What is Dirac equation: Definition and 169 Discussions

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.
The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.
Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-1⁄2 particles.
The Dirac equation appears on the floor of Westminster Abbey on the plaque commemorating Paul Dirac's life, which was unveiled on 13 November 1995.

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  1. S

    A Covariant four-potential in the Dirac equation in QED

    Under the entry "Quantum electrodynamics" in Wikipedia, the Dirac equation for an electron is given by $$ i\gamma^{\mu}\partial_{\mu}\psi - e\gamma^{\mu}\left( A_{\mu} + B_{\mu} \right) \psi - m\psi = 0 ,\tag 1 $$ or $$ i\gamma^{\mu}\partial_{\mu}\psi - m\psi = e\gamma^{\mu}\left( A_{\mu} +...
  2. J

    I Understanding the Equations of Motion for the Dirac Lagrangian

    I'm having trouble following a proof of what happens when the Dirac Lagrangian is put into the Euler-Lagrange equation. This is the youtube video: and you can skip to 2:56 and pause to see all the math laid out. I understand the bird's eye results of the Dirac Lagrangian having an equation of...
  3. PhysicsRock

    I How does one compute the Fourier-Transform of the Dirac-Hamiltonian?

    Greentings, I've dealt with Quantum Theory a lot, but there's one thing I don't quite understand. When deriving the Fermion-Propagator, say ##S_F##, all the authors I've read from made use of the Fourier-Transform. Basically, it always goes like $$ \begin{align} H_D S_F(x-y) &= (i \hbar...
  4. D

    Landau Energy Spectrum in the non-relativistic limit

    At non-relativistic limit, m>>p so let p=0 At non-relativistic limit m>>w, So factorise out m^2 from the square root to get: m*sqrt(1+2w(n+1/2)/m) Taylor expansion identity for sqrt(1+x) for small x gives: E=m+w(n+1/2) but it should equal E=p^2/2m +w(n+1/2), so how does m transform into p^2/2m?
  5. snoopies622

    I What do the psi_3 and psi_4 components of the Dirac equation represent?

    Forgive me if you've heard this song before, but I don't understand how to interpret the \psi_3 and \psi_4 components of the Dirac equation. For instance, at 8:27 of this video we see that while an electron at rest can be in a state like [1,0,0,0], the same electron as viewed from a...
  6. L

    A Interaction between matter and antimatter in Dirac equation

    I'm new to relativistic quantum mechanics and quantum field theory and was trying to learn about the Dirac equation. Unfortunately, I got a little stumped by the interaction between matter and antimatter. It seems like the time derivative of matter is dependent on the spatial derivative of...
  7. M

    I Dirac equation in the hydrogen atom

    Hello! I went over a calculation of the hydrogen wavefunction using Dirac equation (this one) and I am a bit confused by the angular part. The final result for the wavefunction based on that derivation is this: $$ \begin{pmatrix} if(r) Y_{j l_A}^{m_j} \\ -g(r)...
  8. Haorong Wu

    The representation matrix for alpha and beta in Dirac equation

    In the 4-dimensional representation of ##\beta##, ## \beta=\begin{pmatrix}\mathbf I & \mathbf 0 \\ \mathbf0 & -\mathbf I\end{pmatrix} ,## and we can suppose ## \alpha_i=\begin{pmatrix}\mathbf A_i & \mathbf B_i \\ \mathbf C_i & \mathbf D_i\end{pmatrix} ##. From the anti-commutation relation...
  9. snypehype46

    Exercise involving Dirac fields and Fermionic commutation relations

    I'm trying to the following exercise: I've proven the first part and now I'm trying to do the same thing for fermions. The formulas for the mode expansions are: What I did was the following: $$\begin{align*} \sum_s \int d\tilde{q} \left(a_s(q) u(q,s) e^{-iq \cdot x}+ b_s^\dagger(q) v(q,s)...
  10. thaiqi

    I What is the relation between the Dirac equation and QED?

