The h Bar

01:06

@TobiasFünke He's trying to point out that you must be new because you are dutifully rediscovering why some commenters here are getting the cold treatment; for reasons that you are identifying, that is simply showing themselves to be immutable properties.

Silly Goose

01:59

Did you mean to @ tobias or grover 0:

Relativisticcucumber

02:34

@GroveRover i feel that separating the tasks into 'research related' and 'not research related' implies that there exists such a meaningful distinction. id say being able to convince people to fund you and your ideas/being able to realize your ideas in a political society is equally as important as having good idea/education. calling this damaging is strange to me

naturallyInconsistent

02:53

@SillyGoose oh lol yeah sorry; @GroveRover

Debanjan Biswas

03:17

Can anyone provide a general proof for reduced mass where external force is present i.e an external force acting on both bodies in a two body system?

Ryder Rude

08:48

hi

GroveRover

09:25

@naturallyInconsistent got it

@Relativisticcucumber The ability to realize your ideas in a political society has nothing to do with pure knowledge advancement, which is what I care about

Academia has a lot of pathological aspects that damage researchers, some depending on the specific region

I don't think that is my personal opinion, pretty much everyone I talked about tells me the same

to*

Tobias Fünke

09:45

@GroveRover well, but as a group leader you need to have more skills than being good at (e.g.) physics research. so to a certain extend I agree with them

hello everyone :)

eeeehm. I accidentally clicked on "hide posts of this user"--how can I undo this??x(

okay, I found some starred message for which I could click on the user profile. :d

qwerty

I also neither fully agree or disagree with either stance.

Humans do research, research does not happen independent of humans, and where there are humans there is politics. It is simultaneously true that there are some researchers who are nothing more than self-promoters at best and grifters at worst, and this is encouraged by the system (most prominently publish or perish). This is to say nothing of the other aspects.

some random person

10:42

hello uh what is all this?

GroveRover

@TobiasFünke I agree, but I was referring primarily to other skills that are dictated by the current academic practices (such as the funding dynamics) and not by inherent necessities of research (such as leading a group)

DIRAC1930

12:20

Why is reflection symmetry (in the Poincare group) treated as something we can just throw away when it comes to QM?

ACuriousMind

@DIRAC1930 What do you mean? The parity and time reversal operators are crucial parts of statements like the CPT theorem.

DIRAC1930

By this I mean that the full Poincare group includes reflections so shouldn't we force all Lagrangians to be invariant under this (which we don't for the weak interaction as an example)

ACuriousMind

@DIRAC1930 Why? What's the reason we should require that symmetry?

DIRAC1930

Well $t^2 - x^2 - y^2 - z^2$ is invariant under it

So why can we just throw it away

DIRAC1930

12:45

What exactly am I missing

ACuriousMind

@DIRAC1930 I'm not following your argument - sure, the metric is invariant under reflections. The Lagrangian is not the metric. Why would it need to be invariant under reflections?

DIRAC1930

I'm not sure

Isn't that how even Lorentz boosts and rotations are defined

So the metric being invariant and the laws of nature being invariant are two seperate questions?

GroveRover

@ACuriousMind I think they're referring to the fact that the lagrangian is a scalar on the bundles so it can only depend on fields and the metric, without considering the fact that we also assume an orientation structure

This is my naive guess since I've not taken QFT yet

@GroveRover and the orientation structure comes up when you do things like spinors

Ryder Rude

@DIRAC1930 i think so

ACuriousMind

@DIRAC1930 It's certainly not part of the definition that all Lagrangians should be invariant under the full Poincaré group, no.

Ryder Rude

12:59

@GroveRover how to reply to ur own msg

DIRAC1930

But why isn't it? It seems arbritary to consider rotations and boosts and just throw away reflections

GroveRover

@RyderRude you take your permalink and copy the code at the end, then past it as a normal reply (source: @ACuriousMind)

ACuriousMind

@DIRAC1930 Again, I don't know what you mean by "throw away" - the parity operator exists, just the Lagrangian isn't necessarily invariant under it

Ryder Rude

chat.stackexchange.com/transcript/message/66743848#66743848 hi

GroveRover

@RyderRude not like that lol

Ryder Rude

13:00

1 min ago, by Ryder Rude

@GroveRover how to reply to ur own msg

1 min ago, by Ryder Rude

@GroveRover how to reply to ur own msg

@GroveRover what am I doing wrong

GroveRover

:codehere with the code you find at the end of the link

Ryder Rude

66743848#66743848: hi

ACuriousMind

almost :P

GroveRover

@RyderRude just one copy of the code t.t

DIRAC1930

I thought it was an axiom that the laws of nature are invariant under the Poincare group?

