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14

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 $$\... 13 Let us generalize from four space-time dimensions to a d-dimensional Clifford algebra C. Define$$\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 ... 13 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 ... 13 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 ... 13 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, ... 13 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 ... 12 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 ... 12 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 ... 11 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 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 ... 9 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).$$Note that both groups have 6 ... 9 What you have is a good start. If we make the usual assignments that {\partial\over{\partial t}} \to -iE and \nabla \to i{\bf p} then we get$$(E - e\Phi)\psi = (\alpha \cdot ({\bf p} - e{\bf A}) + m\beta)\psi.$$Now, pick a particular representation$$\beta = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},\text{ }\alpha_i = \begin{pmatrix} 0 & \...

8

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 ...

8

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 ...

8

You have no other choice than to use $4\times 4$ matrices. All these "representations" are different realizations (related by similarity transformations) of the only possible irreducible representation of the Clifford algebra that is spanned by the abstract $\gamma^\mu$. This representation, in a way, is the definition of what a "Dirac spinor" is, and it is ...

7

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 ...

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 (...

7

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 ...

7

As per Rob's suggestion, I decided to make this an answer. (Addendum: I've been meditating on this very topic for some time, and have been directed to some interesting literature referenced on Streater's webpage. As per Rococo's comment, I've updated my answer, but kept the old version for posterity.) The Answer To talk about "spin-1/2 particles", we ...

7

The question puts the cart before the horse. It is not that you derive that particles described by the Dirac equation have spin $\frac 1 2$. Rather, the Dirac equation is found as the equation for spin $\frac 1 2$ particles. A Dirac spinor $\psi$ is an element of the representation $(0,\frac 1 2) \oplus (\frac 1 2, 0)$ of the Lorentz group.1 In both ...

7

There isn't a good definition of chirality in (2+1)D or any other odd dimension. This is because the $\gamma_5$ matrix can't be defined usefully in a Clifford algebra with an odd number of generators. For instance try to define $\gamma_5 = \gamma^0\gamma^1\gamma^2$. This commutes (not anti-commutes) with $\gamma^0,\gamma^1,\gamma^2$ and thus commutes with ...

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: $$\left( \gamma_{5}\right) _{ij}=\left( \psi_{+}\right) _{i}\left( \psi _{+}^{\star}\right) _{j}-\left( \psi_{-... 6 Dirac's explanation of the emergence of antiparticles such as positrons out of the Dirac sea, and the Dirac sea itself, is completely valid and legitimate, and you have described some non-quantitative aspects of it and differences between it and some condensed-matter situations. Dirac just began with the assumption that the Dirac spinor field \Psi is a ... 6$$(\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 \partial_\mu\psi^\...

6

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 ...

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

The four-component wave function $\Psi$ in the Dirac equation may be viewed as a counterpart of $\psi(x)$ in non-relativistic Schrödinger's equation. The Dirac equation may be written (and, in fact, was originally written by Dirac) in the Schrödinger's form $$i\hbar \frac{\partial}{\partial t} \Psi = H \Psi$$ where $H=\vec\alpha\cdot \vec p + m\beta$ where ...

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

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

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 ...

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