What is "spin" in particle physics?
Each time I ask a physicist, they start off with confidence but end up confusing themselves.
>>7837947
Are you dumb or what?
Is the direction of which a particle is spinning.
Imagine when you spin a ball, or how the earth spins, it's exactly the same.
>>7837947
The Spin(n) group is a double cover of SO(n).
>>7837988
either wrong or I don't understand it or both.
>>7837989
Spin(n) is usually constructed as a subgroup of a clifford algebra. In physics we are interested in the irreducible representations of the spin groups. These representations characterize particles of different spin.
>>7837996
how does spin physically happen to a particle?
>>7838007
tiny politicians
>>7838007
The Spin Goblin twists and pushes the particles.
>>7837947
spin is one of the two casimirs of the poincare lorentz group.
spin has nothing to do with QM, it is from GR.
>>7838015
what? no?
What about quantum entanglement and the whole spin issue?
They keep using "spin" as a property of particles in various experiments and explanations but I can't get my head around it.
>>7838017
This is very wrong. Spin is a result of special relativity and in Wigner's classification, quantized spin values are explained via the irreducible unitary representations of the Poincare group.
MASS IS NOT FUNDAMENTAL
homogeneity and isotropy of space => poincare lorentz group => two casimirs of this groups => ANYTHING IN SPACE is labelled by AT LEAST a momentum+spin or no-mass+helicity
>The Casimir invariants of this algebra are P?P? and W? W? where W? is the Pauli–Lubanski pseudovector; they serve as labels for the representations of the group.
>The Poincaré group is the full symmetry group of any relativistic field theory. As a result, all elementary particles fall in representations of this group. These are usually specified by the four-momentum squared of each particle (i.e. its mass squared) and the intrinsic quantum numbers JPC, where J is the spin quantum number, P is the parity and C is the charge-conjugation quantum number. In practice, charge conjugation and parity are violated by many quantum field theories; where this occurs, P and C are forfeited. Since CPT symmetry is invariant in quantum field theory, a time-reversal quantum number may be constructed from of those given.
Reminder that ONE particle is ONE irreducible unitary representation of the symmetry group involved in the study. REMINDER that WE CHOOSE to have IRREeducible representations whereas
>One way to think of the allocation into irreducible representations is that our universe is
>clearly filled with different kinds of particles, and different spin states. By doing things, such as
>shoving an electron in a magnetic field, or sending a photon through a polarizer, we manipulate
>the spin states. Some states will mix with each other under these manipulations and some will
>not. We could treat all the particles in the universe together in one big fat representation. But
>since some states never mix with other states, this would be silly, and maybe overconstraining.
>So we look at the irreducible representations because those are the building blocks with which
>we can construct the most general description of nature.
>>7838021
like I said, the spin and the mass are nothing but labels.
What you are asking is where does these label come from ?
since physics is mathematics for many, the answer will be mathematical.
from their experiences, the scientists claim that, in natural language, the concept of space and time are indeed relevant and constitute a/the reality, plus from our faith in the principle of induction, they also claim that the experiments lead the same results when they are observed in inertial frames of reference, ie whether they are done in such or such location in space, which is called homogeneity of space, in such or such direction of space, which is called the isotropy of space, in such or such location in time, which is called the homogeneity of time.
THis is called the principle of relativity
https://en.wikipedia.org/wiki/Principle_of_relativity#Basic_relativity_principles
From this, you ask a mathematician and he says that, once the concepts of space, time, points [in spacetime], events, particles/things in natural language have been mathematized [or even formalized in logic], then you obtain a mathematical object called a '' group of transformations of inertial referential frames''. Each mathematical element of this group is a transformation, turning each mathematized inertial frames of reference into another one. each inertial frame of reference is concept, in natural language, for a set of rulers and one clock, because the humans have faith in that space has three dimensions of space and one of time [even though, they hav eno clue what time is nor even space].
bored biofag here, i watched 10 minutes of youtubes on the subject and now i'll explain to you what spin is in simple language.
Think of a magnet with a north or south pole. Throw it in the air and spin it as you throw it so it's rotating around in the air. Take a photo when it's in midair so the north and south poles axis is pointing in some random direction. This orientation is an important characteristic of your typical everyday magnets; another magnet will exert a varying degree of force on your midair magnet, depending on the orientation of the magnets' north and south poles. North facing directly to south will exert maximum attractive force, north facing north will exert max repulsive force, and anything in between is possible.
If you replace the magnet with a charged particle, e.g. an electron, all that rotational variation disappears. if you have a magnetic field acting on an electron, the electron either is oriented such that it exerts maximum attractive force, or a maximum repulsive force. the strength of the field is the only thing that matters, there is no orientation of the "north or south poles" of the electron relative to the field.
a particle's "spin" is the name we give to the number that characterises whether it is attracted or repelled by the same magnetic field. think of it as an orientation of a classical magnet relative to a magnetic field, but instead of a whole range of possibilities, the orientation is either "towards" or "away".
>>7838025
the work of the mathematician leads him to claim then that there exists several of these groups.
>From the principle of relativity alone without assuming the constancy of the speed of light (i.e. using the isotropy of space and the symmetry implied by the principle of special relativity) one can show that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum.[17][18]
many claim that the principle of finite velocity for the light is not necessary to refine the group mathematizing the isotropy of the space, the homogeneity of the space and the homogeneity of the time.
>>7838028
Nonetheless,
>an amusing example of the fallacy in an easily accessible secondary source is in Rindler's "Essential Relativity" (1969, 1977), which contains a section entitled "Special Relativity without the Second Postulate". After giving the group theoretic derivation, the section concludes with "The relativity principle itself necessarily implies that either all inertial frames are related by Galilean transformations or by Lorentz Transformations with positive c^2. The role of a "second postulate" in relativity is now clear: it has to isolate one or the other of these transformation groups... However, in order to determine the universal constant c^2 the postulate must be quantitative. For example, a statement like "simultaneity is not absolute", while implying A Lorentz group fails to determine c^2... We shall see later that relativistic mass increase, or the famous formula E = mc^2, and others, could all equally well serve as second postulates." The important thing to notice is that Rindler has just explained why we NEED a second postulate, either the light speed postulate or something equivalent to it ... even though the heading of the section is "special relativity without the second postulate". So it goes.
Reference https://www.physicsforums.com/threads/derivation-of-space-time-interval-without-lorentz-transform.477029/
so we need another mathematical assumption derived from empirical knowledge [after we have faith in induction and the relevance of the concepts of space, time etc.] that somehow the velocity of the light is finite.
most people are rationlist, this means that most of people believe that the deductions, from empirical observations, done according to some logical rules chosen by the scientists lead to knowledge/truths about what we perceive as reality [space, time] and mathematized as ''spacetime'', these people claim that the mathematical object that is the ''spacetime'' has an associated mathematical object called the ''poincaré-lorentz group (of inertial frames of reference in space-time)'' and that each element of this group models one transformation acting on the inertial frames of reference which live in the mathematized notion of space+time [the ''spacetime''].
>Space translations, time translations, rotations, and boosts, all taken together, constitute the Poincaré group. The group elements are the three rotation matrices and three boost matrices (as in the Lorentz group), and one for time translations and three for space translations in spacetime. There is a generator for each. Therefore the Poincaré group is 10-dimensional.
https://en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics
then the mathematician does his task and claims that this group has two casimirs.
the casimirs for the poincaré-lorentz group are the two quantum numbers : the mass-squared and the spin.
>The Casimir invariants of this algebra are P?P? and W? W? where W? is the Pauli–Lubanski pseudovector; they serve as labels for the representations of the group.
>From a fundamental physics perspective, mass is ONE OF the numbers describing under which the representation of the little group of the Poincaré group a particle transforms. In the Standard Model of particle physics, this symmetry is described as arising as a consequence of a coupling of particles with rest mass to a postulated additional field, known as the Higgs field.
https://en.wikipedia.org/wiki/Particle_physics_and_representation_theory
https://en.wikipedia.org/wiki/Representation_theory_of_the_Poincar%C3%A9_group
the mathematical object that is the spacetime is nothing but a mathematical ''set of points''+plus some topology. The mathematized entities that the rationalist believes model ''the paraticles'' are the irreducible representations of the poincaré-lorentz group. In natural language, ''the things [that we see] IRL'' are concepts of course and the rationalist-realist thinks that they are made of smaller things, called '' [elementary] particles''. these particles are mathematized as these irreducible representations.
