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Questions tagged [quantum-field-theory]

Quantum Field Theory (QFT) is the theoretical framework describing the quantisation of classical fields which allows a Lorentz-invariant formulation of quantum mechanics. QFT is used both in high energy physics as well as condensed matter physics and closely related to statistical field theory. Use this tag for many-body quantum-mechanical problems and the theory of [tag:particle-physics]. Don’t combine with [tag:quantum-mechanics].

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Elementary particles & their reactions: literature

I would like to find a book which contains lot of examples of different reactions with elementary particles (for instance, electron-electron scattering, electron-positron annihilation, electron ...
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170 views

Explicit computation of singular part of two-loop sunrise diagram

For $\phi^4$, there is two-loop self-energy contribution from sunrise (sunset) diagram. The integration is $$ I(p)=\int\frac{d^D p_1}{(2\pi)^D}\frac{d^Dp_2}{(2\pi)^D}\frac{1}{(p_1^2+m^2)(p_2+m^2)[...
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Fermionic ghost path integral results in $\delta$ function?

This is related to a statement in pg 20 of hep-th/9408074 formula (2.39). Suppose $$\mathcal{L}\sim\frac{i}{\lambda^{\prime}}\bar{\eta}^xg_{ij}U_x{}^i\psi^j+\cdots \tag{2.35}$$where $\bar{\eta}$ to ...
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A few basic questions on SUSY algebra

Consider SUSY generator taking the form {$Q_a^A,Q_{\dot{b}B}$}$=\delta_{a\dot{b}}\delta^A_B$ and {$Q_a^A,Q^B_b$}={$\bar{Q}_{\dot{a}A},\bar{Q}_{\dot{b} B}$}=0 where $a,b$ label spinor index and $A,B$ ...
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Calcluating the photon propagator with gauge fixing parameter

I'm trying to calculate the photon propagator via the functional integral, with lagrangian (plus source) $L = -\frac{1}{4}F^{ab}F_{ab} - \frac{\lambda}{2}\left(\partial^aA_a\right)^2 + J^aA_a $ ...
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Lattice model realization of $SU(2)$ WZW model at level $k$?

Is there any lattice model realization of the following model: $c=1$ boson at the self-dual radius, or the $SU(2)$ WZW model at $k=1$. This is a question inspired by: Orbifolds of the $c =1$ ...
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QFT Why do in and out states have a non-trivial overlap?

Im trying to follow chapter 4 about interacting fields in Peskin and Schröder. They define the S matrix by $_{out}<p_1 p_2 | k_a k_b>_{in} = <p_1 p_2 | S | k_a k_b>$, where $S = \lim_{T\...
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1answer
157 views

Faddeev-Popov determinant and topology of the worldline

I am studying the path integral quantization of relativistic particles, using the BRST quantization method. I have to compute the integral \begin{equation} Z\sim \int Dx \det(\partial_\tau)e^{-\int_0^...
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Problems reading Ramond's book [duplicate]

I am currently going through Ramond's book Field Theory: A Modern Primer. I've found it however extremely difficult to solve the problems with the material presented. Does anyone have any ...
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1answer
83 views

${\cal N} = 1$ SUSY Non-renormalization theorem

In Ref. 1, on Page 53, the ${\cal N} = 1$ SUSY non-renormalization theorem is derived. One first specifies the symmetries of the general ${\cal N} = 1$ SUSY action in the superspace formalism, and ...
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Why the imaginary part of green function is the spectral function?

In the zero temperature Green's function, $$G_{\alpha\beta}(xt,x't')=-i\langle \Psi_H|\hat{T}[\psi_{H\alpha}(xt)\psi^{\dagger}_{H\beta}(x't')]|\Psi_H\rangle .$$ In Lehmann Representation, $$G_{\...
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159 views

Non-local field redefinition and effects on path-integral measure

Consider the partition function $$ Z[0] = \int \left[\mathcal{D}A_\mu\right]\left[\mathcal{D}\pi\right] e^{-i \int d^4x \left(-\frac{1}{2}(\partial\pi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{a}{M^2}...
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Meson scattering amplitude in the linear sigma model

I am trying to calculate scattering amplitudes with linear sigma model Lagrangian, given as $$\mathcal L= \frac{1}{2}(\partial_{\mu}\sigma)^2+\frac{1}{2}(\partial_{\mu}\vec{\pi})^2-\mathcal U(\sigma,\...
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Are vacuum-fluctuations a consequence of causality?

I'n new to QFT, and recently lerned about the propagator of a free scalar field theory in Minkowski-space, which according to our lecture notes looks like $$G(p, q) = \frac{1}{q^\mu q_\mu + M^2} \...
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1answer
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How to understand the two-point correlation function in momentum space?

