Quantum Mechanics for Mathematicians
to annihilation and creation operators satisfying [aj(p),a†k(p′)]+=δjkδ^3 (p−p′) The theory of non-relativistic electrons is som ...
Chapter 38 Symmetries and Non-relativistic Quantum Fields In our study (chapters 25 and 26) of quantization using complex struct ...
38.1 Unitary transformations onH 1 The single-particle state spaceH 1 of non-relativistic quantum field theory can be parametriz ...
The quadratic functions we will consider will be “local”, multiplying elements parametrized by the same points in position space ...
• zz→−ia†a This is the normal ordered form, with eigenvalues−in. With either choice, we get a number operator N= 1 2 (a†a+aa†), ...
An important property ofN̂that can be straightforwardly checked is that [N,̂Ĥ] = [ N,̂ ∫+∞ −∞ Ψ̂†(x)−^1 2 m ∂^2 ∂x^2 Ψ(̂x)dx ] ...
Recall (theorem 25.2) that fornbynmatricesXandY ∑n j,k=1 a†jXjkak, ∑n j,k=1 a†jYjkak = ∑n j,k=1 a†j[X,Y]jkak So, for each ...
{μX,Ψj(x)}=−XjkΨj(x), {μX,Ψj(x)}=XkjΨk(x) After quantization these become the operator relations 38.6 and 38.7. Note that the fa ...
andE(3) will act on Ψ(ψ) by Ψ(ψ)→(a,R)·Ψ(ψ) = ∫ R^3 Ψ(x)ψ(R−^1 (x−a))d^3 x = ∫ R^3 Ψ(Rx+a)ψ(x)d^3 x (using invariance of the int ...
Note that these are not the momentum operatorsPthat act onH 1 , but are operators in the quantum field theory that will be built ...
acting onH 1. Just as for energy and momentum, we can construct angular momentum operators in the quantum field theory as quadra ...
(hereRis theSO(3) rotation corresponding to Ω). This action can be thought of as an action on a tensor product ofC^2 and a space ...
38.4 Fermionic fields It is an experimentally observed fact that elementary particles with spin^12 behave as fermions and are de ...
giving a Lie algebra representation on the fermionic state space. We get the same formulas for operatorsN̂(equation 38.3),X̂(equ ...
Chapter 39 Quantization of Infinite dimensional Phase Spaces While finite dimensional Lie groups and their representations are r ...
For general interacting quantum field theories, one must choose among in- equivalent possibilities for representations of the ca ...
This is ford= 1, for arbitrarydone gets states parametrized by a vector α∈Cd, and |〈α| 0 〉|^2 =e− ∑d j=1|αj|^2 In the infinite d ...
an orthogonal state. This introduces the possibility of inequivalent representa- tions of the commutation relations, built by ap ...
holds in the orthogonal group case). Elements ofsp(2d,R) corresponding to Bogoliubov transformations (i.e., with non-zero commut ...
unitary Lie algebra representation does not exist (due to extra scalar terms in the commutation relations). Recall from section ...
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