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It is commonly to define an antilinear or antiunitary operator $\Theta$ acting on the ket state $|\alpha\rangle$ such as $$\Theta|\alpha\rangle.$$

It is commonly avoid to pursue a definition an antilinear or antiunitary operator $\Theta$ acting on the bra state $\langle \beta|$ such as $$\langle \beta| \Theta.$$ See Modern Quantum Mechanics (2nd Edition) Chap 4.4 by J. J. Sakurai, Jim J. Napolitano commenting on this issue below in the image file.

My question is that could we in fact define antilinear or antiunitary operator $\Theta$ acting on the ket state $|\alpha\rangle$ and on the bra state $\langle \beta|$ consistently simultaneously? If you can or if you can not, how does it reveal the (1) antilinear or antiunitary operator, (2) inner product $\langle \beta|\alpha\rangle$ constraint, and (3) structure of Hilbert space?


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Without spelling any mathematical detail which gives foundation to the Dirac's braket formalism, we define the "sandwich braket" $\langle \beta\vert\Theta\vert\alpha \rangle$ in a proper way iff $\Theta$ is linear and selfadjoint.

Since linear unitary or anti-linear (anti) unitary operators are not self-adjoint, the sandwich braket is defined only in one direction, customarily chosen to the right (i.e. on the ket), because the braket is linear "to the right" and antilinear "to the left".

Using the braket formalism in the absence of selfadjoint and linear operators is one of the shortcomings of the Dirac's notation. People say that the advantages of using it are more important, but this is only a matter of opinion.

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  • $\begingroup$ "Using the braket formalism in the absence of selfadjoint and linear operators is one of the shortcomings of the Dirac's notation." --> What are other notations that you recommend? ;) $\endgroup$ Commented May 14, 2021 at 23:34
  • $\begingroup$ The mathematical ones $(a,b)$ or $\langle a,b \rangle$. This way, it is all of a sudden very important on which side of the comma the operator is placed :) $\endgroup$
    – DanielC
    Commented May 14, 2021 at 23:46

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