Perturbative quantum field theory

Perturbative quantum field theory approximates an interacting quantum field theory by expanding around a solvable reference theory, usually a free field theory. Powers of interaction couplings organize the contributions to correlation functions and scattering amplitudes. Feynman diagrams encode the terms, their internal propagators and their combinatorial factors.

At leading order, tree diagrams often reveal the main process; loop diagrams give quantum corrections. Divergent intermediate expressions require a specified regularization and renormalization prescription before comparison with physical quantities. A small coupling is helpful but is not the only validity condition: large logarithms, thresholds or long evolution can invalidate a fixed-order approximation. Some phenomena, including certain bound states and tunnelling effects, require resummation or methods beyond an ordinary power series.

A Hermitian matrix model replaces a spacetime field by a finite-dimensional Hermitian matrix. Its normalized integral supplies correlation functions and a zero-dimensional perturbative quantum field theory expansion. For an action built from traces of matrix powers, unitary matrix conjugation is a symmetry and the eigenvalues describe the invariant degrees of freedom. A quartic real integral needs a nonnegative highest-order coupling for convergence; a formal power series can be studied independently of this convergence issue.

Imposing vanishing off-diagonal entries with auxiliary integrations and an off-diagonal Grassmann Gaussian integral gives a determinant of the commutator map on the off-diagonal matrix units. Its eigenvalues are ${\lambda }_i-{\lambda }_j$, $i\ne j$. Up to a constant sign, their product is the squared Vandermonde determinant. It contributes $-2{\sum }_{i<j}\log |{\lambda }_i-{\lambda }_j|$ to the eigenvalue action, encoding repulsion between distinct eigenvalues. Diagonal zero modes and constant normalization factors must be treated separately.

For $S=a\mathrm{tr}(M^2)/2+b\mathrm{tr}(M^4)/4$ with $a>0$, the Wick theorem givesThere are eight adjacent external-attachment contractions with one free matrix-index sum and four opposite attachments with no free sum. The second tensor structure is needed at finite $N$; omitting it is an additional large-$N$ approximation, not merely restricting to connected Feynman diagrams.

A radiative correction is a quantum correction beyond the classical or tree approximation, represented in perturbative quantum field theory by Feynman diagrams of positive loop order. Such corrections can alter a scalar potential and its derivatives; symmetries and renormalization constrain which changes are allowed.

A perturbation series expands a correlation function or scattering amplitude in powers of interaction couplings. In quantum field theory it is commonly asymptotic because diagram counts grow factorially and nonperturbative effects are invisible to every finite order.

Weak coupling means a suitably dimensionless interaction strength is small. In BCS theory, ${\nu }_{s}V\ll 1$ gives a gap exponentially small compared with the pairing-shell energy cutoff; the expansion parameter is not the dimensional quantity $V$ alone.

The Dyson series solves interaction picture evolution by keeping noncommuting Hamiltonian operators in chronological order. With $\hslash =1$, integrating $i{\partial }_t{U}_I=H_I(t)U_I$ and iterating gives

Time ordering puts later times on the left. Integrating over all time labels instead of the ordered region introduces $1/n!$. The formal scattering operator limit isAn ordinary matrix exponential suffices when all the $H_I(t)$ commute. For quantum fields, the displayed expansion requires a prescription for singular products and need not converge. ^[ref]

For bounded norm-continuous Hamiltonian operators on $[t_0,t]$, the Dyson series has an $n$-th term bounded by $M^n|t-t_0{|}^n/n!$, so it converges in operator norm. In its nested-integral form, differentiating the outer endpoint puts $H_I(t)$ on the left of the $(n-1)$-fold term. Summing yields ${\partial }_{t}U=-i{H}_I(t)U$ and $U(t_0,t_0)=I$. If $H_I$ is Hermitian, differentiating $U^{†}U$ proves unitarity. For unbounded field operators the same algebra is a formal perturbative identity unless domain and convergence assumptions or a regulator justify the steps. Evolution backward uses inverse, anti-time-ordered evolution.

Wick theorem expresses a time-ordered product of free fields as its normal-ordered product plus the sum over every possible contraction. It turns the Dyson series into Feynman diagrams.

