Energy-band and grand quantum preparations

Fix the positive Boltzmann constant $k_B>0$ in J/K and adopt two preparation rules within the normal-state model of Normal-state quantum measurement interpretation.

Energy-band preparation. Let $H=H^{\ast }$ have spectral PVM $E_H$, and let $P=E_H(B)$ be a nonzero finite-rank projection for a specified Borel energy band $B\subseteq \mathbb{R}$. When the model declares preparation in this band without additional constraints, adopt the density $\rho =P\mathrm{/}\mathrm{Tr}P$. This is a uniform preparation over the finite-dimensional range of $P$; it is not asserted for infinite-rank projections.

Grand preparation. Let $\mathcal{H}\mathrm{\ne }\{0\}$ be a complex Hilbert space with a specified nonempty finite or countably infinite orthonormal basis $(u_j{)}_{j\in J}$, so its finite linear span is dense in $\mathcal{H}$. For each $\psi ={\sum }_{j\in J}{c}_j{u}_j\in \mathcal{H}$, define the diagonal self-adjoint energy and particle-number operators by $D(H)=\{\psi :{\sum }_j\mathrm{\mid }{E}_j{c}_j{\mathrm{\mid }}^2<\mathrm{\infty }\},H\psi ={\sum }_j{E}_j{c}_j{u}_j;D(N)=\{\psi :{\sum }_j\mathrm{\mid }{n}_j{c}_j{\mathrm{\mid }}^2<\mathrm{\infty }\},N\psi ={\sum }_j{n}_j{c}_j{u}_j,$ where $E_j\in \mathbb{R}$ and $n_j\in {\mathbb{N}}_0$. For chemical potential $\mu \in \mathbb{R}$, define the diagonal sum $K_{\mu }=H-\mu N$ by $D(K_{\mu })=\{\psi :{\sum }_j\mathrm{\mid }(E_j-\mu n_j)c_j{\mathrm{\mid }}^2<\mathrm{\infty }\},K_{\mu }\psi ={\sum }_j(E_j-\mu n_j)c_j{u}_j\mathrm{.}$ This specifies the self-adjoint operator sum used in this postulate; on $D(H)\cap D(N)$ it agrees with the algebraic difference. Require ${\inf }_j(E_j-\mu n_j)>-\mathrm{\infty }$. Fix $\beta >0$ in J${}^{-1}$ and assume $0<\mathrm{\Xi }(\beta ,\mu )={\sum }_{j\in J}{e}^{-\beta (E_j-\mu n_j)}<\mathrm{\infty }\mathrm{.}$ Adopt the normal density ${\rho }_{\beta ,\mu }={\mathrm{\Xi }}^{-1}{e}^{-\beta K_{\mu }}$. For response claims, also require the uniform joint-moment envelopes in Grand-response derivatives are covariances. Here $E_j,\mu$ are in J, $n_j$ is dimensionless, $\beta =1\mathrm{/}(k_{B}T)$ is in J${}^{-1}$, and $\mathrm{\Xi }$ is dimensionless.

These are adopted preparation assumptions, not consequences that a physical system is actually prepared in these states. They do not assert dynamical thermalization, an infinite-rank microcanonical state, or a grand ensemble for noncommuting $H,N$. The mathematical variational results to which they refer do not prove the physical postulate.

The self-adjointness assertions above are mathematical properties of the specified maximal diagonal domains. For a real sequence $(a_j{)}_{j\in J}$, let $M_a\psi ={\sum }_j{a}_j{c}_j{u}_j$ on $D(M_a)=\{\psi :{\sum }_j\mathrm{\mid }{a}_j{c}_j{\mathrm{\mid }}^2<\mathrm{\infty }\}$. This domain contains the dense finite linear span. By definition, $\varphi \in D(M_a^{\ast })$ exactly when some $\eta \in \mathcal{H}$ satisfies $⟨M_a\psi ,\varphi ⟩=⟨\psi ,\eta ⟩$ for every $\psi \in D(M_a)$, and then $M_a^{\ast }\varphi =\eta$. Testing $\psi =u_j$ gives ${\eta }_j=a_j{\varphi }_j$; hence $\eta \in \mathcal{H}$ forces ${\sum }_j\mathrm{\mid }{a}_j{\varphi }_j{\mathrm{\mid }}^2<\mathrm{\infty }$, so $\varphi \in D(M_a)$. Conversely, if $\varphi \in D(M_a)$, then for every $\psi \in D(M_a)$ the two inner products converge by Cauchy–Schwarz and satisfy $⟨M_a\psi ,\varphi ⟩=⟨\psi ,M_a\varphi ⟩$, since every $a_j$ is real. Thus $D(M_a^{\ast })=D(M_a)$ and $M_a^{\ast }=M_a$. This applies to $a_j=E_j$, $a_j=n_j$, and $a_j=E_j-\mu n_j$.