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12:16 AM
here's hoping we can get that fixed soon, b/c having that on Springer is a bit embaressing :P
Again, this is a confusion I have. Doesn't spin naturally arise in the quantum theory if there is an $SO(3)$ symmetry in the classical theory since the projective representation of $SO(3)$ is $Spin(3)$?
Or $SU(2)$, I can't remember which one
If the quantum system you're studying possesses rotational invariance, it is possible that some of the quantities involved (like electrons) are described by things that transform under representations of the orthogonal group which are not vectors or tensors of the orthogonal group, but are referred to as spinors. That usually arises because the classical limit possesses rotational symmetry
Sorry I meant to say that if you quantize a vector field that classically transforms under $SO(3)$ don't you automatically get spin?
Because $SU(2)$ is the projective representation of $SO(3)$
No, what are you quantizing and how does it become a spinor, are you talking about promoting a position vector to a position operator
I'm talking about promoting a field to a field operator
I'm not sure how it becomes a spinor
12:31 AM
If you quantize a classical field (i.e. commuting fields in a Lagrangian) which is a spin representation of the orthogonal group, you will get an operator which transforms as a spinor from the point of view of it's indices, but such a quantity doesn't exist in classical physics it's just a mathematical construction in standard physics
there's not really a de-quantization of spin 1/2
@Semiclassical Looks good
In QM, the states carry a projective representation of the classical representation I thought
@bolbteppa "Peneralization" :3
'A much-needed book-length philosophical defense of the information-theoretic interpretation of quantum mechanics'
12:34 AM
if you're wondering how i can simultaneously be an author on that while still being fond of pilot wave stuff
the old Walt Whitman quote comes to mind :P
(though nowadays my sympathy for pilot-wave is founded not so much on it as a viable interpretation of QM, so much as a source of perspective/intuition for the formalism)
So the classical representation is $SO(3)$, the projective representation is $SU(2)$
You can't get a spinor from a vector, you started from a 'classical' spinor field and just turned it into an operator, you started from a 'classical' projective representation, and it's as if you placed a position operator at each point of space(-time) and then interpreted them as transforming into one another as spinors as you perform rotations (Lorentz transformations).
But the 'classical' field is really just a linear combination of solutions of the Schrodinger equation interpreted as a classical field, i.e. spinor wave function solutions of the Schrodinger equation, you're really working with quantum wave functions, so the classical nature of it is illusory
the notion of a classical phase space doesn't make a lot of sense for spin
The statistical ensemble of a system, are copies of the system in different microstates?
simply due to how the conjugacy works
12:39 AM
I'm not saying spin exists classically
I'm saying that it's a consequence of the quantum states carrying a projective representation. i.e. spin is a quantum effect
that might get you as far as being able to pick the spinor representation as an option
But the only alternative is $SO(3) \oplus SO(3)$ or something because $SU(2)$ covers $SO(3)$ twice
So you naturally get a 2 component wave function right?
Classically we're always beginning from position vectors and talking about how they move, that's all we're doing. In QM, the fact that you're even using representation theory and the fact that there's no reason why you can't exclude the possibility of projective representations is basically why they occur as an option, beyond that it's 'spin-statistics + mystery'
mathematically, spin 100 is just as valid an option as spin 1/2
For a one-particle state? So it's a choice we make to use the fundamental representation os $SU(2)$ rather than some tensor product rep?
12:49 AM
It's just an experimental fact that we find particles with spin 0, half, and one mainly. Above that it's all question marks and hopes
This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators.In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles of arbitrary spin j, an integer for bosons (j = 1, 2, 3 ...) or half-integer for fermions (j = 1⁄2, 3⁄2, 5⁄2 ...). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. They are named after Valentine Bargmann and Eugene Wigner. == History == Paul Dirac first published the Dirac equation in 1928, and later (1...
you do see larger representations showing up experimentally, e.g., there's a particular iron molecular cluster (Fe8) which acts like it has spin 10
but that's only an effective description. the underlying interactions are just between electrons
@Semiclassical could you help me with something regarding canonical ensemble?
just ask; don't ask to ask
In the MCE for a system in eq. the eq. state, which is a macrostate, is a mixed state and is characterised by a certain multiplicity, that represents the nr. of microstates that the system can be in in eq, am I correct?
