- #1
Loro
- 80
- 1
I'm trying to fit together my understanding of quantum mechanics, quantum field theory, given my lacking maths education.
In quantum mechanics we have a time displacement operator and a space displacement operator, which are respectively:
[itex]
\hat{T}(t) = e^{-i\hat{H}t}
[/itex]
[itex]
\hat{D}(\underline{x}) = e^{-i\hat{\underline{p}}\cdot\underline{x}}
[/itex]
In quantum field theory there is a spacetime displacement operator:
[itex]
\hat{U}(a_{\mu}) = e^{-i\hat{P}^{\mu}a_{\mu}}
[/itex]
So as I understand this can be written out as:
[itex]
\hat{U}(a_{\mu}) = e^{-i\hat{P}^{0}a_{0}-i\hat{P}^{j}a_{j}}
[/itex]
Or, depending on the metric convention:
[itex]
\hat{U}(a^{\mu}) = e^{-i\hat{P}^{0}a^{0}+i\hat{P}^{j}a^{j}}
[/itex] , or:
[itex]
\hat{U}(a^{\mu}) = e^{+i\hat{P}^{0}a^{0}-i\hat{P}^{j}a^{j}}
[/itex]
Now while the 1st expression is in agreement with what I know from quantum mechanics, the latter two have sign differences, and also what's surprising is that there are sign ambiguities depending on the convention of the metric - as I understand it - depending on it, either time is displaced forwards and space backwards; or time backwards and space forwards.
I'm guessing that for some reason we should only take the first expression, but I don't understand why, and what is the significance of the latter two? Why are t and x in the quantum mechanics formulae for D and T necessarily covariant? Is it that when we multiply vectors like in these formulae - one quantity must be covariant, and the other contravariant?
In quantum mechanics we have a time displacement operator and a space displacement operator, which are respectively:
[itex]
\hat{T}(t) = e^{-i\hat{H}t}
[/itex]
[itex]
\hat{D}(\underline{x}) = e^{-i\hat{\underline{p}}\cdot\underline{x}}
[/itex]
In quantum field theory there is a spacetime displacement operator:
[itex]
\hat{U}(a_{\mu}) = e^{-i\hat{P}^{\mu}a_{\mu}}
[/itex]
So as I understand this can be written out as:
[itex]
\hat{U}(a_{\mu}) = e^{-i\hat{P}^{0}a_{0}-i\hat{P}^{j}a_{j}}
[/itex]
Or, depending on the metric convention:
[itex]
\hat{U}(a^{\mu}) = e^{-i\hat{P}^{0}a^{0}+i\hat{P}^{j}a^{j}}
[/itex] , or:
[itex]
\hat{U}(a^{\mu}) = e^{+i\hat{P}^{0}a^{0}-i\hat{P}^{j}a^{j}}
[/itex]
Now while the 1st expression is in agreement with what I know from quantum mechanics, the latter two have sign differences, and also what's surprising is that there are sign ambiguities depending on the convention of the metric - as I understand it - depending on it, either time is displaced forwards and space backwards; or time backwards and space forwards.
I'm guessing that for some reason we should only take the first expression, but I don't understand why, and what is the significance of the latter two? Why are t and x in the quantum mechanics formulae for D and T necessarily covariant? Is it that when we multiply vectors like in these formulae - one quantity must be covariant, and the other contravariant?