Each element of this poincaré-lorentz group act also on these irreducible representations, because each irreducible representation depends on the spacetime points. [because the rationalist thinks that what he descriminates as ''one thing [in itself] IRL'' [say ONE pen] depends on the coordinates from what he conceptualizes as the frame of reference wherein in thinks he lives]
the poincare group G is of rank 2, that is to say that it has two casimirs.
the casimirs for the poincaré-lorentz group are the two quantum numbers : the squared mass and the spin.
the casimirs are operators belonging to the Lie algebra of the poincaré group.
the lie algebra is a collection of matrices which gives back, once inserted into an exponential of matrices, the elements of the initial group.
In fact, EACH casimir depends on the representation of the group but is always a multiple of the idenity:
>By definition any member of the center of the universal enveloping algebra commutes with all other elements in the algebra. By Schur's Lemma, in any irreducible representation of the Lie algebra, the Casimir operator is thus proportional to the identity. This constant of proportionality can be used to classify the representations of the Lie algebra (and hence, also of its Lie group). Physical mass and spin are examples of these constants, as are many other quantum numbers found in quantum mechanics.
by definition of the casimir operators, they commute with all the generators of the poincaré groups.
>In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.
https://en.wikipedia.org/wiki/Casimir_element
then a representation of the poincaré group G is morphism of groups rho: G ---> GL(V)
from G to the group of matrices/operators acting on a linear space/hilbert space V.
The determination of the dimension of the hilbert space V stems from both the experimental determination of the degrees of freedom that the elementary ssytem has and the work of the mathematician because the dimension of V is connected to the quantum numbers [what values these numbers can take].
so the rationalist-realist claims that One [elementary] particle/field is One such irreducible representation (V, rho) of G. The states of the ONE elementary particle/field are the elements of the vector space V, on which the poincaré group G acts, because the rationalist-realist believes in the concept of states phi associated to the concept of ''things IRL'', states phi which are mathematized as elements of some hilbert space [in QM] V, concept [in natural language] of ''things IRL'' which is itself mathematized in terms of [more or less elementary] particles/fields and are labelled by the points (t, x, y, z) of the mathematization, the spacetime, of the concept [in natural language] of space+time.
Any scientist consider that things IRL are in space and time [the reality], which means that once they are rationalist-realist, they work with at least the group of symmetry which is the poincaré group G.
what does it means that a particle has ''mass-squared'' ?
it means for the rationalist that the the particle is in spacetime, since the particle/field is the irreducible representaiton, it means that there is an operator ''mass-squared'' which is nothing but the ''mass-squared''9as a number) times idenity operator, which returns,once applied on the states [the wave function] phi in the hilbert space V, the mass-squared and returns the state phi.
>In today's theories, elementary particles are irreducible representations of the FULL symmetry group, which is not just the Poincare group. If the mass transforms nontrivially under the internal symmetry group, it is a matrix rather than a number.
http://www.physicsoverflow.org/21958/elementary-particles-as-irreducible-representations
For instance, according to the rationalist-realist, the Isopsin is another quantum number which stems from another casimir operator which stems from another group of symmetry of the studied system.
https://en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics
there is an elementary introduction to this:
Group Theory and Physics
Symmetry is important in the world of atoms, and Group Theory is its mathematics
http://mysite.du.edu/~jcalvert/phys/groups.htm
And a good big book:
Reflections on Relativity Kevin Brown
http://mathpages.com/rr/rrtoc.htm
now the spin seems to derive from the Spacetime itself, and therefore has nothing quantum in it
https://en.wikipedia.org/wiki/Relativistic_angular_momentum
>In special relativity alone, in the rest frame of a spinning object; there is an intrinsic angular momentum analogous to the "spin" in quantum mechanics and relativistic quantum mechanics, although for an extended body rather than a point particle. In relativistic quantum mechanics, elementary particles have spin and this is an additional contribution to the orbital angular momentum operator, yielding the total angular momentum tensor operator. In any case, the intrinsic "spin" addition to the orbital angular momentum of an object can be expressed in terms of the Pauli–Lubanski pseudovector.[1]
all of this connects to the noether theorem which you must know, but sadly many crucial nuances are not taught explicitly in courses. so watch this video for physicist
https://www.youtube.com/watch?v=2ndYHCqSKIM
=>the conclusion is that spin and mass are labels from our formalization of what we think is space and time.
>>7837952
But what if the particle is upside down? How would your theory describe the spin of an electron then? Think about such before you type such stupid things on the internet.
>>7838045
I understand none of the details but I do understand that arbitrary nature of physics and how our intuition tries to make sense of it.
In a similar vain, every term in physics is a label from our formalization of what we think is space and time.
>>7837947
spin is a case of unfortunate naming of a physical phenomenon .its not a literal spin of a particle because saying a particle literally spins is meaningless.
its kinda hard to explain this to someone if i dont know what his background in physics is .im sure theres plenty of popsci books\utub vid that can tell you more or less what it is kind of.
>>7839130
... and also how they're not quite "particles"
>>7838007
The mathematical representation is all you can really have.
Humans don't really have an intuition for things occurring on this level, there's nothing to "get".
>>7838007
also, spin is a quality particles possess, it doesn't happen to them
>>7837947
My understanding is that spin is the intrinsic angular momentum of particles.
>>7840265
correct, but we have to keep in mind that this momentum plays out differently in the quantum world. The math of two spinning basketballs interacting would not translate to two quantum objects
>>7840265
>intrinsic angular momentum
What I was about to say.
>>7840278
Both the spinning basketballs and the quantum objects have angular momentum. One difference between the basketball and the spin of an electron is that the basketball has a very large angular momentum, too large for you to notice the quantum mechanical effects. The other difference is that you can see that the basketball is made of particles moving in a circle. An electron having spin doesn't mean anything's moving in a circle, but you could convert the spin of a bunch of electrons to the angular momentum of something moving in a circle. Angular momentum, like energy, can take different forms, but is a conserved quantity.
>>7840300
I would also add that the quantum objects will behave in radically different ways according to the value of their spin, because this is what determines if they are bosons or fermions ... nothing remotely comparable with basketballs
>>7840328
The difference being because there isn't a feasible way to make identical basketballs.
>>7840333
but if there was a way... my gut tells me still nothing but ive got just one course in qm.... maybe shit would be crazy
>>7840349
It's not feasible with basketballs, but people have done it for smaller composite particles such as atoms:
https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate
>>7840265
>intrinsic angular momentum of particles
But they're not actually spinning, they're just flipping up and down.
The angular momentum of a particle, which is fucking really small.
>>7838025
>nothing but labels
>physics is mathematics for many
Right. Not OP but I didn't really follow either.
Could you explain how wavefunction symmetry/antisymmetry under the exchange of two bosons/fermions emerges as a consequence of their integer and half-integer spins (or vice versa?)
so what is spin and isopsin ?
>>7837988
>>7837996
>>7838015
>>7838051
These guys basically hit the nail on the head, OP, but it can be difficult to understand if you've never studied theoretical physics at any level.
I suggest you go read up on the relationship between the SO(3) and SU(2) groups. Then Weyl's theory of spinors.
Here's a pretty good introduction: http://www.weylmann.com/weyldirac.pdf
>>7842517
Isospin (I assume you mean weak isospin) is a quantity associated with rotations in a complex 2-dimensional vector space. In the standard model, isospin exists to identify the lepton and quark families (electron, electron neutrino)(up quark, down quark)(and the higher mass versions of them) as being a pair of states that are really unrecognizable from one another.
The Higgs field goes about and fucks all that up by way of spontaneous symmetry breaking. Hence why isospin is not something that you can directly observe like electric charge.
>>7837952
No, not even close. Either know what the fuck you are talking about or get the fuck out.
>>7841224
are you asking about the origin of the spin statistics ?
