B Photon Continuity in Double-Slit Experiment

[DarMM]

Well according to QFT in general states don't have a particle decomposition. Of course there are regimes where it is an excellent approximation, that's why we can use QM where particle number is well-defined.

However according to work on field theory by Haag, Ruelle and others in general this picture is not possible and in general we can only work with the notion of number expectation values for asymptotically placed probes. Steinmann in his monograph "Perturbative Quantum Electrodynamics and Axiomatic Field Theory" discusses this, as does Haag in his "Local Quantum Physics", as well as Araki in Chapter 5 of his "Mathematical Theory of Quantum Fields". Particles are only really well defined as clicks in appropriate detectors. Araki uses the phrase "Particle Counter Observable", Steinmann "Particle Probe".

Even then the asymptotic particle states won't definitively lead to $N$ clicks. So in the most general case it is hard to decompose quantum field theoretic states into being combinations of particle states.

Ultimately this just means QFT doesn't have particles as a fundamental notion. I don't see anything that would make you say a bacteria is just a click in a detector, although that combined with "empiricism" sounds like it's heading into a philosophical discussion which isn't my aim. It's just a statement about how particles are defined in QFT in the most general case. Going from "In QFT particles seem to be associated with asymptotically placed detectors, resulting in a count distribution centered around $N$, an integer we call the particle number" to "bacteria aren't real" seems like a long mostly philosophical discussion.

 

[charters]

I'm not really talking about an exactly valid number operator though. I'm thinking about what I quoted in #58.

I agree Fock states are only truly asymptotically valid in QFT (though I am unsure if this still holds in string theory, so perhaps this is a symptom of underlying pathologies in QFT and should not be trusted). But I don't take this to be so relevant to the question of photons having trajectories through spacetime, versus only believing in detector click correlations.

I'm saying: whenever something like an individual photon (or EM coherent state) can be identified fairly well - which in reality is a huge percentage of the time, but definitely not all the time in QFT - then it does appear to follow a superposition of paths through spacetime. There does seem to be a wavepacket that traverses the intervening spacetime between detectors. It doesn't only exist *as* detector clicks per se, ie as merely an effect that jumps discontinuously from source to detector with no presence along a path in the meantime.

I only mentioned bacteria to anticipate the response of: "well we only access information about photons through detector clicks, so who is really to say where the photon is between clicks". I see this as a more general view about observing anything microscopic, which is different from the question of photons in particular not having paths. The latter is specifically related to the absence of a formal position operator for (spin 1) massless fields. The question is what ontological moral should be drawn from this mathematical result in light of experimental measurements in which photon paths can be reconstructed just as well as electron or molecule paths.

Even then the asymptotic particle states won't definitively lead to N clicks

Is this a Reeh Schlieder argument, ie the asymptotic detector is compactly supported but the asymptotic N particle states are global, and these don't commute? Or are you saying something else here?

 

[DarMM]

Is this a Reeh Schlieder argument, ie the asymptotic detector is compactly supported but the asymptotic N particle states are global, and these don't commute? Or are you saying something else here?

Basically that. It's a corollary of the Reeh-Schlieder theorem.

though I am unsure if this still holds in string theory, so perhaps this is a symptom of underlying pathologies in QFT and should not be trusted

I don't think we should not trust a feature of a stringently tested physical theory (QFT) because it's not present in a incompletely developed theory with no experimental support (String Theory).

then it does appear to follow a superposition of paths through spacetime. There does seem to be a wavepacket that traverses the intervening spacetime between detectors

This is going into interpretation stuff, about the quantum state being a "real" wave.

Ignoring this, even for two photons we don't have this notion of a wavepacket traversing intervening spacetime since it isn't a function on spacetime.

The point is in QED particulate behavior only manifests in specific detector types at asymptotic times. If you'd used the wrong types of detectors you don't get particle behavior even at asymptotic times. Really this is because QFT promotes "particle" to having the same counterfactual difficulties that spin and other observables have in non-relativistic QM.

source to detector with no presence along a path in the meantime

Even in fixed particle QM though talking about "paths in the meantime" is fraught with difficulty.