Split a free scalar field into its annihilation and creation parts. For $x^0>y^0$, changing the product into normal ordering adds $[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$, the positive-frequency two-point function. For the opposite ordering interchange the two points. Their combination is the Feynman propagator, equal to the vacuum expectation of the time-ordered product. Iterating this identity gives the scalar Wick theorem.

For a free real scalar field, split $\varphi ={\varphi }^{(+)}+{\varphi }^{(-)}$ into annihilation and creation parts. Moving the creation operators left gives $\varphi (x)\varphi (y)=:\varphi (x)\varphi (y):+[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$. The commutator is the Wightman function. Choosing either time order gives the displayed identity with the Feynman propagator. It is an identity of operator-valued distributions; coincident-point products need a suitable prescription.

Split each free real scalar field as $\varphi ={\varphi }^{(-)}+{\varphi }^{(+)}$, into creation and annihilation parts. For chronologically ordered arguments, the first field has the latest time. Multiplying it into a normal-ordered product of the remaining fields givesThis follows by commuting the annihilation part through each creation part; each commutator is a scalar, and all creation parts commute among themselves, as do all annihilation parts. The scalar commutator is the Feynman propagator for the chronological pair. Induction on the number of fields therefore lists every partial pairing exactly once: either field 1 remains unpaired or it pairs with one $j$. This proves the Wick theorem for free bosonic fields, including the empty pairing. Vacuum expectation retains only complete pairings. The identities are understood for regulated or smeared operator-valued distributions.

For a centered free real scalar field, its vacuum time-ordered product has expectation $⟨T{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4⟩=G_{12}{G}_{34}+G_{13}{G}_{24}+G_{14}{G}_{23}$, where $G_{ij}$ is the Feynman propagator between insertions $i,j$. These are the three complete pairings in the Wick theorem. All remaining normal-ordered products have zero vacuum expectation. No fermionic interchange signs occur for a bosonic field.

A centered free field in a Gaussian vacuum has only pair contractions, so every odd correlator vanishes by Wick theorem. Equivalently, number parity reverses the field while preserving the vacuum. This statement applies to smeared or regulated fields; it does not assert that unregulated coincident products exist. Interactions or a nonzero field mean can invalidate it.

Time ordering places operators with later time arguments to the left. Interchanging fermionic operators also contributes the sign of the required permutation.

For two odd fermionic operators, time ordering contributes a minus sign whenever the chronological rearrangement exchanges them. For several fields the sign is the sign of a permutation restricted to the odd fields. This convention makes the free Dirac propagator a Green function of the Dirac equation with its contact term fixed by the canonical anticommutation relations.

The time-ordered exponential solves $i\hslash {\partial }_{t}U=H(t)U$ with initial condition $U(t_0,t_0)=I$. Its expansion is the Dyson series. For bounded continuous Hamiltonian operators the series converges, as in differentiation of the Dyson time-ordered exponential. If the generators commute at all times, time ordering is unnecessary and an ordinary operator exponential suffices.

For $\dot{U}=A(t)U$ and $U(0)=I$, the Magnus expansion writes the logarithm of the propagator as iterated integrals of commutators. Its first terms are ${\Omega }_1={\int }_0^{T}A(t)dt$ and ${\Omega }_2=\frac{1}{2}{\int }_0^{T}dt{\int }_0^{t}ds[A(t),A(s)]$. For a Quantum Hamiltonian, $A=-iH/\hslash$, so a noncommuting time-dependent control has corrections beyond the exponential of its integrated Quantum Hamiltonian. This formula is a local convergent expansion under suitable smallness conditions, or a formal expansion; it must not be assumed globally convergent for arbitrary pulse durations.

For $H(t)={\Omega }_1(t)A+{\Omega }_2(t)B$ with fixed noncommuting Hermitian operators, $[H(t),H(s)]=[{\Omega }_1(t){\Omega }_2(s)-{\Omega }_2(t){\Omega }_1(s)][A,B]$. A constant ratio of the envelopes is sufficient to remove time ordering. Independently changing that ratio generally produces commutator corrections in the Magnus expansion, and a product of successive pulses need not equal the exponential of their summed areas.

