A Realism from Locality? Bell's Theorem & Nonlocality in QM

[vanhees71]

vanhees71, aren't you shooting superposition between the eyes.

?

 

[ftr]

?

Ok I will elaborate ( I guess I am summarizing while multitasking my life ). I thought the whole enigma of the entangled particles is that the spin(whatever) are in superposition before the measurement. I have heard about the "socks" model, but it seems so unconvincing, also the conservation momentum camp ...etc.

 

[Lord Jestocost]

I always thought that "minimal statistical" and "ensemble" interpretations are just the name of the same interpretation. If not, what's the difference. Is this again one of these unnecessary confusions due to (unnecessary?) philosophical sophistry?

Hilary Putnam in “Philosophical Papers: Volume 1, Mathematics, Matter and Method”, Second Edition, 1979, p. 147:
“To put it another way, it is a part of quantum mechanics itself as it stands today that the proper interpretation of the wave is statistical in this sense: the square amplitude of the wave is the probability that the particle will be found in the appropriate place if a measurement is made (and analogously for representations other than position representation). We might call this much the minimal statistical interpretation of quantum mechanics, and what I am saying is that the minimal statistical interpretation is a contribution of the great founders of the CI— Bohr and Heisenberg, building, in the way we have seen, on the earlier idea of Born — and a part of quantum mechanical theory itself. However, the minimal statistical interpretation is much less daring than the full CI. It leaves it completely open whether there are any observables for which the principle ND is correct, and whether or not hidden variables exist. The full CI, to put it another way, is the minimal statistical interpretation plus the statement that hidden variables do not exist and that the wave representation gives a complete description of the physical system.”
[CI means “Copenhagen Interpretation”, italic in original, principle ND: see footnote **, LJ] [bold by LJ]

** Hilary Putnam in “Philosophical Papers: Volume 1, Mathematics, Matter and Method”, Second Edition, 1979, p. 140:
“Principle ND says that an observable has the same value (approximately) just before the measurement as is obtained by the measurement; the CI denies that an observable has any value before the measurement.”

Oh come on, statistical and probablistic is really synonymous if it comes to the application of probability theory

When reasoning about random outcomes of measurements, one can now question how this randomess emerges: In a statistical way (“classical randomness”, that's what the ensemble interpretation is yearning for) or in a probabilistic way (“quantum randomness” ).

Richard D. Gill in “Statistics, Causality and Bell’s Theorem”:
“In classical physics, randomness is merely the result of dependence on uncontrollable initial conditions. Variation in those conditions, or uncertainty about them, leads to variation, or uncertainty, in the final result. However, there is no such explanation for quantum randomness. Quantum randomness is intrinsic, nonclassical, irreducible. It is not an emergent phenomenon. It is the bottom line. It is a fundamental feature of the fabric of reality.”
[italic in original, LJ]

 

[Jimster41]

The correlations are not caused by the measurements but are due to the correlation following from the preparation in an entangled state.

You make it sound like entanglement is some rare artificially induced thing, and therefore non-locality ("a-causality" is a term I have heard applied) between entangled elements just an exotic oddity - philosophically curious.

But I've always been confused about where entanglement is naturally found. Is it natural and ubiquitous in addition to being an isolated laboratory phenomenon? I mean isn't it natural and ubiquitous?

Curious if this paper is credible. To my mind, entanglement must be ubiquitous, uniform even, the Cauchy surface of causality everywhere. What machinery is there anywhere in nature that is not microscopically (i.e. fundamentally) evolving according to the phenomena of QM and/or QFT?

Except maybe stuff in the... distant past? Even with that I keep wondering, how far back? If the pilot wave, or QFT wave function is here (now) and extends out over some space-time region where else did, does, do, will it go? how far? why that far? and to what, and why that?

https://arxiv.org/abs/1106.2264v3

Entanglement thresholds for random induced states
Guillaume Aubrun, Stanislaw J. Szarek, Deping Ye
(Submitted on 11 Jun 2011 (v1), last revised 15 Oct 2012 (this version, v3))

For a random quantum state on H=Cd⊗Cd obtained by partial tracing a random pure state on H⊗Cs, we consider the whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold s0=s0(d) of order roughly d3. More precisely, for any a>0 and for d large enough, such a random state is entangled with very large probability when s<(1−a)s0, and separable with very large probability when s>(1+a)s0. One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold k0=k0(N)∼N/5 such that two subsystems of k particles each typically share entanglement if k>k0, and typically do not share entanglement if k<k0. Our methods work also for multipartite systems and for "unbalanced" systems such as Cd⊗Cd′, d≠d′. The arguments rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces; some of the auxiliary results may be of reference value. A high-level non-technical overview of the results of this paper and of a related article arXiv:1011.0275 can be found in arXiv:1112.4582.

 

[microsansfil]

Its pretty tough only if one does not accept that there is nothing more about a particle than a click in a detector.

Nothing more about a particle than those who are brought to our consciousness. We human beings, let us be aware of the physical phenomena through our senses: f(r,t), g(sound, t), ...

And guess what, many experts in the field do not accept it.

it's not surprising: QBism and the Greeks

/Patrick

 

[Auto-Didact]

I'd say that "suggests" and "makes plausible" are synonymous. But there is nothing certain in a plausibility argument. The appropriate wording in the sentence would have been ''suggests to me'',
since plausibility is in the eye of the beholder.

As I said, this definitely seems to be a foreign language speaker issue: the confusion arises from the stem 'certain-'; the word 'certainly' in this context has nothing directly to do with 'certainty', i.e. "certainly suggests" means "it is true that it suggests", which clearly is in contrast to "suggests (with) certainty", which means "suggests that it is true".

The statements "it is true that it suggests" and "suggests that it is true" are converse to each other, i.e. certainly not synonymous (pun intended). I know, this is literally arguing semantics and linguistics, but with you being a mathematician with a strong urge for absolute precision in reasoning, I suspect that you (secretly) enjoy such subtleties.

Oh come on, statistical and probablistic is really synonymous if it comes to the application of probability theory to real-world problems, and QT is also a kind of probability theory.

Statistical and probabilistic are not synonymous. Statistics is an empirical methodology based largely on (certain forms of) probability theory, while probability theory is a field in mathematics such as geometry.

In the simplest cases of applications of probability theory in the form of mathematical models - i.e. how statistics is mostly used in the practice of physics - the two cases tend to be the same, but this is purely contingent upon the simplicity of the phenomena of physics and their idealizeable nature, strongly contrasting the mathematical descriptions of other phenomena studied by the other sciences.

I always thought that "minimal statistical" and "ensemble" interpretations are just the name of the same interpretation. If not, what's the difference. Is this again one of these unnecessary confusions due to (unnecessary?) philosophical sophistry?

This is not confusion due to philosophical sophistry, but more confusion due to a wrongly assumed equivalence relation between two different classes/sets: the relation between the elements of the set of statistics and the elements of the set of applications of probability theory is not bijective; the latter set is far larger than the former and moreover, the former set has relations with other formal domains as well, e.g. logic.

In other words, statistics (the textbook subject) based on probability theory in fact is based on a very small subset of probability theory, while the rest of probability theory specifically does not feature in it. Applications coming from the rest of probability theory which have the same form and intended use as statistics are models of non-standard statistics, usually invented and studied by mathematical statisticians.

In practice, applied statisticians and non-physical scientists do not acknowledge such non-standard models as statistics, but see them more as alternate theories, such as all alternative speculative theories without any experimental validation from theoretical physics (e.g. string theory) are seen by most physicists. This is also similar to how physicists view falsified physical theories from the history of physics, i.e. as "theories made and used by physicists historically which are today not anymore part of physics".

To conclude, this is partly because probability theory is a far bigger subject than just the textbook subject, in exactly the same way that mechanics is in actuality a far bigger subject than 'just classical mechanics' and also a far bigger subject than 'classical mechanics plus quantum mechanics'.

