Relativistic Normalization

A normalization convention changes the numerical value of a momentum-space matrix element. It also changes the measure used to convert that element into a probability. This page carries the factors through state rescaling, box-normalized Dirac waves, and external scattering legs. Relativistic Phase Space owns the mass-shell measure and its Lorentz-invariance derivation.

Required background. Relativistic Phase Space supplies the measure and completeness relation; Free Dirac Spinors supplies the spinor norms. Helpful background. Wigner Classification explains how spin labels and normalized states transform under boosts.

Momentum bases and their measures

Use $\hbar=c=1$, $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}>0$, and $d\Pi_p=d^3p/[(2\pi)^3\,2E_{\mathbf p}]$. The spin index $s$ is discrete. Write the covariant basis as $|p,s\rangle_C$, with

$${}_C\langle p',r|p,s\rangle_C =2E_{\mathbf p}(2\pi)^3\delta^3(\mathbf p'-\mathbf p)\delta_{rs}.$$

Two useful rescalings are

$$\begin{aligned} |p,s\rangle_N&=\frac{|p,s\rangle_C}{\sqrt{2E_{\mathbf p}}},\\ |p,s\rangle_\delta&=\frac{|p,s\rangle_C} {\sqrt{(2\pi)^3\,2E_{\mathbf p}}}. \end{aligned}$$

Their overlaps are respectively $(2\pi)^3\delta^3\delta_{rs}$ and $\delta^3\delta_{rs}$. Consequently the same one-particle identity is

$$\begin{aligned} I_1&=\sum_s\int d\Pi_p\,|p,s\rangle_C{}_C\langle p,s|\\ &=\sum_s\int\frac{d^3p}{(2\pi)^3} |p,s\rangle_N{}_N\langle p,s|\\ &=\sum_s\int d^3p\,|p,s\rangle_\delta{}_\delta\langle p,s|. \end{aligned}$$

The subscript $N$ names a convention, not a nonrelativistic approximation: it is legitimate at any momentum. All three bases are distributional. None has a finite norm as a plane-wave vector in infinite volume.

For $|f\rangle=\sum_s\int d\Pi_p f_s(p)|p,s\rangle_C$, the same packet has $N$-basis coefficient $a_s=f_s/\sqrt{2E_{\mathbf p}}$ and $\delta$-basis coefficient $b_s=f_s/\sqrt{(2\pi)^3\,2E_{\mathbf p}}$. Then

$$\|f\|^2=\sum_s\int d\Pi_p|f_s|^2 =\sum_s\int\frac{d^3p}{(2\pi)^3}|a_s|^2 =\sum_s\int d^3p|b_s|^2.$$

The spin rotation in a Lorentz transformation is unitary on the spin indices; it does not remove these scalar rescaling factors.

Dirac spinors in a finite normalization box

Use a periodic cubic box of volume $\mathcal V=L^3$ as a normalization regulator, with $\mathbf p=2\pi\mathbf n/L$. This regulator singles out a frame; physical rates are obtained after its volume cancels and the continuum limit is taken. A positive-frequency mode with one unit of Dirac Hilbert norm is

$$\psi_{p,s}(t,\mathbf x)= \frac{u_s(p)e^{-iEt+i\mathbf p\cdot\mathbf x}} {\sqrt{2E\mathcal V}}.$$

Because $u^\dagger u=2E$ and $\bar u\gamma^\mu u=2p^\mu$,

$$\rho=\frac1{\mathcal V}, \qquad \mathbf j=\frac{\mathbf p}{E\mathcal V}.$$

For a beam directed at a stationary source, the incident number flux is $v/\mathcal V$, where $v=|\mathbf p|/E$. In SI units the group velocity is $|\mathbf p|c^2/E$; the box wave still has total probability one.

The density of final momentum modes is

$$\sum_{\mathbf n}\longrightarrow \mathcal V\int\frac{d^3p}{(2\pi)^3}.$$

Equivalently, continuum normalization corresponds to $|p,s\rangle_C=\sqrt{2E\mathcal V}\,|p,s\rangle_{\rm box}$ when the delta distribution is discretized by $(2\pi)^3\delta^3(\mathbf p'-\mathbf p)\to \mathcal V\delta_{\mathbf n'\mathbf n}$.

