Quantized field

For a one-particle Hilbert space $H$, the bosonic or fermionic Fock space is ${\mathcal{F}}_{\pm }(H)={⨁}_{n=0}^{\infty }{H}^{{\otimes }_{\pm }n}$, using symmetric or antisymmetric tensor powers. Its zeroth summand is the one-dimensional vacuum sector. Creation operators and annihilation operators change particle number. The bosonic Fock space and fermionic Fock space implement the corresponding exchange statistics.

A one-particle state belongs to the one-excitation sector of a Fock space. In a momentum basis with $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, the wavepacket $\int d^{3}pf(\mathbf{p})a_{\mathbf{p}}^{†}|0⟩/(2\pi {)}^3$ has norm squared $\int d^{3}p|f(\mathbf{p}){|}^2/(2\pi {)}^3$. Sharp-momentum states have distributional rather than unit normalization; factors of $\sqrt{2E_{\mathbf{p}}}$ convert to relativistic normalization of a one-particle state.

The eigenvalue of a mode number operator counts excitations in that mode. Bosonic occupation numbers range over all nonnegative integers; fermionic occupation numbers are zero or one.

For $\{f,f^{†}{\}}_{+}=1$ and $f^2=0$, the occupation number operator $f^{†}f$ has eigenvalues zero and one. This gives the single-mode exclusion principle.

For $[a,a^{†}]=1$, the occupation number operator $a^{†}a$ has eigenvalues $0,1,2,\dots$ on the bosonic oscillator Fock space.

An operator-valued distribution assigns a linear operator on a common dense domain to each smooth test function of compact support $f$, continuously and linearly in the appropriate weak sense. A quantized field is generally of this type: $\varphi (x)$ is formal point notation, while the smeared $\varphi (f)$ has operator meaning. Equal-time canonical commutation relations and momentum oscillators are interpreted after smearing, which avoids treating a Dirac delta distribution or a coincident field as a finite ordinary operator. Products at coincident points can need further regularization even when individual smeared fields are defined.

A free field is expanded in positive- and negative-frequency plane waves multiplied by annihilation and creation operators.

For a free field in the annihilation-operator vacuum, the coincident two-point function is its mode-power integral. Unsmeared vacuum products generally have an ultraviolet divergence. A finite band or spatial smearing defines a regulated variance without altering the meaning of a mode-by-mode dimensionless power spectrum.

For a free field, its positive-frequency part consists of the annihilation terms multiplying $e^{-i\omega t}$ with $\omega >0$; its negative-frequency part consists of the creation terms. The sign refers to positive energy, even though the exponential has a minus sign. Imposing the Lorenz gauge only through $(\partial \cdot A{)}^{(+)}|\mathrm{phys}⟩=0$ is the subsidiary condition in Gupta-Bleuler quantization, not an operator identity for the whole divergence.

The canonical commutation relation fixes the antisymmetric product of a mode function $f$ and its canonical-momentum mode $g$. In the convention $[a,a^{†}]=1$, one commonly has $f{g}^{\ast }-f^{\ast }g=i$.

A Fock state has specified occupation numbers. Applying creation operators to the vacuum constructs multiparticle momentum eigenstates.

If creation operators obey $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, set $|\mathbf{p}⟩=\sqrt{2E_{\mathbf{p}}}{a}_{\mathbf{p}}^{†}|0⟩$. Their inner product then has the displayed value. The associated completeness measure is $d^{3}p/((2\pi {)}^{3}2E_{\mathbf{p}})$, the invariant positive-energy mass shell measure. Specifying this normalization prevents hidden external energy factors in a scattering amplitude.

The oscillator vacuum is annihilated by all annihilation operators. Acting with creation operators constructs Fock states. In single-string quantization this is the unexcited string, not the empty spacetime state.

Intrinsic parity is the phase by which a field transforms under spatial inversion after its spacetime argument is reflected. For a Hermitian scalar field it is $+1$ or $-1$.

In the Dirac representation of the gamma matrices, the parity symmetry in quantum field theory action ${\psi }_P(x)={\gamma }^0\psi (x_P)$ and the charge conjugation action ${\psi }_C=i{\gamma }^2{\psi }^{\ast }$ obey $({\psi }_P{)}_C(x)=-{\gamma }^0{\psi }_C(x_P)$, since ${\gamma }^0$ is real and anticommutes with ${\gamma }^2$. At rest, positive-frequency Dirac spinors have nonzero upper components and negative-frequency Dirac spinors have nonzero lower components; they are respectively $+1$ and $-1$ eigenvectors of ${\gamma }^0$. With the common conventional real intrinsic-parity phase, particle and antiparticle therefore have opposite intrinsic parity. The overall sign is conventional, whereas their relative sign is fixed.