The pragmatic restriction of mechanics to mean 'only CM and QM' is actually not named mechanics, but canonical mechanics; but most physicists don't use or respect this terminology anymore because they lack the adequate philosophical training, making them terrible at the reasoning required for foundational research in stark contrast to mathematicians, logicians and philosophers.

 

[Auto-Didact]

You make it sound like entanglement is some rare artificially induced thing, and therefore non-locality ("a-causality" is a term I have heard applied) between entangled elements just an exotic oddity - philosophically curious.

But I've always been confused about where entanglement is naturally found. Is it natural and ubiquitous in addition to being an isolated laboratory phenomenon? I mean isn't it natural and ubiquitous?

Your suspicions are of course warranted: entanglement is ubiquitous, almost all quantum states in Nature are entangled states, but decoherence of course breaks these entanglements, which of course is why building a quantum computer is such an engineering challenge.

But to make the argument even stronger, in textbook QM the description of $\psi$ is non-local whether or not entanglement is involved, i.e. even for a single particle wavefunction non-locality is already present in the following example given by Penrose about a decade ago or earlier:

Imagine a photon source and a screen some distance away and single photons are detected (or measured) as single points on the screen; in between source and screen is where the wavefunction is. Now imagine that there is a detector at each point of the screen; once the photon is detected at a single point on the screen we can call it a detection event.

Each single detection event on the screen instantaneously prohibits the photon from being seen anywhere else on the screen i.e. once a detection event takes place by a single detector, all other detectors are effectively instantaneously prohibited from detecting the photon; if the detector had to communicate this detection event to all the other detectors it would need to convey that information faster than light.

In other words, detection i.e. measurement itself breaks the non-locality of the wavefunction; this can be mathematically described in detail as measurements effectively removing the first cohomology element of single photon wavefunctions (NB: these photon wavefunctions are usually Fourier transformed wavefunctions, and together with their Fourier transforms reside in a larger abstract complex analytic mathematical space).

 

[kent davidge]

event A can only be the cause of event B if it is time- or light-like separated from B and B is within or on the future light cone of A

but wouldn't that make it possible to have a frame in which the two events occurs simultaneously? even though the two frames can't be Lorentz connected?

 

[RUTA]

They can accept its truth, but not its completeness. They want to know what happens behind the curtain.

On the other hand, those who are satisfied with the purely epistemic interpretation either
(i) don't care about things behind the curtain, or
(ii) care a little bit but don't think that it is a scientific question, or
(iii) claim that there is nothing behind the curtain at all.
Those in the category (i) have a mind of an engineer, which would be OK if they didn't claim that they are not engineers but scientists. Those in the category (ii) often hold double standards because in other matters (unrelated to quantum foundations) they often think that questions about things behind the curtain are scientific. Those in the category (iii) are simply dogmatic, which contradicts the very essence of scientific way of thinking.

I believe there is nothing behind the curtain, not because it's irrelevant, but because there really is no thing behind the curtain and there is no dynamical story to tell. QM is providing an adynamical constraint on the distribution of momentum exchange in spacetime without a dynamical counterpart. How is that unscientific? Must all scientific explanation be dynamical?

 

[Auto-Didact]

I believe there is nothing behind the curtain, not because it's irrelevant, but because there really is no thing behind the curtain and there is no dynamical story to tell. QM is providing an adynamical constraint on the distribution of momentum exchange in spacetime without a dynamical counterpart. How is that unscientific? Must all scientific explanation be dynamical?

All other scientific accounts of phenomena ever given so far have ultimately turned out to be dynamical. Because the question is still mathematically wide open, i.e. the correct mathematical description to fully describe the problem has not yet been found or proven to not exist, the question is currently an open question in mathematical physics.

The very fact that Bohmian mechanics even exists at all and may even be made relativistic, is proof that this is a distinctly scientific open problem of theoretical physics within the foundations of QM, let alone the existence of alternate theories waiting to be falsified experimentally and the existence of the open problem of quantum gravity.

It is therefore by all accounts vehemently shortsighted and extremely premature to decide based upon our best experimental knowledge that a dynamical account is a priori impossible; the experimental knowledge itself literally indicates no such thing, instead this is a cognitive bias coming from direct extrapolation of our effective models to arbitrary precision.

 

[RUTA]

All other scientific accounts of phenomena ever given so far have ultimately turned out to be dynamical. Because the question is still mathematically wide open, i.e. the correct mathematical description to fully describe the problem has not yet been found or proven to not exist, the question is currently an open question in mathematical physics.

The very fact that Bohmian mechanics even exists at all and may even be made relativistic, is proof that this is a distinctly scientific open problem of theoretical physics within the foundations of QM, let alone the existence of alternate theories waiting to be falsified experimentally and the existence of the open problem of quantum gravity.

It is therefore by all accounts vehemently shortsighted and extremely premature to decide based upon our best experimental knowledge that a dynamical account is a priori impossible; the experimental knowledge itself literally indicates no such thing, instead this is a cognitive bias coming from direct extrapolation of our effective models to arbitrary precision.

If constraint-based, adynamical explanation only resolved the mysteries of QM, then it might be crazy to consider it. But, it also does so for GR, as we show in our book (Beyond the Dynamical Universe). I think this is precisely why “Einstein’s double revolution” remains unfinished (Smolin’s lingo). Modern physics is complete (minus QG) and self-consistent, it’s us physicists who haven’t realized the Kuhnian revolution for what it is, i.e., “ascending from the ant’s-eye view to the God’s-eye view of physical reality is the most profound challenge for fundamental physics in the next 100 years” (Wilczek).

 

[PeterDonis]

in my understanding, causality implies a specific time-ordering

Yes, but that's a statement about your preferred use of ordinary language, not about physics. We all agree on the physics: we all agree that spacelike separated measurements commute and that such measurements on entangled quantum systems can produce results that violate the Bell inequalities. It would be nice if the discussion could just stop there, but everyone insists on dragging in vague ordinary language terms like "causality" and "locality" and arguing about whether they are appropriate terms to use in describing the physics that we all agree on.

In other words event A can only be the cause of event B if it is time- or light-like separated from B and B is within or on the future light cone of A.

Where does this requirement show up in QFT? QFT is time symmetric.

It's not clear to me, how you define causality to begin with.

The same way you've been defining it (you sometimes use the term "microcausality", but sometimes not): that spacelike separated measurements commute. But now you seem to be shifting your ground and giving a different definition of "causality" (the one I quoted above), the basis of which in QFT I don't understand (which is why I asked about it).

due to microcausality there is no cause-effect relation between space-like separated measurement events

So microcausality means no causal relationship? That seems like an odd use of language.

In this case the entanglement is due to selection (or even post-selection!) of a subensemble out of an before (in the correct relativistic sense!) created system of two entangled (but not among them entangled) photon pairs. Note however that each of these pairs have been created in an entangled state by causal local interaction (SPDC of a laser photon in a BBO crystal).

The way I would describe all this is not that entangled pairs do not have to be causally connected. The way I would describe it is that QFT, strictly speaking, does not admit the concept of "an entangled pair", because it does not admit the concept of the "state" of an extended system at an instant of "time". It only admits measurement events and correlations between them, and it predicts the statistics of such correlations using quantum field operators that obey certain commutation relations. Each individual such operator is tied to a specific single event in spacetime: that's what makes it "local". Any talk about "entangled systems" measured at spacelike separated events is just an approximation and breaks down when you try to look too closely.

Again, this is all about how to describe things in ordinary language, not about physics. Basically many people do not like the extreme viewpoint I just described, which is IMO the proper consistent way to describe what QFT is saying. Many people do not want to give up the notion of "entangled systems" containing multiple spatially separated particles. But that notion IMO is a holdover from non-relativistic physics and needs to be given up in a proper account of what QFT says, if we are going to talk about how best to describe the physics in ordinary language and we aren't willing just to stop at the point of describing the physics in its most basic terms (which I gave at the start of this post).