    Hello, everyone. Need I understand Dirac equation if I plan to learn QED?
  11. peguerosdc

    A Getting particle/antiparticle solutions from the Dirac Equation

    Hi! I am studying Dirac's equation and I already have understood the derivation. Following Griffiths, from factoring Einstein's energy relation with the gamma matrices: ## (\gamma^\mu p_\mu + m)(\gamma^\mu p_\mu - m) = 0 ## You take any of the two factors, apply quantization and you arrive to...
  12. M

    I Condtion on transformation to solve the Dirac equation

    The problem is given in the summary. My attempt: Assume that ##\psi^\prime (x^\prime)## is a solution of the Dirac equation in the primed frame, given the transformation ##x\mapsto x^\prime :=\Lambda^{-1}x## and ##\psi^\prime (x^\prime)=S\psi(x)##, we have $$ \begin{align*} 0&=(\gamma^\mu...
  13. J

    I Changing the effective mass of an electron using electric potentials?

    The Dirac equation for an electron in the presence of an electromagnetic 4-potential ##A_\mu##, where ##\hbar=c=1##, is given by $$\gamma^\mu\big(i\partial_\mu-eA_\mu\big)\psi-m_e\psi=0.\tag{1}$$ I assume the Weyl basis so that $$\psi=\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix}\hbox{ and...
  14. redtree

    I Relativistic quantum mechanics

    Given that the Minkowski metric implies the Lorentz transformations and special relativity, why do the equations of relativistic quantum mechanics, i.e., the Dirac and Klein-Gordon equations, require a mass term to unite quantum mechanics and special relativity? Shouldn't their formulation in...
  15. JD_PM

    I Rewriting Feynman amplitudes and the Dirac equation

    I was studying the photon polarization sum process (second edition QFT Mandl & Shaw,https://ia800108.us.archive.org/32/items/FranzMandlGrahamShawQuantumFieldTheoryWiley2010/Franz%20Mandl%2C%20Graham%20Shaw-Quantum%20Field%20Theory-Wiley%20%282010%29.pdf) and got stuck in how to get certain...
  16. T

    A Evaluating Matrix Spin Dependent Term in Dirac Quadratic Equation

    I derive the quadratic form of Dirac equation as follows $$\lbrace[i\not \partial-e\not A]^2-m^2\rbrace\psi=\lbrace\left( i\partial-e A\right)^2 + \frac{1}{2i} \sigma^{\mu\nu}F_{\mu \nu}-m^2\rbrace\psi=0$$ And I need to find the form of the spin dependent term to get the final expression $$g...
  17. park

    I The Dirac Equation: Understanding Spinors and Approximations

    I'm studying about dirac equation and it's solution. When we starts with the equation (2.75), I can understand that it is possible to set 2 kinds of spinor. But my question is... 1. After the assumption (2.100), how can we set the equation like (2.101) 2. I can't get (2.113) from (2.111)...
  18. sakh1012

    A Dirac Field quantization and anti-commutator relation

    Can anyone explain while calculating $$\left \{ \Psi, \Psi^\dagger \right \} $$, set of equation 5.4 in david tong notes lead us to $$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{-iqy} b_p^s u^s(p)e^{ipx}].$$ My question is how the above mentioned terms can be written as...
  19. Y

    Primary calculation involving the Dirac gama matrices

    When working on the exercise 3.2 of Peskin's QFT, I find one of the calculating steps confused for me. I read the solution, which is showed in the picture. I just don't understand the boxed part. I know it involved the Dirac equation, and the solution seems to treat the momentum as a operator...
  20. S

    Klein Gordon 4-current from Dirac Equation

    The left side of the equality of ##(5)## is obvious from ##(4)##, however the rest of the terms are still unknown to me. I have tried adding and subtracting terms similar to the rest of the terms so as to produce a commutator and use ##(3)##, but I can't seem to figure out how to get ##(5)##...
  21. A

    I Born's rule, causality, and the Dirac equation

    [Moderator's note: Spin off from a previous thread due to topic change.] Actually, the form of the Hamiltonian does matter. Hegerfeldt admits that his results are not correct for the Dirac Hamiltonian unless one considers only positive energy solutions. And why should we do that? It is clear...
  22. V

    I Why does the Dirac equation lead to spin 1/2?

    Why does the derivation of the Dirac equation naturally lead to spin ½ particles? The equation is derived from very general starting assumptions, so which of these assumptions has to be wrong to give us a spin-0 or spin-1 particle? I have tried to search for an answer and got as far as this...
  23. A

    I Is the Chirality Projection Operator Misused in This Scenario?