In my understanding of the micro canonical ensemble ?
ah, ensembles. can't help you much here beyond the obvious points
12:58 AM
maybe this is an obvious point
microcanonical ensemble = you know what the system energy is
but I am talking statistically not in thermodynamic terms
And I need an answer regarding the microstates of the canonical ensemble
(which is why microcanonical doesn't go over too well to QM: if energy is quantized, then saying you can choose the energy doesn't make a lot of sense)
whether they are the same as in the mce or not
What are the same
1:00 AM
I think they are not
the microstates in both ensembles
microstates of canonical ensemble vs. microstates of microcanonical ensemble
while for the MCE the microstates have same energy as the macrostate that they represent
in CE the macrostate is composed from states with TOTAL energy that is different
and they cannot be microstates, because microstates have roughly the same energy
$E$ is the macrostate, knowing the values $(n_1,n_2,n_3)$ in $E = n_1 E_1 + n_2 E_2 + n_3 E_3$ gives one of the microstates, one may have many different $(n_1,n_2,n_3)$'s giving the same overall $E$
Yes I know that
but it's not my question
I'd be inclined to say that, yes, they have the same microstates
at least classically. (for QM the micro/macro distinction is a bit weird)
1:04 AM
how is that possible
look the def in wiki: The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.
macrostates or microstates?
states differ in total energy
that's the point
these states with different total energy
sure? but if all they're saying is that the macrostates have different energy, then there's no implication for the microstates
are the microstates for the canonical
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature.The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.
Again, I see nothing about microstates there
1:06 AM
what other then microstates
can you have in eq?
is unique
in eq
in eq. a system
is in a particular macrostate, that has a certain energy (state variable)
and this single particular, energy characterized macrostate of the system
has a multiplicity
in other words, a bunch of microstates that the system can be in
this is the case for MCE
1:08 AM
and whether you look at the macrostate, or microstates
in MCE they have roughly the same energy, only with an infinitesimal
fair. so maybe it's better to say that a particular MCE picks out a subset of the microstates of the CE
namely, those microstates which have the correct energy
not exactly
Let me give this example
MCE : football field = macrostate, people inside the stadium = microstates
the stadium doesn't change but people can change positions
and velocity
this is the multiplicity
and this is 100% true for MCE
well. what's a microstate of an ideal gas with a given energy?
1:12 AM
Now for CE : you have (as I understand) a bunch of stadiums, and each stadium it's own people
presumably it's some specification of the positions/velocities which happens to have the specified total energy
bunch of stadiums = different total energy
You can apply the microcanonical distribution to a closed system consisting of subsystems which exchange energy to derive the canonical distribution, so the microstates in the canonical distribution are ultimately just the microcanonical microstates, if that's what you're asking
that's what I am asking
but now why?
The total system is a closed system
1:13 AM
In case you read my reasoning above, why am I wrong?
but it changes energy no?
Every system described by the microcanonical distribution can be interpreted as being composed of subsystems which exchange energy or even particle numbers etc between the subsystems
I am trying to understand this
@bolbteppa well, in that case the canonical ensemble for a subsystem will only consist of microstates for that subsystem
it's not like knowing a microstate for one of the subsystems tells you the microstate of the other subsystem
1:17 AM
I see
ok that is another pov that I didn't consider
so that to me cuts against taking them as literally the same thing
But this is so complicated now
the problem, though, is that this perspective naturally has CE as describing subsystems of MCE
once I focus on a subsystem of the MCE (I don't even know how one can do thatr)
rather than comparing CE and MCE of a particular ssystem
1:18 AM
can we talk about microstates and macrostates of the sub system of the MCE?
L&L's derivation gives the canonical distribution from the microcanonical distribution of a closed system (the whole book is from this pov), I would suggest reading that
that said, suppose you start with two subsystems which together are described by a microcanonical ensemlbe
but how do you do the differentiation
and then you separate the two systems so that they can't exchange energy anymore
that's a gigantic problem to try and understand or picture
1:20 AM
i don't think that separation should change the microstates of the two subsystems
but you need to ask yourself
and if it doesn't, then you've gone from CE of a subsystem to MCE of that subsystem without changing the microstates
you had a system, which has microstates and now you look up a sub, how is the microstate of it?
@bolbteppa what's the name of he book?
