Exploring the Connection Between Quantum Mechanics and Quantum Field Theory

[Demystifier]

Also there is no difference between the Schrödinger and the Heisenberg picture (at least not as far as I'm aware of, because I've not heard about problems like with the interaction picture in the case of relativistic QFT, where the latter strictly speaking doesn't exist due to Haag's theorem). It's just two equivalent mathematical descriptions of the same theory. They are just related by a unitary time-dependent transformation, and observables (including correlation functions of gauge invariant observables) thus do not depend on which picture you use to evaluate them.

As long as you only study unitary evolution of matrix elements, there is no difference between the Schrödinger and the Heisenberg picture. But when you attempt to say something more specific about the measurement problem, then, depending on the interpretation you use, some subtle differences between the two pictures may occur.

 

[stevendaryl]

Indeed they say something new [compared to QM foundations, but very old in terms of the physics] on measurement, and they don't miss the opportunity to say it!

In books on nonequilibrium statistical mechanics it is very obvious that whatever they compare with experiment has nothing at all to do with the kind of idealized measurements considered in QM. They talk about field expectations (such as the energy density and mass density) and certain coefficients in the formula for the state of the macroscopic system (such as local temperature and local chemical potential), and relate them to thermodynamic observables, which are measured in the ordinary engineering way. If mentioned at all, Born's rule with its assumption of external measurement is eliminated in the very first few pages of the books in favor of the formula $\langle X\rangle = \mbox{tr}~\rho X$. This formula is a much more general and much more useful axiom for quantum physics! It doesn't have the problematic baggage that the traditional, ill-defined foundations of QM have.

I don't see how the treatment of measurement by QFT is any different, conceptually, than the treatment in QM. In QFT, the field is an operator, and we can get a statistical interpretation by considering expectation values of the field. How is that different (conceptually) from saying, in non-relativistic QM, that position $\hat{x}$ is an operator, and we can get a statistical interpretation by considering expectation values of $x$? That's fine as far as it goes, but in NRQM, there are other operators, as well, such as $\hat{p}$ and various combinations of $\hat{p}$ and $\hat{x}$. We can't simultaneously give a statistical interpretation to all such operators (that would violate the Kochen-Specker theorem), so we have to limit our statistical interpretation to those variables that are actually measured in an experiment. That's how measurement comes in.

I don't see how the situation is any better in QFT. We similarly have incompatible field operators (in scalar field theory, $\hat{\phi}$ and the conjugate field momentum $\hat{\pi}$, for example).

 

[stevendaryl]

In the Schroedinger picture, which is the basis of the usual axiomatization of QM, this object doesn't exist.

Such quantities as $\langle A(t_1) A(t_2) \rangle$ can be computed in the Schrodinger picture: It's just

$\langle \psi| e^{i H t_1} A e^{i H (t_2 - t_1)} A e^{-i H t_2}|\psi\rangle$

 

[stevendaryl]

As long as you only study unitary evolution of matrix elements, there is no difference between the Schrödinger and the Heisenberg picture. But when you attempt to say something more specific about the measurement problem, then, depending on the interpretation you use, some subtle differences between the two pictures may occur.

There is something especially nice about the Heisenberg picture in relativistic quantum field theory, though. In the Schrodinger picture, the state of the universe is described by a wave function, which is an amplitude function on configuration space (configuration of fields), rather than a function in physical 4-D spacetime. So it's hard to understand what it would even mean for QFT to be "local", since the states don't exist in the physical world. In the Heisenberg picture, however, the equations of motion describe the field operators, which are (or can be, if you choose a position basis) localized operators existing in each point in space. They evolve in a purely local way, affected only by other operators in the neighborhood. So it's clear that the field operators are local. There is still a wave function, or state, in the Heisenberg picture, and it's as nonlocal (or "a-local"--the word "local" doesn't even apply to it) as the wave function in the Schrodinger picture. But the state in the Heisenberg picture is constant. It doesn't evolve. So who cares whether it's local or not?

