Axiomatic Quantum Field Theory

Derivation. Derives Proposition 109.1. In Equation (109.1) the wave four-vector is \(k^{\mu}=(\omega_{\vect{k}}/c,\vect{k})\), of dimension \(/\mathrm{m}\), so \(k\cdot(x-y)\) is a pure number (Notation 100.1); the integral \(\int\dd^{3}k/(2\omega_{\vect{k}})\) carries \(\mathrm{s}/\mathrm{m}^{3}\) and the prefactor \(\hbar c^{2}\) turns that into \(\mathrm{J}/\mathrm{m}\), which is \([\phi]^{2}\) for a canonically normalized scalar in this book (Equation (105.101)). Setting \(y=x\),

and for \(k\gg mc/\hbar\) the integrand is \(k^{2}/(2ck)=k/(2c)\), so the integral diverges as \(\Lambda^{2}\) with an ultraviolet cutoff \(\Lambda\) on \(\abs{\vect{k}}\). The divergence is quadratic and no subtraction of a constant removes it.

Smearing repairs it. With \(f\) real and \(\tilde{f}(k)=\int\dd^{4}x\,f(x)\ee^{\ii k\cdot x}\),

which converges because \(\tilde{f}\) is Schwartz and therefore decays faster than any power. The physical content is that the divergence in Equation (109.2) is a statement about resolution: the number of modes available below a wavelength \(\lambda\) grows as \(\lambda^{-3}\), each carries a zero-point amplitude, and a measurement confined to no region at all samples all of them.

Derivation. Derives Theorem 109.4. Step 1: the vacuum goes to the vacuum. Let \(E\) denote the Euclidean group of space and \(U_{i}(R,\vect{a})\) its representation on \(\mathcal{H}_{i}\). By hypothesis \(V U_{1}V^{-1}=U_{2}\). Then \(V\ket{0_{1}}\) is invariant under \(U_{2}\), because \(U_{2}V\ket{0_{1}}=VU_{1}\ket{0_{1}}=V\ket{0_{1}}\). Uniqueness of the invariant vector in theory 2 forces

Step 2: all equal-time correlators coincide. Let \(A_{1}\) be any polynomial in the smeared \(\phi_{1}(0,\cdot)\) and \(\pi_{1}(0,\cdot)\), and \(A_{2}\) the same polynomial in \(\phi_{2},\pi_{2}\). Conjugating Equation (109.5) term by term gives \(A_{2}=VA_{1}V^{-1}\), and with Equation (109.6) the phases cancel between bra and ket:

This holds to all orders and is the whole content of unitary equivalence: a unitary preserves every expectation value, so the two theories are indistinguishable by any equal-time measurement.

Step 3: the equal-time two-point function knows the mass. For a free field of mass \(m\), set \(x^{0}=y^{0}\) in Equation (109.1):

The right-hand side is the Fourier transform of \(\hbar c/(2\sqrt{\vect{k}^{2}+m^{2}c^{2}/\hbar^{2}})\), and two different masses give two different functions of \(\vect{k}\) — they already differ at \(\vect{k}=0\), where the value is \(\hbar^{2}/(2m)\) — of dimension \(\mathrm{J}\,\mathrm{m}^{2}\), as it must be to produce \(\mathrm{J}/\mathrm{m}\) against the measure \(\dd^{3}k/(2\pi)^{3}\). Fourier inversion is injective, so Equation (109.8) determines \(m\).

Step 4: from equal times to all times. By Step 2 the equal-time correlators of \(\phi_{2}\) are those of a free field of mass \(m\), and by Step 3 that mass is \(m_{1}\). A free field is Gaussian: all its correlators are sums of products of two-point functions, so all equal-time correlators of \(\phi_{2}\) agree with the free ones. The remaining step promotes this to all times. Poincaré covariance determines \(\phi_{2}(x^{0},\vect{x})\) from its restriction to \(x^{0}=0\) together with \(\pi_{2}\), because a boost mixes the time argument into the spatial ones and the Wightman functions are boundary values of functions analytic in the tube domain of Theorem 109.26, which are fixed by their values on a real neighbourhood. Hall and Wightman carried this out [Hall:1957]; the conclusion is that every \(W_{n}\) of theory 2 is the free one, and by the reconstruction theorem Theorem 109.21 the theory itself is free.

Derivation. Derives Corollary 109.5. Immediate from Theorem 109.4: if such a \(V\) existed the interacting theory would be free. The second sentence is the same statement in representation-theoretic language — Stone–von Neumann uniqueness holds for finitely many canonical pairs and fails for infinitely many, and quantum field theory has infinitely many. The mechanism is developed for the free field itself in Section 99.9.1, whose warning about the choice of Fock space this corollary makes precise.

Derivation. Derives Proposition 109.20. (i) is Axiom 109.12 applied to the vector \(\phi(f_{2})\cdots\phi(f_{n})\ket{0}\).

(ii) Insert \(U^{-1}U\) between every pair of fields in Equation (109.13), use Equation (109.11) \(n\) times, and use \(U\ket{0}=\ket{0}\) from Axiom 109.11.

(iii) Translation invariance is the \(\Lambda=\identity\) case of (ii) and gives the dependence on differences alone. For the support statement, write \(\phi(x_{j})=U(x_{j})\phi(0)U(x_{j})^{-1}\) with \(U(a)=\exp(\ii a\cdot P/\hbar)\) from Equation (109.10); then

using \(P^{\mu}\ket{0}=0\) at both ends. Each factor \(\ee^{-\ii\xi\cdot P/\hbar}\) contributes, by the spectral theorem for the commuting family \(P^{\mu}\), a Fourier transform supported where the spectral measure lives, and Axiom 109.10 confines that to \(\overline{V}^{+}\). Explicitly, with \(\tilde{W}_{n}\) the transform in the variables \(\xi_{j}\),

(iv) is Axiom 109.14 sandwiched between the vacuum and the remaining fields.

