# How does an operator transform under time reversal?

We know that a time-reversal operator $T$ can be represented as $$T=UK$$ where $U$ is some unitary operator and $K$ is the complex conjugation operator.

Then under time-reversal operation, a quantum state $|\psi\rangle$ will transform as the following: $$|\psi^R\rangle=T|\psi\rangle=UK|\psi\rangle=U|\psi^*\rangle$$ If we require time-reversal symmetry to the system, then we need to have $$\langle\psi^R|O^R|\phi^R\rangle=\langle\psi|O|\phi\rangle$$ where $|\psi\rangle$ and $|\phi\rangle$ are some arbitrary quantum states and $O$ is some operator. From the above equation, we would have $$\langle\psi^*|U^{\dagger}O^RU|\phi^*\rangle=\langle\psi|O|\phi\rangle$$ So based on this equation, how do we obtain the result (given in the book "random matrices" by Mehta) that $$O^R=UO^TU^{\dagger}$$ where $O^T$ means the transpose of $O$.

My second question is that, what if we do NOT assume time reversal symmetry?

• I am reading your question, but I have a doubt. Time reversal is not only complex conjugate, what it has to do is also to transpose the items on which it acts (vectors, matrices). Dec 3, 2014 at 22:33
• Also, for reference \langle and \rangle produce nice looking brackets, $\langle \varphi | \psi \rangle$ instead of $< \varphi | \psi >$, if you care about that sort of thing.
– user12029
Dec 4, 2014 at 0:17
• @Sofia also I don't think time reversal operator needs to transpose the operand Dec 4, 2014 at 7:00

One problem with your formula that $$T$$ factorizes as the product of a unitary operator $$U$$ and complex conjugation $$K$$ is that 'complex conjugation' is meaningless in a Hilbert space a priori.

Let me be more formal about this; consider a vector $$\psi \in H$$, with $$H$$ a $$n$$-dimensional Hilbert space, that is, isomorphic to $$\mathbb{C}^n$$. How is an (abstract) vector space isomorphic to the canonical Hilbert space $$\mathbb{C}^n$$? In the following way: pick a basis $$\{e_i\}_{i=1,...,n}$$ of $$H$$. Then decompose $$\psi$$ in this basis:

$$\psi = \sum_i \psi_i e_i \ .$$

Finally map $$\psi$$ on the $$n$$-tupel $$(\psi_1,...,\psi_n)$$.

As you can see, this isomorphism, lets denote it by $$E$$, very strongly depends on the chosen basis.

Let's ignore this anyway. On $$\mathbb{C}^n$$ we can define complex conjugation $$K$$, it is simply the map

$$K(\psi_1,...,\psi_n) = (\overline{\psi_1},...,\overline{\psi_n}) \ .$$

Hence we can define a complex conjugation $$K_E$$ on $$H$$ simply by

$$K_E = E^{-1} \circ K \circ E \ .$$

Now let's see what happens if we change a basis; consider another basis $$E' = \{e'_i\}_{i=1,...,n}$$ with $$e_i = \sum_j M_{ij} e'_j$$. Then considering $$M_{ij}$$ as the matrix of an operation on $$\mathbb{C}^n$$:

$$K_{E} = E^{-1} \circ K \circ E = E'^{-1} \circ M^{-1} \circ K \circ M \circ E' = E'^{-1} \circ \overline{M}M^{-1} K \circ E' \ ,$$

i.e. unless $$\overline{M} = M$$, $$K_E \neq K_{E'}$$. Hence there is no invariant notion of complex conjugation in a complex Hilbert space.

The statement is true that if we fix a basis $$E$$, we may write complex conjugation w.r.t. any other basis as

$$K_{E'} = U_{E,E'} K_E \ ,$$

where $$U_{E,E'}$$ is a unitary. Note that complex conjugation depends on $$E$$ only through the equivalence class $$[E]$$ of basis connected to $$E$$ through real transformations.

In this language one could say that a time-reversal operation is a choice of preferred equivalence class of basis. That is, at least, if $$T^2 = 1$$. If $$T^2 = -1$$, we should map $$H$$ to a $$\mathbb{H}^{n/2}$$, where $$\mathbb{H}$$ are the quaternions.

This leaves you with a recipe for computing the time-reversed operator, for $$T^2=1$$: simply represent $$O$$ in a real basis (a basis invariant under $$T$$), then take the complex conjugate of this matrix.

Notice that neither of these manipulations depend on $$T$$ being a symmetry.

Time reversal is not only complex conjugate, what it does is also to transpose the items on which it acts (vectors, matrices).

$$T\langle \phi|\hat{O}|\psi\rangle = \langle \psi T|\hat{O}|T \phi\rangle.$$

Notice the change of places of the functions in the right wing with respect to the left wing. Also, I used the fact that $\hat{O}$ is unchanged at time-reversal.

