QFT example units, metric, gauge, and pairing conventions

These are conventions for constructed and formal calculations in this page pair. They are not empirical claims, physical postulates, or existence claims about an interacting continuum theory.

Work on a nonempty open coordinate region $U\subseteq {\mathbb{R}}^4$, with coordinates $x^0=ct,x^1,x^2,x^3$ in metres and ${\mathrm{\partial }}_a=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a$ for $a\in \{0,1,2,3\}$. Use the geometric spacetime metric $\eta =\mathrm{diag}(-1,1,1,1)$ and the time coordinate $x^0=ct$ in metres. For Clifford algebra and momentum contraction use the separate algebraic metric $h=-\eta =\mathrm{diag}(1,-1,-1,-1)$; a slash or a square of an energy four-vector uses $h$, never $\eta$.

A physical energy four-vector is $p=(E,c{\mathbf{p}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})$, with each component in joules. Write $M=m{c}^2$ for rest energy in joules and $p^2=h(p,p)=E^2-c^2\mathrm{\mid }{\mathbf{p}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{\mathrm{\mid }}^2$. For two such vectors, $s=(p_1+p_2{)}^2$ is in ${\mathrm{J}}^2$. Natural momentum coordinates are the energy-vector components divided by $\mathrm{\hslash }c$, in ${\mathrm{m}}^{-1}$; they are not substituted into SI formulas without restoring this factor. Here $m\ge 0$ is a mass parameter in kilograms, so $\mu =mc\mathrm{/}\mathrm{\hslash }$ is in ${\mathrm{m}}^{-1}$. The natural-unit field convention below concerns smooth test fields, not pointwise quantum operators. A real scalar field is a map $\phi \in C_c^{\mathrm{\infty }}(U;\mathbb{R})$; a Dirac test field is $\mathrm{\Psi }\in C_c^{\mathrm{\infty }}(U;{\mathbb{C}}^4)$. Fix dimensionless matrices ${\gamma }^a\in M_4(\mathbb{C})$ satisfying ${\gamma }^a{\gamma }^b+{\gamma }^b{\gamma }^a=2h^{ab}{I}_4$, and put $\bar{\mathrm{\Psi }}={\mathrm{\Psi }}^{†}{\gamma }^0$. Adopt dimensionless action normalizations with kinetic terms $\frac{1}{2}{\int }_U{h}^{ab}({\mathrm{\partial }}_a\phi )({\mathrm{\partial }}_b\phi )d^{4}x$ and ${\int }_{U}i\bar{\mathrm{\Psi }}{\gamma }^a{\mathrm{\partial }}_a\mathrm{\Psi }{d}^{4}x$, summing repeated indices from $0$ to $3$. The coefficients are dimensionless and introduce no additional length scale. These conventions assign $[\phi ]={\mathrm{m}}^{-1}$ and $[\mathrm{\Psi }]={\mathrm{m}}^{-3\mathrm{/}2}$: the integrands have length degrees $2[\phi ]-2$ and $2[\mathrm{\Psi }]-1$, while $d^{4}x$ has degree $4$. No dynamics or continuum existence assertion is adopted.

Take SI $e>0$ in coulombs, $\mathrm{\hslash }>0$ in joule-seconds, $c>0$ in metres per second, ${\mu }_0>0$ in $\mathrm{H}{\mathrm{m}}^{-1}=\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{C}}^{-2}$, and ${ϵ}_0=({\mu }_0{c}^2{)}^{-1}$ in $\mathrm{F}{\mathrm{m}}^{-1}$. Define $\alpha =\frac{e^2}{4\pi {ϵ}_0\mathrm{\hslash }c},g=e\sqrt{\frac{{\mu }_{0}c}{\mathrm{\hslash }}}=\sqrt{4\pi \alpha }>0.$ Here $g$ is a dimensionless coupling. It is not the metric and not a gyromagnetic factor.

For gauge notation take $\varphi \in C^1(U;\mathbb{R})$ in volts and $\mathbf{A}\in C^1(U;{\mathbb{R}}^3)$ in tesla-metres. Write $A_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}=(-\varphi \mathrm{/}c,\mathbf{A})$ for the components of the real one-form $A={\sum }_{a=0}^3{A}_{a}d{x}^a$, so every $A_a$ is in tesla-metres. Take $\chi \in C^2(U;\mathbb{R})$ in webers and a charged test field $\psi \in C^1(U;{\mathbb{C}}^N)$, where $N\ge 1$ is fixed. With a constant real charge parameter $q$ in coulombs, define $D_a\psi ={\mathrm{\partial }}_a\psi -iq{A}_a\psi \mathrm{/}\mathrm{\hslash }$ componentwise. The transformed fields are again $C^1$, with $d\chi ={\sum }_a({\mathrm{\partial }}_a\chi )d{x}^a$. In operator notation the conventions are $D_a={\mathrm{\partial }}_a-\frac{iq{A}_a}{\mathrm{\hslash }},\psi \mapsto e^{iq\chi \mathrm{/}\mathrm{\hslash }}\psi ,A\mapsto A+d\chi \mathrm{.}$ For the electron parameter use $q=-e$. The equivalent natural-unit coordinates are $A_{\mathrm{n}\mathrm{a}\mathrm{t}}=A_{\mathrm{S}\mathrm{I}}\mathrm{/}\sqrt{{\mu }_0\mathrm{\hslash }c}$ and $q_{\mathrm{n}\mathrm{a}\mathrm{t}}=q_{\mathrm{S}\mathrm{I}}\sqrt{{\mu }_{0}c\mathrm{/}\mathrm{\hslash }}$.

Finite-matrix examples act on ${\mathbb{C}}^d$ with the operator norm. For Hilbert spaces, use the first-variable-linear pairing $(u,v{)}_1=v^{†}u$; a displayed coordinate contraction $u^{†}v$ is second-variable-linear and equals $\overline{(u,v{)}_1}$. Thus norms, traces, and modulus-squared amplitudes agree under conversion, although the unsquared pairing is conjugated. No amplitude is assigned a probability by these conventions.