@DIRAC1930 stat mech can be mapped to QFT by means of path integrals though :P

I mean, path integrals are a way

Is it somehow correct if I see and interpret electric sparks as how electric field lines will look like if we can see them

Also, what do equipotential lines represent? Another electric field?

Do any singularities actually exist, or are they all just errors in the math/model?

@WaveInPlace They may exist but I'm not sure they can ever become a part of physics since physics stuff need to be measurable. Infinite stuff can appear in the mathematical models but that doesn't mean they're real either... maybe a better theory can predict the same stuff without any infinite objects... there are several questions about that subject in Physics SE

Are point-like charges/masses considered singularities? Physics SE mostly seems to be talking about black holes.

@WaveInPlace No, there's also a lot of posts like: physics.stackexchange.com/questions/600512/…

Ugh, my google-fu is wholly inadequate. Thanks!

:)

@WaveInPlace They also sometimes indicate that the theory in question is being applied outside of its scope, eg. the "Ultraviolet Catastrophe".

Yeah, that checks out.

Are there any reasonable non-point-like charge models for the electron/proton/neutron/etc? That seems like the obvious fix.

Presumably String Theory fits that description

You can try to pick any dimension as models even, but higher dimensions than a string causes bad divergences too IIRC

I forget the reason why but @ACuriousMind knows it IIRC

Though to be fair even in QFT and standard quantum mechanics the particles are defined by their wavefunctions which aren't (generally) localised to a single point. The requirement of the existence of "being at a point" states actually complicates a fully rigorous definition of those theories a fair bit

Well there is still some sense in which it is "at a point"

They're still pointlike when the wavefunction collapses, no?

although it is a bit complicated

Yes, when you "measure position" the state of a single particle "collapses" to a "position eigenstate". The mathematical (read: pedantic) issue is that the position eigenstates aren't elements of the same well-behaved space that you would be able to use without them.

Though other (non position eigenstate) states also cause issues ofc

@Slereah Is this just semantic depending on the length scale you're looking at?

No it is not just semantics

But it's a little complicated

There's some duality between field theories and theories of point particles

which is not present in theories of strings

Ah, ok

Are there any theories that postulate larger charges? Things closer to the measurable scale of charge radii, etc.

I guess they'd have to be pretty marginal, since I don't think that's compatible with QM.

you can try to have particles that are little shapes instead of points yes

They don't work very well from what I can remember

When you say larger charges do you mean the charge on the particles being larger or physically the particles themselves are larger?

In some sense that is sort of what QFT is trying to be since the field operators are defined on compact sets, not on points

Particles with larger defined sizes.

But we still write a lot of QFT as if it was defined on points

which makes things complicated

There are still vertices in the diagrams

As far as I know the size of your particles is a degree of freedom in (all?) theories. No reason you couldn't make the particles in any model "larger" by simple redefinition of parameters.

The idea presumably being that we know the kind of scale we're looking for. But actually measuring the "radius" of a particle in reality is a strikingly difficult job and as of yet we haven't found a point we can call "the" radius of say the electron. We can just put an "upper bound" on the size and say "we aren't sure what the exact number is, but to be consistent with experiment it must be below X size".

Scattering results could be interpreted to give a minimum size, no?

see physics.stackexchange.com/q/119732/50583 and its linked questions

Shapes of particles is one of those question that can only be answered by other questions

What does it mean for a particle to have a shape

@Slereah do u mean the worldline formulation of QFT?

Roughly so yes

there is an equivalent theory for the duality between theories of higher dimensions and more complicated kinds of fields

Yeah. Even string theory corresponds to a String field theory

This is the only sense in which particles can b point or extended that i know of

What about cosmic strings

And it's still only a feature of perturbation theory, rather than something measurable like "shape of a particle"

So i dont think this stuff has physical significance

I guess I'm coming at it from the other end. If they can't be infinitely small points then they must have some kind of shape.

I mean in the actual theory, the state vector is what exists ontologically

There is no concept of "shape of particle" there. To compute the time evolution, we may use perturbation theory in which points or strings show up

Shape implies a geometry, maybe we don't have the right geometry for this scale

Why would state vectors correspond to particles having shapes?