(v) The left-hand side of Equation (109.14) is \(\norm{\Psi}^{2}\) for \(\Psi=\sum_{n}\int\dd^{4n}y\,f_{n}(y)\phi(y_{1})\cdots\phi(y_{n})\ket{0}\), a vector in \(\mathcal{H}\) by Axiom 109.12. A squared norm is non-negative; that is the whole derivation, and it is worth stressing that positivity of the Wightman functions is the positivity of the Hilbert-space metric and nothing else. The same statement reappears in Euclidean dress as reflection positivity in Section 109.5.1.

(vi) follows from \(\phi(f)^{\dagger}\supset\phi(\bar{f})\) and \(\overline{\bra{0}A\ket{0}}=\bra{0}A^{\dagger}\ket{0}\).

Derivation. Derives Theorem 109.21. The construction is the same one that will reappear as the GNS construction in Section 109.4.2, and it is worth doing once by hand.

The algebra. Let \(\mathfrak{B}\) be the set of terminating sequences \(\mathsf{f}=(f_{0},f_{1},f_{2},\ldots)\) with \(f_{0}\in\C\) and \(f_{n}\in\mathcal{S}(\R^{4n})\), made into an associative algebra by

with the involution \((\mathsf{f}^{*})_{n}(x_{1},\ldots,x_{n}) =\overline{f_{n}(x_{n},\ldots,x_{1})}\). This is the Borchers tensor algebra; the unit is \((1,0,0,\ldots)\).

The linear functional. Define \(w(\mathsf{f}):=\sum_{n}\int\dd^{4n}x\;W_{n}(x)f_{n}(x)\), which converges because the sequence terminates. Property (vi) says \(w(\mathsf{f}^{*})=\overline{w(\mathsf{f})}\), and property (v) says exactly

The Hilbert space. Put \(\gen{\mathsf{f},\mathsf{g}}:=w(\mathsf{f}^{*}\mathsf{g})\). This is a sesquilinear form, non-negative by Equation (109.18), hence obeying the Cauchy–Schwarz inequality \(\abs{\gen{\mathsf{f},\mathsf{g}}}^{2} \leq\gen{\mathsf{f},\mathsf{f}}\gen{\mathsf{g},\mathsf{g}}\). Therefore \(\mathfrak{N}:=\set{\mathsf{f}:w(\mathsf{f}^{*}\mathsf{f})=0}\) is a linear subspace, and by Cauchy–Schwarz it is a left ideal: \(\gen{\mathsf{g}\,\mathsf{f},\mathsf{g}\,\mathsf{f}}=0\) whenever \(\mathsf{f}\in\mathfrak{N}\). Let \(\mathcal{H}\) be the completion of \(\mathfrak{B}/\mathfrak{N}\) in the induced inner product, and \(D\) the image of \(\mathfrak{B}\), dense by construction.

The field. For \(f\in\mathcal{S}(\R^{4})\) let \(\mathsf{f}^{(1)}=(0,f,0,\ldots)\) and define \(\phi(f)[\mathsf{g}]:=[\mathsf{f}^{(1)}\mathsf{g}]\). This is well defined because \(\mathfrak{N}\) is a left ideal, maps \(D\) into \(D\), is linear in \(f\), and satisfies \(\gen{\mathsf{g},\phi(f)\mathsf{h}} =\gen{\phi(\bar{f})\mathsf{g},\mathsf{h}}\) by the definition of the involution — which is Axiom 109.12. The vacuum is \(\ket{0}:=[(1,0,\ldots)]\), and cyclicity Axiom 109.15 holds because \(\mathfrak{B}\) is generated by the \(\mathsf{f}^{(1)}\).

The symmetry. Property (ii) says the form \(\gen{\cdot,\cdot}\) is invariant under the action of \((a,\Lambda)\) on test functions given by Equation (109.11). An invariant form induces an isometry of the quotient, and an isometry of a dense subspace extends to a unitary \(U(a,\Lambda)\) on \(\mathcal{H}\); the group law is inherited from the action on test functions. Continuity follows from continuity of that action in \(\mathcal{S}\) together with temperedness (i), and Stone's theorem then supplies the self-adjoint generators \(P^{\mu}\) of Equation (109.10). Property (iii) says the spectral measure of \(P^{\mu}\) is supported in \(\overline{V}^{+}\), which is Axiom 109.10; property (iv) gives Axiom 109.14.

Uniqueness. If two constructions \((\mathcal{H},\phi,\ket{0})\) and \((\mathcal{H}',\phi',\ket{0}')\) have the same \(W_{n}\), the map \(\phi(f_{1})\cdots\phi(f_{n})\ket{0}\mapsto \phi'(f_{1})\cdots\phi'(f_{n})\ket{0}'\) preserves inner products by hypothesis, is defined on a dense set by cyclicity, and therefore extends to a unitary intertwining the two theories.

Derivation. Derives Proposition 109.23. Write \(A=\phi(x_{1})\cdots\phi(x_{m})\) and \(B=\phi(y_{1})\cdots\phi(y_{n})\), so the left-hand side of Equation (109.19) is \(\bra{0}A\,U(\lambda a)\,B\ket{0}\) with \(U(\lambda a)=\exp(\ii\lambda a\cdot P/\hbar)\). Let \(E(\dd^{4}p)\) be the joint spectral measure of \(P^{\mu}\) and let \(E_{0}\) be the projector onto the eigenvalue \(p=0\). Then

Two things must now be established: that the only atom of the spectral measure sits at \(p=0\), and that the remaining integral vanishes in the limit. The first is a derivation and is where the physics is.

The only eigenvalue of \(P^{\mu}\) is zero. Suppose \(P^{\mu}\ket{\chi}=p^{\mu}\ket{\chi}\) with \(p\neq0\) and \(p\in\overline{V}^{+}\). For every \(\Lambda\) in the proper orthochronous group the vector \(U(0,\Lambda)\ket{\chi}\) is again an eigenvector, with eigenvalue \(\Lambda p\), and two eigenvectors of a self-adjoint family with different eigenvalues are orthogonal. The orbit \(\set{\Lambda p}\) of a non-zero \(p\in\overline{V}^{+}\) is an entire mass hyperboloid or half of the light cone — in either case an uncountable set — so \(\mathcal{H}\) would contain an uncountable orthonormal family and could not be separable, contradicting Axiom 109.9. Hence the spectral measure has no atom away from the origin, and \(E_{0}\) projects onto the translation-invariant vectors.

