The canonical formulation of E6(6) exceptional field theory

We investigate the canonical formulation of the (bosonic) E6(6) exceptional field theory. The explicit non-integral (not manifestly gauge invariant) topological term of E6(6) exceptional field theory is constructed and we consider the canonical formulation of a model theory based on the topological two-form kinetic term. Furthermore we construct the canonical momenta and the canonical Hamiltonian of the full bosonic E6(6) exceptional field theory. Most of the canonical gauge transformations and some parts of the canonical constraint algebra are calculated. Moreover we discuss how to translate the results canonically into the generalised vielbein formulation. We comment on the possible existence of generalised Ashtekar variables.

Quantum statistical physics: Functional integration formalism

The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.

Hamiltonian Formulation

The Hamiltonian formulation of relational particle dynamics unveils its equivalence with modern gauge theory, which admits exactly the same canonical formulation. Both are constrained Hamiltonian systems with nonhonolomic constraints, for which Dirac’s analysis, made popular by his lectures, is necessary. Dirac’s analysis is briefly summarized in this chapter for readers unfamiliar with it. The Hamiltonian formulation of the kind of systems we’re interested in is nontrivial. In fact the standard formulation fails to be predictive, precisely because of the relational nature of our dynamics.