Ryder Rude

13:02

66743848: hi

GroveRover

@RyderRude I'm crying

Ryder Rude

#66743848: hi

think_meaning_buildß

😭

ACuriousMind

@DIRAC1930 I have never heard of such an axiom, and it would be obviously false due to the parity violation of the weak interaction

Ryder Rude

what am.i doing wrong

GroveRover

13:04

@RyderRude first : then the code

DIRAC1930

So what is it an axiom that the laws of nature are invariant under?

ACuriousMind

The Wightman axioms say that you have to have a representation of the Poincaré group on your space of states, but they say nothing about symmetry properties of "the laws of nature" (whatever that is) or anything else

Ryder Rude

hi : 66743848

hi : #66743848

ooh I got it

@RyderRude hi

DIRAC1930

Didn't Einstein say that the laws of nature are invariant under a change in inertial frame?

ACuriousMind

@DIRAC1930 what kind of appeal to authority is this supposed to be :P

think_meaning_buildß

13:08

The Authority of Einstein. a la d u f f i e l d

DIRAC1930

I'm just trying to figure out what the axioms actually are

ACuriousMind

the axioms for what?

DIRAC1930

For the symmetries nature

ACuriousMind

I...don't know what that means

DIRAC1930

Well people discard any field theory that isn't Lorentz invariant (rotations and boosts)

Ryder Rude

13:11

we just fit observations to Lagrangians. it turns out that observations are not invariant under full Poincaire group @DIRAC1930

ACuriousMind

@DIRAC1930 Do they?

Ryder Rude

also note that the Poincaire group is irrelevant for QFT in curved spacetime

so it is not like these symmetries r a fundamental property of nature, at least not globally

DIRAC1930

@ACuriousMind Yes

ACuriousMind

@DIRAC1930 No.

I'm sorry, but if you can't formulate your question in a self-contained way without vague assertions about what "people" do that I just don't consider to be true, I can't give you any useful answer to your question

DIRAC1930

Klein Gordon equation, Dirac equation, Maxwells equations etc.

The first two were entirely motivated by Lorentz invariance

In the latter it was found later

Ryder Rude

13:16

people do this because Lorentz invariance violation hasn't been observed in flat spacetime experiments. so it is a good way to develop theories

but if the violation is observed, people would not throw away non Lorentz invariant lagrangians

GroveRover

@ACuriousMind How do you make a non-Lorentz invariant theory without altering the metric or dropping background independence

DIRAC1930

Ok so if special relativity is true, then one must look for lorentz invariant lagrangians

GroveRover

genuine question, I'm curious

ACuriousMind

@DIRAC1930 Sure, so those are Lorentz invariant, but they're equations for free particles - the argument is that with a free particle you don't have anything that could break Poincaré invariance, there is nothing to pick a preferred direction or point in space, so its equations should be invariant under the Poincaré group

But it turns out that the weak interaction has a "preferred side of the mirror".

@GroveRover What do you mean by "background independence" that it would not be synonymous to being Lorentz invariant?

GroveRover

@ACuriousMind I mean no preferred FOE/invariant under diffeomorphism

Ryder Rude

13:20

i think non Lorentz invariant theories can be consistent with SR

ACuriousMind

@GroveRover I was hoping you would not say "diffeomorphism" :P

DIRAC1930

This doesn't seem very logical

GroveRover

@ACuriousMind why t.t

Ryder Rude

e.g. one can have space translational invariant theory in Galilean spacetimes

Minkowski spacetime is no different

The spacetime structure only tells u how to switch between frames

it does not necessitate dynamical symmetries

But yes, Einsteins principle about all inertial frames being equivalent would be violated

Slereah

Invariant under diffeomorphism would be a pretty strong condition :p

Ryder Rude

13:23

this just means that SR is more general than Einsteins principle

ACuriousMind

@GroveRover 1. It's a running gag in this chat that the topic of "diffeomorphism invariance" haunts me. 2. The problem is that "diffeomorphism" is too vague to pin down what you mean. The Einstein-Hilbert action actually does have diffeomorphisms as its symmetries. Other actions do not.

