Relativistic Normalization

A state overlap, its completeness measure, and its expansion coefficient form one normalization convention. Changing only one changes the physical packet or its computed probability. This table collects the translations derived from Relativistic Phase Space and applied in the scattering normalization discussion.

Required background. Relativistic Phase Space defines the positive-energy measure. Helpful background. Relativistic Normalization derives the amplitude conversions, and Free Dirac Spinors defines the column norms.

States, measures and coefficients transform together

Use $\hbar=c=1$ and $E_p=\sqrt{\mathbf p^2+m^2}>0$. Let

$$d\mu_p=\frac{d^3p}{(2\pi)^3}, \qquad d\Pi_p=\frac{d\mu_p}{2E_p}.$$

In the overlap column below, $\delta^3$ means $\delta^3(\mathbf p'-\mathbf p)$; each row also has a spin factor $\delta_{rs}$ when spin is present. The coefficient is the function multiplying the ket in an expansion using that row’s measure.

Basis	Ket relative to covariant basis	Overlap
Covariant $C$	$\lvert p,s\rangle_C$	$2E_p(2\pi)^3\delta^3$
$N$	$\lvert p,s\rangle_C/\sqrt{2E_p}$	$(2\pi)^3\delta^3$
Bare delta $\delta$	$\lvert p,s\rangle_C/\sqrt{(2\pi)^3\,2E_p}$	$\delta^3$

Use each basis with its matching measure and coefficient:

Basis	Completeness measure	Packet coefficient
Covariant $C$	$d\Pi_p$	$f_s(p)$
$N$	$d\mu_p$	$a_s=f_s/\sqrt{2E_p}$
Bare delta $\delta$	$d^3p$	$b_s=f_s/\sqrt{(2\pi)^3\,2E_p}$

In each row the identity is $I_1=\sum_s\int(\text{measure})\, |p,s\rangle\langle p,s|$. The same normalized packet obeys

$$\begin{aligned} 1&=\sum_s\int d\Pi_p\,|f_s(p)|^2\\ &=\sum_s\int d\mu_p\,|a_s(p)|^2\\ &=\sum_s\int d^3p\,|b_s(p)|^2. \end{aligned}$$

The label $N$ is a normalization choice, valid at relativistic momenta. All three plane-wave bases are distributional; none is a finite-norm state in infinite volume. A square-integrable packet is the appropriate object for the norm test.

For a spinless particle under the passive momentum change $p'=\Lambda p$, the covariant coefficient transforms as $f'(p')=f(p)$. Thus

$$a'(p')=\sqrt{\frac{E_p}{E_{p'}}}\,a(p), \qquad d^3p'=\frac{E_{p'}}{E_p}\,d^3p.$$

The factors cancel in the norm. Spin introduces a unitary rotation of the spin labels; it does not remove the measure factor. See Wigner Classification for the representation convention.

Spinor columns and unit box waves

The accepted spinor conventions use $u^\dagger u=v^\dagger v=2E$, $\bar uu=2m$, and $\bar vv=-2m$. These are column identities, distinct from momentum-state overlaps.

In a periodic cube of volume $\mathcal V=L^3$ and $\mathbf p=2\pi\mathbf n/L$, a positive-frequency wave of unit Dirac Hilbert norm is

$$\psi_{p,s}(x)= \frac{u_s(p)e^{-ip\cdot x}}{\sqrt{2E_p\mathcal V}}.$$

Its density and current are $\rho=1/\mathcal V$ and $\mathbf j=\mathbf p/(E_p\mathcal V)$. The required denominator uses $u^\dagger u$, not $\bar uu$.

Conversion	Rule
Momentum sum to integral	$\sum_{\mathbf n}\longrightarrow\mathcal V\int d\mu_p$
Delta distribution to box delta	$(2\pi)^3\delta^3(\mathbf p'-\mathbf p)\longrightarrow\mathcal V\delta_{\mathbf n'\mathbf n}$
Covariant ket to unit box ket	$\lvert p,s\rangle_C=\sqrt{2E_p\mathcal V}\,\lvert p,s\rangle_{\rm box}$
Spinor rescaled by real positive $r(p)$	Replace $u$ by $r u$ and the box denominator by $r\sqrt{2E_p\mathcal V}$

The last row leaves the wave unchanged. For example, $\widehat u=u/\sqrt{2m}$ requires a denominator $\sqrt{(E_p/m)\mathcal V}$, for $m>0$. It is not a massless normalization. A finite periodic box selects a frame; it is a regulator for state counting.

External amplitudes and final-state measures

For translation-invariant scattering, with every external state covariantly normalized, define

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

If all $N_{\rm ext}$ external legs use the $N$ basis instead, the coefficient of the same delta distribution is

$$\mathcal A_{N,fi} =\frac{\mathcal M_{fi}} {\prod_{\ell=1}^{N_{\rm ext}}\sqrt{2E_\ell}}.$$

For bare-delta external states it is $\mathcal A_{\delta,fi} =(2\pi)^{-3N_{\rm ext}/2}\mathcal A_{N,fi}$. Do not insert either rescaled coefficient into a rate formula derived for $\mathcal M$ while retaining that formula’s original measures and flux.

For a fixed normalized initial state, let $B_C(p,s)={}_C\langle p,s|O|i\rangle$. The conversion has a particularly direct check:

$$\sum_s\int d\Pi_p\,|B_C|^2 =\sum_s\int d\mu_p\, \left|\frac{B_C}{\sqrt{2E_p}}\right|^2.$$

The projected transition norm is unchanged. See Relativistic Normalization for the corresponding two-packet kernel.

For reference, the covariant multiparticle phase-space convention is

$$d\Phi_n(P) =(2\pi)^4\delta^4\!\left(P-\sum_{a=1}^n k_a\right) \prod_{a=1}^n d\Pi_{k_a},$$

and the two-beam incident flux is

$$\mathcal F=4\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}.$$

The $(2\pi)^4$ belongs inside this definition of $d\Phi_n$. Spin sums, initial-state population averages, and identical-final-state counting are additional operations, not normalization rescalings.

For a prescribed static localized source, energy is conserved while momentum can be transferred to the background. The amplitude usually carries an energy delta alone. Translate that problem using its fixed-source scattering convention; do not reinterpret it as a dynamical $2\to2$ invariant amplitude without a recoil and state-normalization analysis.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, chapters 5–6. doi:10.1017/9781139540940. Covariant amplitudes and scattering rates.
• Tong, David. Lectures on Quantum Field Theory. University of Cambridge, 2006, sections 2.4.1 and 5.1. Scalar state normalization and Dirac field normalization.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995, chapters 2–3. doi:10.1017/CBO9781139644167. State normalization and Lorentz transformations.