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4:47 AM
in a field theory and in natural units where $\hbar = c = 1$, are fields always unitless?
 
 
1 hour later…
6:01 AM
@SillyGoose never. $1=\int\psi^\dagger\psi\,\mathrm d\vec r$ means that $\psi$ has square root of reciprocal length cube units.
 
 
1 hour later…
7:25 AM
@naturallyInconsistent this is not correct. the lhs need not be 1. SillyGosse isn't talking about the Schrodinger equation wavefunction
 
8:07 AM
does anyone know a reference for a discussion on defining the spacetime underlying non-rel quantum mechanics and its role in quantum mechanics?
 
Do you mean classical spacetime?
 
sure but i guess particularly in the context of quantum mechanics. including thorough/careful discussions of say what does it mean for a system to be localized/delocalized in spacetime (rather than just working with states in abstract Hilbert space all the time and supposing this position operator has, heuristically, to do with the system's position and so on)
in field theory, the fields truly "live on" the spacetime manifold
in quantum mechanics the states do not (to me) really have anything to do with the spacetime manifold when looking at the textbook formalism
 
u can re-write it in the non rel field formalism
 
@RyderRude what do you mean by this
 
non relativistic quantum.field.theory
 
8:11 AM
I think you technically could define non-relativistic QM in the same way
But it would be a little weird
 
oh u mean write non-rel QM in field theoretic terms?
 
yes
$\psi^{\dagger}|0\rangle=|x\rangle$ holds there in the exact sense
 
The exterior of the light cone is lower dimensional so I'm not entirely sure if it would work okay
 
hm well I guess I should look for some references on that
i guess I would have thought it more preferable, on physical grounds, to make explicit the relationship between spacetime and quantum state space
 
@SillyGoose this says theyre not completely equivalent
17
Q: Is non-relativistic quantum field theory equivalent with quantum mechanics?

user26143Related post Can we "trivialize" the equivalence between canonical quantization of fields and second quantization of particles? Some books of many-body physics, e.g. A.L.Fetter and J.D.Walecka in Quantum theory of many-particle systems, claimed that at non-relativistic level, quantum mechanics (...

 
8:17 AM
@SillyGoose what do you mean, "heuristically"?
the position operator is what measures position in QM
 
i guess my overall want is to develop a better intuition of what a physical theory is and in particular sharpen the procedure of taking an "object in reality" and abstracting it to a mathematical structure. as discussed previously, a first order answer to this is to have some spacetime, define some symmetries of spacetime (whatever this means), then define an object as the orbit of a single element under all symmetries of spacetime.
 
i think the standard non rel QM formalism is also incomplete. it has no mechanism for local measurement operators
while measurements in reality are ofc local
one must define fields to have a complete description of reality
 
@SillyGoose Why would "an object" be the orbit of a single element under all symmetries of spacetime???
 
you seem to have abstracted away all physics in your quest for abstraction :P
 
8:21 AM
@ACuriousMind i thought this is what is done in textbook QFT
 
pretty sure it's not
because the orbit of a single point under the Poincaré group of Minkowski space is all of Minkowski space :P
 
i mean for the element to be some abstraction of the given state of the system of interest
 
are you talking about Wigner's classification?
 
yes
but trying to avoid representation theory language
 
this classification is for the free particle space
 
8:23 AM
the point of Wigner's classification is simply that going into another frame should not change the particle content of the universe
so particles need to form irreps
 
isn't this equivalent to " a one particle state of a certain species is an equivalence class of states related by poincare transformations"
 
no, because we're not quotienting by the transformations
there's no "equivalence class" here
the particle with momentum $p$ is different from the particle with momentum $\Lambda p$
 
well I mean when we say ah that's a mass $m$ spin-$1/2$ particle, we are implicitly quotienting
and we don't classify particles by what momentum they have
 
you do in scattering
all the scattering amplitudes are functions of the incoming momenta
just because you sometimes don't care about the momentum doesn't mean it's not there
 
i think this is unnecessary philosophising of what an "object" is. the "things" in any quantum theory are the states of the hilbert space
 
