I How Does Quantum Mechanics Relate to Quantum Field Theory in Particle Physics?

[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.

 

[fxdung]

Schroedinger equation is one in infinite configurations to contruct QFT,so QFT can be reduced to QM?

 

[atyy]

Schroedinger equation is one in infinite configurations to contruct QFT,so QFT can be reduced to QM?

As many have said throughout this thread, including bhobba and vanhees71: QM is the general framework.

Relativistic QFT is a specific type of QM in which there is a classical Minkowskian spacetime, and measurement outcomes are classical relativistic events.

 

[fxdung]

And there is not the ''difference'' between quantum measure theory and the making average by statistical ensemble.The saying about the ''collapse'' of eigenstate in processes of measurement is equivalent with saying about statistical ensemble?I think the saying about statistical ensemble more general than the saying about the collapse of eigenstate in a process of measurement.The later is a special case of the former.

 

[atyy]

And there is not the ''difference'' between quantum measure theory and the making average by statistical ensemble.The saying about the ''collapse'' of eigenstate in processes of measurement is equivalent with saying about statistical ensemble?I think the saying about statistical ensemble more general than the saying about the collapse of eigenstate in a process of measurement.The later is a special case of the former.

I am not sure exactly what A. Neumaier means, reading through the thread it is clear his view is extremely unconventional, whereas Demystifier, bhobba, Orodruin, vanhees71 have all agreed on the conventional view. QM is the overarching framework. Relativistic QFT is a type of QM. You can also see meopemuk's post #45 - there is a slight difference in terminology (meopemuk's terminology might be better), but his idea is also the conventional view.

To get from relativistic QFT to non-relativistic QM, we note that non-relativistic QM of many-identical particles can be formulated exactly as non-relativistic QFT. This key point is found in condensed matter books about "many-body physics", and the two equivalent forms of many-particle physics are "first quantization" and "second quantization". "Second quantization" is a misleading name - its correct meaning is that it allows you to write the usual non-relativistic QM of many identical particles as a non-relativistic QFT.

So as Demystifier pointed out earlier in post #23, one can do relativistic QFT -> non-relativistic QFT -> non-relativistic QM.

 

[fxdung]

The ''collapse'' of eigenstate is a result of the making many measurements on the same particle due to the probability character.Then in QFT if we consider many times on the same quantum of field , the Born's rule (meaning the ''collapse'')will appear.Is that right?

 

[A. Neumaier]

I am not sure exactly what A. Neumaier means, reading through the thread it is clear his view is extremely unconventional

This is because I interpret what people actually do when doing statistical physics and QFT, rather than what they say in the motivational introduction. It is very easy to verify that my view is the correct one for statistical mechanics and finite time QFT, no matter how unconventional it may sound on first sight.

So as Demystifier pointed out earlier in post #23, one can do relativistic QFT -> non-relativistic QFT -> non-relativistic QM.

But as I had pointed out in post #31, during this apparent ''derivation'' one has to introduce in an ad hoc way

• (i) particle position and momentum operators by hand - via a nonphysical extension of the Hilbert space, and
• (ii) an external classical reality that collapses the probabilities to actualities.

This makes the difference between the ontologies.

The predictions of QFT (field values, correlation functions, semiconductor behavior, chemical reaction rates) are valid for each single macroscopic system, without needing any foundational blabla on eigenvalues, probability, or collapse.

While QM, if strictly based on the traditional axioms, is valid only for measuring discrete observables exactly, and predicts for an individual system nothing at all, for almost all observables.

I should add that most practitioners in QM and QFT get useful results since they don't care about the traditional, far too restrictive axioms or postulates of QM. They apply whatever is needed in any way that is convincing enough for their colleagues. The foundations are not true foundations but post hoc attempts to put the mess on a seemingly sounder footing.

 

[A. Neumaier]

in QFT if we consider many times on the same quantum of field , the Born's rule (meaning the ''collapse'')will appear.Is that right?

