Relativistic Quantum Mechanics Symbol Index

A familiar symbol can name a matrix, a scalar, a differential operator, or a distribution. This index identifies the meaning from the calculation and links the defining page. The local definition takes precedence: this is a map of usage, not a demand that every source use the same letters. The Common Symbols Index covers notation across quantum mechanics.

Helpful background. Metric and Units explains the constants and index conventions used in the entries below.

Spacetime, units and motion

Unless a page explicitly chooses natural units, distinguish $p^0=E/c$ from $E$ and $x^0=ct$ from $t$.

Symbol	Meaning and collision to check	Defining page
$\eta_{\mu\nu}$	Mostly-minus Minkowski metric; raising or lowering a spatial index changes a sign	Metric and Units
$x^\mu,\ \partial_\mu$	Contravariant coordinates and covariant coordinate derivatives; $\partial_0=c^{-1}\partial_t$ with constants explicit	Spacetime Notation
$p^\mu,\ E_{\mathbf p}$	On-shell momentum label and positive energy; canonical and kinetic quantities differ in a background	Energy–Momentum Relation
$\Lambda,\ \xi$	Lorentz matrix and rapidity; the active/passive convention fixes the boost sign	Lorentz Transformations
$\gamma,\ \beta$	Scalars $\gamma=(1-\beta^2)^{-1/2}$ and $\beta=v/c$ here; not Dirac matrices	Lorentz Transformations
$\epsilon^{\mu\nu\rho\sigma}$	Oriented antisymmetric symbol/tensor in the declared Cartesian convention; upper and lower components must be distinguished	Metric and Units
$\bar\lambda_C$	Reduced Compton length $\hbar/(mc)$; differs from $h/(mc)$ by $2\pi$	Natural Units
$P^\mu,\ \mathbf J,\ \mathbf K$	Translation, rotation and boost generators; $\mathbf K$ has units of action in this convention	Poincaré Group

In natural units a mass parameter has energy units, while an inverse energy can represent a length or a time after restoring the appropriate $\hbar,c$. The conversion worksheet keeps these operations distinct.

Potentials, currents and wave amplitudes

Symbol	Meaning and collision to check	Defining page
$q,\ e$	Signed particle charge and positive elementary-charge magnitude; $q=-e$ for an electron	Minimal Coupling
$A^\mu,\ \Phi,\ \mathbf A$	Four-potential, electric scalar potential and vector potential; $\Phi$ is not a phase-space measure	Minimal Coupling
$\mathbf E,\ \mathbf B$	Electric and magnetic fields; bold $\mathbf E$ is not the energy eigenvalue $E$	Dirac in EM Fields
$V,\ \boldsymbol\pi$	Potential energy $q\Phi$ and kinetic momentum $\mathbf p-q\mathbf A$ in SI conventions	Gauge Covariance
$D_\mu$	Gauge-covariant derivative; do not identify it with the first-order Dirac operator $D$ in a kernel calculation	Minimal Coupling
$\phi,\ \psi$	Scalar or spinor amplitude in wave mechanics; in a field expansion the same letter can denote an operator	Why Fields Replace Wavefunctions
$j^\mu,\ \rho$	Current and its density; specify probability, normalized KG charge, or physical charge current before comparing factors of $q,c$	Relativistic Currents
$\alpha$	Fine-structure constant in coupling formulas; its expression in $e$ depends on EM rationalization	Natural Units

The same canonical energy can shift under a time-dependent gauge change while the kinetic energy is unchanged. A bare energy sign therefore does not identify a physical particle sector in an arbitrary gauge.

Spinor matrices, labels and conjugations

Symbol	Meaning and collision to check	Defining page
$\gamma^\mu,\ \gamma^5$	Clifford matrices and the chirality matrix; not the scalar Lorentz factor	Gamma-Matrix Conventions
$\alpha^i,\ \beta$	Dirac Hamiltonian matrices $\gamma^0\gamma^i,\gamma^0$; not fine-structure coupling or speed	Gamma-Matrix Conventions
$\sigma^i,\ \sigma^{\mu\nu}$	Pauli matrices versus $i[\gamma^\mu,\gamma^\nu]/2$; neither is a cross section	Gamma-Matrix Conventions
$\bar\psi,\ p\!\!\!/$	Dirac adjoint $\psi^\dagger\gamma^0$ and slash $\gamma^\mu p_\mu$	Gamma-Matrix Conventions
$u_s(p),\ v_s(p)$	Positive- and negative-frequency columns with future-directed label $p$; the $v$ wave has canonical momentum $-p$	Spinor Conventions
$s,\ h$	A spin label versus a helicity eigenvalue; a massive particle’s helicity need not be boost invariant	Helicity and Chirality
$P_L,\ P_R,\ P_\pm$	Chirality projectors versus free Hamiltonian energy projectors; these are different decompositions	Spinor Conventions
$S(\Lambda)$	Finite-dimensional Lorentz matrix on spinor components; not the unitary one-particle Hilbert representation or scattering matrix	Dirac Spinors
$B,\ C_D$	Matrices in $\psi^c=B\psi^*=C_D\bar\psi^{\mathsf T}$; complex conjugation is part of the operation	Symmetry Conventions
$U_T,\ \Theta$	Matrix part and full antiunitary time-reversal operation; matrix multiplication alone omits conjugation	Time Reversal
$S,\ P_5,\ V^\mu,\ A_5^\mu,\ T^{\mu\nu}$	Bilinears; here $S$ is a scalar and $A_5$ is an axial current, not an EM potential	Dirac Bilinear Table