The oscillatory term vanishes. The remaining integral is the Fourier transform of a finite complex measure supported in \(\overline{V}^{+}\setminus\set{0}\), evaluated along the spacelike ray \(\lambda a\). That it tends to zero is the content of the cluster theorem of [Streater:1964]: the measure has been shown to be non-atomic, so a Riemann–Lebesgue argument gives the vanishing of its transform in the mean, and analyticity in the tube of Theorem 109.26 upgrades the mean convergence to a genuine limit, exponential with rate \(\Delta/(\hbar c)\) when the spectrum has a gap \(\Delta\) above the vacuum and power-law otherwise. That upgrade is the imported half of this proposition; the atom analysis above is the half that carries the physical content.

Granting it, if the vacuum is unique then \(E_{0}=\ketbra{0}{0}\) and the surviving term is \(W_{m}W_{n}\), which is Equation (109.19). Conversely, if Equation (109.19) holds then \(\bra{0}AE_{0}B\ket{0}=\bra{0}A\ket{0}\bra{0}B\ket{0}\) for all local \(A,B\); since the vacuum is cyclic these exhaust a dense set, so \(E_{0}=\ketbra{0}{0}\), which says precisely that the only translation-invariant vector is \(\ket{0}\).

Derivation. Derives Lemma 109.25. Since \(\eta\in V^{+}\) has \(\eta\cdot\eta>0\) and \(\eta^{0}>0\), choose the frame in which \(\vect{\eta}=0\), so \(\eta=(\eta^{0},\vect{0})\) with \(\eta^{0}>0\); this is possible by a proper orthochronous transformation, which preserves the pairing. Then \(q\cdot\eta=q^{0}\eta^{0}\), and \(q\in\overline{V}^{+}\) gives \(q^{0}\geq\abs{\vect{q}}\geq0\). Hence \(q\cdot\eta\geq0\), and it vanishes only if \(q^{0}=0\), which with \(q^{0}\geq\abs{\vect{q}}\) forces \(q=0\).

Derivation. Derives Theorem 109.26. Write the inverse transform of Equation (109.16),

With \(\zeta_{j}=\xi_{j}-\ii\eta_{j}\) the exponent is

whose real part is \(-\hbar^{-1}\sum_{j}q_{j}\cdot\eta_{j}\). By Lemma 109.25 each term is non-negative on the support of \(\tilde{W}_{n}\) and grows linearly with \(\abs{q_{j}}\) when \(\eta_{j}\) is in the open cone, so the exponential decays exponentially in \(\abs{q_{j}}\). Since \(\tilde{W}_{n}\) is a tempered distribution — of at most polynomial growth — the integral converges absolutely, and it converges together with all its \(\zeta\)-derivatives; the integrand is holomorphic in each \(\zeta_{j}\) in the region where the decay holds, so Equation (109.22) defines a holomorphic function there. Letting \(\eta_{j}\to0\) recovers \(W_{n}\) in the sense of distributions.

Two remarks on rigour. That a tempered distribution supported in a cone has a Fourier–Laplace transform holomorphic in the dual cone is the Paley–Wiener–Schwartz theorem, imported here and not proved; what Lemma 109.25 supplies is the identification of the dual cone, and that is the geometric content. The factors of \(\hbar\) are forced: \(q\cdot\zeta\) has dimension \(\mathrm{kg}\,\mathrm{m}^{2}/\mathrm{s}=\mathrm{J}\,\mathrm{s}\), so it is \(q\cdot\zeta/\hbar\) that is a phase, and each four-dimensional momentum measure carries \((2\pi\hbar)^{-4}\) exactly as in the loop integrals of Quantum Electrodynamics and Renormalization.

Status. Derives Theorem 109.28. Not proved here. The content is that real Lorentz covariance, Proposition 109.20(ii), plus holomorphy in the tube force covariance under the complexified group, and that the extension is single-valued — the delicate point, since \(L_{+}(\C)\) is not simply connected and the naive continuation could be multiple-valued. The proof is in [Hall:1957] and [Streater:1964].

Derivation. Derives Proposition 109.31. Take all \(\xi_{j}\) equal to a fixed purely spatial vector, \(\xi_{j}=(0,\vect{d})\) with \(\vect{d}\neq0\). Then \(\sum_{j}\lambda_{j}\xi_{j}=(0,(\sum\lambda_{j})\vect{d})\) is spacelike whenever \(\sum\lambda_{j}>0\), so the configuration is a Jost point. Openness: the condition in Definition 109.29 need only be checked on the compact simplex \(\sum\lambda_{j}=1\), \(\lambda_{j}\geq0\), where \((\sum\lambda_{j}\xi_{j})^{2}\) is a continuous function of \((\lambda,\xi)\) attaining a strictly negative maximum; a small perturbation of \(\xi\) keeps it negative.

Derivation. Derives Theorem 109.34. (i) Choose an open \(\mathcal{O}_{0}\) whose closure lies inside \(\mathcal{O}\), and \(\epsilon>0\) so small that translating \(\mathcal{O}_{0}\) by any \(a\) with \(\abs{a^{\mu}}<\epsilon\) keeps it inside \(\mathcal{O}\). Suppose \(\Psi\in\mathcal{H}\) is orthogonal to \(\mathcal{P}(\mathcal{O})\ket{0}\). Fix \(n\) and test functions \(f_{1},\ldots,f_{n}\) supported in \(\mathcal{O}_{0}\) and consider

For all real \(a_{j}\) with \(\abs{a_{j}^{\mu}}<\epsilon\) every translated support is still in \(\mathcal{O}\), so \(F\) vanishes on that open real neighbourhood of the origin by hypothesis.