 

[PeterDonis]

wouldn't that make it possible to have a frame in which the two events occurs simultaneously?

No. If two events are timelike or null (lightlike) separated, there is no frame in which they are simultaneous.

even though the two frames can't be Lorentz connected?

I have no idea what you mean by this.

 

[zonde]

I believe there is nothing behind the curtain, not because it's irrelevant, but because there really is no thing behind the curtain and there is no dynamical story to tell. QM is providing an adynamical constraint on the distribution of momentum exchange in spacetime without a dynamical counterpart. How is that unscientific? Must all scientific explanation be dynamical?

There are more than one point on which your position is unscientific.
First, you believe in one true explanation. Just because you have an explanation that fits observations does not mean that there can't be other explanations.
Second, the process of gaining scientific knowledge is ... well a process, a dynamical story as you call it. What is the point of denying value of dynamical approach and then seeking justification for that from perspective of dynamical approach. It's stolen concept fallacy.
So answering your question: "Must all scientific explanations be dynamical?" - yes, all scientific explanations must be dynamical because only testable explanations are scientific and the process of testing is dynamical, you have initial conditions and then you observe what happens and if your observations agree with predictions.

 

[Demystifier]

I believe there is nothing behind the curtain, not because it's irrelevant, but because there really is no thing behind the curtain and there is no dynamical story to tell. QM is providing an adynamical constraint on the distribution of momentum exchange in spacetime without a dynamical counterpart. How is that unscientific? Must all scientific explanation be dynamical?

It's scientific to say: Maybe there is nothing behind the curtain, it seems very likely to me that it is so.
But it's not scientific to say: There is nothing behind the curtain, period.

 

[vanhees71]

You make it sound like entanglement is some rare artificially induced thing, and therefore non-locality ("a-causality" is a term I have heard applied) between entangled elements just an exotic oddity - philosophically curious.

Where did I say this? Entanglement is the rule rather than the exception. Alone from the fact that we have indistinguishable particles and thus Bose or Fermi symmetric/anti-symmetric Fock spaces leads to a lot of entanglement.

 

[vanhees71]

but wouldn't that make it possible to have a frame in which the two events occurs simultaneously? even though the two frames can't be Lorentz connected?

No, for time- or lightlike events the time ordering is the same in any frame (of course in SRT; in GR it's more complicated and it holds only in a local sense).

 

[vanhees71]

Yes, but that's a statement about your preferred use of ordinary language, not about physics. We all agree on the physics: we all agree that spacelike separated measurements commute and that such measurements on entangled quantum systems can produce results that violate the Bell inequalities. It would be nice if the discussion could just stop there, but everyone insists on dragging in vague ordinary language terms like "causality" and "locality" and arguing about whether they are appropriate terms to use in describing the physics that we all agree on.

Causality is not vague but a fundamental assumption underlying all physics. Locality is another case since there a lot of confusion arises from the fact that too often people don't distinguish between causal effects and (predetermined) correlations. This becomes particularly problematic when it comes to long-range correlations between entangled parts of a quantum system.

Where does this requirement show up in QFT? QFT is time symmetric.

Indeed, ignoring weak interactions the Standard Model is T-invariant. Nevertheless the S-matrix provides a time ordering. You define an initial state (usually two asymptotic free particles) and then look for the transition probability rate to a given final state. This reflects how we can do experiments, and there's always this time ordering: Preparation of a state and then measuring something. T invariance then just means that (at least in principle) the "time-reversed process" is also possible and leads to the same S-matrix elements.

The same way you've been defining it (you sometimes use the term "microcausality", but sometimes not): that spacelike separated measurements commute. But now you seem to be shifting your ground and giving a different definition of "causality" (the one I quoted above), the basis of which in QFT I don't understand (which is why I asked about it).

I don't understand, what's unclear about this. To motivate the microcausality constraint, which is usually also called locality of interactions, you need the causality principle, and in Q(F)T it's even a weak one, i.e., you need to know only the state at one point in time to know it, given the full dynamics or Hamiltonian of the system, to any later point in time, i.e., it's causality local in time. You don't need to know the entire history before one "initial point" in time.

So microcausality means no causal relationship? That seems like an odd use of language.

Microcausality ensures that there are no faster-than-light causal connections. Given the general causality assumption that's a necessary consequence: Two space-like separated events do not define a specific time order and thus one event cannot be the cause of the other.

The way I would describe all this is not that entangled pairs do not have to be causally connected. The way I would describe it is that QFT, strictly speaking, does not admit the concept of "an entangled pair", because it does not admit the concept of the "state" of an extended system at an instant of "time". It only admits measurement events and correlations between them, and it predicts the statistics of such correlations using quantum field operators that obey certain commutation relations. Each individual such operator is tied to a specific single event in spacetime: that's what makes it "local". Any talk about "entangled systems" measured at spacelike separated events is just an approximation and breaks down when you try to look too closely.

Of course QFT admits entangled states. We write them down all the time discussing about photons. Measurements are just usual interactions between entities described by the fields, and due to microcausality they are local, i.e., there cannot be any causal influence of one measurment event on another measurement event that is space-like separated. I.e., if A's detector clicks this measurement event can not be the cause of anything outside of the future light cone of this event.

Again, this is all about how to describe things in ordinary language, not about physics. Basically many people do not like the extreme viewpoint I just described, which is IMO the proper consistent way to describe what QFT is saying. Many people do not want to give up the notion of "entangled systems" containing multiple spatially separated particles. But that notion IMO is a holdover from non-relativistic physics and needs to be given up in a proper account of what QFT says, if we are going to talk about how best to describe the physics in ordinary language and we aren't willing just to stop at the point of describing the physics in its most basic terms (which I gave at the start of this post).

Of course, there are entangled states and there are the correspondingly observed strong correlations between far distant measurements, and all that is describable by relativistic QFT. It's also clear that the localizability also of massive particles, which have a position observable, is much more constrained in relativstic QFT than in non-relativistic QM since rather than localizing a particle better and better by "squeezing" it somehow in an ever smaller region in space you tend to create new particles.

Note that in general the location of an entity described by QFT is determined by the location of a measurement device with a finite spatial resolution; it's not necessary that the measured systems have position observables, as the example of photons shows: All observable there is is that a detector located in some spatial region registers a photon or not.

If QFT couldn't describe the observed entanglement between, e.g., photons and the corresponding violation of Bell and other related inequalities, it wouldn't be complete. A theory must describe all known observational facts, and entanglement is obviously an observational fact.

 

[vanhees71]

So answering your question: "Must all scientific explanations be dynamical?" - yes, all scientific explanations must be dynamical because only testable explanations are scientific and the process of testing is dynamical, you have initial conditions and then you observe what happens and if your observations agree with predictions.

Of course, QT provides a description of the dynamics of the system and the measurable quantities related with it. That's what QT is all about. I may be buried in many introductory courses, because students tend to get the impression that rather all there is are stationary states (i.e., eigenstates of the Hamiltonian), but that's only "statics" in a sense. Also in hydrodynamics or classical electrodynamics you can stick with static or stationary special cases, but still hydro as well as Maxwell theory are indeed descriptions of the dynamics of the described system (fluids and charges and the em. field, respectively).

 

[julcab12]

Note that in general the location of an entity described by QFT is determined by the location of a measurement device with a finite spatial resolution; it's not necessary that the measured systems have position observables, as the example of photons shows: All observable there is is that a detector located in some spatial region registers a photon or not.