The thread is originally about photons, but my comments are more about particles in general in QFT. In QFT both electrons and photons are associated with asymptotic detectors, which due to Reeh-Schlieder are necessarily always noisy, i.e. no state definitely causes $N$ clicks.

The noise in a detector is related to its size, i.e. the standard deviation of clicks is proportional to the detector volume. However for electrons since they are massive the standard deviation decays as $\mathcal{O}\left(e^{-mV}\right)$ and for photons it is only something like $\mathcal{O}\left(\frac{1}{V^{a}}\right)$ for some $a$.

Ultimately this is related to photons not really having a non-relativistic limit. Since position operators of particles aren't really a "native" notion to QFT, being able to construct one is in essence asking if you can form a well-defined single particle non-relativistic quantum theory. For electrons you can, since errors in detectors fall off exponentially and they have a non-relativistic limit. Photons don't satisfy either of these so you can't.

 

[charters]

Ignoring this, even for two photons we don't have this notion of a wavepacket traversing intervening spacetime since it isn't a function on spacetime.

Why not? If this concerns particle indistinguishability or higher dimensional configuration space, I think this position confuses Fock space indices with the localized wavepackets that actually matter/interact in the real world. See https://arxiv.org/abs/1002.2544

If you'd used the wrong types of detectors you don't get particle behavior even at asymptotic times.

What is a (non-theoretical, physically constructed) example of such a detector?

In QFT both electrons and photons are associated with asymptotic detectors, which due to Reeh-Schlieder are necessarily always noisy, i.e. no state definitely causes NNN clicks.

The noise in a detector is related to its size

No question this is correct under the assumption that a detector can be compactly supported, even though they are made of nucleons and electrons, which cannot be compactly supported. Obviously I'm skeptical about this assumption in general, but even granting it, I don't see how it is relevant here.

My issue is: even when allowing for some detector noise (both Reeh Schlieder and generic background), nondemolition experiments are able to quite reliably mark out the path a photon or multiple photons took, the same as can be done for massive particles. So, why isn't this good enough or trustworthy enough to evidence that a photon wavepacket travels along a continuous lightlike trajectory? Why doesn't this suggest the mathematical result is something of an artefact?

Why is the mathematical absence of a photon position operator more compelling than the experimental evidence, which seems to allow us to mark off time ordered photon positions? Or what experiment could be done with massive particles but not massless particles, such that the formal lack of a position operator would have real implications?

 

[DarMM]

Why not? If this concerns particle indistinguishability or higher dimensional configuration space, I think this position confuses Fock space indices with the localized wavepackets that actually matter/interact in the real world. See https://arxiv.org/abs/1002.2544

The general understanding in QM is that a wavefunction of let's say $N$ particles which individually move on a manifold $Q$ is a function in $\mathcal{L}^{2}\left(Q^{n}\right)$. This can't be regarded as a function on $Q$.
I noticed many of the papers citing that one are philosophical, arguing about metaphysics of identity. I don't want to go down this line.
Again the conventional understanding is that multiparticle wavefunctions cannot be seen as functions on the classical configuration space of a single particle and that this blocks regarding them as functions on spacetime.

No question this is correct under the assumption that a detector can be compactly supported, even though they are made of nucleons and electrons, which cannot be compactly supported

Well in QFT things aren't fundamentally made of particles. Electrons and nucleons are particles and thus a late time idealization. That they can't be localized is not really a problem for the detector. My attitude to this is similar to @vanhees71 here:
https://www.physicsforums.com/threa...ctron-violates-causality.975594/#post-6216222

nondemolition experiments are able to quite reliably mark out the path a photon or multiple photons took, the same as can be done for massive particles

Even in nonrelativistic QM we can make multiple position measurements in quick succession and they sort of trace a path. However from complementarity we know this doesn't mean we can ascribe a classical notion like a path to them. We can perform position measurements of electrons in hydrogen but we know it doesn't move along orbits.