Choose canonical spin states by applying the rotationless Lorentz boost $L(p)$ to rest-frame spin states. Spatial inversion $\Pi =\mathrm{diag}(1,-1,-1,-1)$ commutes with rest-frame rotations, so Schur lemma makes its action on one irreducible rest spin multiplet a phase ${\eta }_P$. Since $\Pi L(p)\Pi =L(\Pi p)$, the formula follows. Canonical spin labels are unchanged; helicity reverses. With $P^2=I$, the intrinsic parity is ${\eta }_P=\pm 1$.

// Target: lie-theory.bigb

A ghost field in the general dynamical sense has a kinetic term with the wrong sign and makes the Hamiltonian unbounded below or produces negative-norm states. This use is distinct from a Faddeev-Popov ghost field, which is an auxiliary Grassmann field for gauge fixing.

A vector field can be decomposed into a divergence-free part and a gradient, $V_{\mu }=A_{\mu }+{\partial }_{\mu }\pi$ with ${\partial }_{\mu }{A}^{\mu }=0$, subject to boundary conditions and zero-mode qualifications.

A constraint equation follows by varying a field component that enters an action without its own time derivative. It restricts initial data or determines a nondynamical field instead of supplying an independent propagating degree of freedom.

A Gaussian functional integral has an exponent quadratic in a bosonic field. Completing the square expresses its source dependence through the inverse quadratic operator, the propagator.

For $S[\varphi ]=\frac{1}{2}\int (\partial \varphi {)}^2$ and a source appearing as $-\int J\varphi$ in the exponent, the positive quadratic operator is $A=-\Delta$. Completing the square gives $Z[J]=\mathcal{N}\exp [\frac{1}{2}J{A}^{-1}J]$. If one writes it with a negative quadratic source exponent, its kernel must be $G=-A^{-1}={\Delta }^{-1}$. In the plane $G(z,w)=(2\pi {)}^{-1}\log (|z-w|/L)$, up to harmonic additions and infrared conventions. The physical covariance is its negative. Zero modes are regulated or separated; oscillatory sources in a string path integral produce momentum conservation from the constant mode.

For a positive differentiable covariance $C_t$ and a sufficiently regular, integrable $W$, differentiating the normalized Gaussian integral and integrating by parts twice gives the displayed identity. An unnormalized integral has the additional field-independent term $\mathrm{Tr}(C_t^{-1}{\dot{C}}_t)/2$ times the integral. A finite-mode regulator makes the analogous Gaussian functional integral identity precise before taking a continuum limit.

Completing the square in the free scalar path integral gives a quadratic exponential in the source. If ${\Delta }_F$ includes the factor $i$ in the momentum-space Feynman propagator, the normalized Minkowski functional is $\exp [-\int J{\Delta }_{F}J/2]$. Two source derivatives with their $1/i^2$ factor recover ${\Delta }_F$, while higher derivatives give Wick contractions.

A Gaussian path integral over a bosonic field with quadratic operator $A$ produces a factor proportional to $(\det A{)}^{-1/2}$. When $A$ depends on background fields, this functional determinant contributes to their quantum effective action.

For a change of field variables $\varphi \mapsto F[\varphi ]$, the functional Jacobian is the functional determinant of the linearized change. At a finite regulator it is an ordinary Jacobian determinant on the finitely many field coordinates; the continuum expression denotes the regulated limit. It changes the functional integral measure and can depend on the background field. The Faddeev-Popov determinant is the example obtained by replacing the integration coordinate along a gauge orbit by a chosen gauge fixing condition.

A Grassmann Gaussian integral satisfies $\int D\bar{\psi }D\psi e^{-\bar{\psi }A\psi }=\det A$. Adding Grassmann sources and completing the square produces an exponent containing $A^{-1}$.

For an antisymmetric matrix $A$, a Berezin integral extracts the top coefficient of the exponential. Expansion into pairings gives the Pfaffian, with a constant sign depending on the order of the differentials. This is a polynomial in the entries of $A$, not the positive square root of its determinant: in dimension two it is proportional to $A_{12}$ and changes sign with that entry. The identity $\mathrm{Pf}(A{)}^2=\det A$ determines its square but does not discard its orientation.