8:26 AM
I don't really understand what you're trying to do here
 
and the hilbert space of the interacting theory is unknown
i think of the "objects" as wavefunctionalss but the eigenspace-structure is unknown
but the asymptotic "objects" can be thought of as fock space elements
but the states of a fundamental theory must have some relation to spacetime. this is why fields are important
a general quantum theory can have qubit states that have nothing to do with spacetime
in some sense, the hilbert space structure of the fundamental theories is derived from spacetime
this is why the CCR of field theories references spacetime
 
8:50 AM
@ACuriousMind are you saying that when someone says "that is a photon", then they are really talking about a photon with particular momentum $p$
@RyderRude well I think it would be an interesting exercise to pursue / find if there is a reversible procedure to construct a position operator from spacetime manifold $M$ and so construct a spacetime manifold $M$ from a position operator
or something along those lines
 
@SillyGoose there is no localization on "spacetime"; there is no relativistic position operator
 
I guess what I mean about the "object in reality" thing is: empirically, my laptop is not a different object at any point in its trajectory if it is jettisoned across the room. my laptop is (processed to be) an object in my mind.
 
@SillyGoose I mean, every concrete electron (let's not talk about massless particles) you encounter will be a wavepacket of concrete shape in momentum space
 
position operators are ultimately ill defined in general. it's only the field CCR that determine the connection of hilbert space to spacetime
 
@ACuriousMind well say then in galilean relativity, or is what you are saying still true then
 
8:54 AM
@SillyGoose note this notion of concrete objects does not exist in qft because the number of particles is frame dependent
the only object that's concrete is the state vector
 
@SillyGoose if you want to do QM on arbitrarily shaped space, you probably need to do geometric quantization
remember that this is physically irrelevant: All of ordinary non-rel QM happens in $\mathbb{R}^3$ (or maybe some kind of interval or circle), and the position operators on $L^2(\mathbb{R}^3)$ are straightforward (and so are those on $[a,b]$ or $S^1$) multiplication operators
 
@RyderRude This is invoking thermodynamics arguments, thermodynamic limit, you can't use this as an argument here
 
@bolbteppa oh
 
@SillyGoose ...which is why physics talks about the trajectory of your laptop in time as the worldline of a single object - but we don't talk about the orbit of your laptop under e.g. translations, and this is why I am puzzled by the idea of "quotienting" by all Poincaré transformations
 
@ACuriousMind well I guess I see the underlying reason for why the laptop along its trajectory remains a laptop as this technical definition of an object as the spacetime symmetry orbit of a "state"
this definition lets you formally say ah yes the laptop is a laptop along its trajectory. how else do you formally (not in an ad hoc manner) make such an identification?
 
9:01 AM
I'm deeply puzzled as to why I need to make this identification "formally"
 
even in a fixed frame, particles can get destroyed
no such identification exists for particles
the fundamental notion is state, not particle
 
When we start treating a system with the physical formalism, we usually start by saying "here's a space of possible states $Q$ for this system" and then all points in there are states of the same system
 
well i guess I'd like to consider the minimal data to construct the physical theory of a system
 
I don't need to "identify them formally", the whole construction begins with says what system (and hence what states) I want to deal with
 
The action of $\hat{H}$ on some state $|n>$ will send it into other $|m>$ states, if we write $|n>$ in terms of creation operators via $|n> = \hat{a}_n^{\dagger}|0>$, then $\hat{H}$ admits an expression in terms of creation and annihilation operators which lets us get the same $|m>$ states, that's all that's going on at the end of the day
 
9:02 AM
@SillyGoose you seem to have a much more formal idea of what a "physical theory" is than reality provides
people just write down stuff
 
well i know there is no general definition i can write down (probably ;) ) to fit all known physical theories, but there seem to at the least be common threads
 
I would like to say $|n> = \hat{a}_n^{\dagger}|0>$ is a 'state-operator correspondence', but maybe that is pushing it too far
 
like all physical theories that I know have some notion of spacetime, which is usually some sort of mathematical structure with some notion of geometry (and/or topology)
 
@SillyGoose the common threads i wud say are "states", "observables" and "evolution equation"
 