Let's take the Ising model as an example and study the two point spin spin correlation function: $$\langle s_0 s_r\rangle = \frac{\sum_{\{s_i\}}e^{K\sum_{\langle i ,j\rangle}s_i s_j} s_0 s_r}{\sum_{\{...
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Dimensional regularization - Expansion of powers of $\epsilon$ turns into logarithms

Looking into Schwartz's book on QFT at the appendices, it seems that when doing a dimensional regularization, one expands around $\epsilon=0$ and usually obtains $$ x^\epsilon=\log x+O(\epsilon), $$ ...
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A puzzle about Green's function and S-matrix

From the discussion in some posts example1, example2, we know that the S-matrix is the residue of the corresponding Green's function. On the other hand, S-matix is a physical observable in QFT, but ...
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106 views

Are any quantum field theories mathematically convergent?

I know for example that theories like QED and QCD when expended perturbatively in terms of Feynman diagrams produce asymptotic series that don't converge after a large number of terms and in fact ...
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What is the publication through which Zinn-Justin published what has come to be known as the “Zinn-Justin equation”?

does anybody know which publication contains the introduction of what has come to known as the Zinn-Justin equation?
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Gauge fixing: Overcounting vs Inversion of Operator

In my studies (various books, and Lectures by Tobias Osborne) I've been told we gauge fix to stop the naive overcounting in the path integral. User @Marmot pointed out in a comment that if this was ...
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Does the Dirac sea have any mass or gravitational effect?

Dirac sea is a model for vacuum which considers the empty space as a sea full of negative-energy particles. Anti-particles are holes in this sea. Dirac sea is used to model Quantum field theory in the ...
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Conformal symmetry, Weyl symmetry, and a traceless energy-momentum tensor

I'm trying to drill down the exact relation between conformal symmetry, Weyl symmetry, and tracelessness of the energy-momentum tensor. However, I'm getting quite confused because every book I can ...
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Weinberg formula in chapter 6

I am trying to understand how Weinberg from this formula arrived to this formula Could anyone help me?
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63 views

Can a general many-body Hamiltonian with quadratic and biquadratic terms be diagonalized?

Can an arbitrary many-body hamiltonian in second quantization form with quadratic and biquadratic terms $$H=\sum_{v_1,v_2} \alpha_{v_1 v_2}\ c_{v_1}^{\dagger}c_{v_2}+ \sum_{v_1,v_2,v_3,v_4}\beta_{v_1 ...
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222 views

What is the actual definition of conformal invariance?

I've seen a large variety of slightly different definitions of conformal invariance. For simplicity I'll only consider scale invariance, which is already confusing enough. Some of the definitions are: ...
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1answer
44 views

Electromagnetic current operator using Feynman rules

In the calculus of the electron anomalous magnetic moment some text books usually calculates the forms factors from $\Gamma_{\mu}$ in the expression bellow $\left<p',s'|J^{em}|p,s\right> = \...
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What is the difference between many body theory and quantum field theory methods in condensed matter?

I am starting to studying condensed matter theory and I do not understand if Many-Body Quantum Mechanics and Quantum Field Theory are just synonyms or are two different methods. It seems to me that ...
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1answer
62 views

How to define the Hamiltonian properly in quantum field theory

In a rigorous fashion, how does one define the Hamiltonian of QFT as $$\hat{H}(t) = \int d^3x \hat{\mathcal{H}}(x, t)$$ For now I'm ignoring the fact that $\hat{\mathcal{H}}$ itself may be ill-...
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An involved Feynman integral

Working with QCD, I have found the following integral from Feynman diagrams to solve $$ I(p)=\int\frac{d^4p_1}{(2\pi)^4}\int\frac{d^4p_2}{(2\pi)^4}\frac{1}{p_2^2-m_0^2} \left(\frac{p\cdot p_1-p_1\cdot ...
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1answer
48 views

Does the external leg contraction of gluon in QCD carry group generator index?

