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The Dirac equation for a particle with charge $e$ is $$\left[\gamma^\mu (i\partial_\mu - e A_\mu) - m \right] \psi = 0$$ We want to know if we can construct a spinor $\psi^c$ with the opposite charge from $\psi$. This would obey the equation $$\left[\gamma^\mu (i\partial_\mu + e A_\mu) - m \right] \psi^c = 0$$ If you know about gauge transformations $$... 11 The interpretation of the Dirac equation states depend on what representation you choose for your \gamma^\mu-matrices or your \alpha_i and \beta-matrices depending on what you prefer. Both are linked via \gamma^\mu=(\beta,\beta\vec{\alpha}). Choosing your representation will (more or less) fix your basis in which you consider the solutions to your ... 10 We know that we can describe a spin 1/2 massless particle using only a single Weyl field (lets say left-handed \psi_{L}). To introduce a mass term we have to use two spinor fields (one left-handed and one right-handed) and this gives the Dirac mass term. The question is now that if we can describe a massive particle with a single Weyl field. Well yes, ... 10 This is standard theory. Try Birrell, N. D., & Davies, P. C. W. (1982). Quantum Fields in Curved Space. Cambridge: Cambridge University Press. Bog standard Curved space QFT text. Don't remember how much is said specifically about spinors though. Brill, D., & Wheeler, J. (1957). Interaction of Neutrinos and Gravitational Fields. Reviews of Modern ... 10 Spin is a property of the representation of the rotation group SO(3) that describes how a field transforms under a rotation. This can be worked out for each kind of field or field equation. The Klein-Gordon field gives a spin 0 representation, while the Dirac equation gives two spin 1/2 representations (which merge to a single representation if one also ... 9 The Zitterbewegung is more of a relic of the early Dirac equation days. It does not exist in the standard position, velocity and acceleration operators of the single particle field, only in alternatively derived versions. These alternative versions were developed because people thought the standard operators were wrong. In fact they didn't understand the ... 9 The mistake you are making is in "daggering" the object \omega_{\mu\nu}. For each \mu, \nu = 0,\dots 3, the symbol \omega_{\mu\nu} is a real number, so its dagger (which is really just complex conjugation in this case) does nothing; (\omega_{\mu\nu})^\dagger = \omega_{\mu\nu}. When we say that \omega_{\mu\nu} is an antisymmetric real matrix, we ... 8 Dear rubenb, yes, what your professor says is surely based on solid maths. The reason is that the 4-component Dirac spinor is actually composed of two separate 2-component pieces. The elementary "spinors" for 3+1 dimensions have two complex components. That results from the isomorphism between groups$$SL(2,C) \sim Spin (3,1).$$Note that both groups have 6 ... 7 Symmetric under charge conjugation (which gives us positrons) and symmetric under the sign of the energy are two different things, which is where I think you are getting confused. Negative energy electrons aren't positrons, they are negative energy electrons. The absence of a negative energy electron in the "sea of charge" can be viewed as a positive ... 7 Let us generalize from four space-time dimensions to a d-dimensional Clifford algebra C. Define$$\tag{1} p~:=~[\frac{d}{2}], $$where [\cdot] denotes the integer part. OP's question then becomes Why must the dimension n of a finite dimensional representation V be a multiple of 2^p? Proof: If C\subseteq {\rm End}(V) and V are both ... 7 The expression A^{\mu}B_{\mu} simply means that$$A^{\mu}B_{\mu}=A^{0}B_{0}+A^{1}B_{1}+A^{2}B_{2}+A^{3}B_{3}$$Using the Minkowski metric with signature (+---) you write this as$$A^{\mu}B_{\mu}=A^{\mu}\eta_{\mu\nu}B^{\nu}=A^{0}B^{0}-A^{1}B^{1}-A^{2}B^{2}-A^{3}B^{3}$$The metric simply tells you have how the components of a vector and its dual vector ... 6 At the risk of telling you how to "suck eggs" (your level in these things is not altogether clear), here goes. Ingredients: The essential ingredients to this explanation are: A physical "system" which evolves in and whose "events" happen in some space \mathcal{U} (ordinary Euclidean 3-space or Minkowsky spacetime, for example); in physics this space is ... 6 For massless particles, helicity coincides with chirality thus you ask to find the basis such that$$ \psi_{\pm}=\left( \psi_{\mp}\right) ^{\star},\quad\gamma_{5}\psi_{\pm}% =\pm\psi_{\pm}. $$Using the decomposition of hermitian operator:$$ \left( \gamma_{5}\right) _{ij}=\left( \psi_{+}\right) _{i}\left( \psi _{+}^{\star}\right) _{j}-\left( ...

6

Let's review how the KG equation is recovered from the Dirac: (in natural units where $\hbar=c_0=1)$ $$(i\gamma^\mu \partial_\mu - m)\Psi = 0$$ $$(-i \gamma^\mu \partial_\mu - m)(i \gamma^\mu \partial_\mu - m) = 0$$ $$(\gamma^\nu \gamma^\mu \partial_\nu \partial_\mu + m^2) \Psi = 0$$ $$(\partial^2+m^2)\Psi = 0.$$ In order for us to recover KG, we had to ...

