Is the Heisenberg picture propagating operators “backwards in time”?

A recent arXiv post ignited an interesting discussion with students and colleagues, demonstrating once more how the Heisenberg picture in quantum mechanics can easily be misunderstood to the point of becoming almost paradoxical. Here I intend to briefly summarize what I think may be the crux of the problem (or problems). The argument below follows a discussion on the topic that I had with Masanao Ozawa few years ago; however, any error or misunderstanding in it is to be entirely attributed to me.

One-step evolutions

Suppose that we are following the evolution of a quantum system from an initial time t=t_0 to a later time t=t_1\ge t_0 , and that the unitary operator evolving the state of the system is U(t_1,t_0), so that

\rho(t_1)=U(t_1,t_0) \rho(t_0) U(t_1,t_0)^\dagger.

The latter is called the Schrödinger picture of the evolution. In this picture, states evolve in time, while observables (like the Hamiltonian) do not.

The Heisenberg picture is meant to do the opposite: it keeps states “freezed”, while observables evolve. It can be also understood as a “pullback” operation: very much like when one looks at a rotation from the viewpoint of vectors (Schrödinger picture) or the viewpoint of the coordinate system (Heisenberg picture).

For the two pictures to give consistent predictions, that is, Tr[\rho(t_1)\ H(t_0)]=Tr[\rho(t_0)\ H(t_1)], it is prescribed that, if an observable at time t_0 is denoted as H(t_0), the same observable at the later time will be H(t_1)=U(t_1,t_0)^\dagger H(t_0) U(t_1,t_0). From this relation, we see that the state evolves according to U(t_1,t_0), while the observable evolves according to U(t_1,t_0)^\dagger .



It is quite tempting at this point to interpret this by saying that “states evolve forward in time, while observables evolve backwards in time”. If only two times are considered, that seems just a curious though innocuous way of phrasing it. Indeed I have heard a lot of researchers explaining the Heisenberg picture this way. I myself would have nodded my head hearing this some years ago. However, I now see why this interpretation can be in fact very confusing, potentially leading to wrong calculations, when more than two times are considered.

Two-step evolutions: the wrong approach

Imagine now to fix three instants in times, t_0\le t_1\le t_2 and two unitary operators:  one, U(t_1,t_0), describing the evolution of states from t_0 to t_1 as before; and another one, U(t_2,t_1), propagating states from t_1 to t_2 . The problem is: how should one model the evolution of an observable H from t_1 to t_2 ? A naive guess based on the “backwards-in-time evolution” intuition would suggest a scheme like the following:


But what should be the evolution operator describing the box denoted by question marks? As the arXiv post mentioned at the beginning of this post argues, one could be tempted to say that the right evolution operator is U(t_2,t_1)^\dagger, probably by symmetry with the Schrödinger’s branch evolving forward in time. This naive guess leads to the equation H(t_2)= U(t_2,t_1)^\dagger H(t_1) U(t_2,t_1) = U(t_2,t_1)^\dagger U(t_1,t_0)^\dagger H(t_0) U(t_1,t_0) U(t_2,t_1) .

Problem is, this is of course wrong! The correct thing to do is to understand that the total evolution of the state from t_0 to t_2 is given by the unitary operator U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0). Consequently, one has that

H(t_2)= U(t_1,t_0)^\dagger U(t_2,t_1)^\dagger H(t_0) U(t_2,t_1) U(t_1,t_0).

This is the correct description of H(t_2) in the Heisenberg picture.

Another, more subtle, source of confusion

We have seen how the naive “backwards in time” interpretation is wrong. However, at this point, another structure emerges that still suggests some kind of “time-reversal”. I am speaking now of the fact that, in the correct equation, that is, H(t_2)= U(t_1,t_0)^\dagger U(t_2,t_1)^\dagger H(t_0) U(t_2,t_1) U(t_1,t_0), the order of the propagators is reversed with respect to the one that is used for states, that is \rho(t_2)= U(t_2,t_1) U(t_1,t_0) \rho(t_0)U(t_1,t_0)^\dagger U(t_2,t_1)^\dagger.

Given that the equation itself is correct, in what follows I am simply criticizing its interpretation. I would like to argue, in particular, that, even though the evolution operators act in reverse order on the observable, the Heisenberg picture should not (or, at least, need not) be interpreted or explained as “backwards in time” evolution.

