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A ground state is a normalized energy eigenstate that attains the lowest possible energy of a quantum system. Precisely, let the Hamiltonian operator $H$ be a self-adjoint operator bounded below on a Hilbert space, and write $E_{\ast }=\inf \sigma (H)$ for the infimum of its spectrum. A ground state is a unit vector ${\psi }_0$ in the operator domain $D(H)$ satisfyingThus a ground state exists exactly when the spectral bottom $E_{\ast }$ is an energy eigenvalue. The lowest energy eigenvalue among some listed bound states is insufficient if other parts of the spectrum lie below it. The ground-state subspace collects all minimizing energy eigenstates; it can have ground-state degeneracy. The quantum variational principle characterizes $E_{\ast }$ as the infimum of normalized energy expectation values. ^[ref].

For the one-dimensional quantum harmonic oscillator, the ground state has energy $\hslash \omega /2$ above the minimum of the potential energy. This zero-point energy is compatible with being the lowest allowed energy: a localized wavefunction cannot simultaneously have zero kinetic energy. For a free particle on the whole line, the spectral bottom is zero but is unattained by any normalizable wavefunction; see ground-state energy. A lower bound therefore need not describe a physically preparable energy eigenstate.

The ground state supplies a reference for excitation costs and for the quantum variational principle. A ground state is also a stationary state under a time-independent Hamiltonian operator: its quantum phase changes, while its position probability density stays fixed. A nonzero ground-state energy does not imply motion along a classical trajectory, and ground-state degeneracy means there need not be a unique lowest quantum state.

A photon is a quantum of the electromagnetic field with energy $E=h\nu =\hslash \omega$.

Parity is the eigenvalue $+1$ or $-1$ of a state under spatial reflection.

A path integral represents a quantum amplitude or correlation function as an integral over field histories weighted by the exponential of the action.

The large-N expansion organizes an $O(N)$-symmetric theory in powers of $1/N$. At leading order, an auxiliary constraint field is determined by a self-consistent saddle-point or gap equation.

Finite-temperature quantum field theory represents thermal expectation values by evolution through an imaginary-time interval of length $\beta =1/T$.

A gauge-theory soliton is a finite-energy localized classical field configuration stabilized by topology or a balance among differently scaling energy terms.

Quantum field theory in curved spacetime quantizes matter fields on a prescribed classical spacetime geometry.

The Standard Model is a gauge theory with gauge group $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$. Its Higgs field breaks the electroweak subgroup to the electromagnetic $U(1{)}_Q$.

Spontaneous symmetry breaking occurs when the action and equations have a symmetry but a chosen ground state is invariant only under a proper subgroup. Acting with the broken symmetry generates a degenerate vacuum manifold.

An effective field theory describes phenomena below a chosen energy scale by retaining the relevant light fields and organizing local interactions by increasing suppression at that scale.

This is an orthogonal projection interpretation of linear least squares. The best affine predictor has slope covariance divided by predictor variance and intercept chosen to match means.

Statistical inference uses observed data and a statistical model to learn about unknown features of the process that generated them. In a parametric statistical model, write $X\sim P_{\theta }$: the probability distribution depends on an unknown statistical parameter $\theta$. An estimand specifies the target, such as a mean or a difference of means. A statistic is a function of the observations; an estimator is a statistic used to estimate the target. Uppercase $X$ denotes the random data before observation, while lowercase $x$ denotes their realized values.

Three questions lead to different outputs. Statistical estimation asks which target value the observations suggest. A confidence interval asks for a range produced by a procedure with calibrated coverage probability. A hypothesis test asks whether the observations depart sufficiently from a specified null hypothesis to enter its rejection region. These outputs complement one another: an estimated effect size describes magnitude, while a standard error describes the sampling distribution of an estimator. A p-value describes a tail probability under the null hypothesis.

The appeal of statistical inference is that it turns variable observations into reproducible uncertainty statements. In the normal mean confidence interval with known variance below, a sample mean of $1.2$ has standard error $0.5$: both the estimated shift and its precision matter. The calibration depends on the statistical model, including independence and the specified probability distribution. Increasing sample size improves precision under those assumptions; it does not repair a systematically misleading sampling procedure. Bayesian inference provides another framework, using a prior distribution and a likelihood function to obtain a posterior distribution.

A Monte Carlo method approximates a deterministic quantity using averages of simulated random variables and controls the approximation with probabilistic limit theorems.