If you have a closed system composed of a subsystem you're trying to describe with the CE, the rest of the body is then the medium ('heat bath') it's interacting with, and since you know the total energy you know that once you find the subsystem properties it constrains the rest of the system, so you can just ignore the medium and focus on the subsystem due to the constraints
the constraints being
1:23 AM
what i have in mind is: you start with a closed system and divide it into two subsystems in such a way that energy can be exchanged but not particles
closed or isolated ?
because MCE are isolated right?
i'm using the words interchangeably. energy can't get out
The total energy is the sum of the energy of the subsystem and the medium, where the total energy is fixed because it's a closed system described by a MCE
now one question
and then you gradually turn on a barrier between the two subsystems which makes them unable to exchange energy
1:24 AM
Also you should look at their derivation of the MCE, they don't argue it's actually the density matrix of the system, they say it's a replacement for it, avoiding all this ergodic stuff
once you've done that, the two subsystems are isolated and thus can be described by MCE
the MCE is in eq. and is characterized by a macrostate and a set of microstates. And now we jump into a sub system of it. What can we say about the macro and micro states of the sub system?
so we have a scenario where we have MCE for the subsystem as a particular limit of the CE for the subsystem
I will look at their derivation, but I need to name of the book
If the subsystem is a closed system then it's just another MCE, if it's not a closed system you need to use something like the CE or the GCE
1:25 AM
L&L's nothing came on google
landau lifschitz
The Course of Theoretical Physics is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s. It is said that Landau composed much of the series in his head while in an NKVD prison in 1938-1939. However, almost all of the actual writing of the early volumes was done by Lifshitz, giving rise to the witticism, "not a word of Landau and not a thought of Lifshitz". The first eight volumes were finished in the 1950s, written in Russian and translated into English in the late 1950s...
presumably it's the Statistical Physics volume
my gut instinct is that, since you can obtain the MCE in this setting as a limiting case of the CE
It's to much damn it, and I am in University, where you learn nothing and you waste time. ANd we are moving on into another topic next week, after just 1 lecture on this
that the microstates for the former can't be different than the latter
1:27 AM
It;s not realistic
tbf, they're not expecting you to learn everything at once. they expect you to come back to it later and learn it again, rinse and repeat
Welcome to the world of stat mech
you see electromagnetism in intro physics. then you see it again in upper-division physics, and finally you see it again in graduate-level physics
each time you get more and more of the story
@bolbteppa Look I understood everything you said, but you didn't gave me the answer to one thing that I asked, that is
even though I understand that depending on how you
while still inevitably having to say that a lot of this would require to learn more
1:29 AM
define a sub system
I simply want to know the answer to this @bolbteppa
System = eq.=macrostate=multiplicity= bunch of microstates. Now we "zoom in" and consider a subsystem of the system that is in eq.
this sub system
whether MCE, CE,GNE not important
this sub ssytem should have it's own micro and macro states
yes or no?
They define a subsystem on page 2, use that as a motivation to give it a first read
Ok, I will try to find the correct volume and search the answer to this
I'm on stage 8 of the process in that quote
1:32 AM
The thing is, it depends how your brain is wired
For example, whenever I read a new topic regardless of it
my goal is to search the things I don't understand
and I get frustrated when I understand everything
because I think, that something is missing, there's no way that I got everything. I am slacking somewhere in my understanding
And that results in that quote
@bolbteppa one final thing. For the internal energy in case of CE $U=\Sigma P_i E_i$
Now this energy this averaged energy is that of the sub system or of the whole system (which is a MCE)
It's very late here. I need to go, but thx for the help both of you.
$E = U + E'$ where $E$ is the MCE total energy, $E'$ is the heat bath, and $U$ is the subsystem
life gets easier when you do QM, since then you just do Dirac notation and pretend that's enough :P
I'll get to that as well
@bolbteppa ok and which of the 3 is this $U=\Sigma P_i E_i$ ?
you are multiplying probabilities, with energies
are these probabilities, the probability P_i that the subsystem i has an energy E_i ?
Or am I wrong again?
well. that'd work for microcanonical ensemble as well: $\sum_i P_i E_i = (\sum_i P_i) E = E$
the difference is that now the E_i's allowed are only those with the same energy
1:42 AM
Can I interpret this
What is $E_i$
that in the phase space, we do not see anymore an infinitesimal energy difference but an interval?
energy of ith microstate, presumably?
@bolbteppa are these probabilities, the probability P_i that the subsystem i has an energy E_i ?
But $U$ is the energy of the subsystem, what subsystem of what
1:44 AM
and in my notes : $U=<E>=Tr(\rho H)=\Sigma_i P_i E_i$
it doesn't say anything
@imbAF if you're doing traces, then that's QM and you're probably not doing microcanonical
Full expression in my notes : $U=<E>=Tr(\rho H)=\Sigma_i P_i E_i$
microcanonical ensemble doesn't make a lot of sense in QM
but you can do it tho
The $E_i$ are presumably the spectrum of possible energies for the subsystem, and $P_i$ is the probability of finding a given energy for the subsystem?