A time-ordered product sorts operator insertions from latest time on the left to earliest time on the right. Reordering odd fermionic operators contributes the sign of their permutation. For bosonic $A,B$, $T[A(t)B(t^{\prime })]=\theta (t-t^{\prime })A(t)B(t^{\prime })+\theta (t^{\prime }-t)B(t^{\prime })A(t)$. Vacuum expectations of free time-ordered products are evaluated by the Wick theorem.

Normal ordering places creation operators to the left of annihilation operators. For free fields it removes vacuum self-contractions from a composite operator.

For a massive real scalar field in two spacetime dimensions with a polynomial potential and no derivative interactions, every connected subgraph has superficial degree of divergence $\omega =2(1-V)$, where $V$ counts interaction vertices. The only divergent loop subgraphs with external legs have one vertex and consist of self-contractions. Normal ordering removes them. With $T_{\Lambda }={\int }_{|k|<\Lambda }{d}^{2}k/[(2\pi {)}^2(k^2+m^2)]$, the ordinary-field potential representing the normal-ordered interaction is $e^{-T_{\Lambda }{\partial }_{\varphi }^2/2}V(\varphi )$. The exponential terminates for a polynomial. After these subtractions all remaining loop subgraphs have negative superficial degree of divergence, so no additional divergent field-dependent counterterms occur. Constant vacuum energy subtractions and finite on-shell renormalization scheme conditions are separate.

This correlation counts distinct detection events without the delta-function self-counting term in an un-ordered product of densities. It is nonnegative as the expectation of $A^{†}A$ with $A={\psi }_n\cdots {\psi }_1$. For a single occupied bosonic mode with sharp occupation $N$, the coincident $n$-point value contains the falling factorial $N(N-1)\cdots (N-n+1)$, rather than $N^n$.

A single-component chaotic Gaussian bosonic field has normalized coincident normally ordered correlation $n!$: every pairing contributes the same product of densities. A same-component fermionic field instead has zero coincident correlations for $n\ge 2$ by exclusion. These are state-dependent statements: a coherent bosonic field has factor one, and a single-mode number state has $N^{\underline{n}}/N^n$. Ratios to distant unnormalized correlations additionally require the compared densities to agree and distant correlations to factorize.

For a number-conserving Gaussian pairing state, six permutations give this expression. Each transposition has sign $ϵ$, positive for bosons and negative for fermions; the two three-cycles have positive sign for either statistics. Equivalently it is the permanent or determinant of the three-by-three one-body density matrix. For bosonic number states, repeated-mode factorial corrections must be included instead of assuming that Wick theorem.

For a product of bosonic number states, direct and exchange contractions correctly count distinct modes. They overcount repeated-mode terms because $⟨a^{†2}{a}^2⟩=N(N-1)$, while Wick theorem would give $2N^2$. Subtracting the excess yields the displayed exact two-point correction. At higher order the occupation decrements carry falling factorials. This distinction is essential when comparing a fixed-number condensate with independent thermal modes.

For the quantum harmonic oscillator $H=\hslash \omega (a^{†}a+1/2)$, its normal symbol is $H_N=\hslash \omega (\bar{\psi }\psi +1/2)$. The Weyl ordering symbol of $a^{†}a$ is $\bar{\psi }\psi -1/2$, so $H_W=\hslash \omega \bar{\psi }\psi =p^2/(2m)+m{\omega }^2{q}^2/2$. Switching from adjacent-label coherent-state time slicing to a midpoint phase-space path integral must retain this ordering correction. Both correctly normalized constructions give the thermal partition function of a quantum harmonic oscillator $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}$; retaining the normal-symbol constant after switching prescriptions would count the zero-point shift twice.

A normal-ordered product of free-field oscillator operators places all creation operators to the left of all annihilation operators, including the fermionic sign required to reorder fermions. Its vacuum expectation vanishes if any nontrivial oscillator factors remain. The Wick theorem expresses a free-field time-ordered product through these products and all Wick contractions.

A Wick contraction replaces a selected pair of free fields by their two-point function. Summing pairings computes Gaussian correlation functions and supplies the propagator lines and combinatorial factors of a Feynman diagram.

