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I would like to understand how the creation and annihilation operators can be represented as matrices in the single-particle picture, rather than in Fock space.

Assume a single particle can occupy one of $N$ orthonormal states: $ |1\rangle, |2\rangle, \dots, |N\rangle. $

This defines an $N$-dimensional Hilbert space. My questions are:

Can we define creation $( a_i^\dagger )$ and annihilation $( a_i )$ operators in this space as $( N \times N )$ matrices? If yes, what is their explicit matrix form in this basis? What is their physical interpretation in this single-particle setting? How (if at all) does this relate to the usual ladder operators or the second-quantized formulation?

To be clear, I am not referring to the action of these operators on Fock space, where the number of particles can vary. I am only interested in how to define and interpret them in a finite-dimensional single-particle Hilbert space.

cconsta1
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M.J
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1 Answers1

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  1. The bosonic canonical commutation relation (CCR) $$[\hat{a}_i,\hat{a}_j^{\dagger}]~=~\delta_{ij}\hat{\bf 1}, \qquad i,j~\in~\{1,\ldots,N\}, \tag{1}$$ (and other CCRs vanish) does not have a finite-dimensional square matrix representation. Just try to take the trace on both sides of eq. (1), cf. e.g. this related Phys.SE post.

  2. The fermionic canonical anticommutation relation (CAR/Clifford algebra) $$\{\hat{a}_i,\hat{a}_j^{\dagger}\}_+~=~\delta_{ij}\hat{\bf 1}, \qquad i,j~\in~\{1,\ldots,N\}, \tag{2}$$
    (and other CARs vanish) does have a finite-dimensional square matrix representation, cf. e.g. this Phys.SE post.

Qmechanic
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