For an even-dimensional antisymmetric matrix $M$, put $\theta =(u+iv)/\sqrt{2}$ and $\bar{\theta }=(u-iv)/\sqrt{2}$, with independent Grassmann variables $u,v$. Anticommutation cancels the mixed terms, giving ${\bar{\theta }}^{T}M\theta =(u^{T}Mu+v^{T}Mv)/2$. The constant Grassmann change-of-variables formula changes only the normalization. The two real Pfaffians multiply to $\mathrm{Pf}(M{)}^2=\det M$, agreeing with the complex Grassmann Gaussian integral. In odd dimension $\det M=0$ and both top-degree integrals vanish.

Use the Berezin integral orientation $\int {\prod }_{i}d{\bar{\theta }}_{i}d{\theta }_i{\theta }_1{\bar{\theta }}_1\cdots {\theta }_n{\bar{\theta }}_n=1$. For an invertible matrix $B$, the normalized weight $e^{-\bar{\theta }B\theta }$ has $⟨{\theta }_i{\bar{\theta }}_j⟩=(B^{-1}{)}_{ij}$. Differentiate its determinant with respect to $B_{ji}$ to obtain this formula; two derivatives give the displayed four-point formula. The second pairing carries a fermionic sign. These are ordered contractions, so exchanging two odd factors changes their sign.

For $C=B^{-1}$, ordered Grassmann Gaussian contractions give $⟨\bar{\theta }M\theta ⟩=-\mathrm{tr}(MC)$ and $⟨(\bar{\theta }M\theta )(\bar{\theta }N\theta )⟩=\mathrm{tr}(MC)\mathrm{tr}(NC)-\mathrm{tr}(MCNC)$. The disconnected contribution has two fermion loops; the connected contribution has one, giving its minus sign. This covariance is an algebraic normalized Berezin integral, rather than a positivity statement about a probability measure.

Integrating finite Grassmann variables out of a quadratic action gives $\det [B-gC(x)]$. If $B$ is invertible, divide by $\det B$ and expand its logarithm to obtain $V_{\mathrm{eff}}={\sum }_{m\ge 1}{g}^m\mathrm{Tr}[B^{-1}C(x){]}^m/m$. Since the bosonic variables commute, an $m$-leg effective Feynman rule uses the fully symmetrized trace tensor, with weight $-g^m(m-1)!\mathrm{Sym}\mathrm{Tr}[B^{-1}{C}^{(k_1)}\cdots B^{-1}{C}^{(k_m)}]$. The minus sign comes from $e^{-V_{\mathrm{eff}}}$. This organizes closed fermion loops into bosonic vertices. It is a formal perturbative identity; a singular $B$ prevents the normalized expansion, even though the unfactored determinant polynomial still exists.

For a free Dirac field with action $\int \bar{\psi }K\psi$ and independent odd sources coupled as $\bar{\psi }J+\bar{J}\psi$, translation invariance of the Berezin integral and completion of the square give $Z[J,\bar{J}]=Z[0,0]\exp (-i\bar{J}{K}_F^{-1}J)$. Here $K_F^{-1}$ has Feynman i-epsilon prescription boundary conditions, and each bilinear includes its spacetime integrations. A left Grassmann derivative with respect to $\bar{J}$, followed by a right Grassmann derivative with respect to $J$, has normalized value $-i{K}_F^{-1}$ at zero sources. Multiplying by $-1$ removes the two insertion factors $i^2$ and yields the Dirac propagator $S_F=i{K}_F^{-1}$. Keeping source order fixed prevents a spurious fermionic sign.

A Grassmann field takes values in a Grassmann algebra, so its components anticommute. A closed loop of Grassmann propagators acquires a minus sign in perturbation theory.

A propagator is a two-point kernel describing propagation and, in a free theory, a Green function for the quadratic kinetic operator. Its boundary prescription determines its type: the time-ordered Feynman propagator uses an $iϵ$ prescription, while retarded and advanced propagators impose causal boundary conditions.

The Feynman propagator is the time-ordered Green function selected by the $iϵ$ prescription.

With ${\Delta }_F=⟨0|T\{\varphi (x)\varphi (y)\}|0⟩$, for a mode of positive energy $E>0$ the Feynman i-epsilon prescription puts poles $+E-i0$ and $-E+i0$ in opposite half-planes. The energy contour integral closes clockwise below for positive time separation and counterclockwise above for negative separation. The residue theorem gives $e^{-iE|t|}/(2E)$ in either case. Thus the four-dimensional Fourier integral has numerator $i$ and denominator $p^2-m^2+i0$. The integral is understood as a distribution limit, not an ordinary absolutely convergent integral; changing the definition to $i{\Delta }_F$ would change the displayed normalization.

