can anyone explain this by giving an example

@PrateekMourya Diff geo course? Where is the problem from?

I think what it means is $\mathrm{M}_{n \times m}(\mathbb{F})$ is the set of all m × n matrices with elements taken from some finite set $\mathbb{F}$, and $e_{i,j}$ is the set of all n × m matrices with only a single element set to 1.

If m = n = 2 then $\mathrm{M}_{n \times m}(\mathbb{F})$ is the set of all matrices of the form:

$$\left(\begin{matrix} a & b \\ c & b \end{matrix}\right)$$

(1) is the spanning condition and (2) is the linear independence condition

@Slereah This message is now the shortest GR book :P

@Slereah volume 2

there's a few extra conditions you may want but that's the basics really

I was visiting my family this morning. Good afternoon @Amit maybe in next few hours I add one or two comments more, but I don't want to bother here. I wondered if some friends of the chat would like to know the momument to Santiago Ramón y Cajal, search in Google the Spanish article of Wikipedia "Santiago Ramón y Cajal" and look at the illustration/photo at the right of the section Correspondencia de Santiago Ramón y Cajal, this moument is incredible (...FONS VITAE, FONS MORTIS).

Can one show $\hat{a}|0\rangle=0$ from the assumption that the vacuum is Poincare invariant?

Does anyone ave a more rigorous account of unstable/decaying states on the Greens function and S matrix level

@Amit I guess it cud b used for universe simulation research :P

@ManasDogra what's your definition of $a$?

I hav been having lucid dreams lately. I wonder if dreams cud b an alternate reality

Some dreams have an extremely consistent plot. They feel like someone else's life

I mean some of them r very grounded in reality. No non-sensical stuff. And they hav a lasting plot.

@ACuriousMind Let's say I define $\hat{a}$ via mode decomposition of a real scalar field?

Then it follows from. Harmonic oscillator analysis

Cuz when u define them using mode expansion, they satisfy the oscillator commutation relation

People neglect the imaginary part in the denominator and only include the exponential decay in the time evolution of the GF but what makes this approximation correct

@ManasDogra and then your question is whether any Poincaré invariant state is automatically the vacuum in the sense of being annihilated by the annihilation operator?

note that translation invariance is $P^\mu\lvert 0\rangle = 0$ and $P^\mu = \int p^\mu a^\dagger(p)a(p) \mathrm{d}^3p$

so the two ways in which $\lvert 0\rangle$ can have zero momentum is either by having no particles or by having exactly the same number of each particle with momenta $p^\mu$ and $-p^\mu$

hello, i am wondering if there is a general law or mathematical principle for which gauss' law for charge and mass are a special case?

I feel like you can extend this kind of argument via Lorentz invariance to conclude that the Poincaré invariant vacuum either has no particles or infinitely many

and HONK

which is consistent with the "Dirac sea" view of the vacuum :P

@Relativisticcucumber it's just Stokes' theorem

Also, what are the non-zero parts of the spectral density where there are no poles in the Kallen-Lehmann spectral representation telling us?

@ACuriousMind i read that Dirac sea is just standard QFT. U start calling the anti-particle annihilation operators as "negative energy particle creation operators".

They said the difference is only in english terminology

@ACuriousMind got it thanks

@ACuriousMind Isn't it $P^\mu = \int p^\mu (a^\dagger(p)a(p)+a(p) a^\dagger(p))\mathrm{d}^3p$

@Relativisticcucumber we have a flux of something - a vector field $A$ or equivalently a n-1 form $a$ and then Stokes theorem says that $\int_V \mathrm{d}a = \int_{\partial V} a$ for any volume $V$. The quantity $\int_V \mathrm{d}a$ is hence meaningfully the amount of "something" inside the surface $\partial S$ and we call $\mathrm{d}a$ the density of that something

@ManasDogra up to ordering sure

remember there's an annoying renormalization for $P^0$ as the vacuum energy

@ACuriousMind :D thanks

And the other thing is that any deviation off the real axis will mean that there is no longer a pole on the real axis meaning that most likely nothing will come up using the LSZ formulation of scattering

@ACuriousMind Thanks

I was thinking that their idea was something like : Instead of writing the vacuum as $|0\rangle$, u start writing it as $|p_1, p_2, p_3........ \rangle$. This is to indicate that the vacuum has all the modes filled with negative-energy particles. When a positron of momentum $p_1$ is created, u write the new state as $|p_2, p_3, p_4...... \rangle$. So u knock off a $p_1$ to indicate that a negative energy particle as been annihilated