DIRAC1930

So Einsteins principle is not fully correct?

ACuriousMind

@DIRAC1930 What is "Einsteins principle"

Ryder Rude

@DIRAC1930 in the sense that one can develop non Lorentz invariant theories in minwkoski spacetimes

it is no different from having non rotationally invariant or non translationally invariant theories in Galilean spacetimes

DIRAC1930

Special relativity

In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper, On the Electrodynamics of Moving Bodies, the theory is presented as being based on just two postulates: The laws of physics are invariant (identical) in all inertial frames of reference (that is, frames of reference with no acceleration). This is known as the principle of relativity. The speed of light in vacuum is the same for all observers, regardless of the motion of light source or observer. This is known as t...

Point number 1

GroveRover

13:25

@ACuriousMind I mean that there is no preferred point in space, which is necessary to formulate equivalent theories on arbitrary manifolds

Slereah

To be fair, there is the equivalence principle, where particles have to move in an "invariant way" in some sense

But the specific statement of the equivalence principle is about neutral monopole particles not under any other force

Ryder Rude

Note that, when Einstein derived Lorentz transforms from the principle of relativity, he did not actually use any dynamical equations

ACuriousMind

@Slereah and the equivalence principle famously has so many different meanings that it's impossible to tell if it's true or false if you just say "equivalence principle" :P

Slereah

Which is very much not the case for particles with T breaking

@ACuriousMind I go by Will

Ryder Rude

So, in principle, Einsteins derivation only assumes that the behavior of spacetime is symmetric under Lorentz transforms

ACuriousMind

13:27

@GroveRover What does that mean, technically? What would a theory with a "preferred point" look like?

Ryder Rude

which is equivalent to the metric being invariant under Lorentz transforms

but it does not necessitate the dynamical laws being invariant

that the dynamical laws are invariant is a further experimental fact

Slereah

Very good principle

Typically the condition also include that it's a monopole, ie spin 0

Ryder Rude

@ACuriousMind e.g. if u couple the theory to an external non dynamical vector field, then the laws of physics are different in different reference frames

it is like putting the universe under an external potential in Newtonian mechanics

Slereah

I do think it's a little weird, but tbh most people thought it was weird

Ryder Rude

these theories do not have Noether symmetries cuz the symmetries get broken by the external fields

Slereah

13:30

I think people expected P and T symmetry to hold before the 60's

But that's just how the universe be I guess

fqq

11

think_meaning_buildß

lol

ACuriousMind

@Slereah they sure did, but it's just aesthetics :P

DIRAC1930

It is literally the weirdest thing I have come across in physics

Ryder Rude

e.g. imagine a Galilean spacetime such that an external potential $V(x)=(x-a)^2$ is applied to the whole universe

this universe does not have a conserved momentum

Slereah

13:32

Just shake the universe until it's symmetric again

Ryder Rude

note that the potential is non dynamical. its source isn't matter

DIRAC1930

Murray Gell-Mann - Disagreement among the top physicts (77/200)

►

GroveRover

@ACuriousMind the space is GL(n). You include the distance between the particle and the identity in the lagrangian. Anyway we're straying from the topic

DIRAC1930

Murray Gell-Mann of course says he wasn't shocked at 3:58 lol

think_meaning_buildß

27 mins ago, by ACuriousMind

@DIRAC1930 what kind of appeal to authority is this supposed to be :P

Ryder Rude

13:36

this paper analyses four versions of the equivalence principle link.springer.com/article/10.1007/s13194-010-0009-z

GroveRover

@RyderRude Yes you need additional structure, not Lorentz invariant, breaking the homogeneity. If all you have is a metric, then every theory is lorentz invariant as your lagragrangian can't depend on anything weird

ACuriousMind

stop calling the weak interaction weird :P

Ryder Rude

@GroveRover yes. we should also note that messing with Lorentz symmetry can lead to causality paradoxes. So i think that the restrictions on SR dynamics is much tighter than Galilean dynamics

ACuriousMind

it's bad enough that it's named "weak"

give it a break

Ryder Rude

@GroveRover I'm not sure if one can couple SR to an external field without causality paradoxes. maybe there r restrictions on the external field

Sine of the Time

13:40

chat is crowded today

GroveRover

@ACuriousMind I just checked the parity-violating lagrangian and it depends on the chirality of the spinor, which is defined for a given orientation. So wasn't I right when I said that the violation stems from not preserving the orientation?