A photon for instance. In a visible -UV spectroscopy. A photon is an object that has definite frequency V and definite energy hv. However, Its size and position are unknown or undefined even if it is absorbed and emitted by a molecule. OTOH, a photon in a quantum optics experimenter, detection correlation studies,; it has no definite frequency, but has somewhat defined position and size, looks localized particle when it gets detected in a light detector. The high energy experimenter talks about is a small particle that is not possible to see in photos of the particle tracks and their scattering events, but makes it easy to explain the curvature of tracks of matter particles with common point of origin within the framework of energy and momentum conservation (e. g. appearance of pair of oppositely charged particles, or the Compton scattering). This photon has usually definite momentum and energy (hence also definite frequency), and fairly definite position, since it participates in fairly localized scattering events. At the end of the day, the only common denominator is a mathematical description of EM field and its interaction with or some version of fock states, countable things. One measurable dynamic of reality is Time passes at different rates from place to place. Locality is always an approximation in the dynamical sense. Somewhat frozen image/depiction/description/detection of things that is always formless dynamic in nature --unless it interacts. That is exactly the view of Rovelli.

Scientific Realism



https://arxiv.org/pdf/1508.05533.pdf
"The observed dynamics of the world is time-reversal invariant: a given un-oriented sequence of quantum events does not determine a time arrow. Charging the wave function, (more in general, the quantum state) with a 4 realistic ontological interpretation, leads to a picture of the world where this invariance is broken. It is not broken in the sense that the full theory breaks T-reversal invariance (it does not), but in the sense that the wave function we associate to observed events depends on a choice of orientation of time. "

 

[Demystifier]

Somewhat frozen image/depiction/description/detection of things that is always formless dynamic in nature --unless it interacts. That is exactly the view of Rovelli.

I think it doesn't make sense. To see why, suppose that the universe contains only a hydrogen atom and nothing else. In the hydrogen atom, the electron interacts with the proton. Does it create the "form" of the electron?

 

[Heikki Tuuri]

I think the word "causality", or the phrase "A causes B", is informal, heuristic language.

If there were something called "free will", we could interpret that if I choose to do A, then B necessarily happens. But no one in the past 2,500 years has been able to give a sensible definition of free will.

A more exact phrase is that "from A and other initial conditions we can calculate B and other end results". It is a mathematical problem, and in mathematics we do not use the word "causes".

There was recently a thread where the word "cause" is central:
https://www.physicsforums.com/threads/quantum-interpretations-of-this-optical-effect.974340/

In the above paper, Aharonov et al. claim that photons detected at D2 mystically "cause" a mirror to be pushed to the left. I think they calculated a correlation: observation of photons at D2 correlates with an observed force to the left on the mirror.

 

[julcab12]

I think it doesn't make sense. To see why, suppose that the universe contains only a hydrogen atom and nothing else. In the hydrogen atom, the electron interacts with the proton. Does it create the "form" of the electron?

It doesn't say that. Well according to him. locality of quantum mechanics is by postulating relativity to the observer for events and facts, instead of an absolute “view from nowhere”. The main ontology of “observers”, measurement interactions and relative events. And, it doesn't say any form or becomes meaningless otherwise. Besides the only way to detect/picture a electron(seen as local) is through electron interacting.

 

[Jimster41]

Your suspicions are of course warranted: entanglement is ubiquitous, almost all quantum states in Nature are entangled states, but decoherence of course breaks these entanglements, which of course is why building a quantum computer is such an engineering challenge.

But to make the argument even stronger, in textbook QM the description of $\psi$ is non-local whether or not entanglement is involved, i.e. even for a single particle wavefunction non-locality is already present in the following example given by Penrose about a decade ago or earlier:

Imagine a photon source and a screen some distance away and single photons are detected (or measured) as single points on the screen; in between source and screen is where the wavefunction is. Now imagine that there is a detector at each point of the screen; once the photon is detected at a single point on the screen we can call it a detection event.

Each single detection event on the screen instantaneously prohibits the photon from being seen anywhere else on the screen i.e. once a detection event takes place by a single detector, all other detectors are effectively instantaneously prohibited from detecting the photon; if the detector had to communicate this detection event to all the other detectors it would need to convey that information faster than light.

In other words, detection i.e. measurement itself breaks the non-locality of the wavefunction; this can be mathematically described in detail as measurements effectively removing the first cohomology element of single photon wavefunctions (NB: these photon wavefunctions are usually Fourier transformed wavefunctions, and together with their Fourier transforms reside in a larger abstract complex analytic mathematical space).

I’m familiar with the two slit experiment etc. But to me it seems just as sensible to interpret the vave-like interference pattern seen by the detector(s) as “confirming” or “realizing” the non-locality of the quantum field, especially if new entangled states ensue - as opposed to describing it as “decoherence”. But that may be what you were getting at.

I mean is Cauchy surface reduced? Or is the lab just - moved along with it - even if the detectors are saying “we just decohered that thing didn’t we” isn’t it (the entangled surface of “now”) just sitting there all up in them?

I get that the “prepared” entanglement is decohered. But what I’m curious about is the dynamics of natural uncontrolled, un-prepared entanglement. If the former realizes non-locality and faster than light superposition rule enforcement, how are those manifest in the natural evolution of the Cauchy surface.

Is there a specific notion of entanglement conservation?

 

[Heikki Tuuri]

The thought experiment of Roger Penrose, as mentioned above, shows that a single photon makes the photon detectors D1,... in its neighborhood "entangled" in the sense that if one detector clicked for the photon, then the others do not click for that same photon. Nature is obviously full of entanglement. In the Penrose example, entanglement is a result of "the photon A could have interacted with detectors D1,...".

How can one erase the Penrose entanglement? The detectors D1,... should forget what they might have measured. It is impossible with macroscopic objects.

 

[RUTA]

There are more than one point on which your position is unscientific.
First, you believe in one true explanation. Just because you have an explanation that fits observations does not mean that there can't be other explanations.
Second, the process of gaining scientific knowledge is ... well a process, a dynamical story as you call it. What is the point of denying value of dynamical approach and then seeking justification for that from perspective of dynamical approach. It's stolen concept fallacy.
So answering your question: "Must all scientific explanations be dynamical?" - yes, all scientific explanations must be dynamical because only testable explanations are scientific and the process of testing is dynamical, you have initial conditions and then you observe what happens and if your observations agree with predictions.

There may be other explanations, but as a physicist I have to stake my approach on just one. It took my math colleague and I three months to modify and apply Regge calculus to the SCP Union2 data and it took us four months to modify and program a fit to the Planck 2015 CMB power spectrum data. These are just two examples of the many papers I have written based on my one approach. Maybe a philosopher can write one paper on a particular approach this month and turn around and write another paper on another approach next month, but we don't have that luxury in physics.

It is true that physics is done dynamically, but that doesn't mean an explanation of what we find has to be ultimately dynamical. It is also true that we do astrophysics and cosmology from Earth, seeing the sky rotate about us, but we long ago abandoned geocentricism.

Changing from dynamical to adynamical explanation is revolutionary. As Skow said in his review, "It really is necessary to understand how radical this idea is. ... You can't explain A because B and B because A." But, in adynamical explanation (such as Einstein's equations of GR), it is precisely the case that "A (the spacetime metric) because B (the stress-energy tensor) and B because A." You can't input the SET to solve EE's for the metric unless you already know how to make spatiotemporal measurements, i.e., you already have the metric. And vice-versa of course. Solutions to EE's are self-consistent sets of the spacetime metric, energy, momentum, force, etc. on on the spacetime manifold, where "self-consistent" means "satisfies the constraint, i.e., EE's." That's why our proposed approach constitutes a Kuhnian revolution. When I started in foundations 25 years ago, I too was convinced that GR and/or QM were flat out wrong. Now, I believe (base my research approach on the fact that) they are in fact both right and beautifully self-consistent.

Every physicist has to stake their research on a particular model of "the real external world." I'm very happy now with mine because it shows modern physics is in fact complete (minus QG) and consistent, i.e., it is amazingly comprehensive and coherent ... as long as you're willing to give up your anthropocentric dynamical bias.

 

[Demystifier]

It's stolen concept fallacy.

Interesting.

 

[PeterDonis]

Causality is not vague

Not if you give it a precise definition, such as "operators at spacelike separated events commute", no. Which is why I've repeatedly suggested that we should all stop using vague ordinary language and start using precise math, or at least precise definitions that refer to precise math, as the one I just gave in quotes does. Then we can stop arguing about words and start talking about physics.

