In quantum statistical mechanics one has the fundamental object ρ(β; x a ,x b ), a matrix entry which defines the density matrix ρl, given in the Feynman formulation by the Euclidean path integral of the preceding chapter: $$\begin{array}{*{20}{c}} {\rho {{{(\beta ;{{x}_{a}},{{x}_{b}})}}^{{\underline{\underline {def}} }}}\int_{{{{x}_{a}} = x(0)}}^{{{{x}_{b}} = x(\beta \hbar )}} {\exp ( - {{S}_{E}}(x(t))/\hbar )D[x(t)]} } \\ { = {{I}_{E}}(0,\beta \hbar ,{{x}_{a}},{{x}_{b}})} \\ \end{array}$$ (14.1.1) where
β=kT1 is the inverse temperature
T1 up to a constant
k1, k being Boltzmann’s constant. From ρ one obtains the all-important partition function Ζ as a trace: $$\begin{array}{*{20}{c}} {Z = \int_{{ - \infty }}^{\infty } {\rho (\beta ;x,x)dx} } \hfill \\ { = \int_{{ - \infty }}^{\infty } {\int_{{x(0) = x}}^{{x(\beta \hbar ) = x}} {\exp ( - {{S}_{E}}(x(t))/\hbar )D[x(t)]dx.} } } \hfill \\ \end{array}$$ (14.1.2)
There is no conservation of probability in the Euclidean formulation of the acceleration Lagrangian. A narrow definition of a Hamiltonian, and the one that holds in quantum mechanics but is not suitable for our case, is that the Hamiltonian defines a probability conserving unitary evolution. A broader definition of the Hamiltonian is adopted that is consistent with both statistical mechanics and quantum mechanics. Namely, the Hamiltonian – which is equivalent to the transfer matrix in statistical mechanics – is the generator of infinitesimal translations in time for both the Minkowski and Euclidean cases. The Hamiltonian derived for the acceleration Lagrangian in Chapter 13 is studied in this chapter using the concepts of the state space and operators. The Hamiltonian for unequal and real frequencies ω 1 ,ω 2 is pseudo-Hermitian and is explicitly mapped to a Hermitian Hamiltonian using a nontrivial differential operator. An explicit expression is obtained for the matrix element of the differential operator. The mapping fails for a critical value of the coupling constants. The Hamiltonian and state space are shown to have real and complex branches that are separated by the critical point. At the critical point, the Hamiltonian is inequivalent to any Hermitian operator, and is shown in Chapter 15 as being equal to an infinite-dimensional block diagonal matrix, with each block being a Jordan matrix.
In quantum theory, equilibrium statistical mechanics is usually formulated through the canonical ensemble, whose privileged status is tied to the Euclidean continuation of time evolution. The microcanonical ensemble, by contrast, is commonly introduced as a separate spectral construction. In this work we show that this asymmetry is representational rather than structural. We formulate the system in an extended Hilbert space in which time is promoted to an auxiliary clock degree of freedom and physical states are selected by a reparametrization-invariant constraint operator
C^=P^T+H^. The corresponding projector
δ(C^) provides a single unified object from which both canonical and microcanonical ensembles emerge as complementary projections in the clock sector. In the clock-time representation, a purely imaginary clock separation yields the Euclidean kernel and the canonical partition function. In the conjugate clock-energy representation, the same projector reduces to the spectral operator
δ(H^−E) and hence to the microcanonical density of states. The main consequence is structural: canonical and microcanonical statistics need not be introduced as independent constructions, since both are already encoded in the same constrained quantum dynamics.