Tube Domains, Complex Lorentz Covariance, and Analyticity

The spectrum condition is an analyticity statement in disguise. After passing to relative coordinates, a Wightman distribution whose Fourier transform is supported in future cones is the boundary value of a holomorphic function when each coordinate acquires a past-directed imaginary part. Complex Lorentz covariance can enlarge this primitive tube, but only along orbits on which the continued representation and the holomorphic function are well-defined.

Required background. Wightman functions and spectral support supplies the cone-supported distributions; holomorphic functions and Cauchy theory supplies several-complex-variable analyticity; and tempered distributions and Fourier calculus supplies Fourier–Laplace boundary values.

Helpful background. Branches, sheets, continuation, and monodromy helps distinguish a local continuation from a globally single-valued formula.

From spectral support to a tube

Use relative coordinates $\xi_j=x_j-x_{j+1}$ and the convention $w(\xi)=\int e^{-iq\cdot\xi}\widehat w(q)\,\mathrm d^{4(n-1)}q$. If $\operatorname{supp}\widehat w_n\subseteq(\overline V_+)^{n-1}$, set

$$z_j=\xi_j-i\eta_j, \qquad \eta_j\in V_+.$$

Then $e^{-iq_j\cdot z_j}=e^{-iq_j\cdot\xi_j}e^{-q_j\cdot\eta_j}$. For future-directed $q_j$ and strictly future timelike $\eta_j$, the last factor damps the Fourier–Laplace transform. The resulting function is holomorphic on the primitive tube

$$\mathcal T_{n-1}=M^{n-1}-i(V_+)^{n-1}.$$

As every $\eta_j$ tends to zero within a closed subcone of $V_+$, the holomorphic function approaches $w_n$ as a tempered-distribution boundary value. Polynomial bounds near the boundary replace pointwise convergence. The sign is convention-dependent: authors using $e^{+iq\cdot\xi}$ for the inverse transform call the corresponding region the opposite tube. The invariant content is damping of the spectrum-supported exponential.

The Fourier–Laplace theorem and its distributional boundary-value converse are treated in Streater and Wightman 2016, §§ 2-2–2-3, pp. 43–62. The converse requires the appropriate growth bounds; arbitrary holomorphic functions on a tube need not have tempered boundary values.

The proof is local on compact subsets of the tube. If each $\eta_j$ stays in a compact subcone bounded away from the light cone, $q_j\cdot\eta_j$ controls a positive multiple of the relevant momentum norm. Exponential damping then dominates the polynomial growth allowed for a tempered distribution. Differentiating with respect to $z_j$ inserts powers of $q_j$, which remain dominated, so all complex derivatives exist. Near the real boundary the estimates deteriorate only polynomially; this is the condition that permits convergence in $\mathcal S'$, rather than at each real point.

Complex Lorentz extension

Real Lorentz covariance identifies values at $z$ and $\Lambda z$ for real proper orthochronous $\Lambda$. Holomorphy and the identity theorem continue this relation to complex Lorentz transformations connected to the identity whenever the orbit starts in the primitive tube. The union

$$\mathcal T'_{n-1}=\bigcup_{\Lambda_c\in L_+(\mathbb C)} \Lambda_c\mathcal T_{n-1}$$

is the extended tube, more precisely understood on the relevant connected cover for spinorial fields. The Bargmann–Hall–Wightman theorem supplies this continuation and single-valuedness on the appropriate domain; see Streater and Wightman 2016, § 2-4, pp. 63–73 and the original invariant-analytic-function theorem of Hall and Wightman 1957, pp. 1–41.

This does not mean that an arbitrary correlator is entire in all complexified spacetime variables. Singular hypersurfaces remain, and different orderings begin as boundary values of different tubes. Nor does complex Lorentz covariance alone provide crossing symmetry for scattering amplitudes; that conclusion needs additional reduction, particle, and analyticity hypotheses.

Massive two-point function

This boundary-value calculation is the local-field prototype for the more demanding analytic continuations in analyticity and crossing of amplitudes; no amplitude-level crossing claim is assumed here.

For the free scalar,

$$W_2(z)=\int\frac{\mathrm d^3\mathbf p}{(2\pi)^3\,2E_{\mathbf p}}e^{-ip\cdot z}, \qquad z=x-i\eta,\quad\eta\in V_+.$$

Because $p\cdot\eta>0$ on the positive mass shell, the factor $e^{-p\cdot\eta}$ gives exponential damping at large momentum. Differentiation under the integral is valid on compact subsets of the tube, proving holomorphy. Its distributional boundary at $\eta\downarrow0$ is $\Delta_+(x;m^2)$. Lorentz-invariant expressions involving a square root or a Bessel function are representations of this same analytic object; the positive-energy boundary prescription selects the branch.

The invariant combination $z^2$ helps locate, but does not remove, the singular geometry. The light-cone locus $z^2=0$ and its continued cuts obstruct entire continuation. A closed-form Bessel expression must therefore be accompanied by the tube from which it is approached; selecting a square-root branch without the positive-energy boundary prescription loses physical information.

Add an equal negative-energy mass-shell contribution. For $p\in-V_+$, $e^{-p\cdot\eta}$ grows exponentially, so the same tube integral no longer defines a tempered holomorphic function. This is the required adversarial check: real Lorentz invariance survives, but the positive-energy tube does not.

An independent check uses a purely imaginary point $z=-i(\tau,\mathbf0)$ with $\tau>0$. The integrand becomes $e^{-E_{\mathbf p}\tau}$, manifestly convergent. Choosing $z=+i(\tau,\mathbf0)$ instead produces $e^{+E_{\mathbf p}\tau}$ and reveals the sign error immediately.

Exercises

Show that $p\cdot\eta>0$ for every nonzero $p\in\overline V_+$ and every $\eta\in V_+$.

Solution

Go to the rest frame of the timelike vector $\eta$, where $\eta=(\eta^0,\mathbf0)$ with $\eta^0>0$. Then $p\cdot\eta=p^0\eta^0$. A nonzero future causal vector has $p^0>0$, so the product is positive. Lorentz invariance gives the result in every frame.

References

• Hall, David, and Arthur S. Wightman. 1957. “A Theorem on Invariant Analytic Functions with Applications to Relativistic Quantum Field Theory.” Matematisk-fysiske Meddelelser, Det Kongelige Danske Videnskabernes Selskab 31 (5): 1–41. Catalog record.
• Streater, Raymond F., and Arthur S. Wightman. 2016. PCT, Spin and Statistics, and All That. Princeton Landmarks in Physics. Princeton University Press. DOI.