    Hello everybody! I have a doubt in using the chiral projection operators. In principle, it should be ##P_L \psi = \psi_L##. $$ P_L = \frac{1-\gamma^5}{2} = \frac{1}{2} \begin{pmatrix} \mathbb{I} & -\mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} $$ If I consider ##\psi = \begin{pmatrix}...
  24. K

    I The 4 solutions to the Dirac equation

    Hello! I understand that the free Dirac equations has spinors as solutions, of dimension 4, and one can't discard the negative energy solutions (as one needs a complete basis to span the Hilbert space of solutions), and these negative energy particles are interpreted as positive energy...
  25. M

    Proving Even Parity for this Expression

    My idea was straight forward calculation: $$\begin{align*}\bar { \psi }' ( x' ) \gamma ^ { \mu } \partial _ { \mu }' \psi ( x' ) &= \psi^\dagger\gamma^{0\dagger}\gamma^0\gamma^\mu \partial_\mu'\gamma^0\psi = \bar\psi\underbrace{\gamma^0\gamma^\mu\gamma^0}_{=\gamma^{\mu\dagger}=-\gamma^\mu}...
  26. M

    Understanding solutions of Dirac equation

    some notes: There was actually no proof given why ##u^s(p)## or ##v^s(p)## should solve the Dirac equation, only a statement that one could prove it using the identity $$(\sigma\cdot p)(\bar\sigma\cdot p)=p^2=m^2.$$ We were using the Wely-representation of the ##\gamma##-matrices, if this should...
  27. M

    Calculating field transformation

    Homework Statement Let ##\psi(x)=u(p)e^{-ipx}##, where $$ u((m,0)) = \sqrt{m}\begin{pmatrix} \xi\\\xi \end{pmatrix}\quad\text{where}\quad \xi = \sum_{s\in \{+,-\}}c_s\xi^s\quad \text{and}\quad \xi^+\equiv\begin{pmatrix} 1\\ 0 \end{pmatrix}\quad \xi^-\equiv\begin{pmatrix} 0\\ 1 \end{pmatrix}, $$...
  28. N

    A Perturbation solution and the Dirac equation

    I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$ The solution up to first order is $$ \psi(x) =...
  29. P

    A Dirac's solution to the Klein-Gordon equation

    Dirac wanted to fix the problems with the Klein-Gordon equation by seeking a new solution to it. He wanted a relativistic solution so it makes sense that the solution needed to satisfy Einstein's energy-momentum relation. But why did it need to be of first order in time- and...
  30. A

    I Dirac equation as one equation for one function

    Previously (see, e.g., https://www.physicsforums.com/threads/klein-gordon-equation-and-particles-with-spin.563974/#post-3690162), I mentioned my article in the Journal of Mathematical Physics where I showed that, in a general case, the Dirac equation is equivalent to a fourth-order partial...
  31. C

    I Spin conservation in the Dirac equation

    Here I am considering the one particle free Dirac equation. As is known the spin operator does not commute with the Hamiltonian. However, the solutions to the Dirac equation have a constant spinor term and only an overall phase factor which depends on time. So as the solution evolves in time...
  32. It's me

    Dirac hydrogen atom vs spin symmetry

    Homework Statement Exact spin symmetry in the Dirac equation occurs when there is both a scalar and a vector potential, and they are equal to each other. What physical effect is absent in this case, that does exist in the Dirac solution for the hydrogen atom (vector potential = Coulomb and...
  33. E

    I Dirac equation, one question

    In https://quantummechanics.ucsd.edu/ph130a/130_notes/node45.html after "Instead of an equation which is second order in the time derivative, we can make a first order equation, like the Schrödinger equation, by extending this equation to four components." it is evident that the solution is...
  34. B

    I Spin Angular Momentum Dirac Equation

    In the Dirac equation, the wave-function is broken into four wave-functions in four entries in a column of a matrix. Since there are four separate versions of the wave-function, does each version have the spin angular momentum of h-bar/2? This seems overly simplistic. How does spin angular...
  35. neilparker62

    I Use the Dirac Equation to calculate transition frequencies in Hydrogen

    I would just like to understand how to use the above Dirac energy equation to calculate (for example) the 1s-2s transition frequency in hydgrogen. Does one substitute n=1, j=0 for 1s energy level and n=2, j=0 for 2s energy level ? From previous reading I understand the mass referred to in the...
  36. LarryS

