Poincaré Covariance and the Spectrum Condition

Poincaré covariance and positive energy constrain different parts of a relativistic quantum theory. Covariance specifies how states, fields, observables, and correlation functions transform. The spectrum condition adds that the joint spectrum of the translation generators lies in the closed forward cone. With an invariant vacuum, those two inputs make the scalar vacuum two-point function translation and Lorentz covariant and give its Fourier transform future-cone support. The free real scalar makes that support visible on one positive-energy mass shell; proof-level domain, analyticity, and reconstruction statements remain outside this page.

Required background. One-Particle States: Mass, Spin, and Relativistic Normalization supplies future mass shells, the unitary particle action, and the invariant momentum measure used in the scalar check.

Helpful background. Lorentz Field Representations and Poincaré Particle Representations separates finite-dimensional field representations from unitary state representations. It uses $U_{\mathrm{linked}}(a)=e^{-iP\cdot a}$ for the active configuration translation $x\mapsto x-a$; at fixed future-directed $P$, the convention here is $U_{\mathrm{here}}(a)=U_{\mathrm{linked}}(-a)=e^{+iP\cdot a}$ and gives the conjugated-field translation $x\mapsto x+a$. Spectra, Resolvents, Spectral Measures, and Functional Calculus supplies the joint spectral-measure language.

Covariant states, fields, and translations

Work in four-dimensional Minkowski spacetime and initially restrict the spacetime symmetry to the connected proper orthochronous Poincaré group, or its cover when spin requires it. Assume that the physical state space carries a strongly continuous unitary representation $U(\Lambda,a)$. Its translation subgroup is abelian, so the spectral theorem supplies four strongly commuting self-adjoint generators $P^\mu$ and a joint projection-valued measure $E$:

$$U(a)=e^{iP\cdot a} =\int_{\mathbb R^{1,3}} e^{ip\cdot a}\,E(\mathrm dp).$$

This is the translation convention fixed by the chapter. Together with the site Fourier pair, it makes a momentum-$p$ contribution to a field correlation carry the phase $e^{-ip\cdot x}$. The opposite exponential convention is equally possible, but changing it requires changing the generator and Fourier crosswalk together.

For a real scalar field, the corresponding covariance law is

$$U(\Lambda,a)\,\phi(x)\,U(\Lambda,a)^{-1} =\phi(\Lambda x+a).$$

A field multiplet may instead carry a finite-dimensional Lorentz matrix on its indices. That matrix is generally not unitary, because it acts on finitely many field components rather than on the physical Hilbert space. By contrast, $U(\Lambda,a)$ is unitary on physical states. Conflating those two representations obscures both probability conservation and the role of field intertwiners. The distinction and the translation generators are developed pedagogically in Schwartz 2014, § 8.1, pp. 109–112.

Parity and time reversal are not part of the connected group used here. Their implementation, when present, is additional structure.

The joint spectrum selects the forward cone

The joint spectrum is the support of the projection-valued measure,

$$\operatorname{sp}P:=\operatorname{supp}E.$$

In the site’s $(+---)$ convention, the spectrum condition is

$$\boxed{ \operatorname{sp}P\subset\overline V_+, \qquad \overline V_+ =\{p\in\mathbb R^{1,3}:p^0\ge0,\ p^2\ge0\}. }$$

Covariance and this inclusion do separate jobs. Covariance under the connected Lorentz group makes the joint spectrum Lorentz invariant. It does not choose the future cone over the past cone. Positive energy supplies that time orientation.

There is a useful qualification behind the phrase “positive energy.” Without Lorentz covariance, a bound $P^0\ge0$ refers only to one chosen frame and does not control the spatial generators. With exact connected Lorentz covariance and an invariant joint spectrum, however, nonnegativity of $P^0$ in the whole representation rules out both past-directed and spacelike spectral points: a suitable boost would give a spacelike point negative energy. Under those additional hypotheses, the Hamiltonian bound and the forward-cone condition are two forms of the same invariant restriction.

The cone is closed under addition. If $p,q\in\overline V_+$, then

$$(p+q)^0=p^0+q^0 \ge |\mathbf p|+|\mathbf q| \ge |\mathbf p+\mathbf q|,$$

so $p+q\in\overline V_+$. This elementary fact is why sums of positive-energy particle momenta remain compatible with the spectrum condition. It does not prove that a particular interacting theory exists or that its spectrum contains isolated particles.

