# What does it mean that physics it time/CPT symmetric?

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Posted (edited)

Time/CPT symmetry is at heart of many models of physics, like unitary evolution in quantum mechanics, or Lagrangian formalism we use from classical mechanics, electromagnetism, up to general relativity and quantum field theories.
In theory we should be able to decompose any scenario (history of the Universe?) into ensemble of Feynman diagrams, apply CPT symmetry to all of them, getting CPT analogue of entire scenario (?)

There are many QM-based experiments which kind of use time symmetry (?), for example (slides with links) :
Wheeler experiment, delayed choice quantum eraser (DCQE), “asking photons where they have been”, “photonic quantum routers”, Shor algorithm as more sophisticated DCQE.

However, this symmetry is quite nonintuitive, very difficult to really accept – mainly due to irreversibly, thermodynamical counterarguments (are there other reasons?)
Can e.g. this conflict with 2nd law of thermodynamics be resolved by just saying that symmetry of fundamental theories can be broken on the level of solution, like throwing a rock into symmetric lake surface?
Are all processes reversible? (e.g. wavefunction collapse, measurement)

So is our world time/CPT symmetric?
What does it mean?

Personally I interpret it that we live in 4D spacetime, (Einstein's) block universe/eternalism: only travel through some solution (history of the universe) already found in time/CPT symmetric way, like the least action principle or Feynman path/diagram ensemble - is it the proper way to understand this symmetry?
Are there other ways to interpret it?

Edited by Duda Jarek

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1 hour ago, Duda Jarek said:

However, this symmetry is quite nonintuitive, very difficult to really accept – mainly due to irreversibly, thermodynamical counterarguments (are there other reasons?)
Can e.g. this conflict with 2nd law of thermodynamics be resolved by just saying that symmetry of fundamental theories can be broken on the level of solution, like throwing a rock into symmetric lake surface?
Are all processes reversible? (e.g. wavefunction collapse, measurement)

CPT is often applied to single particles,  which are in 1 state. There’s no way for entropy to change, so there’s no application of the 2nd law.

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This is a really good question, and, I think, one that we don’t have the final answer to. I have been thinking about this for some time as well (see what I did here...)
The situation is complicated, because what seems to be happening is that, while CPT symmetry and T-symmetry are formal properties of the laws themselves, they don’t appear to necessarily apply to physical systems described by these laws. For example, each individual microscopic interaction between elementary particles is CPT invariant, but a large thermodynamic ensemble of the same particles isn’t (due to the second law of thermodynamics). Or take the GR field equations - they are trivially T-symmetric, but some solutions of these equations (spacetimes that contain event horizons) are not.

I think the basic problem is that we do not have a unified description of the fundamental interactions with gravity, so it is very difficult to see just what is really going on, on a fundamental level.

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CPT symmetry is not limited to single particle, cannot we transform any Feynman diagram this way?

Couldn't we imagine decomposition of the History of the Universe into Feynman diagrams and applying CPT symmetry to all of them?

Regrading entropy, this is effective picture - having some concrete configuration of particles, there is no entropy (it is zero) ...

To get entropy we need to go to statistical picture by some smoothing, mean field approximation - in which there is entropy and it has a general tendency to grow ... and this tendency might be time symmetric (?)

Related question: under hypothetical assumption that our Universe will finally collapse (true or not), what should be entropy of such Big Crunch?

Entropy should depend on situation (densities, energies) in a given moment - isn't Big Bang and Big Crunch similar from entropy perspective?

"Proofs" of entropy growth e.g. of Boltzmann H-theorem for time symmetric systems seem a nonsense (we could also prove entropy growth after time symmetry - contradiction) - they always use some uniformity assumption "stoßzahlansatz", kind of mean field approximation.