>The theorem states that:
>The wave function of a system of identical integer-spin particles has the same value when the positions of any two particles are swapped. Particles with wave functions symmetric under exchange are called bosons.
>The wave function of a system of identical half-integer spin particles changes sign when two particles are swapped. Particles with wave functions antisymmetric under exchange are called fermions.
https://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem
or are you asking why we anti-symmetrize the wave function of fermion, under the operation of permutation (of labels), and symmetrize the wave function of the bosons, under the operation of permutation (of labels) ?
if this is the one, it is because we cannot distinguish what particle is what particle with respect to the initial particles. so we must consider all assignments of labels that we can do. it means that there is no preferred labels for the wave functions. so mathematically, we consider individual labelisations, then we group these labelisations into one big entity. how to group wave functions ? well in QM, the novelty is that a sum of WV remains a WV. Plus, there is only two usual operation: addition or multiplication. multiplication of WV refers to grouping systems together, which we already did since we consider a system which is the product of plenty of subsystems. so it cannot be multiplication twice. so that is nice : we sum all the different labelisations that we can think of.
how does this relate to spin ? the answer is the theorem of spin statistics.
well first, two particles are called identical when they have the same properties [=labels, in QM, the quantum numbers, such as mass, spin, other charges (stemming from symmetry group of the system thanks to noether theorem and our faith in group theory to get out of the mess that is the induction through space and time (=the physical equations are the equations which are stable under whatever group is the symmetry group of your system))] (in physics, it means the same values of these quantified properties).
of course, since every physicists today thinks his system in terms of space and time, his system is always in space-time and will always be labelled by position, time and spin+mass or helicity+no mass.
let's begin by the beginning.
so the labels that we use today are, at the very least, the labels of spatial position and temporal position, because we choose to take seriously our wish to describe the world through space and time (even though we have no idea what space and time are).
so let's say that we have at least two particles. these two particles can be different or the same.
what we learn from statistical physics, is that there is either a change in the state of the system or there is no change, when we swap the ''spatial position'' of the particles. states are judged as relevant today (even in instrumentalist interpretation of QM, even though in this interpretation, states are not real things, only a human device to get knowledge) and the typical ''physical property tied to a state'' is the energy. so, in classical mechanics, you ask ''is the energy changing when we swap labels (of spatial position) ?'' the answer is yes or no, the answer is no.
let's stick to CM a bit: in CM, we have faith in spatial labels and we can follow spatially each particles from the initial state to the final state (the final state is explicitly the spatial positions + velocities). it means that, at each time, you can identify one subsystem, in knowing at least its spatial position. so for instance, you have tennis ball 1 and tennis ball 2 at the beginning. you can identify tennis ball 1 until the end of your experiment.
this fact holds for identical sub-systems: it holds for two tennis balls. the thing is that , in QM, it no longer holds for identical systems: ex two electronic wave functions.
why do we have faith in spatial position ? because we are able to track the path followed by one system. this holds in CM, but not in QM. so two electronicWF, spatially distinct can interfere, but at the end of the interference, once you break the system into to non-interacting systems, you do not know which sub-systems corresponds to the initial sub-system. you could distinguish sub-system when they are different ''in nature'' which means when they have different properties (like charges) which means, by the spin statistics, bosons and fermions.
why can we not have faith in spatial labels during the interaction of identical sub-systems ?
because the WF during the interaction of two identical systems, leads to the physical entity which is the ''probability of spatial presence of the system'' so, for each spatial position and each temporal position, when you compute the modulus square of your WF during the interaction, you have at best a probability (in the interpretations which take the modulus square as probability) of finding your system. you have a probability of a finding the two systems in some spatial area. nothing more, nothing less.
why can we not separate identical systems in QM ?
because our operations that we do in physics, exactly touch the physical characteristic of the particles , which is the mass , the spin, the other charges. there is nothing else in physics than the modification of these parameters/labels. when we apply a field to a systems with identical sub-systems, like a magnetic field, the fields touches at least one label (here the spin), but each subsystem ''obeys'' (for the rationalist) the field, each subsystem changes, under the field, like the other subsystem because they are the same
why are they the same ? because we experientially find that they do change identically under the field, which lead us to conclude that the two subsystems have the same abstract characteristics (at least with respect to this operation.)
with the mass for instance, in CM, you have two balls of the same mass: if you push the two balls with the same energy, then each ball will behave like the other one (each ball will go as far as the other ball).
on the contrary, if one ball go farther than the other one, you would say that the two balls are not identical (here, they have not the same mass).
let's recall that in physics, in any field called a science, you have
-first step is inductive; with what you see, you fix a system, then you discriminate between systems which behave like your system, and systems which do not behave like you system [the definition of a system is bogus of course, since the system is literally putting, at least, spatial and temporal boundaries to get an ''event'' (people love to take seriously space and time, they cannot think outside space and time)+ giving this event other qualities that it is supposed to bear]
induction serves, at the very least, to tie things/events/phenomena together through the concept of identity (or its opposite, of difference). instead of induction here, you can talk about abstractions, but they are the same things : to group things together and/or to differentiate between things.
-you continue your induction/abstraction (and frankly, you cannot even anything else in your life; it is too difficult to stop having faith in your inductions), in saying that, since two systems behave the same so far, they must have a few qualities which are the same
-then you apply deductive reasoning borrowed from math/logic: you quantify your qualities above and get new formulas from deductive rules (deductive rules are got by induction/abstraction just as above, why do you have faith in the modus ponens ? because you want to which leads you to see the world through logical causation. Rationalists like Quine who think of themselves as empiricists say that we are wired to see the world through classical logic (kant says that we are wired to see the world through space and time...)
-then you go back to induction in telling the experimental physicist (a complete stranger) to check statistically your deductive predictions
-then you get the result and you ask people what degree of statistical significance they like ? (the famous p-value or the n-sigma (n is number like 3 or 5 today))
if the null hypothesis is rejected (p-value of 0.05 or any other socially accepted level to reject officially the null hypothesis) then your predictions are officially accepted (by whom ? nobody really knows)
and then you can claim that your deductive formulas ''describe the world'' (if you are a good rationalist-realist).
the underlying fantasy under this endeavour is ''motion brings knowledge''. the error is to think that ''immobility does not bring knowledge, or at least less knowledge than the study of motion''. but of course, this falls outside of physics like the rationalists have been perpetually painfully doing.
When you try not to move, physically and mentally, things happen too, and they yield certainty, contrary to studies through induction.
let's recap:
take the system composed of plenty of electronic wavefunctions,
you have sequence quantum numbers, A, B, C. the sequence A gives you a state of a system (the eWF). B gives a state of the eWF and so on.
the knowledge of the ''state of the system'' is to say:
-there is number a ''particles'' in state A
-there is a number b of particles in state B
-there is a number c of particles in state C
and so on for each possible sequence of quantum numbers.
in QM, WE CANNOT SAY ''the FIRST 30 particles are in state A, the second 20 particles are in state B, the third 64 particles are in state C and so on''.
in Qm, we can indeed calculate the probability of presence of ONE electron of a given spin, from the complete wave function of the system, BUT you do not know what electron you are talking about, wrt to the electrons of the initial state. all you know is that you can separate one electron from your tensor product which is the complete WF, but is it the third electron in your initial state ? you cannot say.
let's recall that the concept of particles is heavily flawed. but I continue to used it casually.
CONCLUSION: for identical particles without interaction, the good quantum variables are the occupation numbers of the state you wish to look at (= for the energy, at energy 3ev, how many particles to you have ?) [the energy is believed to be a quality which is relevant macroscopically). Only one fermion can occupy a given quantum state at any time, while the number of bosons that can occupy a quantum state is not restricted.
You then POSTULATE :
for a system of N identical particles, the only ''physical'' states are either the the symmetric states (under the operation of swapping the labels (typically the labels of position or the alphabetic labels)) or the anti-symmetric states (under the same operation).
so you have wither
\psi(1, 2, 3, 4, 5, 6...) = + \psi(2, 1, 3, 4, 5, 6, ....)
