Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2 in the reading of Assigned sharp observables and the outcome law of the framework, and the autonomous-dynamics postulate P4 in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution for a declared isolated system with state space $H$ (The state space of the framework: preparations, pure rays and density operators) and assigned energy observable $(H_{\mathrm{o}\mathrm{p}},u_H)$ with spectral projection-valued measure $E_H$ and flow $U(t)=\exp (-it{H}_{\mathrm{o}\mathrm{p}}\mathrm{/}\mathrm{\hslash })$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution). Units are those of Dimensions, SI unit conventions and quantity types of the framework.

(b) Symmetry of the dynamics. A symmetry of the declared dynamics, or simply a symmetry, is a unitary operator $S\in \mathcal{B}(H)$ with $SD(H_{\mathrm{o}\mathrm{p}})=D(H_{\mathrm{o}\mathrm{p}}),S{H}_{\mathrm{o}\mathrm{p}}=H_{\mathrm{o}\mathrm{p}}S\text{ on }D(H_{\mathrm{o}\mathrm{p}})\mathrm{.}$ The reading is the model's: a symmetry is a unitary that preserves the assigned energy domain and commutes with the assigned energy operator on that domain; the equivalent formulation by commutation with the flow $U(t)$, and the concomitant conservation statements, are proved results of this page and are not part of this definition. Neither the energy operator nor its potential or coupling terms are assumed to possess any symmetry here.

(c) Unitary symmetry group. A unitary symmetry group of the declared system is a group $G$ together with a map $g\mapsto S_g$ from $G$ into the unitary operators of $H$ such that $S_e=I$, $S_{gh}=S_g{S}_h$ and every $S_g$ is a symmetry in the sense of (b). When $G$ carries a topology, the group is strongly continuous when $g\mapsto S_{g}f$ is continuous for every $f\in H$; for the one-parameter case the concrete condition is that of Strongly continuous one-parameter unitary group. The model's standard candidate spatial symmetry groups and additional unitary symmetry actions, which must satisfy clause (b) before any one of them is called a symmetry, include:

1. the spatial translations $T_a$ of Componentwise position and momentum operators on the component-valued wave-function space clause (c), forming the group ${\mathbb{R}}^3$, with generator data supplied by Componentwise position and momentum operators on the component-valued wave-function space clause (d);
2. the spatial rotations $U(R)$ of The spatial rotation action on the component-valued wave-function space, forming the group $SO(3)$ with its Euclidean topology, and their coordinate one-parameter subgroups, whose generators are the operators $L_i$ of Coordinate-rotation generators on the Schwartz classes and their commutators;
3. the spatial parity action $\mathrm{\Pi }$ of The spatial parity action on the component-valued wave-function space, forming the two-element group $\{I,\mathrm{\Pi }\}$;
4. in the spinless scalar case $H=L^2({\mathbb{R}}^3;\mathbb{C})$, the gauge phase multipliers $G_{\chi }(t)$. No action on an internal factor is defined here. Given an open interval $I$, a real Borel function $\chi :{\mathbb{R}}^3\times I\to \mathbb{R}$ with unit $\mathrm{J}\mathrm{s}{\mathrm{C}}^{-1}$, a charge parameter $q\in \mathbb{R}$ with unit C, and $\mathrm{\hslash }>0$ with unit $\mathrm{J}\mathrm{s}$, define $(G_{\chi }(t)\psi )(x):=\exp \text{\hspace{0.17em}⁣}(iq\chi (x,t)\mathrm{/}\mathrm{\hslash })\psi (x)\text{for almost every }x,\psi \in L^2({\mathbb{R}}^3;\mathbb{C})\mathrm{.}$ For each fixed $t$, this is a unitary operator. The real coefficients in the defining power series give $\overline{\exp (i\theta )}=\exp (-i\theta )$ for real $\theta$; by absolute convergence, the Cauchy product for $\exp (i\theta )\exp (-i\theta )$ groups the coefficient of degree $k$ as $\frac{1}{k!}{\sum }_{j=0}^k\left(\genfrac{}{}{0px}{}{k}{j}\right)(-1{)}^{k-j}$; the finite sum is $(1-1{)}^k$, hence is $1$ for $k=0$ and $0$ for $k\ge 1$. Therefore $\mathrm{\mid }\exp (i\theta )\mathrm{\mid }=1$ and the inverse multiplier is $\exp \text{\hspace{0.17em}⁣}(-iq\chi (\cdot ,t)\mathrm{/}\mathrm{\hslash })$. This definition asserts only its action on $L^2$; it assumes no regularity of $\chi$ and does not imply preservation of an unbounded energy domain. The gauge-covariance result treats only the additional regularity class stated by the prescribed-coupling postulate.