At fixed spatial momentum with mode energy $E>0$, $g_E(t)=e^{-iE|t|}/(2E)$ is continuous, with derivatives $-i/2$ and $+i/2$ at zero from opposite sides. Hence its second distributional derivative has a $-i\delta (t)$ term, while away from zero $g_E^{\prime \prime }+E^2{g}_E=0$. Therefore $({\partial }_t^2+E^2)g_E=-i\delta (t)$ and, after the spatial Fourier transform, $(\square +m^2){\Delta }_F=-i{\delta }^4$. This checks the propagator numerator, contour signs and equal-time canonical commutation relations independently of residue calculation.

The Feynman $iϵ$ prescription displaces poles or tilts an integration contour by an infinitesimal imaginary part. It selects vacuum boundary conditions and damps oscillatory integrals at infinite time.

A Wightman function is an unordered vacuum correlation function. For a free scalar mode $f_k(t)$, the positive-frequency two-point function is $G^{>}(t_1,t_2;k)=f_k(t_1)f_k^{\ast }(t_2)$.

The free Dirac propagator is $S_F(p)=i(/p+m)/(p^2-m^2+iϵ)$ and inverts the Dirac kinetic operator with Feynman boundary conditions.

For action $-\int \bar{\psi }(\gamma \cdot \partial +m)\psi$, choose gamma matrices with $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$ and signature $(-,+,\dots ,+)$. The Fourier kernel of $D=\gamma \cdot \partial +m$ is $i\gamma \cdot p+m$. Its algebraic inverse is $(-i\gamma \cdot p+m)/(p^2+m^2)$. The Grassmann Gaussian integral for $e^{-i\bar{\psi }D\psi }$ has covariance $-i{D}_F^{-1}$, giving the displayed Dirac propagator. Its denominator has positive-energy poles below and negative-energy poles above the real energy axis by the Feynman i-epsilon prescription. Both the metric and the factor of $i$ in the derivative kernel must be specified when comparing propagator formulas.

In a covariant gauge, the photon propagator is the inverse of the gauge-fixed Maxwell kinetic operator.

At non-null four-momentum, these complementary projectors satisfy $P_L^2=P_L$, $P_T^2=P_T$ and $P_L{P}_T=0$. The gauge-fixed Maxwell kinetic operator is $-k^2(P_T+{\xi }^{-1}{P}_L)$; its inverse gives propagator $-i(P_T+\xi P_L)/k^2$. Transversality here removes one four-dimensional longitudinal direction, leaving rank three; it does not by itself project onto the two physical photon polarizations. The pole requires the Feynman i-epsilon prescription.

Define the inclusive photon-field tensor for hadronic final states by $W_A^{\mu \nu }=2\pi {\rho }_h(s)(-g^{\mu \nu }+q^{\mu }{q}^{\nu }/s)$ for timelike $q$ with $s=q^2$ and $q^0>0$. Here each final state has relativistic normalization and the sum includes the four-momentum delta function. This hadronic photon spectral density has mass dimension $-2$. At leading electromagnetic order, ${\rho }_h=e^2{\rho }_J/s$, with ${\rho }_J$ the dimensionless hadronic electromagnetic-current spectral density. Stating the tensor convention prevents mismatched powers of $s$ or $e$ in an inclusive cross-section.

For $q=(q^0,\mathbf{q})$ with $\mathbf{q}\ne 0$, the Coulomb gauge propagator hasThe temporal component is instantaneous; the spatial part projects onto transverse photon polarizations. Although these components are not individually Lorentz covariant, their contraction with conserved external currents equals the covariant answer. Zero-spatial-momentum limits should be taken after combining the current contractions.

For $q\cdot J=q\cdot K=0$, the instantaneous term in the Coulomb-gauge photon propagator combines with its spatial longitudinal subtraction:The identity follows from $\mathbf{q}\cdot \mathbf{J}=q^0{J}^0$ and $q^2=(q^0{)}^2-|\mathbf{q}{|}^2$. It is equality after contraction with conserved currents, not equality of the uncontracted propagator tensors. It is especially useful in showing that tree scattering amplitudes computed in Coulomb gauge are Lorentz invariant.

Time-ordering the canonical transverse photon field gives a propagator with numerator equal to the spatial transverse projector and denominator $k^2+i0$. Its two energy poles encode forward propagation of the positive-frequency term and backward propagation of the negative-frequency term. This is the reduced free-field propagator; it is distinct from the covariant four-potential propagator.