Quantum Mechanics for Mathematicians
that preserves the canonical commutation relations. Here the use of normal ordered operators means thatUA′ is a representation o ...
Quadratic combinations of the creation and annihilation operators give rep- resentations onHof three subalgebras of the complexi ...
These will be linear combinations of pairs of either creation or annihilation operators, so will change the eigenvalue ofHor :H: ...
Note that this is a differentSO(2) action than the one with moment map the Hamiltonian, it acts separately on positions and mome ...
To construct the angular momentum operators in the Bargmann-Fock rep- resentation, recall that in the Schr ̈odinger representati ...
for a complex symmetric matrixBwith matrix entriesBjk. There is no normal ordering ambiguity here, and quantization will give th ...
in annihilation and creation operators is of fundamental importance in quantum field theory, wheredis infinite. This topic is ho ...
Chapter 26 Complex Structures and Quantization The Schr ̈odinger representation ΓSofH 2 d+1uses a specific choice of extra struc ...
we introduced complex coordinates on phase space, making the choice zj= 1 √ 2 (qj−ipj), zj= 1 √ 2 (qj+ipj) Thezjwere then quanti ...
The pair (V,J) can be thought of as givingVthe structure of a complex vector space, withJproviding multiplication byi. Similarly ...
Mis identified withM+J 0 by equation 26.1, with basis elementqjgoing tozj, andpjtoizj. Mis identified withM−J 0 by equation 26. ...
and preserves the compatibility condition since, ifJ∈Sp(2d,R), so isgJg−^1. A complex structureJcan be characterized by the subg ...
for some non-zerov∈M, so 〈u 1 ,u 2 〉=iΩ(u 1 ,u 2 ) =i 1 2 Ω(v 1 +iJv 1 ,v 2 −iJv 2 ) = 1 2 (−Ω(Jv 1 ,v 2 ) + Ω(v 1 ,Jv 2 )) + i ...
where theRcomponent is the constant functions. The Lie bracket is the Poisson bracket. Complexifying gives h 2 d+1⊗C= (M⊕R)⊗C= ( ...
For more general choices ofJwe start by taking Γ′J(0,c) =−ic 1 (26.14) which is chosen so that it commutes with all other operat ...
Proof.An outline of the construction goes as follows: Define a positive inner product onMby (u,v)J = Ω(u,Jv) onM. By Gram-Schmi ...
quantization of the theory, while in the relativistic case it does not, and in that case a different sort of complex structureJi ...
and one has all the elements needed for the standard Bargmann-Fock quantiza- tion. Note that the Hermitian inner product here is ...
by replacing theiby an arbitrary complex numberτ. Then the condition that q−τpbe inM+J and its conjugate inM−J is J(q−τp) =J(q)− ...
which in the Schr ̈odinger representation is the differential equation (Q−τP)ψ(q) = ( q+iτ d dq ) ψ(q) = 0 which has solutions ψ ...
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