Now repeat the argument of Theorem 109.26 on \(F\). Writing \(\phi(f^{a})=U(a)\phi(f)U(a)^{-1}\) and using \(P^{\mu}\ket{0}=0\) at the right end, the exponentials collapse pairwise and Equation (109.24) becomes

in the differences \(b_{j}=a_{j}-a_{j+1}\). One step more is needed than in Equation (109.15), which used \(P^{\mu}\ket{0}=0\) at both ends: \(\Psi\) is not translation invariant, so a residual \(U(a_{1})\) survives on the left. Absorb it, holding \(a_{1}\) fixed and replacing \(\Psi\) by \(\Psi_{a_{1}}:=U(a_{1})^{-1}\Psi\), which is again orthogonal to \(\mathcal{P}(\mathcal{O})\ket{0}\) for the correspondingly translated region and is a legitimate state. What remains is of exactly the form of Equation (109.15) with \(\bra{\Psi_{a_{1}}}\) in place of \(\bra{0}\) at the left end. The spectrum condition applies unchanged to the right of each field, so for each fixed \(a_{1}\), \(F\) is the boundary value of a function holomorphic in the forward tube in the variables \(b_{j}\).

A function holomorphic in a tube whose distributional boundary value vanishes on a non-empty open set of the real boundary vanishes identically. (This is the edge-of-the-wedge theorem, imported; in the one-variable case it is the Schwarz reflection principle, and it is the statement that a real open set is a set of uniqueness for such functions.) Hence \(F\equiv0\) for all real \(a_{j}\), so \(\Psi\) is orthogonal to \(\phi(g_{1})\cdots\phi(g_{n})\ket{0}\) for all test functions \(g_{j}\) whatever — translates of a fixed \(f\) span a dense set of \(\mathcal{S}\) in the relevant sense, and \(n\) was arbitrary. By Axiom 109.15, \(\Psi=0\).

(ii) Let \(\mathcal{O}'\) be an open region spacelike to \(\mathcal{O}\), and let \(B\in\mathcal{P}(\mathcal{O}')\). Since \(A\in\mathcal{P}(\mathcal{O})\) and \(B\) commute by Axiom 109.14,

By part (i) applied to \(\mathcal{O}'\), the vectors \(B\ket{0}\) are dense in \(\mathcal{H}\); an operator vanishing on a dense subset of its domain is zero.

Derivation. Derives Corollary 109.35. Part (ii) of Theorem 109.34 with \(\mathcal{O}\) any bounded region, whose spacelike complement is never empty. A local projector \(E\) with \(E\ket{0}=0\) would violate it, and so would a number operator, since a number operator annihilates the vacuum by definition.

Status and mechanism. Derives Theorem 109.38. The full proof requires the asymptotic construction of Section 109.3.4 and is not carried out here. The mechanism is that the asymptotic fields are built from a field only through its matrix elements between the vacuum and the one-particle states, together with its commutation properties; two relatively local fields have the same one-particle overlaps up to normalization and the same commutation properties, so they generate the same in and out states, and the \(S\) matrix is the map between those.

Derivation. Derives Lemma 109.42. Consider the one-parameter family, written in the block form \((x^{0},x^{1})\oplus(x^{2},x^{3})\),

The upper block is a boost with rapidity \(\chi=\ii\pi s\), since \(\cosh(\ii\pi s)=\cos(\pi s)\) and \(\sinh(\ii\pi s)=\ii\sin(\pi s)\); it is therefore symmetric, with both off-diagonal entries equal to \(\ii\sin(\pi s)\), as a boost matrix always is. The lower block is a rotation by \(\pi s\) in the \((x^{2},x^{3})\) plane. Each block separately preserves the corresponding part of \(\eta_{\mu\nu}=\diag(+1,-1,-1,-1)\) — the first because a boost does for any complex rapidity, the second because a rotation does — so \(\Lambda(s)\transpose\eta\Lambda(s)=\eta\) for every \(s\), and \(\det\Lambda(s)=1\) because each block has unit determinant: \(\cos^{2}(\pi s)-\left(\ii\sin(\pi s)\right)^{2} =\cos^{2}(\pi s)+\sin^{2}(\pi s)=1\) for the boost, and likewise for the rotation. Hence \(\Lambda(s)\in L_{+}(\C)\) throughout. At the endpoints

so the path is continuous, lies in \(L_{+}(\C)\), and joins the identity to the total inversion.

Over the reals the same construction is unavailable: a real boost has \(\cosh\chi\geq1\), so the \(00\) entry of a real proper orthochronous transformation never reaches \(-1\). Indeed \(-\identity\) has \(\Lambda^{0}{}_{0}=-1\), so it fails the condition \(\Lambda^{0}{}_{0}\geq1\) that Definition 39.5 imposes and lies on the non-orthochronous branch; \(\det(-\identity)=+1\) in four dimensions, so \(-\identity\in L^{\downarrow}_{+}\).

Derivation. Derives Corollary 109.43. Apply Equation (109.23) with \(\Lambda=-\identity\), which lies in \(L_{+}(\C)\) by Lemma 109.42; the extended tube is \(L_{+}(\C)\)-invariant by Definition 109.27, so both sides are defined. Restricting to the real points, which are the Jost points by Theorem 109.30, gives Equation (109.28).

Derivation. Derives Theorem 109.44. Take the scalar case; the index bookkeeping in the general case multiplies both statements by the same \((-1)^{J}\) and cancels.

(\(\Leftarrow\)) Assume WLC. At a Jost point, combine Equation (109.26) with Equation (109.28):

the first step being legitimate because the reversed configuration \((x_{n},\ldots,x_{1})\) is a Jost point whenever \((x_{1},\ldots,x_{n})\) is — the Jost condition of Definition 109.29 is invariant under reversal, since reversing the order replaces each difference \(\xi_{j}\) by \(-\xi_{n-j}\) and the defining condition is even. So Equation (109.25) holds on the open set of Jost points. Both sides of Equation (109.25) are boundary values of functions holomorphic on the extended tube, which is connected, and they agree on a real open subset of it; by the identity theorem for holomorphic functions of several variables they agree everywhere, so Equation (109.25) holds as an identity of distributions.

