Measurement in QFT: Mapping Fields to Theory's Math Formalism

[Jdeloz828]

TL;DR Summary

How do we map experimental measurements of quantum fields, such as those seen in accelerators, to the theory's mathematical formalism?

How do we map experimental measurements of quantum fields, such as those seen in accelerators, to the theory's mathematical formalism? When we see images of particle tracks produced in accelerators such as the LHC, I think it's safe to say a measurement (or series of measurements) has been performed, but what linear operator are the fields now in an eigenstate of? In some sense we can interpret these images as measuring the positions of particles, but I'm told that position is not an operator in QFT, but rather a parameter for the field operators. In addition to this, when such a measurement on the field is performed, does it then collapse the field throughout the entire universe? In Sean Carroll's book Something Deeply Hidden, he resolved this issue by noting that we can think of the state of quantum fields as a series of entangled patched throughout spacetime, where the greater the distance between the patches, the less they are entangled. How do we represent a quantum field state (I'm thinking of something like the occupation number basis) as such an entangled network of patches? I''ve never seen anything in the textbooks related to this. If anyone has some insight on this, or could point me to the relevant literature it would be much appreciated.

 

[PeroK]

The predictions for particle scattering experiments are calculated using cross-sections, amplitudes and the S-matrix. There are some notes here:

https://www.ippp.dur.ac.uk/~krauss/Lectures/IntroToParticlePhysics/2010/Lecture6.pdf

 

[Jdeloz828]

The predictions for particle scattering experiments are calculated using cross-sections, amplitudes and the S-matrix. There are some notes here:

https://www.ippp.dur.ac.uk/~krauss/Lectures/IntroToParticlePhysics/2010/Lecture6.pdf

I'm not sure this really answers my questions. I understand the idea of computing amplitudes by performing time evolution on a state resembling incoming free particles and projecting on an outgoing state of (possibly different) free particles, but when a measurement has been performed, what operator are the fields now in a eigenstate of? Is the idea of the state not being an eigenvector after a measurement has been performed not viewed as a postulate within QFT?

 

[PeroK]

I'm not sure this really answers my questions. I understand the idea of computing amplitudes by performing time evolution on a state resembling incoming free particles and projecting on an outgoing state of (possibly different) free particles, but when a measurement has been performed, what operator are the fields now in a eigenstate of? Is the idea of the state not being an eigenvector after a measurement has been performed not viewed as a postulate within QFT?

The states before and after scattering are taken to be approximately the free states for (for example) a single particle of a given momentum - so, eigenstates of the momentum operator. The scattering itself requires an interaction picture, which is modeled by perturbations, Feynman diagrams and all the other mathematical paraphernalia, as presented in those notes.

 

[Jdeloz828]

The states before and after scattering are taken to be approximately the free states for (for example) a single particle of a given momentum - so, eigenstates of the momentum operator. The scattering itself requires an interaction picture, which is modeled by perturbations, Feynman diagrams and all the other mathematical paraphernalia, as presented in those notes.

Thanks for the reply! Hmm okay, so they're in an eigenstate of the momentum operator when the measurement is performed, but then what do we make of the particle tracks such as those seen in the image below:

It seems as if a series of very precise position measurements has been performed on the fields involved. If the fields are collapsing in momentum eigenstates, wouldn't this very precise position data violate the uncertainty principle?

 

[PeroK]

Thanks for the reply! Hmm okay, so they're in an eigenstate of the momentum operator when the measurement is performed, but then what do we make of the particle tracks such as those seen in the image below:

View attachment 293309It seems as if a series of very precise position measurements has been performed on the fields involved. If the fields are collapsing in momentum eigenstates, wouldn't this very precise position data violate the uncertainty principle?

That looks like a much more complicated scattering event where jets of particles emerge in all directions. The HUP is a statistical law. To violate it, you would need a state where the range of position measurements and the range of momentum measurements for a single particle both have sufficiently small standard deviations. I see no evidence of that at all in that diagram!