 

[Demystifier]

Such quantities as $\langle A(t_1) A(t_2) \rangle$ can be computed in the Schrodinger picture: It's just

$\langle \psi| e^{i H t_1} A e^{i H (t_2 - t_1)} A e^{-i H t_2}|\psi\rangle$

I think he might want to give an ontological status to either $A(t)$ or $|\psi(t)\rangle$, but not to both. From such an ontological point of view, which may be relevant in the context of measurement problem, the two pictures are not equivalent.

 

[atyy]

There is something especially nice about the Heisenberg picture in relativistic quantum field theory, though. In the Schrodinger picture, the state of the universe is described by a wave function, which is an amplitude function on configuration space (configuration of fields), rather than a function in physical 4-D spacetime. So it's hard to understand what it would even mean for QFT to be "local", since the states don't exist in the physical world. In the Heisenberg picture, however, the equations of motion describe the field operators, which are (or can be, if you choose a position basis) localized operators existing in each point in space. They evolve in a purely local way, affected only by other operators in the neighborhood. So it's clear that the field operators are local. There is still a wave function, or state, in the Heisenberg picture, and it's as nonlocal (or "a-local"--the word "local" doesn't even apply to it) as the wave function in the Schrodinger picture. But the state in the Heisenberg picture is constant. It doesn't evolve. So who cares whether it's local or not?

Don't we still have to collapse the operators when a measurement is done?

 

[stevendaryl]

Don't we still have to collapse the operators when a measurement is done?

Not if we go the Many-Worlds route.

 

[atyy]

Not if we go the Many-Worlds route.

I don't think that's possible in the Heisenberg picture.

 

[Demystifier]

I don't think that's possible in the Heisenberg picture.

Good point! More generally, in the realm of interpretations the choice of the picture matters a lot.

 

[stevendaryl]

I don't think that's possible in the Heisenberg picture.

Why not? I haven't actually tried to develop a Many Worlds theory in the Heisenberg picture (I'm not 100% sure I understand it in the Schrodinger picture, either), but it seems to me that you could take the wave function in Many-Worlds and distribute the information to the field operators in a Heisenberg picture, which would give an equivalent description of the same state.

 

[A. Neumaier]

Then you should have said that at the beginning, to avoid all the misunderstandings that this non-standard terminology caused.

I couldn't do this in the beginning, as I found it out only during the discussion. I wouldn't spend so much time in discussing these things here if everything were already crystal clear in my mind. It is being clarified through the attempt to express myself clearly and seeing through the responses how well I succeeded.

 

[A. Neumaier]

In QFT, the field is an operator, and we can get a statistical interpretation by considering expectation values of the field. How is that different (conceptually) from saying, in non-relativistic QM, that position ^xx^\hat{x} is an operator, and we can get a statistical interpretation by considering expectation values of xxx?

It would not be so different if one would consider the expectation of operators in QM as something measurable to a certain accuracy, which is how the field expectations are interpreted in statistical mechanics. But in the QM foundations, measuring is something completely different! There one measures by collapse to an eigenstate (or its statistical version), which is completely foreign to measurement in statistical mechanics.

This is why the formal structure of Hilbert spaces and states is similar in QFT and QM, but the ontology is not.

 

[A. Neumaier]

In all cases the Born rule is used to associate formal quantities with real-world observables.

In all cases?

Then please explain for the following two explicit examples, the first from relativistic QFT, the second from nonrelativistic statistical mechanics:

• (i) How is the Born rule used to associate poles of the renormalized propagators with observable masses?
• (ii) How is the Born rule used in case of a real-world observation of temperature of a bucket of water?

 

[atyy]

Why not? I haven't actually tried to develop a Many Worlds theory in the Heisenberg picture (I'm not 100% sure I understand it in the Schrodinger picture, either), but it seems to me that you could take the wave function in Many-Worlds and distribute the information to the field operators in a Heisenberg picture, which would give an equivalent description of the same state.