@SillyGoose not sure where the "spacetime" is in thermodynamics :P
 
9:04 AM
this is what u need to write down
 
right and also some notion of state space, observables, and a dynamic relation as Ryder says
@ACuriousMind well where do your system's live in thermodynamics :P
 
in NVT space
or whatever specifies my macrostate
 
i feel like, perhaps this is a wrong idea, accompanying any physical theory no matter how abstract, it needs to come with some sort of map that embeds your abstraction into the "real world", i.e. spacetime structure again
 
note that only fundamental theories need to reference spacetime. there r many theories which dont reference spacetime. e.g. u dont reference spacetime to probabilistically model a coin toss
 
i mean the temperature of a system of interest is not living off in no where land, it is a quantity you have measured (or hope to measure or whatever) in the lab. and the lab exists in spacetime
 
9:07 AM
@SillyGoose I feel this is a very formal objection: The thing that really makes contact with the real world is that temperature is what thermometers measure, for instance
 
so even though you have abstracted temperature from your thermometer into a number, it is still coming from the world and not an independent entity in a constructed formalism
 
of course all that happens "in spacetime", but that's accidental - the core content of the physical theory here isn't about spacetime at all, only insofar as that's where thermometers exist
 
yes. any theory must be connected to a fundamental theory, so some connection to spacetime must be there but the non-fundamental theory need not reference it
u can write down some derivation connecting it to the fundamental theory
 
you sound to me dangerously close to completely crossing the line into philosophy ;)
 
9:09 AM
this is y qfts are the most fundamental theories. they can model local measurements
while non rel QM is not as fundamental
in non rel QM, operators are not bound to spacetime
so non rel QM is an example of an "abstracted" theory
 
@ACuriousMind so in this statement you are saying a physical theory is really just a blackbox that takes in certain values, which are measurable values, and then produces outcomes that have to do with the thing from which the measurable values came from. So as to circumvent any discussion of spacetime.
well i guess nonrel QM just seems weird structurally. CM, even Hamiltonian, Lagrangian, Symplectic formalism are all explicitly and directly connected to the underlying configuration space, which is directly connected to spacetime (to my understanding). QM doesn't have this nice link. Or perhaps I am just unfamiliar with the position operator
and then field theories don't have this problem to begin with by construction
 
@SillyGoose Yes, but then again I generally dislike making any strong claims ontology. I usually take an operational stance: A physical theory models the real world if and only if its predictions (or postdictions or whatever, you don't always need to compute this before you measure) match experiment to an acceptable degree.
 
u can do CM in any co oedinates, just like u can do QM in any basis
 
@SillyGoose this for thermodynamics may be overkill. we usually deal with 1 or 2 systems, and their positions dont rly matter. If youre talking about doing thermodynamics for the atmosphere or something then it might make sense to talk about $T(x)$
 
i dont see anything weird about QM in that sense
 
9:13 AM
well it's not about the coordinates
 
@SillyGoose Just say that QM happens in $L^2(\Sigma)$, where $\Sigma$ is the configuration space (by which I mean the "spatial half" of phase space, the terminology here is terrible).
 
but that the state spaces of CM (in the three formalisms i wrote) are constructed in a natural way from configuration space, which is constructed in a natural way from spacetime
 
geometric quantization formalizes this by having you pick a polarization on phase space, which is a "choice of positions", essentially
 
hence, the state space of CM is constructed in a natural way from spacetime
 
it depends on what kind of QM and CM we r doing. in most QMs, [X,P]=i is part of the definition of the theory
 
9:14 AM
maybe i can finally understand the leinaas and myrheim paper :P
 
it is possible to obtain inequivalent quantizations from different polarizations, so in a sense this structural riddle isn't solved by this - but at least then it has formalized that the "right positions" really are a choice during the quantization process and not baked into the structure of QM
 
@ACuriousMind i optimistically hope for (platonic) forms to exist, but I feel they probably do not. but I think that hope motivates my want to discuss such as the above :)
 
Feb 4, 2023 at 16:06, by ACuriousMind
oh no not another platonist :P
 
i am no longer a platonist personally
i realised plato himself isnt a platonist
look up plato's realm of being and realm of becoming
 
well perhaps I should weaken my claim: I would hope that mathematics allows the language to obtain approximations to the form (if it exists) of a physical theory of a system :P
 
9:22 AM
but the approximation is what we call the physical theory of the system
in Plato's terms, the system just exists in the realm of being and its measurable properties in the realm of becoming can be approximately mapped to math.
 
