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For a free complex scalar field, independent particle annihilators and antiparticle creators giveThe invariant mass shell integral vanishes at spacelike separation: transform to equal time, then reverse the spatial momentum in one term. Retaining only the particle-annihilation term would leave a Wightman function, which is generally nonzero even at spacelike separation. Thus both frequency branches are needed for a local relativistic field. For a real scalar field, the creation term supplies the same cancellation but creates the same self-conjugate particle species.

Microcausality requires local bosonic observables at spacelike separation to commute. For a free real scalar field, the field commutator is the difference of its two Wightman functions; it vanishes at equal time and, by Lorentz invariance, at every spacelike separation. Nonzero spacelike vacuum correlations do not violate this condition, since a correlation is different from the response to a local intervention. Odd fermionic fields satisfy a corresponding spacelike anticommutation condition, while physical even local observables commute.

Use units $\hslash =c=1$ and the Minkowski metric $g=\mathrm{diag}(1,-1,-1,-1)$. The canonical momentum of the real scalar field is $\pi =\partial \mathcal{L}/\partial \dot{\varphi }=\dot{\varphi }$. Canonical quantization replaces the classical fields by operator-valued distributions and imposes the equal-time canonical commutation relationsThese relations are imposed on an initial slice, with smeared fields understood. In the Heisenberg picture, simultaneous unitary time evolution of both operators preserves their commutators. Since the momentum is $\dot{\varphi }$, the requested equal-time commutator is $[\varphi (t,\mathbf{x}),\dot{\varphi }(t,\mathbf{y})]=i{\delta }^{(3)}(\mathbf{x}-\mathbf{y})$.

The Legendre transform in mechanics, applied pointwise, gives the canonical Hamiltonian density of a real scalar field. ThusTo obtain its dynamics rather than assume them, use the Heisenberg equation of motion $\dot{O}=i[H,O]$. The momentum-square term gives $[H,\varphi ]=-i\pi$. The field-square term gives $[H,\pi ]=i{m}^2\varphi$, while differentiating the Dirac delta function in the gradient term and integrating by parts gives $[H,\pi ]=-i{\nabla }^2\varphi +i{m}^2\varphi$. ConsequentlyBoundary terms vanish for suitably decaying smeared fields. This is the Klein-Gordon equation as an operator identity.

A spatial Fourier transform reduces the Klein-Gordon equation to ${\ddot{\varphi }}_{\mathbf{p}}+E_p^2{\varphi }_{\mathbf{p}}=0$, where $E_p=\sqrt{{\mathbf{p}}^2+m^2}$. Its two frequency branches and the Hermitian adjoint condition giveThe invariant-normalized scalar mode extraction can be written particularly simply asSubstituting the mode expansion of a free field proves these inversion formulas: the opposite-frequency coefficient is proportional to $E_p-E_q$ and vanishes after spatial Fourier inversion. Alternatively they are the corresponding Klein-Gordon inner products. Taking their commutators using the equal-time fields gives $(E_p+E_q)(2\pi {)}^3{\delta }^{(3)}(\mathbf{p}-\mathbf{q})$ in the mixed case. In the two equal-type cases the delta function enforces $\mathbf{q}=-\mathbf{p}$, multiplying $E_p-E_q=0$. ThereforeConversely these mode commutators reproduce $[\varphi ,\dot{\varphi }]=i{\delta }^{(3)}$: the two frequency contributions each supply half the spatial Dirac delta function.

For the covariant oscillator Hamiltonian of a real scalar field, insert the mode expansion of a free field into $H$. Spatial integration sets the two momenta equal in the mixed terms and opposite in the two-creator or two-annihilator terms. The latter have coefficient $-E_p^2+{\mathbf{p}}^2+m^2=0$. The mixed terms giveThe constant $E_0$ is the divergent zero-point energy; a finite box and a momentum cutoff make its meaning explicit. Normal ordering sets the free vacuum energy to zero and leaves all commutators unchanged. Directly applying the mode commutators to the mixed expression yieldsChoose the Fock vacuum by $a(p)|0⟩=0$. Each creation operator adds a boson of energy $E_p$ and four-momentum $p$, while each annihilation operator removes one. Products of commuting creation operators give bosonic statistics from commuting creation operators and unrestricted occupation numbers. The real scalar field has one spin-zero species, identical to its antiparticle. Its one-particle states obey $⟨q|p⟩=(2\pi {)}^{3}2E_p{\delta }^{(3)}(\mathbf{p}-\mathbf{q})$, precisely the relativistic normalization of a one-particle state.

