Postulates of Quantum Field Theory

Quantum field theory (QFT) extends quantum mechanics to systems with infinitely many degrees of freedom and incorporates special relativity. The following postulates are stated in the Wightman axiomatic style, which provides a mathematically precise framework for relativistic QFT in Minkowski spacetime ${\mathbb{M}}^4$ with metric ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

See Definitions and Preliminaries for the underlying mathematical objects, and Remarks and Open Issues for the Wightman reconstruction theorem, Haag's theorem, gauge theories, and the status of rigorous construction.

Postulate 1 — Relativistic State Space

The states of the system are unit rays in a complex separable Hilbert space $\mathcal{H}$. There exists a strongly continuous unitary representation $U(\Lambda ,a)$ of the (proper, orthochronous) Poincaré group ${\mathcal{P}}_{+}^{\uparrow }$ acting on $\mathcal{H}$:

$$U({\Lambda }_1,a_1)U({\Lambda }_2,a_2)=U({\Lambda }_1{\Lambda }_2,a_1+{\Lambda }_1{a}_2).$$

Physical predictions (probabilities, expectation values) are independent of the choice of inertial frame.

Postulate 2 — Spectrum Condition

The generators of spacetime translations form the energy–momentum operator $P^{\mu }$, defined by

$$U(1,a)=e^{i{a}_{\mu }{P}^{\mu }}.$$

Its joint spectrum lies in the closed forward light cone:

$$\mathrm{spec}(P^{\mu })\subseteq {\overline{V}}_{+}=\{p^{\mu }:p^0\ge 0,p^{\mu }{p}_{\mu }\ge 0\}.$$

This guarantees positive energy in every inertial frame and forbids tachyonic or negative-energy states.

Postulate 3 — Unique Poincaré-Invariant Vacuum

There exists a unique (up to phase) state $|0⟩\in \mathcal{H}$, the vacuum, that is invariant under all Poincaré transformations:

$$U(\Lambda ,a)|0⟩=|0⟩,P^{\mu }|0⟩=0.$$

Uniqueness implements the absence of spontaneous symmetry breaking of the Poincaré group; it is essential for the cluster decomposition property.

Postulate 4 — Field Operators

Quantum fields ${\varphi }_i(x)$ are operator-valued tempered distributions on Minkowski spacetime: for every test function $f\in \mathcal{S}({\mathbb{M}}^4)$, the smeared field

$${\varphi }_i(f)=\int d^{4}xf(x){\varphi }_i(x)$$

is a (generally unbounded) operator on a common dense domain $\mathcal{D}\subset \mathcal{H}$ that contains the vacuum and is stable under the action of all ${\varphi }_i(f)$ and $U(\Lambda ,a)$.

Fields are not pointwise operators because ${\varphi }_i(x)|0⟩$ generally has infinite norm; smearing with $f(x)$ regularizes this.

Postulate 5 — Poincaré Covariance of Fields

Under a Poincaré transformation, the fields transform according to a finite-dimensional representation $S(\Lambda )$ of the Lorentz group (e.g. scalar, spinor, vector):

$$U(\Lambda ,a){\varphi }_i(x)U(\Lambda ,a{)}^{-1}=\sum _j{S}_{ij}({\Lambda }^{-1}){\varphi }_j(\Lambda x+a).$$

This ties the algebraic structure of the fields to the geometry of spacetime.

Postulate 6 — Microcausality (Local Commutativity)

Fields at spacelike-separated points either commute or anticommute, according to their spin–statistics:

$$[{\varphi }_i(x),{\varphi }_j(y){]}_{\pm }=0\text{whenever }(x-y{)}^2<0,$$

with the commutator ($[,{]}_{-}$) for bosonic (integer-spin) fields and the anticommutator ($[,{]}_{+}$) for fermionic (half-integer-spin) fields. This expresses relativistic causality: measurements at spacelike separation cannot influence each other.

Postulate 7 — Spin–Statistics

In a Lorentz-invariant local QFT with positive energy, fields of integer spin must be quantized as bosons (commutators) and fields of half-integer spin as fermions (anticommutators). Any other choice leads to either a non-positive Hilbert space, violation of microcausality, or violation of the spectrum condition. (This is a theorem — the spin–statistics theorem — in the Wightman framework, but it is often listed alongside the postulates.)

Postulate 8 — Cyclicity of the Vacuum

Polynomials in the smeared field operators acting on the vacuum yield a dense subspace of $\mathcal{H}$:

$$\overline{\mathrm{span}\{{\varphi }_{i_1}(f_1)\cdots {\varphi }_{i_n}(f_n)|0⟩:n\ge 0\}}=\mathcal{H}.$$

Equivalently, every state can be approximated by acting with local field operators on the vacuum. This ensures the field algebra is large enough to describe all physical states.