Fields are space-time dependent. If you look at a field at different times or different places you look at different observables. Thus, strictly speaking, it is impossible to measure anything repeatedly. (It can be done only under an additional stationarity assumption.)

 

[atyy]

This is because I interpret what people actually do when doing statistical physics and QFT, rather than what they say in the motivational introduction. It is very easy to verify that my view is the correct one for statistical mechanics and finite time QFT, no matter how unconventional it may sound on first sight.

But as I had pointed out in post #31, during this apparent ''derivation'' one has to introduce in an ad hoc way

• (i) particle position and momentum operators by hand - via a nonphysical extension of the Hilbert space, and
• (ii) an external classical reality that collapses the probabilities to actualities.

This makes the difference between the ontologies.

The predictions of QFT (field values, correlation functions, semiconductor behavior, chemical reaction rates) are valid for each single macroscopic system, without needing any foundational blabla on eigenvalues, probability, or collapse.

While QM, if strictly based on the traditional axioms, is valid only for measuring discrete observables exactly, and predicts for an individual system nothing at all, for almost all observables.

I should add that most practitioners in QM and QFT get useful results since they don't care about the traditional, far too restrictive axioms or postulates of QM. They apply whatever is needed in any way that is convincing enough for their colleagues. The foundations are not true foundations but post hoc attempts to put the mess on a seemingly sounder footing.

Yes, there are some mathematical difficulties in introducing position operators, for example, but they are at the level of mathematical physics. At the non-rigourous level of ordinary physics, one can simply start with lattice QED, which is already non-relativistic, and get everything in QM. This is the same as the Wilsonian paradigm, and if one wants to argue that the Wilsonian paradigm is not properly justified in rigourous mathematics, that is fine.

However, it is definitely not true that QFT solves the foundational problems. QFT has all the same postulates as QM (state is vector in Hilbert space, probabilities given by Born rule, collapse of the wave function etc), including the need for the classical apparatus, with all the problems that entails. One way to see this is that a QFT like QED is really just non-relativistic QM because it is lattice QED.

 

[A. Neumaier]

QFT has all the same postulates as QM (state is vector in Hilbert space, probabilities given by Born rule, collapse of the wave function etc), including the need for the classical apparatus, with all the problems that entails.

You didn't understand. Statistical mechanics can start with Hilbert spaces, unitary dynamics for operators, density operators for Heisenberg states, the definition of

(EX)$~~~~~\langle A\rangle:=\mbox{tr}~\rho A$

as mathematical framework, and the following rule for interpretation, call it (SM) for definiteness:

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.''

Everything deduced in statistical mechanics about macroscopic properties follows from this without ever invoking ''probabilities given by Born rule, collapse of the wave function etc), including the need for the classical apparatus, with all the problems that entails''. Look into an arbitrary book on statistical physics and you'll never find such an invocation, except in the beginning, where the formula $\langle A\rangle:=\mbox{tr}~\rho A$ is derived! Thus one can skip this derivation, make this formula an axiom, and has a completely self-consistent setting in which the classical situation is simply the limit of a huge number of particles.

Note that it is impossible to deduce the Born rule from the rules (EX) and (SM) without introducing the notion of external measurement which is not present in the interpretation of quantum theory based upon (EX) and (SM) alone. This shows that the ontologies are indeed different!

 

[atyy]

You didn't understand. Statistical mechanics can start with Hlbert spaces, unitary dynamics for operators, density operators for Heisenberg states, the definition of $\langle A\rangle:=\mbox{tr}~\rho A$ as mathematical framewok, and the following rule for interpretation:

Everything deduced in staistical mechanics about macroscopic properties follows from this without ever invoking ''probabilities given by Born rule, collapse of the wave function etc), including the need for the classical apparatus, with all the problems that entails''. Look into an arbitrary book on statistical physics and you'll never find such an invocation, exept in the beginning, where the formula $\langle A\rangle:=\mbox{tr}~\rho A$ is derived! Thus one can skip this derivation, make this formula an axiom, and has a completely self-consistent setting in which the classical situation is simply the limit of a huge number of particles.