But i think this idea is very hand-vawy. Among other reasons, writing $|p_1, p_2, p_3..... \rangle$ doesn't make sense becuz $R^3$ is an uncountable set

It looks like standard QFT with just convoluted notation. Becuz this is just a new notation for the Fock space states to make creation operators look like annihilation operators

I was really just point out that the idea of a state of infinitely many particles being Lorentz invariant has "prior art" in the idea of the Dirac sea

I didn't have anything formal or technical in mind

hello again, i have a question about the einstein field equations. i am trying to read a conceptual derivation of how einstein worked these out, but i cant find something that really gets to the heart of the issue. for example, in this image, they seem to gloss over the insertion of the energy momentum tensor, but my entire confusion is where the connection between curvature and this tensor originates :PPPP

@Slereah do you have an example of this? i am curious. because i also hear a lot about GR being incompatible with quantum and whatnot, but i havent seen an explicit example of something in GR that is known to be paradoxical or missing something in the same way i have seem for QM, tho i am not that well versed in either.

@Relativisticcucumber Slereah is just saying that the "foundations" of GR are simple: "Spacetime is curved, yo", there's not that much of philosophical issues with it like there are with quantum mechanics

ok ok that i can see -- so no problems with the theory but just still stuff to develop further?

and GR is only "incompatible" with QM if you want it to go to arbitarily high energies, see physics.stackexchange.com/a/399/50583

@ACuriousMind i see - thanks !

@Relativisticcucumber I think this answer gives a really neat heuristic for why we need to relate curvature and stress-energy

your screenshot seems be for the vacuum EFE

if you want a very handwaving idea: Once you have the idea that gravity should be mediated by curvature, you need to make it depend on the mass. Mass is energy (special relativity!) and essentially the only tensor you have that involves mass/energy is the stress energy tensor

so what is classically $\phi\sim m$ becomes $R_{\mu\nu}\sim T_{\mu\nu}$ in tensor land

I don't think there's that much more to it

@ACuriousMind GR, in some special spacetimes, can be thought of as an external field $h_{\mu \nu}$ on top of standard minkowski spacetime. This external field can be thought of as an interacting field on top of minkowski spacetime, just like the electromagnetic field. Is this all correct?

that's Pauli-Fierz theory and how you get to the usual non-renormalizable QFTs with gravity, yes

By special spacetimes, I mean when space-like foliations exist and the topology is flat space

@ACuriousMind ok so i was just thinking about this idea today. And I realized a serious problm with developing the quantum theory this way

it's also usually what you use to derive stuff like gravitational waves

i.e. this is not exclusively a quantum approach, thinking about the metric as Minkowski + perturbation is a very general idea

@ACuriousMind okay i think i see so the major jump is actually moving to the idea of a curved spacetime? and this idea of some sort of spatial field and mass being related is something we have seen experimentally for quite some time, is this correct?

the major jump that these equations made i mean*

in particular because normal coordinates guarantee you can always find coordinates for the neighbourhood of a point where this is actually a good approximation

I understand its use as a perturbation approach. But i saw something very troublin when using it in the quantum theory @ACuriousMind

@Relativisticcucumber Yes, after we have realized in special relativity that it's spacetime and not just space, the crucial step is arguing that gravity should be mediated by the actual curvature of spacetime, which is what all the waffling about the equivalence principle and whatnot is about

I'm sure @Slereah can cite like 50 philosophical treatises on how to arrive at that point :P

The problm I see is that this underlying Minkowski spacetime is actually unobservable. Any clock would measure the time that is given by the full metric solution. I found this problematic becuz the quantum theory interprets this underlying Minkowski metric as the time measured by the clock!

But this metric is suposd to b UNOBSERVABLE

I have no idea what you mean

QFT doesn't say anything about time on clocks :P

you get scattering amplitudes at infinity, $t=\infty$ is the same on any clock!

