Weyl's spinor and Dirac's spinor What is the difference between the two from a mathematical point of view? So are there different mathematical definitions of spinor? Is it correct to say that the Weyl spinor appears in the Weyl equation and the Dirac spinor in the Dirac equation? What is the physical meaning of the two equations instead? The Dirac equation describes the relativistic electron (spin 1/2) while the Weyl equation describes the relativistic fermions (including spin 1/2).  What is the difference? Is it correct to say that a Weyl spinor is an element of the vector space C2 on the field C? (C is the complex field)
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To get to the point - what's the defining differences between them? Alas, my current understanding of a spinor is limited. All I know is that they are used to describe fermions (?), but I'm not sure why?   Although I should probably grasp the above first, what is the difference between Dirac, Weyl and Majorana spinors? I know that there are similarities (as in overlaps) and that the Dirac spinor is a solution to the Dirac equation etc. But what's their mathematical differences, their purpose and their importance?  (It might be good to note that I'm coming from a string theory perspective. Plus I've exhausted Wikipedia here.)
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Rather predictably, I'm having a hard time understanding quantum spin. I know that it has units of angular momentum and relates to rotation, but that it can't be thought of as the actual spin of a particle with real size. Also, I understand that it's quantized.  What I'm wondering is, what happens to quantum spin in a rotating frame of reference?  Specifically, are there (rotating) frames of reference in which (for instance) an electron has spin that isn't half-integer? Is there a frame of reference rotating about a single electron such that that one electron acts like a spin-0 particle?
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Could someone please provide an explanation for the origin of ? (Ideally someone who I have been speaking with on the h-bar)  Any advanced maths beyond basic calculus will most probably leave me at a loss, though I do not mind a challenge! Please assume little prior knowledge, as over the past few days I have discovered that much of my understanding surrounding the process as virtual particle pairs is completely wrong.
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It seems to me that we could change all the current spin values of particles by multiplying them by two. Then we could describe Bosons as even spin particles and Fermions as odd spin particles. Is there some consequence or reason why we can't simply change 1/2 as the smallest unit of spin into 1?  It just seems prettier this way. In fact, there's almost an interesting relation to even and odd functions in that if you switch any two bosons around the wave function stays the same while if you switch any two fermions around the wave function becomes negative.
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If we do the wick rotation such that τ = it, then Schrödinger equation, say of a free particle, does have the same form of heat equation. However, it is clear that it admits the wave solution so it is sensible to call it a wave equation.   Whether we should treat it as a wave equation or a heat equation with imaginary time, or both? If it is a wave equation, how do we express it in the form of a wave equation? Is there any physical significance that Schrödinger equation has the same form of a heat equation with imaginary time? For example, what is diffusing?
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I got the difference between polar vectors and axial vectors (pseudovectors).   An example of pseudovector is $\mathbf{B}$. But why exactly the magnetic field is a pseudovector and its components parallel to an coordinate axis do no change signs if I reverse that axis?
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I'm a beginner of QFT. Ref. 1 states that      [...] The Lorentz group $SO(1,3)$ is then essentially $SU(2)\times SU(2)$.    But how is it possible, because $SU(2)\times SU(2)$ is a compact Lie group while $SO(1,3)$ is non-compact?   And after some operation, he says that the Lorentz transformation on spinor is complex $2\times2$ matrices with unit determinant, so Lorentz group becomes  $SL(2,\mathbb{C})$. I'm confused about these, and I think there must be something missing.   References:   L.H. Ryder, QFT, chapter 2, p. 38.
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I can't seem to connect these two stories. Can you please help?  I have heard that you have to turn electron by 720 degrees in order to get the same spin state. Has this been seen experimentally? How do you turn an electron?  I know that Bloch sphere is used to explain spin states. I sort of understand how it works and how matrices can be constructed. However there you turn the vector by 360 degrees in order to get the same state. Why so? (Is that in reality you have to turn twice as much?)
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While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices:     $$\tag{16.1.2} (\gamma^0)^2 = I , (\gamma^j)^2 = -I \ (j=1,2,3) $$    $$\tag{16.1.3}  \gamma^0\gamma^j + \gamma^j \gamma^0 = 0 $$   $$\tag{16.1.4}  \gamma^j \gamma^k + \gamma^k \gamma^j = 0, \ j\neq k$$    In looking for solutions of these equations in terms of matrices, one finds that they must have as order a multiple of 4, and that there exists a solution of order 4.   Obviously the word order here means dimension. In my QM classes the lecturer referenced chapter 5 from Advanced Quantum Mechanics by F. Schwabl, especially as regards the dimension of Dirac $\gamma$ matrices. However there it is stated only that, since the number of positive and negative eigenvalues of $\alpha$ and $\beta^k$ must be equal, $n$ is even. Moreover, $n=2$ is not sufficient, so $n=4$ is the smallest possible dimension in which it is possible to realize the desired algebraic structure.  While I got that the smallest dimension is 4, I fail to find any argument to reject the possibility that $n=6$ could be a solution. I also checked  Phys.SE post, but I didn't find it helpful at all.  Can anyone help me?
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I'm a huge fan of mathematical physics and I know what the formal definitions of those  two areas are, I've seen them. But I still get completely baffled when someone asks me to explain it simply. The difference is obvious to me, but I just can't seem to put it into words in a satisfying enough manner. So I'm asking you to help me...  What is the essential difference between  and ?  or if you prefer the rephrased version...  What was the motivation for introducing the name "mathematical physics" as a separate entity?  (It doesn't matter if you're one of those people who don't like labels, the thing is that we do have these two separated very often in academia.)