It appears that you are treating QFT as some sort of add-on to non-relativistic QM. This is not the case at all. QFT is a different ball-game altogether. You can, of course, recover non-relativistic QM, the SDE and the HUP in the appropriate limits. But, QFT itself is significantly more mathematically and conceptually sophisticated than QM. And, it's not always obvious how QM emerges from QFT.

 

[Jdeloz828]

That looks like a much more complicated scattering event where jets of particles emerge in all directions. The HUP is a statistical law. To violate it, you would need a state where the range of position measurements and the range of momentum measurements for a single particle both have sufficiently small standard deviations. I see no evidence of that at all in that diagram!

It appears that you are treating QFT as some sort of add-on to non-relativistic QM. This is not the case at all. QFT is a different ball-game altogether. You can, of course, recover non-relativistic QM, the SDE and the HUP in the appropriate limits. But, QFT itself is significantly more mathematically and conceptually sophisticated than QM. And, it's not always obvious how QM emerges from QFT.

I suppose I'm still somewhat confused, but I think we're getting somewhere. Going back to the image above, I'm wondering how we break the image up into measurements where collapse is taking place. Can we break up each track into equally time spaced blips and all such collections of blips constitute a collapse of the fields involved? And are these collapses measurements of position or momentum?

 

[PeroK]

I suppose I'm still somewhat confused, but I think we're getting somewhere. Going back to the image above, I'm wondering how we break the image up into measurements where collapse is taking place. Can we break up each track into equally time spaced blips and all such collections of blips constitute a collapse of the fields involved? And are these collapses measurements of position or momentum?

Okay, but this is a more fundamental question about how particles can exhibit approximately classical trajectories despite the HUP. The answer is that the tracks in a cloud chamber, for example, are simply not precise enough measurements of position to disrupt the approximately classical trajectories of circles and straight lines. If you analyse the uncertainties of position in a picture like that, they may be several orders of magnitude too coarse for the HUP to be significant.

Ultimately, we know that electrons can follow approximately classical trajectories unless we try to pin them down too closely, at which point the HUP and the constraints of QM show themselves.

 

[Jdeloz828]

Okay, but this is a more fundamental question about how particles can exhibit approximately classical trajectories despite the HUP. The answer is that the tracks in a cloud chamber, for example, are simply not precise enough measurements of position to disrupt the approximately classical trajectories of circles and straight lines. If you analyse the uncertainties of position in a picture like that, they may be several orders of magnitude too coarse for the HUP to be significant.

Ultimately, we know that electrons can follow approximately classical trajectories unless we try to pin them down too closely, at which point the HUP and the constraints of QM show themselves.

Gotchya. So am I correct in assuming the image can be broken up in a series of position measurements? Or should it be thought of as a series of momentum measurements?

 

[PeroK]

Gotchya. So am I correct in assuming the image can be broken up in a series of position measurements? Or should it be thought of as a series of momentum measurements?

Technically, we have a series of position measurements. From which the (classical) momentum of a particle is inferred. For example, presumably there is a magnetic field present, from which the charge and momentum of the particles are inferred. As before, the HUP does not prevent charged particles being observed moving (approximately) in a circle of a given radius in a magnetic field. For practical purposes we may see apparently classical trajectories following the laws of classical electrodynamics! But, critically, it's QFT and modern particle physics that explains which particles or jets of particles emerge from high-energy interactions. That's where QFT does its job.

 

[Jdeloz828]

Okay, but this is a more fundamental question about how particles can exhibit approximately classical trajectories despite the HUP. The answer is that the tracks in a cloud chamber, for example, are simply not precise enough measurements of position to disrupt the approximately classical trajectories of circles and straight lines. If you analyse the uncertainties of position in a picture like that, they may be several orders of magnitude too coarse for the HUP to be significant.

Ultimately, we know that electrons can follow approximately classical trajectories unless we try to pin them down too closely, at which point the HUP and the constraints of QM show themselves.

