7 · relativistic quantum fields from symmetry and locality

asymptotic states and the $S$-matrix

The Hilbert space of a relativistic theory contains subspaces of multi-particle states transforming as tensor products of the irreducible Poincare representations classified in Chapter 5. In and out asymptotic states $|\alpha\rangle_{\mathrm{in}}$ and $|\beta\rangle_{\mathrm{out}}$ are identified with free-particle Fock states at $t\to\pm\infty$. The scattering operator $S$ is the unitary map relating the two bases, with matrix elements $$ S_{\beta\alpha}=\langle\beta|S|\alpha\rangle $$ the observable transition amplitudes.

definition (cluster decomposition). $S$-matrix elements factorize when a subset of particles is translated by a large spacelike vector. this factorization is the quantum expression of the independence of distant experiments.

example (two distant scatterings). if process $A$ occurs near $x$ and process $B$ near $y$ with $(x-y)^2<0$ and $|x-y|$ large compared with all intrinsic scales, the amplitude for the combined process approaches the product of the separate amplitudes, up to disconnected vacuum bubbles.

creation and annihilation operators

On the free Fock space one introduces creation and annihilation operators $a^\dagger(\mathbf{p},\sigma)$ and $a(\mathbf{p},\sigma)$ that add or remove a single particle of momentum $\mathbf{p}$ and spin (or helicity) $\sigma$. They satisfy the CCR or CAR dictated by the statistics of the representation. Any free-particle state is obtained by acting with a finite product of creation operators on the vacuum $|0\rangle$.

The free Hamiltonian and momentum operators are the corresponding one-particle generators summed over all modes: $$ H=\sum_\sigma\int d^3p\,E_{\mathbf{p}}\,a^\dagger(\mathbf{p},\sigma)a(\mathbf{p},\sigma), $$ and likewise for $\mathbf{P}$. These operators implement the unitary Poincare action on the free Fock space.

remark (normalization). continuum-normalized operators satisfy $[a(\mathbf{p}),a^\dagger(\mathbf{p}')]=\delta^3(\mathbf{p}-\mathbf{p}')$ (bosons) or the analogous CAR (fermions). wave-packet smearing returns to the Hilbert-space setting of Chapter 3.

construction of covariant fields

A quantum field is an operator-valued distribution $$ \phi:f\mapsto\phi(f)=\int\phi(x)f(x)\,d^4x,\qquad f\in\mathcal{S}(\mathbb{R}^{1,3}), $$ that creates and destroys particles. More concretely, one seeks operator-valued distributions of the form $$ \psi_\ell(x)=\sum_\sigma\int d^3p\Bigl( u_\ell(\mathbf{p},\sigma)a(\mathbf{p},\sigma)e^{-ip\cdot x} +v_\ell(\mathbf{p},\sigma)a^{c\dagger}(\mathbf{p},\sigma)e^{ip\cdot x} \Bigr) $$ whose coefficient functions $u_\ell$ and $v_\ell$ are chosen so that $\psi_\ell(x)$ transforms covariantly under the Poincare group: $$ U(\Lambda,a)\psi_\ell(x)U(\Lambda,a)^{-1} =\sum_{\ell'}D_{\ell\ell'}(\Lambda^{-1})\psi_{\ell'}(\Lambda x+a). $$ Here $D(\Lambda)$ is a finite-dimensional representation of the Lorentz group (or of $\operatorname{SL}(2,\mathbb{C})$).

The requirement that $\psi_\ell(x)$ be local (fields at spacelike separation either commute or anticommute) fixes the relative normalization of the $u$ and $v$ coefficients and forces the spin-statistics connection: integer-spin fields commute, half-integer-spin fields anticommute.

example (real scalar). for a free hermitian scalar of mass $m$, $u(\mathbf{p})=v(\mathbf{p})=(2E_{\mathbf{p}})^{-1/2}$ (in a standard continuum convention) and $a^c=a$, yielding the familiar mode expansion of Chapter 6 as a special case of the general Weinberg construction.

locality and causality

Locality is expressed by the vanishing of the commutator (or anticommutator) at spacelike separation: $$ \bigl[\phi(x),\phi(y)\bigr]_{\mp}=0\qquad\text{for }(x-y)^2<0. $$ This condition guarantees that measurements performed in spacelike-separated regions do not interfere, and it is equivalent to the cluster-decomposition property of the $S$-matrix once the fields are used to construct the interaction.