Yeah, but we arrive at that after first developing the quantum theory of finite time, which has the clock time in its evolution parametwr

not really

We don't NEED to go this route. We can just begin with the partition function. But then the foundations get shaky, i think

I mean you can complain about the whole Hamiltonian thing in QFT without GR, too: Nothing in this derivation where we muck about with the interaction picture etc. is Lorentz invariant

but in the end we get expressions that are invariant

and they work

I feel I'm saying this a lot lately, but not every intermediate expression in a derivation has to be physically meaningful

when I write down an interaction operator $U(t)$ during the LSZ derivation I don't care whether that $t$ is something you can measure on a clock or not

for all I care that $t$ can only be measured by one specific alien

or by no one at all

what matters is that the math is correct and the formula at the end doesn't contain any questionable things anymore

@ACuriousMind i think the justification for the $t\rightarrow \infty$ limit is given using the theory of finite time. In case of GR, the theory of finite time looks non-sensical. So i think we shudnt expect the infinite time partition function thing to give correct predictions, even if its mathematically sound

This is becuz we take scattering situations into account when we justify the $t\rightarrow \infty$ limit of the finite time theory

This limit is not valid in general situations

@RyderRude I don't understand what you think this has to do with GR

again, using some special $t$ coordinate is already breaking Lorentz invariance in SR

Yeah

if you buy normal relativistic QFT I don't understand what your problem here is

and if you don't buy normal relativistic QFT then you shouldn't be talking about GR here :P

I'm thinking that the finite time QFT does give correct predictions, even if it doesnt look lorentz invariant. Specifically, what do we mean by it not being lorentz invariant? Do we mean that the choice of space-like foliations do not remain space-like after a lorentz transformation @ACuriousMind

But idk why we shud care about that. A different observer will just describe a different experiment using different start and end foliations

@RyderRude Careful: Observers and their personal time are different from picking a complete foliation!

A full foliation is more data than just the proper time of the world line of an observer

I don't really understand the problem you see here with respect to QFT: Consider how one combines classically e.g. pure GR and electromagnetism: You just add the EM Lagrangian to the E-H Lagrangian

you could complain there, too, that EM was derived in an SR setting and its evolution equations "didn't consider" the proper times of observers in curved space time

I think you first need to figure out why or why not you object to this classical merging of GR with other "SR-derived" theories before I can understand what your problem with this in a QFT setting is

I will think about it. I have a vague idea. I hav to put it into words

@ACuriousMind ok ok this makes sense

thank you

@ACuriousMind The difference that I see is that the solutions of the GR+EM theory do not depend on any clock-time parameter. The solutions to usual EM in SR does not need any mention of clocks either. It's just solutions to the euler langrange equation on a manifold.

But the solutions of quantum mechanics are parametrized wrt to a clock measured parameter

@RyderRude I don't know what that means

None of the formulae for QFT amplitudes involve "clock times"

I have literally never heard anyone talking about a clock in normal QFT

no one cares what the parameter $t$ in some intermediate equations represents

and it's not there in the equations at the end anyway

just do path integrals if you don't want to see any $t$s

@ACuriousMind I mean, not in scattering. But in the underlying theory that we use to justify the scattering limit. This underlying theory wud b useful for more general predictions, no?

I mean for more general situations

I don't think you get how people think about QFT

Idk :P

one of the popular axiomatizations (Wightman) is literally "all we need are the n-point functions"

you want to predict the time evolution of states on some smaller scale? just do (rel) QM

that's not what QFT is for

Yes, u can just start the theory that way. And it gives correct predictions in the domain of scattering experiments.

@ACuriousMind Do u think that the QFT of finite time may be wrong? we can't compute much with it. But, is it not correct in principle?

@RyderRude I don't know what that question means on a technical level

To me QFT is largely a toolset to compute expectation values for operators (often in vacuum, but there is also QFT for specific states)

(in fact that's what I think QM is, too, but that's another discussion)

nothing about e.g. $\langle \phi(x)\phi(y)\rangle$ depends on what I think time is

what would it mean to do "QFT at finite time"?

By "QFT", I mean the generalisation of what you define to be QFT. It evolves wavefunctionals in finite time (say, electric field probabilities) . If we cud test this theory somehow, is it wrong to expect it to be correct?

@RyderRude why would I care about evolving wavefunctionals

I think that is literally the worst, least insightful and most useless way to do QFT

I mean, in principle. Suppose a futuristic civilisation cud test electric field probabilistic evolution

you're asking me to assume a particular interpretation of quantum mechanics

I refuse

Is this seen as an interpretation in the physics community?

