What is Quantum field theory: Definition and 571 Discussions
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles.
QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory in quantum mechanics.
I can understand how ##\phi (x)|0\rangle## represents the wavefunction of a single boson localised near ##x##.I don't understand how the same logic appies to ##A^{\mu}(x)|0\rangle## and ##\psi |0\rangle##. Both of these operators return a four component wavefunction when operated on the vaccuum...
In https://arxiv.org/pdf/1709.07852.pdf, it is claimed in equation (1) and (2) that when we take non-relativistic limit, the following Lagrangian (the interaction part)
$$L=g \partial_{\mu} a \bar{\psi} \gamma^{\mu}\gamma^5\psi$$
will yield the following Hamiltonian
$$H=-g\vec{\nabla} a \cdot...
David Wallace, The sky is blue, and other reasons quantum mechanics is not underdetermined by evidence, Manuscript (2022). arXiv:2205.00568.
From the Abstract:
''I argue that there as yet no empirically successful generalization of''
[Bohmian Mechanics and dynamical-collapse theories like the...
Hello, i need help with the S-matrix. From what i understand, with the S-matrix i would be able to compute the scattering amplitude of some processes, is that correct? If so, how would i be able to do that if i have some field ##\phi(x,t)## in hands? Is that possible?
In quantum field theory, we have the following expansion on a scalar field (I follow the convention of Schwarz's book)
$$\phi(\vec{x},t)=\int d^3 p \frac{a_p exp(-ip_\mu x^\mu)+a_p^{\dagger}exp(ip_\mu x^\mu)}{(2\pi)^3 \sqrt{2\omega_p}} \quad p^{\mu}=(\omega_p,\vec{p})$$
With commutation relation...
Hi there,
In his book "Quantum field theory and the standard model", Schwartz assumes that the canonical commutation relations for a free scalar field also apply to interacting fields (page 79, section 7.1). As a justification he states:
I do not understand this explanation. Can you please...
I came across this upcoming book -- https://press.princeton.edu/books/hardcover/9780691174297/quantum-field-theory-as-simply-as-possible -- peer reviewed as it is published by Princeton University Press, which is due to be published in October. I've already ordered a copy coming from the UK. It...
I have derived the Coulombian potential as an effective potential between two spinless charged particle taking the non-relativitic approach on the scattering amplitude obtained in terms of the Feynman rules in SQED.
The scattering amplitudes are:
I'm using the gauge in which xi = 1.
How could...
I am looking to learn about these topological effects or phases in solids. More specifically, I'm trying to find a set of lecture notes or a textbook or some other text that do not shy away from discussing homotopy classes and the application algebraic topology to describe these materials.
I...
Hi, I'd like to clarify the following terminology
(Fradkin, Quantum Field Theory an integrated approach)
"carry the quantum numbers of the representation of the gauge group":
Does the author basically mean that the wilson loop is a charged operator, in a sense that it transforms non-trivially...
How do we map experimental measurements of quantum fields, such as those seen in accelerators, to the theory's mathematical formalism? When we see images of particle tracks produced in accelerators such as the LHC, I think it's safe to say a measurement (or series of measurements) has been...
Hi there, I've been stuck on this issue for two days. I'm hoping someone knowledgeable can explain.
I'm working through the construction of the quantum path integral for the free electrodynamic theory. I've been following a text by Fujikawa ("Path Integrals and Quantum Anomalies") and also...
Are there any QFT books that use little to no math? If there is a little math that is okay. I don't know much about math. I am looking for good explanations on how it works without math. Any help would be great!
Edward G. Timoshenko
PhD, MSc, EurPhys, CPhys MInstP, CChem MRSC
Web site: https://www.EdTim.live
Bio:
2011- Researcher, TEdQz Research after an early retirement from UCD
2005 - 2011 Senior Lecturer in Physical Chemistry, School of Chemistry and Chemical Biology, UCD
1997 College Lecturer...
A lot of people say that Quantum Field theory (QFT) an Quantum Mechanics (QM) are equivalent. Yet, I've found others who dispute these claims. Among the counter-arguments (which I admittedly do not have the expertise to pick apart and check their validity in full) are the following:
1) While QFT...
-1st: Could someone give me some insight on what a ket-state refers to when dealing with a field? To my understand it tells us the probability amplitude of having each excitation at any spacetime point, but I don't know if this is accurate. Also, we solve the free field equation not for this...
In This wikipedia article is said:
"If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator, or more accurately, the ground state of a...
Also, I have heard from physicists that vacuum energy fluctuation (creation and destruction of virtual particles) violates energy conservation. The reason, they justify, is based on uncertainty principle (energy-time form of uncertainty principle), energy can exist and disappear for a very short...
I tried as first step to find Z_q the renormalization parameter, to do so I did the same procedure to find the renormalization parameter of the gauge field of the gluon A^a_\mu when a is representation index a \in {1,2,...,N^2-1} such that A^{a{(R)}}_{\mu}=\frac{1}{\sqrt{Z_A}}A^{a}_{\mu}...