A state vector is a very abstract object. We r no longer describing Newtonian -like ontology

If the state vector is what exists, then asking for a shape would be a meaningless question. It wud b a categorical error

The state vectors look like cats no? We know their shape... ^_^ But no, srsly, I think the question is reasonable... since the things we are measuring take place in physical space (and time), space is associated with a geometry, and so I think it's reasonable to believe that objects have shape. Even waves have shape classically

The quantum theory lives in the hilbert space tho. There's only time there. The space isn't physical space. It is Hilbert space

That's not a knock against reality though. A model reflects reality, not the other way around.

Oh u mean shapes do exist in our everyday world, yes

@RyderRude voila

But the everyday world is an approximation of the the ontology living on the Hilbert space

Again, I'd say that's got the cart before the horse.

Shapes arise as approximate stuff. They r not associated to particles. They may be associated to a bunch of localised particles

That's putting the bras before the horses isn't it

Lol

@WaveInPlace omg, I didn't notice you wrote it even

lol

@RyderRude But it may be true that objects having shapes is in some way an emergent quality, not necessarily an approximation... maybe the entire geometry is emergent

It would be more interesting actually that way. Otherwise if we need to look at smaller and smaller "shapes" to discover new stuff, we will necessarily hit some kind of practical limit, like we have now with the scale of the electron probably

When the state vector can b approximately factorized u again the ability to talk about isolated objects by themselves. And when the particles r more or elss bound, u get a shape in physical space. This is the approximation stuff i mean

It wont yield u a well defined shape of everyday objects

okay... I know, words don't do justice to this stuff :)

Dirac had a great trick about it, he hardly ever said anything so he minimized the inevitable mistakes we make when we put the models into words lol

If i just give u a collection of points describing a circle, is the shape contained in that information, or is shape something different

I believe the shape is smthing different

the shape is a global quality

Surely the "shape" something is in the real world just depends on what things you consider to be "part" of it and also how much error you're willing to give before you consider it to be a "different" shape

Like, u need the qualia of shape for shapes to be well defined

Math is just a language to describe it

Math doesnt capture the quality of the objects of reality

@RyderRude Yes I didn't mean to preclude language from being useful, but rather saying that using only the every day language is tricky to describe physical models, it almost always leads to misunderstandings for people who have no common reference point

violent flashbacks of discussing tensors in this chat

@RyderRude btw, if the information includes also the relation between the points, e.g. you also tell me they all live in the same space and the information is information relative to that space... then yes I would say that this entire thing also determines a shape

It determines it only becuz u already know what shapes mean

U know how to invert the transformation

@RyderRude I beg to differ. Even if I didn't know, it is something I could define, like we define many things in Math without having met them prior in every day life

But u cudnt define a shape in math

Shape is a qualia

Idk exactly what you mean by that

Immutable concept?

I mean that a blind person woud never understand that the points refer to a shape

So shape isnt contained in the math

he could feel shapes, but I guess you won't let him do that

Idk

We are only disagreeing on the definition of the word I think

Touching is a different qualia

I would agree that you can't define a shape from a series of points. But that's kind of the point (heh). Points aren't what we should be working with.

A blind person could map the information about points to the experience of touch

I can say that "the locus of all points with equal distance from a particular point" already defines a shape, and it doesn't require sight nor touch to grasp this idea. It requires the idea of distance... now it would be interesting if we also imagine a creature with no concept of distance -- then I may agree, no shapes for him I guess

I'm defining shape to b the visual percetion of shape

Yeah but that's not necessary, a lot of geometry as you know is done in math, without visually drawing anything

Yes but math is only useful becuz in the end u can map it to ur experiences

That map isnt mathematical

Math describes volumes pretty well.

So the math doesnt capture what exists

I can give you a weird set of points in 3d space, and only tell you they are "continuous" -- you will already know they have a shape, without having to see what shape is it, in particular

I will know it only becuz im familiar with shapes already

The math is only a temporary placeholder for the things that exist that im familiar with

I'm not 100% on terminology, in general, but I'm pretty sure that a volume of continuous space is a shape.