In general most POVMs don't correspond to the quantization of any classical quantity. They're nameless as such aside from being the POVM representing that device.
https://arxiv.org/abs/quant-ph/0207020
The whole notion of thinking of $N$ photon states as multiple wave packets moving along paths is just not possible.

What is a (non-theoretical, physically constructed) example of such a detector?

Anything that measures coherent states, like teslameters or magnetometers.

 

[charters]

The whole notion of thinking of NNN photon states as multiple wave packets moving along paths is just not possible.

How do you reconcile this with the existence of the worldline formalism?

To be clear, I am not claiming a photon/any particle has *classical* path, I am just claiming it is reasonable to say the photon/any particle was present along the superposition of possible spacetime paths between emission and absorption, after accouting for interference among such paths.

Anything that measures coherent states, like teslameters or magnetometers.

But coherent states are in the global Fock space (can be seen as superpositions of the number operator) and are more easily localized than particle states/have even more classical seeming paths. The question was identifying some real world measurement which undermines the notion that massless particles/excitations sweep out a continuous, microcausal path through spacetime, such that we must restrict our speech to only detector click correlations.

 

[vanhees71]

You appear inflexible by sticking to one and only one description of the photon (one which is shunned by most textbook authors, by your own admission). Your description often seems useless in the context of B level threads. Labeling one description as "right" and another "wrong" - when both are useful in proper context - makes no sense to me.

Just my opinion.

Hm, I meant the conjecture that textbook writers may be lazy not as an insult but as an excuse for not writing about the correct modern picture we have about photons. I'm sure almost all of these textbook writers know the correct theoretical description of photons. That's why I think it's rather laziness to rewrite the introductory chapters of their textbooks than ignorance of the theoretical and empirical facts.

Also at B-level you should not tell the people wrong things. You can, of course, not really explain what a photon is without the use of field operators, but you can at least write correct things in words rather the just copying long outdated wrong pictures from a time, where no consistent theory about the interaction of electromagnetic waves with matter on a fundamental level was available. Nevertheless this theory, QED, is available now since 1926 (Born, Jordan) or better knowns 1927/28 (Dirac)!

 

[vanhees71]

Well according to QFT in general states don't have a particle decomposition. Of course there are regimes where it is an excellent approximation, that's why we can use QM where particle number is well-defined.

However according to work on field theory by Haag, Ruelle and others in general this picture is not possible and in general we can only work with the notion of number expectation values for asymptotically placed probes. Steinmann in his monograph "Perturbative Quantum Electrodynamics and Axiomatic Field Theory" discusses this, as does Haag in his "Local Quantum Physics", as well as Araki in Chapter 5 of his "Mathematical Theory of Quantum Fields". Particles are only really well defined as clicks in appropriate detectors. Araki uses the phrase "Particle Counter Observable", Steinmann "Particle Probe".

Even then the asymptotic particle states won't definitively lead to $N$ clicks. So in the most general case it is hard to decompose quantum field theoretic states into being combinations of particle states.

Ultimately this just means QFT doesn't have particles as a fundamental notion. I don't see anything that would make you say a bacteria is just a click in a detector, although that combined with "empiricism" sounds like it's heading into a philosophical discussion which isn't my aim. It's just a statement about how particles are defined in QFT in the most general case. Going from "In QFT particles seem to be associated with asymptotically placed detectors, resulting in a count distribution centered around $N$, an integer we call the particle number" to "bacteria aren't real" seems like a long mostly philosophical discussion.

Just to put it in a more physical context concerning photons: I think what's measured today in quantum-optics labs is mostly well-described by the appropriate autocorrelation functions of electro-magnetic-field operators, as detailed in any textbook on quantum optics, e.g., in my favorite by Garrison and Chiao. This also automatically provides the necessary space-time information about the "clicks", i.e., that's how in a sense photons get "localized", but not in the sense of position observables but in the sense of space-time information of the "click events".