Building on my previous reply. I suppose we can think of the image as a series of both coarse position and momentum measurements, and what you're saying about the coarse position measurements not disturbing the classical trajectories makes sense if the states are not collapsing into pure position states, but rather some sort of localized state that retains a degree of imprecision. In this way the momentum information the state possessed prior to having it's position measured is no longer lost the moment the position is measured. That would definitely clear things up, but I never see in textbooks examples of particles/fields collapsing into somewhat localized states. Can we form hermitian operators for which the eigenstates are the states resulting after collapse associated with a coarse measurement?

 

[PeroK]

Building on my previous reply. I suppose we can think of the image as a series of both coarse position and momentum measurements, and what you're saying about the coarse position measurements not disturbing the classical trajectories makes sense if the states are not collapsing into pure position states, but rather some sort of localized state that retains a degree of imprecision. In this way the momentum information the state possessed prior to having it's position measured is no longer lost the moment the position is measured. That would definitely clear things up, but I never see in textbooks examples of particles/fields collapsing into somewhat localized states. Can we form hermitian operators for which the eigenstates are the states resulting after collapse associated with a coarse measurement?

I think you have stumbled on the point that measurements in experimental physics are not the explicit measurements of elementary theoretical QM, but more subtle in nature. Let's take the spectrum of hydrogen as an example.

If you have just studied the QM model of the Hydrogen atom, you might imagine the following:

1) Measure the energy of a hydrogen atom. Obtain the result $E_k$.

2) Observe the hydrogen atom emit a photon of a given energy, $E_p$.

3) Measure the energy of the hydrogen atom again and obtain $E_k - E_p$, which should also be a recognised energy eigenstate.

That's not what happens. All you get is 2). Just a photon emission. Nothing else. All the confirmation of the QM model of the hydrogen atom comes from inferring the transition between energy states - including the subtelies of fine and hyperfine spectral lines.

You might talk about measuring the momentum of the electron in a hydrogen atom, or its AM or spin etc. But, ultimately, the only measurement you are likely to do is that of the energy of an emitted photon. Everything is inferred from that.

The Stern-Gerlach experiment similarly involves an inference of the quantisation of spin. There's no direct measurement of spin in the S-G experiment. The only measurements are where the silver atoms impact the screen - from which the theory of quantised spin is inferred indirectly.

We have the same with the quark model and the interactions of particles in collision experiments. All the theoretical structure around spin, isospin, strangeness and charm is inferred from what amounts to classical trajectories in a cloud chamber!

 

[Jdeloz828]

I think you have stumbled on the point that measurements in experimental physics are not the explicit measurements of elementary theoretical QM, but more subtle in nature. Let's take the spectrum of hydrogen as an example.

If you have just studied the QM model of the Hydrogen atom, you might imagine the following:

1) Measure the energy of a hydrogen atom. Obtain the result $E_k$.

2) Observe the hydrogen atom emit a photon of a given energy, $E_p$.

3) Measure the energy of the hydrogen atom again and obtain $E_k - E_p$, which should also be a recognised energy eigenstate.

That's not what happens. All you get is 2). Just a photon emission. Nothing else. All the confirmation of the QM model of the hydrogen atom comes from inferring the transition between energy states - including the subtelies of fine and hyperfine spectral lines.

You might talk about measuring the momentum of the electron in a hydrogen atom, or its AM or spin etc. But, ultimately, the only measurement you are likely to do is that of the energy of an emitted photon. Everything is inferred from that.

The Stern-Gerlach experiment similarly involves an inference of the quantisation of spin. There's no direct measurement of spin in the S-G experiment. The only measurements are where the silver atoms impact the screen - from which the theory of quantised spin is inferred indirectly.

We have the same with the quark model and the interactions of particles in collision experiments. All the theoretical structure around spin, isospin, strangeness and charm is inferred from what amounts to classical trajectories in a cloud chamber!