(\(\Rightarrow\)) Assume the \(PCT\) condition. Reading the same chain backwards, at a Jost point \(W_{n}(x_{1},\ldots,x_{n})=W_{n}(-x_{n},\ldots,-x_{1}) =W_{n}(x_{n},\ldots,x_{1})\), which is Equation (109.26).

Derivation. Derives Corollary 109.45. At a Jost point all \(n\) arguments are mutually spacelike, so Axiom 109.14 permits an arbitrary permutation of the fields inside Equation (109.13), in particular the total reversal. That is WLC, and Theorem 109.44 applies.

Derivation. Derives Theorem 109.46. Define \(\Theta\) on the dense set of vectors \(\Psi=\phi(f_{1})\cdots\phi(f_{m})\ket{0}\) by

extended antilinearly. Two things must be checked: that this preserves inner products in the antilinear sense, and that it is therefore well defined.

Let \(\Phi=\phi(g_{1})\cdots\phi(g_{n})\ket{0}\). Then

after the substitutions \(x\to-x\), \(y\to-y\), whose Jacobian is \(+1\). Applying Equation (109.25) to the \((m+n)\)-point function turns its argument list into \((y_{n},\ldots,y_{1},x_{1},\ldots,x_{m})\), and the result is exactly

which is antiunitarity. Well-definedness follows: if a linear combination of the \(\Psi\) has zero norm, so does its image, because norms are preserved. A norm-preserving antilinear map on a dense domain extends to an antiunitary on \(\mathcal{H}\).

The relation \(\Theta\phi(f)\Theta^{-1}=\phi(\bar{f}^{-})\) is Equation (109.30) read on one factor, and \(\Theta\ket{0} =\ket{0}\) is the case \(m=0\). Applying Equation (109.30) twice returns \(f\), so \(\Theta^{2}=\identity\). Finally, from Equation (109.29) and Equation (109.11), \(\Theta U(a,\identity)\Theta^{-1}=U(-a,\identity)\); writing both sides with Equation (109.10) and using antilinearity of \(\Theta\), which conjugates the \(\ii\), gives \(\Theta P^{\mu}\Theta^{-1}=P^{\mu}\), so \(\Theta\) commutes with the energy.

Derivation, integer spin. Derives Theorem 109.49. Take \(\phi\) a complex scalar field and define the two two-point functions of the difference \(\xi=x-y\),

Note that in both the argument \(x\) stands on the left, so both are covered by Theorem 109.26 with the same tube: each is the boundary value of a function \(\mathcal{F}\), \(\mathcal{G}\) holomorphic on the forward tube \(\mathcal{T}_{1}\) and, by Theorem 109.28, on the extended tube \(\mathcal{T}'_{1}\).

Step 1: evenness. By Corollary 109.43 applied to each of \(\mathcal{F}\) and \(\mathcal{G}\) — these are two-point functions of a scalar field, so \(J=0\) —

Step 2: the wrong bracket at a Jost point. For \(\xi\) spacelike — which for \(n=2\) is exactly the Jost condition — the assumed anticommutation gives \(\phi(x)\phi^{\dagger}(y)=-\phi^{\dagger}(y)\phi(x)\), so

the middle equality because \(G\) was defined with its arguments in the order (dagger, no dagger) and here they appear at \(y\) and \(x\), i.e.\ with difference \(y-x=-\xi\).

Step 3: analytic propagation. \(\mathcal{F}+\mathcal{G}\) is holomorphic on the connected domain \(\mathcal{T}'_{1}\) and, by Step 2, vanishes on the open set of real Jost points contained in it (Theorem 109.30 and Proposition 109.31). A holomorphic function vanishing on a real open subset of its domain vanishes identically, so \(\mathcal{F}=-\mathcal{G}\) on all of \(\mathcal{T}'_{1}\), and taking boundary values from the same tube,

Step 4: positivity kills it. Smear Equation (109.35) with \(\overline{g(x)}\,g(y)\) for an arbitrary \(g\in\mathcal{S}(\R^{4})\). The left side gives \(\bra{0}\phi(\bar{g})\phi^{\dagger}(g)\ket{0} =\norm{\phi^{\dagger}(g)\ket{0}}^{2}\), since \(\phi(\bar{g})=\left(\phi^{\dagger}(g)\right)^{\dagger}\); the right side gives \(-\bra{0}\phi^{\dagger}(\bar{g})\phi(g)\ket{0} =-\norm{\phi(g)\ket{0}}^{2}\). Hence

a sum of two non-negative numbers, so both vanish: \(\phi(g)\ket{0}=0\) for every \(g\).

Step 5: from the vacuum to the operator. Take \(g\) supported in a bounded open region, whose spacelike complement is non-empty. By Theorem 109.34(ii) the vacuum is separating for that region, so \(\phi(g)=0\). Such \(g\) are dense in \(\mathcal{S}\), and \(\phi\) is continuous in \(g\) by Axiom 109.12, so \(\phi=0\).

Derivation, half-integer spin. Derives Theorem 109.49. The only change is in Step 1. A field in the \((A,B)\) representation of \(\SL(2,\C)\) carries \(2A\) undotted and \(2B\) dotted spinor indices, and Theorem 109.28 for such a field multiplies the right-hand side of Equation (109.23) by the analytic continuation of \(S(\Lambda)\). Along the path Equation (109.27) that continuation returns, at \(s=1\), the factor \((-1)^{2A+2B}=(-1)^{2s_{\phi}}\) on the two-point function, where \(s_{\phi}\) is the spin. For half-integer spin this is \(-1\), so Equation (109.33) becomes \(\mathcal{G}(-\zeta)=-\mathcal{G}(\zeta)\). The assumed commutator at spacelike separation gives, in place of Equation (109.34), \(F(\xi)=+G(-\xi)=-G(\xi)\) — the same relation as before. Steps 3 to 5 are unchanged, and again \(\phi=0\).

