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The equal-time canonical commutation relation is $[\varphi (t,\mathbf{x}),\pi (t,\mathbf{y})]=i{\delta }^3(\mathbf{x}-\mathbf{y})$ in units $\hslash =1$, with the two field-field and momentum-momentum commutators zero. The quadratic Hamiltonian operator gives $\dot{\varphi }=\pi$ and $\dot{\pi }={\nabla }^2\varphi -m^2\varphi$, hence the Klein-Gordon equation as an operator identity.

Quantizing a moduli-space approximation gives wavefunctions with measure $\sqrt{\det g}{d}^{d}q$ and, under the minimal scalar ordering, the Hamiltonian operator $-{\hslash }^2{\Delta }_g/2+V$, where ${\Delta }_g$ is the Laplace-Beltrami operator. Global identifications, statistics and regularity supply additional restrictions on wavefunctions. Curvature-dependent ordering terms or quantum corrections are extra choices, not determined by the classical kinetic energy alone.

CPT symmetry equates the diagonal entries of the decay-effective Hamiltonian operator. It does not make this effective matrix Hermitian operator; its imaginary diagonal part describes a total decay width.

In units with $\hslash =1$, the inverse lifetime of an unstable particle. A neutral meson mixing effective Hamiltonian operator incorporates decay through its anti-Hermitian part $-i\Gamma /2$.

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.

An energy eigenstate is a nonzero quantum state represented by an eigenvector of the Hamiltonian operator. Its time evolution changes only its overall phase.

The ground-state energy is the smallest energy eigenvalue of a Hamiltonian operator.

The ground-state subspace is the eigenspace of a Hamiltonian operator at its lowest eigenvalue $E_0$. A computational history state construction can have many valid quantum witnesses, hence a degenerate ground space. Perturbation bounds must optimize on this entire subspace, rather than on a chosen basis of ground states.

For equal particle rates in the GRW model, the next centre falls in $B$ with probability $N^{-1}{\sum }_i{\int }_B\Vert L_i(c)\Psi {\Vert }^{2}dc$. With zero Hamiltonian operator, two successive centres in $B$ have joint probability $N^{-2}{\sum }_{ij}{\int }_{B}dc{\int }_{B}dd\Vert L_j(d)L_i(c)\Psi {\Vert }^2$. Dividing by the first-event probability gives the conditional second event, including repeated particle labels. Spatial probabilities depend on actual packet mass, not solely the nominal packet centres.

Hamiltonian simulation constructs a quantum circuit approximating the time-evolution operator $e^{-itH}$ generated by a Hamiltonian operator $H$, with cost controlled by the evolution time, requested precision, and structure of $H$.

Transitions between a neutral meson carrying flavour and its antiparticle. Decays are described on this two-state subspace by an effective Hamiltonian operator $M-i\Gamma /2$ with Hermitian matrices $M,\Gamma$.

The operator formalism describes a quantum system using states in a Hilbert space and observables represented by self-adjoint operators with suitable domains on that Hilbert space. Evolution and correlation functions are computed from Hamiltonian operators and their time-ordered products. For free quantized fields, the oscillator construction gives a Fock space. The operator-path-integral equivalence relates these matrix elements to integrals over field histories.

The parity operator acts on a spatial wavefunction by $(P\psi )(x)=\psi (-x)$ and satisfies $P^2=I$. For a Hamiltonian operator with an even potential and parity-invariant domain, $HP=PH$. A nondegenerate energy eigenvalue has a one-dimensional eigenspace, so its eigenfunction is an eigenvector of $P$ with eigenvalue $+1$ or $-1$.