Locality is another case

If you define "locality" as "does not violate the Bell inequalities", then QFT is not local. If you define "locality" as meaning the same thing as "causality" does above, then QFT is local, but you've also used two ordinary language words to refer to the same physical concept, plus you've thrown away the usual way of referring in ordinary language to another physical concept.

In any case, once again, can we please stop using vague ordinary language?

the S-matrix provides a time ordering. You define an initial state (usually two asymptotic free particles) and then look for the transition probability rate to a given final state. This reflects how we can do experiments

In other words, it reflects how we experience things. But QFT does not explain why we experience things that way. We don't fully understand why we experience things that way.

Of course QFT admits entangled states. We write them down all the time discussing about photons

Sure, you can write down such states, but they include operators at different spacetime events (which can be spacelike separated events). So they're not "local" in the ordinary sense of the term.

Measurements are just usual interactions between entities described by the fields, and due to microcausality they are local

With your preferred definition of "local", yes. But they violate the Bell inequalities, which means they are not "local" in that sense of the term "local". Which, again, is why I keep saying we should stop using vague ordinary language.

there cannot be any causal influence of one measurment event on another measurement event that is space-like separated

With your preferred definition of "causal influence", yes. But not everybody shares your preferences for definitions of ordinary language terms. Which is why we should stop using them.

 

[PeterDonis]

a lot of confusion arises from the fact that too often people don't distinguish between causal effects and (predetermined) correlations.

Correlations that violate the Bell inequalities can't be "predetermined" locally. That's what Bell's Theorem shows.

 

[A. Neumaier]

Correlations that violate the Bell inequalities can't be "predetermined" locally. That's what Bell's Theorem shows.

They can only not be predetermined Bell-locally, by the latter's definition, as in your previous posts. But this is a tautology and means nothing.

 

[A. Neumaier]

Hm, in my understanding, causality implies a specific time-ordering. In fact it's the only sense you can give to specific time-ordering, and thus causally connected events cannot be space-like separated. In other words event A can only be the cause of event B if it is time- or light-like separated from B and B is within or on the future light cone of A. It's not clear to me, how you define causality to begin with.

The same way you've been defining it (you sometimes use the term "microcausality", but sometimes not): that spacelike separated measurements commute. But now you seem to be shifting your ground and giving a different definition of "causality" (the one I quoted above), the basis of which in QFT I don't understand (which is why I asked about it).

Not if you give it a precise definition, such as "operators at spacelike separated events commute", no. Which is why I've repeatedly suggested that we should all stop using vague ordinary language and start using precise math, or at least precise definitions that refer to precise math, as the one I just gave in quotes does. Then we can stop arguing about words and start talking about physics.

If you define "locality" as "does not violate the Bell inequalities", then QFT is not local. If you define "locality" as meaning the same thing as "causality" does above, then QFT is local, but you've also used two ordinary language words to refer to the same physical concept, plus you've thrown away the usual way of referring in ordinary language to another physical concept.

In any case, once again, can we please stop using vague ordinary language?

Causality is generally defined as the universal law that causes must preceed effects in every frame of reference, and it entails nothing more. In contrast to locality, a notion with multiple, partially conflicting uses, one cannot tamper with the content of the concept of causality without misrepresenting much of classical and quantum physics.

Causality is a precise notion to the extent that cause and effect are precise notions. In physics, cause and effect are made fully precise in the context of dynamical systems. Here changes in the initial conditions of a differential equation at some time $t_0$ are the causes, and the resulting changes in the trajectory for times $t>t_0$ are the effects caused by these changes. Lorentz invariance then implies that the causes of an effect at some spacetime position $x$ must lie in the past cone of $x$, and that the causes of an effect whose definition involves the spacetime positions from some set $X$ must lie in the union of the past cones of the points in $X$.

The operational content of causality are expressed in terms of response functions - which embody the notion of causality in their definition - through the so-called Kramers–Kronig relations and resulting dispersion relations. In this form, the notion of causality extends to dynamical systems with memory. In quantum field theory, causality is rigorously implemented through [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL], where the dispersion relations are the essential tool that ensures a perturbationally well-defined finite and manifestly covariant renormalization process. In the operator apprach to quantum field theory, causality is implemented through the causal commutation relations of fields - i.e., the commutativity or anticommutativity of the field operators at spacelike separation. The fact that cusal commutation rules hold is called microcausality.

Operationally, the causal commutation relations assert (roughly) that states with prescribed field expectations at $x_1,\ldots,x_n$ can be prepared independently by causes near $x_1,\ldots,x_n$ whenever $x_1,\ldots,x_n$ are mutually spacelike separated. This is meant by causal independence.
Simultaneously, they express certain local independence properties. This is the reason why causal commutation relations are - in my opinion somewhat misleadingly - also referred to as local commutation relations, though there is nothing local about them as the relations involve two distinct spacetime points.

 

[vanhees71]

Correlations that violate the Bell inequalities can't be "predetermined" locally. That's what Bell's Theorem shows.

No, I didn't say "locally". In QT a state is given by a statistical operator. That's it. It's neither local nor non-local. It's just given by the preparation procedure. The most intuitive picture of time evolution is the Heisenberg picture, where also in the mathematical formulation the state is time-independent (or only time-dependent if there's explicit time-dependence).

 

[A. Neumaier]

The most intuitive picture of time evolution is the Heisenberg picture, where also in the mathematical formulation the state is time-independent (or only time-dependent if there's explicit time-dependence).

Small correction: In the Heisenberg picture, the state is never time-dependent, as any explicit time dependence is necessarily in the terms of the Hamiltonian.

 

[vanhees71]

Not if you give it a precise definition, such as "operators at spacelike separated events commute", no. Which is why I've repeatedly suggested that we should all stop using vague ordinary language and start using precise math, or at least precise definitions that refer to precise math, as the one I just gave in quotes does. Then we can stop arguing about words and start talking about physics.
If you define "locality" as "does not violate the Bell inequalities", then QFT is not local. If you define "locality" as meaning the same thing as "causality" does above, then QFT is local, but you've also used two ordinary language words to refer to the same physical concept, plus you've thrown away the usual way of referring in ordinary language to another physical concept.

In any case, once again, can we please stop using vague ordinary language?
In other words, it reflects how we experience things. But QFT does not explain why we experience things that way. We don't fully understand why we experience things that way.
Sure, you can write down such states, but they include operators at different spacetime events (which can be spacelike separated events). So they're not "local" in the ordinary sense of the term.
With your preferred definition of "local", yes. But they violate the Bell inequalities, which means they are not "local" in that sense of the term "local". Which, again, is why I keep saying we should stop using vague ordinary language.
With your preferred definition of "causal influence", yes. But not everybody shares your preferences for definitions of ordinary language terms. Which is why we should stop using them.

In my field, i.e., relativistic QFT, these words have a clear meaning: Causality is implemented by the microcausality constraint, and locality means the locality of interactions, i.e., the Hamiltonian density is a local polynomial of the fields and their canonical momenta, i.e., it's depending only on one space-time argument. Also proper orthochronous Poincare transformations are realized on the field operators locally, i.e., the field operators transform under the unitary representations of the proper orthochronous Poincare group as the analogous classical fields do. Together with microcausality this makes the standard QFTs successfully used to describe real-world observations consistent with the space-time structure and thus the induced meaning of causality of special relativity.

It's also clear that we have states which are describing non-local correlations as entanglement. Nothing in the formalism described above prevents this, and it's necessary to describe the very experiments we are discussing here. I've never stated otherwise. The only thing which clearly contradicts the locality of interactions, which is a very clear concept and defined above, is the nonsensical assumption as if a local measurement on one part of an entangled system leads to acausal interactions at distance or causal influence. That's indeed only put by unclear language into what some people call interpretation, and it's just contradicting the sharply defined mathematical concepts underlying the mathematical construction of the theory.