9:38 AM
@bolbteppa yeah its pushinng to far, the point of the state-operator map in conformal field theory is that states are in one-to-one correspondence with local field operators
@ACuriousMind great, you can always drop the algebro-geometric language with me :P
 
9:49 AM
What does the cracked pot make as noise?
 
clearly, it quacks
 
Ah
Q U A C K
then
 
10:05 AM
"From the nPOV we may think of a local net as a co-flabby copresheaf of algebras on spacetime "
How can a sheaf be flabby
 
10:37 AM
do u think physicists like bohr, einstein remained top contributers in their 50s?
 
@RyderRude it seems like something that could be measured more than an opinion
 
yeah but i dont know much about it
it seems like young people take over in physics in a few decades just like sports
 
11:17 AM
im just saying: search for the data!
 
11:49 AM
the epr paradox was published somewhat later in Einstein's life . it seems like he remained a good contributer
there is a wikipedia article on physics in general en.wikipedia.org/wiki/Physics
any attempt at defining physics comes across as awkward
 
@RyderRude Bohr, Pauli etc. are severely underrated
Also, Schrodigner is underrated
And also perhaps Dirac
Their understanding of physics is way greater than the discovery they are most well known for
 
12:09 PM
these people are already held in the highest regard in physics. i dont think they can be underrated at this point @DIRAC1930
 
Yes you are right
 
Schrodinger could maybe be overrated lol
 
Shows that it wasn't just a case of him being in the right place at the right time
He really was brilliant
 
einstein called his equation beautiful in a letter. i guess einstein didnt initially catch the non realism aspects
 
The Schrodinger equation or the equation of GR?
 
12:18 PM
the former
@DIRAC1930 it seems like a GR text
he has given the meaning of the metric too
theres also an alternative derivation of GR
this book is from 1950
 
That's why i think some of these people are underrated
Like Bohr isn't known for QFT but IIRC, Feynman was put into a room with Bohr and some others in 1948 to try and convince them that it worked
Their impact and input spanned all areas of fundamental physics
As I get older, I get less interested in Dirac
However for anyone interested, here they have a tonne of Dirac's handwritten pages of notes etc diginole.lib.fsu.edu/islandora/object/fsu%3Adiraccol
 
12:35 PM
This golem.ph.utexas.edu/category/2019/12/… posted the other day talks about that Schrodinger book and the crazy story behind it
 
@DIRAC1930 wow that's super cool
unimaginable that he predicted anti particles
mustve felt unreal for him to see it confirmed
 
The golden age of physics. Enough mathematics to keep you satisfied, enough things still left to be discovered and no existential worries about your work not being physically relevant
Paul Dirac in a cap and sunglasses diginole.lib.fsu.edu/islandora/object/fsu%3A115951
lol
'Paul Dirac in audience, eyes closed and hand on head' is the caption
 
@bolbteppa "It seems undesirable to me to present such preliminary attempts to the public in any form. It is even worse when the impression is create that one is dealing with definite discoveries concerning physical reality. Such communiqués given in sensational terms give the lay public misleading ideas about the character of research. The reader gets the impression that every five minutes there is a revolution in science, somewhat like the coup d’état in some of the smaller unstable republics."
einstein was ahead of his time in 1947
@DIRAC1930 maybe Feynman's time traveler electron hypothesis
it seems like Dirac never worked on black holes
and einstein doesnt seem to have worked on qft. he kept doing classical field theory
 
1:34 PM
How do people toe line between I) Working on a project with interesting math but little physical relevance versus II) Working on a project that is not fundamental however with immediate physical relevance?
 
Whichever you prefer
 
I have an existential crisis everyday
 
1:46 PM
I know that if I switched to a PhD in AMOP that I would feel stiffled
 
2:13 PM
phd is always suffocating i guess
 
@ACuriousMind btw I think I solved that problem I had a while back but I would prefer to check
When they say $dS(x)$, do they mean "The variation of the action, as a one-form evaluated at configuration $x$", and when they say $dS(v)$, do they mean "The scalar meant by the one-form of the variation of the action applied to the transformation field $v$"
 
@Slereah yeah, I think we'd usually write $\mathrm{d}S_x$ for the first one
 
My least favorite collision of notation
Well, not my least, but certainly the most common one
Least favorite might be that time when I saw $\sigma^2$ and I wondered why there was a square there
[it was the second Pauli matrix]
 
upper indices suck when they're actual numbers
the square of the second gamma-matrix $(\gamma^2)^2$ is an abomination
 
2:34 PM
Whats everyones favourite equation?
 
i think schrodinger equation
i think i dont like equations too much, so there isnt a high bar
 
In which form?
 