Looking at your first example, wouldn't we say that when the photon is emitted, the photon and hydrogen atom state are entangled so that by measuring the photon energy we can reliably infer the energy of the hydrogen atom? Or, with the S-G example, that the silver atom's position degree of freedom is entangled with the spin degree of freedom of the silver atom's valence electon, resulting in a correlation between the two measurements? Going back to cloud chamber example, are you saying that we're not measuring the position or momentum directly, but rather indirectly via, say, a photon energy, but that this photon is entangled with the particle states in such a way that we can infer an approximation of the particle positions and momenta? What about what I said about directly measuring a quantum degree of freedom with a bit of imprecision, is that possible, could we form a hermition operator corresponding to such a measurement?

 

[PeroK]

Looking at your first example, wouldn't we say that when the photon is emitted, the photon and hydrogen atom state are entangled so that by measuring the photon energy we can reliably infer the energy of the hydrogen atom? Or, with the S-G example, that the silver atom's position degree of freedom is entangled with the spin degree of freedom of the silver atom's valence electon, resulting in a correlation between the two measurements?

Yes, exactly. But, there is no direct measurement of energy levels or spin. That's my point.

If we extend this to QFT, there is no way to measure directly a quantum field. Ironically, measurements are of classical things. That's why I think you are on the wrong track by asking what quantum measurements (of your quantum field) are being done when you observe a particle's track through a cloud chamber or a dot on the screen or the wavelength of a photon on some device.

What you are seeing is significantly removed from the elementary QM/QFT measurement to eigenstate that underpins the theory. You cannot directly relate what are essentially macroscopic classical measurements to what the fields are doing. It's an indirect inference.

QM/QFT is the underlying theoretical mechanism, but it's not what we directly measure. Not the way we directly measure the amplitude of a classical pendulum or the Earth's angular speed about its axis of rotation.

 

[Jdeloz828]

Yes, exactly. But, there is no direct measurement of energy levels or spin. That's my point.

If we extend this to QFT, there is no way to measure directly a quantum field. Ironically, measurements are of classical things. That's why I think you are on the wrong track by asking what quantum measurements (of your quantum field) are being done when you observe a particle's track through a cloud chamber or a dot on the screen or the wavelength of a photon on some device.

What you are seeing is significantly removed from the elementary QM/QFT measurement to eigenstate that underpins the theory. You cannot directly relate what are essentially macroscopic classical measurements to what the fields are doing. It's an indirect inference.

QM/QFT is the underlying theoretical mechanism, but it's not what we directly measure. Not the way we directly measure the amplitude of a classical pendulum or the Earth's angular speed about its axis of rotation.

Okay, that all makes sense. So we can think of the fields position and momentum degrees of freedom as being entangled without classical measurement device (whatever that may be) in such a way that when we make an observation, we can infer that the field has collapsed, not into a pure position or momentum state, but a sort of state representing a coarse measurement of both. In this way enough position/momentum information is preserved between collapses to make HUP happy, and allow for the emergence of approximate classical particle tracks. But how do we represent such localized states of the quantum fields. Using the occupation number basis I can wrap my head around how we could represent a coarse momentum state, but what about position? How do we represent a somewhat spacialy localized state of a quantum field as a linear combination in the occupation number basis, or any basis for that matter? It's pretty obvious in regular QM where we have a position operator and position eigenstates to project our state vector onto, but what is the analgous procedure in QFT where we've demoted position to a field parameter?

 

[PeroK]

How do we represent a somewhat spacialy localized state of a quantum field as a linear combination in the occupation number basis, or any basis for that matter? It's pretty obvious in regular QM where we have a position operator and position eigenstates to project our state vector onto, but what is the analgous procedure in QFT where we've demoted position to a field parameter?