Concerning the violation of Bell's inequalities I always stressed as the main point that this doesn't prove non-locality but just the existence of long-range correlations.

I don't understand, why you think, we don't understand something. We never understand why things are as they are but we can only investigate how things are and describe our findings as accurately as possible. So far everything is described by relativistic QFT (except the unsolved question, how to describe gravitation). You cannot expect more from physics.

 

[vanhees71]

[Edit: Corrected the equation of motion for the Heisenberg-picture stat. op.]

What I meant is the partial time derivative in the Heisenberg-picture equation of motion,
$$\frac{\mathrm{d}}{\mathrm{d} t} \hat{\rho}(t)=\partial_t \hat{\rho}(t)+\frac{1}{\mathrm{i} \hbar} [\hat{\rho}(t),\hat{H}(t)]=0.$$
Here the first term refers to the dependency of $\hat{\rho}$ on the "fundamental operators" of the theory which by definition are not explicitly time-dependent. These are in QFT the field operator, from which all the other operators are built as appropricate functions/functionals. The partial derivative includes a possible explicit time dependence.

An example is that sometimes you like to describe a QFT system with some time-dependent "classical background field" present. This background field brings in explicit time-dependence.

[The following is WRONG as @A. Neumaier pointed out in #187]
Another important example for an explicitly time-dependent state is local thermal equilibrium. In the grand-canonical description it reads
$$\hat{\rho}=\frac{1}{Z} \exp[-(\hat{p} \cdot u(x)-\mu(x))/T(x)], \quad Z=\mathrm{Tr} \exp[-(\hat{p} \cdot u(x)-\mu(x))/T(x)].$$
Here the explicit time dependence comes from the depencence of the local temperature and chemical potential(s) and the four-flow field $u=\gamma(1,\vec{v})$ on $x=(t,\vec{x})$; $\hat{p}$ is the operator for total four-momentum which as a functional of the field operators is by definition not explicitly time dependent.

 

[zonde]

There may be other explanations, but as a physicist I have to stake my approach on just one. It took my math colleague and I three months to modify and apply Regge calculus to the SCP Union2 data and it took us four months to modify and program a fit to the Planck 2015 CMB power spectrum data. These are just two examples of the many papers I have written based on my one approach. Maybe a philosopher can write one paper on a particular approach this month and turn around and write another paper on another approach next month, but we don't have that luxury in physics.

In physics you always have to be ready that your model will be falsified by observation in some domain where your model has not yet been tested.

It is true that physics is done dynamically, but that doesn't mean an explanation of what we find has to be ultimately dynamical.

Any scientific explanation has to give predictions that are testable within dynamical process. So even if you believe that adynamical view can explain observation better you still have to be able to translate your adynamical view into dynamical story and point out unique features that show up in dynamical story.

It is also true that we do astrophysics and cosmology from Earth, seeing the sky rotate about us, but we long ago abandoned geocentricism.

Almost all observations are still geocentric. So non geocentric model still have to express it's predictions for geocentric observer.

Changing from dynamical to adynamical explanation is revolutionary. As Skow said in his review, "It really is necessary to understand how radical this idea is. ... You can't explain A because B and B because A." But, in adynamical explanation (such as Einstein's equations of GR), it is precisely the case that "A (the spacetime metric) because B (the stress-energy tensor) and B because A." You can't input the SET to solve EE's for the metric unless you already know how to make spatiotemporal measurements, i.e., you already have the metric. And vice-versa of course. Solutions to EE's are self-consistent sets of the spacetime metric, energy, momentum, force, etc. on on the spacetime manifold, where "self-consistent" means "satisfies the constraint, i.e., EE's." That's why our proposed approach constitutes a Kuhnian revolution. When I started in foundations 25 years ago, I too was convinced that GR and/or QM were flat out wrong. Now, I believe (base my research approach on the fact that) they are in fact both right and beautifully self-consistent.

There is big difference between facts A and B and components of the model A and B. It seems you are mixing them together.

Every physicist has to stake their research on a particular model of "the real external world." I'm very happy now with mine because it shows modern physics is in fact complete (minus QG) and consistent, i.e., it is amazingly comprehensive and coherent ... as long as you're willing to give up your anthropocentric dynamical bias.

Every scientist has to operate within common generally accepted framework. Within that framework you have a lot of freedom with your explanations, but there is one condition - your explanation has to produce predictions that are testable even for those who do not believe in your explanation. So your condition "as long as you're willing to give up your anthropocentric dynamical bias" takes you out of that framework.

 

[A. Neumaier]

An example is that sometimes you like to describe a QFT system with some time-dependent "classical background field" present. This background field brings in explicit time-dependence.

No. It changes only the Hamiltonian by a time-dependent term.

Another important example for an explicitly time-dependent state is local thermal equilibrium. In the grand-canonical description it reads
$$\hat{\rho}=\frac{1}{Z} \exp[-(\hat{p} \cdot u(x)-\mu(x))/T(x)], \quad Z=\mathrm{Tr} \exp[-(\hat{p} \cdot u(x)-\mu(x))/T(x)].$$

No. The local equilibrium density operator is not in the Heisenberg picture, which would result in a covariant formula. The formula you give is covariant but in correct since the right hand side depends on $x$, not on $t$. You need to integrate over a Cauchy surface to get a valid exponent to which to apply the standard cumulant expansion. This shows that the expression is frame-dependent, hence constitutes a Schrödinger picture.

 

[vanhees71]

The right-hand side depends on $t$ and $x$, and it's manifestly covariant, $u$ is a four-vector field, $\hat{p}$ is a four-vector, $T$ and $\mu$ are four-vector fields.

How else would you write this standard statistical operator?

 

[A. Neumaier]

The right-hand side depends on $t$ and $x$, and it's manifestly covariant, $u$ is a four-vector field, $\hat{p}$ is a four-vector, $T$ and $\mu$ are four-vector fields.

How else would you write this standard statistical operator?

I admitted that your right hand side is covariant, but it hasn't the correct form, hence is nonsense.

In nonrelativistic QFT, local equilibrium is given by the Schrödinger picture density operator $\rho(t)=Z^{-1}e^{-S(t)/\hbar}$, where, with 3-position $x$, 3-momentum operator density $\hat p(t,x)$, Hamiltonian density $\hat H(t,x)$, and number operator density $\hat N(t,x)$,
$$S(t)=\int dx \frac{\hat p(t,x) \cdot u(t,x)+\hat H(t,x)-\hat N(t,x) \mu(t,x)}{T(t,x)}$$
and $Z=Tr~e^{-S(t)/\hbar}$.

This does not become your formula when transformed to the Heisenberg picture.

 

[vanhees71]

Ok, I have to think about this, though I don't understand why your $\hat{\rho}$ describes local thermal equilibrium.

 

[A. Neumaier]

Ok, I have to think about this, though I don't understand why your $\hat{\rho}$ describes local thermal equilibrium.

I corrected my formula, which was also nonsense. The new formula describes local equilibrium since if you discretize the integral into a sum over a number of mesoscopic cells, you get the formula that you would get from regarding the cells as independent and in equilibrium by taking a tensor product.

You can also get this formula from Jaynes' maximum entropy principle, assuming the local densities at fixed time to be the set of relevant variables.

 

[vanhees71]

Argh. Aside the maybe wrong idea of a local thermal equilibrium stat. op. I quoted a wrong equation of motion for the statistical operator. Of course, in the Heisenberg picture the covariant total time derivative is the the usual total time derivative, and thus it must read (von Neumann equation)
$$\mathring{\hat{\rho}(t)}=\frac{\mathrm{d}}{\mathrm{d} t} \hat{\rho}(t)=\frac{1}{\mathrm{i}} [\hat{\rho},\hat{H}(t)]+\partial_t \hat{\rho}(t)=0.$$
So in the Heisenberg picture the correct EoM for the statistical operator reads
$$\partial_t \hat{\rho}(t)=\frac{1}{\mathrm{i}} [\hat{\rho}(t),\hat{H}(t)].$$
That's valid also for time-dependent Hamiltonians (that's why I wrote $\hat{H}(t)$).