For two identical even-degree $n$ scalar Feynman vertices, choose $r$ half-edges on each and pair them across the vertices. The remaining $n-r$ half-edges pair locally. This gives the displayed number of Wick contractions, with even $r$ and $(-1)!!=1$. The coefficient in the unnormalized partition function is $N_{n,r}/[2!(n!{)}^2]$. The $r=0$ contribution is disconnected and is removed from the logarithm, but remains in the partition function itself. This rule produces the quartic and sextic oscillator vacuum graph coefficients without drawing-dependent symmetry-factor guesses.

At second order in the coupling constant of a real phi cubed theory, the two-point numerator has eight fields, hence $7!!=105$ Wick contractions. Of these, six give a free external line times a vacuum theta graph, nine give a free line times a vacuum dumbbell, 36 give a two-line bubble, 36 give a connected tadpole attachment, and 18 give two separate one-point tadpoles. Dividing by $2!(3!{)}^2=72$ gives weights $1/12,1/8,1/2,1/2,1/4$. The normalized vacuum generating functional cancels the first two contributions, while the last remains in the full two-point correlation function as the product of one-point functions. Thus cancellation of vacuum bubbles does not, by itself, take the connected part.

A fermionic sign is a factor $-1$ produced whenever two Grassmann-odd fields or fermionic creation and annihilation operators are interchanged. Relative signs between diagrams must be computed after putting every external fermion operator into one common order.

In a Dirac field mode expansion, the antiparticle contribution to $\bar{\psi }\Gamma \psi$ first appears with operator order $d{d}^{†}$. Normal ordering changes it to $-d^{†}d$ using the fermionic anticommutation relations. Thus its scattering coefficient is $-{\bar{v}}_{\mathrm{in}}\Gamma v_{\mathrm{out}}$. Pair-annihilation and creation terms have the operator orders $db$ and $b^{†}{d}^{†}$. Keeping these orders and a fixed external-state convention determines interference signs, especially between exchange and annihilation graphs in a Yukawa interaction.

A Wick rotation deforms a time or energy contour from real to imaginary values, converting suitable Minkowski integrals into Euclidean integrals while respecting pole prescriptions.

Analytic continuation from real time to $t=-i\tau$ turns unitary evolution into $e^{-\tau H/\hslash }$. A thermal trace has extent $\beta \hslash$, or $\beta$ when $\hslash =1$. This converts oscillatory action phases into an Euclidean path integral where convergence permits it; it is not the actual laboratory time of a dissipative process.

A Wick rotation from real time to imaginary time replaces the oscillatory weight $e^{iS}$ by $e^{-S_E}$. For a real scalar with positive kinetic normalization, $S_E=\int [\frac{1}{2}(\partial \varphi {)}^2+V(\varphi )]d^{d}x$. Positivity of its kinetic term does not ensure convergence if the potential is unbounded below. Fermionic integrals instead use independent Grassmann fields and need not define a positive measure.

A generating functional is a path integral with a source $J$ coupled to a field. Its functional derivatives with respect to $J$ generate time-ordered correlation functions.

With source term $J\varphi +\bar{\eta }\psi +\bar{\psi }\eta$ and left functional derivatives, a free functional written as $Z_0\propto \exp (iJ\Delta J/2+i\bar{\eta }S\eta )$ has physical time-ordered contractions $D=-i\Delta$ and $F=-iS$. The kernels in this exponent must not be identified with contractions already carrying factors of $i$. Four contractions in a second-order Yukawa scalar two-point graph introduce $(-i{)}^4=1$, leaving its kernel formula's overall coefficient unchanged.

The zero-source generating functional is the vacuum state persistence amplitude, conventionally written $Z[0]=\int \mathcal{D}\varphi e^{iS[\varphi ]}$ with a suitable Feynman i-epsilon prescription. Dividing by $Z[0]$ removes disconnected Vacuum Feynman diagrams from normalized Green functions. Adding a constant vacuum energy density to the Lagrangian density changes $Z[0]$ by a spacetime-volume phase but leaves these normalized Green functions unchanged.