$i\frac{d|\psi\rangle}{dt}=H|\psi\rangle$
wait , $a_p^{\dagger}|0\rangle=|p\rangle$ is deep too
this is mostly a definition, but it's deep
@DIRAC1930 what about you
 
I'm not quite sure
 
@ACuriousMind i also like $dS |_x$ but its kinda heavy notation
 
2:44 PM
do u think qft will become equally rigorous to qm after we can do functional integration
 
I think I prefer the Schrodinger equation written as $-\frac{\hbar^2}{2m} \partial_ i \partial^i \psi+ V(X) \psi = \imath \hbar \partial_t \psi$
 
@DIRAC1930 oh.
@DIRAC1930 this is less fundamental so i like it less
 
It hints at the fact that the whole of non-rel QM is essentially a representation of the Einstein energy-momentum relation on the space of DeBroglie waves
 
@DIRAC1930 yes, but non rel QM wavefunctions usually work on higher dimensional spaces than 3D
 
That is true
 
2:50 PM
so i think we shudnt think about it in terms of waves
 
i'm a big fan of 4d though
 
one thing i cant make out exactly. i read that the minus sign in $e^{-iEt+ipx}$ is an anticipation of relativity
it looks like a Minkowski product
but this is just from the free non rel Schrodinger eqn
 
Have you done QFT yet?
 
a small bit
i havent done interacting qft
 
There is a nice historical introduction in Weinberg's book on QFT
It is very deep and not too complicated. I wish I had found it earler
 
2:56 PM
yeah. i remember reading it once
 
I might read it again tonight
 
it mentioned einstein's model of discrete radiation
@DIRAC1930 it was really interesting as far as i remember
 
I think the best way to learn QFT is the historical way but I think many people disagree
 
i believe modern ways are much smoother
but whatever works for u i guess
it depends on the individual
i get too annoyed at historical ways now because of Newton's laws
mechanics isnt complicated. laws make it look spooky
 
@SillyGoose no no...It was for a friend who got selected for a Ph.D. in physics in the west coast.
 
3:13 PM
@DIRAC1930 i wouldnt say i have one
i like relations between special functions
like Euler's reflection formula $$\Gamma(1/2 - z)\Gamma(1/2 + z) = \frac{\pi}{\cos(\pi z)}$$
just kidding special functions are annoying to remember
 
i so hoped u were kidding lol
 
3:31 PM
@DIRAC1930 EFE, the most elegant equation ever written
In the most beautiful theory ever conceived
 
4:09 PM
@lucabtz there is a reformulation of the standard special functions so that they are symmetric in the arguments and so forth. Very alien, but very pretty. The symmetry makes it very nice to compute too, though.
 
@SirCrackpot I was going to learn GR properly from that Schrodinger book
Seems like a bad idea
 
What about Dirac's little GR book
 
@DIRAC1930 are u also interested in quantum gravity
 
Before worrying about quantum gravity, I need to get my understanding of gravity in proper order
 
yeah. same
 
4:19 PM
@bolbteppa What are your thoughts on this book strangebeautiful.com/other-texts/schrodinger-st-struc.pdf
 
I tried to read it one time and gave up pretty fast
 
Might just stick to L&L 2
 
u can also try Carroll
 
The problem is it's extremely difficult and really (I'd guess) few people finish it, there is simply some crazy stuff in there, Dirac's book is more readable and short, it has stupid stuff in it and a stupid approach starting from a higher dimensional theory but it's not that bad
 
I wasn't a big fan of Dirac's book. Felt like it didn't go into much detail
 
4:24 PM
Zee's GR book is definitely worth the time
They are the closest thing to L&L I've ever found, he has a book on GR and it basically follows/quotes L&L on the basic stuff
 
Thanks for the link
Peter Woit mentioned mentioned this article recently which was written by that lecturer arxiv.org/pdf/1712.06605.pdf
 
He even gives his own version of the invariance of $ds^2$ argument in L&L that really upsets people and wikipedia slanders
 
lol
 
4:47 PM
@bolbteppa I was about to mention that bit but I didn't expect to be anticipated :(
 
