B.13 Chapters 27 and 28

B.13 Chapters 27 and 28

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · irreducible representation

Problem 1:

Consider the fermionic oscillator, for $d=3$ degrees of freedom, with Hamiltonian

$$H=\frac{1}{2}\sum _{j=1}^3(a_{Fj}^{†}{a}_{Fj}-a_{Fj}{a}_{Fj}^{†})$$

• Use fermionic annihilation and creation operators to construct a representation of the Lie algebra $\mathfrak{u}(3)=\mathfrak{u}(1)+\mathfrak{su}(3)$ on the fermionic state space ${\mathcal{H}}_F$ . Which irreducible representations of su $.(3)$ occur in this state space? Picking a basis $X_j$ of $\mathfrak{u}(3)$ and bases for each irreducible representation you find, what are the representation matrices (for each $X_j)$ for each such irreducible representation?

• Consider the subgroup $SO(3)\subset U(3)$ of real orthogonal matrices, and the Lie algebra representation of $\mathfrak{so}(3)$ on ${\mathcal{H}}_F$ one gets by restriction of the above representation. Which irreducible representations of $SO(3)$ occur in the state space?

Problem 2:

Prove that, as algebras over $C$,

$\mathrm{Cliff}(2d,\mathbf{C})$ is isomorphic to $M(2^d,\mathbf{C})$

$\mathrm{Cliff}(2d+1,\mathbf{C})$ is isomorphic to $M(2^d,\mathbf{C})\oplus M(2^d,\mathbf{C})$

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原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

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