Oh then maybe people really don't expect this theory to b correct :P

I was thinking that the popular view was that it was correct, in principle

But yeah, people can also take the "fields are calculational tool to predict particle behavior" approach

when you assume that someone could "test" the wavefunction evolution, you're saying that the wavefunction is real

that is an interpretation

e.g. $\psi$-epistemic interpretations disagree and think the wavefunction is merely a representation of the knowledge of an observer, rather than a feature of reality

you can't "test" such representations of knowledge regardless of how advanced you are

Oh no. I meant that an advanced civilisation prepared an Em field in an initial wavefunctional state, and evolves it for finite time, and measures the expectation values of electric field. I was only talking about testing expectation values

No the wavefunction itself

if you want the probabilities to measure specific values those are merely the expectation values of the projectors onto the eigenstates corresponding to these values

I don't need to think about time evolution of a state for this, really

there's some set of projectors $P_i$ associated with your measurement, we compute the $\langle P_i\rangle$

I mean that we want to make predictions after evolving it for a finite time

Then we cud use the QFT of finite time

that's not what we do in QFT

like, you're right that one might expect QFT to do that based on what we do in QM, but that's just not what happens

the interacting space of states is a scary place where no one lives

Lol

So these experiments r not very well-defined in the first place

see all the discussions about bound states and whatnot in the last few days here: The answer is just that QFT doesn't do that, just take a limit and escape to rel . QM

@RyderRude I mean what are you even trying to set up here? "Evolution of the electric field", what does that even mean - electric fields are either EM waves or generated by charges

asking about electric fields in specific situations is well-defined, there's quantum optics after all

but if you look into that you'll find it does very much not involve generic QFT methods or wavefunctionals :P

I think you're stuck in the reductionist trap that because QFT is supposed the "most fundamental" theory it must be able to do everything satisfactorily

I disagree

I hope it does everything, at least in principle tho

But we can never test that :)

I just like having a reductionsit view of the universe :P

For thinking purposes

Otherwise, the universe seems like a mess

In a reductionist view, u can just visualize a manifold or a wavefunctional. It makes for intuitive thinking for me :P

there is a subfield that cares about time-evolving states: non-equilibrium QFT

I don't know a lot about it but as far as I understand it's mainly based in path integral approaches like the Schwinger-Keldysh formalism

This looks pretty niche. Theres no wikipedia on it

And i hate statistical mech stuff :P

Becuz of my reductionist tendencies

That is why GR is great

@RyderRude The problem is always, how do you ensure that you are not projecting some unconscious desire into this experience, seeing as it is totally subjective and no one else can access it first hand. If two or more people can interact in this alternate reality and provide consistent information it could be interesting :)

@Amit I agree. It's the verification from two idependent people that's missing. I don't think it'll come

@Amit In fact, astral projections got disproved this way by scientists :pyr

My dumb ass was about to put dish soap on my pancakes instead of the syrup bottle right next to it

@RyderRude Even there it's tricky, because I guess they coined the term "collective unconscious" for a reason, so even that needs to be taken into account

Were they disproved? Interesting, I didn't know

The astral projectors cud not tell what was actually happening in the room they claimed 2 b projecting in @Amit

Oh, cool

I used to believe in that stuff until i saw this on tv :)

Maybe they used stereographic projection and couldn't see what's up with the pole

Lol

Yeah it's easy to get caught in that stuff in your "mental childhood" if that makes sense

I used to believe in school. But it was very scary cuz people always reported meeting bad ghosts who wanted to take over their body

believe in school?

Oh, at the time, I see

Yeah

Those ghosts r apparently always floating in the neighbourhood :P

Belief is a very strong thing, it can create all sorts of experiences

And affect the state of the body

Yeah, it leads to confirmation baises. Seeing faces in stuff

Yep. But when you realize how strong that is, it's also kind of clear it probably had a very important role in human evolution

Let alone human culture

Yeah, belief is necesary too

@SirCumference i once drank vinegar from the fridge instead of water

I mean, i spit it

In Physics belief is interesting too. When you think about Einstein and what he did... both the achievements and also his ultimate disconnection from the community was kind of the result of a very strong belief in how a correct theory must look like

He studied a lot of philosophy

I think it's great thing to learn from

Really? I know he has a philosophical book but I thought he was kind of independent in that regard

In absence of all experimental evidence, u can let philosophy guide u

Oh and I know he did admire Spinoza, if I remember rightly

@Amit i read he was inspired a lot by Mach's principle

Even tho Mach's principle is eventually wrong even in GR :P

Oh? Why is it wrong in GR?