OR
\psi(1, 2, 3, 4, 5, 6...) = - \psi(2, 1, 3, 4, 5, 6, ....)
then
you call bosons the symmetric wavefunction
fermions the anti-symmetric wavefucntion
then , in 2016, you experimentally notice that fermions have half integer spins, bosons have integer spins.
>The operator (- 1)^F arises because of the invariance of the results of experiment under rotation through an angle of 2.\pi around any axis. If \psi_1 is a state of half odd integer spin, and \psi_2 one of integer spin, then a rotation through an angle 2\pi takes a\psi_1 + b \psi_2 into -a\psi_1 + b \psi_2. These two, which are physically indistinguishable, must belong to the same ray, which is possible only if a = or b= 0.
is it possible to offer a deductive sequences of formulas to connect bosons/fermions with the integer/half-integer spins ?
people say that the theorem of the spin statistics is the answer.
this theorem is developed in algebraic quantum field theory. AQFT is QFT done through algebras of operators (C* algebras).
>All experimental evidence indicates that systems with integer spin obey the laws of Bose-Einstein statistics, and systems with half-odd integer spin those of Fermi-Dirac statistics. Although there are' perfectly. respectable laws of statistics different from either Bose- Einstein or Fermi-Dirac, so far no system has been seen to follow the (Ref. 28). A natural way to arrive at Bose-Einstein statistics is to describe the system in question by a field which commutes for space-like separations, while the analogous way for Fermi-Dirac statistics is to use a field which anti-commutes for space-like separations. The theorem on the connection of spin with statistics or, as we shall say for brevity, the spin-statistics theorem, is an assertion that in quantum field theory a non-trivial integer spin field cannot have an anti- commutator vanishing for space-like separations, and a non-trivial half-odd integer spin field cannot have a commutator vanishing for space-like separations. If one puts aside the possibility of laws of statistics other than Bose-Einstein or Fermi-Dirac, the spin statistics theorem then accounts for the experimental results. When one turns from the commutation relations for a given field to those between different fields the situation becomes more complicated. It turns out that "abnormal" commutation relations in which two integer spin fields or an integer spin and a half-odd integer spin field anti-commute, or two half-odd integer spin fields commute, can be realized but, in general, the resulting theories possess special symmetries.
>By virtue of these symmetries it turns out that there always exists another set of fields, satisfying normal commutation relations and related to the original fields by a so-called Klein transformation. The original theory can equally well be regarded as a theory of this set of fields. In this sense, a theory with abnormal commutation relations is a special ease of a theory with normal commutation relations, one which possesses a set of symmetries.
In passing, you note that only the phase changes under the swapping, which means that, since you favourite interpretation says that only the modulus square matters, you have the same information (about the interaction) whether you swap or not. this information concerns only the interaction. the swapping provides information about the nature of the system (bosonic or fermionic).
when we study interactions, we look at the Hamiltonian and the Hamiltonian is precisely invariant under the swapping [which leads us to say that the swapping does not change over time)
in passing, I promote the books of
http://www.motionmountain.net/
which explain physics from A to Z
>>7837952
g8 b8 m8
>>7838051
Just ignore dumb shits like >>7840214, to pick out one example of the fucking pretentious losers in this thread. You're a terrible physicist if you're going to say "there's nothing physical to get" and forever fall back on your precious mathematical jargon to help hide the fact that you don't have any intuition for the shit you think you're describing.
If you want a good physical analogy for "exotic" things like spin 1/2, Dirac's belt trick or the Mobius strip is an easy way to see examples of when you have to go around 720 degrees to get back to where you started. 360 degree rotational symmetry would be spin 1, 720 is spin 1/2.
The way we give particles their quantum numbers is just a property of the symmetries a certain field (excitations of which are particles) enjoys, in particular spin corresponds to its rotational symmetry, hence the example above.
Keynes once quipped that public figures who think they are expressing their original thoughts are usually echoing the words of some dead economist. The same might be said of the dead Kant in respect to science. While his thought provides a comprehensive modern framework for science, most practicing scientists have never read him and large swaths of the philosophy of science ignore or reject what they take to be muddled Kantianism.
Bertrand Russell and E.G. Moore were especially hostile to Kant, convicting him of logical errors and supposing that transcendental idealism rested on a mistaken faith in the inviolability of Euclidean Geometry, which Kant presented as the very model of the "synthetic a priori." These dubious charges stuck all the way through logical positivism and continue in the analytic tradition. Certainly Popper's demarcation criteria seem to reject Kantian approaches, or at least so Popper claimed.
Today, however, it seems that more philosophers of science may be open to Kant. Henry Allison offers detailed refutations of the above calumnies, and Kuhn was explicitly influenced by Kant in his anti-Popperian demarcation by "paradigms." He claimed that reading Kant while studying physics utterly altered his naive realism, though those are not his words.
As to the bigger picture. Bacon first described the emerging rift between the rationalists, "the spiders," who weave webs of theory (coherence theory, we might say) and the new empirical naturalists, "the busy ants," who gather bits of data (correspondence theory, roughly). The rift grew into Leibnizian mathematical rationalism and Humean skepticism. It was Kant's great project to merge and mutually limit the two on a firm metaphysical basis, in part to secure the basis of Newtonian physics.
His unique amalgam of coherence and correspondence theory is in many ways, and intentionally, a kind of philosophical version of the hypothetic-deductive framework of science, an expansion of knowledge by rational (conceptual) methods and experimental (intuitional) confirmation. And indeed science does undertake continuous active synthesis based on a priori assumptions of necessity and universality. Even the "experiment" is somewhat Kantian, in that it is hardly passive reception of sense data... the experiential confirmation in "common sense" is quite artificial and actively constructed.
Most working scientists tend to believe they are following analytical rules and discovering "correspondences" with hard "reality." So they look askance at Kantian idealism, regarding it as akin to Berkeleyan empirical idealism or some sort of structuralism. Such assurance was, of course, shaken by statistical mechanics in thermodynamics and then quantum mechanics. But such "shaking" may only register with a few...how many physicists, after all, have time to read Kant? Yet when pressed on theoretical issues many physicists will admit they are constructing models and cannot speak about the "ultimate reality," mere speculative metaphysics, unaware that they are defaulting to a near Kantian position.
Meanwhile, cosmology today seems to be weaving "theoretical webs" well outside any Kantian remit, moving far beyond experimental range and tumbling headlong into the Antinomies. In CPR B511, for example, we see hints of Copenhagen in "you never come face to face with anything unconditioned...." or "...neither a simple appearance [i.e., final particle] nor an infinite composition [i.e., material universe] can ever come before you." One might say Kant attempted to work out a physics that included the observer.
Of course, this is very general and it is easy to contrive this sort of cozy compatibility. I don't know enough about Kant yet to know where he might be in serious conflict with scientific practices, especially in theory of evolution... or perhaps in the evolution and "selection" of theories, where Hegel's criticism of Kant begins. I hope other will offer more specific references to Kant in current philosophy of science, since I'd like to know as well.
>>7837947
I'm just a pleb but as far as I know there are no good physical interpretations of what spin is. So, just think of spin so:
struct particle {
float spin;
float pos_x;
float pos_y;
float pos_z;
float charge;
};
>>7840265
>intrinsic angular momentum of particles.
Wait is this from their movement through time?
All particles are going moving through time right?
Quantum that quantum those. Su(x) supersymmetry bla bla bla.
Nobody of you can explain it.
>>7837947
Spin is the fundamental quantum of rotation. You can think of it as the particale is spining around it's center, but it is not actually spinning. It's just a fundamental property of the particle. You can deduce it from applying classical laws of rotation to quantum mechanics as Griffiths does, but don't be fooled by the classical analogue of a spinning ball or something along those lines.
>>7843082
How it can rotate if it doesn't rotate?
angular momentum
>>7838010
underrated post
>>7842805
how can a point particle have rotational symmetry
>>7844191
google some examples of fields, and you'll see examples of how certain fields look; the Higgs field for instance looks like a sombrero
>>7843084
>How it can rotate if it doesn't rotate?
It's not rotating in space, but time.