For the Feynman propagator, put $z=x-y$ and contract the two fields in the Fock vacuum. Only $a{a}^{†}$ survives, giving the Wightman functionThe stipulated time-ordered product convention is $i{\Delta }_F(z)=\theta (z^0)W(z)+\theta (-z^0)W(-z)$. In a contour integral over $p^0$, the Feynman i-epsilon prescription places the positive-energy pole just below the real axis and the negative-energy pole just above it. For $z^0>0$ close below, clockwise, obtaining $-i{e}^{-i{E}_p{z}^0}/(2E_p)$; for $z^0<0$ close above, obtaining $-i{e}^{i{E}_p{z}^0}/(2E_p)$. Changing $\mathbf{p}\mapsto -\mathbf{p}$ in the second contribution therefore provesActing with $\square +m^2$ multiplies the integrand by $-(p^2-m^2)$. The remaining factor tends to $-1$ as a distribution, so Fourier inversion givesThe sign also follows from the derivative jump of a free scalar time-ordered two-point function: the first time derivative of $i{\Delta }_F$ jumps by $⟨[\dot{\varphi },\varphi ]⟩=-i{\delta }^{(3)}$, so that of ${\Delta }_F$ jumps by $-{\delta }^{(3)}$.

Use units $\hslash =c=1$ and the Minkowski metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$, so $p\cdot x=E_{p}t-\mathbf{p}\cdot \mathbf{x}$. The canonical momentum of the real scalar field is $\pi =\dot{\varphi }$, and the Hamiltonian operator isCanonical quantization of a real scalar field promotes $\varphi$ and $\pi$ to operator-valued distributions satisfying the equal-time canonical commutation relationProducts and operator commutators can first be interpreted after spatial smearing. An additive vacuum constant in $H$ does not affect the following equations. In the Heisenberg picture, $\dot{O}=i[H,O]$. Applying the canonical commutation relation to the quadratic Hamiltonian operator, and integrating the gradient term by parts, givesThus the Klein-Gordon equation is an operator equation. The spatial Fourier transform reduces it to ${\ddot{\varphi }}_{\mathbf{p}}+E_p^2{\varphi }_{\mathbf{p}}=0$, with $E_p=\sqrt{{\mathbf{p}}^2+m^2}$. Its two frequency branches, together with the Hermitian adjoint condition ${\varphi }^{†}=\varphi$, give the mode expansion of a free fieldThe measure $d\mu (p)$ is Lorentz invariant; its normalization fixes that of the creation and annihilation operators rather than imposing any extra physical assumption.

For invariant-normalized scalar mode extraction, take the Klein-Gordon inner product of $f_p$ with the field. The positive-frequency contribution has coefficient $E_p+E_q$ and spatial integral $(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$. The negative-frequency contribution has coefficient $E_p-E_q$ and spatial integral $(2\pi {)}^3{\delta }^3(\mathbf{p}+\mathbf{q})$, so it vanishes because $E_{-p}=E_p$. ConsequentlyTaking the Hermitian adjoint, with the order of the scalar factors immaterial, yieldsEquivalently $a(\mathbf{p})=\int d^{3}x{f}_p^{\ast }(E_p\varphi +i\pi )$ and $a^{†}(\mathbf{q})=\int d^{3}y{f}_q(E_q\varphi -i\pi )$. At a common time their operator commutator therefore isThe same calculation gives $[a(\mathbf{p}),a(\mathbf{q})]=[a^{†}(\mathbf{p}),a^{†}(\mathbf{q})]=0$: the surviving factor is an energy difference supported at opposite spatial momenta. These are the bosonic canonical commutation relations in relativistic normalization of a one-particle state.

Choose the Fock vacuum by $a(\mathbf{p})|0⟩=0$ and $⟨0|0⟩=1$. Only the annihilator-creator product contributes to the Wightman function, givingUsing time ordering and the reverse product for $y^0>x^0$, the Feynman propagator, with the convention $D_F=i{\Delta }_F$ in this calculation, isIn particular the denominator in the spatial measure is $2E_p$ in both terms.