Covariance and the positive-energy condition are stated as distinct hypotheses in Fewster and Rejzner 2019, arXiv v2, §§ 4.1 and 5.1, printed pp. 13–15 and 24–25 (PDF).

Vacuum two-point functions inherit spectral support

Now add a normalized invariant vacuum $|\Omega\rangle$ and a Hermitian scalar field understood as an operator-valued distribution:

$$U(\Lambda,a)|\Omega\rangle=|\Omega\rangle, \qquad P^\mu|\Omega\rangle=0.$$

Vacuum existence is an additional assumption. The equations do not imply that the vacuum is unique, separated by a gap, or contained in the same representation as every physically interesting state.

For the scalar field, covariance directly gives

$$W(x,y):=\langle\Omega|\phi(x)\phi(y)|\Omega\rangle =W(x-y), \qquad W(\Lambda z)=W(z),$$

where $z=x-y$ and $\Lambda$ belongs to the connected Lorentz group. After smearing so that the vectors and products are defined, translation covariance then gives the schematic but useful spectral calculation

$$\begin{aligned} W(z) &:=\langle\Omega|\phi(z)\phi(0)|\Omega\rangle \\ &=\langle v|e^{-iP\cdot z}|v\rangle \\ &=\int_{\overline V_+}e^{-ip\cdot z}\,\mathrm d\mu_v(p), \end{aligned} \qquad \mu_v(\Delta) :=\langle v|E(\Delta)|v\rangle,$$

where $|v\rangle$ is formal shorthand for the vector created from the vacuum by the smeared field. Positivity of $\mu_v$ uses the physical Hilbert inner product. A nonzero one-point function contributes an allowed atom at $p=0$; centering the field removes that disconnected term.

Because the site Fourier transform uses $e^{+ik\cdot z}$, the last line implies

$$\operatorname{supp}\widetilde W\subset\overline V_+.$$

This is a support statement about the non-time-ordered, fixed-order Wightman function. A time-ordered Feynman two-point function is a different distribution and does not have one-sided forward-cone support. For higher Wightman functions, cumulative momentum support and the associated complex analytic domains require a careful statement of field domains and smearing; those belong to the rigorous continuation.

The free scalar sits on one forward mass shell

For the required first application, take a free real scalar with $m>0$ and

$$E_{\mathbf p}=\sqrt{|\mathbf p|^2+m^2}, \qquad p=(E_{\mathbf p},\mathbf p).$$

Its vacuum Wightman function is

$$\begin{aligned} W_0(z) &=\int\frac{\mathrm d^3\mathbf p}{(2\pi)^3\,2E_{\mathbf p}} e^{-ip\cdot z} \\ &=\int\frac{\mathrm d^4p}{(2\pi)^3} \theta(p^0)\delta(p^2-m^2)e^{-ip\cdot z}. \end{aligned}$$

The equality is a normalization check, not a mnemonic. Since

$$\delta\!\left((p^0)^2-E_{\mathbf p}^2\right) =\frac{\delta(p^0-E_{\mathbf p})+\delta(p^0+E_{\mathbf p})} {2E_{\mathbf p}},$$

$\theta(p^0)$ selects the positive root and supplies the factor $1/(2E_{\mathbf p})$. The four-dimensional measure

$$\mathrm d^4p\,\theta(p^0)\delta(p^2-m^2)$$

is invariant under the proper orthochronous Lorentz group: $p^2$ and $\mathrm d^4p$ are invariant, while the chosen group preserves the time orientation of a nonspacelike momentum.

Fourier transforming the four-dimensional representation gives

$$\boxed{ \widetilde W_0(k) =2\pi\,\theta(k^0)\delta(k^2-m^2), \qquad \operatorname{supp}\widetilde W_0 =\mathscr O_m^+\subset\overline V_+. }$$

The coefficient follows from $(2\pi)^4/(2\pi)^3=2\pi$. In four dimensions $[W_0]=2$ and $[\widetilde W_0]=-2$, consistent with the mass dimension $-2$ of $\delta(k^2-m^2)$. These checks fix the Fourier normalization independently of the support argument.

