1
$\begingroup$

I would be glad if someone can help me understand the argument in the first paragraph of page 4 of this paper.

Especially I don't understand their first sentence,

"Using N bosons (fermions) distributed over m states, one can construct completely symmetric (antisymmetric) irreducible representations of the group U(m) associated with Young tableaux with N boxes in a row (column)"

(I am quite familiar with the Quantum Statistics concepts being alluded to but not so much with the Young-Tableux technology being used)

All I can see is that $U(n)$ can act on the space of $c_i$ and keep the operators defined in their equation 2.9 and 2.10 unchanged.

Also on page 4 is there a typo in Equation 2.13? It doesn't seem to follow from the line previous to it and the line previous to it makes no sense to me. I guess there should have been a "=" between the $\lambda^N$ and $exp$ in the line just before 2.13.

Even if I make the above "correction" I don't see how 2.13 follows from it.

  • 1
    Let V be the defining representation of U(m). The symmetric power S^N(V) describes N bosons and it is an irreducible representation; the exterior power \Lambda^N(V) describes N fermions and it is also an irreducible representation. The Young tableaux are ways to label these representations among all representations.2011-01-25
  • 0
    @Qiaochu: why not an answer?2011-01-25
  • 0
    I didn't know which part he was confused about, and I also didn't look at the rest of the paper.2011-01-25
  • 2
    @Qiaochu: I hate it when you ruin a perfectly good rhetorical question with a reasonable response.2011-01-25
  • 0
    @Willie Now I am worried! I at least didn't intend my question to be rhetorical.2011-01-26
  • 0
    @Anirbit: I think Willie was referring to his own question to Qiaochu, not to your question.2011-01-26

1 Answers 1

2

I see this paper was written by a physicist. Here's what he's trying to say. Let $T$ be a linear operator $V \to V$ with eigenvalues $\lambda_1, ... \lambda_n$. $T$ defines linear operators $\Lambda^N T : \Lambda^N V \to \Lambda^N V$ and $S^N T : S^N V \to S^N V$ for all $N$. Let us suppose that $T$ is diagonalizable with eigenvectors $v_1, ... v_n$. Then a basis of eigenvectors for $\Lambda^N T$ is given by the set of all exterior products of $N$ distinct eigenvectors, and a basis of eigenvectors for $S^N T$ is given by the set of all symmetric products of $N$ eigenvectors.

It follows that $\text{tr } \Lambda^N T$ is the $N^{th}$ elementary symmetric polynomial in the eigenvalues (this determines the character of the corresponding representation of $U(n)$) and $\text{tr } S^N T$ is the $N^{th}$ complete homogeneous symmetric polynomial in the eigenvalues (same). If $T$ is the time evolution operator $U(t)$ for a quantum system, then the eigenvalues of $T$ are complex exponentials of the eigenvalues of the Hamiltonian.

I can't help you with the rest; I don't understand the notation. As far as I can tell, he's just using the identity described in this blog post:

$$\frac{1}{\det(I - Tz)} = \frac{1}{\sum_{n \ge 0} \text{tr } \Lambda^n T z^n} = \sum_{n \ge 0} \text{tr } S^n T z^n = \exp \left( \sum \frac{\text{tr } T^n}{n} z^n \right).$$

This identity expresses, in a way I don't really understand, boson-fermion duality.

  • 0
    @Quachu Thanks for your efforts. I haven't yet read your blog post but the interesting identity that you mention doesn't seem to be getting used in that paper. Though I may be wrong. (1) Can you tell me about the connection to representations of $U(n)$ that you mention? (You started with a general T to begin with) (2) In that paper can you tell me why the equation 3.2 must hold? (3) Equation B1 in that paper again seem to a representation theory identity that I am not familiar with. Can you help with that or give references.2011-01-28
  • 0
    My understanding is that the whole paper is basically showing applications in physics coming from the interplay between equation 3.2 and B1.2011-01-28
  • 0
    1) Let T vary over all elements of U(n). 2) Like I said, I don't understand the notation here. 3) That is the second Jacobi-Trudi identity. There are references at the Wikipedia article: http://en.wikipedia.org/wiki/Schur_polynomial2011-01-28
  • 0
    @Qiachu I didn't get your first comment. If $T \in U(n)$ then are you saying that $Tr(T)$ is somehow related to $Tr(\Lambda ^n T)$. I am not aware if representations on $\Lambda ^n V$ and $Sym^n V$ are somehow specially related to representations of $U(n)$. In the equation numbered $3.2$ can you tell me what property should the $Z$ on the LHS satisfy for that expansion to hold? Is that some known kind of expansion (has a name?) which I can try to track down in representation theory books?2011-01-28
  • 0
    @Anirbit: I am confused about what you are confused about. If T varies over the elements of U(n), then the computations I do above describe the characters of the U(n)-representations Lambda^n V and S^n V. Again, I still don't understand the notation in this paper.2011-01-28