The expectation values have to be taken with the appropriate states describing the preparation of the electromagnetic field being detected. Some usual ones nowadays range from thermal radiation, described by the Statistical Operator $\propto \exp(-\beta \hat{H})$, coherent states (e.g., Gaussian beams from a laser), and finally also Fock states with a definite photon number (like single-photon states, entangled pairs from parametric downconversion).

Qualitatively photons are the prime example for the fact that "particle properties" occur in the detection process, i.e., without detection there's not even a well-defined notion of localizability of a photon as some "point-like object". Formally that's seen from the fact that one cannot construct a proper position operator for photons.

In the relativistic realm it's even difficult for massive particles since due to the constraint by the relativistic "speed limit" localizability of massive particles is although constrained though a position observable is always constructable, but make usually "particle sense" only for asymptotic free states. The reason is also heuristically simple: If you want to localize a massive particle like an electron you somehow have to force it into a small region in space, which rather leads to the creation of new particles like electron-positron pairs than a better localization as intended.

Consequently also a "particle-number observable" is available only for asymptotic free states. The interpretation of "transient states" in scattering processes in terms of "particle states" is at least problematic. If I understand @DarMM right, it's even mathematically impossible within mathematically more rigid formulations of the theory!

 

[DarMM]

How do you reconcile this with the existence of the worldline formalism?

To be clear, I am not claiming a photon/any particle has *classical* path, I am just claiming it is reasonable to say the photon/any particle was present along the superposition of possible spacetime paths between emission and absorption, after accouting for interference among such paths.

A path integral is a way of computing the analytic continuation of quantum theoretic correlation functions, that is computing Schwinger functions. Thinking of this in terms of "being in a superposition of paths" isn't valid to me because it's a decomposition only possible after Wick Rotation where the Schrodinger equation becomes a Heat equation and thus can be recast as a modified Brownian motion.

But coherent states are in the global Fock space (can be seen as superpositions of the number operator) and are more easily localized than particle states/have even more classical seeming paths. The question was identifying some real world measurement which undermines the notion that massless particles/excitations sweep out a continuous, microcausal path through spacetime, such that we must restrict our speech to only detector click correlations.

Coherent states are just a simple example. As you said they're not particle states and their operators don't commute with particle operators. So we have non-particle states that are complimentary to particle ones, thus you can't really think of particles as being fundamental since there are observables that are complimentary to them.

Of course they are very classical in a sense, one can have non-particle states that are highly quantum. They're just a simple example of a very "non-particle" state, the field like states with minimum uncertainty. The actual Hilbert space is not a Fock space, so that fact that "field like" coherent states can always be expressed as a superposition of particle states in a free theory doesn't matter in the more general interacting case.

@vanhees71 has interesting details above about photons and detectors.

 

[vanhees71]

Why not? If this concerns particle indistinguishability or higher dimensional configuration space, I think this position confuses Fock space indices with the localized wavepackets that actually matter/interact in the real world. See https://arxiv.org/abs/1002.2544

I've already have a hard time to understand the arguments of the introduction. How can the authors claim that "all electrons in the universe are in the same state"? As fermions they are obviously not. The Fock space basis wrt. the single-particle position-spin basis shows that of course no two electrons can be in the same single-particle state since $[\psi^{\dagger}(\vec{x},\sigma_z)]^2=0$ (I argue with non-relativistic electrons to avoid complications with relativistic particles, but in principle the arguments are all the same).

