# Interpretation of the Heisenberg picture in QM

#### atyy

Again! A particle is not (and in no picture of time evolution whatsoever) a Hilbert-space vector and its observables are no self-adjoint operators on the Hilbert space. There are real things in the lab called measurement apparati and a particle source but no mathematical abstract objects. The link to associate the mathematical abstract objects of the theory and the real-world equipment is Born's Rule, associating the modulus squared of a wave function with the probabilities and/or probability densities of the occurance of measurement results, and these probailities (and derived quantities from them like expectation values of observables or scattering cross sections and the like) are independent of the picture chosen. Any attempt to associate real-world equipment directly with the abstract objects of the QT formalism leads to confusion and are thus of no help!
Can you show an actual error if such an association is made?

#### vanhees71

Gold Member
I can't not even imagine how you'd make such and association, let allone which possible errors this might imply. Since physics should be independent of the representation, particularly the choice of the time-evolution picture, the only thing you could somehow think to associate with the particle (or any other entity under consideration) is the probability distribution of observables, given by the statistical operator of the system (determined by an appropriate preparation procedure), but then you run into the well-known problems which lead Born to his probabilistic interpretation of the wave function in the first place.

#### atyy

I can't not even imagine how you'd make such and association, let allone which possible errors this might imply. Since physics should be independent of the representation, particularly the choice of the time-evolution picture, the only thing you could somehow think to associate with the particle (or any other entity under consideration) is the probability distribution of observables, given by the statistical operator of the system (determined by an appropriate preparation procedure), but then you run into the well-known problems which lead Born to his probabilistic interpretation of the wave function in the first place.
Well, one just adds "FAPP". First, we make a classical/quantum cut, so we already say the wave function is not necessarily real. But once we have done that, I don't see any problem with saying that the observable to represent the apparatus FAPP, and the wave function to represents the system FAPP.

It's the same as simultaneity in classical special relativity. Simultaneity is not absolute, but it is absolute relative to a family of observers. So having chosen the family of observers, we can treat simultaneity as real FAPP - this merely involves using Newtonian-like intuition in a particular inertial coordinate system.

Similarly, the wave function can be taken to be real relative to a particular Heisenberg cut.

#### Sonderval

@Jilang
So consider the particle as the operator rather than the state.
O.k., I'm happy with that.
What I'm unhappy with is that in the standard phrasing the word "state" is used although the state does not fully describe the system at any point in time (except at the initial point). As far as I'm aware, everywhere else in physics (classical mechanics, thermodynamics), a "state" of a system describes the system at a certain time in point.
So I find it more clear to say that the "state" meant in the Heisenberg picture is the "initial state" (or initial condition), that the observables (operators) evolve (and the nice thing about the heisenberg picture is that they evolve independently of the initial state, i.e., I can calculate the evolution of teh ooperators once and for all and then apply them to different initial states) and that to actually calculate any measurable quantity I of course need to apply the operator to the initial state. Would you say that this interpretation/phrasing is incorrect?
PS: I do agree that the association of the observable with the measurement apparatus only was highly problematic (if not so say, wrong...)

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#### atyy

@vanhees71,all I mean is that in physics we do indeed have a intuitive notion of reality, so I agree with you that only that level is "truly real". But I think it is good to have the option to have some fake realities that help our intuition, which anyway may turn out to be real. For example, if it turns out that Lorentz invariance is broken at high enough energy, then although within special relativity, the absolute frame is observer dependent, in the new theory, the absolute frame could be real and reproduce all predictions that made us consider special relativity to be a superb theory.

Also, from mathematics and physics, we usually assume the natural numbers to "exist", whatever that means. But in fact all our prediction to date can be made with computers with finite memory, so it isn't obvious that we really need a theory in which infinitely large integers are possible. Maybe we could live with a theory with a largest integer (100^100^100^100). So in this sense the integers may not exist, but it's ok for us to act as if they do exist FAPP.

#### vanhees71

Gold Member
Well, but #29 just before your posting shows the confusion coming out of the idea to (wrongly) associate abstract mathematical entities of the theory with the real things in the lab. As a theoretician myself, I feel, it's quite important to look at a real lab from time to time and how "quantum things" are prepared and observed.

#### atyy

Well, but #29 just before your posting shows the confusion coming out of the idea to (wrongly) associate abstract mathematical entities of the theory with the real things in the lab. As a theoretician myself, I feel, it's quite important to look at a real lab from time to time and how "quantum things" are prepared and observed.
But that is in the Heisenberg picture, I think the OP does understand how to apply the Schroedinger and Heisenberg pictures to experiment, and also how to "interpret" the Schroedinger picture in a naive but helpful way in which the state is real FAPP.