It is nicely seen in this this trivial Kac ring model - just dots on ring rotating by one position per time unit, switching color if crossing marked position (diagram below). It is time symmetric, we can "prove" entropy growth using such assumption (percentage of white balls = percentage of white balls before marked position). However, we can also have cyclic evolution of entropy (all white balls after 2n steps are again all white balls): https://pdfs.semanticscholar.org/aa48/764e372033dcd45feb8f487a3d10e2420088.pdf

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6 minutes ago, Duda Jarek said:

CPT symmetry is not limited to single particle, cannot we transform any Feynman diagram this way?

I did not claim otherwise.

But where would entropy come into play in a multi-particle Feynman diagram?

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There is no entropy in this fundamental level like QFT (it is zero).

For entropy we need to get to effective picture, like replacing positions of particles with densities - what is kind of mean field approximation.

Fundamental symmetries can be violated on the level of solution, like throwing a rock to symmetric lake surface.

Assuming there is a (time/CPT symmetric) tendency for entropy growth, low entropic Big Bang should cause 2nd law of thermodynamics.

We know that Big Bang had low entropy, need to understand why - a natural explanation is that because everything was localized(?)

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15 minutes ago, Duda Jarek said:

CPT symmetry is not limited to single particle, cannot we transform any Feynman diagram this way?

Couldn't we imagine decomposition of the History of the Universe into Feynman diagrams and applying CPT symmetry to all of them?

I think the issue with this is that CPT symmetry is a local symmetry, just like is the case with Lorentz invariance. From what I remember (and I am by no means an expert on QFT, it is the one area of physics I am least comfortable with), to calculate the scattering matrix of an interaction, you perturbatively sum all possible Feynman diagrams of all intermediate states into what is called a Dyson series, which then allows you to obtain the overall S-matrix. It is obvious that each individual Feynman diagram is CPT invariant, but it is not obvious whether or not a large ensemble of resulting S-matrices necessarily has this symmetry, especially not after renormalisation. So I think CPT invariance holds only locally, same as Lorentz invariance can and does only hold locally. I think it is precisely this failure of CPT symmetry to hold over larger ensembles that gives rise to the second law of thermodynamics.

The other problem is that from what I remember, renormalisation is a procedure that in itself really only works locally. In any case, it would be practically impossible to actually do the maths for anything more than a few particles at a time, in a small region of space.

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Renormalization is mathematical trick to ignore infinities, e.g. of infinite energy of electric field of point charge (which seem not to have experimental basis) - it indeed works locally.

But what do you mean that CPT symmetry is only local? It consists of C symmetry which indeed can be seen local, but how to imagine P and T symmetry as local?

Considering scattering e.g. of topological solitons, we also need ensemble of scenarios: Feynman diagrams, but I cannot imaging how P, T symmetry could be local?

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40 minutes ago, Duda Jarek said:

Fundamental symmetries can be violated on the level of solution, like throwing a rock to symmetric lake surface.

+1.

Very interesting, meaningful and inspiring conversation going on here.

I only wish to emphasize observation by @Duda Jarek that charge symmetry can indeed be formulated either locally or globally.

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While P and T symmetries are rather global, in theory we can make C locally: changing just one charge ... but it would switch direction of all Coulomb forces for it, making it a bit different situation - it seems the safest to think of all 3 CPT as global (?)

ps. slides with links, diagrams (e.g. Wheeler, delayed choice): https://www.dropbox.com/s/0zl18yttgnpc52w/causality.pdf

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19 hours ago, Duda Jarek said:

But what do you mean that CPT symmetry is only local?

What I mean is that CPT symmetry implies and requires Lorentz invariance, and vice versa. We know that Lorentz invariance is a purely local symmetry - it holds across small enough patches of spacetime that can be taken as Minkowskian. However, an ensemble of many such patches of spacetime taken together is generally not guaranteed to remain Minkowskian. At the very least, the same must hence be true for CPT symmetry as well. Just exactly what “local” encompasses will depend on the circumstances - every time you add another particle to the ensemble, the failure of spacetime to be Minkowskian becomes a little less negligible by degrees, and eventually there comes a point where one can’t ignore it anymore, at least not on the characteristic length scales of the interactions in question. Determining where that point is would be quite a difficult undertaking, since the gravitational effects of each particle do not add linearly. One might argue that a very large number of particles is needed, but I am not so sure, because it is also a matter of scale - what is negligible on our scales may already be very significant on QCD and electroweak scales.

