Quantum field theory Definition and 587 Threads

I was reading about the renormalization of ##\phi^4## theory and it was mentioned that in order to renormalize the 2-point function ##\Gamma^{(2)}(p)## we add the counterterm : \delta \mathcal{L}_1 = -\dfrac{gm^2}{32\pi \epsilon^2}\phi^2 to the Lagrangian, which should give rise to a...

I usually don't read papers on philosophy of quantum field theory, but this one is really good: http://philsci-archive.pitt.edu/8890/ In particular, the prelude which I quote here is a true gem: "Once upon a time there was a community of physicists. This community be- lieved, and had good...

Hello, I'm trying to figure out where the term (3) came from. This is from a textbook which doesn't explain how they do it. ∂_μ(∂L/(∂(∂_μA_ν)) = ∂L/∂A_ν (1) L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν (2) Here is Eq (1) the...

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 am still rather new to renormalising QFT, still using the cut-off scheme with counterterms, and have only looked at the ##\varphi^4## model to one loop order (in 4D). In that case, I can renormalise with a counterterm to the one-loop four-point 1PI diagram at a certain energy scale. I can...

I already know this quantity diverges, however I was wondering where to go from there. Any resource would be appreciated. Thank you. Useful Information: $$\phi=\int\frac{d^3k}{2\omega_k (2\pi)^3}(\hat{a}(\overrightarrow{k})e^{-ikx}+\hat{a}(\overrightarrow{k})^{\dagger}e^{ikx}))$$...

All physical laws have to be Lorentz invariant according to a lecture I just watched. Why is general covariance (which is more general than Lorentz invariance) not a requirement for all laws of physics? Are there any quantum gravity theories that take the approach of adding general covariance to...

From my understanding the Seiberg-Witten map is a way to convert a non-commutative field theory into a commutative field theory. For example for the commutative relation between positions [x,y]=i*n, the common SW map I see in the literature for non commutative quantum mechanics is x->...

Hi, I have been studying Quantum Field Theory this semester! It seems that Shwartz and Peskin are the most popular choices when it comes to studying QFT. But apparently my professor have another "old" preference. He strongly suggested that we learn QFT from Zuber's book. I have looked at the...

Suppose we have a Hamiltonian containing a term of the form where ∂=d/dr and A(r) is a real function. I would like to study this with harmonic oscillator ladder operators. The naïve approach is to use where I have set ħ=1 so that This term is Hermitian because r and p both are.*...

Summary:: What are the prerequisites to study quantum field theory? What are the prerequisites to study quantum field theory?

In many QFT textbooks, we usually see the calculations of vertex function, vacuum polarization and electron self-energy. For example, one calculates the vacuum polarization to correct photon propagator $\langle{\Omega}|T\{A_{\mu}A_{\nu}\}|\Omega\rangle$, where $|\Omega\rangle$ is the ground...

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...

In the following I will try to deduce the scattering amplitude for a specific interaction. My question is at the bottom, the entire rest is my reasoning to explain how I came to the results I present. My working Let's assume I would like to calculate the second order scattering amplitude in ##...

I understand how do 3 no. equation come from 1 & 2 no. equation. But I am struggling to understand how do 4 no. equation come from 3 no. equation. Will anyone do the steps between 3 no. equation and 4 no. equation, please ?

I'm currently working my way through Griffith's Elementary Particles text, and I'm looking to understand what's going on with the underlying Hilbert space of a system described using a Feynman diagram. I'm fairly well acquainted with non relativistic QM, but not much with QFT. In particular, I'd...

Hi all, - an initial apology - there are a large number of threads on virtual particles on the site, and I apologize for adding another one. I had two questions - on a related note, the guidance provided by @A. Neumaier's FAW on virtual particles has been highly valuable for a novice . 1) Upon...

Silly question but could someone explain why a real, 'observable' particle is said to be 'off-shell' in an external field? @A. Neumaier 's excellent FAQ notes that the mass shell constraints ceases to have meaning in this case. I'm just not fully clear on why (probably obvious) given that energy...