>>7843046
This just isn't true
To me what spin "actually is" is pretty irrelevant. The only time i've ever seen it used is when you say "this particle is spinning this way, and this one is spinning the other way". In my head I only think of spin as a relative value, the actual direction of spin is meaningless, it only matters how this spin compares to that spin.
Welp, it's safe to say that none of you are going to be particularly good educators if by some miracle you manage to get a professorship.
So an electron doesnt actually spin in place?
>>7837996
In pleb terms pls
>>7840214
> there's nothing to get
> implying
Check this get
>>7837947
The simplest way to identify where spin comes from is to say that, in the Dirac equation (which is Schrodinger's equation in Minkowski spacetime), there needs to be two distinct solutions a.k.a. wave-functions for each particle and antiparticle due to the different geometric framework of special relativity. These two solutions are the spin-up and spin-down portions of the particle in question.
It is "interpreted" as an intrinsic angular momentum because the type of energy these particles obtain through interaction with an external electromagnetic field is the same as if the charged particle was spinning.
Not sure how useful that was, but hopefully better than spouting stuff about Grassman. It's a tough and interesting question.
>>7844665 here, figured out a simpler explanation.
Because of special relativity, a charged particle gains/loses a small amount of energy through direct interaction with a magnetic field and in proportion to it (which doesn't happen if you just formulated the energy of a charged particle classically). This is just because of special relativity; classically it would only depend on the magnetic vector potential.
The energy the wavefunction/particle gains/loses is just as if the charged particle were spinning in a particular orientation in this external field. You can imagine spin-up and spin-down as being the two components of the wavefunction/particle that distinctly either gained or lost the most energy as a result of that interaction, and are (through hand-waving) the eigenvectors and only measurable spin states.
>>7844555
>mad about complicated words
The first three guys you quoted knew what they were talking about. Do you get mad when people talk about UV rays? Those are also outside the range of natural human understanding and so we use mathematics.
>>7844444
CHECKED
>>7844731
I didn't say those statements were false, I'm saying they're lousy explanations for someone who's just asking "what is spin"?
Saying Spin(n) is a double cover of SO(n) and talking about Clifford algebras provides no physical insight to someone asking that question.
In physics maths is a tool for getting physical results, not your own intellectual fleshlight to sound clever while not saying anything of value.
>>7844444
Go on then?
Spin is just a number, scientists are shit so they need a bunch of extra dimensions for their crappy little equations to work.
>>7837947
You can't explain it if you want it explained to you in terms of classical mechanics.
That's why you don't get it and it's a very confusing concept for everybody.
>>7844305
That's not rotational symmetry; that's an internal symmetry (transforming the field value).
>>7844191
Wouldn't it make more sense to ask how can a point not have rotational symmetry? After all, a simple geometric point should be a point no matter what direction you look at it in. But some particles aren't just points. They have not just position but spin.
>>7844305
It would be better here to point out the difference between, for example, the Higgs field, which doesn't have a direction, and the electromagnetic field, which does.
>>7846934
>which doesn't have a direction,
in what space
>>7847392
Ordinary space. I'm aware it has a direction in SU(2).
>>7846934
does the higg bosons give mass to every other particles?
>>7846918
they didn't say it had rotational symmetry they just gave an example of how fields can look
>>7847503
Technically it's the Higgs field that does that. The potential energy as a function of the Higgs field is such that even in its ground state (i.e. no Higgs bosons present), the Higgs field has a nonzero value on average. Many particles have a mass that is proportional to the value of the Higgs field.
(Another technicality is that hadrons would still have mass without the Higgs.)
To avoid misunderstanding, it's not that the Higgs field is an explanation for mass. It's just a mechanism by which certain particles can have mass without violating the weak force's equivalent of charge conservation.
>>7847657
>hadrons would still have mass without the Higgs
and black holes too
https://youtu.be/JqNg819PiZY?t=40m
>>7837988
Just understood what people mean by this today when I decided to finally figure out why physicists care about quaternions. They're my favorite noncommutative real banach division algebra now.
>>7847657
I heard that only bosons W and another one are given masses through the higgs. and this is even in a special case.
>>7838010
overrated post
if you have faith in physics, you believe that there is a the renormalization group floating around, also known as GOD, always renormalizing bare masses and other abstract parameters of elementary particles.
>science is truth, r-r-right guize ?
>>7847966
you don't understand the implications of the rg flow
>>7844919
Posts like this are a breath of fresh air. Maybe there is hope for humanity?
It's what makes our NMRs work, so we can see what the hell we're doing in the lab.
>>7849948
the flow of couplings upon renormalization. to see what I meant above read on universality and renormalization. our "most fundamental" theory is an effective theory.
![neil-degrasse-tyson-heres-how-long-you-could-survive-on-every-planet-in-our-solar-system[1].jpg](https://i.imgur.com/BL1wIgh.jpg "neil-degrasse-tyson-heres-how-long-you-could-survive-on-every-planet-in-our-solar-system[1].jpg")
>you don't need to know math to do physics
>all you have to do is imagine colorful balls spinning around
>just think of a torus orbifolded by a hot dog
>your physical intuition trumps everything else
>>7844919
It's meant to provide models allowing you to predict behavior. If the behavior is complicated, I don't see how is masturbatory to use complicated models. It's not as if we have any intuitive physical idea of how things the size of particles behave.
Spin in QM is just an assigned attribute. It's poorly named. It doesn't even represent all we would think about a spin normally.
>>7850853
>It's poorly named.
thank the uneducated anglos.
>>7850822
the behavior is the model itself though.
>>7850764
you did not meant anything.
>>7849105
Are you saying that the re-normalization group is not used in physics ?
>>7850853
It works exactly the way you'd expect an intrinsic angular momentum to, in regards to generating a magnetic moment.
It's just that, classically, for an object that 'size' to generate that magnitude of magnetic moment with that quantity of charge means it would be rotating such that its surface is moving faster than c.
>>7838027
So in the same magnetic field, one electron will move towards the source, but a second electron, with opposite spin, will move away. And that's just a fundamental property of a particle, with no simple, non-mathematical reason behind it? Just checking I've understood.
(this is what I learned from history of time, it is almost 30 years so I might be wrong)
When a particle spins it generally doesn't look the same in the process
A particle with spin 0 means that it looks the same all the time when spinning
A particle with spin 1 looks the same every half-spin
A particle with spin 2 looks the same every one entire spin
This particles form gravitons, electromagnetic waves, strong and weak nuclear force and some shit I'm too ignorant to know
Then you have the normal matter particles that look the same every two spins and are named "particles with spin 1/2"
I could be wrong or confused, I read the book in English (not my main language) and long time ago
>>7842805
Your examples don't help intuit concepts like spin at all. All you're doing is shrouding mathematical truths in metaphors.
>>7853640
do you think that there is an explanation of spin which is less mathematical ?
>>7853914
Not in the sense that any degree of understanding is gained over the mathematical description. Nor is there a reason to really believe there should be one, or that we should have the capabilities for this kind of intuition.
Logical manipulation of symbols is the closest we can get to truth.
>>7852353
yes, and there are stranger behavioral differences in whole and half number spins, namely, that whole number spin particles (bosons) will tend to assume quantum states identical to nearby bosons, while half number spin particles (fermions) will tend towards opposite states. A pair of fermions will act like a boson... reminds me of multiplying two negative numbers. All of this occurs with no outside force and it appears that quantum objects simply have very different and unique natures.
I think it is part possible to have "intuition" about these particles... but possible only to an extent. The same way that these particles appear to have wave nature sometimes and particle nature other times. You can't say either fully represents what they "are", but by understanding wave physics and particle physics, we certainly have a better sense of the quantum world than we would otherwise. In the case of spin, angular momentum and rotational symmetry are concepts that are useful, but will never get us to the point of saying "oh, thats what spin is".
>>7853640
>All you're doing is shrouding mathematical truths in metaphors.
That's kind of what intuition is
>>7838066
>but I do understand that arbitrary nature of physics and how our intuition tries to make sense of it.
oh god, these guys.
>>7853631
Pretty much, you have half-spin and entire spin reversed, but I think most people could notice that and figure that out.
>>7838049
There is no "upside down" in physics, just like there's no left, right, up and down. All movements are measured relative to the viewer.
angular momentum
>>7838027
this is good
>>7854046
Are you ok dude?