Perhaps - and I am just speculating here - the second law emerges precisely due to the failure of background spacetime in that region to be exactly Minkowskian. It would seem too strong a correlation to be a mere coincidence.

Purely technically, if one was to be really strict, there is no such thing as a pure CPT symmetry in the real world, because the presence of even a single particle already makes the spacetime region non-Minkowskian - we just choose to ignore this, because the deviation is vanishingly small by all practical standards. So maybe it would be better to say that what is CPT symmetric isn’t the physical system as such, but rather its dynamics, i.e. its Lagrangian.

19 hours ago, joigus said:

I only wish to emphasize observation by @Duda Jarek that charge symmetry can indeed be formulated either locally or globally.

Question to you, since you know much more about QFT than I do - is there some version of the CPT theorem that holds in curved spacetime QFTs as well? I don’t see how it could be possible, since in general curved spacetimes there won’t be any time-translation symmetry.

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General relativity at least in theory allows to directly perform T symmetry by non-time-orientable spacetime.

For example imagine wormhole which glues like in Klein bottle - traveling through it should apply T or P symmetry.

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Ordinary spacetimes for simple ensembles of particles don’t have a topology of this nature, so I’m not sure how this is relevant...?

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Personally I don't think wormholes are realized in nature, but assuming general relativity is true, in theory it allows for non-orientable spacetime, so such possibility needs to be included in considerations.

And it gives very valuable thought experiment - for better understanding of nature of time.

Another question is if there are possible CPT analogues of any phenomena?

For example of laser - which stimulated emission causes excitation of target later.

Is CPT analogue of laser possible? If so, shouldn't its stimulated absorption cause deexcitation of (constantly excited like lamp) target earlier?

For free electron laser building its CPT analogue seems quite straightforward ...

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2 hours ago, Markus Hanke said:

Question to you, [...] - is there some version of the CPT theorem that holds in curved spacetime QFTs as well? I don’t see how it could be possible, since in general curved spacetimes there won’t be any time-translation symmetry.

Very good question. +1 A quick scan of,

(The Global Approach to Quantum Field Theory, Volume 1

By Bryce Seligman DeWitt)

Allows you to find only a couple of paragraphs where space and time inversions are introduced on a coordinate-patch basis. Nothing like the predictive power and generality of CPT in flat space-time is suggested. For all I can remember, the context where CPT is really powerful is the S-matrix approach. And defining assymptotic states in a curved space time is problematic, to say the least.

Searches on other more modern books, or on Quantum Field Theory in Curved Space-Times, also by Brice DeWitt, haven't produced anything that remotely resembles "CPT". Not even mentioned AFAIK.

I do not think there is anything like CPT valid for curved ST that is remotely as robust as it is in flat ST. But I would be very thankful if anybody knows.

This is very interesting, because it connects with my question following up on a suggestion by you on the thread about "What is time?"

Would you have some licence to consider signature-preserving continuous transformations that re-shuffled the space-time coordinates, of which our T, P transformations were a discrete version?

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20 hours ago, joigus said:

Allows you to find only a couple of paragraphs where space and time inversions are introduced on a coordinate-patch basis. Nothing like the predictive power and generality of CPT in flat space-time is suggested. For all I can remember, the context where CPT is really powerful is the S-matrix approach. And defining assymptotic states in a curved space time is problematic, to say the least.

Searches on other more modern books, or on Quantum Field Theory in Curved Space-Times, also by Brice DeWitt, haven't produced anything that remotely resembles "CPT". Not even mentioned AFAIK.