Causality further requires that the support properties of the commutator functions be compatible with relativistic propagation. The resulting analyticity of time-ordered products yields dispersion relations and the CPT theorem.

example (pauli-jordan function). for a free scalar the commutator $[\phi(x),\phi(y)]$ is a $c$-number distribution supported in the light cones, vanishing for spacelike $x-y$. that support property is the distributional form of microcausality.

spin-statistics and cpt

The demand that a covariant local field realize a given Poincare representation forces a rigid relation between spin and statistics. If $D$ is the finite-dimensional Lorentz representation carried by the field indices and $s$ is the spin of the associated particle, locality $$ \bigl[\psi_\ell(x),\psi_{\ell'}(y)\bigr]_{\mp}=0\qquad\text{for }(x-y)^2<0 $$ is compatible with a nontrivial $u,v$ expansion if and only if one uses the commutator when $s\in\mathbb{Z}$ and the anticommutator when $s\in\tfrac12+\mathbb{Z}$.

The same hypotheses imply the existence of an anti-unitary operator $\Theta$ implementing CPT, $$ \Theta\psi_\ell(x)\Theta^{-1}=\eta_{\ell\ell'}\psi_{\ell'}^\dagger(-x) $$ (up to standard phase conventions), which intertwines each particle representation with its antiparticle representation and forces equality of particle and antiparticle masses $m=\overline{m}$.

remark (where locality enters). without microcausality one can invent covariant particle theories that violate spin-statistics. the support condition on $[\psi(x),\psi(y)]_{\mp}$ is the additional geometric constraint that selects the physical quantization.

free-field propagators

The vacuum expectation value of the time-ordered product of two free fields is the Feynman propagator $$ \langle 0|T\phi(x)\phi(y)|0\rangle=i\Delta_F(x-y). $$ In momentum space it is the familiar meromorphic distribution $$ \Delta_F(p)=\frac{1}{p^2-m^2+i\varepsilon} $$ (with appropriate numerator polynomials for fields of nonzero spin). The propagator is the basic building block of the perturbative expansion and encodes both the particle poles and the $i\varepsilon$ boundary conditions required by causality.

example (dirac propagator). the numerator becomes $(\not p+m)$, recovering the electron propagator of Chapter 6 as the spin-$\tfrac12$ case of the same construction.

interactions and the dyson expansion

An interacting theory is defined by adding to the free Hamiltonian a local interaction density $\mathcal{H}_{\mathrm{int}}(x)$ constructed as a polynomial in the fields and their conjugates. The $S$-operator admits the Dyson series $$ S=T\exp\Bigl(-i\int\mathcal{H}_{\mathrm{int}}(x)\,d^4x\Bigr). $$ Each term in the expansion is a time-ordered product of interaction densities. Wick's theorem reduces these products to sums of normal-ordered products multiplied by contractions; the contractions are precisely the free propagators.

The resulting expressions are represented graphically by Feynman diagrams whose external lines correspond to the asymptotic particles and whose internal lines are propagators.

example (local monomials). for a scalar field, $\mathcal{H}_{\mathrm{int}}=\lambda\phi^4/4!$ is a local density. its Dyson expansion generates the standard $\phi^4$ Feynman rules, with a four-scalar vertex of strength $-i\lambda$.

perturbative scattering amplitudes

For a given process the Feynman rules translate every diagram into a momentum-space integral. External wave functions are the coefficient functions $u_\ell(\mathbf{p},\sigma)$ and $v_\ell(\mathbf{p},\sigma)$ of the asymptotic fields; vertices are the numerical coefficients appearing in $\mathcal{H}_{\mathrm{int}}$; internal lines contribute propagators.

After integration over loop momenta one obtains the invariant amplitude $\mathcal{M}$. The differential cross-section or decay rate follows by squaring $\mathcal{M}$, averaging over initial spins, summing over final spins, and multiplying by the appropriate phase-space measure. At tree level the procedure is finite; at loop level ultraviolet divergences appear and must be absorbed by renormalization (Chapter 9).

the connection between fields and particles

The asymptotic fields create and destroy the free particles whose representations were classified by Mackey theory. The same fields, when inserted into a local interaction, generate the nontrivial $S$-matrix that converts in-states into out-states. Because the fields transform covariantly and commute (or anticommute) at spacelike separation, the resulting theory automatically satisfies Poincare invariance, cluster decomposition, and causality.

Thus the operator-valued distributions $\phi(f)$ are the unique objects that mediate between the representation-theoretic description of particles and the local composition of relativistic experiments.

summary

Relativistic quantum field theory is obtained by requiring that the Hilbert space of Poincare representations also carry a set of operator-valued distributions transforming covariantly under the Poincare group and satisfying microcausality. Creation and annihilation operators build the free fields; locality forces the spin-statistics connection and the CPT theorem; the Dyson expansion of a local interaction produces the perturbative $S$-matrix whose matrix elements are computed by Feynman diagrams.

The construction therefore supplies the unique operator-valued distributions that mediate between Poincare particle representations and locally composable relativistic experiments, and it is the structural foundation on which gauge theories, renormalization, and the operator-algebraic formulation of quantum field theory are subsequently erected.