(d) One-parameter symmetry group and generator. A one-parameter symmetry group is a strongly continuous one-parameter unitary group $t\mapsto S_t=e^{-itA\mathrm{/}\mathrm{\hslash }}$ with every $S_t$ a symmetry in the sense of (b), where $A$ is the self-adjoint generator furnished by Stone's theorem (Stone's theorem: unitary groups and self-adjoint generators); its declared unit is that of the spectral coordinate of $A$. When $S_t=U(R_i(t))$ for a coordinate rotation subgroup, $A$ is the operator $L_i$ above; when $S_t=T_{t{e}_j}$, $A$ is the momentum component $P_j$ in the reading of Componentwise position and momentum operators on the component-valued wave-function space clause (d).

(e) Conserved sharp observables. Let $(A,u_A)$ be an assigned sharp observable of the framework (Assigned sharp observables and the outcome law of the framework). It is conserved along the flow when its outcome law is independent of time in every preparation: for every preparation $\rho$ (The state space: density operators, pure and mixed states, and purity) and every Borel $B\subseteq \mathbb{R}$, ${\mu }_{\rho (t)}^A(B)={\mu }_{\rho (0)}^A(B),\rho (t)=U(t)\rho (0)U(t{)}^{\ast }\mathrm{.}$ The conserved object is thus the outcome law of the assigned observable — a probability measure on the Borel sets — and, where the moments exist, its moments; the definition does not say that a single numerical value is revealed at each time, and it identifies no laboratory quantity with the pair $(A,u_A)$.

(f) Status and scope. Clauses (a)-(e) fix the vocabulary of this page within P1, P2 and P4 and the cited readings. They assert no symmetry of any concrete model: whether $T_a$, $U(R)$ or $\mathrm{\Pi }$ is a symmetry of a given bounded-potential dynamics is decided by the conditional results of this page. The gauge definition asserts only an $L^2$ unitary; a symmetry claim must separately verify the energy-domain and commutation conditions of clause (b). Antiunitary symmetries, superselection sectors, symmetries of preparations as opposed to dynamics, and any empirical claim about which symmetries a laboratory system possesses are outside this definition. The parameter $a$ of a translation has the unit of length (metre) of Dimensions, SI unit conventions and quantity types of the framework; the rotation group and parity act on the spatial factor only, in the readings of The spatial rotation action on the component-valued wave-function space and The spatial parity action on the component-valued wave-function space, and the prescribed-coupling reading is that of Bounded-coefficient magnetic Schrodinger operator.

• Why the energy domain appears. For an unbounded energy operator, the statement "$S$ commutes with $H_{\mathrm{o}\mathrm{p}}$" needs the domain equality $SD(H_{\mathrm{o}\mathrm{p}})=D(H_{\mathrm{o}\mathrm{p}})$ to make the operator identity meaningful; this is the same convention under which the covariance theorems of the kinematics page are stated.
• Candidate actions are not symmetries by definition. The translations, rotations, parity and gauge multipliers are defined as unitaries or unitary families; the definition records them as the model's candidate symmetry actions, and the theorems decide for which potentials or couplings each candidate is actually a symmetry.