Ok thanks, I expected that. My understanding is that CPT invariance always implies Lorentz invariance and vice versa (a duality?), so it can only ever be a local symmetry. That has interesting implications though, because what it really means - at least in my reasoning - is that one can only meaningfully speak of ‘future-oriented time’ for larger ensembles of particles. For individual particle interactions, I don’t see how the notion of ‘past-to-future ordering’ could possibly be intrinsic. So another technical question to you @joigus: when calculating the Dyson series in order to get to a S-matrix, how does one know the correct order of the terms? It looks like this is a series of nested integrals, so the order of the terms is not in general arbitrary. I know there is such a thing as a time-ordering operator, but my question is - does the ordering somehow follow from the theory itself, or is it imposed ad-hoc? Hopefully this makes sense.

20 hours ago, joigus said:

Would you have some licence to consider signature-preserving continuous transformations that re-shuffled the space-time coordinates, of which our T, P transformations were a discrete version?

I think this would be problematic, because it leaves electric charge unchanged. There will certainly be a large number of scenarios that are TP invariant, but some systems are not, notably anything that explicitly relies on electroweak interactions.
Trouble is, unlike T and P, C is not spatiotemporal, in the sense that it doesn’t directly arise from coordinate symmetries.
But perhaps C/P/T could be connected to topological invariants somehow? This way one might be able to treat them under a common framework that is independent of particular coordinates or metrics. It would mean though that vacuum spacetime would have to have a very non-trivial topology, even on scales of the Standard Model, which is potentially difficult to reconcile with GR.

Edited by Markus Hanke

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Posted (edited)
2 hours ago, Markus Hanke said:

Ok thanks, I expected that. My understanding is that CPT invariance always implies Lorentz invariance and vice versa (a duality?)

Lorentz invariance says that intuition of "objective time direction", "current situation" (hyperspace of constant time) are wrong - they literally change with velocity of observer.

So it says something about time, but at most time symmetry (as in the CPT theorem requiring it for QFT), but special/general relativity alone don't need to combine it with P, they don't see charge C.

Quote

But perhaps C/P/T could be connected to topological invariants somehow?

Topological charge is great to understand spin/charge quantization, e.g. defining EM field as curvature of some vector field, Gauss law counts winding number/topological charge which has to be integer (Gauss-Bonnet theorem) - we get electromagnetism with built-in charge quantization (Faber's model, slides with links: https://www.dropbox.com/s/aj6tu93n04rcgra/soliton.pdf ).

While they are P, T symmetric, the C symmetry changes the situation:

Edited by Duda Jarek

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Posted (edited)
6 hours ago, Markus Hanke said:

So another technical question to you @joigus: when calculating the Dyson series in order to get to a S-matrix, how does one know the correct order of the terms?

I'm a bit hazy about this, because it's been a while. There's a lot that has to do with prescriptions you adopt just because you want your fields to propagate causally. After some investigation, you find out that your Fourier expansion of the fields must contain both,

$e^{-ip_{\mu}x^{\mu}}$

and,

$e^{ip_{\mu}x^{\mu}}$

with the ordering prescription given by,

which amounts to prescribing the "positive energies" to propagate forwards in time, and the "negative ones" to propagate backwards. I don't think this is a big deal: After all, you're interpreting what you energy-dimensional parameter E is doing in your physics.

So far you're kind of forcing your amplitudes to behave causally (microcausality). If you do all that, you get amplitudes that commute outside of their causal cones (anti-commute, if they're fermions):

$\left[\varphi\left(x\right),\varphi\left(x'\right)\right]=i\delta^{\left(3\right)}\left(\boldsymbol{x}-\boldsymbol{x}'\right)$

provided that,

$\left(x-x'\right)^{2}<0$

(depending on signature criterion). Then you proceed to solve Heisenberg's evolution eq. in the Dirac or interaction picture.

