Hilbert–Schmidt and Trace Class Operators

Light–Matter Response in Nonrelativistic Quantum Electrodynamics

“…In quantum mechanics we then also have generalized eigenstates, so-called scattering states, which are not square-integrable and that constitute the continuous spectrum of such a Hamiltonian. 88 The simplest example is the free electronic Hamiltonian which in infinite space has a purely continuous spectrum consisting of non-normalizable plane-waves. 89 The physical interpretation of such scattering states - as already the name indicates - is that particles propagate to infinity and do not stay bound anywhere.…”

“…If we excite a matter system from its ground state into such an excited state, it will remain in this state as long as we do not perturb it. In quantum mechanics we then also have generalized eigenstates, so-called scattering states, which are not square-integrable and that constitute the continuous spectrum of such a Hamiltonian . The simplest example is the free electronic Hamiltonian which in infinite space has a purely continuous spectrum consisting of non-normalizable plane-waves .…”

Understanding Polaritonic Chemistry from Ab Initio Quantum Electrodynamics

“…We note that eq implies that we think about the flat (Euclidean) space R 3 or its extension including time, the Minkowski space. , Its homogeneity , i.e., that no point is special, and its isotropy , i.e., that no direction is special, are very important since these symmetries determine the basic building blocks of our theories. These symmetries are connected directly to the position-momentum and energy-time uncertainty relations, ,, i.e., the translations in space are connected to momentum operators and the translations in time to the energy operator. Thus, the basic building blocks are (self-adjoint realizations of) the momentum–i ℏ bold∇ and position r operators and the energy i ℏ∂ t and time t operators (see Appendices A.1 and A.2 for more details).…”

“…That is, the common argument that a Coulomb-gauged field can be multipole expanded and in this way connected to the Power-Zienau-Woolley gauge only holds perturbatively and not on the level of operators in ab initio QED (see also Appendix A.3 for further details). In the context of working with operators instead of with perturbation theory we note that we have implicitly assumed that we are on R 3 and instead of boundary conditions on the matter wave functions we have imposed normalizability to have self-adjoint operators. , This is the standard setting of ab initio quantum physics , (see also Appendix A.2). If we would restrict the matter domain, e.g., choose genuine periodic boundary conditions in the velocity gauge, the length gauge transformation changes these boundary conditions as well in a nontrivial manner, , again highlighting subtle differences when working with different gauges.…”

“…Quantum theories are commonly formulated on infinite-dimensional complex Hilbert spaces . , The necessity for infinite dimensions in quantum physics is relatively easy to understand. If we assumed that the Heisenberg uncertainty principle (up to ℏ ), x̂ p̂ − p̂ x̂ = i l̂ should hold on a finite dimensional complex Hilbert space, which is equivalent to C N , then we would find by the cyclic property of the trace the contradiction 0 = tr false( x̂ p̂ false) − tr false( p̂ x̂ false) = itr false( double-struckl̂ false) = i N This contradiction leads to the conclusion that we cannot represent the basic commutation relations in finite dimensions . Commonly one further assumes in ab initio quantum physics and constructive quantum field theory separable infinite-dimensional Hilbert spaces, , i.e., they allow for a countably infinite basis, such that all of these Hilbert spaces are isometrically isomorphic to the fundamental sequence space l 2 false( double-struckN false) , which contains all infinite sequences {ψ 1 , ψ 2 , ...} with ψ n ∈ C that have finite square norm, i.e., ∑ n = 1 ∞ true| ψ n | 2 < ∞.…”