Acceleration is locally detectable

I mean proper acceleation. It's absolute

All observers agree if something's not going along a geodesic

Oh I see, I probably don't remember it correctly. I thought Mach was on about the spinning Bucket filled with water

Think about earth. U know this is a non-inertial frame

It doesn't behave like a freely falling lift

@Amit it's a very related problm, i guess

Yes, so you're saying Mach wouldn't have approved of the equivalence principle?

Idk. Mach's principle was all frames are equivalent

He didnt comment on gravity specifically

Einstein was initiallu very confused by diffeomorphism invariance too, for philosophical reasons

He initially thought diffeomorphism invariance meant GR wasn't deterministic

Yes I think I heard Schuller mention that in one of his lectures... that Einstein took a lot of time to realize coordinates have nothing to do with distances

Yeah

There's this very interesting page, perhaps you've seen it, about the interaction between him and his Mathematician friend Grassman, a lot of relations to that

Grossman*

cantorsparadise.com/…

I've never even heard of Grossman

What is it about

He's the guy that hooked him up with Riemann I think, and also helped him personally with the Math

No, not Riemann. He wasn't alive by then

This is very interesting. Thanks

He says "Please help me. Or ill go crazy" :P

Good to know even geniuses suffer like that

I think Pauli once said he wished he was never introduced to physics

For mental suffering reasons :P

lol. I didn't know that. Einstein said similarly "I wish I've been a plumber"

Lol

I sometimes feel that physics causes me suffering :P

Any kind of ambition can lead to suffering

True

It's what I like about that Feynman anecdote... that only when he relaxed after the war and allowed himself to play with physical problems that weren't "important" that suddenly all the good stuff started to pour out

Very important to take breaks. It's like ur mind keeps working in the background

The problms that made u suffer a month ago seem easy

Yeah if you're not enjoying what you're doing anymore that's the first warning sign. But this warning sign is usually ignored 'cause people need to make money and do some dealz to pay the billz :) It's hard to tell someone to act differently in that situation

Life-hacks are great, if someone actually uses them

Even a 5 mins break can save u frm suffering

It means you always need to set up "filters" to prevent yourself from reaching certain situations. But too many filters lead to an optimization problem: how much do you invest into the filters vs. how much time do you actually get to benefit from the filters :)

But I do agree, a few good habits can take you a long way

Solve this optimization problm recursively :)

Just start somewhere and improve that

lol, recursion is a bad word in some circles

Yeah, I agree, I am only wary of becoming obsessive about being ultra productive

Which is why I hang out in chatrooms? ^_^

Ive picked physics as a hobby. But i think im might b thinking about physics too much :)

I will divide the time. 1hr physics max :P

The mathematical foundations subject is even greater suffering

I found that if I write down some of what I'm thinking about / studying, it makes it easier for me to move on to other things later in the day

I guess it gives one the feeling that "ok, I wrote it down so I can continue from this point later on, no need to rush"

I shud keep a diary too

Yeah. You can even try to convert your "1 hour per day" condition above to a "1 page per day" or "5 pages per day" (however many you decide), after all length is time :)

I will divide 1 hr by the speed of light :)

lol now I'm confused

Sorry, multiply

oh, I hope you're working in natural units (then both are the same also lol)

Id been keeping my stackexchange questions as my diary lol

But i can't post all the shit that i think

I wud get banned probably :P

That's nice. But there's some kind of magic in actually writing on a piece of paper, I can't fathom it

Yeah that's just a rules thingy

I hated writing in college. But guess ill like it if i write for myself, and not bcause i have to

I will try ur life hack. Thanks

Yeah that's another thing that school destroys, natural love for writing lol

No problem I hope it'll be fun

@ACuriousMind Is there a fundamental difference between the frame bundle/tangent bundle and an isomorphic GL(n) bundle/Rn vector bundle

@Slereah the soldering?