1:45 AM
using multiplicity
yes @bolbteppa that's what I was assuming
you basically divide the system in parts, and each part has a probability of having a certain energy value
Wikipedia has some applicable remarks: en.wikipedia.org/wiki/…
I read that, and I need translator to get that last part
Or maybe I am to tired
"The microcanonical ensemble is defined by taking the limit of the density matrix as the energy width goes to zero [but then problems]"
i wonder what you'd call "microcanonical except with an energy interval $\Delta E$"
1:48 AM
basically a spherical shell of the hypersppere with thickness 0
I don't know. But i gotta sleep so I can wake up early and start again. Maybe I find you guys here tmrw
Thanks again
And if you have anything to say to this topic, please tag me, so I can see it tmrw
"Based on the above definition, the [classical] microcanonical ensemble can be visualized as an infinitesimally thin shell in phase space, centered on a constant-energy surface."
that makes it sound like the microstates would just be phase space points, albeit only those lying in a specific surface
so "microstates of a given MCE" = subset of microstates for CE
yes because that's how microstates are represented in phase space
as points
1:51 AM
and as 6n dimensional squares in quantum
in which case this starts to feel semantic.
the microstates of a given MCE will also be microstates for CE, and every microstate of a CE is a microstate of some particular MCE
so i guess we could say that CE = ensemble of MCEs
"every CE microstate is also a microstate of some MCE" seems good enough for me
so i don't see much to be gained in thinking of them as basically different things. they're the same options, just with different restrictions
3 hours later…
5:23 AM
Only two weeks till the JWST launches
6:20 AM
Hi All...
Hello @JohnRennie Sir...
6:54 AM
Q: Why does it show "Queue has been cleared", every time I check 3 question?

Ishaan ManishI recently got the Review Questions privilege, so I know very little stuff about how to review questions. I was recently reviewing some questions, while after checking 3 questions, it says "This queue has been cleared". I have experienced this on Space Exploration S.E. and Astronomy S.E. also, bu...

3 hours later…
9:38 AM
It's always hard to know where to start reading in a paper
Sometimes if you skip the transition is like "A group is a little set with those properties (...) now consider the $(\infty, 1)$-topos on the category of twisted homotopies"
In QM degenerate perturbation theory I get some matrix equation as shown here: quantummechanics.ucsd.edu/ph130a/130_notes/node334.html It almost looks like a Schrödinger's equation. But it is not, because there is no wave function present, only these vectors of coefficients.
I struggle to formulate a question, but I would like to construct an effective Hamiltonian and a Schrödinger equation from my perturbation theory, but I don't know how this is done rigorously
I think what confuses me is that an "effective Hamiltonian" seems to consist of numbers or matrix elements instead of operators. Then again, a matrix is sort of an operator.
10:07 AM
@B.Brekke it's just written strangely
take $\lvert\phi\rangle = \sum_i \alpha_i\vert\phi^{(i)}\rangle$, then that equation is $\langle \phi^{(j)}\vert H\vert \phi\rangle = \langle \phi^{(j)}\vert E \vert \phi\rangle$
so this is just the eigenvalue equation $H\lvert \phi\rangle = E\lvert \phi\rangle$ restricted to the $\mathcal{N}$ subspace
10:22 AM
@ACuriousMind What is $\vert \phi\rangle$? Is it the unperturbed wave function?
@B.Brekke it's just a state in the $\mathcal{N}$ subspace
note that I wrote $\lvert \phi\rangle = \sum_i \alpha_i\lvert \phi^{(i)}\rangle$ in the message above that
I don't know what this means for you in terms of "wavefunctions", I'm just pointing out that that equation isn't an equation of "matrix elements instead of operators" - matrices and operators, and lists of coefficients and vectors are equivalent, after all
@ACuriousMind Yes, I understand. What confuses me is really a phrase for effective mass theory for electrons. After doing some perturbation theory, they say that the result is equivalent to free electrons with an "effective mass". These free electrons were not part of our initial subspace $\mathcal{N}$. However, I think this is rather an observation instead of some rigorous result
I guess such an observation is that the dispersion relation or eigenvalues are identical, so it must be equivalent.
10:47 AM
@ACuriousMind In the Canonical Ensemble it's microstates, are the same ones we have in the micro canonical ensemble?