>>7854340
>Does spin have a frequency?
No.
>>7853631
this
spin refers to how a field transforms, under the spacetime transformations
https://en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics#Transformations_of_spinor_wavefunctions_in_relativistic_quantum_mechanics
>>7853631
To clear up some confusion, the particle is not spinning. It's not that a spin 0 particle looks the same all the time when spinning. It's that if you rotate your perspective (or rotate the particle), the particle looks the same.
Another small caveat is that instead of
>A particle with spin 2 looks the same every half-spin
it should be
>A particle with angular momentum 2 about the z-axis looks the same at every half-turn about the z-axis
A particle with spin 2 can have angular momentum +/- 2 about an axis (for all but spin 0 we can only measure one axis at a time without disturbing the others), but if the spin 2 particle has mass it can also have angular momentum -1, 0, or +1 about the axis.
More generally, angular momentum describes how something transforms under rotations, and spin is the intrinsic part of angular momentum. There's also angular momentum from the motion of particles. For example, in an atom, you have s orbitals (orbital angular momentum number l=0) that look the same from any direction, p orbitals (l=1) that can have angular momentum -1, 0, or +1, d orbitals (l=2), f orbitals (l=3), and so on. Here's a picture of the orbitals' wavefunctions, where you can see how some are unchanged by turning and how some look the same after a full turn or a half-turn or a third of a turn.
>>7837952
>Imagine when you spin a ball, or how the earth spins, it's exactly the same.
kek
>>7856808
Also, I should add that the more precise statement is that if you have angular momentum [math]m \hbar[/math] about an axis, rotating by [math]\theta[/math] multiplies the quantum state by [math]e^{-im\theta}[/math]. That gives you looking the same every [math]m[/math]th-turn.
bunp
This may be regarded as a continuation of the Klein Erlanger Program, in the sense that a geometrical space with its group of transformations is generalized to a category with its algebra of mappings. (Eilenberg and MacLane 1945, p. 237)
Commenting on this passage Marquis says:
Klein’s fundamental idea was that to study a geometry, one had to look at its group of transformations and, furthermore, the geometric properties of that geometry are those which are invariant under the group of transformations. This seems to be the core of the generalization that Eilenberg and Mac Lane had in mind. (Marquis 2009, p. 10]
In the 1920s, evidence from hydrogen spectra and the Stern-Gerlach experiment were consistent with the idea that particles have an intrinsic angular momentum and magnetic moment. The name is due to a classical phenomenon where a spinning charged object acquires a magnetic field, but the particle is not actually spinning, because if it were, it would have to be spinning faster than light in order to produce the observed magnetic moment.
It just has an intrinsic angular momentum, don't think too hard about it.
At the time where spin was discovered, the quantization of orbital angular mometnum was already known. Assuming the spin behaved in the same way, it can be shown that an intrinsic spin quantum number s=1/2 would explain experimental observation.
The mathematics of angular momentum can be worked out using the representation theory of SO(3), the rotation group in 3 dimensions, as people have already said ITT.
>>7844731
An alleged scientific discovery has no merit unless it can be explained to a barmaid.
>>7858688
but barmaid do not care about science
>>7844555
>implying stackexchange isn't better than this piece of shit low quality board
No one said that they were doing physics, they may just know which maths is linked to it.
Name dropping is -extremely- useful fyi so OP can go ahead and search up the relevant tools himself.
Not only that, OP has provided no background information on what he already knows.
>>7837988
Don't you mean the Mp(n)?
>>7838013
this, basically, according to modern science and math
>>7838010
controversial post
>>7859500
>doesn't know all modern science must be vetted by barmaids or be consigned to the trash
>thinking he know more about theoretical physics than a barmaid
>thinking anyone does
sage goes in all memes.
Literally OP here.
10 fucking days, I don't remember /sci/ being this fucking slow.
To update you: No I haven't understood what exactly spin is but I know better what things it isn't.
I narrowed it down by a lot and it's obvious to me in order for further narrowing I'd have to understand more of the physics/math, until then I'm ok.
Thanks for the replies
>>7863942
let it gooooo
>>7837947
so no more spin ?
what are spinors exactly ?
So if spin is just a label for the Poincaré group ireps, why does it show up in QM, where special relativity is not taken into account?
On that note, why does it show up when doing GR or QFT on curved spaces, where one is using an arbitrary Lorentzian metric which does not necessarily have the Poincaré group as isometry group?
>>7866350
elements of the vector space of a spin representation
>>7866754
Not quite. They're sections of a spin bundle, which reduces to what you said in case you're working in flat space.
>>7866503
>So if spin is just a label for the Poincaré group ireps, why does it show up in QM, where special relativity is not taken into account?
is it not about the solutions of the schrodinger equations, which are parametrized by various quantum numbers ?
https://en.wikipedia.org/wiki/Spin_quantum_number
then, it appears that spin operators have special commutations, which through the group theory, is tied to symmetries
https://en.wikipedia.org/wiki/Angular_momentum_operator#Angular_momentum_as_the_generator_of_rotations
>>7866503
>which does not necessarily have the Poincaré group as isometry group
what are the symmetry group that you think of ?
>>7866503
>So if spin is just a label for the Poincaré group ireps,
Calling it "just a label" is misleading; it's a real difference in the way those irreps behave under transformations.
>why does it show up in QM, where special relativity is not taken into account?
Because QM has rotation symmetry.
>>7844731
I hate fucks like you. Billy asks how rockets work and an autist like you spews out the Tsiolkovsky rocket equation instead of just using the balloon analogy. Protip: if you can't understand the physics you're doing without maths then you aren't good at physics. Any trained monkey can push symbols around equations, only a talented person can explain what it all means in layman's terms.
Here's a test for you autists; I was doing calculations on aircraft fuel mass, m required to lift an aircraft. I got a cubic equation which gave three positive real roots and one negative root. An autist such as you would just trash the negative result because negative mass doesn't exist right? But I found a meaning to it, what was it?
>>7868825
>>7868875
But from the point of view of the Poincaré group, the irreps are classified by eigenvalues of the Casimirs (i.e., quantum numbers), with the mass being associated to P^2 and the spin associated to W^2. But the SO(3) subgroup of the Poincaré group doesn't have W^2 as Casimir, right? So how come the spin in QM is related to SO(3) representations, whereas in relativity one has to deal with this W^2 Casimir instead?
>what are the symmetry group that you think of ?
When thinking about QFT on curved spaces, the symmetry / isometry group of whatever metric I'm taking. When thinking about GR, I don't even know since the metric is a dynamical variable of the theory.
>>7868956
>>7868956
yes GR has a symmetry group, it is the group of diffeomorphisms leaving the metric invariant, in one word, the symmetry group is the isometries of your manifold.
this is the representation theory of Diff(M)
https://en.wikipedia.org/wiki/Representation_theory_of_diffeomorphism_groups
you are right that the spin is not explicit for general manifold. this is why people forces a spin structure on the manifold, in asking ''what does it take to have a spin structure on my manifold'?'':
https://en.wikipedia.org/wiki/Spin_structure#Spin_structures_on_Riemannian_manifolds
acting on your bundle of frames
https://en.wikipedia.org/wiki/Frame_bundle#Orthonormal_frame_bundle
they want to see when we have the spin showing up in your math. of course, they do not really know, so they mimic the famous double cover, by SU(2), of SO(3) and they generalize what they learnt from the study of the spin group and pin group and other clifford algebras.
this is pure math and they generalize because they like it.
the theories in physics which are trendy today include spin, so of course, sooner or later, when you begin on the side of spacetime in order to go on the Qm side, with plenty of particles, you better seek a spin structure on the spacetime that you like. if you do not have a spin structure, today everybody will laugh at you.
so how to connect the ''spin in QM'' with the casimir of the P-L group [which is the real spin] ?
first, the link
https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group
says explicitly that you have a W2 even in CM, in dim 3+1, but it deals with the angular momentum.
