Dirac Definition and 900 Threads

Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century.Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions and predicted the existence of antimatter. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger "for the discovery of new productive forms of atomic theory". He also made significant contributions to the reconciliation of general relativity with quantum mechanics.
Dirac was regarded by his friends and colleagues as unusual in character. In a 1926 letter to Paul Ehrenfest, Albert Einstein wrote of Dirac, "I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful." In another letter he wrote, "I don't understand Dirac at all (Compton effect)."He was the Lucasian Professor of Mathematics at the University of Cambridge, was a member of the Center for Theoretical Studies, University of Miami, and spent the last decade of his life at Florida State University.

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  1. 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...
  2. PeroK

    I Dirac Lagrangian and Covariant derivative

    This is from Griffiths particle physics, page 360. We have the full Dirac Lagrangian: $$\mathcal L = [i\hbar c \bar \psi \gamma^{\mu} \partial_{\mu} \psi - mc^2 \bar \psi \psi] - [\frac 1 {16\pi} F^{\mu \nu}F_{\mu \nu}] - (q\bar \psi \gamma^{\mu} \psi)A_{\mu}$$ This is invariant under the joint...
  3. F

    A Relation between Dirac's equation density matrix and current with spin

    After computind dirac 1D equation time dependant for a free particle particle I get 2 matrixs. From both,them I extract: 1) the probablity matrix P =ps1 * ps1 + psi2 *psi2 2) the current matrix J = np.conj(psi1)*psi2+np.conj(psi2)*psi1 I think that current is related to electricity, and...
  4. 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...
  5. abivz

    I Obtaining the Dirac function from field operator commutation

    Hi everyone, I'm new to PF and this is my second post, I'm taking a QFT course this semester and my teacher asked us to obtain: $$[\Phi(x,t), \dot{\Phi}(y,t) = iZ\delta^3(x-y)]$$ We're using the Otto Nachtman: Elementary Particle Physics but I've seen other books use this notation: $$[\Phi(x,t)...
  6. evinda

    MHB Interval with Dirac function in a finite interval

    Hello! (Wave) I want to calculate the integral $\int_{-1}^2\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt$. I have done the following so far: $$\int_{-\infty}^{+\infty}\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt=\int_{-\infty}^1\sin \left (\pi (t-1)\right )\delta (-t+1)\...
  7. 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)...
  8. G

    Integral Involving the Dirac Delta Function

  9. Ayoub Tamin

    A Dirac Propagator: Learn to Reach 8.2

    wanna know how to get to 8.2
  10. Z

    Writing a squared observable in Dirac notation

    Edited after post below: Hi, I need to show that the square of the expectation value of an observable takes a certain form in Dirac notation. I know in wave notation that the expectation value is a sandwich integral which looks like this: ##<A>=\int_{-\infty}^\infty \Psi^*(x) \hat A \Psi (x)...
  11. 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...
  12. 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...
  13. M

    Find the probability of a particle in the left half of an Infinite Square well

    Attempt: I'm sure I know how to do this the long way using the definition of stationary states(##\psi_n(x)=\sqrt{\frac {2} {a}} ~~ sin(\frac {n\pi x} {a})## and ##\int_0^{{a/2}} {\frac {2} {a}}(1/5)\left[~ \left(2sin(\frac {\pi x} {a})+i~ sin(\frac {3\pi x} {a})\right)\left( 2sin(\frac {\pi x}...
  14. giveortake

    Engineering Dirac Delta Function in an Ordinary Differential Equation

    1.) Laplace transform of differential equation, where L is the Laplace transform of y: s2L - sy(0) - y'(0) + 9L = -3e-πs/2 = s2L - s+ 9L = -3e-πs/2 2.) Solve for L L = (-3e-πs/2 + s) / (s2 + 9) 3.) Solve for y by performing the inverse Laplace on L Decompose L into 2 parts: L =...
  15. S

    Understand the Outer Product of two qubits

    Hi, I'm trying to understand an outer product |1>_a<1| where |1>_a is the ket for one qubit (a) and <1| is the bra for another qubit. Does this make sense and is it possible to express it in terms of tensor products or pauli matrices?
  16. RicardoMP

    A Trace of a product of Dirac Matrices in a Fermion loop

    I'm working out the quark loop diagram and I've drawn it as follows: where the greek letters are the Lorentz and Dirac indices for the gluon and quark respectively and the other letters are color indices. For this diagram I've written...
  17. electrogeek