In a finite-dimensional integral, $u_i{e}^{b\cdot u}={\partial }_{b_i}{e}^{b\cdot u}$. Expand $e^{V(u)}$ as a formal perturbation series to obtain $\int du{e}^{-u^{T}Au/2+b\cdot u+V(u)}=e^{V({\partial }_b)}{I}_0(b)$. With a finite-mode regulator, a Minkowski source insertion satisfies $\varphi (x)e^{i\int J\varphi }=(1/i)\delta e^{i\int J\varphi }/\delta J(x)$; this gives the displayed interacting generating functional. The substitution is a perturbative differential-operator identity, not an assertion that the interaction exponential defines a convergent series. Divide the resulting functional by $Z[0]$ for normalized correlation functions.

An external source is a prescribed field coupled to a dynamical field or operator in the action, for example $S_J=S_0+\int J_i(x){\varphi }_i(x)d^{d}x$. Functional derivatives of the generating functional with respect to $J$ insert the corresponding fields into correlation functions. The source is varied to generate these insertions, rather than integrated over as a dynamical degree of freedom. Transforming it in the appropriate representation maintains an invariant source coupling when deriving a Ward identity.

A functional derivative of the source term in a Minkowski scalar field path integral inserts $i\varphi (x)$. Time-slicing identifies the resulting product with the time-ordered product of field operators. Dividing by the zero-source vacuum functional removes disconnected vacuum diagrams. The normalized formula has one factor $1/i$ for every insertion.

For a vacuum generating functional $Z[J]$, normalization gives $\hat{Z}[J]=Z[J]/Z[0]$ and $\hat{Z}[0]=1$. Functional derivatives of $\hat{Z}$ give normalized vacuum expectation values. The linked-cluster theorem shows that this ratio cancels Vacuum Feynman diagrams, while $-i\log \hat{Z}[J]$ gives connected correlators. Components attached to external insertions are not removed merely by the normalization.

A translation-invariant connected generating functional with a constant external source is extensive in spacetime volume ${\mathcal{V}}_d$. Its total logarithm is dimensionless, while its density has mass dimension $d$. Consequently its dimensionless representation at fixed physical volume must retain ${\mu }^d{\mathcal{V}}_d$. For a free massive scalar with source coupling ${\mu }^{-ϵ/2}h\varphi$, completing the square gives $F={\mathcal{V}}_d{\mu }^{-ϵ}{h}^2/(2m^2)$. This already disproves a literal rewriting of the full uniform-source logarithm using only $\lambda ,m^2/{\mu }^2,h/{\mu }^3$ at fixed volume. In infinite volume the total functional is generally infinite; use the density, a source of compact support, or an explicit volume regulator. After writing the density as ${\mu }^{d}f$, its renormalization group equation has the engineering term $+df$. Ultraviolet subtraction alone does not establish infrared finiteness for a massless constant source.

Completing the square in $S[q]=\frac{1}{2}qKq+Jq$ with $K=-{\partial }_t^2-m^2+iϵ$ gives the displayed source factor, where $G=K^{-1}$. In the frequency convention $e^{i\omega t}$, $\tilde{G}=({\omega }^2-m^2+iϵ{)}^{-1}$ and $G(t)=-i{e}^{-im|t|}/(2m)$ for $m>0$. The physical time-ordered oscillator quantum field theory propagator is $iG$. Sources are first taken as test functions so the pairings are defined.

The logarithm of a generating functional generates connected correlation functions because its cumulant expansion removes products of disconnected vacuum components.

For a normalized Minkowski source functional $Z=e^{iW}$ with vanishing one-point function, $C_n=i^{1-n}{\delta }^{n}W/\delta J^n$ andDifferentiating $e^{iW}$ assigns the four labels to connected blocks. Blocks of size one vanish, leaving one block of four or one of the three pair partitions. Thus $G_4=i{W}_4-\sum W_2{W}_2$ in this source convention.

