In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s. Major advances in the theory were made in the 1940s and 1950s, and led to the introduction of renormalized quantum electrodynamics (QED). QED was so successful and accurately predictive that efforts were made to apply the same basic concepts for the other forces of nature. By the late 1970s, these efforts successfully utilized gauge theory in the strong nuclear force and weak nuclear force, producing the modern standard model of particle physics.
Efforts to describe gravity using the same techniques have, to date, failed. The study of quantum field theory is still flourishing, as are applications of its methods to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to several different branches of physics.
I want to compute the equations of motion for this theory in terms of the functions ##f## and ##a##. My plan was to apply the Euler-Lagrange equations, but it got confusing very quickly.
Am I right that we'll have 3 sets of equations? One for each of the fields ##\phi,\phi^\dagger, A_\mu## ...
The first step seems easy: computation of the $\theta$ and $\overline{\theta}$ integrals give
$$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \det(\partial_j w_i(x)) \exp{\left(-\frac{1}{2}w_i(x)w_i(x)\right)}.$$
From here, I tried using that $$\det(\partial_j w_i (x)) = \det\left(\partial_j w_i...
My understanding is:
$$\phi (\mathbf{k})=\int{d^3}\mathbf{x}\phi (\mathbf{x})e^{-i\mathbf{k}\cdot \mathbf{x}}$$
But what is ##\phi (\mathbf{x})## in Qft?
In quantum mechanics,
$$|\phi \rangle =\int{d^3}\mathbf{x}\phi (\mathbf{x})\left| \mathbf{x} \right> =\int{d^3}\mathbf{k}\phi...
In Peskin P85:
It says the Time-ordered exponential is just a notation，in my understanding, it means
$$\begin{aligned}
&T\left\{ \exp \left[ -i\int_{t_0}^t{d}t^{\prime}H_I\left( t^{\prime} \right) \right] \right\}\\
&\ne T\left\{ 1+(-i)\int_{t_0}^t{d}t_1H_I\left( t_1 \right)...
I have this Lagrangian for a free massless left Weyl spinor, so it’s just the kinetic term, that can be written embedding the field into a larger Dirac spinor and then taking the left projector in this way:
$$i \bar{\psi} \cancel{\partial} P_L \psi$$
Srednicki says that the momentum space...
Hey all,
I am currently having trouble understanding equation (12.52) in Peskin's QFT book. Specifically the term for external leg corrections, in which they tack on an additional prefactor of ##(-ig)##. Normally with external leg prefactors, we don't see the coupling constant multiplied onto...
I'm trying to apply an operator to a massless and minimally coupled squeezed state. I have defined my state as $$\phi=\sum_k\left(a_kf_k+a^\dagger_kf^*_k\right)$$, where the ak operators are ladder operators and fk is the mode function $$f_k=\frac{1}{\sqrt{2L^3\omega}}e^{ik_\mu x^\mu}$$...
What is the big picture of QFT?
I have studied quantum mechanics from:
-Griffiths
-the first few chapters of Sakurai
-Ballentine
I have studied electrodynamics from Griffiths and General Relativity from Carroll
I have assigned level I to the question, but any answer is welcome
I am sorry but I can't seem to find the actual estimated value of the cosmological constant that is predicted by quantum field theory. Can anyone help me and tell me the approximation of that value and/or the value of the approximate observed cosmological constant that physicists today think...
I noticed that ##V(\phi)## has nonzero minima, therefore I found the stationary points as ##{{\partial{V}}\over{\partial\phi}}=0##, and found the solutions:
$$\phi^0_{1,2}=-{{m}\over{\sqrt{\lambda}}}\quad \phi^0_3={{2m}\over{\sqrt{\lambda}}}$$
of these, only ##\phi^0_3## is a stable minimum...
Reading the introduction to path integrals given in the latest edition of Zee's "Quantum field theory in a nutshell", I have found a remark which I don't really understand. The author is evaluating the free particle propagator ##K(q_f, t; q_i, 0)##
$$\langle q_f\lvert e^{-iHt}\lvert q_i...
I have a question related to the uncertainty principle in QFT and if it is related to the early universe conditions.
Do we still have four-vector momentum and position uncertainty relation in relativistic quantum theory?
I have been following the argument related to the early universe and the...
