An operator $A$ is anti-Hermitian if it satisfies the following condition:
$$ A^\dagger = -A $$
where the dagger stands for the conjugate transpose.
This means that the Hermitian conjugate of the operator is equal to the negative of the operator itself.
Properties of anti-Hermitian operators:
1. Eigenvalues: The eigenvalues of an anti-Hermitian operator are purely imaginary or zero. This is because the Hermitian conjugate of an operator has the complex conjugate eigenvalues, and the negative of the operator has the negative eigenvalues. Since $A^\dagger = -A$, the eigenvalues must satisfy this relation.
2. They constitute the Lie algebra of the Lie group of unitary operators. (also in the infinite-dimensional case??)
3. Role in Quantum Mechanics: In quantum mechanics, anti-Hermitian operators are the generators of unitary operators, which describe the time evolution and symmetries in quantum systems. Specifically, the exponential of an anti-Hermitian operator is a unitary operator.
4. Physical Interpretation: When multiplied by $i$, they get converted into Hermitian operators, so they are related to observables.
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Author of the notes: Antonio J. Pan-Collantes
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