$\varphi_{\textrm{int}}=e^{-iH_{\textrm{int}}t}\varphi e^{iH_{\textrm{int}}t}$

Then you substitute this expression into the Heisenberg evolution equation in the Dirac picture and discover that the solution must include the time ordering given by Dyson's formula:

$\varphi_{\textrm{int}}\left(t\right)=\left[T\exp\int_{0}^{t}dt'H\left(t'\right)\right]\varphi_{\textrm{int}}\left(0\right)$

So far, so good. It's complicated, you have implemented what you know about the world, as well as used the room that the quantum formalism gives you to represent the states (change picture to a unitarily equiv. one). The really weird step, IMO, comes now. If you try to expand this as a Fourier series in harmonic oscillators, you have an infinite sequence of differently-ordered powers of creation and annihilation operators, so you (again, IMO) kind of pull a rabbit out of a hat by re-defining your formal series as,

$:\varphi_{\textrm{int}}\left(t\right):=:\left[T\exp\int_{0}^{t'}dt'H\left(t'\right)\right]\varphi_{\textrm{int}}\left(0\right):$

The colon-bracketing means that everything that has differently-ordered power of creators and annihilators, is re-ordered so that all the creators are to the left (and conv. for the annihilators). When you do that, you don't end up with the same operator. It's a different one!

Then comes the use of Wick's theorem, by using the vacuum state. The re-ordering that you've imposed proves now very useful, because the annihilators to the right kill the vacuum, so that you remove a lot of junk. I think, or vaguely remember, that the steps are justified.

This is not the way most people learn QFT. In the old days people invested a lot of time in understanding the gradual steps. Today, everything is considered justified and people tend to jump as swiftly as possible to Feynman diagrams, so they can do calculations.

I just want to add (and sorry for a lengthy and perhaps obscure explanation) that in order to rigourously get to Feynman graphs, there are quite many (mainly combinatoric) steps farther ahead. Basically you must remove over-counting due to your re-ordering, because, obviously, when you identify expressions like,

$a^{\dagger}aa^{\dagger}$

and,

$aa^{\dagger}a^{\dagger}$

with,

$:a^{\dagger}aa^{\dagger}:=:aa^{\dagger}a^{\dagger}:=a^{\dagger}a^{\dagger}a$

you must keep track of how many times this last term appears by re-ordering operators.

Sorry for such a lengthy attempt at an answer, I may not have been very helpful. Take it just as an appetiser, and feel free to ignore it. Sorry if you know many of these things.

---------------------------------------------------------------------

I suppose my succinct answer to your question would be: Dyson's time ordering appears to me as quite natural, because it's a step for you to make your solution formally satisfy the evolution eq. But steps come later that, although immensely useful and allegedly "rigorous" by many people, do present fuzzy areas, at least to me. I'd love to understand them better.

For me it's a work in progress, maybe a lifetime-long project, to get to understand the fundamentals satisfactorily enough.

PD: Both @Duda Jarek and you have made comments about topology that I think are very interesting and point in the direction that I would like the theory to go. AAMOF, it was Gerard 'tHooft, Polyakov, among others, one of the first pioneers to try to develop a more geometric language for QFT. I can't say that's the ticket, but it sounds to me like a much more promising scope.

Other things are going on in QFT. Have you guys heard of MHV amplitude calculations? It's a very quickly-developing subject.

Edited by joigus

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Posted (edited)

@joigus, time ordering is required for performing calculations, but just to understand time/CPT symmetry it is sufficient to imagine e.g. history of the Universes as the result of ensemble of all possible Feynman diagrams throughout this history ... for unification with general relativity it should include ensemble of shapes of spacetime, what leads to infinity which is too large for standard renormalization techniques.

Regarding topology, I usually use the below diagram:

Coupled pendula can be defined through "classical" evolution, but we can also go to normal modes where evolution becomes just rotations - "unitary/quantum".

Then we have lattice of pendula e.g. as crystal, its normal modes are plane waves ... and in QFT are treated as real particles - phonons.

