Relativistic Phase Space

The invariant momentum measure for a free particle is proportional to $d^3p/(2E_{\mathbf p})$, not $d^3p$ alone. The energy denominator is the Jacobian of restricting four-momentum to its mass shell. This page derives that one-particle measure and explains why it appears in state normalization; it does not identify it with a full six-dimensional classical phase-space volume or with a scattering probability by itself.

Required background. The Energy–Momentum Relation supplies the mass shell, and Four-Vectors supplies momentum transformations. The calculation also uses delta distributions at simple roots.

Restricting the invariant four-momentum measure

First retain constants. Write $p^0=E/c$ and $p^2=(p^0)^2-\mathbf p^2$. For a positive-energy particle of mass $m$, consider a scalar test function $F$ and the integral

$$I[F]=\int d^4p\,\theta(p^0)\, \delta(p^2-m^2c^2)F(p).$$

The four-volume measure and delta argument are Lorentz invariant. The step function is invariant on the massive or nonzero massless shell under proper orthochronous Lorentz transformations. It selects a time orientation; time reflection does not preserve that choice.

Let $\omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2c^2}=E_{\mathbf p}/c$. The roots in $p^0$ are $\pm\omega_{\mathbf p}$, so

$$\delta\big((p^0)^2-\omega_{\mathbf p}^2\big) =\frac{\delta(p^0-\omega_{\mathbf p})+ \delta(p^0+\omega_{\mathbf p})}{2\omega_{\mathbf p}}.$$

Performing the $p^0$ integral gives

$$I[F]=\int\frac{c\,d^3p}{2E_{\mathbf p}} F(E_{\mathbf p}/c,\mathbf p).$$

The factor $c$ occurs because the integration coordinate was $p^0$, not $E$. Multiplying an invariant measure by a fixed constant preserves invariance, so $d^3p/(2E_{\mathbf p})$ is also invariant. Numerical normalization must still be declared when using the measure in Fourier transforms.

Checking the boost Jacobian directly

For a passive boost along $x$,

$$p'_x=\gamma(p_x-vE_{\mathbf p}/c^2), \qquad p'_y=p_y,\quad p'_z=p_z.$$

On the mass shell, $\partial E_{\mathbf p}/\partial p_x=c^2p_x/E_{\mathbf p}$. The Jacobian is triangular in the transverse rows, giving

$$\det\frac{\partial\mathbf p'}{\partial\mathbf p} =\gamma\left(1-\frac{vp_x}{E_{\mathbf p}}\right) =\frac{E_{\mathbf p'}}{E_{\mathbf p}}.$$

It follows immediately that $d^3p'/E_{\mathbf p'}=d^3p/E_{\mathbf p}$. This check also shows why boosting a distribution while keeping its momentum volume element fixed gives the wrong normalization.

Covariant normalization of a scalar state

For the remainder of this page use $c=\hbar=1$ and $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. Define

$$d\Pi_p=\frac{d^3p}{(2\pi)^3\,2E_{\mathbf p}}.$$

A convenient scalar momentum basis obeys

$$\begin{aligned} \langle\mathbf p'|\mathbf p\rangle &=(2\pi)^3\,2E_{\mathbf p} \delta^{(3)}(\mathbf p'-\mathbf p),\\ I_1&=\int d\Pi_p\,|\mathbf p\rangle\langle\mathbf p|. \end{aligned}$$

The right-hand side of the overlap is invariant because the delta distribution transforms with the inverse momentum Jacobian. A normalized wave packet $|f\rangle=\int d\Pi_p f(\mathbf p)|\mathbf p\rangle$ has

$$\langle f|f\rangle=\int d\Pi_p\,|f(\mathbf p)|^2=1.$$

For a spin-zero particle the amplitude can transform by pullback, $f'(p')=f(p)$, with no additional square-root Jacobian. In a basis normalized instead by $\delta^{(3)}(\mathbf p'-\mathbf p)$, such factors move into the state transformation. These are equivalent descriptions after consistent rescaling. Spin adds a momentum-dependent rotation on the spin indices; it does not change the mass-shell measure.

This construction is the one-particle part of the normalization used in Tong’s free-field notes and Weinberg’s representation treatment. It neither defines a local position probability density nor performs field quantization.

Limits and uses

For $|\mathbf p|\ll m$,

$$d\Pi_p=\frac{d^3p}{(2\pi)^3\,2m} \left(1-\frac{\mathbf p^2}{2m^2}+\cdots\right).$$

The leading constant can be absorbed into a nonrelativistic wavefunction’s normalization. The momentum-dependent correction cannot be discarded when working to the corresponding relativistic order.

For a massless particle in spherical momentum coordinates,

$$d\Pi_p=\frac{p\,dp\,d\Omega}{2(2\pi)^3}.$$

The apparent $1/E$ singularity does not make a bounded radial integral diverge at $p=0$: $\int_0^\epsilon p\,dp$ is finite. Singular amplitudes or additional propagator factors can still produce infrared divergences. The measure alone does not decide the convergence of a scattering calculation.

Multiparticle kinematics uses products of these measures together with a four-momentum conservation delta distribution. A cross section additionally requires an incident flux and a dynamical transition amplitude. Counting available final momenta is only one part of the prediction.

Exercises

1. With constants restored, integrate the mass-shell delta distribution over $p^0$ without the $\theta(p^0)$ factor. What changes?

Solution

Both roots contribute with positive Jacobian weights:

$$\int\frac{c\,d^3p}{2E_{\mathbf p}} \left[F(E_{\mathbf p}/c,\mathbf p)+F(-E_{\mathbf p}/c,\mathbf p)\right].$$

This is a sum over two sheets, not a signed charge or KG norm.

2. In natural units compute the invariant measure of the massless momentum ball $0\leq p\leq K$ in one fixed frame. Is the ball itself invariant?

Solution

The integral is $4\pi\int_0^K p\,dp/[2(2\pi)^3]=K^2/(8\pi^2)$. A boost maps the ball to a different region. Invariance equates the integral over the original region to the integral over its transformed image, not to an independently imposed $p'\leq K$ cutoff.

3. Define $|\mathbf p\rangle_N=|\mathbf p\rangle/\sqrt{2E_{\mathbf p}}$. Find its overlap and completeness relation.

Solution

The overlap is $(2\pi)^3\delta^{(3)}(\mathbf p'-\mathbf p)$, and $I_1=\int d^3p\,|\mathbf p\rangle_N{}_N\langle\mathbf p|/(2\pi)^3$. Factors of $2E$ have moved to the basis vectors, not disappeared physically.

References

• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940 — state normalization and scattering phase space.
• D. Tong, Lectures on Quantum Field Theory, University of Cambridge, 2006, section 2.4.1 — relativistic normalization of one-particle states.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, chapter 2 — invariant one-particle representations and measures.