Conditions
Cosmolgy
Early universe
Principle
Qft
Quantum field theory
Space and time
Uncertainity principle
Uncertainty
Uncertainty principle
Universe
Vacuum
Take the Klein-Gordon equation:
##\Box^2 = m^2##
Say we want to linearize this equation, we try to come up with a new operator that squares into ##\Box^2##.
##(A\partial_t - B\partial_x - C\partial_y - D\partial_z)^2 = \Box^2##
So we need ##-A^2=B^2=C^2=D^2=I## as this gives back the 2nd...
Hello,
I was wondering what math/ math courses I should study or take before QFT. I've taken courses in linear algebra (one course), diff. equations, partial diff. equations.
Thanks!
I'm just starting to get into QFT as some self study. I've watched some lectures and videos, read some notes, and am trying to piece some things together.
Take ##U(1)_{EM}: L = \bar{\psi}[i\gamma^{\mu}(\partial_{\mu} - ieA_{\mu}) - m]\psi - 1/4 F_{\mu\nu}F^{\mu\nu}##
This allegedly governs spin...
What I have done is the following:
\begin{equation}
\braket{\eta_k | \eta_k}=|N|^2\sum_{n=0}^{\infty}\dfrac{1}{n!}\bra{0}(A^{\dagger})^nA^n\ket{0}=|N|^2\sum_{n=0}^{\infty}\dfrac{1}{n!}\int...
I've learned that in canonical quantization you take a Lagrangian, transform to a Hamiltonian and then "put the hat on" the fields (make them an operator). Then you can derive the equations of motion of the Hamiltonian.
What is the reason that you cannot already put hats in the QFT Lagrangian...
I was a denizen of this forum some 15 years ago during undergrad. However I since joined the dark side working in software on ML, AI, and distributed data processing.
Every now and then I pick up a physics textbook to get into the weeds of a topic I would have missed due to skipping out on grad...
I found a copy of David McMahon's "Quantum Field Theory Demystified" and I'm already confused on page 4 where he says, " . . in order to be truly compatible with special relativity, we need to discard the notion that \phi and \psi in the Klein-Gordon and Dirac equations respectively describe...
Summary: Suppose that observer ##\mathcal{O}## sees a ##W## boson (spin-1 and ##m > 0##) with momentum ##\boldsymbol{p}## in the ##y##-direction and spin ##z##-component ##\sigma##. A second observer ##\mathcal{O'}## moves relative to the first with velocity ##\boldsymbol{v}## in the...
Zee, in his QFT in a nutshell, tells that beautiful story about a "wise guy" who, through his annoying questions to the professor, actually describes a fundamental principle of quantum mechanics, essential to Feynman's approach to quantum phenomena (pp. 9 in Zee's). Now, Zee appears to imply...
My attempt/questions:
I use ##T^{0i} = \dot{\phi}\partial^i \phi##, ##\dot{\phi} = \pi##, and antisymmetry of ##Q_i## to get:
##Q_i = 2\epsilon_{ijk}\int d^3x [x^j \partial^k \phi(\vec{x})] \pi(\vec{x})##.
I then plug in the expansions for ##\phi(\vec{x})## and ##\pi(\vec{x})## and multiply...
I'm using a Peskin & Schroeder's copy that looks like it has all typos corrected and I wonder if the following is an undetected typo:
On pp. 103 and on the RHS of the bra expression just after (4.68), shouldn't the ##\phi_f({\mathbf p}_f)## be complex conjugated?
We know that we need to go to 5th order in perturbation theory to match 10 decimals of g-2 for electron, theory vs. experiment. But let us not assume QED is pure and independent, but it's a lower energy limit of GSW (not Green-Schwartz-Witten from superstrings) electroweak theory. Has anyone...
What is entanglement in QM and QFT?
I understood that it only corresponds to the concept of linear combination of states with multiple particles. Seeing lectures on YB it seems to me that it is something much deeper than that. What did I miss? How is it treated in QFT?
I am studying NRQM from...
What is acceleration in QFT at the fundamental level?
What causes it?
Is it quantized?
Is there a connection between acceleration in QFT and the equivalence principle?
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?
Hello, recently I'm learning about correlation functions in the context of QFT. Correct me with I'm wrong but what i understand is that tha n-point correlation functions kinda of describe particles that are transitioning from a point in space-time to another by excitations on the field. So, what...