I'm reading peskin QFT textbook. In page 196 eq. (6.58) it says $$F_2(q^2=0)=\frac{\alpha}{2\pi}\int ^1_0 dx dy dz \delta (x+y+z-1) \frac{2m^2z(1-z)}{m^2(1-z)^2}\\=\frac{\alpha}{\pi}\int ^1_0 dz\int ^{1-z}_0 dy \frac{z}{1-z}=\frac{\alpha}{2\pi}$$ I confirmed the conversion from the first line...

Hi! In QFT we are usually interested in actions that are hermitian. Say we are looking at scattering of Dirac fermions with a real coupling constant g, whose Lagrangian is given by: $$L= \bar{\psi}(i \gamma_{\mu} \partial^{\mu} -m) \psi - \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi -...

I want to know what are the QFT topics that I need to understand in order to proceed in reading papers on entanglement entropy such as, Entanglement Entropy and Quantum Field Theory Entanglement entropy in free quantum field theory Entanglement entropy: holography and renormalization group An...

Hello, I have been following Tong's notes on QFT and have found them to be a great introduction. I am almost at the end and am trying to figure out how to proceed. I have seen recommendations on David Skinner's notes, but I think I want to use a textbook either with Skinner's notes or maybe...

Does this mean that the expression for the above vertex is $$ -\frac{g}{2}\epsilon^{abx}\epsilon^{cdx}\int d\tau \langle A_{a} (\tau) A_{c} (\tau)\rangle \langle Y^{i}_{b} (\tau)Y^{i}_{d}(\tau) \rangle $$

This is the paper that I refer to. I'm trying to figure out the ghost action (Equation 2.16) in the background field gauge. I am attempting to use Srednicki's (chapter 78) expression for the ghost field in the background gauge. However, I am missing out on a √g coefficient in front of the term...

Trying to better understand quantum field theory, I've read that particles are created when it becomes an exitation of its quantum field. Would it then be right to think of a particle as the manifested kinetic energy of its field?

I've been reading about Quantum Field Theory and what it says about subatomic particles. I've read that QFT regards particles as excited states of underlying quantum fields. If this is the case, how can particles be regarded as objective? It seems to me that this also removes some of the...

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...

I don't quite understand how he got the line below. By using discrete time approximation, we can get the second order time expression. But i don't see how by combining terms he is able to get such expression.

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...

This is one problem from Robin Ticciati's Quantum Field Theory for Mathematicians essentially asking us to find Cartan subalgebras for the matrix algebras ##\mathfrak{u}(n), \mathfrak{su}(n),\mathfrak{so}(n)## and ##\mathfrak{so}(1,3)##. The only thing he gives is the definition of a Cartan...

While writing out the Dyson series due to the time ordering above I encountered the two expressions $$T(\mathcal{L}_{int}(x))\quad \text{and}\quad T(\mathcal{L}_{int}(x)\mathcal{L}_{int}(y))$$ I was able to write out the first term in terms of contractions using Wick's theorem and then finally...

I'm currently working out quantities that include the vector and axialvector currents ##j^\mu_B(x)=\bar{\psi}(x)\Gamma^\mu_{B,0}\psi(x)## where B stands for V (vector) or A (axialvector). The gamma in the middle is a product of gamma matrices and the psi's are dirac spinors. Therefore on the...

I'm having a hard time understanding how exactly to evaluate the expression} $$\partial_t \mathcal{T}\exp\left(-i S(t)\right)\quad \text{where}\quad S(t)\equiv\int_{t_0}^tdu \,H(u) .$$ The confusing part for me is that if we can consider the following: $$\partial_t \mathcal{T}\exp\left(-i...