Now we do the following change which is allowed under the integral if the two functions vanish at infinity:

$$\langle UK\psi|\hat{O}|UK\phi\rangle = \langle \phi|U\hat{O}U^\dagger|\psi\rangle.$$

So, we got the time-reversed of $\hat{O}$.

• I don't think we can assume that the operator is unchanged under time-reversal Dec 4, 2014 at 6:55
• @Timo: you may think or not, but this is the assumption in the text of the question, see the 3rd equation. Dec 4, 2014 at 21:32
• I'm sorry but your argument seems contradictory. We are supposed to find out how the operator transform under time-reversal, but you assumed that $O$ is unchanged under time-reversal. My assumption is that that the inner product is invariant instead of the operator. Dec 5, 2014 at 2:19
• @Timo: when reversing the time, we see that the particles reverse their movement. One starts from the final state, (we introduce the change ⟨ϕ|†), we apply on it Ô†, and as the exercise hints, the vector obtained should be now projected on the initial state (which therefore should appear as |ψ⟩†. (I am not sure which are the initial and final states, but if I trust the LHS of the 3rd equality, i.e. that he took the transposed of the product of components, it seems that |ψ⟩ is the initial state. But, let me return to you some later. Dec 5, 2014 at 10:39
• @Timo: please see why I thought that Ô is a symmetrical operator. Before the 3rd equality in the question, is written "if we require time-reversal symmetry, ..." Dec 5, 2014 at 14:17

I am going to use the notation used by Mehta instead of using Dirac brackets because I think it makes understanding the equations easier.

\begin{align} (\Phi, A\Psi) =& (\Psi^{R}, A^{R}\Phi^{R})\\ =& (T\Psi, A^{R}T\Phi)\\ =& ((A^{R})^{\dagger}T\Psi, T\Phi)\\ =& (T^{\dagger}(A^{R})^{\dagger}T\Psi, \Phi)^{*}\\ =& (\Phi, T^{\dagger}(A^{R})^{\dagger}T\Psi) \end{align}

Third and fourth lines are considering that $$A^{R}$$ is not an antiunitary operator, so its adjoint is taken normally, while since $$T$$ is an antiunitary operator so it follows: $$(x, Ty)=(T^{\dagger}x,y)^{*}$$.

Now the last equation implies that: $$A=T^{\dagger}(A^{R})^{\dagger}T \Rightarrow A^{\dagger}=T^{\dagger}A^{R}T$$ which implies: \begin{align} A^{R} =& (T^{\dagger})^{-1}A^{\dagger}T^{-1}\\ =& (T^{-1})^{\dagger}A^{\dagger}T^{-1}\\ =& (KU^{\dagger})^{\dagger}A^{\dagger}T^{-1}\\ =& UK^{\dagger}A^{\dagger}KU^{-1} \end{align}

where I have used the relations: $$T=UK\Rightarrow$$ \begin{align} T^{-1}=& K^{-1}U^{-1}\\ =& KU^{-1}\\ =& KU^{\dagger} \end{align}

Now I need to prove that $$K^{\dagger}A^{\dagger}K=A^{T}$$.

For complex conjugation operator $$K$$ we have the relation: $$O^{*}=KOK^{-1}$$ for any operator $$O$$, for example here, https://www.its.caltech.edu/~xcchen/img/Ph129b2020/lecture/lecture0312.pdf . This implies: \begin{align} O^{*}=& K^{-1}OK^{-1}\\ =& K^{-1}KK^{\dagger}OK^{-1}\\ =& K^{\dagger}OK^{-1}. \end{align}

Now if we replace $$O=A^{*}\Rightarrow O^{*}=A$$ which implies: \begin{align} &A=K^{\dagger}A^{*}K^{-1}\\ \Rightarrow &AK=K^{\dagger}A^{*}\\ \Rightarrow &(AK)^{\dagger}=(K^{\dagger}A^{*})^{\dagger}=(A^{*})^{\dagger}K=A^{T}K\\ \Rightarrow & K^{\dagger}A^{\dagger}=A^{T}K\\ \Rightarrow & A^{T}=K^{\dagger}A^{\dagger}K^{-1}\\ \Rightarrow & A^{T}=K^{\dagger}A^{\dagger}K. \end{align}

Edit: You can find a much better explanation in Appendix A of the book written by Kota https://books.google.co.in/books/about/Embedded_Random_Matrix_Ensembles_in_Quan.html?id=BLK5BQAAQBAJ&redir_esc=y.