6

Dirac's derivation of the existence of positrons that you described was a totally legitimate and solid argument and Dirac rightfully received a Nobel prize for this derivation. As you correctly say, the same "sea" argument depending on Pauli's exclusion principle isn't really working for bosons. Modern QFT textbooks want to present fermions and bosons in a ...

6

Neutrinos interact in the Standard Model only through their left-handed component, via electroweak interactions. However, the propagating neutrinos, which are mass eigenstates, are described by a field that is a Dirac spinor, i.e. with both chiralities $$\nu=\nu_L+\nu_R.$$ Therefore, when neutrinos are created or measured, the Dirac spinor is projected ...

5

I think that the first volume of the series "The Quantum Theory of Fields", by Steven Weinberg, is a good text to understand the origin of Dirac equation, QFT, and all these kind of topics. Maybe Weinberg's books are not the best for a first course in QFT (or in General Relativity, he has also a great book on this topic), but his great coverage and unique ...

5

For the details of the physics involved in the two ways of interpreting the Dirac wave equation I recommend chapters XI and XII of Dirac's "Principles of Quantum Mechanics" 4th edition, and chapters XX and XXI of Messiah's "Quantum Mechanics", vol. II. For the more historical details I recommend chapters 5 and 6 of Crease and Mann's "The Second Creation", ...

5

What you've written down is the spatial part of the electron wavefunction. The spin state is not included. The full wavefunction of the electron involves both the spatial part and the spin part. Sometimes in quantum mechanics books the full electron wavefunction is written as the tensor product of the spatial and spinor parts, sometimes you'll just see it ...

5

In QFT, the Dirac spinor will also be promoted to a field, whose oscillation mode coefficients are creation and annihilation operators. BUT: For the Dirac spinor it is possible to well-define a probablility density and current: $$\rho^\mu \propto \bar\psi \gamma^\mu \psi$$ This current's zero component is positive definite and using the Dirac equation one ...

5

Spin-1/2 admits first order equations simply because $(\mathbf{1/2,1/2})\otimes (\mathbf{0,1/2})$ contains the representation $(\mathbf{1/2,0})$ so that a linear equation for free particles can be written (i.e. it contains a derivative acting on one field and returning one field). The first term in the product is the derivative that transforms as a ...

5

Hints : a) Find a representation for the gamma matrices for a space-time with one spatial dimension, from their defining relation $\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu= 2 g^{\mu \nu}$ b) Remember the Dirac equation in presence of an electromagnetic field : $[i \gamma^\mu(\partial_\mu + ie A_\mu) - m] \psi=0$ c) Think to the potential $U$ as a ...

5

$$(\psi^\dagger \gamma^0 \psi)^* = \psi^\dagger \gamma^0 \psi$$ because $\gamma^0$ is hermitian. Also, \begin{align} (\psi^\dagger i \gamma^0 \gamma^\mu \partial_\mu \psi)^* &= -i \partial_\mu\psi^\dagger \gamma^{\mu\dagger} \gamma^0 \psi\\ &= -i \partial_\mu\psi^\dagger (\gamma^0 \gamma^\mu \gamma^0)\gamma^0 \psi\\ &= -i ... 5 Non-conservation of charge in Majorana terms The Dirac mass term is m\bar\psi \psi where one field-factor \bar\psi is complex conjugated (aside from other transpositions included in the Dirac conjugation) and the other is not. So one may assign a fermion number 1 to \psi which means that \bar\psi automatically carries -1 and in the product, the ... 5 Think it with an example, Einstein's field equations are much more precise than Newton's law of gravity, but it's much more complicated to solve a Classical Mechanics problem with General Relativity. More fundamental and precise doesn't mean that it will give easier calculations. If it did, then then chemistry, medicine, etc... wouldn't exist because they ... 5 This particular extra term may be removed by a field redefinition\psi\to \psi' = \psi - K \cdot \gamma^\mu \partial_\mu \psi $$for an appropriate value of K\sim 1/\Lambda, up to terms that are even higher dimension operators. This also modifies the mass. This field redefinition is an explicit off-shell way to realize Vibert's comment that one is just ... 5 The Lagrangian density for a Dirac field is$$ \mathcal{L} = i\bar\psi\gamma^\mu\partial_\mu\psi -m \bar\psi\psi $$The Euler-Lagrange equation reads$$ \frac{\partial\mathcal{L}}{\partial\psi} - \frac{\partial}{\partial x^\mu}\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\right] = 0 $$We treat \psi and \bar\psi as independent dynamical ... 5 Yes. You are missing the fact that he is using the convention$$ \nabla = (\partial_1, \partial_2, \partial_3) $$as opposed to$$ \nabla = (\partial^1, \partial^2, \partial^3)  The first convention is by far the most common in my experience.

4

How about just testing the two different cases? I.e. if $\mu\not=0$ then the LHS becomes $$(\gamma^\mu)^\dagger= (\gamma^i)^\dagger= -\gamma^i \tag{see below}$$ while the RHS becomes $$(\gamma^\mu)^\dagger=\gamma^0\gamma^i\gamma^0 = -\gamma^0\gamma^0\gamma^i=-\gamma^i~~~~~~~~ (\text{OK}).$$ For ...

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