Derivation. Derives Corollary 109.50. The first sentence is the contrapositive of Theorem 109.49. For the second, let \(a^{\dagger}\) denote the creation part of the field in the sense of Section 109.3.4; the exchange symmetry of a two-quantum state \(a^{\dagger}(f)a^{\dagger}(g)\ket{0}\) is the sign in the bracket relating \(a^{\dagger}(f)\) and \(a^{\dagger}(g)\), which is inherited from the field's spacelike bracket by the asymptotic construction. That is the symmetrization postulate assumed at Section 83.1.4, which is a postulate there because none of the hypotheses used here — Lorentz covariance, positivity of the energy, locality — is available in nonrelativistic quantum mechanics.

Derivation. Derives Theorem 109.54. Insert a complete set of states between the fields. By Axiom 109.10 every state has four-momentum in \(\overline{V}^{+}\); group the states by their sector label \(\lambda\) and their invariant mass \(m_{\lambda}\), and use Notation 109.52. The vacuum contributes \(\abs{\bra{0}\phi(0)\ket{0}}^{2}=0\) by hypothesis, so

Translation covariance, Equation (109.10) with \(\phi(x)=U(x)\phi(0)U(x)^{-1}\) and \(P^{\mu}\ket{0}=0\), gives \(\bra{0}\phi(x)\ket{\lambda_{\vect{k}}} =\ee^{-\ii k\cdot x}\bra{0}\phi(0)\ket{\lambda_{\vect{k}}}\), so the \(x\) and \(y\) dependence collapses to \(\ee^{-\ii k\cdot(x-y)}\) and the matrix elements enter only through \(\abs{\bra{0}\phi(0)\ket{\lambda_{\vect{k}}}}^{2}\).

Lorentz covariance, Equation (109.11), says that this modulus squared is invariant: acting with \(U(\Lambda)\) on both sides sends \(\ket{\lambda_{\vect{k}}}\) to a state of the same sector with boosted wave vector, and Equation (109.37) is constructed to be invariant, so \(\abs{\bra{0}\phi(0)\ket{\lambda_{\vect{k}}}}^{2}\) depends on \(\vect{k}\) only through the invariant \(\hbar^{2}k\cdot k=m_{\lambda}^{2}c^{2}\) — that is, not at all within a sector. Call it \(c_{\lambda}\).

Now insert \(1=\int_{0}^{\infty}\dd s\,\delta(s-m_{\lambda}^{2}c^{2})\) and define

whose positivity is manifest, each \(c_{\lambda}\) being a squared modulus. Exchanging the order of the \(s\) and \(\vect{k}\) integrations and comparing with Equation (109.38) gives Equation (109.39). The prefactor is the ratio of the two measures, not of the two prefactors: Equation (109.41) carries \(\dd^{3}k\,c/\left[(2\pi)^{3}2\omega_{\vect{k}}\right]\) while Equation (109.38) carries \(\hbar c^{2}\,\dd^{3}k/\left[(2\pi)^{3}2\omega_{\vect{k}}\right]\), and \(c/(\hbar c^{2})=1/(\hbar c)\). The extra power of \(c\) is supplied by the Lorentz-invariant state measure of Notation 109.52, and it is what makes the bookkeeping close: that normalization gives \(\ket{\lambda_{\vect{k}}}\) the dimension \(\mathrm{m}\), so \(c_{\lambda}=\abs{\bra{0}\phi(0)\ket{\lambda_{\vect{k}}}}^{2}\) carries \(\mathrm{J}\,\mathrm{m}=[\hbar c]\), the quotient \(c_{\lambda}/(\hbar c)\) is a pure number, and \(\rho\) inherits the dimension \(\mathrm{s}^{2}/\mathrm{kg}^{2}/\mathrm{m}^{2}\) of the delta function alone — that is, \(1/[s]\), so that Equation (109.40) is a statement between pure numbers. For a free field of mass \(m\) the single sector has \(c_{\lambda}=\hbar c\) and \(\rho(s)=\delta(s-m^{2}c^{2})\), as it must.

For the sum rule, subtract Equation (109.39) from its \(x\leftrightarrow y\) transpose:

with \(\Delta\) the Pauli–Jordan function of Equation (105.101) for squared momentum \(s\). Now \(\Delta(\xi;s)\) vanishes at equal times and

which is independent of \(s\): differentiating Equation (105.101) brings down \(\mp\ii\omega_{\vect{k}}\) from the two exponentials, which cancels the \(1/(2\omega_{\vect{k}})\) and leaves \(-\int\dd^{3}k\,\ee^{\ii\vect{k}\cdot(\vect{x}-\vect{y})} /(2\pi)^{3}\) regardless of the mass. Therefore

using \(\pp_{t_{y}}=-\pp_{t_{x}}\) on a function of \(x-y\). Comparing with the canonical commutator \(\comm{\phi}{\pi}=\ii\hbar\,\delta^{3}\identity\) gives Equation (109.40).

Derivation. Derives Corollary 109.55. Substitute the decomposition into Equation (109.40). Both \(Z\) and \(\int\sigma\) are non-negative because \(\rho\) is, and they sum to one, which gives Equation (109.45). For the equality case, \(Z=1\) forces \(\int\sigma=0\), and a non-negative measure of vanishing total mass is the zero measure, so \(\sigma\equiv0\); conversely \(\sigma\equiv0\) gives \(Z=1\) by the same sum rule.

Status and mechanism. Derives Theorem 109.59. The mechanism is a competition of rates. A single \(B_{f}(t)\) applied to the vacuum is \(t\)-independent: its one-particle component is exact, because \(\tilde{f}\) meets only the one-particle shell and Equation (109.46) is the Klein–Gordon inner product, which is conserved. For a product of \(n\) such operators the \(t\)-dependence comes from the overlap of the wave packets, and cluster decomposition Equation (109.19) makes connected contributions fall off faster than \(t^{-3/2}\) per pair while the number of pairs is fixed; the Cauchy criterion then gives norm convergence. What must be supplied is the quantitative decay estimate, which uses the mass gap of Definition 109.58 and stationary phase.

