^\mu,x^\nu]=+ig^{\mu\nu}$?

Is there any cheatsheet (paper/textbook) with sign conventions for momentum operators, EM field tensor, etc?

nox
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stanislav-iablokov
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1 Answers1

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  1. First of all, note that $$ \partial_{\mu}~:=~\frac{\partial}{\partial x^{\mu}} , \qquad \mu~\in~\{0,1,2,3\},\tag{A}$$ and $$ \partial^{\mu}~:=~g^{\mu\nu}\partial_{\nu}, \qquad \mu~\in~\{0,1,2,3\}.\tag{B}$$ More generally, we use the metric to raise and lower indices. Eq. (B) explains OP's first eq.

  2. The standard CCR reads $$ [\hat{x}^j,\hat{p}_k]~=~i\hbar\delta^j_k \mathbb{1}, \qquad j,k~\in~\{1,2,3\}. \tag{C}$$ In the Schrödinger representation $$\hat{x}^j~=~x^j, \qquad \hat{p}_j~=~\frac{\hbar}{i}\frac{\partial}{\partial x^j}, \qquad j~\in~\{1,2,3\}. \tag{D}$$ OP's CCR and momentum operator should agree with eqs. (C) & (D).

  3. Note that time $x^0$ is a parameter not an operator in quantum mechanics, so that $$ [x^0,\hat{p}_0]~=~0, \tag{E}$$ cf. e.g. this & this Phys.SE posts.

Qmechanic
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