Is there a difference between the adjoint and conjugate? I have recently started some work for a quantum field theory module and I'm wondering if there is a difference between the adjoint or conjugate of an equation such as the Dirac equation.

  • $\begingroup$ complex conjugate and transposed versus just conjugate? $\endgroup$ – Žarko Tomičić May 15 '16 at 9:16
  • $\begingroup$ I'm unsure, my lecture notes just refer to the conjugate of the Dirac equation and also its adjoint. $\endgroup$ – bidby May 15 '16 at 9:24

The adjoint of an operator is obtained by taking the complex conjugate of the operator followed by transposing it.

i.e., $(A)^\dagger_{ij}=\left((A)^T_{ij}\right)^*=\left((A_{ij})^*\right)^T=A_{ji}^*$

You can do it in any order. The adjoint of an operator is the infinite dimensional generalization of conjugate transpose, where you find the transpose of an operator (in matrix form this is done by $A_{ij}^T=A_{ji}$) and then take the complex conjugate of it.

Now, the complex conjugate of an operator is obtained by reversing the sign of the imaginary term in the operator representation (or in it's Matrix form). It's like finding the conjugate of the complex number $a+ib$ as $a-ib$. The complex conjugate of an operator $A$ in this discussion is represented as $A^*$ (some textbooks show other notations also).

In quantum mechanics, the operators corresponding to (real) physical state are Hermitian in nature since the eigen values corresponding to the eigen kets of a Hermitian operator are real. Hermitian operators are called self-adjoint operators since the operator and it's adjoint are the same. i.e., if $A$ is Hermitian, then $A^\dagger=A$. Now, in a complex vector space, the adjoint and complex conjugate of an operator has the same meaning.


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