Derivation. Derives Theorem 109.60. The pole structure is Theorem 109.54: a leg carrying momentum \(p\) has, by Corollary 109.55, an isolated pole of residue proportional to \(Z\) at \(p^{2}=m^{2}c^{2}\) and a branch cut starting at \(s_{\text{thr}}\), and multiplying by \((p^{2}-m^{2}c^{2})\) before taking the limit isolates the first and discards the second. The identification of what survives with a matrix element between Haag–Ruelle states is Theorem 109.59. The algebra is carried out in Theorem 106.48; what this chapter adds is that its two ingredients — the analytic structure of the two-point function and the existence of asymptotic states — are theorems of the axioms rather than assumptions of the path integral.

Derivation. Derives Theorem 109.66. Put \(\gen{A,B}_{\omega}:=\omega(A^{*}B)\) on \(\mathfrak{A}\) regarded as a vector space. This is sesquilinear and non-negative, hence satisfies Cauchy–Schwarz, \(\abs{\omega(A^{*}B)}^{2}\leq\omega(A^{*}A)\,\omega(B^{*}B)\).

Let \(\mathfrak{N}:=\set{A:\omega(A^{*}A)=0}\). By Cauchy–Schwarz, \(A\in\mathfrak{N}\) implies \(\omega(B^{*}A)=0\) for all \(B\), so \(\mathfrak{N}\) is a subspace; and for \(A\in\mathfrak{N}\), \(\omega\!\left((BA)^{*}(BA)\right)=\omega(A^{*}B^{*}BA)\leq \norm{B^{*}B}\,\omega(A^{*}A)=0\) using the \(C^{*}\) inequality \(A^{*}CA\leq\norm{C}A^{*}A\) for \(C\geq0\). So \(\mathfrak{N}\) is a left ideal.

Let \(\mathcal{H}_{\omega}\) be the completion of \(\mathfrak{A}/\mathfrak{N}\) in the induced — now definite — inner product, write \([A]\) for the class of \(A\), and set \(\pi_{\omega}(B)[A]:=[BA]\), well defined because \(\mathfrak{N}\) is a left ideal and bounded because \(\norm{[BA]}^{2}=\omega(A^{*}B^{*}BA)\leq\norm{B}^{2}\norm{[A]}^{2}\). That \(\pi_{\omega}\) is a \(*\)-homomorphism is immediate from the definitions. Take \(\ket{\omega}:=[\identity]\): then \(\bra{\omega}\pi_{\omega}(A)\ket{\omega}=\omega(\identity^{*}A\identity) =\omega(A)\), which is Equation (109.48), and cyclicity holds because \(\pi_{\omega}(\mathfrak{A})\ket{\omega}=\mathfrak{A}/\mathfrak{N}\) is dense by construction. Uniqueness: two such triples are intertwined by \(\pi_{\omega}(A)\ket{\omega}\mapsto\pi'_{\omega}(A)\ket{\omega}'\), which preserves inner products by Equation (109.48) and is densely defined by cyclicity.

Derivation. Derives Proposition 109.74. The KMS condition characterizing a thermal state at temperature \(T\) with respect to a time evolution \(\alpha_{t}\) is that \(t\mapsto\omega\!\left(A\,\alpha_{t}(B)\right)\) extends analytically to the strip \(0<\im t<\hbar/(k_{B}T)\) with \(\omega\!\left(A\,\alpha_{t+\ii\hbar/k_{B}T}(B)\right) =\omega\!\left(\alpha_{t}(B)\,A\right)\); equivalently the correlation functions are periodic in imaginary time with period \(\hbar/(k_{B}T)\). Tomita–Takesaki delivers exactly this for \(\sigma_{\tau}\) with period \(1\) in \(\tau\), which is a general property of the modular group; by Equation (109.50) the flow is the boost with rapidity \(\eta=-2\pi\tau\), so the vacuum correlation functions are periodic in imaginary rapidity with period \(2\pi\).

Now convert to the proper time of the accelerated observer. A worldline of constant proper acceleration \(a\) in \(W\) is an orbit of the boost, and the relation \(\eta=a\tau_{p}/c\) is not quoted from elsewhere but follows in two lines. Parametrize the orbit by the boost rapidity \(\eta\) of Definition 38.21, so that the four-velocity of Definition 40.2 is

which has \(u^{2}=c^{2}\) as Proposition 40.3 requires. The four-acceleration is then \(a^{\mu}=\dd u^{\mu}/\dd\tau_{p} =c\left(\sinh\eta,\cosh\eta,0,0\right)\dd\eta/\dd\tau_{p}\), whose invariant square is \(a^{\mu}a_{\mu}=-c^{2}\left(\dd\eta/\dd\tau_{p}\right)^{2}\). The proper acceleration is the magnitude of this spacelike vector, \(a=\sqrt{-a^{\mu}a_{\mu}}=c\,\dd\eta/\dd\tau_{p}\), so \(a\) constant integrates to \(\eta=a\tau_{p}/c\), and \(c/a\) is the length scale of the resulting hyperbola. Periodicity in imaginary \(\eta\) with period \(2\pi\) is therefore periodicity in imaginary \(\tau_{p}\) with period \(2\pi c/a\), and equating this to \(\hbar/(k_{B}T)\) gives Equation (109.51). Dimensional check: \(\hbar a\) has \(\mathrm{J}\,\mathrm{s}\times \mathrm{m}/\mathrm{s}^{2}=\mathrm{J}\,\mathrm{m}/\mathrm{s}\), and dividing by \(c\) leaves \(\mathrm{J}\), so \(\hbar a/(2\pi c k_{B})\) is a temperature.

Derivation. Derives Proposition 109.79. Let \(\mathcal{E}_{+}\) be the space of functionals of the field supported in \(x^{4}>0\), and \(\theta\) the reflection \(x^{4}\mapsto-x^{4}\). Reflection positivity says that \(\gen{F,G}:=\avg{\overline{\theta F}\,G}\) is a non-negative sesquilinear form on \(\mathcal{E}_{+}\). Quotient by its null space and complete, exactly as in Theorem 109.66: the result is a Hilbert space \(\mathcal{H}\) with a positive-definite inner product, which is property (v).