Slereah

Also on the plus side is, the particles do behave the same if they're free?

ACuriousMind

@GroveRover Yes, that's right

Slereah

So no worries with the equivalence principle even then

DIRAC1930

After QED, everything become illogical

ACuriousMind

13:43

It's not illogical when the world does not conform to your expectations

2

parity violation is a proven phenomenon, if your logic says it shouldn't happen, it's your logic that's wrong, not nature

DIRAC1930

lol

GroveRover

@ACuriousMind So @DIRAC1930 the mystery's solved: if your lagrangian depended only on the metric, you would have full Poincarè symmetry. But since the weak lagrangian depends on the chirality of spinors, which is defined using a given orientation of your space-time, then a transformation that breaks orientation (such as a reflection) is not a symmetry anymore

DIRAC1930

@ACuriousMind My point is that clearly something must be confusing

Murray Gell-Mann - Disagreement among the top physicts (77/200)

►

see 1:40

Other people had the same confusion

Slereah

Also like you have to remember that which symmetries are Good is entirely up to individual opinions

If you look at the Machian people they would say that even full Poincaré symmetry isn't enough :p

qwerty

Machian people?

Slereah

13:50

Mach's principle

In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Albert Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The hypothesis attempted to explain how rotating objects, such as gyroscopes and spinning celestial bodies, maintain a frame of reference. The proposition is that the existence of absolute rotation (the distinction of local inertial frames vs. rotating reference frames) is determined by the large-scale distribution of matter, as exemplified by this anecdote...

qwerty

yes, I thought there's many versions of it floating around

Slereah

there are

qwerty

and not that much interest in it these days

Slereah

There's still some, although it's pretty niche

GroveRover

@Slereah this is the vaguest thing I've ever read in physics

Slereah

13:52

It has many interpretations indeed :p

GroveRover

No wait I forgot everett

qwerty

Machian sounds too much like Martian. I can imagine a green alien with a rotating bucket.

GroveRover

@Slereah How

I mean, some symmetry violations can be reabsorbed in dynamics, e.g. the inertial structure in SR is then dynamic in GR. But I can't see how you can make the orientation dynamic, being non-smooth

Slereah

Orientation is just defined by an n-form

GroveRover

@Slereah I know(?)

But the sign is what matter

And you can't change it smoothly

Slereah

13:59

So what

GroveRover

So you can't make it dynamic

Slereah

Why would you need to make it dynamic

GroveRover

To reabsorb the parity violation in a bigger theory, hence restoring poincarrè symmetry as Machians'd like

Slereah

The Machians do not decide what happens :p

and why do you need a bigger theory

GroveRover

This was my objection to taking the poincarrè symmetry as true

Slereah

14:01

Lagrangian just fails to be reversal invariant

Well the full Poincaré group certainly :p

GroveRover

@Slereah To reabsorb the parity invariance that is required by the full poincarrè symmetry

Slereah

Unfortunate but them's the break

In 1+1 dimensions, there's a discrete symmetry interchanging time and space, you wouldn't expect it to be true either

I can come up with many many symmetries that people tried and that are not invariant in physics :p

GroveRover

@Slereah Yes I'm trying to figure out how Machians try to restore the poincarrè since you can't make it dynamical

Slereah

Forget Mach

They're not concerned with QFT

or about discrete symmetries

GroveRover

@Slereah Lmao that's a small detail to leave out of their considerations

Slereah

14:06

There's no model of particle physics that's invariant under the full Poincaré group

you took the wrong lesson from my words :p

GroveRover

@Slereah mine was sarcasm

Ryder Rude

i think Machians believe that the structure of spacetime near earth is decided by configuration of cosmos

Einstein showed that it is instead decided by configuration of energy near earth

Slereah

the short answer is that yes it's surprising that it's not a symmetry of all physical theories but that's just like that

Many surprising things in physics

Ryder Rude

Mach imagined rotating a bucket. The water inside it experiences a centrifugal force. Then Mach thought that anything that rotates wrt the frame of the stars must experience a centrifugal force

