- #1

RedX

- 970

- 3

[tex]\mathcal L= \partial_\mu \phi \partial^\mu \phi - m^2 \phi \phi [/tex]

So in order to make all the terms have the same units, you can try either:

[tex]\mathcal L=\frac{\hbar^2}{c^2} \partial_\mu \phi \partial^\mu \phi - m^2 \phi \phi [/tex]

or

[tex]\mathcal L= \partial_\mu \phi \partial^\mu \phi - \frac{c^2}{\hbar^2}m^2 \phi \phi [/tex]

But once both terms are the same units, you can multiply it by any other unit, for example, 1/c:

[tex]\mathcal L=\frac{1}{c} \partial_\mu \phi \partial^\mu \phi - \frac{c}{\hbar^2}m^2 \phi \phi [/tex]

Once both terms are of the same units, [tex]\phi [/tex] takes on whatever units required to make the Lagrangian density have units of Planck's constant (units of the action) divided by the units of a volume of space (i.e., depends on how many dimensions of spacetime you specify).

Once you specify units of the field [tex]\phi(x) [/tex], you can find the units of the source current for the field [tex]J(x) [/tex].

Also, what are the units of the propagator [tex]1/(p^2-m^2) [/tex]?

I want to follow the [tex]\hbar[/tex]'s really closely, because they are small quantities and 1-loop diagrams are smaller than tree diagrams by a factor of [tex]\hbar[/tex], but they're not really that much smaller (they're 1/137 smaller for QED, not [tex]\hbar [/tex] smaller), so something has to happen.