I'm not sure MWI in Schroedinger works either. In the Heisenberg picture, one would have all possible observables evolving in time, including the simultaneous evolution of non-commuting observables. In MWI one has to (roughly) pick a preferred basis, and then let those branch. Picking a preferred basis in the Heisenberg picture would be like choosing a set of commuting observables. Since in a number versions of MWI decoherence picks the preferred basis, maybe we can avoid the difficulties of MWI by trying to discuss:

Can decoherence be formulated in the Heisenberg picture?

 

[A. Neumaier]

Such quantities as $\langle A(t_1) A(t_2) \rangle$ can be computed in the Schrodinger picture: It's just

$\langle \psi| e^{i H t_1} A e^{i H (t_2 - t_1)} A e^{-i H t_2}|\psi\rangle$

This is just the Heisenberg picture.

In terms of the Schroedinger picture this is a meaningless mess, evaluated for the state at time $t=0$. Given only the conventional axioms of QM, one can dream up this expression. But one cannot interpret it as something related to measurements at times $t_1$ and $t_2$ without silently leaving the interpretation framework defined by the axioms.

 

[stevendaryl]

I'm not sure MWI in Schroedinger works either. In the Heisenberg picture, one would have all possible observables evolving in time, including the simultaneous evolution of non-commuting observables. In MWI one has to (roughly) pick a preferred basis, and then let those branch.

I always thought that the description of MWI as different "branches" is just a subjective interpretation. In MWI, there is just the universal wave function, evolving smoothly, and we are free to think of it as a superposition of "possible worlds", but that's not inherent.

 

[A. Neumaier]

Can decoherence be formulated in the Heisenberg picture?

I think you should do this in an independent threat, not to overload this one.

 

[atyy]

I always thought that the description of MWI as different "branches" is just a subjective interpretation. In MWI, there is just the universal wave function, evolving smoothly, and we are free to think of it as a superposition of "possible worlds", but that's not inherent.

Yes, or at least that's my understanding of Wallace's approach. That's why I put in "roughly" in my statements. That was just the motivation for getting to rephrasing the question in a more technical way:

Can decoherence be formulated in the Heisenberg picture?

 

[atyy]

I think you should do this in an independent threat, not to overload this one.

Yes, will do that.

 

[vanhees71]

As long as you only study unitary evolution of matrix elements, there is no difference between the Schrödinger and the Heisenberg picture. But when you attempt to say something more specific about the measurement problem, then, depending on the interpretation you use, some subtle differences between the two pictures may occur.

How can that be? The different pictures are just equivalent mathematical formulations of the QT formalism. How can the physical interpretation be different for the very same theory in different mathematical formulations?

 

[vanhees71]

In all cases?

Then please explain for the following two explicit examples, the first from relativistic QFT, the second from nonrelativistic statistical mechanics:

• (i) How is the Born rule used to associate poles of the renormalized propagators with observable masses?
• (ii) How is the Born rule used in case of a real-world observation of temperature of a bucket of water?

Ad (i). The definition of masses as poles of the propagators is derived from unitarity of the S-matrix. The S-matrix is defined as transition-probability amplitudes from the asymptotic into the asymptotic out states. The probabilities are evaluated via Born's rule.

Ad (ii). Temperature is not an observable in the quantum-theoretical sense. You measure a temperature by putting a thermometer in thermal contact with the heatbath whose temperature you want to measure (more precisely for the relativistic case comoving with the corresponding "fluid cell"). The temperature is a "coarse-grained macroscopic quantity" making sense as an average of some macroscopic quantity (e.g., the average energy density of an ideal gas).

 

[Demystifier]

How can that be? The different pictures are just equivalent mathematical formulations of the QT formalism. How can the physical interpretation be different for the very same theory in different mathematical formulations?

