Quantum Mechanics for Mathematicians
taking the boundary value asτ→0. This fixes the branch of the square root in 23.12 and one finds (see for instance section 7.6.7 ...
The Bargmann transform is then given by (Bψ)(z) = ∫+∞ −∞ 〈δz|q〉〈q|ψ〉dq = (ω π ) (^14) e− z^2 2 ∫+∞ −∞ e− ω 2 q 2 e √ 2 ωzqψ(q)dq ...
23.6 For further reading Coherent states and spin coherent states are discussed in more detail in chapter 21 of [81] and in [66] ...
Chapter 24 The Metaplectic Representation and Annihilation and Creation Operators, d = 1 In section 22.4 we saw that annihilatio ...
24.1 The metaplectic representation ford= 1 in terms ofaanda† Poisson brackets of order two combinations ofzandzcan easily be co ...
For the case ofzz(which is real), in order to get thesl(2,R) commutation relations to come out right (in particular, the Poisson ...
representation Γ′BFcomes from a Lie group representation ΓBFofSL(2,R), we have ΓBF(eθZ) =eθΓ ′BF(Z) where Γ′BF(Z) =−i ( a†a+ 1 2 ...
Lie algebra calculations, since the Lie algebras ofMp(2,R)andSL(2,R)can be identified. Another aspect of the metaplectic represe ...
For group elements,gθ=eθZ∈SO(2)⊂SL(2,R) and the representation is given by unitary operators Ugθ= ΓBF(eθZ) =e−i θ 2 (aa†+a†a) wh ...
and intertwining operators Ugr=eΓ ′BF(rμ) =e r 2 (a^2 −(a†)^2 ) which satisfy Ugr ( a a† ) Ug−r^1 =e r 0 1 1 0 ( a a† ) = ...
With this choice the distinguished state| 0 〉will be an eigenstate ofHwith eigenvalue^12. The choice of the coordinatezgives a ...
24.4SU(1,1) and Bogoliubov transformations Changing bases in complexified phase space fromq,ptoz,zchanges the group of linear tr ...
Applied not toz,zbut to their quantizationsa,a†, suchSU(1,1) trans- formations are known to physicists as “Bogoliubov transforma ...
Chapter 25 The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d In this chapter we’ll turn from t ...
on the harmonic oscillator state space. This gives the same construction of all SU(2)⊂U(2) irreducible representations that we s ...
where one should keep in mind that each degree of freedom can be rescaled separately, allowing different parametersωjfor the dif ...
A Lie subalgebra with basis elementszjzk, which has dimensiond^2. Com- puting Poisson brackets one finds {zjzk,zlzm}=zj{zk,zlz ...
Proof.Using 25.2 one has {μA,μA′}=− ∑ j,k,l,m {zjAjkzk,zlA′lmzm} =− ∑ j,k,l,m AjkA′lm{zjzk,zlzm} =i ∑ j,k,l,m AjkA′lm(zjzmδkl−zl ...
one confusing point about these. Recall that in chapter 16 we found the moment mapμL=−q·Apfor elementsL∈sp(2d,R) of the block-di ...
unitary representation ofu(d) on Fock spaceFd, but after exponentiation this is a representation not of the groupU(d), but of a ...
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