I feel like you are trying to fit things into a preconceived model of what QFT says. QFT predicts, for example, what comes out of particle experiments. If you want to explain the helical path of a charged particle in a magnetic field, there is a thread about that:

https://www.physicsforums.com/threa...rons-circulating-in-a-magnetic-field.1008856/

 

[PeroK]

How do we represent a somewhat spacialy localized state of a quantum field as a linear combination in the occupation number basis, or any basis for that matter?

That's central to the mathematics of QFT. There's no problem there, other than the compexity of the integrals and functional analysis.

 

[vanhees71]

Look for LSZ-reduction formalism. A good account is given in the good old textbook by Bjorken and Drell (of course the quantum-field theory volume, not the obsolete relativistic QM volume though!).

 

[Jdeloz828]

This discussion has been very helpful, so thanks a lot! I suppose I always thought that the microscopic states that became entangled with the measurement apparatus had to be pure eigenstates. I suppose this doesn't have to be the case with a more coarse measurement? When we consider something like the S-G experiment though, why does it seem like the microscopic states of the measurement apparatus are entangled with either spin-up spin-down states with no in between? In other words, why do we see the electrons deflected up or down and never off to the side somewhat? In this case it seems like the microscopic states of the measurement apparatus have become separated based on how they're entangled with the pure z-component of spin eigenstates. What exactly is the difference between these two scenarios?

 

[PeroK]

When we consider something like the S-G experiment though, why does it seem like the microscopic states of the measurement apparatus are entangled with either spin-up spin-down states with no in between? In other words, why do we see the electrons deflected up or down and never off to the side somewhat? In this case it seems like the microscopic states of the measurement apparatus have become separated based on how they're entangled with the pure z-component of spin eigenstates. What exactly is the difference between these two scenarios?

If you analyse the S-G experiment, each atom enters the magentic field in some unknown superposition of spin-up and spin-down. The state evolves in the magnetic field in such a way that it emerges with a different spatial component associated with each spin component. I.e. it emerges in the state:
$$|\Psi \rangle = a|\psi_1 \ + \rangle + b|\psi_2 \ - \rangle$$Where the spatial wavefunctions $\psi_1$ and $\psi_2$ have momentum in the positive and negative lateral directions determined by the orientation of the magnetic field.

The "measurement" takes place when the atom reaches the screen and the physical location of the interaction must effectively in one of two places.

Unless you analyse things like the S-G experiment yourself to this level of detail, you are never going to understand QM.

 

[A. Neumaier]

I suppose we can think of the image as a series of both coarse position and momentum measurements, and what you're saying about the coarse position measurements not disturbing the classical trajectories makes sense if the states are not collapsing into pure position states, but rather some sort of localized state that retains a degree of imprecision. In this way the momentum information the state possessed prior to having it's position measured is no longer lost when the moment the position is measured. That would definitely clear things up, but I never see in textbooks examples of particles/fields collapsing into somewhat localized states. Can we form hermitian operators for which the eigenstates are the states resulting after collapse associated with a coarse measurement?

To describe the measurement of a particle track one cannot use eigenstates at all. For a correct treatment, see Section 3.4 of my paper here.

 

[vanhees71]

The standard explanation in terns of standard QT is by Mott in 1929

https://www.google.com/url?sa=t&sou...4QFnoECAYQAQ&usg=AOvVaw3xAhnEpBejIBilVEcWUyNq

 

[A. Neumaier]

The standard explanation in terns of standard QT is by Mott in 1929

https://www.google.com/url?sa=t&sou...4QFnoECAYQAQ&usg=AOvVaw3xAhnEpBejIBilVEcWUyNq

This nicely explains why one gets tracks, but not in which sense it is a joint position and momentum measurement of the particle causing the track.

 

[vanhees71]

But isn't this obvious simply by looking at the track?

 

[A. Neumaier]

But isn't this obvious simply by looking at the track?

In Mott's paper, the track is not caused by a particle but by a rotationally invariant outgoing wave.

There is no explanation why this wave may be interpreted as a particle with fairly definite position and momentum. Born's rule cannot be applied to deduce this!