The point is that U(t_2,t_1), on its own, has no meaning in the Heisenberg picture. In the Heisenberg picture, all operators must be evolved consistently. In particular, the operator U(t_2,t_1), which is defined formally at t=t_0 , when applied at time t_1 , must also be consistently evolved before being applied on anything. (Better said, the Hamiltonian generating the unitary evolves in time and, with it, the unitary operator it generates.)

Hence, in the Heisenberg picture, the propagator of observables from t_1 to t_2 is not U(t_2,t_1)^\dagger but its evolved version, that is,

\tilde U(t_2,t_1)^\dagger=U(t_1,t_0)^\dagger  U(t_2,t_1)^\dagger U(t_1,t_0).

If we substitute this into the initial formula, then we indeed obtain that H(t_2)=\tilde U(t_2,t_1)^\dagger H(t_1) \tilde U(t_2,t_1)=U(t_1,t_0)^\dagger  U(t_2,t_1)^\dagger H(t_0) U(t_2,t_1) U(t_1,t_0) , as it was computed at point 2 above.


However, once written as above, it gives us a very clear understanding of what is going on in the Heisenberg picture.

Summarizing, the Heisenberg picture is indeed a pullback transformation, but a pullback that happens forward in time. After all, both Heisenberg and Schrödinger pictures provide equivalent representations of exactly the same process, which of course happens forward in time.

Birkhoff-von Neumann Prize

I was delighted to learn that I was awarded with the “Birkhoff-von Neumann Prize” by the International Quantum Structures Association. I feel very honored and humbled — at once! — to join a list including such superb colleagues. Thank you very much!

Francesco Buscemi is Associate Professor at the Department of Mathematical Informatics of Nagoya University, Japan. His results solved some long-standing open problems in the foundations of quantum physics, using ideas from mathematical statistics and information theory. He established, in a series of single-authored papers, the theory of quantum statistical morphisms and quantum statistical comparison, generalizing to the noncommutative setting some fundamental results in mathematical statistics dating back to works of David Blackwell and Lucien Le Cam. In particular, Prof. Buscemi successfully applied his theory to construct the framework of “semiquantum nonlocal games,” which extend Bell tests and are now widely used in theory and experiments to certify, in a measurement device-independent way, the presence of non-classical correlations in space and time.

In such an occasion, it is impossible not to remember Professor Paul Busch, gentleman scientist, President of IQSA until his sudden death, of which I learned almost simultaneously with my award.

Popper against the ideas of dignity, wholeness, real truth, and essentiality in science


Thus I freely admit that in arriving at my proposals I have been guided, in the last analysis, by value judgments and predilections. But I hope that my proposals may be acceptable to those who value not only logical rigour but also freedom from dogmatism; who seek practical applicability, but are even more attracted by the adventure of science, and by discoveries which again and again confront us with new and unexpected questions, challenging us to try out new and hitherto undreamt-of answers.

Karl Popper, The Logic of Scientific Discovery. 2nd Edition (Routledge, 1999), p.38.

Saint Lucy’s Day: the longest night of the year

In Piacenza, the city where I was born, children receive their presents not for Christmas, but on Saint Lucy’s day (Santa Lucia). They say that Saint Lucy’s Day, the 13th day of December, has the longest night of the year:

“Santa Lucia, la notte più lunga che ci sia.”

This, in a way, makes perfect sense, as the Saint Little Girl needs time to go around, house by house, delivering the presents.

However, everyone knows that the longest night of the year (for the Boreal Hemisphere) is not the one between the 12th the 13th of December, but that between the 21st and the 22nd, aka the Winter Solstice! That’s why I always assumed that such a saying was to be meant as a sort of `poetic license,’ possibly suggested by the rhyming words `Lucia‘ and `sia.’

Today, however, I discovered another interesting, plausible reason for the saying. The discrepancy between Saint Lucy’s day and the Winter Solstice could also be due to the introduction of  the Gregorian calendar, in AD1582, which shifted the calendar back of ten~ish days. And so, everything fits together again: how nice!

Happy Saint Lucy’s Day!

Edit 2014-12-16: Richard Gill on Google+ points out that, as a matter of fact, the current St Lucy’s Day has the earliest (though not the longest) night.