    I Dirac notation and calculations

    Hello everyone, I'm stuck on the question which I have provided below to do with Dirac notation: In these questions |a>, |b> and |c> can be taken to form an orthonormal basis set Consider the state |ξ> = α(|a> − 2|b> + |c>). What value of α makes |ξ> a normalised state? I'm brand new to Dirac...
  18. 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)##...
  19. Arman777

    Solve $$\int_{∞}^{∞}dxf(x)\delta((x-x_1))$$: Dirac Delta Function

    If the question was $$ \int_{∞}^{∞}dxf(x)δ((x - x_1)) = ? $$ The answer would be ##f(x_1)## So the delta function has two roots, I searched the web and some books but I am not sure what approach should I use here. I guess there's sometihng happens when ##x_1 = -x_2##. So I am not sure what...
  20. S

    I Could fundamental laws change in Dirac's Large Numbers Hypothesis?

    Paul Dirac proposed a hypothesis called "Large Numbers Hypothesis" (https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis), where he basically stated that, if he was correct, laws of physics would change with time. But what about fundamental laws and constants? (Not only 'effective'...
  21. JorgeM

    I Is this Dirac delta function integral correct?

    I have to integrate this expression so I started to solve the delta part from the fact that when n=0 it results equals to 1. And the graph is continuous in segments I thought as the sumation of integers $$ \int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx $$ From the fact that actually $$ δ(sin(x))=...
  22. S

    Did Paul Dirac believe in multiple universes?

    Prominent physicist Paul Dirac proposed a hypothesis that indicated that constants and laws of physics would evolve with time into different constants and laws of nature (https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis) This hypothesis was used by Robert Dicke...
  23. Q

    A Explicit form of annihilation and creation operators for Dirac field

    I'm unclear on what exactly an annihilation or creation operator looks like in QFT. In QM these operators for the simple harmonic oscillator had an explicit form in terms of $$ \hat{a}^\dagger = \frac{1}{\sqrt{2}}\left(- \frac{\mathrm{d}}{\mathrm{d}q} + q \right),\;\;\;\hat{a} =...
  24. K

    I Time ordering for Dirac spinors

    Hello! The time ordered product for Dirac spinors is defined as: $$<0|\psi(x)\bar{\psi}(y)|0>-<0|\bar{\psi}(y)\psi(x)|0>$$ Can someone explain to me how should I think of the dimensionality of this. For a Dirac spinor, ##\psi(x)## is a 4 dimensional column vector, so the first term in that...
  25. D

    Quantum Should I Get Both of Dirac's Quantum Mechanics Books?

    Hello, I remembered once hearing of a must-have quantum mechanics book by Paul Dirac. I don't remember if it was his Principles of QM or Lectures on QM. Based on the table of contents, I believe it was the Principles of QM book; however, looking at both I was thinking about getting his Lectures...
  26. SisypheanZealot

    Dirac Delta using periodic functions

    I know it is something simple that I am missing, but for the life of me I am stuck. So, I used the identity ##sin(a)sin(b)+cos(a)cos(b)=cos(a-b)## which gives me $$\int^{\infty}_{-\infty}dx\:f(x)\delta(x-y)=\int^{\infty}_{-\infty}dx\:f(x)\frac{1}{2L}\sum^{\infty}_{n=-\infty}\lbrace...
  27. peguerosdc

    I Confusion with Dirac notation in the eigenvalue problem

    Hi! I am studying Shankar's "Principles of QM" and the first chapter is all about linear algebra with Dirac's notation and I have reached the section "The Characteristic Equation and the Solution to the Eigenvalue Problem" which says that starting from the eigenvalue problem and equation 1.8.3...
  28. 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...
  29. W

    Equivalent representations for Dirac algebra

    One thing I was thinking about doing was to consider these representations as a basis for the gamma matrices vector space, then try to determine what the change of basis from one to the other would be. However I'm unsure if it's correct to treat the representations as a basis, or whether this is...
  30. 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...
  31. 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}...
  32. J

    I Advanced Dirac Notation Question

    Hello everyone, I have been working through some research papers on a topic that really interests me, but I believe I am misunderstanding a few things about Dirac Notation. I have expressions that read: \begin{align*} &< \psi_n \mid g(H - E_{n+1}) \mid \psi_n> \text{,} \\ &< \psi_n \mid (H -...
  33. 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...
  34. mishima

    I Dirac Delta, higher derivatives with test function

    Hi, I am curious about: $$x^m \delta^{(n)}(x) = (-1)^m \frac {n!} {(n-m)!} \delta^{(n-m)}(x) , m \leq n $$ I understand the case where m=n and m>n but not this. Just testing the left hand side with m=3 and n=4 and integrating by parts multiple times, I get -6. With the same values, the...
  35. 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...
  36. amjad-sh