To verify the four-dimensional Fourier transform representation, let $\tau =x^0-y^0$ and integrate the energy variable first:The Feynman i-epsilon prescription puts the positive-energy pole at $E_p-i0$ and the negative-energy pole at $-E_p+i0$. For $\tau >0$, the contour integral closes in the lower half-plane with clockwise orientation. The residue theorem multiplies the residue $i{e}^{-i{E}_p\tau }/(2E_p)$ by $-i$, giving $e^{-i{E}_p\tau }/(2E_p)$. For $\tau <0$, it closes counterclockwise above: multiplying the residue $-i{e}^{i{E}_p\tau }/(2E_p)$ by $i$ gives $e^{i{E}_p\tau }/(2E_p)$. Thus $I(\tau )=e^{-i{E}_p|\tau |}/(2E_p)$. After inserting the spatial exponential, reversing $\mathbf{p}$ in the negative-time term gives exactly the time ordering expression above. HenceThis equality is understood as a distribution limit; the same prescription gives $(\square +m^2)D_F(x-y)=-i{\delta }^4(x-y)$.

The normalized Fock vacuum obeys $a_{\mathbf{p}}|0⟩=0$ for every momentum and $⟨0|0⟩=1$. Taking adjoints gives $⟨0|a_{\mathbf{p}}^{†}=0$. Of the four products in the mode expansion of $\varphi (x)\varphi (y)$, only $a_{\mathbf{p}}{a}_{\mathbf{q}}^{†}$ has a nonzero vacuum expectation value. The commutator impliesConsequentlyThis unordered Wightman function differs from the time-ordered Feynman propagator when the temporal ordering is reversed.

Antiparticles are positive-energy quanta of relativistic quantum fields, not physical particles with negative energies. The negative-frequency part of a quantum field creates these quanta; the distinction between its classical frequency and a state's excitation energy is central to their interpretation.

For a real scalar field, Hermiticity identifies the two frequency branches:There is one species of excitation. Its particle is its own antiparticle; a single Hermitian scalar cannot carry the nonzero continuous phase charge of a complex field. This does not prevent self-conjugate particles from having other quantum numbers.

A complex scalar field is not Hermitian. Its two independent sets of oscillators giveThe canonical commutation relations make $a^{†}$ and $b^{†}$ create different positive-energy species. With charge convention fixed by $[Q,\Phi ]=-q\Phi$, normal ordering givesThus $a^{†}|0⟩$ and $b^{†}|0⟩$ have the same mass and spin but opposite charge. Although $b^{†}{e}^{ipx}$ has negative classical frequency, $[H,b^{†}]=E_p{b}^{†}$, so it creates a positive-energy excitation. Two real scalar fields can be combined into one complex field; the particle–antiparticle basis diagonalizes the conserved phase charge. Neutrality alone does not require a particle to be self-conjugate.

Locality explains why both branches occur. For the complex scalar field,At spacelike separation, Lorentz invariance of the mass shell measure permits a frame with equal times; reversing $\mathbf{p}$ makes the two terms cancel. This is microcausality. If the antiparticle creation term were omitted, the commutator would be just a Wightman function, generally nonzero at spacelike separation. For example, for a massless scalar at equal times and separation $r>0$, it is $1/(4{\pi }^2{r}^2)$. The cancellation is the role of antiparticle modes and spacelike commutativity. In a real scalar field its self-conjugate creation modes supply the same cancellation.

A nonrelativistic particle field obeys a first-order time equation and need not contain an antiparticle-creation branch. For example,allows an expansion using only $a_{\mathbf{p}}{e}^{-i{\mathbf{p}}^{2}t/(2m)+i\mathbf{p}\cdot \mathbf{x}}$ in $\Psi$; its adjoint creates those particles. The basic theory conserves particle number and does not impose a Lorentz-invariant light cone locality condition. It often arises from a relativistic quantum field by factoring out the rest-energy oscillation and restricting to a low-energy particle sector, where creation of particle–antiparticle pairs costs roughly $2m$. An antiparticle species can certainly also be treated nonrelativistically with its own field; antiparticles are simply not forced by this nonrelativistic quantum field equation.

Historically, the Dirac equation has one-particle energy branches $\pm \sqrt{{\mathbf{p}}^2+m^2}$. Treating both as ordinary electron energies would leave no lowest-energy one-particle vacuum: an electron could fall through arbitrarily negative levels. The Dirac sea interpretation fills every negative-energy electron level. The Pauli exclusion principle then blocks further occupation. A missing negative-energy electron is a hole: removing energy $-E$ increases energy by $E$, and removing charge $-e$ leaves charge $+e$. This predicts the positron's mass, spin and opposite electric charge.