I guess that there's no canonical isomorphisms between the two, so for instance you couldn't map a given tangent vector to it uniquely?

hence the soldering

you basically just need to choose an isomorphism to solder them?

like the point of those geometric bundles isn't just the bundle itself, it's the behaviour that connects them to curves and such

I guess a basic example is that without some musical isomorphism, you can't really speak of a tangent dual vector of a curve

like the tangent bundle is fundamentally defined by those tangents and not by a bundle structure

which is unfortunate because every textbook seems to think that the tangent bundle is a good basic example of a vector bundle

@Slereah I think I agree that the tangent bundle is a very bad example of a vector bundle because it is the most special bundle there is. But it is also a very good example of a vector bundle because it feels very natural :P

quite literally a natural bundle

I am having big thoughts about it bc I am trying to figure out what happens if you try to do gauge GR without any soldering

Khalili from their book Life on the Edge: The Coming of Age of Quantum Biology Bantam Press, 2014 (I know the Spanish edition of the book). Thinking in the mentioned talk by professor Hossenfelder I think yes, biologists and physicists have discovered a new continent, it is the cell and the examples of quantum biology that the authors cite (how life tamed quantum mechanics)

What if you just have a random GL bundle and a random associated vector bundle

I'm guessing a lot of the same things, but then you can't do much with it because it doesn't relate to curves at all

@Slereah without the soldiering, how are you defining the Ricci scalar that you need to write into the action?

@ACuriousMind A good question and my current strategy is to have it be a Yang-Mill-ish action

the Yang-Hilbert action

although I don't care that much about the action, more about the geometry

From what I can find it seems that if you have some isomorphic principal bundles, then the associated vector bundles are also gonna be the same

So you indeed will have some fake tangent bundle, but it is completely desoldered from any tangent interpretation

Other curiosity (about a device or technology, and about journeys) is that I've mentioned yesterday, the Project Orion (nuclear propulsion), this seems interesting for me and if there were implications of Gott's copernican principle or from Andy Grove's quote about his fundamental rule in technology.

@Slereah I mean if you just do Yang-Mills with GL(n) then you just get Yang-Milsl with GL(n)

won't really resemble GR much I think

For me here there is something: we never build these increadible spaceships which we could have exlored our solar system. Regards @Amit and all users of Physics Stack Exchange. I'm disconnected in next hours.

@ACuriousMind yeah but it's more of a demonstrative sort of thing

Like "Why GR is like it is"

Why we need soldering, what is geometry, etc

why we can't just slap a gauge theory and call it a day

because GR is about the curvature of spacetime, not the curvature of some random bundle over it :P

well yes, but then what does that mean!

I feel it's probably because mass is special?

by being both gravitational and inertial

also true, alas

that makes it unlike any other charge

mass has that weird thing of being like

conformal symmetry breaking as well???

although that's not 100% relevant since you can have gr without mass

so mass is weird since inertia ties it to all motion, and this means the theory it's the charge of is also weird and has to be tied to spacetime itself

idk what kind of isomorphism you'd need for a vector bundle to not be a fake bundle like that, natural bundle isomorphism, maybe?

Like you also need the isomorphism to preserve the diff to aut map

what's a natural bundle morphism

they only seem to define natural bundles as functors

I guess it's just the bundle isomorphisms in the image of this functor

stoklasa-eu.com/grain-natural-bundle-x151306

Google isn't helping

@Slereah Some things you say remind me of ideas from "Gravitoelectromagnetism"

does it

Yes they tried to make mass analogous to charge basically, make GR on equal footing as Maxwell equations, as far as I understand, it doesn't really work all the way

Gravity has kind of a lot of weird degrees of freedom

mass is not enough for it

Quoting wikipedia: "While Maxwell's equations are invariant under Lorentz transformations, the GEM equations are not. The fact that ρg and jg do not form a four-vector (instead they are merely a part of the stress–energy tensor) is the basis of this difference." but I honestly don't know enough to understand why that's true

Yes, it's weird that when you start GR you think that the metric tensor encodes the entire geometry, and when you get to EFE you see that the Weyl tensor apparently doesn't play a role... and yet it is a tensor that relates to geometry in some way, I still don't completely get it

the common way of showing that gravity and EM aren't that different is Kaluza-Klein in 5d

Gravity and electromagnetism used to be considered quite differently

Gravity was due to objects falling in the center of the universe because it was their natural place while magnets attract objects because they have a soul

I guess the thing that's disturbing about it is why we get a theory that is so deeply related to mass without having to postulate the existence of mass, isn't it?

But is soul quantizable

You'll have to ask Plotinus

I'll see if he can squeeze me in between eternal inflation and purgatory

If you have a functor, can you define a category of that functor by looking at the image of it?

@Slereah the image of a functor is not in general a (sub)category

cf. e.g. math.stackexchange.com/q/413138/143136

unfortunate

Time travel is officially conducted today in some places

time travel?

DST :)

ahh heheh

I wish it was something a bit more concrete lol