Mathematical definition of annihilation and creation operators

Question

I am self-studying quantum field theory and have gotten to creation and annihilation operators, respectively denoted $A^\dagger$ and $A$. Conceptually I understand what these objects are, at least on a surface level, which is they increase/decrease the number of particles present by one. In terms of maps: $$A_n: H_s^{\otimes n} \rightarrow H_s^{\otimes (n-1)}\\ A_n^\dagger: H_s^{\otimes n} \rightarrow H_s^{\otimes (n+1)} $$ where $H_s^{\otimes n}$ is the $n$-fold symmetric tensor product of a Hilbert space $H$.

The formula for these operators are: $$A_n(\xi)(\beta)_{i_1,~\ldots,~ i_{n-1}} \equiv \sqrt{n}\sum_{i=1}^\infty \xi_i^* \beta_{i_1,~\ldots,~i_{n-1}, i} \quad \quad \xi \in H, \beta \in H_s^{\otimes n}\\ A_n^\dagger(\eta)(\alpha)_{i_1,~\ldots,~ i_{n+1}} \equiv \frac{1}{\sqrt{n+1}}\sum_{\ell=1}^{n+1} \eta_{i_\ell} \alpha_{i_1,~\ldots,~\hat i_{\ell}, \ldots,~ i_{n+1}} \quad \eta \in H, \alpha \in H_s^{\otimes n}.$$

I have been having trouble breaking down the above formulas. Why is $\eta$ subscripted by $i_\ell$ and why is $\xi$ is just subscripted by $i$?

Also, $\alpha$ and $\beta$ are both $n$-tensors, so how are we taking $\alpha$ to be subscripted by $n+1$ indices in the creation operator, and what is the logic behind attaching an $i$ for $\beta$? Mathematically it seems after the summation the right hand side of both operators will still be $n$-tensors. I must be misunderstanding something with the indices here.

Answer 1

• $\eta$ is subscripted by $i_\ell$ because the indices $i_1,...,i_{n+1}$ are given on the left hand side and we are summing over their sub indices

• Talagrand gives on p. 79 the example $$\tag{A} A^\dagger_2(\eta)(\alpha)_{i_1i_2i_3}=\frac{1}{\sqrt{3}}(\eta_{i_1}\alpha_{i_2i_3}+ \eta_{i_2}\alpha_{i_1i_3}+\eta_{i_3}\alpha_{i_1i_2}) $$ in which $\ell$ runs through $1,2,3$.

• To get out of that subindex business perhaps it is helpful to write (A) equivalently as $$\tag{A'} A^\dagger_2(\eta)(\alpha)_{\mu\nu\rho}=\frac{1}{\sqrt{3}}(\eta_{\mu}\alpha_{\nu\rho}+ \eta_{\nu}\alpha_{\mu\rho}+\eta_{\rho}\alpha_{\mu\nu})\,. $$

• $\alpha,\beta$ are both $n$-tensors. Note hovewer that the $\hat{i_\ell}$ is omitted. We subscript $\alpha$ always only with $n$ indices. See again (A) where $\alpha$ is a $2$-tensor.

• To $\beta$ we are not attaching an index $i$. We are contracting the $n$-tensor with a vector which results in an $(n-1)$-tensor -as it must because one particle gets destroyed.

• The $i$ is just a dummy index which for which one could use any symbol that was not used yet. It would be clumsy to use a subscript for $i$.