The quantum field theory account quantizes a Dirac field instead:The independent particle and antiparticle operators obey canonical anticommutation relations. Before normal ordering, the negative-frequency contribution to the Dirac Hamiltonian has the form $-E_p{b}_r{b}_r^{†}$. Anticommutation rewrites it as $+E_p{b}_r^{†}{b}_r$ plus a vacuum constant. Removing the constant leaves positive particle and antiparticle energies. The charge has opposite signs on the two occupation numbers, and interactions permit pair creation and annihilation while conserving the relevant charges.

There is therefore no need for a literal infinitely populated material sea. The hole picture is a useful fermionic reinterpretation, but it cannot explain bosonic antiparticles through filled-level exclusion, since bosons can multiply occupy a state. Canonical quantization of a complex scalar field supplies them just as consistently. More generally charge conjugation relates field representations with opposite internal charges; real scalar fields and Majorana fermions are self-conjugate exceptions. The relativistic Fock space description handles these cases, statistics and changing particle number within one framework.

Split the free scalar field into its annihilation part ${\varphi }^{(+)}$ and creation part ${\varphi }^{(-)}$. Normal ordering moves every creation operator to the left of every annihilation operator, without adding the commutators produced by that rearrangement. For two fields,Its vacuum expectation is zero. Time ordering instead places the field at the later time on the left:The equal-time convention is immaterial away from coincident singularities; these expressions are understood as operator-valued distributions.

The only nonzero vacuum contraction uses $a_{\mathbf{p}}{a}_{\mathbf{q}}^{†}$. The oscillator commutator therefore gives the Wightman functionPut $\tau =x^0-y^0$ and $\mathbf{r}=\mathbf{x}-\mathbf{y}$. Combining the two time orders, and reversing $\mathbf{p}$ in the second when necessary, givesTo recover the four-dimensional Fourier transform, use the Feynman i-epsilon prescription in the energy variable:The positive-energy pole lies just below the real axis, and the negative-energy pole just above it. For $\tau >0$, close below clockwise: the residue at $E_{\mathbf{p}}-i0$ is $i{e}^{-i{E}_{\mathbf{p}}\tau }/(2E_{\mathbf{p}})$, and the clockwise factor $-i$ after dividing by $2\pi$ gives the required positive coefficient. For $\tau <0$, close above counterclockwise: the residue at $-E_{\mathbf{p}}+i0$ is $-i{e}^{i{E}_{\mathbf{p}}\tau }/(2E_{\mathbf{p}})$, and the factor $+i$ gives the other time order. Equivalently the energy contour passes above the positive-energy pole and below the negative-energy pole. HenceThe limit is distributional. The scalar Feynman propagator pole prescription is essential; omitting it from the denominator must be accompanied by the specified contour.

Finally, moving the annihilation part of $\varphi (x)$ past the creation part of $\varphi (y)$ givesThe normal-ordered two-field product is symmetric in $x,y$, because the two creation parts commute and the two annihilation parts commute. Applying the two time orders thus proves the two-field scalar Wick identity

The CPT theorem is a structural consequence of relativistic quantum field theory. In flat spacetime, assume a positive physical state space, the positive-energy spectrum condition, Lorentz invariance, microcausality, the usual Spin-statistics theorem, and a Hermitian dynamics. For ordinary field theories these are realized by a local Hermitian Lorentz-scalar Lagrangian density. There is then an antiunitary operator $\Theta$ implementing the combined charge conjugation, parity symmetry in quantum field theory and time-reversal symmetry, with a CPT-invariant vacuum. The combined symmetry is required even when the three separate symmetries fail.

Under the combined spacetime transformation, $x\mapsto -x$. A complex scalar field can be taken to transform into its adjoint at $-x$; a Hermitian vector field transforms into minus itself at $-x$. A derivative contributes a minus sign through differentiation of its reflected argument. For a Dirac field the transformation supplied in the paper may be writtenField phases are conventional and do not affect the physical conclusions. Crucially, antiunitarity means $\Theta i{\Theta }^{-1}=-i$ and conjugates every numerical coefficient. The reversal of time is compatible with positive energies because this conjugation changes $e^{-iHt}$ to $e^{iHt}$ while leaving the Hamiltonian invariant. Treating time reversal as a unitary change of coordinates would instead give incorrect signs.