    Dirac delta function of a function of several variables

    Form solid state physics, we know that the volume of k-space per allowed k-value is ##\triangle{\mathbf{k}}=\dfrac{8\pi^3}{V}## ##\sum_{\mathbf{k}}F(\mathbf{k})=\dfrac{V}{(2\pi)^3}\sum_{\mathbf{k}}F(\mathbf{k})\triangle{\mathbf{k}}##...
  37. Clara Chung

    I Question about divergence theorem and delta dirac function

    How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)
  38. topsquark

    MHB Graded Algebra: Understanding Color Dirac Spinors in Space-Time

    I just read through a paper on a \mathbb{Z} _ 3 graded Algebra. In this instance we are talking about color Dirac spinors in space-time. It looks like the author is talking about \left ( SU(3) \otimes L^4 \otimes \mathbb{Z}_2 \otimes \mathbb{Z} _2 \right ) \otimes \mathbb{Z} _3. ( SU(3) is...
  39. 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) =...
  40. W

    I Is quantum mechanics formulated from 1st principles?

    I was surprised recently to learn that one of the reasons both Newton and Einstein were so revolutionary was that they performed a neat mathematical trick. For Newton, it was the mathematical derivation of Kepler's laws from Newton's laws of gravitation and motion. For Einstein, it was the...
  41. Dewgale

    Anti-commutation of Dirac Spinor and Gamma-5

    Homework Statement Given an interaction Lagrangian $$ \mathcal{L}_{int} = \lambda \phi \bar{\psi} \gamma^5 \psi,$$ where ##\psi## are Dirac spinors, and ##\phi## is a bosonic pseudoscalar, I've been asked to find the second order scattering amplitude for ##\psi\psi \to \psi\psi## scattering...
  42. P

    I Confusion about Dirac notation

    Using that ##\hat{a} =a = \sqrt{\frac{mw}{2 \hbar}} \hat{x} +\frac{i}{\sqrt{2mw \hbar}} \hat{p}## and ## a \dagger = \sqrt{\frac{mw}{2 \hbar}} \hat{x} -\frac{i}{\sqrt{2mw \hbar}} \hat{p}## We can solve for x in term of the lowering and raising operator. Now, recently I read a derivation of...
  43. George Keeling

    Covariant coordinates don't co-vary

    Homework Statement I am studying co- and contra- variant vectors and I found the video at youtube.com/watch?v=8vBfTyBPu-4 very useful. It discusses the slanted coordinate system above where the X, Y axes are at an angle of α. One can get the components of v either by dropping perpendiculars...
  44. Another

    I How Are the Kronecker Delta and Dirac Delta Related?

    I want to know if these functions are related? for example. I can write Dirac delta in term Delta Kronecker from? Where can I learn these?
  45. D

    A Lorentz invariance from Dirac spinor

    I have a really naive question that I didn't manage to explain to myself. If I consider SUSY theory without R-parity conservation there exist an operator that mediates proton decay. This operator is $$u^c d^c \tilde d^c $$ where ##\tilde d## is the scalar superpartner of down quark. Now...
  46. Rabindranath

    A Lagrange multipliers on Banach spaces (in Dirac notation)

    I want to prove Cauchy–Schwarz' inequality, in Dirac notation, ##\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2##, using the Lagrange multiplier method, minimizing ##\left|\left<\psi\middle|\phi\right>\right|^2## subject to the...
  47. A

    A Measuring the spin of a moving Dirac spinor particle

    Hello, I would like to ask about the process of measuring the Spin of a Dirac 4-spinor Ψ that is not in the rest frame. Note that even though there is plenty of information about what a Dirac spinor is, what reasoning lead to its discovery and how it can be expressed in terms of particle and...
  48. 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...
  49. enter

    Why did Dirac strongly pursue mathematical beauty?

    In everyday language, beauty is an emotional concept. How would you mix that with quantum physics and the mathematics behind it? Or is what he refers to as "beauty" is more like simplicity? I mean, I agree with the man, the Standard Model feels redundantly complex, but I feel like there is...
  50. A

    I Liouville equation with Dirac delta as probability density

    I would like to know the solution to Liouville equation ∂ρ/∂t=-{ρ,H} given the initial condition ρ(t=0)=δ(q,p) where δ(q,p) is a dirac delta centered in some point (q,p) in phase space. I have the feeling, but I'm not sure, that the solution is of the form ρ(t)=δ(q(t),p(t)) where q(t) and...
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