While I am trying to compute the amplitude for the following Feynman diagram I realized that the external leg contraction of $g$ should carry group generator index $A$ or $B$, is that right? If so, ...
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Transition amplitude in scalar QED from a point-like charge

I have a problem for you to compute. I have a classical source, $A_\mu = \delta_\mu^0 \frac{Q}{r}$, representing the Coulomb potential generated by a point-like charge. I want to compute the ...
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2answers
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Global $U(1)$ transformation properties of gauge fields

What are the Global gauge transformations of gauge bosons in Standard Model? To elaborate: Initially, we consider the global $U(1)$ transformations of scalars ($\phi$) and fermions ($\psi$) as $$\...
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Why are infinitesimal shifts sufficient to prove that a symmetry holds

Why are infinitesimal shifts in the Lagrangian sufficient to prove that a symmetry holds? Couldn't a lot of things happen at higher orders? Especially when I am introducing an infinitesimal shift of a ...
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Self-energy that does not obey sum rule

Analytically, I calculated a self-energy $\Sigma(\omega)$, for which I verified that 1) $\text{Im}\big[\Sigma(\omega)\big] \leq 0$ for all $\omega$ and specifically $\text{Im}\big[\Sigma(0)\big] = 0$,...
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1answer
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Fields or Particles [duplicate]

What is more fundamental, particles or fields? I keep reading what appears to be conflicting answers on this, but I am sure it is just a limitation of my understanding. I have heard some physicists ...
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2answers
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Corresponding particle-antiparticle solutions for Klein-Gordon equation

For free particle solutions in a box, the following 4 solutions are possible(Not all 4 are independent though) as $$\psi_+=A_+ \exp{\frac{i}{\hbar}(px-Et)}\\\psi_+^*=A_+^* \exp{\frac{-i}{\hbar}(px-Et)}...
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For dimensional regularization, why the arbitrary mass scale $\mu$ has the meaning of UV cutoff?

For sharp cut off regularization, we introduce the UV cutoff $\Lambda$. When we need to do momentum integral, we integrate the momentum ball with radius of $\Lambda$. This $\Lambda$ has the explicit ...
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Approximations in QFT and the problem of convergence

I would like to understand the problem of the convergence/divergence of the series appearing in QED: my question is inspired by a great article of John Baez, which can be found online. On the page 23 ...
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62 views

Is there a difference of sign conventions of Dirac Index between mathematics and physics?

In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by $$\mathrm{Ind}(D\!\!\!\!/_{A})=\frac{-1}{8\pi^{2}}\int_{M}\mathrm{Tr}(F\wedge F)+\frac{...
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20 views

Continuum limit of Hubbard model

I would like to obtain continuum limit of Hubbard model for 1+1 case. Starting from the following hamiltonian $$H=-t\sum_{\langle i,j\rangle\sigma}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_{i}n_{\uparrow ...
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Feynman diagrams: How to find factors for external particles

In class we learned how to compute Feynman rules from the Lagrangian. I know how to find propagators and vertices. But is there a general rule for external particle lines?
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QFT cross section: missing information?

So I'm having a hard time understanding how the (QM) cross section fits into the general picture, of e.g. collider experiments. So we can calculate cross sections (for one reaction) exactly in QFT ...
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The kinematic region for the operator product expansion

In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.613 the operator product expansion (OPE) is introduced $$\mathcal{O}_1(x)\mathcal{O}_2(0)\to \sum_n C_{...
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1answer
122 views

Temperature-dependence of quark potential in Abelian lattice gauge theory

I am working with Kapusta's "Finite-Temperature Field Theory" textbook, and am working through the first part of chapter 10. When building the correlator of the two quarks a distance $R$ apart in the ...
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1answer
69 views

What is the content of an occupied QFT fermionic state?

A simple non-interacting quantum field is constructed by analogy to a harmonic oscillator, with $\hat{x}$ & $\hat{p}$ replaced by $\hat{ \phi}$ & $\hat{\pi}$ & with a separate oscillator ...
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1answer
57 views

Supercharge transformation rules

Consider ${\cal N}=2$ supersymmetry with $SU(2)$ global symmetry group. Then both supercharges $Q_{ai},\bar{Q}_{\dot{a}\dot{j}}$ transform by 2 dimensional representation of $SU(2)$. Denote $SU(2)_I$ ...
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What are the anti-commutation relations needed to be fulfilled by fermionic Bogoliubov transformation?

Bogoliubov transformations are a well used tool to get rid off interaction terms in second quantized Hamiltonians. I'm interested in the fermionic Bogoliubov transformation used in BCS theory, e.g. \...
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1answer
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Can someone Tong got this equation in his QFT notes

Can someone explain how D.Tong got equation 2.18 in his QFT notes in chapter 2? I am lost from equation 2.5, can someone explain? Link to notes: http://www.damtp.cam.ac.uk/user/tong/qft.html Can ...
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2answers
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E.L. Equations in QFT

In QFT, we use the Lagrangian to construct the Hamiltonian, and in the Interaction Picture (with regards to the Free Field Hamiltonian) use the full Hamiltonian to calculate the changes in the field (...
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55 views

Why is color confinement a difficult problem?

Assuming color force follows a constant rule of force instead of an inverse square rule of force. And that red, green and blue are all attracted to each other. Why is color confinement considered a ...