One can see explicitly how the CPT transformation of Dirac bilinears works. Let $B_{\Gamma }(x)=:{\psi }^{†}{\gamma }^0\Gamma \psi :$ be a Hermitian fermion bilinear, with ${\Gamma }^{†}={\gamma }^0\Gamma {\gamma }^0$. The colons indicate normal ordering so that the graded reordering below has no vacuum contact term. Using the reflected fields and conjugating ${\gamma }^0\Gamma$, then interchanging the two fermion factors, givesThe last step uses the Dirac-Hermiticity condition and $\{{\gamma }^0,{\gamma }_5\}=0$. The identities for the gamma matrices then giveThe pseudoscalar's sign is particularly instructive: its explicit $i$ must also be conjugated. The minus sign from exchanging fermions is equally essential. These signs depend on the number of free vector indices, rather than on a field's separate parity assignment.

This supplies a local Lagrangian argument for CPT invariance. Hermitian local field combinations with $r$ vector indices transform with sign $(-1{)}^r$, including the appropriate reordered derivative terms. A Lorentz scalar contracts those indices in pairs or with four-index invariant tensors, giving a positive total sign. For charged terms, the reflected conjugate fields and conjugated coefficients exchange a term with its Hermitian-conjugate partner. ThusThe integration variable $x\mapsto -x$ has unit absolute Jacobian. The general field-index argument is discussed in Srednicki's treatment of discrete symmetries, section 40.

For example, the scalar mass term $\overline{\psi }\psi$ is even. The Dirac electromagnetic current and vector potential are both odd, making their contracted interaction even. The Hermitian Dirac kinetic term $(i/2)\overline{\psi }{\gamma }^{\mu }{\overleftrightarrow{\partial }}_{\mu }\psi$ is also even: reflection and reordering interchange the two derivative placements, while antiunitarity conjugates its $i$. A complex Yukawa coupling and its Hermitian conjugate are exchanged rather than requiring the coupling itself to be real. Hence CP violation in the CKM matrix is compatible with exact CPT. Renormalizability is not necessary: local Hermitian higher-dimensional Lorentz-scalar operators satisfy the same argument.

The axiomatic form of the CPT theorem does not start by assuming a polynomial Lagrangian density. The positive-energy spectrum condition provides analytic continuations of Wightman functions, and complexified Lorentz transformations relate configurations to their total spacetime inversions. At suitable spacelike configurations, microcausality permits reversal of the order of field operators with the signs dictated by the Spin-statistics theorem. Uniqueness of analytic continuation extends this relation to the correlators generally, from which the Hilbert-space antiunitary operator is obtained. This explains why locality and positive energy are genuine hypotheses, not incidental features of a particular interaction. The elementary argument above displays the mechanism for the scalar, spinor and vector fields relevant to the Standard Model; it is not a substitute for all the analytic steps of the axiomatic proof.

For one-particle states, CPT reverses additive charges and spin projections. The spatial momentum is unchanged by the combination: parity symmetry in quantum field theory and time reversal each reverse it. Thus, up to phases,Since the energy-momentum spectrum is preserved, the CPT constraints on particle properties give equal particle-antiparticle masses, equal spin magnitudes, and opposite conserved additive charges. Relating the corresponding two-point functions also fixes equal pole positions, so unstable conjugate particles have equal total decay widths, hence equal lifetimes. Magnetic moments at the same chosen spin orientation have opposite signs. These are precise experimental tests of CPT, not tests that every individual discrete symmetry is respected.

For scattering, CPT interchanges incoming and outgoing boundary conditions. The antiunitary operator identity $⟨a|b⟩=⟨\Theta b|\Theta a⟩$ relates $⟨f,\mathrm{out}|i,\mathrm{in}⟩$ to the amplitude with transformed $f$ incoming and transformed $i$ outgoing. Thus the reversed process with antiparticles and reversed spins has the same transition probability. This is not, in general, equality with the forward CP-conjugate process. In particular, CP violation can make corresponding partial decay widths different while their complete sums remain equal. With CPT preserved, genuine CP violation requires corresponding time-reversal symmetry violation.

