Concepts and notation

Units and recurring symbols

Unless an equation explicitly restores constants, relativistic particle and field formulas use $c=\hbar=1$. Then energy, mass, inverse length, and inverse time have the same units. Temperature also has energy units when $k_B=1$. The text restores $k_B$ or $\hbar$ where a physical time, temperature, or experimental phase benefits from it. Logarithms are natural, so entropies without $k_B$ are measured in nats; division by $\ln2$ converts them to bits.

$G$ is Newton's constant and $M_{\rm P}=(8\pi G)^{-1/2}$ the reduced Planck mass in natural units. In cosmology $a(t)$ is the scale factor and $H=\dot a/a$ the expansion rate. In the strong-CP chapter $a$ instead denotes the axion field, explicitly declared there. Lattice spacing is written $a_{\rm lat}$ where ambiguity could arise. $T$ denotes temperature except where declared to be a Euclidean time extent. Symbols are local to each model; identical letters do not imply identical physical quantities across chapters.

Quantum language

State and density operator. A state assigns expectation values to observables. A pure Hilbert-space state is a ray represented by a unit vector; a density operator is a positive operator with trace one that can also represent mixtures. Its eigenvalues are nonnegative weights. A reduced state can be mixed because of entanglement, even when the total state is pure.

Observable and operator. An observable encodes a measurable quantity or measurement statistics. In elementary quantum mechanics a self-adjoint operator represents an ideal observable. In field theory it is often safer to formulate local operations with bounded operators rather than assume every pointwise field is an ordinary finite matrix.

Entanglement. A joint state contains correlations that cannot be described by a probabilistic mixture of independent local states. For a pure bipartite state, equal nonzero eigenvalues of the two reduced density operators explain why their entropies agree.

Coherence and dephasing. Coherence is the phase-sensitive part of a state relative to a specified basis and set of measurements. Dephasing suppresses off-diagonal terms in that basis, either by averaging phases or correlating them with inaccessible degrees of freedom. It need not destroy global quantum information.

Quantum channel. A linear, completely positive, trace-preserving map representing an allowed unconditioned state transformation. “Completely positive” ensures positivity also when the system is part of a larger entangled system. A selected measurement outcome generally uses a trace-decreasing map; its trace is the outcome probability.

Relative entropy. A measure of distinguishability between states, not a symmetric distance. It is nonnegative and decreases under an information-discarding channel. It can be infinite when the reference state excludes possibilities present in the other state. Continuum definitions require algebraic care.

Correlation function. An expectation value of products of observables, often at different positions or times. Two-point functions describe pair correlations; higher-point functions can distinguish states with identical two-point data. Connected correlations remove contributions already explained by lower-order averages.

Gaussian state or statistics. A distribution or quantum state determined by first and second moments in the appropriate variables. Higher connected correlations vanish in the corresponding Gaussian description. Gaussianity is a restrictive property, not a synonym for weak interactions in every setting.

Theory and approximation

Symmetry. A transformation leaving the relevant physical predictions unchanged. A symmetry can forbid interactions. It can be explicitly broken by terms in the equations or spontaneously broken when a state fails to share the symmetry of those equations.

Gauge symmetry. A redundancy in the variables used to describe a physical state. Gauge-related descriptions should not be mistaken for different measurable states. Gauge-invariant observables, such as closed Wilson loops, do not depend on this redundancy.

Effective field theory. A systematic description at a specified resolution. Effects of inaccessible scales are represented by allowed interactions, ordered by their expected size. Predictivity comes from a controlled expansion and a finite number of inputs at a chosen accuracy.

Renormalization and matching. Renormalization specifies how regulated quantities and parameters combine into finite predictions. Its arbitrary scale must not affect an exact observable. Matching chooses an effective theory's coefficients to reproduce a more detailed theory in their common regime.

Regulator and continuum limit. A regulator removes troublesome arbitrarily short distances or large momenta, for example by introducing a lattice. A continuum limit removes it while keeping specified physical quantities meaningful. Refinement without identifying what is held fixed is not a well-defined physical limit.

Controlled approximation. An approximation with an organizing parameter or error estimate. Small coupling, widely separated scales, and a large number of field species can provide control in particular regimes. Agreement with one numerical value is not itself an error estimate.

Semiclassical. A treatment keeping some variables classical while retaining specified quantum effects. Quantum matter on a classical spacetime is one example. Which corrections are omitted must be stated before applying the approximation to entropy, backreaction, or a singular region.

Toy model. A deliberately simpler system preserving selected mechanisms or constraints. It can settle a question exactly inside its setting. Transfer to a material, continuum theory, or universe requires a separate comparison.

Universality. Insensitivity of specified large-scale behavior to some microscopic details. It is a statement about a class, observable, and regime; it does not mean all mechanisms or all observables become identical.

Collective behavior

Order parameter. A quantity that distinguishes phases, often by revealing a broken symmetry. Not every phase has a conventional local order parameter; the appropriate diagnostic depends on the system.

Critical point and correlation length. Near a continuous phase transition, fluctuations can become correlated over a very large distance. The correlation length quantifies how far a disturbance or correlation persists. Its growth can justify scale-based descriptions and also defeat simple perturbation theory.

Quasiparticle. A sufficiently long-lived excitation described approximately as a particle with renormalized properties. The approximation requires a suitable relation between its lifetime and energy. Failure of quasiparticles does not by itself specify the replacement transport mechanism.

Transport and relaxation. Transport concerns flows of conserved quantities, such as charge or energy. Relaxation describes approach toward a reference state or decay of a disturbance. Different disturbances can have different rates because they overlap different conserved or nearly conserved quantities.

Phase stiffness. The free-energy cost of a slow spatial twist of an order-parameter phase. It measures collective coherence against distortions and need not be large merely because local pairs are strongly bound.

Thermodynamic limit. System size and particle number tend to infinity at fixed density or other declared intensive parameters. True phase-transition singularities often require this limit. A finite cluster can reveal precursors without possessing the same phase transition.

Observation and inference

Standard Model. The quantum field theory of known strong, weak, and electromagnetic interactions with specified matter fields. Its minimal renormalizable form leaves neutrinos massless. It does not include a microscopic quantum theory of gravity or identify the cosmological dark component.

Cosmic microwave background. The relic electromagnetic radiation released when the early plasma became transparent. Temperature and polarization patterns carry information about earlier perturbations through a calculable transfer history, rather than presenting a direct photograph of inflation.

Confidence, significance, and model comparison. A confidence interval is constructed by a procedure with stated coverage assumptions. A significance quantifies how unusual a statistic is under a tested hypothesis. Neither is automatically a probability that a theory is true. Parameter priors, model flexibility, multiple comparisons, and uncertain backgrounds matter.

Local versus global significance. Local significance tests a particular location or parameter choice. Global significance accounts for searching across a specified range of possibilities. Selecting the most unusual result after many comparisons makes its local significance an incomplete summary.

Degeneracy of an inference. Distinct parameters or mechanisms predict indistinguishable values for the available observables. Breaking the degeneracy requires an additional observable or an explicitly justified assumption. More precise measurement of the same insufficient quantity need not solve the problem.