Correct is of course that in any $N$-indistinguishable-particle Fock state, which is what the authors seem to discuss in the introduction, the particles cannot be individualized. Writing the state down in position-space-spin representation you can only give the probability distribution for finding the $N$ particles around any given place in $3N$-dimensional with given spin components, but it's impossible to somehow specify anyone individual particle. This is of course even more the case for more general states, i.e., there's no way to prepare a collection of any number of particles (or some state where the particle number is not determined either) such that one can individualize one or more particles. That's why one should not talk about "identical particles" but about "indistinguishable particles". You cannot say if you measure an electron, whether it's the same, you measured some time before or if it's another one since there's no way to individualize any electron against any other. They are indistinguishable but not necessarily identical. The authors are of course right in saying that particles are distinguishable whenever they differ in at least one intrinsic property (within the standard model by mass, spin, and various charge-like quantities like electric charge, baryon number, lepton number, etc.). E.g., an electron and a muon is distinguishable by their masses (but nothing else, being both charged leptons!).

The good thing is that we don't have to care much about this indistinguishability descriptions by just using the QFT formulation (no matter whether you deal with a relativistic or non-relativistic model). Then you don't even fall into the trapp of trying to individualize particles by their position and spin (or momentum and spin or whatever single-particle basis you like to use in an application). E.g., instead of writing down antisymmetrized $N$-electron wave functions ("Slater determinants") you write down the corresponding particle-number eigenstates, saying I prepare a state $|\{N(\vec{x},\sigma_z) \}_{\vec{x} \in \mathbb{R}^3,\sigma_z \in \{\pm \hbar/2\}}$, which just tell me the number of particles at each position and spin with $N(\vec{x},\sigma_z) \in \{0,1 \}$ for all $\vec{x} \in \mathbb{R}^3$ and $\sigma_z \in \{\pm \hbar/2 \}$.

 

[ftr]

To be clear, I am not claiming a photon/any particle has *classical* path, I am just claiming it is reasonable to say the photon/any particle was present along the superposition of possible spacetime paths between emission and absorption, after accouting for interference among such paths.

I think the problem is that QM is silent on the question of path (because of the nature of the model) that does not mean the (single)particle doesn't have a path. But if you have model that does predict the path( electron or photon) you are welcome to publishing it.

 

[dsaun777]

Well, with there's always some probability that a photon is not registered, but if it's registered then at one point and only one point of the screen.

When you say one point what does that mean, that the photon is absorbed by a single atom?

 

[vanhees71]

The closest thing to a "path" is imho a sequence of observations as produced in a cloud chamber and is well understood in a now classical paper by Mott:

https://doi.org/10.1098/rspa.1929.0205

 

[charters]

Coherent states are just a simple example. As you said they're not particle states and their operators don't commute with particle operators. So we have non-particle states that are complimentary to particle ones, thus you can't really think of particles as being fundamental since there are observables that are complimentary to them.

I think the issue I was trying to focus on got lost yesterday- I really was not at all trying to say particle states as eigenstates of the number operator are fundamental. So, above you said:

People are lead to think photons are something that hits the camera and causes the excitation of a pixel, where as under QED it is more the case that a photon is the excitation of a pixel in a camera and QED gives rules for the probability of a given pixel being excited a given amount.

I didn't read this claim as having anything to do with the EM/photon field being in specifically a particle state or not. It seems to apply just as much to coherent states. I read this as a rejection that anything "hits" detectors, or travels between detectors, and so all we have are the correlations among macro detectors that spontaneously click due to some notion of causally delayed direct action.

But are you saying instead that you do think that EM field excitations in some general form traverses the spacetime between source and detector, but that the discreteness of detector responses, naively attributed to the field being in a pre-existing discrete n-particle state, is in fact just a feature of the field-detector (or more generally field-atom) interaction?

 

[DarMM]

I read this as a rejection that anything "hits" detectors, or travels between detectors, and so all we have are the correlations among macro detectors that spontaneously click due to some notion of causally delayed direct action

It's not a rejection that something interacts with the detectors, or a proposal of direct action. It's a just a statement of what seems to happen in QFT. When a light source (or more generally a matter source) causes an effect in a given detector, if the detector is of an appropriate design and placed far enough away from areas of interaction then the effects on the detector can be understood in particulate terms using asymptotic particle states. If not, then it can't.