>>7869720
first, recall that
in Cm, we have the classical angular momentum, L
the conservation of L is the invariance of the hamiltonian H under 3d spatial rotations. [ISOTROPY]
in Qm, we have
- the ORBITAL angular momentum = L (a vector)
- the SPIN angular momentum = S (a vector)
-the TOTAL angular momentum = J = L + S [a vector]
MATHEMATICALLY, the pauli lubanski vector in SR is, A PRIORI, about the TOTAL angular momentum J, BUT the contribution of L when you apply the (quadrivector) operator W^\mu to your field disappears ( for instance, when we choose a massive irreducible representation and act with W\mu on your vectors). so remains ONLY the SPIN, and the number ''s'' that classifies the ''particles'' is really the SPIN, NOT the total s+\ell.
[this is not so obvious, but it is easy mathematically].
each L, S, J generates a rotation.
which ones?
how to connect TOTAL angular momentum with rotations ?
>>7869721
>In simpler terms, the total angular momentum operator characterizes how a quantum system is changed when it is rotated. The relationship between angular momentum operators and rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics, as discussed further below.
-L generates the rotation R_spatial in (what a few people formalize mathematically as the) 3d space
-S generate the rotation R_internal in whatever is the mathematical space for the spin
[the mathematical space of the spin is INTERNAL, which means not SPATIAL]
-J generates the rotation R in 3d space and the (mathematical space of the spin)
a few properties of L:
>Mathematically, L2 is a Casimir invariant of the Lie algebra so(3) spanned by L.
>L (unlike J and S) cannot have half-integer quantum numbers.
let's leave the rotations and ask why L, S, J are quantified in QM, whereas L is not quantified in CM.
if you define a vector angular momentum L, by
L = r x p [like in CM]
and you apply the usual commutation rules for [r, p] = 1ih that we have in QM, then you obtain
L x L = ih L
>>7869723
GENERALISATION:
>IN QM, ANY OPERATOR Q which satisfy [Q, Q]= ih Q is called a ANGULAR MOMENTUM [WITH QUANTIFIED VALUES].
It means that Q^2 is the casimir of the lie algebra generated by Q_x, Q_y, Q_z.
then with the famous Q_+ and Q_-, we build the eigenvectors [j, m> of Q^2 and Q_z and we find that
-j can be half integer => Q is in fact S, you work with SU[2]
-j can be integer and not half-integer => Q is in fact L, you work with SO[3]
is this number j the spin ''s'' from the symmetry group of SR [which is the real spin like you have always heard of] ? the answer is yes.
before answering this, there is a great subtlety here, because we would think that the 3d SPATIAL rotations are really quantum. this is false.
3d rotations are not at all quantum. IT TURNS OUT that, IN PURE MATH, the infinitesimal rotations are exactly a group and that a group is defined through it multiplication table.
https://en.wikipedia.org/wiki/Rotation_group_SO%283%29#Lie_algebra
in fact, when you start from PURE MATH, you still have the possibility to have the number j as integer or half integer. in CM, we choose for j to be integer, while in QM, j can be both.
the fact that the values of j are discrete means that the group is COMPACT [which has nothing to do with QM, once more]
>>7869725
>Starting with a certain quantum state |\psi_0\rangle, consider the set of states R(\hat{n},\phi) |\psi_0\rangle for all possible \hat{n} and \phi, i.e. the set of states that come about from rotating the starting state in every possible way. This is a vector space, and therefore the manner in which the rotation operators map one state onto another is a representation of the group of rotation operators.
>>When rotation operators act on quantum states, it forms a representation of the Lie group SU(2) (for R and R_internal), or SO(3) (for Rspatial).
>From the relation between J and rotation operators,
>>When angular momentum operators act on quantum states, it forms a representation of the Lie algebra SU(2) or SO(3).
>>(The Lie algebras of SU(2) and SO(3) are identical.)
the difference between QM and Cm lies in the usefulness of SO[3] and SU[2]. you take SO[3] when you work with the usual L 9no spin S), when you want to be classical.
you work with SU[2] when you are clearly in QM. why ? because of the phase in QM. we know that the phase does not matter in QM. this leads us to study the PROJECTIVE representations of our symmetry group. but then the projective representation of SO[3] are the STANDARD, NORMAL, USUAL representation of SU[2], precisely because SU[2] is the double cover of SO[3]. BUT in CM, we work directly with SO[3], since we do not even need to think of projective representations, but only of usual representations..
All of this means that in fact, it is far more natural to work with SU[2] in QM.
>>7869728
in the following, λ is the j above, as in [j, m>.
>As stated above, representations of SU(2) describe non-relativistic spin due to double covering of the rotation group of Euclidean 3-space. Relativistic spin is described with representation theory of SL2(C), a supergroup of SU(2), which in the similar way covers SO+(1;3), the relativistic version of rotation group. SU(2) symmetry also supports concepts of isobaric spin and weak isospin, collectively known as isospin.
>λ = 1/2 gives the 2 representation, the fundamental representation of SU(2). When an element of SU(2) is written as a complex 2×2 matrix, it is simply a multiplication of column 2-vectors. It is known in physics as the spin-½ and, historically, as the multiplication of quaternions (more precisely, multiplication by a unit quaternion).
>λ = 1 gives the 3 representation, the adjoint representation. It describes 3-d rotations, the standard representation of SO(3), so real numbers are sufficient for it. Physicists use it for the description of massive spin-1 particles, such as vector mesons, but its importance for the spin theory is much higher because it anchors spin states to the geometry of the physical 3-space. This representation emerged simultaneously with the 2 when William Rowan Hamilton introduced versors, his term for elements of SU(2). Note that Hamilton did not use standard group theory terminology since he preceded Lie group developments.
>>7869731
>The λ = 3/2 representation is used in particle physics for certain baryons, such as the Δ.
how to connect the P-L group and its W2 and its number ''s'' with the number ''j'' of any operator Q verifying [Q, Q]= i (h) Q ????
well this is easy by the commutation relation of the P-L group. you take the pauli-lub. quadrivector W\mu, you show that it can be expressed, at least for the massive case, as a angular momentum, defined as [Q, Q]= i (h) Q , and that its eigenvector are of the form [j, m>.
then you show that the ''j'' is indeed a half integer.
https://en.wikipedia.org/wiki/Pauli%E2%80%93Lubanski_pseudovector#Massive_fields
for this, you go back to the definition of P^\mu and J^\mu\nu. you take a representation of P^\mu as the famous derivative -i\partial^\mu and J as the TOTAL angular momentum : J_µν = L_µν + S_µν, with L_µν = i(x_µ ∂_ν − x_ν ∂_µ).
Then, the quadrivector W^\mu is composed indeed of only P and S, no longer of L !!!
but this L that we ejected from W is exactly, in this representation for functions, the orbital angular momentum !
there remains only the spin.
the number s is ENTIRELY determined by the very first relation that initiated all this group theory: the link between a state S[x] at position ''x'' and a state S'[x'] at position x':
S'[x'] = some matrix [depending on the metric g] S[x]
[we say explicitly that we want a matrix, because we want the relation to be LINEAR, between the new fields/states, at a new position x', and the old fields/states at the old position.
>>7869735
this was all math.
the spin in Qm comes far more form physics than from math. the spin is really a feature that the physicists discovered and then tried to formalize. then the mathematicians came along and said that the work of the physicists can be deductively got thanks to group theory.
So how to get spin in QM ? We must look at the history of the spin, which means the history of QM.
In short, it is because of the zeaman effect and the landé factor [g =2] and Stern and Gerlach. they posited a INTRINSIC angular momentum for the electrons of the atoms that they studied, BECAUSE, the ORBITAL angular momentum L used in the usual equations, for the hamiltonian, coupling the magnetic field with the atom FAILED to explain the various fine levels of energy that appear once the atom is in a magnetic field.
I give a very good article on the group theory and all these classification through casimirs, in order to derive the wave equation of the system.
The unitary representations of the Poincare group in any spacetime dimension [it begins at section 2]
http://arxiv.org/abs/hep-th/0611263
>>7838007
It's just one of the elements that you're going to have to take about the world, and from that brute fact, we can explain many other things which might look like they're unrelated.
https://www.youtube.com/watch?v=MO0r930Sn_8
It's complicated but here's the simplest answer I can give you that will give you a full idea of what it means.