Do not replace $2E$ by $2m$ just because $\bar uu=2m$. The latter uses the Dirac adjoint and is a Lorentz scalar; it is not the positive Hilbert norm used to normalize a wave in the box. The convention with $\bar uu=1$ also fails as a direct massless normalization, whereas $u^\dagger u=2E$ remains useful at $E>0$.

Rescaling an amplitude and its probability

Let $O$ be an operator whose one-particle momentum kernel is $O_C(p',p)$. Basis rescaling gives

$$O_N(p',p)=\frac{O_C(p',p)}{\sqrt{2E_{\mathbf p'}\,2E_{\mathbf p}}}.$$

For example, the transition amplitude from a normalized packet $f$ to a normalized packet $g$ is

$$\langle g|O|f\rangle =\sum_{r,s}\int d\Pi_{p'}\,d\Pi_p\, g_r(p')^*O_C(p',r;p,s)f_s(p).$$

Substituting $f=\sqrt{2E}\,a$, $g=\sqrt{2E'}\,a_g$ and the expression for $O_N$ gives the same result with two measures $d^3p/(2\pi)^3$. This is an exact change of representation.

An even shorter probability check fixes a normalized initial state $|i\rangle$ and sets $B_C(p,s)={}_C\langle p,s|O|i\rangle$. Since $B_N=B_C/\sqrt{2E}$,

$$\sum_s\int d\Pi_p\,|B_C|^2 =\sum_s\int\frac{d^3p}{(2\pi)^3}|B_N|^2.$$

These are the same projected transition norm. Inserting $B_N$ into the left-hand measure would count the factor $1/(2E)$ twice.

For a multiparticle amplitude with covariant normalization on every external leg, each external ket or bra contributes its own square root. In a translation-invariant scattering problem define

$${}_C\langle f|S-I|i\rangle_C =i(2\pi)^4\delta^4(P_f-P_i)\mathcal M_{fi}.$$

Using $N$-normalized external states changes the coefficient to

$$\mathcal A_{N,fi} =\frac{\mathcal M_{fi}} {\sqrt{\prod_{\text{all external legs }a}2E_a}}.$$

It would be inconsistent to substitute $\mathcal A_N$ into a cross-section formula derived for $\mathcal M$ without also translating the flux and final-state measures. Spin sums and identical-particle counting are additional operations; neither follows from this square-root rule alone.

A fixed external potential is a different kinematic problem. A static localized source conserves energy but can absorb momentum, so its matrix element normally contains a single energy delta, not a four-momentum delta. The S-matrix bridge places the invariant-amplitude convention in its asymptotic-state setting.

Exercises

1. A scalar packet has constant covariant coefficient $f=C$ on a finite momentum region $\mathcal R$ and vanishes outside. Find its normalization and its $N$-basis coefficient. Does the latter remain constant?

Solution

$|C|^{-2}=\int_{\mathcal R}d\Pi_p$ and $a(\mathbf p)=C/\sqrt{2E_{\mathbf p}}$ on the region. It is generally not constant. Equal coefficient functions in two differently normalized bases describe different packets.

2. Rescale massive spinors to $\widehat u=u/\sqrt{2m}$. Find their spin sum and the denominator needed for a unit box wave.

Solution

The covariant sum becomes $\sum_s\widehat u_s\overline{\widehat u}_s=(p\!\!\!/+m)/(2m)$, and $\widehat u^\dagger\widehat u=E/m$. The unit box wave is $\widehat u e^{-ip\cdot x}/\sqrt{(E/m)\mathcal V}$. It is exactly the earlier wave. Changing the spinor sum while retaining the earlier denominator would change the norm.

3. For a $2\to2$ amplitude with all four energies equal to $E$ in the center-of-momentum frame, how are $\mathcal A_N$ and $\mathcal M$ related?

Solution

$\mathcal A_N=\mathcal M/(4E^2)$. Its squared modulus is smaller by $16E^4$, compensated by the corresponding normalization factors in the probability and flux. The rescaling is not a dynamical suppression.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, chapters 5–6. doi:10.1017/9781139540940. External-state normalization and scattering rates.
• Tong, David. Lectures on Quantum Field Theory. University of Cambridge, 2006, sections 2.4.1 and 5.1. Relativistic normalization and Dirac field normalization.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995, chapters 2–3. Particle-state conventions and scattering.