This elementary free field creates a one-particle state from the vacuum, so its two-point function has one mass shell and no multiparticle continuum. Multiparticle sums still illustrate closure of $\overline V_+$, but a continuum appears in two-point data for suitable composite or interacting operators. Schwartz 2014, § 24.2.1, pp. 467–468 develops the positive spectral support and the free mass-shell measure. The Källén–Lehmann Representation treats the later isolated-atom-plus-continuum structure and its positivity assumptions.

What the spectrum condition does not guarantee

• Not vacuum uniqueness or a mass gap. The origin may contain more than one invariant vacuum sector, and continuous support may approach it or a nominal particle mass.
• Not an isolated stable particle. A forward-cone spectrum can contain only continuum support in a channel. Resonances and infraparticles require more refined spectral or analytic diagnoses.
• Not locality or clustering. Energy–momentum support does not make spacelike observables commute and does not determine the long-distance decay of connected correlations.
• Not an interacting construction. A formally covariant Lagrangian with positive classical energy does not by itself produce a Hilbert-space QFT satisfying the joint spectrum condition.
• Not asymptotic completeness. Even isolated particle shells do not show that all physical states are built from asymptotic particles.
• Not Feynman support. The momentum-space Feynman distribution $i/(p^2-m^2+i0)$ is not supported only on $\mathscr O_m^+$; its boundary prescription answers a different ordering question.
• Not a ban on negative-frequency field factors. In a free-field expansion the factor $e^{+ip\cdot x}$ accompanies a creation operator. The state $a^\dagger(\mathbf p)|0\rangle$ still has future-directed momentum $p$, so a phase label is not a negative-energy state.

The spectrum condition is an input to later spin–statistics and CPT theorems, but it is not either theorem by itself.

Check your understanding

Check 1: recover the invariant shell measure

Start from $\theta(p^0)\delta(p^2-m^2)$. Show that integrating over $p^0$ produces $1/(2E_{\mathbf p})$.

Answer. Factor the argument as $(p^0-E_{\mathbf p})(p^0+E_{\mathbf p})$. The delta-function Jacobian is $2E_{\mathbf p}$ at either root, and $\theta(p^0)$ retains only $p^0=E_{\mathbf p}$. Thus

$$\int\mathrm dp^0\,\theta(p^0)\delta(p^2-m^2)f(p^0) =\frac{f(E_{\mathbf p})}{2E_{\mathbf p}}.$$

Check 2: separate covariance from time orientation

Why does Lorentz invariance of a spectral set not by itself imply future-cone support?

Answer. The past cone is also invariant under the connected proper orthochronous group, and the union of future and past cones is invariant as well. A positive-energy or time-orientation input is needed to select $\overline V_+$. Conversely, once the joint spectrum is invariant and $P^0$ is nonnegative on the whole representation, spacelike and past-directed points are excluded.

Check 3: distinguish Wightman and Feynman support

Which free two-point distribution has Fourier support exactly on the positive mass shell?

Answer. The non-time-ordered, fixed-order vacuum Wightman function has $\widetilde W_0(k)=2\pi\theta(k^0)\delta(k^2-m^2)$. The Feynman distribution $i/(k^2-m^2+i0)$ is an off-shell boundary value with both energy signs; it should not be substituted into the support claim.

Where to continue

• Separate spectral support from probability conservation: Hilbert Positivity and Unitary Evolution develops the physical inner product, self-adjoint dynamics, and gauge-fixed qualification without treating them as consequences of the forward cone.
• Continue to the rigorous support theorem: Wightman Functions and Spectral Support develops smeared-field domains, cumulative momentum support, and proof-level analytic consequences.
• Generalize the two-point measure: The Källén–Lehmann Representation separates isolated spectral atoms from continuum support under its positivity assumptions.
• Return to the chapter comparison: Structural Principles and Axiom Maps places covariance and the spectrum condition beside positivity, locality, clustering, spin–statistics, CPT, and their failure tests.

References

• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” arXiv:1904.04051v2, 2019, 47 pp.; published in Progress and Visions in Quantum Theory in View of Gravity, edited by Felix Finster, Domenico Giulini, Johannes Kleiner, and Jürgen Tolksdorf, 1–61. Birkhäuser, 2020. DOI. Open manuscript PDF, v2.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.