By Seeger R. J.
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Extra info for A Critique of Recent Quantum Theories
In the following we distinguish two cases: Either it is possible to achieve Im χ(x), ∇x χ(x) = 0 by a smooth gauge transformation χ(x) → χ(x) = eiθ(x) χ(x) or not. 46) with domain H 2 (Rd ). Thus A acts similar to an external magnetic vector potential. However, while A comes from a connection of a line bundle over R3 , A comes from a connection of a line bundle over Rd . Although A and A ε with an ε in front only, and therefore are not retained in the appear in HBO semiclassical limit to leading order, they do contribute to the solution of the Schr¨ odinger equation for times of order ε−1 .
However, for small ε the commutator is small and one can show that spectral subspaces corresponding to spectrum which is separated by a gap are approximately invariant. 2 in a straightforward way to the case where H0 (x) is a family of self-adjoint operators on 40 2 First order adiabatic theory some Hilbert space Hf depending on the parameter x ∈ Rd and the Hamiltonian is given through H ε = f (−iε∇x ) ⊗ 1 + ⊕ Rd dx H0 (x) . 16) The Hamiltonian H ε now acts on H = L2 (Rd ) ⊗ Hf = L2 (Rd , Hf ) and f (−iε∇x ) with f : Rd → R is the perturbation which in a sense replaces iε∂t .
As a second step the approximately invariant subspace, which is a rather complicated and ε-dependent object, is mapped to a reference subspace which is simple and adapted to the problem. 2. The eﬀective Hamiltonian for the slow degrees of freedom is deﬁned as the restriction of the full Hamiltonian to the almost invariant subspace mapped to the reference subspace. This procedure allows to compute the eﬀective Hamiltonian as acting on the reference subspace to arbitrary order in ε. 4, the leading orders of this expansion provide very relevant information about the dynamics of the slow degrees of freedom.