 

[vanhees71]

The outgoing spherical wave is the initial state. It's interpretation is the fundamental point of Born's rule: It gives the position-probability distribution at the initial state of the $\alpha$ particle just emitted from the nucleus. The point of the paper is that after a view interactions of the $\alpha$ particle with the vapour in the cloud chamber this probability distribution is quite sharply peaked around a straight trajectory with a quite sharply peaked momentum. This must be described by a reduced statistical operator, formally by tracing out the vapour-degrees of freedom (a prescription also derived from Born's rule). Maybe it's even a good example for the POVM formalism. Of course the observed trajectory is far from violating the Heisenberg momentum-position uncertainty relation. The position resolution and thus also the momentum (velocity) resolution is rather limited.

 

[A. Neumaier]

The point of the paper is that after a few interactions of the $\alpha$ particle with the vapour in the cloud chamber this probability distribution is quite sharply peaked around a straight trajectory with a quite sharply peaked momentum.

This is quite inaccurate. Born's rule does not give a probability distribution for trajectories, but only for either position or momentum.

This must be described by a reduced statistical operator, formally by tracing out the vapour-degrees of freedom

This does not help to get a distribution for trajectoies.

Maybe it's even a good example for the POVM formalism.

Indeed, it must be treated the POVM formalism. The reference I gave was precisely to a description of how to view it in this way.

Of course the observed trajectory is far from violating the Heisenberg momentum-position uncertainty relation. The position resolution and thus also the momentum (velocity) resolution is rather limited.

No objection here.

 

[vanhees71]

This is quite inaccurate. Born's rule does not give a probability distribution for trajectories, but only for either position or momentum.

This does not help to get a distribution for trajectoies.

Indeed, it must be treated the POVM formalism. The reference I gave was precisely to a description of how to view it in this way.

Which reference?

I'd try to use the Wigner distribution of the (real-time) Green's function and the gradient expansion for "coarse-graining" it to a proper positive definite phase-space distribution function. Maybe that's formalizable to a POVM formalism. My problem with this formalism is that I don't find its connection to the physics, while the real-time Green's function technique (a la Kadanoff and Baym) has a pretty straightforward heuristics.

The usual way is of course to apply and additional magnetic field and measure the momentum of the $\alpha$ particles through the curvature of the "trajectory".

No objection here.

 

[A. Neumaier]

Which reference?

see post #21 in this thread.

I'd try to use the Wigner distribution of the (real-time) Green's function and the gradient expansion for "coarse-graining" it to a proper positive definite phase-space distribution function. Maybe that's formalizable to a POVM formalism.

Maybe, I haven's seen it, but there are too many papers on POVMs to scan them all...

The usual way is of course to apply and additional magnetic field and measure the momentum of the $\alpha$ particles through the curvature of the "trajectory".

How this fits into the POVM formalism is described in the above reference.

 

[vanhees71]

But in your reference I don't see a concrete description of a TPC with the POVM formalism. I doubt that my experimental colleagues actually building such instruments use the POVM formalism at all ;-)).

 

[A. Neumaier]

But in your reference I don't see a concrete description of a TPC with the POVM formalism.

This would be a separate paper on its own. I just describe how track measurements actually performed with a TCP (no matter how it is built) can be interpreted as a POVM measurement of position and momentum.

I doubt that my experimental colleagues actually building such instruments use the POVM formalism at all ;-)).

Neither did the spectroscopists before 1925 use the modern quantum formalism.

None of your statements mean that one cannot get correct descriptions in terms of POVMs respective energy levels.

 

[vanhees71]

I don't doubt the POVM formalism in any way. Of course spectroscopists before 1925 didn't use modern quantum formalism. To the contrary before at least 1913 with Bohr's "old quantum mechanics" it was an enigma how to explain the discrete spectra observed by Fraunhofer as well as Kirchhoff and Bunsen in the 19th century. It's only with modern QM that a complete understanding, including intensities, from a generally valid theory has been achieved.