Spin is an intrinsic amount of angular momentum a particle has. it's always some multiple of half the reduced plank constant.
Now imagine a ball that's spinning. From whatever angle you look at it from, you can tell that its spinning on a very clear axis in a very clear direction. With particle spin, things are a bit different. No matter what axis you look at a particle from, it will always seem to spinning along that axis, however, there is a random chance that it will be spinning in one direction and another chance it will be spinning in the opposite
Spinning objects have an arrow that points in the direction they are seen spinning counter-clockwise. spinning particles have that same arrow, but instead of telling you what direction it's spinning, it will tell you the relationship between the direction you measure it's spin from and what the chances are that you'll measure it spinning clockwise or counter-clockwise. The closer that axis's arrow is to the spin arrow of the particle, the more likely it is that you'll see the particle spinning counter-clockwise
There's some more to it than that however. each particle has a spin number: 0, 1/2, 1, 3/2, 2, ...
what I was describing was a particle whose spin is 1/2. When you measure a spinning particle, the angular momentum it has also varies, but there are different "steps" between it's spin number and the negative of that number each one apart. An electron which has a spin of 1/2 can only be measured with a spin of 1/2 or -1/2. A proton which has a spin of 1 can be measured with a spin of 1, 0 or -1. Something with a spin of 2 can be 2, 1, 0, -1, or -2. That number corresponds to how hard the particle is spinning, with negative numbers meaning it's spinning clockwise.
>>7869735
>for this, you go back to the definition of P^\mu and J^\mu\nu. you take a representation of P^\mu as the famous derivative -i\partial^\mu and J as the TOTAL angular momentum : J_µν = L_µν + S_µν, with L_µν = i(x_µ ∂_ν − x_ν ∂_µ).
in fact, you can just say that L is antisymmetric, so when you multiply it by P and the levicita symbol, it vanishes.
also, with SR, you can assign a spin to a whole system, not just point-like particles.
also, to come back to the mass. the mass has several definition in GR. so do not take mass seriously in GR.
>>7837947
http://www.scientificamerican.com/article/what-exactly-is-the-spin/
>>7870980
This pop sci shit doesn't answer the question at all.
I was just reading about the Stern-Gerlach experiment in Walker's intro (baby calc. based) textbook.
I understand how magnetic spin dipoles are measured and how we determined they are quantized.
But why doesn't orientation matter? It's easy to think of atoms as symmetrical and so maybe valence electron alignment does not matter, but suppose you had a non-symmetrical molecule with only one atom having a magnetic dipole, would you still detect quantized measurements along the same axis?
Please don't copy-paste paragraphs from your textbook like the other posts ITT. I have obviously not read requisite material and will not understand it, I'd rather you say that you can't explain it layman's terms or give references and tell me to fuck off.
>>7872919
You detect quantized measurements along whatever axis you measure.
>>7872947
But why? Not all molecules should be aligned in the same spatial rotations before entering the magnetic field and thus statistically some of the magnetic dipole moments will not have the same spatial alignments and the forces should be slightly different instead of exactly quantized.
>>7872959
I think you might have the impression that if an electron has spin +1/2 along the z-axis, that means the electron's spin is aligned with the z-axis. That isn't correct; all it tells you is the component along the z-axis. After such a measurement, you are uncertain about the components along the x-axis and y-axis.
If you measure an electron to have spin +1/2 along the z-axis, its spin component along the x-axis has an equal chance of being +1/2 or -1/2 if measured. For other axes not perpendicular to the z-axis, the probabilities aren't 50/50 but the options are always +1/2 or -1/2.
>>7872975
Alright thanks.
>>7872979
and once you measure along z-axis, let's say you find -1/2, then measure along x-axis, let's say you find +1/2, then re-measure along the z-axis, the spin can have changed, which means that you can get +1/2 along z.
>ask someone a question that involves reasonably high levels of math to explain
>they try and dumb it down but do a poor job
>say they are confusing themselves
>>7869737
Merci beaucoup for the explanation. I've spent some time thinking about it, think I understand most of what you wrote. I never really realized the importance of the Galilean group in this context.
>>7874382
in what university are you and at what level ?
>>7874394
n-th grade of anomos uni
>>7874401
if you speak french, you can go on libgen and search these two books
Jean-Pierre Derendinger-Théorie quantique des champs-Presses Polytechniques et Universitaires Romandes (PPUR) (2001)
Texier, Christophe-Mécanique quantique _ cours et exercices corrigés-Dunod (2015)
these two plus the article from arxiv explains everything concisely.
zuber also explains things well, but on the side of statistical physics, in an article
Sur les symétries exactes d’espace-temps
J.-B. Zuber
http://www.lpthe.jussieu.fr/~zuber/MesPapiers/lapp94.pdf
http://www.lpthe.jussieu.fr/~zuber/UPMC.html
>>7837947
It's the same thing as spin in pool.
>yfw this thread has been up for 14 days and still no one understands spin
>>7875043
no one understands spin, because it is not expressible in terms of what people think is the universal categories of space and time.
space & time, aka the formalization consisting in position X and velocity V (or P), which then leads you to ask what symmetries you have in your fantasies of spacetime and this leads you to the group of poincaré-lorentz (or the galilean group, but let's be honest, the PL group is far more natural, since the analysis, done by einstein, of the synchronisation of clocks is far less retarded than the deliriums which happen in newtonian physics; to call special relativity a revolution shows how less people are empiricists in their mathematical formalizations of their concepts),
then you think that the concept of states is a serious one and you make your states depend on position X and time T.
then you apply the transformations of the PL group and you find that there is one quantity, the pauli-lubanski vector, which appear and is not connected to the usual angular momentum, L = r x p, that you express in terms of space and velocity.
So the spin is just a tensor which is attached to every state which depend on position X and time T and this state transforms under the PL group that you constructed in classical logic.
this spin tensor has the property of a angular momentum with quantized values, but so far nobody is able to shows that it is a quantity expressible in terms of space and time. this is why people say that there is no equivalent of the spin in classical mechanics, since classical mechanics is all about expressions of quantity in terms of space and velocity.
so far, the spin is defined only mathematically and is not reduced to concepts that are called physical.
ok
The spin of a particle is its intrinsic angular momentum, the part of its angular momentum that doesn't have anything to do with its motion. It doesn't necessarily mean the particle is made of subparticles moving around the center of mass, although that is one way for a particle to have spin. Point particles can also have spin. They have spin because they arise from fields that have polarization. Why does that give the particle angular momentum? A common tactic for dumbing down quantum theory is to invoke the uncertainty principle, so let's do that. The more you know about something's orientation, the less you know about its angular momentum. Anything that carries orientation information, such as polarized light, necessarily also has to be able to carry angular momentum, such as photons which have spin.
>>7878522
>The more you know about something's orientation, the less you know about its angular momentum.
do you mean that there is coupling between the orientation with angular momentum, just like there is a coupling between energy and time ?
>>7879513
Yes, angular momentum is to orientation what energy is to time and momentum is to position. All conserved quantities that arise from Noether's theorem.
>>7879571
oh yes, that is a nice way to phrase it.
>a photon is a state of the free electromagnetic field which is an eigenstate of the photon number operator with eigenvalue 1.
>>7879571
But then, what does it mean to ''know its angular momentum'' if it is not to know the orientation already ?
>>7882017
The idea applies to each component of angular momentum individually. Knowing what the z component of the angular momentum is prevents you from knowing how far anything has been rotated about the z-axis. That includes the x- and y- components of the angular momentum. That's why you can only know one component of the angular momentum at once (except for spin-0 particles for which all components are zero).
>>7844654
>654
as he said: no get
>>7843046
spin takes on integer or half-integer values, charge takes on integer values (unless you're looking at quarks and shit but I'm not a fucking particle physicist fuck that shit)
>>7882730
>charge
what charge though?
>>7878522
>A common tactic for dumbing down quantum theory is to invoke the uncertainty principle, so let's do that.
What is the explanation that is not dumb downs of quantum mechanics ?
>>7837947
It's a form of angular momentum