 

[vanhees71]

I corrected my formula, which was also nonsense. The new formula describes local equilibrium since if you discretize the integral into a sum over a number of mesoscopic cells, you get the formula that you would get from regarding the cells as independent and in equilibrium by taking a tensor product.

You can also get this formula from Jaynes' maximum entropy principle, assuming the local densities at fixed time to be the set of relevant variables.

Sure, you are right. But if I write the statistical operator for relativistic quantum fields in the Heisenberg picture, why should it then be the statistical operator in the Schrödinger picture all of a sudden? I'd say one can just take your formula an write it down using the relativistic (canonical) energy-momentum-tensor $\hat{\mathcal{T}}^{\mu \nu}(x)$. Then
$$\hat{\rho}=\frac{1}{Z} \exp(-\hat{S}),$$
where
$$\hat{S}=\int \mathrm{d}^3 \Sigma_{\mu} [u_{\nu}(x) \hat{\mathcal{T}}^{\mu \nu}(x)-\mu(x) \hat{\mathcal{J}}^{\mu}(x)]/T(x),$$
where $\hat{\mathcal{J}^{\mu}}$ is the four-current operator of a conserved charge (baryon number, electric charge for instance). The integral is over some spacelike hypersurface. With the operators in the Heisenberg picture this should be the stat. op. in the Hiesenberg picture too, right?

However, I must admit I've never seen this idea used in non-equilibrium QFT. As you well know, there usually one works with a general ansatz, derives the Kadanoff-Baym equations und does further approximations from there.

 

[A. Neumaier]

With the operators in the Heisenberg picture this should be the stat. op. in the Heisenberg picture too, right?

No. Apply your recipe to the Schrödinger state of a free relativistic scalar particle instead of a quantum field, and you'll see that one cannot transform from the Schrödinger to the Heisenberg picture by a relabeling of the kind you do. It still remains the Schrödinger state. You need to apply the usual unitary transformation to mediate between the pictures.

Moreover, in your ansatz, there is a redundancy in that multplying $u,\mu$ and $T$ by the same field doesn't change the result...

 

[A. Neumaier]

I've never seen this idea used in non-equilibrium QFT.

Well, you'd never claim something you had never seen...

As you well know, there usually one works with a general ansatz, derives the Kadanoff-Baym equations und does further approximations from there.

This is needed in the covariant case since due to renormalization, there are no sensible dynamical equations in interacting QFTs, so one needs to find the hydrodynamic equations from the functional integral rather than via a projection operator formalism.

There should be something covering the above in older nonrelativistic work on nonequilibrium statistical mechanics, perhaps in the book de Groot and Mazur, but I don't have it available to check.

 

[vanhees71]

In the Schrödinger picture the field operators would be time-independent, and I've always a hard time to see the Poincare transformation properties for the Schrödinger-picture operators.

The non-relativistic case is a bit simpler, but if I use Heisenberg fields, why is then the Stat. Op. all of a sudden in another picture of time evolution? In the non-relativstic case, I'd write it in the form
$$\hat{S}=\int \mathrm{d}^3 x \left [\hat{\mathcal{H}}(t,\vec{x}) - \vec{\beta}(t,\vec{x}) \cdot \vec{\mathcal{G}}(t,\vec{x})-\mu(t,\vec{x}) \hat{\rho}(t,\vec{x}) \right]/T(t,\vec{x}).$$

I didn't understand your remark about redundancy. Note that the script T (energy-momentum tensor operator) is different from the usual T (the temperature, which is a c-number field).

 

[A. Neumaier]

In the Schrödinger picture the field operators would be time-independent,

Except possibly for $H$. I was lazy and wrote everywhere a dependence on $t$.

The non-relativistic case is a bit simpler, but if I use Heisenberg fields, why is then the Stat. Op. all of a sudden in another picture of time evolution?

OK, I see now what you mean. Need to think about this...

In the non-relativistic case, I'd write it in the form
$$\hat{S}=\int \mathrm{d}^3 x \left [\hat{\mathcal{H}}(t,\vec{x}) - \vec{\beta}(t,\vec{x}) \cdot \vec{\mathcal{G}}(t,\vec{x})-\mu(t,\vec{x}) \hat{\rho}(t,\vec{x}) \right]/T(t,\vec{x}).$$

What is $\cal G$?

I didn't understand your remark about redundancy.

Sorry; corrected. $u,\mu,T$ are coefficient fields but only the quotients are well-determined. Thus the redundancy. To fix this, there should be no denominator $T$, and $T$ should be computed from your $u$ as its time component.

But your covariant formula also does not have enough intensive fields since the energy-momentum tensor has more coordinates than the momentum vector in my formula. There must be a multiplier field for every field operator component.

 

[PeterDonis]

I don't understand, why you think, we don't understand something.

We don't understand why we experience things a certain way. More precisely, we don't understand how our experiences are produced by our brains (which are in turn connected to the rest of the universe through our senses). But that's not a question of physics; it's a question of neuroscience, cognitive science, etc.

We also don't understand why we experience time to have a particular direction even though the underlying physical laws are time-symmetric (with the minor exception of weak interactions that don't play a part in the operation of our brains and bodies anyway). To some extent that is a question of physics, in that if physics can give an explanation for how time asymmetry can be produced from underlying laws that are time symmetric, we might not need to understand all the details of neuroscience, cognitive science, etc. to understand why we experience time to have a particular direction.

I am aware of only one hypothesis from physics to explain time asymmetry, namely asymmetry of initial conditions: time has an arrow in our universe because our universe started in a state with a very high degree of symmetry and uniformity. But we don't really have any way of testing this hypothesis since we can't run controlled experiments on universes.

 

[vanhees71]

Except possibly for $H$. I was lazy and wrote everywhere a dependence on $t$.

OK, I see now what you mean. Need to think about this...

What is $\cal G$?

Sorry; corrected. $u,\mu,T$ are coefficient fields but only the quotients are well-determined. Thus the redundancy. To fix this, there should be no denominator $T$, and $T$ should be computed from your $u$ as its time component.

But your covariant formula also does not have enough intensive fields since the energy-momentum tensor has more coordinates than the momentum vector in my formula. There must be a multiplier field for every field operator component.

Well, how you choose your thermodynamical parameters is a matter of convention. I also usually prefer $\alpha=\mu/T$ (I don't even know a name for it, but calculational-wise it's often more convenient if it comes to certain quantities like susceptibilities of (conserved) charges and things like that).

It's also for sure that what I wrote down is the statistical operator in the Heisenberg picture. One has to distinguish between dynamical and explicit time dependence. In my idea of a local-thermal-equilibrium stat. op. $\hat{\mathcal{H}}$ is the Hamilton density, $\hat{\vec{\mathcal{G}}}$ the momentum density, and $\hat{\rho}$ the density of some conserved charges (if there are more then you need a separate chemical potential for each of them).

Now concerning the time dependence. That's something not well covered in almost all QM textbooks. The best one I know in this respect is

E. Fick, Einführung in die Grundlagen der Quantenmechanik, Aula-Verlag

It's the only book which clearly writes all the equations in terms of an arbitrary picture of time evolution, and of course you must stay clearly within one picture or cleanly go from one picture to the other by the appropriate time-dependent unitary transformation.