GroveRover

@Slereah I think the surprise just stems from not being used to it. An inertial structure would be a way bigger symmetry breaking but it was the standard until GR

Slereah

14:11

There's a very wide variety of symmetries you can look at in kinematic terms

The Poincaré group is neither the largest nor smallest group people have tried

GroveRover

@RyderRude But that's just the usual inertial structure in Newtonian mechanics

Newton was already talking about this and had a theologic explaination

Slereah

Why isn't physics invariant under the conformal group 😔

or the projective group

Ryder Rude

@GroveRover yes. it was poorly understood in those days, even by Newton

But even Einstein ended up taking some version of Mach's principle seriously

but he had the right idea. That the notion of inertial frames must be non-fixed, determined by matter

GroveRover

@RyderRude this is weird to me since GR is the exact thing that solves the presence of a god-given inertial structure

DIRAC1930

So is it just an experimental fact that the laws of nature are not invariant under reflections and that people just assumed it was?

GroveRover

14:13

What was He possibly looking for in mach's principle

Slereah

@DIRAC1930 Yes

ACuriousMind

@DIRAC1930 yes

Slereah

Discovered in the 60's

Ryder Rude

@GroveRover it is philosophically close to Mech's vague idea. Mach thought the notion of inertial frames wasn't god given, but determined by distribution of stars

DIRAC1930

So why aren't people making a bigger deal out of it

Slereah

14:14

Very strange result and unexpected, from what I remember

So you're not alone feeling confused

Ryder Rude

@GroveRover and in GR, inertial frames are determined by local distribution of matter

GroveRover

@RyderRude yes but if you already have GR, what then?

ACuriousMind

@DIRAC1930 what kind of "deal" do you mean? it's in every textbook that discusses the weak interaction

Ryder Rude

it is only philosophically close

Slereah

Yeah it's not exactly an obscure fact :p

Ryder Rude

14:16

@GroveRover it is just that Einstein was inspired by Mach, but his finally theory isn't actually the realisation of Mach's principle

Einstein initially misunderstood a lot of the aspects of his theory

DIRAC1930

I'm confused why people can be certain that the laws of nature under all circumstances in the future must be invariant under boost and rotations when something like this has happened

Ryder Rude

He misunderstood what co ordinates meant

Slereah

@DIRAC1930 I mean we're not?

ACuriousMind

@DIRAC1930 People aren't certain about that :P

Slereah

It's just an assumption that we hold for now since that seems to be true

Historically being certain of symmetries has not worked out great :p

You have plenty of people working on violations of Poincaré symmetry

Seems to hold so far?

Ryder Rude

14:18

i think Einstein also played with the idea that all frames are equivalent, instead of just inertial frames

but the final theory doesn't have that

it tells us that the philosophy that motivates a person need not be realised in their theory

GroveRover

@RyderRude ofc it's not, so what you're saying is that he promoted this idea before he found the answer in GR, right?

Ryder Rude

@GroveRover I'm not sure if he promoted it. But he himself believed in it

it is a vague idea anyway. we can say that GR makes it precise, and removes its incorrect aspects

GroveRover

@RyderRude did he believe in it before GR or also after?

Ryder Rude

@GroveRover before

@GroveRover I'm not sure about after. in the initial years, he misunderstood how GR was supposed to be interpreted

GroveRover

@RyderRude I understood after so that was the thing bothering me

Makes sense now

Ryder Rude

14:21

e.g. he thought that GR didn't have determinism because he misinterpreted what co ordinates meant

it is weird that he was able to develop the theory without knowing what it meant

GroveRover

@RyderRude that sounds interesting. I have a book with some of his original papers, will read it sooner or later to see the historical developing

Ryder Rude

@GroveRover great

GroveRover

@RyderRude I mean, there are some weird things that are mathematically easy but conceptually subtle like singularities

So I can understand why

Even now you see misinterpretations of coordinates in physicists' divulgation books