The difficulty comes with the continuous limit, but there are many hydrodynamical QM-like phenomena (like Casimir, Aharonov-Bohm, interference, tunneling, orbit quantization) - slides with links: https://www.dropbox.com/s/kxvvhj0cnl1iqxr/Couder.pdf

For varying number of particles we use QFT, but we should be able to also ask for e.g. EM configuration behind each Feynman diagram, e.g. with nearly singularity for charge.

In this picture particles are localized field configurations: solitons, e.g. topological for spin/charge quantization, for their effective description we still need ensemble of scenario - perturbative QFT.

Soliton particle models are slowly getting to mainstream e.g. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.232002 Many these models of particles e.g. Penrose twisters work in this soliton picture, or e.g. from superfluid/superconductor very topological picture e.g. from Volovik's "The Universe in Helium droplet" - if somebody is interested, in 90 minutes we have online talk about it: http://th.if.uj.edu.pl/~dudaj/QMFNoT

Edited by Duda Jarek

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42 minutes ago, Duda Jarek said:

The difficulty comes with the continuous limit, but there are many hydrodynamical QM-like phenomena (like Casimir, Aharonov-Bohm, interference, tunneling, orbit quantization) - slides with links: https://www.dropbox.com/s/kxvvhj0cnl1iqxr/Couder.pdf

There are at least two things connecting with this that I think are worth mentioning. There may be more, but there's only so much one can say.

1) Is the path to building a well-defined unitary S matrix the only sensible approach to quantum field theory? Aren't we leaving out important substance if we use the |in> and |out> unbounded Hilbert space as representation space for all phenomena? Actually I think many physicists are aware of this.

2) Grivov's ambiguity and a proper study/classification of topological sectors. What possible physical variables are (or may be) hidden behind the humongously big arbitrariness that gauge invariance doesn't let us see in the formalism because they don't appear in the gauge-invariant quantities? Are there missing variables to be described in what is normally described as "gauge junk"? Could there be that what we usually throw in the garbage bin has some meaningful invariants in it?

The more likely solution I see for the wave-particle duality of elementary particles is that what we see as quantum formalism is only dealing with the propagating factor of a more complete state that involves as a co-factor a topological Lagrangian that codifies in it the "only particle" aspect and, by suitable hypotheses, can be statistically correlated to the evolution of the linear wave in some kind of à la Bohm solution.

That's what I like to think when the lights go off.

There are other very interesting off-topic subjects in the link that you provided. Thank you.

3 hours ago, Duda Jarek said:

time ordering is required for performing calculations, but just to understand time/CPT symmetry it is sufficient to imagine e.g. history of the Universes as the result of ensemble of all possible Feynman diagrams throughout this history ... for unification with general relativity it should include ensemble of shapes of spacetime, what leads to infinity which is too large for standard renormalization techniques.

Assembling different space-times makes me much more uncomfortable, but that's just me. It's far over my head.

On 8/3/2020 at 9:38 AM, Duda Jarek said:

General relativity at least in theory allows to directly perform T symmetry by non-time-orientable spacetime.

For example imagine wormhole which glues like in Klein bottle - traveling through it should apply T or P symmetry.

<image>

Very nice point. +1. That's more or less what I had in mind when I said T, P (as Markus said, not C) as coming from some kind of homotopy or diff. transformation.

5 hours ago, joigus said:

with the ordering prescription given by,

<image>

Sorry, I didn't make it clear, although I'm sure you know. These diagrams represent the momentum space.

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15 hours ago, joigus said:

I suppose my succinct answer to your question would be: Dyson's time ordering appears to me as quite natural, because it's a step for you to make your solution formally satisfy the evolution eq. But steps come later that, although immensely useful and allegedly "rigorous" by many people, do present fuzzy areas, at least to me. I'd love to understand them better.

Ok, thank you. It would appear to me - and maybe I am wrong and this is just my own ignorance of the subject - that there are a lot of ad-hoc impositions going on that do not necessarily follow from the model/formalism itself. I don't mean boundary conditions, I mean 'doing things to make things fit' kind of prescriptions. This isn't necessarily meant as a criticism of QFT, which evidently works well enough, but rather as an observation that invokes a sense of QFT either not being fundamental, or of us having chosen a mathematical formalism that just doesn't fit the underlying ontology very well, and thus obscures whatever the fundamental physics actually are. Somewhat like Maxwell's equations written with 3-vectors being an unholy mess that obscures the real physics, whereas same become immediately and intuitively obvious when the equations are written in terms of differential forms.