@bolbteppa I'm liking that lecturer a lot
 
He does that argument around 33 minutes in
 
I have a GR exam tomorrow, I'm not in the mood to watch somebody replicate that :P
 
My question from today is #2 result
Probably a bad sign
 
Mad
5:02 PM
https://web.mit.edu/8.333/www/lectures/lec3.pdf
how is he getting equation 1.47?
given a function f(y_1,y_2,y_3 ... ) i know that $df= \sum \partial f /\partial x_j *dx_j $
 
Differentiate (1.46) w.r.t. $\lambda$
 
Mad
5:22 PM
yea but i am not understanding how to do that
He is not even getting the dx_j in the formula i wrote, instead just x_j?
 
@Slereah hi. i saw ur question. is the notion of causality even well defined for general groups
 
It is for kinematic groups
 
They all have a notion of causality baked in
they just have weirdly shaped light cones
 
yeah. in case of Galilean group, it seems like we're gonna have to consider open sets of a lower dimension
so that axiom does not generalise well
 
Mad
5:25 PM
oh is he setting x_j= lambda times whatever, i think i get it.
 
@Slereah judging from past experience, yeah
If that happens, your question will die alone :(
 
I have many such questions
 
@Slereah a kinematic group has an associated metric right?, but not all metrics need induce a kinematic group?. becuz i dont think the (+,+,+,+) metric has a notion of causality
 
Occasionally I answer my own questions
they have some associated tensors
But no not all groups are kinematic
 
yeah so not all metrics give a kinematic group
 
5:28 PM
There's only so many kinematic groups
like 11
 
oh
i think u linked this once..this is extremely fundamental.stuff to physics
i also found this question of urs physics.stackexchange.com/questions/703045/…
maybe u were looking for the notion of kinematic groups then
 
No it's not that either
 
5:57 PM
Okay I looked at Dirac's embedding thing again and it's actually brilliant
 
 
2 hours later…
7:37 PM
is it accurate to say that the representation space of $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$ is precisely the solution space of the Weyl equation
 
 
2 hours later…
9:38 PM
@naturallyInconsistent i wish i knew more about special functions. they are hella boring to study tbh, but it is very useful to know a lot about them
 
Lol truee
 
You don't need to know that many really
If you know Bessel functions and the cool polynomials that should be enough for most of physics
 
@Slereah idk but for the field i am in loads of special functions pop up all the time
 
What are you in, like antenna theory?
 
LOL
integrability
 
9:41 PM
the theory of special functions
 
Antenna people love special functions
 
@Slereah i guess in reality when you have to solve any real problem some weird special function will show up
 
yeah
 
anyhow in my work so far i use often gammas, hypergeometric and local Heun functions
but way more show up often like Painlevé trascendents, polygammas, etc.
 
The exact solution of monochromatic waves in GR uses the integral trig functions
 
9:46 PM
i love and hate them. like sometimes its cool to know the solution to your equation has a name and somebody has studied it. however then you have to study all these people said about these functions
mathematica is often useful because it knows a lot about a lot of functions
 
It is nice to have some exact solutions for weird PDEs
Like there is an exact solution for quartic Klein Gordon in terms of special functions
 
but for example for the local Heun functions it knows basically just the taylor expansion around z=0
 
It can be nice to get a feel of their behaviour
 
@Slereah yeah but it doesnt help much in my case
i tried playing around with them but then i realized that the examples in the mathematica docs is basically all mathematica can do with local heun functions
 
Mathematica is only so much help I fear
 
9:50 PM
@Slereah it can actually do a lot, im not complaining
 
It is certainly good at doing derivatives
And solving equations
Solving PDEs or complicated integrals is a bit trickier
 
@Slereah i dont really do PDEs. for integrals sometimes it can find asymptotic expansions of the solutions or similar and often it is enough
 
10:41 PM
word of advice, don't by physics textbooks on kindle
the math looks like it's written for ants
 
11:08 PM
@SirCumference its been written by the quantum elves
 
@lucabtz I guess it's fitting that a QM book is written in a microscopic font lol
 
@SirCumference the text is subject to quantum effects
 
11:43 PM
physics.stackexchange.com/a/804022/141132 what do you think about my answer?
 

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