U can define that to be a shape, yes.

But then, it's just a discussion of semantics

I'm saying that that math on its own does not contain the quality of real world shapes

I will agree definitely that the concept of shape is motivated by our everyday experience... since we can't avoid experiencing shapes, we are stuck with this "prejudice" lol, but that doesn't mean that this concept isn't extensible beyond our everyday experience, say, to include shapes in extra dimensions maybe?

Maybe qualia just means prejudice? lol

We can mathematically talk about extra dimensional shapes, yes

But again, it's only a placeholder language for actual shapes

@WaveInPlace Yeah, I was only trying to show that at least we can make use of it as a "pure definition" and apply it to cases where we don't directly see any shape

I don't think you even need to add extra dimensions, necessarily. Subatomic particles definitely exist in the usual four.

@Amit, fair. You definitely showed a nice contradiction as to my points =/= volume statement.

We don't know that particles definitely "exist" in four "dimensions" however you may define those terms, string theory for example uses many more

Again, when we talk about worldlines in four dimensions, that math does not capture the ontology

@WaveInPlace I'm not sure where I contradicted you

When I said you couldn't define space as a series of points. You did so like 2 minutes later.

@Charlie, at least four?

Math is always a temporary placeholder for ontology

The ontology has qualities in it

@WaveInPlace Oh, yeah, I did add an extra (continuity) condition though to make it work :)

So mathematical definition of shapes =/= ontological shapes imo

If you just want to say that the Math doesn't contain the entire experiential information I can't disagree

Even logically apparently, math is a very small byproduct of human experience

Yes!

Not just humans experience, math doesnt even capture ontological shapes

These r the things that exist independent of experience

Don't think there is any objective quality in the number of dimensions your theory uses since a) no model is perfect at least as far as we've found so far and there's no reason to believe any model can be perfect and b) the number you consider to be your theory's "dimension" is somewhat arbitrary in the sense that one could embed spacetime into a greater dimensional space without non-trivially altering the theory and claim the world is $N>4$ dimensional

But math is only a placeholder for them

@RyderRude It's too bold for me to talk about independence from experience, I'm kind of a junky of experience

Not ready to give it up just yet

@Charlie As usually is the case however, we may find that this or that $N$ is wrong... and when this kind of thing happens usually the old theory doesn't make a "comeback" :)

You'd need to carefully define what you mean by "wrong", and if your definition is "diverges from experiment at some point" I'd argue there's no reason to believe a "right" theory exists in which case the term is not very useful

It is impossible to know what objectively exists @Amit @Charlie

Yes I'd agree with that statement

@Charlie wrong stuff can be useful. Wrong is not a bad term

But I too tend to define all theories as correct

I don't think those are mutually exclusive

I think wrong is not the approporiate term

Yeah, I am being Popperian here, I'm saying we may refute a theory with a certain $N$, but will never prove that another $N$ is objectively correct

There's an old joke by Mitchel and Webb, about Bertrand Russell spending his morning proving that his chair exists, before he tries to sit down.

If we're taking the position that we can never be correct I'm not sure we'll make much progress.

@WaveInPlace It's more nuanced than that -- we can never be absolutely correct is the important point, we can certainly asymptotically approach "truth" imo

All theories r correct. We can simultaneously just accept that no theory will be the theory of everything @WaveInPlace

How about "useful"? We can certainly have theories that allow prediction of physical properties better than others, or give insights that others don't.

This will not hinder progress

Hard disagree. The earth isn't flat, and a theory that holds that it is will be of limited utility.

If the point of creating new theories is that we asymptotically reach a model that describes reality to arbitrary precision then it only really makes sense to talk about "rightness" or "wrongness" of a theory relative to another theory

@WaveInPlace that's not a theory. It's a disproven hypothesis

The word theory is assigned to tested stuff

In scientific language

Whether or not the earth is flat is totally testable.

Omg but string theory isnt tested

It is tested in the sense that it reproduces GR's predictions i guess

Are you saying that theories are hypotheses we haven't disproven yet?