I was not sufficiently precise. What I meant is that in some interpretation only one of the pictures may be appropriate. For example, in the many-world interpretation only the Schrodinger picture is appropriate.

 

[A. Neumaier]

http://arxiv.org/abs/1208.6565

I couldn't find out where in this paper you are using the Born rule to associate your formal quantities with real-world observables (photoproduction). while you had claimed in your post #70 that this is always the case. Instead I noticed that you use a number operator expectation for evaluating photon number in (41), and you used pair correlators in (49), in accordance with what I had claimed is typical for QFT.

 

[A. Neumaier]

Ad (i). The definition of masses as poles of the propagators is derived from unitarity of the S-matrix. The S-matrix is defined as transition-probability amplitudes from the asymptotic into the asymptotic out states. The probabilities are evaluated via Born's rule.

Ad (ii). Temperature is not an observable in the quantum-theoretical sense. You measure a temperature by putting a thermometer in thermal contact with the heatbath whose temperature you want to measure (more precisely for the relativistic case comoving with the corresponding "fluid cell"). The temperature is a "coarse-grained macroscopic quantity" making sense as an average of some macroscopic quantity (e.g., the average energy density of an ideal gas).

ad (i) The unitarity of the S-matrix is independent of the Born rule and suffices for interpreting masses. That one can interpret the S-matrix elements in terms of the Born rule doesn't contribute anything to this interpretation.
ad (ii) Here your explanation uses expectations but not the Born rule.

Thus nothing is left in your explanation that needs the Born rule.

 

[A. Neumaier]

The different pictures are just equivalent mathematical formulations of the QT formalism.

They are not equivalent. The Heisenberg picture is far more general, as it allows to discuss time correlations. The Schroedinger picture addresses only single-time dynamics.

 

[dextercioby]

The S-matrix comes from requiring that time-evolution is unitary, i.e. it conserves probability. Since Born rule is the existence of a probabilistic interpretation of QM mathematics, it follows that the asking the S-matrix to be unitary cannot be independent from the Born rule, it's a consequence of it.

 

[Demystifier]

I couldn't find out where in this paper you are using the Born rule to associate your formal quantities with real-world observables (photoproduction). while you had claimed in your post #70 that this is always the case. Instead I noticed that you use a number operator expectation for evaluating photon number in (41), and you used pair correlators in (49), in accordance with what I had claimed is typical for QFT.

If $A$ is a projector, then the probability is $P(A)={\rm Tr} \rho A$. In standard terminology this is the Born rule. In this form the Born rule does not depend on the picture (Schrodinger, Heisenberg) or type of theory (QM, QFT, quantum gravity, string theory).

 

[Demystifier]

The Schroedinger picture addresses only single-time dynamics.

There is a generalization of the single-time Schrodinger picture to a many-time Schrodinger picture. See e.g.
http://lanl.arxiv.org/pdf/0912.1938
and Refs. [15, 16, 17, 18, 19] therein.

 

[A. Neumaier]

There is a generalization of the single-time Schrodinger picture to a many-time Schrodinger picture. See e.g.
http://lanl.arxiv.org/pdf/0912.1938
and Refs. [15, 16, 17, 18, 19] therein.

One can generalize everything to weaken arguments aimed at the ungeneralized version. The conventional axioms of QM say how the state of a system changes through a perfect measurement. [See, e.g., Messiah I, end of Section 8.1, or Landau & Lifschitz, Vol. III, Chapter I, Par. 7.] This is a context that makes sense only in the ordinary Schroedinger picture.

 

[A. Neumaier]

The S-matrix comes from requiring that time-evolution is unitary, i.e. it conserves probability. Since Born rule is the existence of a probabilistic interpretation of QM mathematics, it follows that the asking the S-matrix to be unitary cannot be independent from the Born rule, it's a consequence of it.

No. The unitarity of the S-matrix is something that follows from asymptotic completeness alone, without reference to the Born rule. Weinberg gives a proof in Vol. I at the end of Section 3.2 (p.115 in the 1995 edition), long before he invokes the Born rule in (3.4.7) to give an experimental meaning to the absolute values of certain S-matrix elements.

