What is Quantum field theory: Definition and 567 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.

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  1. gremory

    A S-Matrix in Quantum Field Theory

    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?
  2. Tan Tixuan

    I Classical field in quantum field theory?

    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...
  3. E

    I Commutation relations for an interacting scalar field

    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...
  4. StevieTNZ

    B "Quantum Field Theory, as Simply as Possible" upcoming publication

    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...
  5. S

    Relativistic correction to Coulomb Potential in SQED

    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...
  6. D

    Solid State Texts on Topological Effects/Phases in Materials

    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...
  7. P

    A Clarifying Fradkin's Terminology on Quantum Numbers of Gauge Groups

    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...
  8. J

    A Measurement in QFT: Mapping Fields to Theory's Math Formalism

    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...
  9. Wizard

    A Orthogonality of variations in Faddev-Popov method for path integral

    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...
  10. BadgerBadger92

    Non Mathematical Quantum Field Theory Books?

    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!
  11. E

    Who is Edward G. Timoshenko, PhD in theoretical physics?

    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...
  12. J

    A Quantum Field theory vs. many-body Quantum Mechanics

    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...
  13. J

    A Concept of wavefunction and particle within Quantum Field Theory

    -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...
  14. T

    A QFT with vanishing vacuum expectation value and perturbation theory

    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...
  15. Ebi Rogha

    I Vacuum energy and Energy conservation

    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...
  16. DaniV

    The 1-loop anomalous dimension of massless quark field

    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}...
  17. Tianluo_Qi

    Quantum Reading list recommendation for HEP-ph to HEP-th/math-ph transition

    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...
  18. Mirod

    Zee QFT problem I.4.1: inverse square laws in (D+1)-dimensions

    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 }...
  19. J

    Proof involving exponential of anticommuting operators

    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...
  20. DaniV

    RG flow of quadrupole coupling in 6+1 dimension electrostatic problem

    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...
  21. snypehype46

    Exercise involving Dirac fields and Fermionic commutation relations

    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)...
  22. A

    Quantum Resources for learning Quantum Field Theory

    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...
  23. M

    I How to determine matching coefficient in Effective Field Theory?

    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...
  24. M

    Showing that this identity involving the Gamma function is true

    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} =...
  25. M

    How to translate expression into momentum-space correctly

    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...
  26. M

    Mass correction in ##\phi^4##-theory

    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...
  27. S

    I Matrix construction for spinors

    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}...
  28. steve1763

    A Exploring Free and Interaction Terms of L in Quantum Field Theory

    With free part L=-½(∂Φ)^2 -½m^2 Φ^2 and interaction term L=½gΦ^2Any help would be appreciated, thank you.
  29. Haorong Wu

    Quantum Anyone tried "Problem Book in Quantum Field Theory" by Radovanovic?

    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...
  30. Vanilla Gorilla

    I Would Black Holes Be Coherent Under Quantum Mechanics?

    I just want to know if Black Holes would be quantumly coherent.
  31. W

    I Renormalization of scalar field theory

    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...
  32. Demystifier

    A Philosophy of quantum field theory

    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...
  33. fabstr1

    Simplification of the Proca Lagrangian

    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...
  34. M

    I Condtion on transformation to solve the Dirac equation

    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...
  35. tomdodd4598

    I Renormalisation scale and running of the φ^3 coupling constant

    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...
  36. Diracobama2181

    A Exploring Solutions to $\phi$ and $\ket{\overrightarrow{P}}$

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

    A General Covariance in Quantum Field Theory

    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...
  38. Q

    A Is The Seiberg-Witten Map Unique?

    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->...
  39. Phylosopher

    Quantum Is Zuber's Quantum Field Theory textbook any good?

    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...
  40. S

    A Converting between field operators and harmonic oscillators

    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.*...
  41. M

    What are the prerequisites to study quantum field theory?

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

    A Confusions on QED renormalization

    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...
  43. 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...
  44. M

    I Calculating 2nd Order Scattering Amplitude: Feynman Diagrams

    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 ##...
  45. Adwit

    A Quantum Field Theory: 3-4 Equation Steps Explained

    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 ?
  46. J

    A Looking Under the Hood of Feynman Diagrams

    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...
  47. A

    I Frank Wilczek on Virtual Particles and Summing diagrams....

    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...
  48. A

    I 'Off-shell' particle in an external field....

    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...
  49. N

    Calculation of g-factor correction in Peskin p. 196

    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...
  50. John Greger

    A Understanding Hermiticity of Actions in QFT: A Guide for Checking and Confirming

    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 -...
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