@Charlie Hmm, I slightly disagree, you can also talk about absolute wrongness via experiment: certain experiments for example definitely disprove the classical EM theory

No, in official science, the word theory is assigned to stuff that's been experimentally confirmed. The word "hypothesis" or "conjecture" is for the other stuff @WaveInPlace

@WaveInPlace We may never disprove them, but that we can't conclusively prove them is the important point. Popper distinguished corroboration from proof -- in science we can do the former, not the latter

@RyderRude, perhaps I need to read a bit more philosophy. But to my knowledge nothing has been fully confirmed. They're all ultimately hypotheses.

Which is good. Life would be pretty boring, otherwise.

By "confirmed", I mean useful in making predictions

@Amit Not unless you believe measurement to arbitrary position is in principle possible

So I would count Newtonian mechanics as a theory

Even tho it's disproven

A theory is some logical framework for making predictions on the outcome of experiments. A theory does not become "wrong" the moment a more accurate theory is discovered though (not that that's the point you're making above)

@WaveInPlace I mean u can accept that any theory can only b useful, but there is no final theory of everything. Accepting this will not hinder progresss

Becuz usefulness is desired

I define all useful theories to be correct

@RyderRude. There is no theory of everything /yet/.

:)

@Charlie Yes ofc you can also question the "refuting" experiments. But I am only trying to put such experiments on a stronger ground than "corroborating" experiments. Neither of them conclusively prove anything in the mathematical sense. But refuting a theory is akin to finding a single counter example in Math -- once you found one, you only need to make sure you didn't mess up the "proof"(/"experiment"). That's why I think refutation is "stronger" in that way than corroboration...

@Charlie Definitely not. In fact, there's a good case to be made that in application it's often better to use refuted theories because we have a better bound on exactly where their predictive bounds lie :)

There are two key variables though, precision is one of them but scope is the other. Is a theory with wider scope (i.e. able to make predictions within a margin of error on a wider range of experiments) more correct than a theory with less scope that makes more accurate predictions on a smaller range of experiments?

If it were, why would it matter? A theory could be correct under more conditions, but still perform worse relative to another under one particular circumstance.

this is y i define all useful theories as correct.

Depends on whether you ask the physicist or the engineer, no? :)

But at this point, we r just arguing over whether "wrong" is the appropriate term or not lol

There is no disagreement in what our hearts mean

It's just a words issue

Hearts mean??

I mean "what we mean in our hearts"

@WaveInPlace I'm not sure what you mean, that's exactly what I've said

Oh, you don't mean emotionally right?

@Charlie, I was addressing the "more correct" characterization.

No, i only mean that we r just arguing over the semantics of "wrong"

I wud say all useful stuff is correct. But it is also right to call refuted theories as "wrong". It depends on context

Correct isn't a useful metric.

At some point one has to draw a line on what is "useful" though unfortunately

Yeah as already pointed out also by @Charlie, that's kind of why we have categories of both scope and precision, so we can talk about different types of "wrongness" (or "rightness").

Yes!

Newtonian gravity isn't particularly useful to someone building a GPS system but it is useful to someone trying to figure out roughly where the planets are

Oh well

"Useful" changes in different contexts...

The next word needs to be added "Useful for..." and then the rest of the words :)

This is a bit of an endless wormhole, there might be a fundamental set of ideas we could all agree on but it will take too long going back and forth like this to stumble onto it lol

Fair.

But this is time spent properly isn't it? Proper time?