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Absence of magnetic charges is reflected in one of Maxwell's fundamental equations: $$\operatorname{div} \vec B = 0. \tag1$$ This equation allows us to introducte concept of vector potential: $$\vec B = \operatorname{rot} \vec A.$$ Using this concept, it is possible to express Maxwell's equations in a graceful symmetric form: $$\nabla^2 \vec A - \frac{1}{c^2}\frac{\partial^2 \vec A}{\partial t^2} = - \frac{\vec j}{\epsilon_0 c^2}, \tag2$$ $$ \nabla^2 \phi -\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = - \frac{\rho}{\epsilon_0}. \tag3 $$  Noticing, that vector $\vec A$ and scalar $\phi$ potentials, as well as electric current density $\vec j$ and charge density $\rho$, form a 4-vector in Minkovsky space-time. Therefore, Maxwell's equations can be expressed in a covariant form, using d'Alembertian: $$\nabla_{\mu}\nabla^{\mu} A_{\nu} = \frac{j_{\nu}}{\epsilon_0}. \tag4$$  If magnetic monopols exist, Maxwell's equation $(1)$ will look as: $$\operatorname{div} \vec B = \mu_0 c \rho_{\mathrm{magnet}}.$$  As the divergence of $\vec{B}$ isn't equal to zero, it impossible to introduce concept of vector potential. Thus, the equation in the form of $(4)$ will not be possible to express.
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Let us consider Dirac equation $$(i\gamma^\mu\partial_\mu -m)\psi ~=~0$$ as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that Dirac equation becomes Hamiltonian equation   $$\dot{\psi}~=~\{ \psi,H\}_{PB}~?$$  Of course, such Poisson bracket would be graded (super Poisson bracket), but if it exists this would explain on classical level why $\frac{1}{2}$-spinors correspond to fermions.
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I'm trying to understand proof of this inequality. But I have some problems.   So, Shankar starts a proof with definition a new vector $|z \rangle$:  $$ |z \rangle = |v\rangle - \frac{\langle w|v \rangle}{|w|^2} |w \rangle.$$  And here, I would like to know, why we suppose vector in this form? I know that it's projection of one vector on the second one.   My second question is, that we multiply this equation by $\langle z|$.  $$ \langle z|z\rangle = \left\langle v - \frac{\langle w|v \rangle}{|w|^2} w \right|\left.v-\frac{\langle w|v \rangle}{|w|^2} |w \right\rangle\\ = \langle v|v \rangle - \frac{\langle w|v \rangle \langle v|w \rangle}{|w|^2} - \frac{\langle w|v \rangle ^*\langle w|v \rangle}{|w|^2} + \frac{\langle w|v \rangle ^*\langle w|v \rangle \langle w|w \rangle}{|w|^4} $$  What I don't understand is a complex conjugate in the last two expression. Why then, we don't conjugate the first and the second one?  EDIT:  So - we're multiplying ket-z times bra-z. So, if the ket-z is: $$ |z \rangle = |v\rangle - \frac{\langle w|v \rangle}{|w|^2} |w \rangle.$$ and bra-z is: $$ \langle z |= \langle v|- \frac{\langle w|v \rangle}{|w|^2} \langle w|.$$  And what concernes me is this term above: $$ \langle w|v \rangle, $$ because if $ | z \rangle = [(\langle z |^*)^T]. $ So shouldn't bra-z be $$ \langle z |= \langle v|- \frac{\langle v|w \rangle}{|w|^2} \langle w|.$$
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As everybody here knows, Maxwell's equation would look more beautiful if a magnetig charge were present. Beyond the aesthetics question, if a single magnetic Dirac monopole would be found, the consequence is that the electric charge would be quantized.  While, I guess, everybody should be happy with the news, my experimentalist background doesn't allow me to understand the experimental setting. I therefore ask for the following advice -- a rather theoretically oriented question.  A recent paper,     Observation of Dirac monopoles in a synthetic magnetic field. M. W. Ray, E. Ruokokoski, S. Kandel, M. Möttönen and D. S. Hall. .   claims to have observed such monopoles.   Can somebody shed light on the theoretical implicacions of this work? Concretely, should we now take care before saying that it's been proven that the electric charge is quantized?
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I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great.
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When we derive the Dirac equation from the Lagrangian, $$ \mathcal{L}=\overline{\psi}i\gamma^{\mu}\partial_{\mu}\psi-m\overline{\psi}\psi, $$ we assume $\psi$ and $\overline{\psi}=\psi^{*^{T}}\gamma^{0}$ are independent. So when we take the derivative of the Lagrangian with respect to $\overline{\psi}$, we get the Dirac equation  $$ 0=\partial_{\mu}\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\overline{\psi}\right)}=\frac{\partial\mathcal{L}}{\partial\overline{\psi}}=\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi. $$  Now if we include a term with charge conjugation, $\psi^{c}=-i\gamma^{2}\psi^{*}$, into the Lagrangian (like $\Delta\mathcal{L}=\overline{\psi}\psi^{c}$), does this $\psi^c$ depend on $\overline{\psi}$ or $\psi$? Why or why not?  If $\psi^{c}$ depends on $\psi$, why wouldn't the reason that $\overline{\psi}$ and $\psi$ are independent apply for $\psi^{c}$ and $\psi$?  If $\psi^{c}$ depends on $\overline{\psi}$, how should we take derivative of $\Delta\mathcal{L}$ with respect to $\overline{\psi}$?
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To introduce magnetic monopoles in Maxwell equations, Dirac uses special strings, that are singularities in space, allowing potentials to be gauge potentials. A consequence of this is the quantization of charge.  Okay, it looks great. But is this the only way to introduce magnetic monopoles?