One sentence summarization
For a student initially working on a more phenomenological side of the high energy physics study, what is the recommendation of introductory reading materials for them to dive into a more mathematically rigorous study of the quantum field theory.
Elaboration...
I tried to do it for 2+1 D (3+1 is done in the text, by writing the integral in spherical coordinates and computing it directly). In 2+1 D I wrote it as:
E = - \int \frac{d^2 k}{ (2\pi)^2 } \frac{e^{kr cos\theta}}{k^2 + m^2}
= - \int_0^{\infty} \int_0^{2\pi} \frac{d k d\theta}{ (2\pi)^2 }...
For ##N=1##, I have managed to prove this, but for ##N>1##, I am struggling with how to show this. Something that I managed to prove is that
$$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$
which generalizes what I initially...
I tried to do a Euler Lagrange equation to our Lagrangian:
$$\frac{S_\text{eff}}{T}=\int d^6x\left[(\nabla \phi)^2+(\nabla \sigma)^2+\lambda\sigma (\nabla \phi)^2\right]+\frac{S_{p.p}}{T}$$
and then I would like to solve the equation using perturbation theory when ##Q## or somehow...
I'm trying to the following exercise:
I've proven the first part and now I'm trying to do the same thing for fermions.
The formulas for the mode expansions are:
What I did was the following:
$$\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)...
hello :)
i would very much like study some quantum field theorie, but have not previously study any regular quantum mechanic (i am not so interest in regular quantum mechanic, but more the relativistic theories).
so i ask, this is possible or not? to what extent knowledge of regular quantum...
Assume that I have the Lagrangian
$$\mathcal{L}_{UV}
=\frac{1}{2}\left[\left(\partial_{\mu} \phi\right)^{2}-m_{L}^{2} \phi^{2}+\left(\partial_{\mu} H\right)^{2}-M^{2} H^{2}\right]
-\frac{\lambda_{0}}{4 !} \phi^{4}-\frac{\lambda_{2}}{4} \phi^{2} H^{2},$$
where ##\phi## is a light scalar field...
My attempt at this:
From the general result
$$\int \frac{d^Dl}{(2\pi)^D} \frac{1}{(l^2+m^2)^n} = \frac{im^{D-2n}}{(4\pi)^{D/2}} \frac{\Gamma(n-D/2)}{\Gamma(n)},$$
we get by setting ##D=4##, ##n=1##, ##m^2=-\sigma^2##
$$-\frac{\lambda^4}{M^4}U_S \int\frac{d^4k}{(2\pi)^4} \frac{1}{k^2-\sigma^2} =...
This seems rather straight forward, but I can't figure out the details... Generally speaking and ignoring prefactors, the Fourier transformation of a (nicely behaved) function ##f## is given by
$$f(x)= \int_{\mathbb{R}^{d+1}} d^{d+1}p\, \hat{f}(p) e^{ip\cdot x} \quad\Longleftrightarrow \quad...
Before I start, let me say that I have looked into textbooks and I know this is a standard problem, but I just can't get the result right...
My attempt goes as follows:
We notice that the amplitude of this diagram is given by $$\begin{align*}K_2(p) &= \frac{i(-i...
I'm reading the book QFT by Ryder, in the section where ##\rm{SU(2)}## is discussed.
First, he considered the group of ##2 \times 2## unitary matrices ##U## with unit determinant such that it has the form,
$$U =\begin{bmatrix}
a & b \\
-b^* & a^*
\end{bmatrix}, \qquad \xi =
\begin{bmatrix}...
It is a wonderful book for learning QFT. Interesting problems with detailed solutions. I have tried the problems from chapter 1 to chapter 7. In most chapters, I could at least solve some part of the problems. But I got stuck in chapter 4, the Dirac equation. I could not solve any of the...
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}))$$...
All physical laws have to be Lorentz invariant according to a lecture I just watched. Why is general covariance (which is more general than Lorentz invariance) not a requirement for all laws of physics? Are there any quantum gravity theories that take the approach of adding general covariance to...
From my understanding the Seiberg-Witten map is a way to convert a non-commutative field theory into a commutative field theory. For example for the commutative relation between positions [x,y]=i*n, the common SW map I see in the literature for non commutative quantum mechanics is
x->...
Hi,
I have been studying Quantum Field Theory this semester! It seems that Shwartz and Peskin are the most popular choices when it comes to studying QFT. But apparently my professor have another "old" preference.
He strongly suggested that we learn QFT from Zuber's book. I have looked at the...
Suppose we have a Hamiltonian containing a term of the form
where ∂=d/dr and A(r) is a real function. I would like to study this with harmonic oscillator ladder operators. The naïve approach is to use
where I have set ħ=1 so that
This term is Hermitian because r and p both are.*...
In many QFT textbooks, we usually see the calculations of vertex function, vacuum polarization and electron self-energy.
For example, one calculates the vacuum polarization to correct photon propagator $\langle{\Omega}|T\{A_{\mu}A_{\nu}\}|\Omega\rangle$, where $|\Omega\rangle$ is the ground...
I derive the quadratic form of Dirac equation as follows
$$\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...