    I Dirac's Equation vs. QFT

    I’m attempting to learn QFT on my own and would like to get an idea of just how much I still do not know. Consider a system consisting only of electrons and for the purpose of this question, pretend that particle creation and annihilation never occur. QUESTION: Would Dirac’s famous...
  37. Gene Naden

    Positive and negative plane wave solutions of Dirac equation

    I continue to be occupied with the first chapter of Lessons on Particle Physics by Luis Anchordoqui and Francis Halzen. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf. I am on page 24, where they derive equations 1.5.67, which are: ##(\gamma^\mu p_\mu-m)u(p)=0## and...
  38. Gene Naden

    A Transformation of solutions of the Dirac equation

    I am working through "Lessons on Particle Physics." The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf. I am on page 21, equation (1.5.50), which is ##S(\Lambda)=1-\frac{i}{2}\omega_{\mu\nu}\Sigma^{\mu\nu}##. I would like some motivation for this equation. I wonder what the...
  39. Gene Naden

    A Transformation of Dirac spinors

    So I am working through Lessons in Particle Physics by Luis Anchordoqui and Francis Halzen, the link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf I am in the discussion of the Dirac equation, on page 21, trying to go from equation 1.5.49 to 1.5.51. And I get stuck. Equation...
  40. W

    I Why this system has a rotational symmetry in Dirac equation?

    why system has a rotational symmetry in dirac equation? is that a general property of all systems?
  41. B

    I The Relation Between Wavefunctions in Dirac Equation

    Can the wave function in four dimensions be expressed as e^i(kx+ky+kz-wt)?
  42. Milsomonk

    The Dirac equation in Weyl representation

    Homework Statement Compute the antiparticle spinor solutions of the free Dirac equation whilst working in the Weyl representation.Homework Equations Dirac equation $$(\gamma^\mu P_\mu +m)v_{(p)}=0$$ Dirac matrices in the Weyl representation $$ \gamma^\mu= \begin{bmatrix} 0 & \sigma^i \\...
  43. Milsomonk

    I Dirac equation solved in Weyl representation

    Hi guys :) I'm just wondering if anyone knows of a book that has the Dirac equation solved in the Weyl basis in it? I'd like to check my method to make sure I'm on the correct lines. Thanks
  44. B

    I Dirac Matrices and the Pythagorean Theorem

    I understand that momentum, rest mass and energy can be put on the sides of a right triangle such that the Pythagorean Theorem suggests E^2=p^2+m^2. I understand that the Dirac equation says E=aypy+axpx+azpz+Bm and that when we square both sides the momentum and mass terms square while the cross...
  45. Turbotanten

    A What does it mean for the Hamiltonian to not be bounded?

    If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian, see Peskin and Schroeder $$ H = \int\frac{d^3p}{(2\pi)^3}E_p \sum_{s=1}^2 \Big( a^{s\dagger}_\textbf{p}a^s_\textbf{p}...
  46. bleist88

    A The Lagrangian Density and Equations of Motion

    Can Lagrangian densities be constructed from the physics and then derive equations of motion from them? As it seems now, from my reading and a course I took, that the equations of motion are known (i.e. the Klein-Gordon and Dirac Equation) and then from them the Lagrangian density can be...
  47. Z

    I Why the velocity operator commutes with position (Dirac equation)

    ##\hat{v}_i=c\hat{\alpha}_i## commute with ##\hat{x}_i##, ##E^2={p_1}^2c^2+{p_2}^2c^2+{p_3}^2c^2+m^2c^4## But in classical picture,the poisson braket...
  48. M

    Exploring Breit Interactions: Books for Conceptual Understanding and Application

    I am finding references and good books for the conceptual understanding of Breit Interactions. Are there any books which specifically include the topic? The related topics explaining the formulation Breit Hamiltonian and its involvement to the correction to the atomic structure calculation are...
  49. L

    A How the g factor comes from QFT?

    I'm reading the book Quantum Field Theory and the Standard Model by Matthew Schwartz and currently I'm studying the chapter 17 titled "The anomalous magnetic moment" which is devoted to computing the corrections due to QFT to the g factor. My main issue is in the beginning of the chapter, where...