 

[A. Neumaier]

If $A$ is a projector, then the probability is $P(A)={\rm Tr} \rho A$. In standard terminology this is the Born rule.

The mathematical formulas are just shut-up-and-calculate, with no interpretation attached.

The Born rule is the interpretation of certain formulas as a specific statement about measurement. Taking for definiteness the Born rule as stated in wikipedia, the Born rule leaves undefined what to measure an arbitrary orthogonal projector $A$ means in operational terms,but can be specialized to this case.

Thus the traditional foundation of quantum mechanics says:

''Upon measuring an orthogonal projector $A$, the measured result will be 0 or 1, and the probability of measuring 1 will be $P(A)=\langle A\rangle$.''

In contrast, the practice of statistical mechanics says:

''Upon measuring a Hermitian operator $A$, the measured result will be approximately $\bar A=\langle A\rangle$, with an uncertainty at least of the order of $\sigma_A=\sqrt{\langle (A-\bar A)^2\rangle}$. If the measurement can be sufficiently often repeated (on an object with the same or sufficiently similar state) then $\sigma_A$ will be a lower bound on the standard deviation of the measurement results.''

A world of difference in the ontology! To go from one to the other in any direction involves a lot of handwaving arguments, far from constituting a derivation.

 

[bhobba]

I don't see how the situation is any better in QFT.

And its mathematically a lot harder.

It seems to me similar to Zureck's Quantum Darwinian where observations and the Born Rule emerge from quantum states. Its a valid approach but its simply a matter of interpretive preference if it gains you anything.

Thanks
Bill

 

[fxdung]

I think that there is a deep difference between QFT and QM by the number of degree of freedom,so the methods of making the average are very different.Then the ontology of the two approaches are different.On QM we base on the ''collapse'' of eigenfunction,but in statistical mechanics we use the average base on statistical ensemble.Is that right?

 

[atyy]

The mathematical formulas are just shut-up-and-calculate, with no interpretation attached.

The Born rule is the interpretation of certain formulas as a specific statement about measurement. Taking for definiteness the Born rule as stated in wikipedia, the Born rule leaves undefined what to measure an arbitrary orthogonal projector $A$ means in operational terms,but can be specialized to this case.

Thus the traditional foundation of quantum mechanics says:

''Upon measuring an orthogonal projector $A$, the measured result will be 0 or 1, and the probability of measuring 1 will be $P(A)=\langle A\rangle$.''

In contrast, the practice of statistical mechanics says:

''Upon measuring a Hermitian operator $A$, the measured result will be approximately $\bar A=\langle A\rangle$, with an uncertainty at least of the order of $\sigma_A=\sqrt{\langle (A-\bar A)^2\rangle}$. If the measurement can be sufficiently often repeated (on an object with the same or sufficiently similar state) then $\sigma_A$ will be a lower bound on the standard deviation of the measurement results.''

A world of difference in the ontology! To go from one to the other in any direction involves a lot of handwaving arguments, far from constituting a derivation.

http://arxiv.org/abs/1309.0851
"The common feature behind these works is the understanding that closed quantum systems described by pure states can behave, for many practical purposes, like statistical mechanic ensembles at equilibrium."

 

[atyy]

[
For the purposes of foundations, I call QFT that part of quantum theory where only expectations and correlation functions are asserted to have meaning related to experiment, and QM that part of quantum theory where the Schroedinger equation is used and Born's rule relates it to experiments. This naturally divides quantum physics in two nearly disjoint parts with completely different ontologies.

But that is absolutely bizarre terminology. In Copenhagen QM (which includes QFT), it is true that only expectations (which include correlation functions) are asserted to have meaning related to experiment. And it is also true in Copenhagen QM (which includes QFT) that the Schroedinger equation and the Born rule is used to calculate the expectations.