I am not even sure what the fundamental ontology of a quantum field is actually supposed to be - it's clearly not the concept of 'particle',  but neither is it the field itself, since an operator-valued field is only a mathematical abstraction. So I don't really know what to make of it.

16 hours ago, joigus said:

Other things are going on in QFT. Have you guys heard of MHV amplitude calculations? It's a very quickly-developing subject.

I'm afraid not, no. I don't really follow much of what is going in particle physics, since I am more comfortable with GR.

11 hours ago, joigus said:

The more likely solution I see for the wave-particle duality of elementary particles is that what we see as quantum formalism is only dealing with the propagating factor of a more complete state that involves as a co-factor a topological Lagrangian that codifies in it the "only particle" aspect and, by suitable hypotheses, can be statistically correlated to the evolution of the linear wave in some kind of à la Bohm solution.

Hm...I don't even think that the excitations of quantum fields are ontologically fundamental to the world. After all, both the concepts of 'vacuum ground state' and 'number of particles in a given region of spacetime' are observer-dependent and not invariants; so how could they be fundamental in any real sense? I think there has to be more going on here, which we don't see at the moment. Perhaps it is hidden by an archaic and messy formalism.

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I wonder if there is a force similar to the Lorentz force for gravity? Its existence would violate the symmetry. For a proton, the magnetic and mechanical moments are directed in one direction, and for an antiproton, they are directed in the opposite direction.

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Posted (edited)
15 hours ago, joigus said:

Is the path to building a well-defined unitary S matrix the only sensible approach to quantum field theory?

For the scattering matrix as

S_fi = <fin| S |in>

it is good to see it as one amplitude being a result of propagator from -infinity time, second from +infinity, and S as propagator between them.

This time-symmetric view is called TSVF: https://en.wikipedia.org/wiki/Two-state_vector_formalism

This way we get Born rule from time-symmetry, what also works if asking for probability distribution inside sequence from Ising model: using spatial symmetry instead of temporal, Boltzmann path ensemble instead of Feynman:

2 hours ago, SergUpstart said:

I wonder if there is a force similar to the Lorentz force for gravity? Its existence would violate the symmetry. For a proton, the magnetic and mechanical moments are directed in one direction, and for an antiproton, they are directed in the opposite direction.

Sure there is - what Gravity Probe B has directly tested is gravitoelectromagnetism approximation of general relativity, which is second: gravitational set of Maxwell's equations (necessary to make Newton force Lorentz invariant):

Frame dragging, Lense-Thirring effects are gravitational analogues of Lorentz force ( https://en.wikipedia.org/wiki/Lense–Thirring_precession ).

Kepler problem with such correction simulator: https://demonstrations.wolfram.com/KeplerProblemWithClassicalSpinOrbitInteraction/

In hydrodynamics, Coriolis force corresponds to Lorentz force, see e.g. https://www.pnas.org/content/107/41/17515

Edited by Duda Jarek

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26 minutes ago, Duda Jarek said:

Sure there is - what Gravity Probe B has directly tested is gravitoelectromagnetism approximation of general relativity, which is second set of Maxwell's equations (necessary to make Newton force Lorentz invariant):

And how then to be with symmetry? I don't see any other option, except that the universe appeared together with the anti-universe, which consists of antimatter and electrodynamics in it works on the left-hand screw rule, while in our case it works on the right-hand screw rule.

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32 minutes ago, Duda Jarek said:

Sure there is - what Gravity Probe B has directly tested is gravitoelectromagnetism approximation of general relativity

Note the word ‘approximation’ in this. It’s valid only for the low-velocity, low energy domain.

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