Euclidean time translation \(T_{\tau}\) by \(\tau>0\) maps \(\mathcal{E}_{+}\) into itself, and \(\gen{F,T_{\tau}G}=\gen{T_{\tau/2}F,T_{\tau/2}G}\) by translation invariance and the reflection property, so \(T_{\tau}\) descends to a self-adjoint operator on \(\mathcal{H}\), positive by the same identity with \(F=G\).

That \(T_{\tau}\) is a contraction takes one step more than Cauchy–Schwarz alone. Write \(g(\tau):=\norm{T_{\tau}F}\), so that \(g(0)=\norm{F}\). The identity above with \(G=F\) gives \(g(\tau/2)^{2}=\gen{F,T_{\tau}F}\), and Cauchy–Schwarz bounds the right-hand side by \(\norm{F}\,\norm{T_{\tau}F}\), so

This relates two different times and by itself bounds nothing; it must be iterated. Feeding it into itself \(n\) times,

and if \(g\) is bounded uniformly in \(\tau\) — which is what the regularity condition of Remark 109.80 is there to supply — then \(g(2^{n}\tau)^{2^{-n}}\to1\) as \(n\to\infty\) and \(g(\tau)\leq g(0)\), that is \(\norm{T_{\tau}}\leq1\). A self-adjoint contraction semigroup is \(\ee^{-\tau H/\hbar}\) for a unique self-adjoint \(H\geq0\), and the factor \(\hbar\) is fixed by the Euclidean weight \(\ee^{-S_{E}/\hbar}\) from which \(T_{\tau}\) was built. The quantum-mechanical case of this argument, with the operator chain written out explicitly, is Proposition 106.21.

Derivation. Derives Proposition 109.82. Write the Euclidean action as a sum over time slices, \(S_{E}=\sum_{n}\left[V(U_{n})+K(U_{n},U_{n+1})\right]\), in which \(V\) involves only the links within slice \(n\) and \(K\) only the links of two adjacent slices — this is exactly the plaquette structure, since a plaquette either lies in one slice or joins two neighbours. The functional integral then factorizes into a product of kernels, and reflection about the plane between slice \(0\) and slice \(1\) exchanges the two halves while conjugating, so that \(\avg{\overline{\theta F}\,F}\) is the squared norm of the vector obtained by applying the half-chain to the ground state — the argument of Proposition 106.21, with the continuous evolution replaced by a finite product of kernels. Positivity of the transfer operator follows because the kernel \(\ee^{-K}\) is a positive-definite function of the pair of slice configurations, and self-adjointness because \(K\) is symmetric under exchange of the two slices. The exponent \(aH/(\hbar c)\) is dimensionless because \(H\) is an energy and \(a/c\) a time.

The converse warning is Remark 106.22: an action coupling next-to-nearest time slices generally breaks the factorization, and the resulting transfer operator can have negative eigenvalues, whose signature is a correlator oscillating in Euclidean time.

Derivation. Derives Proposition 109.83. By Proposition 109.79, Euclidean time translation is \(\ee^{-\tau H/\hbar}\). Inserting a complete set of energy eigenstates between the two operators and subtracting the vacuum contribution,

which for large \(\tau\) is dominated by the smallest gap. Writing \(\tau=r/c\) for the Euclidean distance \(r\) in metres and \(E_{1}-E_{0}=\Delta\) gives Equation (109.52). With \(\Delta=Mc^{2}\), \(\xi=\hbar c/(Mc^{2})=\hbar/(Mc)\).

Derivation. Derives Proposition 109.88. Both sides of Equation (109.54) must scale the same way under \(x\to\lambda x\). At a fixed point of the renormalization group the operator \(\mathcal{O}_{n}\) scales with the power \(d_{n}+\gamma_{n}\) rather than \(d_{n}\), the correction being generated by the same loop effects that make the coupling run; matching powers gives Equation (109.57). The operator with the smallest \(d_{n}+\gamma_{n}\) dominates as \(\xi\to0\), which is why the expansion is useful: an infinite sum is controlled by its first few terms at short distance, and the accuracy improves as the separation shrinks. The identity operator has \(d=0\) and no anomalous dimension, so it always leads, and the leading singularity is \(\abs{\xi}^{-d_{A}-d_{B}}\) as in Example 109.87, where \(d_{A}=d_{B}=1\) gives \(\abs{\xi}^{-2}\).

Derivation. Derives Proposition 109.95. Write \(\beta=\dd\hat{\lambda}/\dd\ln\mu =24\hat{\lambda}^{2}/(16\pi^{2})=3\hat{\lambda}^{2}/(2\pi^{2})\). Separating variables, \(\dd\hat{\lambda}/\hat{\lambda}^{2}=3\,\dd\ln\mu/(2\pi^{2})\), so

and the right-hand side reaches \(1/\hat{\lambda}(\mu_{0})\) — at which point \(1/\hat{\lambda}(\mu)=0\) — when \(\ln(\mu/\mu_{0})=2\pi^{2}/\left(3\hat{\lambda}(\mu_{0})\right)\), which is Equation (109.58). Numerically \(2\pi^{2}/(3\times0.126)=52.2201\) and \(\ee^{52.2201}=4.7743\times 10^{22}\), so \(\mu_{L}=172.57\times4.7743\times 10^{22} =8.24\times 10^{24}\,\mathrm{GeV}\); the fourth figure of the exponential is carried because rounding it to \(4.77\times 10^{22}\) would move the answer to \(8.23\times 10^{24}\). The Planck energy is \(E_{P}=\sqrt{\hbar c^{5}/G}=1.956\times 10^{9}\,\mathrm{J} =1.221\times 10^{19}\,\mathrm{GeV}\) from \(\hbar=1.054571817\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\), \(c=299792458\,\mathrm{m}/\mathrm{s}\) and \(G=6.67430\times 10^{-11}\,\mathrm{m}^{3}/\mathrm{kg}/\mathrm{s}^{2}\) [Mohr:2025], giving the stated ratio.