Ryder Rude

@GroveRover yes, i think that mathematics is one's primary guide to developing theories. interpretations come later. Schrodinger too had no idea what the amplitudes meant

except some vague interpretation using double slit experiment

GroveRover

@RyderRude the issue right now is that it is our only guide

So things like string theory happen

Ryder Rude

14:26

@GroveRover lol. i just realised I didn't mention experiments as the primary guide :P

tbf GR in particular had no experimental guide

GroveRover

@RyderRude To be fair even now we don't know what amplitudes really mean

Ryder Rude

@GroveRover do u like or dislike string theory

@GroveRover yes :P

GroveRover

@RyderRude dislike

Ryder Rude

i think only string theorists can know what kinda beautiful math the theory has. it is supposed to be extremely intricate

@GroveRover oh

GroveRover

Not in a strong way, I don't know if it was reasonable to pursue it in the past

But I (and pretty much anyone who's not in strings) don't see many results

Ryder Rude

14:30

yes

GroveRover

@RyderRude They won fields so the math must be awesome for sure

Ryder Rude

@GroveRover i think it was supposed to explain all fundamental constants and have unique dynamics in the beginning

would have been great. but then the 11D compactification came long

@GroveRover yes

@GroveRover my problem is that they haven't figured out their parent theory

its just an S-matrix and a web of conjectures

but let's see. maybe they can get something.... i will learn something else for now tho

if they figure out their stuff, I will switch

i also think they don't touch on the measurement problem. but some string theorists think that their parent theory may touch on the measurement problem. but string theory is just a lot of ifs and mays

GroveRover

@RyderRude I think I'll take a summer school in strings anyway. Not going to work on it or learn it further, but I think some basic knowledge could be helpful for anyone working in gravity

@RyderRude ifs and mays that cost 50 years and millions of dollars

Silly Goose

14:48

Here $S$ is a subgroup of some group $G$, $N(S)$ is the normalizer subgroup of $S$ in $G$, and $T$ is a set each individual element of which is stabilized by all of $S$.

I think what 3.7 is supposed to show is that $\langle \psi_j \lvert E \lvert \psi_i \rangle \neq C_{E} \delta_{ij}$ where $C_{E}$ is just some constant dependent on $E \in N(S) \backslash S$.

But I don't see how this is the case necessarily. Naively, to preserve orthogonality $E$ must be proportional to the identity.

ACuriousMind

@SillyGoose All it shows is what the text says, i.e. $E\psi\in T$ for every $\psi\in T$. I don't know where what you think it's "supposed" to show comes from, nor what you mean by "preserving orthogonality"

fqq

15:12

@DIRAC1930 en.m.wikipedia.org/wiki/Modern_searches_for_Lorentz_violation

Silly Goose

16:09

@ACuriousMind it states “is undetectable by the code”. Which precisely means the condition I wrote after “is supposed to show…”

This is page 19 of arxiv.org/pdf/quant-ph/9705052

Where eqn 2.10 on page 14 is what the author defines as an error that is detectable by the code

@ACuriousMind what is an $E$ not proportional to the identity which preserves “orthogonality” $\langle \psi_i \lvert E \lvert \psi_j \rangle = C_E \delta_{ij}$? I thought this would mean that $E\lvert \psi_i = \psi_j$ for some $j\neq i$, but the only consistent $E$ is then proportional to the identity. If $E$ does permute basis vectors, then there exists the situation $\langle \psi_i \lvert E \lvert \psi_j \rangle = \langle \psi_i \lvert \psi_i \rangle$ for at least one $i$.

Sorry “for some $j\neq i$” should be for some $i,j$

ACuriousMind

17:48

@SillyGoose Your $\psi_i$ are not a basis. They are $2^k$ vectors in a larger $2^n$-dimensional space. Your condition 2.10 is just that the operator restricts to a multiple of the identity on the subspace spanned by them.

Silly Goose

19:16

oh i see

think_meaning_buildß

19:40

@fqq that meme is worthy of everyone of those stars ⭐

imbAF

20:00

I have a question.
I am currently trying to somehow change the canonical energy-momentum tensor so that it can be symmetric on the indices:
$$T^{\mu\nu}=F^{\gamma\mu}\partial^\nu A_\gamma +\frac 1 4 F^{\alpha\beta}F_{\alpha\beta}g^{\mu\nu}-\frac{1}{2}m^2A_\alpha A^\alpha g^{\mu\nu}$$

I saw in some notes, that one can consider the non-symetric part and use integration by parts

and use the lorentz gauge, which then solves the problem

My question is, what's the idea behind the integration ?

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