Postulate 9 — Dynamics from a Local Action

The dynamics of the fields are derived from a Poincaré-invariant, local action

$$S[\varphi ]=\int d^{4}x\mathcal{L}(\varphi (x),{\partial }_{\mu }\varphi (x)),$$

where the Lagrangian density $\mathcal{L}$ is a polynomial in the fields and their first derivatives at the same spacetime point. Classical equations of motion follow from $\delta S=0$. The corresponding quantum theory is defined either through:

• Canonical quantization: imposing equal-time (anti)commutation relations on the fields and their conjugate momenta ${\pi }_i=\partial \mathcal{L}/\partial ({\partial }_0{\varphi }_i)$, or
• Path integral quantization: defining correlation functions via $$⟨0|T\varphi (x_1)\cdots \varphi (x_n)|0⟩=\frac{\int \mathcal{D}\varphi \varphi (x_1)\cdots \varphi (x_n)e^{iS[\varphi ]}}{\int \mathcal{D}\varphi e^{iS[\varphi ]}}.$$

Postulate 10 — Asymptotic Completeness and the S-Matrix

In scattering theory, the framework rests on two non-trivial assumptions, with the existence and unitarity of the S-matrix following as a consequence:

(P10a) Existence of asymptotic states. The Møller operators

$${\Omega }_{\pm }\equiv \underset{t\to \mp \infty }{\lim }{e}^{iHt}{e}^{-i{H}_{0}t}$$

exist on a dense subspace of $\mathcal{H}$ (limits taken in a strong-operator-on-wavepackets sense). They map free Fock states to the corresponding dressed in/out states of the interacting theory:

$$|i,\text{in}⟩={\Omega }_{+}|i,\text{free}⟩,|f,\text{out}⟩={\Omega }_{-}|f,\text{free}⟩.$$

This fails when interactions do not switch off enough at large times — e.g. long-range Coulomb potentials, or QED with massless soft photons.

(P10b) Asymptotic completeness. The asymptotic in- and out-spaces both equal the full interacting Hilbert space:

$${\mathcal{H}}_{\text{in}}\equiv \mathrm{ran}{\Omega }_{+}=\mathcal{H},{\mathcal{H}}_{\text{out}}\equiv \mathrm{ran}{\Omega }_{-}=\mathcal{H}.$$

That is, every state of the interacting theory is reachable as some asymptotic in-state, and equivalently as some asymptotic out-state. This fails in confining theories (QCD: quarks/gluons are not asymptotic states; only hadrons are).

Consequence: the S-matrix. Given (P10a) and (P10b), the S-matrix

$$S\equiv {\Omega }_{-}^{†}{\Omega }_{+}:{\mathcal{H}}_{\text{in}}\to {\mathcal{H}}_{\text{out}},|f,\text{out}⟩=S|i,\text{in}⟩$$

is automatically a unitary operator on $\mathcal{H}$. Its matrix elements $⟨f|S|i⟩$ are computed from time-ordered correlation functions via the LSZ reduction formula.

Status of P10. Provable from the Wightman axioms + isolated mass shells via the Haag–Ruelle theorem (1962); held conjectural for realistic 4D theories like QED and QCD (where rigorous construction is an open problem, see Remarks); modified or replaced in IR-divergent and confining cases (Faddeev–Kulish dressing for soft photons; reformulation on hadron Fock space for QCD; abandoned entirely for conformal field theories without a mass gap).

See foundations-modern.md §2.0.1 for the modern derivation and discussion of failure modes.

Implicit Empirical Inputs

Beyond the ten postulates above, the standard QFT framework silently absorbs several empirical inputs that the modern Wigner–Weinberg derivation makes explicit (see foundations-modern.md §1 and its §7.1 reverse mapping):

• Particle = irreducible unitary Poincaré representation. P1 names "Hilbert space + unitary Poincaré action" but does not say particles are irreps. The identification is logically prior to P1 — it is what makes P1 the natural starting point. See foundations-modern.md §1.1.
• Symmetrization postulate. P7 (spin–statistics) packages two distinct claims: the binary "states are symmetric OR antisymmetric under exchange" and the assignment "integer spin → bosons, half-integer → fermions". The first is a separate postulate (an additional input in $d\ge 4$ that can be relaxed to anyons in $2+1$d); the second is a theorem given the first plus M1+M2+M3 + locality. Most treatments collapse the two; see foundations-modern.md §1.3.
• Variable particle number. No postulate above states that particle number is non-conserved; this is empirically motivated (relativity allows pair production via $E=m{c}^2$; decays change $n$; scattering admits different in/out counts) and is silently used by P4 (operator-valued field distributions) and P10 (multi-particle in/out spaces). See foundations-modern.md §1.4.
• Field content (which species, which masses, which couplings). The postulates set the framework; the actual content of any given QFT (QED, QCD, electroweak) is empirical.

These inputs are universally accepted and never controversial in standard QFT — but they are not derivable from the ten postulates alone, and a reader using this page in isolation would not see them flagged anywhere. The modern derivation makes the assumption budget transparent at the cost of more upfront machinery.