Thus particles are only associated with late time detectors placed appropriately.

 

[DarMM]

I think the problem is that QM is silent on the question of path (because of the nature of the model) that does not mean the (single)particle doesn't have a path. But if you have model that does predict the path( electron or photon) you are welcome to publishing it.

QM simply doesn't use paths at all. I think this is "silence" in the same way GR is silent about the universe being embedded in a larger spacetime. Similarly Maxwellian electromagnetism is silent on the notion of the EM field being a limiting case of some more general field.

Really the success of QM shows the notion of paths just doesn't seem to be needed at all for current observations.

I think it should be mentioned that attempting to come up with paths for particles is highly constrained by no-go theorems such as the Kochen-Spekcer theorem and the PBR theorem.

 

[DarMM]

Consequently also a "particle-number observable" is available only for asymptotic free states. The interpretation of "transient states" in scattering processes in terms of "particle states" is at least problematic. If I understand @DarMM right, it's even mathematically impossible within mathematically more rigid formulations of the theory!

Sorry forgot to answer this. Yes indeed, impossible.

 

[charters]

When a light source (or more generally a matter source) causes an effect in a given detector, if the detector is of an appropriate design and placed far enough away from areas of interaction then the effects on the detector can be understood in particulate terms using asymptotic particle states. If not, then it can't.

Sorry for the delay, but I wanted to follow up on this. Suppose we have a macroscopic device which experimentalists would consider a good single photon source. What would be a concrete (can actually be built) detector of appropriate design which we could use to measure this emission in such a way that the effect could *not* be understood in terms of an asymptotic 1 photon state?

 

[DarMM]

Sorry for the delay, but I wanted to follow up on this. Suppose we have a macroscopic device which experimentalists would consider a good single photon source. What would be a concrete (can actually be built) detector of appropriate design which we could use to measure this emission in such a way that the effect could *not* be understood in terms of an asymptotic 1 photon state?

Well if the source is defined in terms of a single photon state, the statistics of experiments are going to be compatible with a single photon state. This is like how if you prepare a momentum (near) eigenstate the statistics of any experiment are going to be compatible with a momentum eigenstate, both position and momentum measurements. Simply because that is the state. However the statistics in the two bases exhibit complimentarity. Thus it is for a single photon state, it can be measured in a non-photon basis. However even ignoring this most states in quantum optics are not single photon states and even calling something a "single photon state" is an idealisation.

Asher Peres has some interesting points (https://arxiv.org/abs/quant-ph/0212023):

Although states with a definite number of particles area useful theoretical concept, a look at quantum optics techniques or at the Table of Particle Properties shows that experimentally accesible quantum states are usually not eigenstates of particle number operators. In general any process that is not explicitly forbidden by some conservation law has a non-zero amplitude (Weinberg, 1995;Peskin and Schroeder, 1995; Haag, 1996). There are multiple decay channels, extra soft photons may always ap-pear, so that the so-called ‘one-photon’ states are often accompanied by soft multiphoton components,
$\alpha|\Omega\rangle + \beta|1_{\omega}\rangle + \gamma|2_{\omega^{′}\omega^{′′}}\rangle +..., |\beta|∼1$ (63)
Thus the physical realization of a single qubit is itself necessarily an idealization.

So a single photon state is a more fraught concept than one would think.

 

[charters]

Thus it is for a single photon state, it can be measured in a non-photon basis

Right, I am just asking how exactly this can be done in practice. Typically (afaik) a coherent state is measured by 1) recording the number of clicks across many runs, 2) treating each click as indicative of an individual photon, 3) confirming the distribution of clicks converges to the Born rule distribution of the number states which describe the coherent state in the number basis. I am not aware of a detector which can directly measure radiation on a non-number basis.

Likely I am just missing a counterexample because I haven't been exposed to all the experimental possibilities but I haven't had luck finding one so far.

 

[DarMM]

Homodyne detectors are probably the simplest examples.