Finally, a proposed CPT violation requires revisiting at least one assumption: for example, a nonlocal interaction, a preferred spacetime direction violating Lorentz invariance, or loss of unitary Hermitian dynamics. A generic curved background need not possess the global spacetime inversion used here. Those observations delimit the theorem; they do not weaken its prediction for the ordinary local Standard Model in Minkowski spacetime.

Use ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and $\hslash =c=1$. In the interaction picture, fields evolve with the free Hamiltonian, while interactions enter the state evolution through the Dyson series. Write the Klein-Gordon field as $\varphi ={\varphi }^{(+)}+{\varphi }^{(-)}$, with the positive-frequency part of a quantum field containing annihilation operators and ${\varphi }^{(-)}=({\varphi }^{(+)}{)}^{†}$ containing creation operators. The mode normalization in the PDF corresponds to $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$ and $E_{\mathbf{p}}=\sqrt{|\mathbf{p}{|}^2+m^2}$.

The time-ordered product places fields with the latest time on the left. For two bosonic insertions,The normal-ordered product $:\varphi (x_1)\cdots \varphi (x_n):$ instead moves every creation operator to the left of every annihilation operator, without adding the commutators generated by that rearrangement. Its vacuum expectation value is zero whenever a nonempty set of free-field factors remains.

The mode commutator gives the positive-frequency Wightman functionDefine the Feynman propagator here to be the two-point function itself, rather than that function divided by $i$. ThenEquivalently,The positive-energy pole is below and the negative-energy pole above the real axis. Closing the energy contour integral below for positive time separation and above for negative time separation gives the two terms above. Thus this is the scalar Feynman propagator pole prescription, and $(\square +m^2){\Delta }_F=-i{\delta }^4$. In particular, $T\{\varphi (x)\varphi (y)\}=:\varphi (x)\varphi (y):+{\Delta }_F(x-y)$: the difference between time ordering and normal ordering is the Wick contraction.

For $n$ free bosonic fields, the Wick theorem stateswhere $P$ runs over all sets of disjoint pairs of indices, including the empty set, and $V(P)$ is the set of paired indices. The empty normal-ordered product equals one. This is an operator identity for free fields, understood with smearing or a regulator when products of operator-valued distributions require it.

Here is an inductive proof of the scalar Wick theorem. Sort the arguments chronologically so that $x_1^0$ is the latest. For any remaining set $I$, commuting the annihilation part of ${\varphi }_1$ through the creation parts in a normal-ordered product givesThe creation part of ${\varphi }_1$ is already in the correct position. Every commutator on the right is a scalar, so no further operator rearrangement is needed for it. Because $x_1$ is latest, that commutator is ${\Delta }_F(x_1-x_j)$. Assume the Wick theorem for the other $n-1$ fields and multiply its expansion by ${\varphi }_1$. The first term of the displayed identity gives every pairing in which index 1 is unpaired; the sum gives every pairing in which 1 is paired with exactly one $j$. Each final partial pairing has exactly one such construction. The one-field identity supplies the base case, proving the theorem by mathematical induction. Applying time ordering restores arbitrary argument order.

Taking the vacuum expectation value retains only complete pairings. Odd free correlators vanish; for example the free scalar four-point function is ${\Delta }_{12}{\Delta }_{34}+{\Delta }_{13}{\Delta }_{24}+{\Delta }_{14}{\Delta }_{23}$. These facts explain why propagators are the edges of Feynman diagrams.

For the interacting quartic scalar field theory, the interaction Hamiltonian density is $\lambda {\varphi }^4/4!$. With a single time ordering acting on external insertions and interaction vertices, the unnormalized numerator expands asApply the proved Wick theorem term by term. Each complete pairing connects external insertions and four-legged vertices by Feynman propagators. There are $4!$ assignments of four distinct incident legs at a vertex, canceling its factorial. Permuting identical integrated vertices cancels the corresponding part of $v!$. Permutations still leaving the labeled graph unchanged give its Feynman-diagram symmetry factor. For example,whereas the two-point tadpole diagram has coefficient $-i\lambda /2$, because choosing the two vertex legs contracted with the labeled external fields gives $4\cdot 3$ possibilities and the other two contract with each other.

Thus the position-space rules attach ${\Delta }_F$ to every edge, $-i\lambda \int d^{4}z$ to every four-legged interaction vertex, and divide by the graph's Feynman-diagram symmetry factor. Fourier transformation gives the momentum-space Feynman rules:Enforce momentum conservation at each vertex, retaining the overall four-momentum delta function. External lines of these correlation functions remain unamputated. The unnormalized expression includes Vacuum Feynman diagrams; dividing by $N_0=⟨0|S|0⟩$ cancels those disconnected vacuum factors. They should not be discarded without that normalization.