I think he is only asking whether the Heisenberg picture also has a naive interpretation which is also helpful to the intuition. (As far as I know, it doesn't. If it does it is something like the system is unchanged, but what you consider a position measurement changes - but I think most people will not consider that helpful. Mathematically, of course the time evolution is just a rotation, and it doesn't matter if the system rotates and you keep still, or you rotate and the system keeps still.)

#### vanhees71

Gold Member
There is no difference in interpretation depending on the picture of time evolution used. So how can it make sense to associate the one or other mathematical element of the theory with the one or other physics piece of reality?

#### atyy

There is no difference in interpretation depending on the picture of time evolution used. So how can it make sense to associate the one or other mathematical element of the theory with the one or other physics piece of reality?
It depends on what one means by interpretation. If in Copenhagen, one is working at the level of true reality, then there is no difference in interpretation. But if in Copenhagen, one is working at the level of FAPP reality, then the state is the state in the sense of classical physics - it is the complete specification of of an individual system within the theory. And if one is trying to solve the measurement problem, then so far the Schroedinger picture seems privileged, since Bohmian Mechanics and Many-Worlds both privilege the Schroedinger picture.

It is a bit like in classical mechanics it doesn't matter whether we use Newtonian, Lagrangian or Hamiltonian formulations. However, in quantum mechanics, the Hamiltonian formulation is privileged. Even if one were to argue for the Lagrangian formulation via path integrals, the interpretation is dramatically changed, since the idea that the particle takes an extremal path is no longer true.

#### Jilang

@Jilang

O.k., I'm happy with that.
What I'm unhappy with is that in the standard phrasing the word "state" is used although the state does not fully describe the system at any point in time (except at the initial point). As far as I'm aware, everywhere else in physics (classical mechanics, thermodynamics), a "state" of a system describes the system at a certain time in point.
So I find it more clear to say that the "state" meant in the Heisenberg picture is the "initial state" (or initial condition), that the observables (operators) evolve (and the nice thing about the heisenberg picture is that they evolve independently of the initial state, i.e., I can calculate the evolution of teh ooperators once and for all and then apply them to different initial states) and that to actually calculate any measurable quantity I of course need to apply the operator to the initial state. Would you say that this interpretation/phrasing is incorrect?
PS: I do agree that the association of the observable with the measurement apparatus only was highly problematic (if not so say, wrong...)
I think you are getting there (although you are still perhaps subconsciously identifying the state with the particle!), however it is misleading to identify observables with operators in any picture (as Vanhees pointed out). Obervables arise as a function of both the operator and the state. If the interpretation works for you, stick with it. It's only maths.

#### atyy

I can calculate the evolution of teh ooperators once and for all and then apply them to different initial states) and that to actually calculate any measurable quantity I of course need to apply the operator to the initial state. Would you say that this interpretation/phrasing is incorrect?
If I understand what you mean by "once and for all", this isn't right. After a measurement, you still have to collapse either the wave function or the observables.

If you want a very shut up and calculate formula, you can try Eq 37 in Laloe's http://arxiv.org/abs/quant-ph/0209123. But the classical/quantum cut is fundamental in the traditional formulation of quantum mechanics, and it is natural to identify the state with the quantum side, and the operators with the classical side. But as with all observations, the physics is relative, and it is only the relative that is absolute (for example, the universe is absolutely expanding relative to a family of observers) , so one can imagine that the system is evolving relative to apparatus, or that the apparatus is evolving relative to the system.

More in this spirit is given in section 1.3 of Wiseman and Milburn's https://books.google.com/books?id=ZNjvHaH8qA4C&printsec=frontcover#v=onepage&q&f=false.

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#### vanhees71

Gold Member
where "collapse" simply means to adapt your state to the gained knowledge from the measurement. It's rather a preparation than a measurement procedure.

#### atyy

where "collapse" simply means to adapt your state to the gained knowledge from the measurement. It's rather a preparation than a measurement procedure.
I would rather say that measurement is a means of state preparation.

#### Sonderval

@Jilang
Thanks for the encouragement.

@atty
The collapse itself is not part of either the Schrödinger or the Heisenberg time evolution, so I don't really think this argument applies (It's what Penrose calls the R-operation and it's not unitary). For calculating expectation values etc. no collapse is needed.

#### atyy

@atty
The collapse itself is not part of either the Schrödinger or the Heisenberg time evolution, so I don't really think this argument applies (It's what Penrose calls the R-operation and it's not unitary). For calculating expectation values etc. no collapse is needed.
The collapse is needed in order to calculate correlations (which are expectation values) between sequential measurement outcomes. For example, it is needed in the Bell test experiments.