Dirac Definition and 859 Threads

If we have the wavefunction ##|ab \rangle##, what do the a and b stand for? In particular, do a and b signify an outcome of some pending or possible measurement, or do they signify some aspect of the wavefunction, and if so, which aspect?

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...

I've been studying Tong's beautiful chapter (pages 106-109; See also Peskin and Schroeder pages 52-58), together with his great lectures at Perimeter Institute, on how to quantize the following Dirac Lagrangian in the wrong way $$\mathscr{L}=\bar{\psi}(x)(i\not{\!\partial}-m)\psi(x) \tag{5.1}$$...

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...

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...

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...

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)...

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)\...

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)...

wanna know how to get to 8.2

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)...

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...

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...

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}...

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 =...

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?

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...

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...

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)##...

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...

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'...

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))=...

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...

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} =...

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...

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...

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...

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...

[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...

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...

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...

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}...

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 -...

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...

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...

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...

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}}##...

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)

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...

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) =...

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...

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...

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...

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...

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?

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...

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...

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...

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...