What is the difference? $\langle A\rangle:=\mbox{tr}~\rho A$ is the Born rule.

Also, there is quantum mechanics without statistical mechanics (eg. T=0).

 

[A. Neumaier]

$\langle A\rangle:=\mbox{tr}~\rho A$ is the Born rule.

Neither Wikipedia nor Dirac nor Messiah calls this the Born rule.

Note that this formula is shut-up-and-calculate since it is a purely mathematical definition. A definition (the left hand side is defined to be an abbreviation for the right hand side), not a postulate or axiom! Hence it cannot represent Born's rule. The interpretation is not in the formula but in the meaning attached to it. The meaning in statistical mechanics is the one given in (SM) of my updated post #114.

The meaning according to Born's probability definition is unclear as it is ''derived'' using plausibility arguments that lack a clear support in the postulates. Born's original paper says only something about the probability of simultaneously measuring all particle positions. One can deduce from this a statistical interpretation of $\langle A\rangle$ only if $A$ is a funcion of the position operators. But even if one generalizes this to arbitrary Hermitian operators, as it is generally done, the derivation says nothing about the individual case but only asserts that if you measure $A$ sufficiently often you'll get on the average $\langle A\rangle$. However, Born's rule says that you always get exact values $0$ or $1$ when you measure a projection operator (whatever this is supposed to mean for an abitrary projection opeator - the fondations are silent about when a measurement measures $A$) - which is statement different from (SM). Thus the interpretations are not equivalent.

Also, there is quantum mechanics without statistical mechanics (eg. T=0).

$T=0$ is an unphysical limiting case that can be derived as such a limit from statistical mechanics. The meaning of the rules (EX) and (SM) remains intact in this limit.

 

[atyy]

The meaning according to Born's probability definition is unclear as it is ''derived'' using plausibility arguments that lack a clear support in the postulates. Born's original paper says only something about the probability of simultaneously measuring all particle positions. One can deduce from this a statistical interpretation of $\langle A\rangle$ only if $A$ is a funcion of the position operators. But even if one generalizes this to aritrary Hermitian operators, as it is generally done, the derivation says nothing about the individual case but only asserts that if you measure $A$ sufficiently often you'll get on the average $\langle A\rangle$. However, Born's rule says that you always get exact values $0$ or $1$ when you measure a projection operator (whatever this is supposed to mean for an abitrary projection opeator - the fondations are silent about when a measurement measures $A$) - which is statemnt different from (SM). Thus the interpretations are not equivalent.

Hmmm, the Born rule should give the complete probability distribution, from which we know the only values are 0 or 1. The complete probability distribution is given by assuming that the Born rule (meaning $\langle A\rangle:=\mbox{tr}~\rho A$) gives the expectation values of all observables that commute with A.

 

[A. Neumaier]

Hmmm, the Born rule should give the complete probability distribution, from which we know the only values are 0 or 1. The complete probability distribution is given by assuming that the Born rule gives the expectation values of all observables that commute with A.

You would have to derive this from the Born rule as given in the official sources. The precise form given depends on the source, though, so you'd be clear about which form you are using.

 

[atyy]

You would have to derive this from the Born rule as given in the official sources. The precise form given depends on the source, though, so you'd be clear about which form you are using.

I think I should be able to get all cumulants from the Born rule, since the cumulants commute with A and are expectation values .. ?

 

[A. Neumaier]

I think I should be able to get all cumulants from the Born rule, since the cumulants commute with A and are expectation values .. ?

Something in ths statement is strange since cumulants are numbers, not operators, so they commute with everything.

I know of different ways to ''get'' the result you want from appropriate versions of the Born rule. But the ''derivations'' in the textbooks or other standard references I know of are all questionable. The challenge is to provide a derivation for which all steps are physically justified.