I'm not even sure whether it can be defined in QFT, but I got this from SE:
Which I don't understand. I'm not mathematically sophisticated enough for that.
Which textbook is recommended for a modern introduction to QFT?
What mathematical topics do I need to know to start studying QFT?
From which textbooks can I learn them?
I know calculus, linear algebra, mathematical methods of physics (the necessary topics for quantum mechanics).
I learned basic...
I need to use hermiticity and electromagnetic gauge invariance to determine the constraints on the constants. Through hermiticity, i found that the coefficients need to be real. However, I am not sure how gauge invariance would come into the picture to give further contraints. I think the...
Hi everyone,
In his book "Quantum field theory and the standard model", Schwartz derives the position-space Feynman rules starting from the Schwinger-Dyson formula (section 7.1.1). I have two questions about his derivation.
1) As a first step, he rewrites the correlation function as
$$...
I know that the Action has units Energy·time or Momentum·position. A second fact is that the derivative of the action with respect to time is Energy and similar with momentum-position, consistent with a units ie. dimensions check.
Is it a coincidence that both are Noether conserved...
In Lagrangian mechanics we learn about generalized forces. However, I haven't seen these explicitly mentioned in books on QFT. Can the Lagrangians of QED or QCD be expressed in terms of generalized forces or is there some connection there, in particular to the Nielsen form.
I had been doing some calculations involving propagators with both a quadratic and a linear power of loop momentum in the denominator. In the context of HQET and QCD with strategy of regions.
The texts which I am following sometimes tend to straightaway use Schwinger and I am just wondering if...
I was reading Diagrammatica by Veltman and he treats the photon field as a massive vector boson in which gauge invariance is disappeared and the propagator has a different expression than in massless photon. After some googling, I found that this is one way to formulate QED which has the...
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 all,
I'm interested in the interplay between condensed matter and high energy theory. I'm a bit more than half-way through peskin and schroeder (done with part II, RG and critical phenomena).
What I find out is that I'm still sorely lacking in ability to read any of the current research in...
There is a draft of Srednicki's QFT book available for free online (here: https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf). I have a book voucher and, since I couldn't find it in the library, was thinking of buying it so long as the content of the book was sufficiently better (i.e...
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...
In Coleman's QFT lectures, I'm confused by equation 7.57. To give the background, Coleman is trying to calculate the scattering matrix (S matrix) for a situation in which the Hamiltonian is given by
$$H=H_{0}+f\left(t,T,\Delta\right)H_{I}\left(t\right)$$
where ##H_{0}## is the free Hamiltonian...
While in QFT we remove infinite energy problem with renormalization procedure, asking e.g. "what is mean energy density in given distance from charged particle", electric filed alone would say $$\rho \propto |E|^2 \propto 1/r^4 $$
But such energy density would integrate to infinity due to...
I know that studying QFT requires understanding Lie Groups and infinitesimal generators as they correspond to symmetry transformations. I want to study or take a course (offered by my university) in QFT in the coming academic year and I have the option to take a abstract algebra course offered...
There are some articles from the 1980s where the authors discuss 1D quantum oscillators where ##V(x)## has higher than quadratic terms in it but an exact solution can still be found. One example is in this link:
https://iopscience.iop.org/article/10.1088/0305-4470/14/9/001
Has anyone tried to...
I tried to do it for 2+1 D (3+1 is done in the text, by writing the integral in spherical coordinates and computing it directly). In 2+1 D I wrote it as:
E = - \int \frac{d^2 k}{ (2\pi)^2 } \frac{e^{kr cos\theta}}{k^2 + m^2}
= - \int_0^{\infty} \int_0^{2\pi} \frac{d k d\theta}{ (2\pi)^2 }...
I have been studying scattering process in QFT, but i am stuck now because i can't understand how this integral was evaluated:
$$\int dp\space \frac{1}{\sqrt{p^2+c²}}\frac{1}{\sqrt{p^2+k²}}\space p² \space d\Omega \space \delta(E_{cm}-E_{1}-E_{2})$$$
Where Ecm = c + k, E1 is the factor in the...
While classical mechanics uses single action optimizing trajectory, QM can be formulated as Feynman ensemble of trajectories.
As in derivation of Brownian motion, mathematically it is convenient to use nonphysical: nowhere differentiable trajectories - should it be so?
Can this connection be...