A gamma-basis change, metric-signature change and redefinition of a spinor’s normalization are three different translations. Apply the matching dictionary before comparing component formulas.

Kernels, states and scattering

Symbol	Meaning and collision to check	Defining page
$L,\ D,\ D_+$	Free operators $\Box+m^2$, $i\gamma^\mu\partial_\mu-m$ and $i\gamma^\mu\partial_\mu+m$ in natural units	Propagator Table
$G_X,\ K_X$	Scalar and Dirac unit-source inverses; the subscript fixes a boundary prescription	Propagator Table
$D_F,\ S_F$	Time-ordered scalar and spinor correlators, with different source factors from unit inverses	Propagator Table
$W,\ C,\ \mathcal A$	Wightman function, scalar commutator and Dirac anticommutator distributions; $C$ is not $C_D$	Propagator Table
$i0$	Distributional boundary-value prescription; a finite numerical regulator obeys a modified equation	Propagator Visualization
$d\Pi_p,\ d\Phi_n$	One-particle mass-shell measure and multiparticle final-state measure; check where $(2\pi)^4$ is included	Invariant Phase Space
$f_s,\ a_s,\ b_s$	Coefficients of the same packet in the explicitly named normalization bases; other field expansions may reuse $a,b$ as operators	Normalization Table
$S,\ T,\ \mathcal M$	Scattering operator, transition operator in $S=I+iT$, and the invariant coefficient after stripping the conservation delta	Optical Theorem
$\sigma,\ \mathcal F$	Cross section and invariant incident flux; $\sigma$ is not a Pauli matrix here	Invariant Phase Space
$Z$	Pole residue in LSZ, or a source generating functional $Z[J]$; these are unrelated uses of a letter	LSZ and Generating Functionals

An $S$ in an action exponent, an $S(\Lambda)$ acting on a spinor, and an $S$ between in/out states are identified by their arguments and domain of action, not by typography alone.

Spectral labels and controlled approximations

Symbol	Meaning and collision to check	Defining page
$\kappa$	Signed Dirac angular quantum number in the Coulomb problem; its sign depends on the angular-operator convention	Hydrogen Fine Structure Revisited
$n,\ n_r,\ N$	Principal, radial or Landau labels as locally defined; a spin-resolved Landau orbital label is not always the physical level index	Relativistic Landau Levels
$\alpha,\ \beta$	Bogoliubov coefficients in mode mixing; not Dirac matrices or fine-structure coupling	Pair Creation
$N_{\mathbf p},\ P_{\rm vac}$	Mean pair occupation per mode and vacuum survival probability; neither determines the other without the statistics and mode distribution	Vacuum Instability
$M,\ \mathcal O,\ V$	Rest energy $mc^2$, odd Dirac operator and even potential in the FW expansion; $M$ can denote a target mass elsewhere	FW Expansion
$F$	Ordered magnetic square $\boldsymbol\pi^2-q\hbar\boldsymbol\Sigma\cdot\mathbf B$ in FW formulas; not the flux $\mathcal F$	FW Expansion
$c_F,\ c_D,\ c_S$	NRQED matching coefficients; the subscript $D$ does not refer to the charge-conjugation matrix $C_D$	Nonrelativistic QED

When copying a formula, record the symbol’s units, operator or scalar character, domain, and normalization. Those four checks usually expose a notation collision before it becomes a sign or dimension error.

References

• Bjorken, James D., and Sidney D. Drell. Relativistic Quantum Mechanics. McGraw–Hill, 1964. A standard source for wave-equation and spinor notation.
• Dreiner, Herbi K., Howard E. Haber, and Stephen P. Martin. “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry.” Physics Reports 494, 1–196 (2010). doi:10.1016/j.physrep.2010.05.002. Detailed comparisons of spinor notation and conventions.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995. doi:10.1017/CBO9781139644167. Distinguishes particle-state, Lorentz-component and scattering representations.