I don't understand the question
a microstate is a point in phase space
(or a cube, if we're coarse-graining)
this notion of microstate never mentions any ensembles, so what's the question?
Well Yesterday I was discussing about the canonical ensemble with semiclassical and another person, and they told me that I can consider a CE as a subsystem of MCE
a subsystem which changes it's energy
that's exactly the opposite of what Semiclassical said
why is that?
9 hours ago, by Semiclassical
so i guess we could say that CE = ensemble of MCEs
10:50 AM
This is what I initially said
i.e. you can find MCEs as subsystems of a CE, not the other way around
but @bolbteppa said otherwise
I've read the discussion and you're oversimplifying
@bolbteppa said : "You can apply the microcanonical distribution to a closed system consisting of subsystems which exchange energy to derive the canonical distribution, so the microstates in the canonical distribution are ultimately just the microcanonical microstates, if that's what you're asking"
bolbteppa talked about deriving the CE by considering a system in the MCE and then thinking about what sort of ensemble one of its subsystems is in
10:53 AM
and in case of only energy exchange, the sub systems would be in CE
I'm still waiting for what this has to do with microstates
you have a habit of asking a question and then immediately going on either a wild tangent or a long explanation :P
Similarly to when we observe a system in eq. for MCE we talk about micro and macro states of the system
I want to do the same with a system in eq. but for CE
Why would that be wrong to do?
an ensemble just is a particular kind of macrostate
the notion of microstate doesn't change
I don't know what you're looking for
10:58 AM
I simply wanted to know whether the microstates in both ensemble are the same. And apparently they are, meanwhile the macrostates are different
I don't know what you mean by "the same"
and depending on what exactly you mean, the answer might be "they're the same" or "they're different", but I can't tell which it is
My faulty understanding of the CE was this initially : "while for the MCE the microstates have same energy as the macrostate that they represent
in CE the macrostate is composed from states with TOTAL energy that is different
and they cannot be microstates, because microstates have roughly the same energy"
And the 2nd part was straight out of Wikipedia: "The system can exchange energy with the heat bath, so that the states of the system will differ in total energy."
oh, by "the microstates" you mean the microstates with non-zero probability
I think this is just confusing language
The MCE is a density function on phase space $\rho(x,p) = \delta(H(x,p)-E)$ for a constant $E$. The CE is a density function on phase space $\rho(x,p) = \frac{1}{Z}\mathrm{e}^{-\frac{H(x,p)}{kT}}$ for a constant $T$. They're functions on the same phase space, they just give different probabilities for the microstates.
by their very construction, the MCE gives non-zero probability only for microstates with a certain fixed energy $E$, while the possible microstates in the CE have all possible energies
11:14 AM
and this is what I was trying to say
Ofc I consider only microstates which belong to the macro state of eq. for MCE
which have an equal proability
I don't see why would I consider also microstates, with 0 probability, which belong to another macrostate, with another energy value
depends on for what purpose you're doing your consideration :P
But I always was going with the standard
I tried to explain my understanding like this:
MCE : Macrostate= football stadium , microstate = different arrangement of the people who are inside.
CE: Macrostate = different stadiums, microstates = different arrangement of the people in each stadium
I know it's silly
but I was trying for a simplistic explanation
11:35 AM
@ACuriousMind if for MCE we consider a fixed energy $E$, would it be wrong that for the CE we consider an finite interval of Energy $\Delta E$
yes, that would be wrong
note the CE probability $\mathrm{e}^{-E/kT}$ is non-zero for all $E$
Yes I understand that for all $E$ you have a certain probability
But realistically a system that exchanges heat, can have a certain range of values for its energy
Unless there is some sort of confusion in my language, similarly to the case for "the microstates"
why are you trying to argue with "realism"
the MCE isn't "realistic" either, very few real-world systems are confined to exactly one value of energy :P
all the ensembles are idealizations, like most other physical models
Because in case of MCE, even though the energy fluctuates between E and E + dE, still you can consider it as fixed macroscopically , and that would be a system, isolated whose energy doesn't change
that was my aim
Or what I was trying to imply
macroscopic ally**
that's an argument why the MCE works as an approximation to any real-world system
11:44 AM
I get it
but the MCE as a model still considers the energy absolutely fixed, it doesn't fluctuate at all
And reflecting back on what I said about MCE and CE not having the same microstates, I understand now that that makes no sense
since both Ensembles are in the same phase space, where we have the same points (microsystems)
The only difference being the range, which for MCE is a fixed E value and for CE are all the possible E values
So for CE we consider more Microstates, that have non-zero probability
12:00 PM
My article on measurements is getting a bug where if I add more information, the text at the end disappears
Has it gotten so long that the database won't accept any longer thing
I hope nothing important got deleted in the process
65532 bytes
that does seem like a suspisciously "size of a thing" number
yeah, that's 2^16
" TEXT – 64KB (65,535 characters)"
some length counter seems to be only 2 byte large
Welp time to convert to MEDIUMTEXT
::squints:: using ASCII and not unicode? do you have to pay for every bit or something? :P
12:06 PM
Only part of the bibliography got squished, hopefully nothing I can't find again
fortunately I save every piece of bibliography I even vaguely look at, but if I want to find them again it's gonna be challenging
What do you even call the compactification of the Lorentz space
I guess properly it's called the Mobius space $S^{n-1,1}$
there are too many structures
Can't we live in one dimension like those guys :
2 hours later…
2:25 PM
So correct me if I'm wrong. A classical point particle has $SO(3)$ symmetry. Therefore the states in our Hilbert states must carry a representation of this group. But we have a choice whether to use the projective representation or the fundamental representation. We choose the projective representation because it agrees with experiement.
