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 $Spin(3)$?
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
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
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)
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
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'
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...
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 ?
(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)
$E$ 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 $E$
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.
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
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
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
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 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
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
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?
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...
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
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?
"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."
I 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...
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.
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
@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.
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
@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"
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
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
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
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.
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
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...
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?