If there was a mic drop here it would definitely reach terminal velocity

lol

I'm deeply confused why this descended into philosophical waffling; what we mean by a shape and size in the context of elementary particles is perfectly meaningful, just not intuitive: physics.stackexchange.com/a/119802/50583 (shape), physics.stackexchange.com/a/277799/50583 (size)

We got bored?

weird platonist ideas about whether or not shapes "exist" are completely irrelevant to the operational definition and usage of the term

Yes, we can define shape of a particle in terms of measurable consequences like "charge distribution"

But, in this definition, this is all there is to the "meaning of shape"

that is all you should reasonably expect to get from QM where things don't have definite positions

QM is definitely a weird funky and not too well lit room in Plato's cave

The Born-Infeld model Lagrangian looks so cool

First time I'm reading an answer on the SE that includes Emojis such as: 😂... my reaction: 😂

Wheres the answer @Amit

Lol I love bad answers

I sorted stackexchange questions by score. There is some comedy gold at the bottom

lol, probably some things that should get deleted too

The bottom question is about moon landing conspiracy i think

There's some flat earth stuff too

we talked about shapes, it's interesting that flat earthers can't define a precise shape for their earth lol

is it a finite slab of material or what, what happens at the edges. shapes are more tricky with sharp edges

It is a frisbee with a tortoise carrying it. But they do accept that other planets r spherical

Most of them r trolling

I'm all for the tortoise part

Lol

Don't forget the four elephants.

What's the weirdest basis for $L^2(\mathbb{C})$?

Is the Fourier basis etc. the only one?

Could there in principle be some field expansion for some of the pre-quantized QFT eon's?

I suppose it would be impossible since the equations are non-linear

Does anyone know where to find the Heisenberg S Matrix paper?

The more I look into QFT, the less crazy it seems

I suppose it's important to differentiate between QFT in general, and what we are currently able to calculate

@DIRAC1930 Seems like they are in german, don't know if it helps you then even if you can find the originals online

Also, when we say that the pole in the 2pt function means that we have a 1 particle state with $P^2=M^2$, is this a stationary state? @ACuriousMind

@Amit Thanks, yeah the ones I could find were in german too

@DIRAC1930 I don't understand the question; $P^2 = M^2$ does not uniquely define a state. If any specific state of mass $M^2$ is additionally an eigenstate of the Hamiltonian $P_0$ it's stationary, otherwise it isn't.

Say if I had a state with $(E,p)=(E,0)$. Therefore $E=\pm M$. If I choose say $E=M$, does this mean that I have a state with eigenvalue $M$ of the full $\hat{H}$?

@DIRAC1930 I mean, what do you mean by the state having "energy $E$" if not that it is an eigenstate of the full Hamiltonian with that eigenvalue?

note that being an eigenstate of $P^2$ does not imply being an eigenstate of any of the components

also note that being an eigenstate of $P^2$ is a Lorentz-invariant property, being an eigenstate of any of the components isn't

Ah yes I didn't think of that

So if I am in the rest frame of a one particle state (corresponding to a pole in the 2pt func.), it's an eigenstate of $\hat{H}$?

even the notion of "rest frame" already assumes it's a full momentum eigenstate

otherwise you don't have a definite velocity for which frame to boost into

Oh yeah

If the state is an eigenstate of $P^2$ doesn't that mean it must also be an eigenstate of $E$ and $p$ since $P^2$ is a number hence $E^2$ and $p^2$ must also be numbers and consequently $E$ and $p$?

No, for example a superposition of (E,p) and (E,-p) is an eigenstate of P^2 but not of P

@DIRAC1930 You must not forget this is still quantum mechanics. You're saying "If the state is an eigenstate of $L^2 = L_x^2 + L_y^2 + L_z^2$, doesn't that mean it must be an eigenstate of each of the $L_i$", which you should know is false in case of angular momentum.

Ah yes thanks

Honk

How do I infer anything about what the one particle excitation above the vacuum is? Do I just assume that it is a one-particle state by the way it transforms under Lorentz transformations i.e. it carries one Lorentz index

My new sencha green tea arrived today muaha

How do I even know it carries one Lorentz index

does the nr. of elements of a point group depend on the dimensionality of the representation?

@DIRAC1930 a state in the Hilbert space must be in a unitary repr of the Lorentz group, so it cannot be in a finite dimensional one. So assuming "carry a Lorentz index" means transforming in the defining representation, no it can't

@fqq But won't a one particle state be $|p^\mu \rangle$ and a 2 particle $|p^\mu,k^\nu \rangle$ etc.?

Here, by "does not completely determine the state ket" does state ket mean the post-measurement state of the system?