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For $d=4-ϵ$, write the bare field as ${\varphi }_0=Z^{1/2}\varphi$ and the renormalized source coupling as ${\mu }^{-ϵ/2}h\varphi$. In bare variables this is $h_0{\varphi }_0$, with the displayed source. At fixed bare parameters, the source running is $\mu d\log h/d\mu =ϵ/2+\frac{1}{2}\hat{\beta }{\partial }_{\lambda }\log Z$. Normalizing the generating functional by its zero-source value cancels source-independent vacuum diagrams and the constant measure factor from field rescaling. It does not remove source-dependent ultraviolet divergences or infrared divergences; the former require the ordinary field, mass and interaction counterterms.

A Feynman diagram records a term in the perturbative expansion: edges represent propagators, vertices represent interactions, and loops represent unconstrained momentum integrals.

A schematic tracking of valence quark and antiquark flavor through a hadronic process. It can explain flavor conservation and required pair creation without computing the nonperturbative hadronization amplitude. A complete elementary perturbative diagram contains additional interaction and color information.

A vertex written in terms of composite hadron fields, with a coupling or form factor that summarizes unresolved Quantum chromodynamics. For a strong Delta baryon decay, a $\Delta p\pi$ vertex describes the external channel but does not assert that the decay is perturbative at constituent-quark scales.

Feynman rules translate a Feynman diagram into a contribution to a scattering amplitude or correlation function in perturbation theory. They specify factors for external states, interaction vertices, internal propagators, momentum-conservation constraints and loop integrals, with statistics-dependent signs and symmetry factors. For quantum electrodynamics, a photon-fermion vertex contains the coupling $e$, while a photon propagator supplies an inverse squared four-momentum. The complete amplitude also contains currents, so propagator powers alone do not determine the energy dependence of a scattering cross-section.

A connected loop Feynman diagram has at least one independent internal loop momentum. With $I$ internal lines and $V$ vertices, the number of independent loops is $L=I-V+1$. A tree-level Feynman diagram has $L=0$. A multi-leg amplitude generated by a loop is not an extra fundamental interaction vertex of the original Lagrangian.

A Feynman diagram with one independent closed cycle and hence one independent loop momentum. An $n$-vertex polygon of Dirac propagators in four dimensions has superficial degree of divergence $4-n$ before symmetry cancellations. For a fermion loop, the closed fermion loop sign is separate from any symmetry factor or pairwise charge conjugation cancellation.

A Feynman integral is the momentum-space integral associated with a Feynman diagram: each independent loop momentum is integrated with measure $d^{d}k/(2\pi {)}^d$, and propagators, vertex numerators and momentum conservation determine the integrand. Its definition also requires a pole prescription and, when divergent, a regularization in quantum field theory. Feynman parameters combine propagator denominators and permit shifts of loop momenta. A massless two-propagator example is $\int d^{d}k/[(2\pi {)}^d{k}^2(p-k{)}^2]$, with its causal prescriptions understood. Subtracting local counterterms turns a regulated diagram into a finite contribution to a renormalized amplitude.

The two-point function Feynman integral with two interaction vertices joined by five parallel scalar propagators has four independent loop momenta:This formula uses Euclidean momenta. In three dimensions the convolution theorem and $G(x)=e^{-m|x|}/(4\pi |x|)$ givewith cutoff regularization required at $\rho =0$. With a lower radial cutoff $a$, the coefficient of $p^2$ has ultraviolet divergence $-\log (1/(ma))/(1536{\pi }^4)$. It therefore generates a divergent wave-function renormalization in three-dimensional sextic scalar field theory. By contrast the three-line sunset diagram has only a momentum-independent logarithmic ultraviolet divergence in three dimensions. This illustrates why a superficial degree of divergence of two allows both a mass counterterm and a kinetic counterterm, while degree zero allows only the former in a two-point function.

The arrow on a charged-fermion line in a Feynman diagram tracks fermion-number flow and fixes the order of the gamma matrices between external Dirac spinors. For an external particle it agrees with the particle direction; for an external antiparticle it is opposite to the antiparticle direction. It therefore need not point along the assigned four-momentum of every external leg.

A box diagram has a one-loop cycle of four propagator edges and four vertices connecting external legs. Its internal particle types depend on the theory. In a charged weak box contribution to kaon mixing, two charged weak-boson propagators connect two up-type quark lines, with flavor-changing charged-current vertices.