You can use a bunch of representations, even
The fundamental representation is the spinor representation, maybe you meant vector representation
The representation of your rotation group is just what kind of system you're dealing with
@bolbteppa Sorry I meant vector representation
QM is just different, we are describing a system by a wave function which transforms as a certain representation of a symmetry group, it's determined by experiment what the spin of the representation is of the wave function of some given system when it transforms in a representation of the rotation group
2:29 PM
You can use a projective rep since states are in the projective Hilbert space
So they are unphased by phases
@Slereah I get that but I'm wondering if it's a choice or not to use it
You can choose a variety of representations, giving you different quantum theory
After that you need to look at what the experiments are like to find out which is correct
sometimes it's the trivial representation, sometimes it's the spinor, sometimes it's vector
Sometimes it's a reducible representation because you're dealing with multiple particles
Okay thanks
And because we use the spinor representation, that automatically means that our state space is composed of spinors?
there are even representations that exist but we don't use because they don't correspond to any real case!
Like what?
2:36 PM
Continuous spin particle (or CSP in short), sometimes called an infinite spin particle, is known as a massless particle never observed before in nature. This particle is one of Poincaré group's massless representations which along with ordinary massless particles was classified by Eugene Wigner in 1939. Historically, a compatible theory that could describe this elementary particle was unknown, however, 75 years after Wigner's classification, the first local action principle for bosonic continuous spin particles was introduced in 2014, and the first local action principle for fermionic continuous...
although that's for the Lorentz group
Okay but we always start off with the classical Lorentz group and look at what the projective representations are to try and form the quantum theory right?
It is certainly the best route for it, yes
What other routes are there?
The standard representation I think it's called
@DIRAC1930 You could define it by its algebra directly
All representations have the same algebra
By $SO(3)$ symmetry classically, do we mean that you can rotate a classical particle around it's center?
2:43 PM
also remember that the Lie groups already exist "in the abstract" as manifolds
@DIRAC1930 It means that physical quantities measured will not change by rotating your frame of reference
Isn't mine just the active transformation and yours the passive?
Well I'm not 100% sure of what you're saying, but the rotation isn't necessarily done at the particle itself
You can rotate your system at any point
How is that a symmetry though?
As in the world doesn't have $SO(3)$ symmetry
Well there are several notions of "symmetry"
Different parts of the theory can be symmetric
If I rotate the reference point the particle will be somewhere completely different
2:47 PM
You're thinking of a case where the configuration is symmetric
Which is also a type of symmetry, yes, but there are others
the usual notion of symmetry (classically) is that the action is symmetric
Where do the two $SO(3)$ symmetries come from classically? We have angular momentum and the other one
Well not all configurations have $SO(3)$ symmetries
For instance if you consider a stick, it is not $SO(3)$ symmetric
But the equations of motion are covariant under SO(3)
But a point particle on its own should be, right?
And measurements are invariant under SO(3) as well
@DIRAC1930 Yes
Although a point particle lacks translational symmetry
But doesn't the rotational symmetry of a particle on it's own one $SO(3)$ symmetry, and the rotated reference frame thing another $SO(3)$ symmetry?
Sorry, the start of the sentence was meant to be 'So is the rotational symmetry of a particle on...'
2:59 PM
there is a PSE post about this if you want
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