Starting from the general formula: $$I_{n,m}=\frac{1}{(4\pi)^2}\frac{\Gamma(m+2-\frac{\epsilon}{2})}{\Gamma(2-\frac{\epsilon}{2})\Gamma(n)}\frac{1}{\Delta^{n-m-2}}(\frac{4\pi M^2}{\Delta})^{\frac{\epsilon}{2}}\Gamma(n-m-2+\frac{\epsilon}{2})$$ I arrived to the following...

I want to calculate transition amplitudes in QCD for processes like ##q(k)q^\prime(p)\rightarrow q(k^\prime)q^\prime(p^\prime)##, where ##q,q^\prime## are quarks. However, I am unsure what to do with the colour indices of the quark spinors upon squaring the matrix element. For the sake of...

This is my first time dealing with scaling symmetry, so I'm sorry if the following is fundamental wrong. My approach was the same as if I was trying to show the same for translation or Lorentz symmetry. We have $$\delta\phi(x)= \phi'(x')-\phi(x)=...

My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from ##\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle##, which in the end is a transition amplitude and can be interpreted as...

Consider, for example, the gluon propagator $$D^{\mu\nu}(q)=-\frac{i}{q^2+i\epsilon}[D(q^2)T^{\mu\nu}_q+\xi L^{\mu\nu}_q]$$ What exactly is the renormalized gluon dressing function ##D(q^2)## and what is its definition? My interest is in knowing if I can then write the bare version of this...

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...

Quantum gravity theories and GUTs are nonrenormalizable theories, but does this actually mean that these theories must be flawed, or does it mean that renormalization must be a flawed concept, or is this not actually a problem? If it is impossible to produce a renormalizable quantum gravity...

So I’m trying to compute the probability amplitude of an electron with momentum p1 and a positron with momentum p2 annihilating into a photons with momenta q1 and q2. My question is how do you use Feynman diagrams to calculate the first and second order expansions (seen in the third image)? I...

Hello everyone, I am stuck in the derivation of the three gauge-boson-vertex in Yang-Mills theories. The relevant interaction term in the Lagrangian is$$\mathcal{L}_{YM} \supset g \,f^{ijk}A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \partial^{\mu} A^{\nu}{}^{(i)} $$ I have rewritten this term using...

Hello everyone, I am stuck in deriving the three gauge-boson-vertex in Yang-Mills theories. The relevant interaction term in the Lagrangian is $$\mathcal{L}_{YM} \supset g \,f^{ijk}A_{\mu}{}^{(j)} A_{\nu}{}^{(k)} \partial^{\mu} A^{\nu}{}^{(i)} $$ I have rewritten this term using the...

For the diagram In scalar field theory, I have obtained an integral which looks like $$\int_{0}^{\Lambda} \frac{d^4 q}{(2\pi)^4} \frac{i}{q^2 - m^2 + i\varepsilon} \frac{i}{(p - q)^2 - m^2 + i\varepsilon}$$ I am required to calculate this and obtain the divergent amplitude $$i\mathcal{M} =...

I'm looking for a book that describes the quantum field theory without going deeply in the theory with formulas or complex description of the mathematics under the theory. I know that this theory is really complex and it needs a deep knowledge of quantum physics in order to be understood. But...

In quantum field theory, a dressed particle is a particle ("bare particle") considered in combination with certain secondary effects that it produces (e.g. the virtual pair creation involved in vacuum polarization). The dressed states are regarded as more physical, hence closer to reality. Axel...

Summary: In the past, physicists talked of the phenomenon of "wave function collapse" very freely, whereas now there seems to be some reservation about it. Why? In reading older popular physics literature, physicists used to talk about "wave function collapse" freely and more often...

Let's for example consider the Z boson. It can't directly be detected; so is it ever really correct to draw it as an external line on a Feynman diagram? I've seen processes involving it before be written as something -> Z + something, then Z -> ... but since unstable particles aren't really on...

Is there anyone on here who could help me fill in my gaps in quantum field theory up to renormalization? I know how to canonically quantize a theory and how to use scalars (spin 0), vectors (spin 1) and spinors (spin 1/2) but lack more advanced knowledge like renormalization which I could...

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...