There is a time-ordering convention in the printed correlation-function notation worth making explicit. The usual interacting correlator uses the single global time ordering in $N_n$ above. If the displayed $T\{{\varphi }_1\cdots {\varphi }_n\}S$ is read literally with $S$ outside that ordering, the external block and the vertex block are separately time ordered: their cross-contractions are $D^{+}(x_i-z)$ because each external field stands to the left of every vertex field. The same pairing construction then uses ${\Delta }_F$ for lines within either block and $D^{+}$ for lines between blocks; in momentum space the latter is $2\pi \theta (p^0)\delta (p^2-m^2)$. This literal product is different from the conventional globally time-ordered interacting correlator, and does not have the ordinary external Feynman propagators derived above.

For the real scalar field, the mostly-plus Minkowski metric gives a positive kinetic energy. Its canonical momentum and Hamiltonian operator areThe Euler-Lagrange field equation is $(\square -m^2)\varphi =0$, or $({\partial }_t^2-{\nabla }^2+m^2)\varphi =0$. Canonical quantization of a real scalar field promotes the fields to Hermitian operators and replaces the equal-time Poisson brackets by canonical commutation relations:The Heisenberg equation of motion then gives $\dot{\varphi }=\pi$ and $\dot{\pi }={\nabla }^2\varphi -m^2\varphi$, so the classical field equation remains an operator identity. Products at coincident points need a regulator; equivalently, the field is an operator-valued distribution.

Resolve the free field into positive- and negative-frequency plane waves. With $E_{\mathbf{k}}=\sqrt{|\mathbf{k}{|}^2+m^2}$ and $k\cdot x=-E_{\mathbf{k}}t+\mathbf{k}\cdot \mathbf{x}$,Reality pairs the two terms by Hermitian conjugation. The equal-time canonical commutation relations are equivalent toFor example, the two mixed terms in $[\varphi ,\pi ]$ each supply half the Dirac delta function. Thus the normalization $1/\sqrt{2E_{\mathbf{k}}}$ is fixed, not optional. Each independent mode is a quantum harmonic oscillator, with $a$ and $a^{†}$ its annihilation operator and creation operator.

Choose the Fock vacuum by $a(\mathbf{k})|0⟩=0$. Repeated creation operators construct the bosonic Fock space. Substitution into the Hamiltonian operator yieldswhere a finite volume $V$ can be used as a regulator. Normal ordering sets the flat-space vacuum reference to zero. The spatial momentum operator isIn particular, $[H,a^{†}(\mathbf{k})]=E_{\mathbf{k}}{a}^{†}(\mathbf{k})$ and $[\mathbf{P},a^{†}(\mathbf{k})]=\mathbf{k}{a}^{†}(\mathbf{k})$. A one-particle state therefore has positive energy, momentum $\mathbf{k}$, and invariant mass $m$. A scalar field transforms in the trivial spin representation, so its particles have spin zero. Because the field is real there is only one set of oscillators: the particle is its own antiparticle, with no distinct conserved particle-minus-antiparticle charge. The free theory does conserve its occupation-number sum, though generic scalar interactions need not.

A momentum eigenstate extends throughout space. A localized particle is described by a superposition such asIts wave packet evolves through the phase $e^{-i{E}_{\mathbf{k}}t}$; a narrow packet has group velocity $\mathbf{v}=\mathbf{k}/E_{\mathbf{k}}$, whose magnitude is at most one. Localization and dispersion concern these superpositions, rather than classical trajectories attached to individual field modes.

Relativistic spacetime behavior is encoded by microcausality. Directly from the oscillator commutators,This is a Lorentz-invariant distribution and vanishes at equal time. Any spacelike separation can be transformed to an equal-time separation, so $[\varphi (x),\varphi (y)]=0$ for spacelike $x-y$. Thus local operations at spacelike-separated points commute. The vacuum Wightman function need not vanish outside the light cone, but that correlation does not transmit a controllable signal. The Feynman propagator iswhich propagates positive energy forward and negative energy backward in the time-ordered product. For causal response one uses the retarded Green function, whose support lies in or on the future light cone.