# Does quantum mechanics only deal with Hermitian/self-adjoint operators?

Whatever matrix I am seeing in quantum mechanics are all Hermitian matrices. We are using their eigenvalues for different types of work. Fortunately all their eigenvalues are real. Have you ever seen any use of eigenvalues of real non-symmetric or non-Hermitian matrix in quantum mechanics? What are they? Is there any such? Please include some references.

• The raising and lowering operators are two very prominent non-Hermitian matrices. – Physics_Plasma Jan 13 '15 at 16:55
• I also think this and the links therein are relevant. – jabirali Jan 13 '15 at 16:58

If you want to describe the way in which a mixed state transforms, this in generally expressed using non-hermitian operators, either the Lindbladian in the continous case, or e.g. the Kraus representation of trace preserving completely positive maps in the discrete case.

• Can you elaborate more? – Dutta Jan 13 '15 at 17:16
• Have you read the wikipedia links? If yes, what do you want to know? – Norbert Schuch Jan 13 '15 at 17:59
• How important the eigenvalues of Kraus operator is? – Dutta Jan 13 '15 at 18:14
• What is important are the eigenvalues of the matrix representation $\sum A_i\otimes \bar{A_i}$ of the completely positive map (with $A_i$ the matrices in the Kraus representation), or the eigenvalues of the Lindbladian: They tell us e.g. how quickly a system thermalizes. --- I think it would be helpful for giving a precise answer if you could explain what exactly you would like to understand. – Norbert Schuch Jan 13 '15 at 18:19

Unitary matrices are (almost) never Hermitian, hence no finite version of the infinitesimal transformations described by the Hermitian observables will ever be Hermitian itself.

The most prominent example is the time evolution $\mathcal{U}(t) = \mathrm{e}^{\mathrm{i}Ht}$, which is unitary, and not Hermitian.

• Anything else than that? – Dutta Jan 13 '15 at 16:58
• @Dutta: Loads: Every finite symmetry transformation. Creation and annihilation operators. Don't be fooled into thinking that only the Hermitian operators matter - they are the observables, but the non-Hermitian ones are not unimportant because of that. – ACuriousMind Jan 13 '15 at 17:02
• I do not know them till now. I shall study. Have you seen any application of their eigenvalues? – Dutta Jan 13 '15 at 17:10
• @Dutta: Of course - for example, the time evolution tells you that each energy eigensate evolves only with a phase $\mathrm{e}^{\mathrm{i}Et}$ in time, and hence doesn't change its state in time at all. – ACuriousMind Jan 13 '15 at 17:28

Algebraic Quantum Field Theory starts from the assumption that a quantum mechanical system can be described by a C*-algebra. But not every element in the C*-algebra has a physical interpretation, but just the self-adjoint part, which has the structure of a Jordan algebra through the product $$a\circ b = \frac{(a+b)^2 - a^2 - b^2}2.$$ The matrices you see arise from a representation of said C*-algebra, and of course representations of self-adjoint elements are self-adjoint (or "hermitian") matrices. Any other matrix has no "definite" physical meaning as something that you can measure.