You find this formalism also in my (German) QM manuscript (however in much shorter form):

https://itp.uni-frankfurt.de/~hees/faq-pdf/quant.pdf
Now concerning dynamical and explicit time dependence: You start to develop a specific quantum theory by a set of "fundamental observables". E.g., in the non-relativistic quantum mechanics of one scalar particle you usually start in the QM 1 lecture with position and momentum $\hat{\vec{x}}$ and $\hat{\vec{p}}$. These "fundamental operators" are by definition not explicitly time-dependent but get their time dependence from the choice of the picture of time evolution. In the Schrödinger picture, which is usually taught first, these operators are completely time-independent. All observables are then built as functions of these fundamental operators. Of course you need some algebra, which in this case is motivated by the fact that momentum should be the generators of spatial translations, leading to the assumption of the usual Heisenberg algebra. Everything else is then built as functions by educated guesses from classical mechanics, including the Hamiltonian of the system, which is the operator defining dynamical time evolution.

Now an observable $\hat{O}$ is a function of $\hat{\vec{x}}$ and $\hat{\vec{p}}$ and, maybe, explicitly on $t$ (note that $t$ is not an observable but a parameter in QT in order to have a stable ground state, i.e., the possibility to write down Hamiltonians bounded from below, an argument brought forward by Pauli in his famous encyclopedia articles on wave mechanics which still is among the best expositions of the theory ever written): $\hat{O}=\hat{O}(\hat{x},\hat{p},t)$.

Now one must distinguish different "time derivatives". First of all there's the mathematical time dependence of the "fundamental operators", defined by an equation of motion
$$\mathrm{d}_t \hat{\vec{x}}(t)=\frac{1}{\mathrm{i} \hbar} [\hat{\vec{x}}(t),\hat{H}_0(\hat{\vec{x}},\hat{\vec{p}},t)]$$
and analogously for $\hat{\vec{p}}(t)$.

For the "state kets" one has the Schrödinger-like equation
$$\mathrm{d}_t |\psi(t) \rangle = \frac{\mathrm{i}}{\hbar} \hat{H}_1(\hat{\vec{x}},\hat{\vec{p}}) |\psi(t) \rangle.$$
The relation to the Hamiltonian of the system is
$$\hat{H}=\hat{H}_0 + \hat{H}_1.$$
The extension to the most general case of a statistical operator (including also the case of pure states of course, for which $\hat{\rho}=|\psi \rangle \langle \psi|$) is
$$\mathrm{d}_t \hat{\rho}(\vec{x},\vec{p},t) = \frac{1}{\mathrm{i} \hbar} [\hat{H}_1(\hat{\vec{x}},t).$$
For an arbitrary observable you get
$$\mathrm{d}_t \hat{O}(\hat{\vec{x}},\hat{\vec{p}},t)= \frac{1}{\mathrm{i} \hbar} [\hat{O}(\hat{\vec{x}},\hat{\vec{p}},t)]+ \partial_t \hat{O}(\hat{\vec{x}},\hat{\vec{p}},t).$$
The partial time derivative refers to the explicit time dependence only.

Of course, the same must hold for the statistical operator as an operator depending on the fundamental operators and explicitly on time,
$$\mathrm{d}_t \hat{\rho}(\hat{\vec{x}},\hat{\vec{p}},t)= \frac{1}{\mathrm{i} \hbar} [\hat{\rho}(\hat{\vec{x}},\hat{\vec{p}},t),\hat{H}_0(\hat{\vec{x}},\hat{\vec{p}},t)]+ \partial_t \hat{\rho}(\hat{\vec{x}},\hat{\vec{p}},t).$$
Together with the above equation for the time dependence of $\hat{\rho}$ one gets the (picture independent!) von Neumann-Liouville equation of motion
$$\frac{1}{\mathrm{i} \hbar} [\hat{\rho},\hat{H}]+\partial_t \hat{\rho}=0.$$
This describes just the time derivative of mathematical formal objects, and you can choose an arbitrary picture of time evolution, just for convenience of calculational treatment of some given problem (e.g., for scattering theory the interaction picture is most convenient, where $\hat{H}_0$ is the Hamiltonian of non-interacting particles, and $\hat{H}_1$ the interaction part of the full Hamiltonian).

Of course, on the other hand, there should also be a description of the "physical time dependence" of observables and a corresponding "covariant time derivative". This answers the question, what is for a given observable $O$ the operator which describes the observable $\dot{O}$, i.e., the time-derivative of $O$. That's given by the picture-independent equation
$$\mathring{\hat{O}}(\hat{\vec{x}},\hat{\vec{p}},t) = \frac{1}{\mathrm{i} \hbar} [\hat{O}(\hat{\vec{x}},\hat{\vec{p}},t),\hat{H}(\hat{\vec{x}},\hat{\vec{p}},t)]+\partial_t \hat{O}(\hat{\vec{x}},\hat{\vec{p}},t).$$
Note that the von Neumann-Liouville equation for the stat. op. then reads in this notation
$$\mathring{\hat{\rho}}=0.$$
The same scheme holds of course for (relativistic or non-relativistic) QFT. There the fundamental operators from which all observables, stat. ops., etc are built up are the fields and the canonical field momenta, and you can have functionals instead of functions of the field operators and their canonical momenta rather than simple functions. Formally there's not much difference.

 

[vanhees71]

We don't understand why we experience things a certain way. More precisely, we don't understand how our experiences are produced by our brains (which are in turn connected to the rest of the universe through our senses). But that's not a question of physics; it's a question of neuroscience, cognitive science, etc.

We also don't understand why we experience time to have a particular direction even though the underlying physical laws are time-symmetric (with the minor exception of weak interactions that don't play a part in the operation of our brains and bodies anyway). To some extent that is a question of physics, in that if physics can give an explanation for how time asymmetry can be produced from underlying laws that are time symmetric, we might not need to understand all the details of neuroscience, cognitive science, etc. to understand why we experience time to have a particular direction.

I am aware of only one hypothesis from physics to explain time asymmetry, namely asymmetry of initial conditions: time has an arrow in our universe because our universe started in a state with a very high degree of symmetry and uniformity. But we don't really have any way of testing this hypothesis since we can't run controlled experiments on universes.

Well, as already said above, natural sciences are one (on purpose limited!) aspect of human knowledge. They restrict themselves to describe what can be accurately observed. For the most simple systems (which are usually described by physics) there's a surprising discovery that we can describe our observations by mathematical theories and understand a lot of phenomena from a very few fundamental principles, which finally cannot be derived from even more fundamental principles and which are an abstraction from our experience. Finally physics, as all of natural sciences, is an empirical science (on purpose). To find descriptions of ever more complex systems (many-body systems) from the fundamental theories, is a creative act. Though there's a big hype about AI, machine learning, and all that, I don't think there's a automatic way to find such descriptions, and thus that will stay a human art for a long time to come.

Now, the more and more complex things get, the more difficult it is to get descriptions based on fundamental theories. Fortunately nature is kind enough to be describable very well also by "effective theories", i.e., approximations to the fundamental theories just cutting out all "irrelevant" details. To describe a container of gas under usual conditions you don't need to describe a mole of molecules in all detail, but some thermodynamic quantities will do. If it's moving, you'd need fluid dynamics or Boltzmann transport theory.

Also some quite humble-looking "complex systems" become pretty easily very complicated. E.g., in my own field of heavy-ion collisions, just smashing two large nuclei together leads to a plethora of phenomena one has to describe with a lot of different effective theories, reaching from relativistic hydrodynamics, transport models to lattice QCD and all that.

As said in some textbook on QFT, physics teaches humility.

Now, if it comes to biological systems, it's even more difficult to get a fundamental description. Though there's a lot of progress to describe some subjects from a fundamental approach using effective models (e.g., ion transport through membranes in cells, protein folding), it's still far from being "understood" in the sense of describing life as such reaching back to the fundamental physical theories as far as they are known, anyway.

Considering our own human brains, it may become even philsophical, whether it's possible to finally "understand" it in this scientific sense at all, because after all it's the human brain itself which processes all the empirical information we can get about it, and the description of a system within itself is already mathematically a quite mind boggling thing. Maybe, you can argue, it's not only one brain which studies itself but the collective endeavor of many scientists to understand it better, maybe one day in a satisfactory way from fundamental physical principles.